Kevin Costello and Owen Gwilliam - Semantic Scholar...2.5. The factorization algebras associated to...

Factorization algebras in quantum field theory

Kevin Costello and Owen Gwilliam

K.C. and O.G. are partially supported by NSF grants DMS 0706945 and DMS1007168, and K.C. is also partially supported by a Sloan fellowship. O.G. is

supported by an NSF postdoctoral fellowship DMS-1204826.

Contents

Chapter 1. Introduction and overview 11.1. Introduction 11.2. The motivating example of quantum mechanics 31.3. A preliminary definition of prefactorization algebras 71.4. Prefactorization algebras in quantum field theory 81.5. Classical field theory and factorization algebras 111.6. Quantum field theory and factorization algebras 121.7. The weak quantization theorem 131.8. The strong quantization conjecture 14

Chapter 2. Prefactorization algebras and basic examples 172.1. Prefactorization algebras 172.2. Structured factorization algebras 192.3. The P0 operad 232.4. The Beilinson-Drinfeld operad 272.5. The factorization algebras associated to free scalar field theories 292.6. Standard quantum mechanics as a prefactorization algebra 422.7. Translation-invariant and holomorphic prefactorization algebras 45

Chapter 3. Factorization algebras: definition and formal aspects 593.1. Factorization algebras 593.2. Factorization algebras from cosheaves 653.3. Locally constant factorization algebras 693.4. The category of factorization algebras 713.5. Pushing forward factorization algebras 753.6. Extension from a basis 753.7. Pulling back factorization algebras 763.8. Descent and gluing 77

Chapter 4. Classical field theory 83

iii

iv CONTENTS

4.1. Introduction 834.2. Elliptic moduli problems and local Lie algebras 864.3. Formal moduli problems and Lie algebras 864.4. Examples of elliptic moduli problems related to gauge theories 904.5. Examples of elliptic moduli problems related to scalar field theories 934.6. Cochains of a local Lie algebra 964.7. D-modules and local Lie algebras 984.8. The classical BV formalism in finite dimensions 1024.9. The exterior derivative of a local action functional 1064.10. A succinct definition of a classical field theory 1124.11. Examples of field theories from action functionals 1144.12. Cotangent field theories 1154.13. The graded Poisson structure on classical observables 1194.14. The Poisson structure for free field theories 1214.15. The Poisson structure for a general classical field theory 1234.16. Symmetries and deformations of a classical field theory 1284.17. Conserved quantities and Noether’s theorem 131

Chapter 5. Quantum field theory 1395.1. Free fields 1415.2. Overview of perturbative quantum field theory 1455.3. Local action functionals 1465.4. The definition of a quantum field theory 1475.5. The simplicial set of theories 1625.6. The theorem on quantization 1665.7. The BD algebra of global observables 1685.8. Global observables 1775.9. Local observables 1795.10. Local observables form a prefactorization algebra 1825.11. Local observables form a factorization algebra 1865.12. Translation-invariant factorization algebras from translation-invariant

quantum field theories 1925.13. Cotangent theories and volume forms 199

Appendix A. Homological algebra with differentiable vector spaces 209A.1. Diffeological vector spaces 209A.2. Differentiable vector spaces from sections of a vector bundle 218

CONTENTS v

A.3. Differentiable cochain complexes 221A.4. Pro-cochain complexes 225A.5. Differentiable pro-cochain complexes 227A.6. Differentiable cochain complexes over a differentiable dg ring 230A.7. Classes of functions on the space of sections of a vector bundle 230A.8. Derivations 237A.9. The Atiyah-Bott lemma 240

Appendix B. Extending a factorization algebra from a basis 243

Bibliography 247

CHAPTER 1

Introduction and overview

1.1. Introduction

This book will provide the analog, in quantum field theory, of the deformationquantization approach to quantum mechanics. In this introduction, we will start byrecalling how deformation quantization works in quantum mechanics.

The collection of observables in quantum mechanics form an associative algebra.The observables of a classical mechanical system form a Poisson algebra. In the defor-mation quantization approach to quantum mechanics, one starts with a Poisson alge-bra Acl , and attempts to construct an associative algebra Aq, which is an algebra flatover the ring C[[h]], together with an isomorphism of associative algebras Aq/h ∼= Acl .In addition, if a, b ∈ Acl , and a, b are any lifts of a, b to Aq, then

limh→0

1h[a, b] = a, b ∈ Acl .

We will describe an analogous approach to studying perturbative quantum fieldtheory. In order to do this, we need to explain the following.

• The structure present on the collection of observables of a classical field theory.This structure is the analog, in the world of field theory, of the commutativealgebra which appears in classical mechanics. This structure we call a com-mutative factorization algebra (section 2.2).• The structure present on the collection of observables of a quantum field the-

ory. This structure is that of a factorization algebra (section 3.1). We view ourdefinition of factorization algebra as a C∞ analog of a definition introducedby Beilinson and Drinfeld. However, the definition we use is very closelyrelated to other definitions in the literature, in particular to the Segal axioms.

1

2 1. INTRODUCTION AND OVERVIEW

• The extra structure on the commutative factorization algebra associated to aclassical field theory which makes it “want” to quantize. This is the analog, inthe world of field theory, of the Poisson bracket on the commutative algebraof observables.• The quantization theorem we prove. This states that, provided certain ob-

struction groups vanish, the classical factorization algebra associated to aclassical field theory admits a quantization. Further, the set of quantizationsis parametrized (order by order in h) by the space of deformations of the La-grangian describing the classical theory.

This quantization theorem is proved using the physicists’ techniques of perturbativerenormalization, as developed mathematically in [Cos11c]. We claim that this theoremis a mathematical encoding of the perturbative methods developed by physicists.

This quantization theorem applies to many examples of physical interest, includ-ing pure Yang-Mills theory and σ-models. For pure Yang-Mills theory, it is shown in[Cos11c] that the relevant obstruction groups vanish, and that the deformation groupis one-dimensional; so that there exists a one-parameter family of quantizations. A cer-tain two-dimensional σ-model was constructed in this language in [Cos10, Cos11a].Other examples are considered in [GG11] and [CL11].

Finally, we will explain how (under certain additional hypotheses) the factoriza-tion algebra associated to a perturbative quantum field theory encodes the correlationfunctions of the theory. This justifies the assertion that factorization algebras encode alarge part of quantum field theory.

1.1.1. Acknowledgements. We would like to thank Dan Berwick-Evans DamienCalaque, Vivek Dhand, Chris Douglas, John Francis, Dennis Gaitsgory, Greg Ginot,Ryan Grady, Theo Johnson-Freyd, David Kazhdan, Jacob Lurie, Takuo Matsuoka, FredPaugam, Nick Rozenblyum, Graeme Segal, Josh Shadlen, Yuan Shen, Stephan Stolz,Dennis Sullivan, Hiro Tanaka and David Treumann for helpful conversations. K.C. isparticularly grateful to thank John Francis and Jacob Lurie for introducing him to fac-torization algebras (in their topological incarnation [Lur09a]) in 2008; and to GraemeSegal, for many illuminating conversations about QFT.

1.2. THE MOTIVATING EXAMPLE OF QUANTUM MECHANICS 3

1.2. The motivating example of quantum mechanics

The model problems of classical and quantum mechanics involve a particle mov-ing in some Euclidean space Rn under the influence of some fixed field. Our main goalin this section is to describe these model problems in a way that makes the idea of a fac-torization algebra (section 3.1) emerge naturally, but we also hope to give mathemati-cians some feeling for the physical meaning of terms like “field” and “observable.”We will not worry about making precise definitions, since that’s what this book aimsto do. As a narrative strategy, we describe a kind of cartoon of a physical experiment,and we ask that physicists accept this cartoon as a friendly caricature elucidating thefeatures of physics we most want to emphasize.

1.2.1. A particle in a box. For the general framework we want to present, the de-tails of the physical system under study are not so important. However, for concrete-ness, we will focus attention on a very simple system: that of a single particle confinedto some region of space. We confine our particle inside some box and occasionally takemeasurements of this system. The set of possible trajectories of the particle around thebox constitute all the imaginable behaviors of this particle; we might write this math-ematically as Maps(I, box), where I ⊂ R denotes the time interval over which weconduct the experiment. In other words, the set of possible behaviors forms a space offields on the timeline of the particle.

The behavior of our theory is governed by the action functional. The simplest caseis the action of the massless free field theory, whose value on a function f : I → box is

S( f ) =∫

I〈 f , ∆ f 〉 .

The aim of this section is to outline the structure one would expect the observables –that is, the possible measurements one can make – should satisfy.

1.2.2. Classical mechanics. Let us start by considering the much simpler case,where our particle is treated as a classical system. In that case, the trajectory of theparticle is constrained to be in a solution to the Euler-Lagrange equations of our the-ory. For example, if the action functional governing our theory is that of the masslessfree theory, then a map f : I → box satisfies the Euler-Lagrange equation if it is astraight line.


We are interested in the observables for this classical field theory. Since the trajec-tory of our particle is constrained to be a solution to the Euler-Lagrange equation, theonly measurements one can make are functions on the space of solutions to the Euler-Lagrange equation.

If U ⊂ R is an open subset, we will let Fields(U) denote the space of fields on U,that is, the space of maps f : U → box. We will let

EL(U) ⊂ Fields(U)

denote the subspace consisting of those maps f : U → box which are solutions to theEuler-Lagrange equation. As U varies, EL(U) forms a sheaf of spaces on R.

We will let Obscl(U) denote the space of functions on EL(U) (the precise class offunctions we will consider will be discussed later). As U varies, the spaces Obscl(U)

form a cosheaf of commutative algebras on R. We will think of Obscl(U) as the spaceobservables for our classical system which only consider the behavior of the particleon times contained in U.

Note that Obscl(U) is a cosheaf of commutative algebras on R.

1.2.3. Measurements in quantum mechanics. The notion of measurement is fraughtin quantum theory, but we will take a very concrete view. Taking a measurementmeans that we have physical measurement device (e.g., a camera) that we allow tointeract with our system for a period of time. The measurement is then how our mea-surement device has changed due to the interaction. In other words, we couple thetwo physical systems, then decouple them and record how the measurement devicehas modified from its initial condition. (Of course, there is a symmetry in this sit-uation: both systems are affected by their interaction, so a measurement inherentlydisturbs the system under study.)

The observables for a physical system are all the imaginable measurements wecould take of the system. Instead of considering all possible observables, we mightalso consider those observables which occur within a specified time period. This pe-riod can be specified by an open interval U ⊂ R.

Thus, we arrive at the following principle.

1.2. THE MOTIVATING EXAMPLE OF QUANTUM MECHANICS 5

Principle 1. For every open subset U ⊂ R, we have a set Obs(U) ofobservables one can make on U.

The superposition principle tells us that quantum mechanics (and quantum fieldtheory) is fundamentally linear. This leads to

Principle 2. The set Obs(U) is a complex vector space.

We think of Obs(U) as being the space of ways of coupling a measurement deviceto our system on the region U. Thus, there is a natural map Obs(U) → Obs(V) ifU ⊂ V is an open subset. This means that the space Obs(U) forms a pre-cosheaf.

1.2.4. Combining observables. Measurements (and so observables) differ qual-itatively in the classical and quantum settings. If we study a classical particle, thesystem is not noticeably disturbed by measurements, and so we can do multiple mea-surements at the same time. Hence, on each interval J we have a commutative mul-tiplication map Obs(J) ⊗Obs(J) → Obs(J), as well as the maps that let us combineobservables on disjoint intervals.

For a quantum particle, however, a measurement disturbs the system significantly.Taking two measurements simultaneously is incoherent, as the measurement devicesare coupled to each other and thus also affect each other, so that we are no longer mea-suring just the particle. Quantum observables thus do not form a cosheaf of commu-tative algebras on the interval. However, there are no such problems with combiningmeasurements occurring at different times. Thus, we find the following.

Principle 3. If U, U′ are disjoint open subsets of R, and U, U′ ⊂ Vwhere V is also open, then there is a map

? : Obs(U)⊗Obs(U′)→ Obs(V).

If O ∈ Obs(U) and O′ ∈ Obs(U′), then O ?O′ is defined by couplingour system to measuring device O for t ∈ U, and to device O′ fort ∈ U′.

Further, these maps are commutative, associative, and compati-ble with the maps Obs(U)→ Obs(V) associated to inclusions U ⊂ Vof open subsets. (The precise meaning of these terms is detailed insection 2.1.)


1.2.5. Perturbative theory and the correspondence principle. In the bulk of thispaper, we will be considering perturbative quantum theory. For us, this means thatwe work over the base ring C[[h]], where at h = 0 we find the classical theory. Inperturbative theory, therefore, the space Obs(U) of observables on an open subset Uis a C[[h]]-module, and the product maps are C[[h]]-linear.

The correspondence principle states that the quantum theory, in the h → 0 limit,must reproduce the classical theory. Applied to observables, this leads to the followingprinciple.

Principle 4. The vector space Obsq(U) of quantum observables is aflat C[[h]]-module that, modulo h, is the space Obscl(U) of classicalobservables.

These simple principles are at the heart of our approach to quantum field theory.They say, roughly, that the observables of a quantum field theory form a factoriza-tion algebra, which is a quantization of the factorization algebra associated to a clas-sical field theory. The main theorem presented in this paper is that one can use thetechniques of perturbative renormalization to construct factorization algebras pertur-batively quantizing a certain class of classical field theories (including many classicalfield theories of physical and mathematical interest).

1.2.6. Associative algebras in quantum mechanics. The principles we have de-scribed so far indicate that the observables of a quantum mechanical system shouldassign, to every open subset U ⊂ R, a vector space Obs(U), together with a productmap

Obs(U)⊗Obs(U′)→ Obs(V)

if U, U′ are disjoint open subsets of an open subset V. This is the basic data of afactorization algebra (section 2.1).

It turns out that the factorization algebra produced by our quantization procedureapplied to quantum mechanics has a special property: it is locally constant (section 3.3).This means that the map Obs((a, b)) → Obs(R) is an isomorphism for any interval(a, b). Let A be denote the vector space Obs(R); note that A is canonically isomorphicto Obs((a, b)) for any interval (a, b).

1.3. A PRELIMINARY DEFINITION OF PREFACTORIZATION ALGEBRAS 7

The product map

Obs((a, b))⊗Obs((c, d))→ Obs((a, d))

(defined when a < b < c < d) becomes, when we perform this identification, aproduct map

m : A⊗ A→ A.

The axioms of a factorization algebra imply that this multiplication turns A into anassociative algebra.

This should be familiar to topologists: associative algebras are algebras over theoperad of little intervals in R, and this is precisely what we have described. (As wewill see later (section 2.5), this associative algebra is the Weyl algebra one expects tofind as the algebra of observables of quantum mechanics.)

One important point to take away from this discussion is that associative algebrasappear in quantum mechanics because associative algebras are connected with the geome-try of R. There is no fundamental connection between associative algebras and anyconcept of “quantization”: associative algebras only appear when one considers one-dimensional quantum field theories. As we will see later, when one considers quan-tum field theories on n-dimensional space times, one finds a structure reminiscent ofan En-algebra instead of an E1-algebra.

1.3. A preliminary definition of prefactorization algebras

Below (see section 2.1) we give a more formal definition, but here we providethe basic idea. Let M be a topological space (which, in practice, will be a smoothmanifold).

1.3.0.1 Definition. A prefactorization algebra F on M, taking values in cochain complexes,is a rule that assigns a cochain complex F (U) to each open set U ⊂ M along with

(1) a cochain map F (U)→ F (V) for each inclusion U ⊂ V;(2) a cochain map F (U1)⊗ · · · ⊗ F (Un) → F (V) for every finite collection of open

sets where each Ui ⊂ V and the Ui are pairwise disjoint;(3) the maps are compatible in the obvious way (e.g. if U ⊂ V ⊂ W is a sequence of

open sets, the map F (U)→ F (W) agrees with the composition through F (V)).


Remark: A prefactorization algebra resembles a precosheaf, except that we tensor thecochain complexes rather than taking their direct sum.

The observables of a field theory (whether classical or quantum) form a prefactor-ization algebra on the spacetime manifold M. In fact, they satisfy a kind of local-to-global principle in the sense that the observables on a large open set are determinedby the observables on small open sets. The notion of a factorization algebra (section3.1) makes this local-to-global condition precise.

1.4. Prefactorization algebras in quantum field theory

The (pre)factorization algebras of interest in this paper arise from perturbativequantum field theories. We have already discussed (section 1.2) how factorizationalgebras appear in quantum mechanics. In this section we will see that this pictureextends in a very natural way to quantum field theory.

The manifold M on which the prefactorization algebra is defined is the space-timemanifold of the quantum field theory. If U ⊂ M is an open subset, we will interpretF (U) as the space of observables (or measurements) that we can make, which only de-pend on the behavior of the fields on U. Performing a measurement involves couplinga measuring device to the quantum system in the region U.

One can bear in mind the example of a particle accelerator. In that situation, onecan imagine the space-time M as being of the form M = A × (0, t), where A is theinterior of the accelerator and t is the duration of our experiment.

In this situation, performing a measurement on some open subset U ⊂ M is some-thing concrete. Let us take U = V × (ε, δ), where V ⊂ A is some small region inthe accelerator, and (ε, δ) is a short time interval. Performing a measurement on Uamounts to coupling a measuring device to our accelerator in the region V, starting attime ε and ending at time δ. For example, we could imagine that there is some pieceof equipment in the region V of the accelerator, which is switched on at time ε andswitched off at time δ.

1.4. PREFACTORIZATION ALGEBRAS IN QUANTUM FIELD THEORY 9

1.4.1. Interpretation of the prefactorization algebra axioms. Suppose that wehave two different measuring devices, O1 and O2. We would like to set up our ac-celerator so that we measure both O1 and O2.

There are two ways we can do this. Either we insert O1 and O2 into disjoint re-gions V1, V2 of our accelerator. Then we can turn O1 and O2 on at any times we like,including for overlapping time intervals.

If the regions V1, V2 overlap, then we can not do this. After all, it doesn’t makesense to have two different measuring devices at the same point in space at the sametime.

However, we could imagine inserting O1 into region V1 during the time interval(a, b); and then removing O1, and inserting O2 into the overlapping region V2 for thedisjoint time interval (c, d).

These simple considerations immediately suggest that the possible measurementswe can make of our physical system form a prefactorization algebra. Let Obs(U)

denote the space of measurements we can make on an open subset U ⊂ M. Then, bycombining measurements in the way outlined above, we would expect to have maps

Obs(U)⊗Obs(U′)→ Obs(V)

whenever U, U′ are disjoint open subsets of an open subset V. The associativity andcommutativity properties of a prefactorization algebra are evident.

1.4.2. The cochain complex of observables. In the approach to quantum field the-ory considered in this paper, the factorization algebra of observables will be a factor-ization algebra of cochain complexes. One can ask for the physical meaning of thecochain complex Obs(U).

It turns out that the “physical” observables will be H0(Obs(U)). If O ∈ Obs0(U) isan observable of cohomological degree 0, then the equation dO = 0 can often be inter-preted as saying that O is compatible with the gauge symmetries of the theory. Thus,only those observables O ∈ Obs0(U) which are closed are physically meaningful.

The equivalence relation identifying O ∈ Obs0(U) with O + dO′, where O′ ∈Obs−1(U), also has a physical interpretation, which will take a little more work todescribe. Often, two observables on U are physically indistinguishable (that is, they


can not be distinguished by any measurement one can perform). In the example ofan accelerator outlined above, two measuring devices are equivalent if they alwaysproduce the same expectation values, no matter how we prepare our system, or nomatter what boundary conditions we impose.

As another example, in the quantum mechanics of a free particle, the observablemeasuring the momentum of a particle at time t is equivalent to that measuring themomentum of a particle at another time t′. This is because, even at the quantum level,momentum is preserved (as the momentum operator commutes with the Hamilton-ian).

From the cohomological point of view, if O, O′ ∈ Obs0(U) are observables whichare in the kernel of d (and thus “physically meaningful”), then they are equivalent inthe sense described above if they differ by an exact observable.

It is a little more difficult to provide a physical interpretation for the non-zerocohomology groups Hn(Obs(U)). The first cohomology group H1(Obs(U)) is the re-cipient of any anomalies to defining observables at the quantum level. For example,in a gauge theory, one might have a classical observable which respects gauge sym-metry. However, it may not lift to a quantum observable respecting gauge symmetry;this happens if there is a non-zero anomaly in H1(Obs(U)).

The cohomology groups Hn(Obs(U)), when n < 0, are best interpreted as sym-metries, and higher symmetries, of observables. Indeed, we have seen that the physi-cally meaningful observables are the closed degree 0 elements of Obs(U). One canconstruct a simplicial set, whose n-simplices are closed and degree 0 elements ofObs(U) ⊗ Ω∗(∆n). The vertices of this simplicial set are observables, the edges areequivalences between observables, the faces are equivalences between equivalences,and so on.

The Dold-Kan correspondence tells us that the nth homotopy group of this simpli-cial set is H−n(Obs(U)). This allows us to interpret H−1(Obs(U)) as being the groupof symmetries of the trivial observable 0 ∈ H0(Obs(U)), and H−2(Obs(U)) as thesymmetries of the identity symmetry of 0 ∈ H0(Obs(U)), and so on.

Although the cohomology groups Hn(Obs(U)) where n > 1 do not have sucha clear physical interpretation, they are mathematically very natural objects and it isimportant not to discount them.

1.5. CLASSICAL FIELD THEORY AND FACTORIZATION ALGEBRAS 11

1.5. Classical field theory and factorization algebras

The main aim of this book is to present a deformation-quantization approach toquantum field theory. In this section we will outline how a classical field theory givesrise to the classical algebraic structure we consider.

We use the Lagrangian formulation throughout. Thus, classical field theory meansthe study of the critical locus of an action functional. In fact, we use the language ofderived geometry, in which it becomes clear that functions on a derived critical locus(section 4.8) should form a P0 algebra (section 2.3), that is, a commutative algebra witha Poisson bracket of cohomological degree 1. (For an overview of these ideas, see thefinal section of this chapter.)

Applying these ideas to infinite-dimensional spaces, such as the space of smoothfunctions on a manifold, one runs into analytic problems. Although there is no dif-ficulty in constructing a commutative algebra Obscl of classical observabes, we findthat the Poisson bracket on Obscl is not always well-defined. However, we show thefollowing.

1.5.0.1 Theorem. For a classical field theory (section 4.10) on a manifold M, there is a sub-commutative factorization algebra Obs

cl of the commutative factorization algebra Obscl on

which the Poisson bracket is defined, so that Obscl

forms a P0 factorization algebra. Further,

the inclusion Obscl→ Obscl is a quasi-isomorphism of factorization algebras.

Remark: Our approach to field theory involves both cochain complexes of infinite-dimensional vector spaces and families over manifolds (and dg manifolds). Insteadof working with topological vector spaces, we use differentiable vector spaces. For acareful discussion, see Appendix A. As a gloss, a differentiable vector space is a vectorspace V with a smooth structure, meaning that we have a well-defined set of smoothmaps from any manifold X into V. In fact, going a bit further, we work with what wecall differentiable vector spaces. This is a differentiable vector space with a flat connec-tion, so that one knows how to take derivatives of smooth maps into the vector space.These notions make it possible to efficiently study cochain complexes of vector spacesin families over manifolds.


1.5.1. A gloss of the main ideas. In the rest of this section, we will outline whyone would expect that classical observables should form a P0 algebra. More details areavailable in section 4.1.

The idea of the construction is very simple: if U ⊂ M is an open subset, we will letEL(U) be the derived space of solutions to the Euler-Lagrange equation on U. Sincewe are dealing with perturbative field theory, we are interested in those solutions tothe equations of motion which are infinitely close to a given solution.

The differential graded algebra Obscl(U) is defined to be the space of functions onEL(U). (Since EL(U) is an infinite dimensional space, it takes some work to defineObscl(U). Details will be presented later (chapter 4).

On a compact manifold M, the solutions to the Euler-Lagrange equations are thecritical point of the action functional. If we work on an open subset U ⊂ M, thisis no longer strictly true, because the integral of the action functional over U is notdefined. However, fields on U have a natural foliation, where tangent vectors lying inthe leaves of the foliation correspond to variations φ→ φ + δφ, where δφ has compactsupport. In this case, the Euler-Lagrange equations are the critical points of a closedone-form dS defined along the leaves of this foliation.

Any derived scheme which arises as the derived critical locus (section 4.8) of afunction acquires an extra structure: it’s ring of functions is equipped with the struc-ture of a P0 algebra. The same holds for a derived scheme arising as the derived criticallocus of a closed one-form define along some foliation. Thus, we would expect thatObscl(U) is equipped with a natural structure of P0 algebra; and that, more generally,the commutative factorization algebra Obscl should be equipped with the structure ofP0 factorization algebra.

1.6. Quantum field theory and factorization algebras

Another aim of the book is to relate quantum field theory, as developed in [Cos11c],to factorization algebras. We give a natural definition of an observable of a quantumfield theory, which leads to the following theorem.

1.6.0.1 Theorem. For a classical field theory (section 4.10) and a choice of BV quantization(section 5.4), the quantum observables Obsq form a factorization algebra over the ring R[[h]].

1.7. THE WEAK QUANTIZATION THEOREM 13

Moreover, the factorization algebra of classical observables Obscl is homotopy equivalent to thequotient Obsq /h as a factorization algebra.

Thus, the quantum observables form a factorization algebra and, in a very weaksense, are related to the classical observables. The quantization theorems will sharpenthe relationship between classical and quantum observables.

1.7. The weak quantization theorem

We have explained how a classical field theory gives rise to a lax P0 factorizationalgebra Obscl , and how a quantum field theory (in the sense of [Cos11c]) gives rise toa factorization algebra Obsq over R[[h]], which specializes at h = 0 to the factorizationalgebra Obscl of classical observables. In this section we will state our weak quantizationtheorem, which says that the Poisson bracket on Obscl is compatible, in a certain sense,with the quantization given by Obsq.

This statement is the analog, in our setting, of a familiar statement in quantum-mechanical deformation quantization. Recall (section 1.1) that in that setting, we re-quire that the associative product on the algebra Aq of quantum observables is relatedto the Poisson bracket on the Poisson algebra Acl of classical observables by the for-mula

a, b = limh→0

h−1[a, b]

where a, b are any lifts of the elements a, b ∈ Acl to Aq.

One can make a similar definition in the world of P0 algebras. If Acl is any commu-tative differential graded algebra, and Aq is a cochain complex flat over R[[h]] whichreduces to Acl modulo h, then we can define a cochain map

−,−Aq : Acl ⊗ Acl → Acl

which measures the failure of the commutative product on Acl to lift to a product onAq, to first order in h. (A precise definition is given in section 2.3).

Now, suppose that Acl is a P0 algebra (that is, a commutative dga equipped witha Poisson bracket of cohomological degree 1). Let Aq be a cochain complex flat overR[[h]] which reduces to Acl modulo h. We say that Aq is a weak quantization of Acl if


the bracket −,−Aq on Acl , induced by Aq, is homotopic to the given Poisson bracketon Acl .

This is a very weak notion, because the bracket −,−Aq on Acl need not be aPoisson bracket; it is simply a bilinear map. When we discuss the notion of strongquantization (section 1.8), we will explain how to put a certain operadic structure onAq which guarantees that this induced bracket is a Poisson bracket.

1.7.1. The weak quantization theorem. Now that we have the definition of weakquantization at hand, we can state our weak quantization theorem.

For every open subset U ⊂ M, Obscl(U) is a lax P0 algebra. Given a BV quantiza-tion of our classical field theory, Obsq(U) is a cochain complex flat over R[[h]] whichcoincides, modulo h, with Obscl(U). Our definition of weak quantization makes sensewith minor modifications for lax P0 algebras as well as for ordinary P0 algebras.

1.7.1.1 Theorem (The weak quantization theorem). For every U ⊂ M, the cochaincomplex Obsq(U) of classical observables on U is a weak quantization of the lax P0 algebraObscl(U).

1.8. The strong quantization conjecture

We have seen (section 1.7) how the observables of a quantum field theory are aquantization, in a weak sense, of the lax P0 algebra of observables of a quantum fieldtheory. The definition of quantization appearing in this theorem is unsatisfactory,however, because the bracket on the classical observables arising from the quantumobservables is not a Poisson observable.

In this section we will explain a stronger notion of quantization. We would like toshow that the quantization of the classical observables of a field theory we constructlifts to a strong quantization. However, this is unfortunately is still a conjecture (exceptfor the case of free fields).

1.8.0.2 Definition. A BD algebra is a cochain complex A, flat over R[[h]], equipped with acommutative product and a Poisson bracket of cohomological degree 1, satisfying the identity

d(a · b) = a · (db)± (da) · b + ha, b.

1.8. THE STRONG QUANTIZATION CONJECTURE 15

The BD operad is investigated in detail in section 2.4. Note that, modulo h, a BDalgebra is a P0 algebra.

1.8.0.3 Definition. A quantization of a P0 algebra Acl is a BD algebra Aq, flat over R[[h]],together with an equivalence of P0 algebras between Aq/h and Acl .

More generally, one can (using standard operadic techniques) define a concept ofhomotopy BD algebra. This leads to a definition of a homotopy quantization of a P0

algebra.

Recall that the classical observables Obscl of a classical field theory have the struc-ture of a P0 factorization algebra on our space-time manifold M.

1.8.0.4 Definition. Let F cl be a P0 factorization algebra on M. Then, a strong quantiza-tion of F cl is a lift of F cl to a homotopy BD factorization algebra F q, such that F q(U) is aquantization (in the sense described above) of F cl .

We conjecture that our construction of the factorization algebra of quantum ob-servables associated to a quantum field theory has this structure. More precisely,

Conjecture. Suppose we have a classical field theory on M, and a BV quantization of thetheory. Then, Obsq has the structure of a homotopy BD factorization algebra quantizing theP0 factorization algebra Obscl .

CHAPTER 2

Prefactorization algebras and basic examples

In this chapter we will give a formal definition of the notion of prefactorizationalgebra, and construct some relatively simple examples, related to free fields and tothe Kac-Moody vertex algebra.

2.1. Prefactorization algebras

Let M be a topological space and let (C,⊗) be a symmetric monoidal category. Weare particularly interested in the case where M is a smooth manifold and C is Vect ordgVect with the usual tensor product as the monoidal product. In this section we willgive a formal definition of the notion of a prefactorization algebra.

2.1.1. The essential idea of a prefactorization algebra. A prefactorization algebraF on M, taking values in cochain complexes, is a rule that assigns a cochain complexF (U) to each open set U ⊂ M along with

• a cochain map F (U)→ F (V) for each inclusion U ⊂ V;• a cochain map F (U1) ⊗ · · · ⊗ F (Un) → F (V) for every finite collection of

open sets where each Ui ⊂ V and the Ui are pairwise disjoint;• the maps are compatible in the obvious way (e.g. if U ⊂ V ⊂ W is a se-

quence of open sets, the map F (U) → F (W) agrees with the compositionthrough F (V)). It resembles a precosheaf, except that we tensor the cochaincomplexes rather than taking their direct sum.

2.1.2. Prefactorization algebras in the style of algebras over an operad.

2.1.2.1 Definition. Let FactM denote the following multicategory associated to M.

• The objects consist of all connected open subsets of M.

17

18 2. PREFACTORIZATION ALGEBRAS AND BASIC EXAMPLES

• For every (possibly empty) finite collection of open sets Uαα∈A and open set V,there is a set of maps FactM(Uαα∈A, V). If the Uα are pairwise disjoint and all arecontained in V, then the set of maps is a single point. Otherwise, the set of maps isempty.• The composition of maps is defined in the obvious way.

Remark: By “multicategory” we mean what is also called a colored operad or a pseudo-tensor category. In [Lei04], there is an accessible discussion of multicategories; inLeinster’s terminology, we work with “fat symmetric multicategories.”

2.1.2.2 Definition. A prefactorization algebra on M taking values in C is a functor (ofmulticategories) from FactM to C.

Remark: In other words, a prefactorization algebra is an algebra over the colored op-erad FactM.

Remark: When the monoidal product on C is the coproduct, then a precosheaf on M de-fines a prefactorization algebra. Hence, our definition broadens the idea of “inclusionof open sets leads to inclusion of sections” by allowing more general monoidal struc-tures to ”combine” the sections on disjoint open sets. Something analogous happenswhen we equip the category of abelian groups with the tensor product as its monoidalstructure.

Note that if F is any prefactorization algebra, then F (∅) is a commutative algebraobject of C.

2.1.2.3 Definition. We say a prefactorization algebra F is unital if the commutative algebraF (∅) is unital.

2.1.3. Prefactorization algebras in the style of precosheaves. Any multicategoryC has an associated symmetric monoidal category SC , which is defined to be theuniversal symmetric monoidal category equipped with a functor of multicategoriesC → SC . Concretely, an object of SC is a formal tensor product a1⊗ · · · ⊗ an of objectsof C . Morphisms in SC are characterized by the property that for any object b in C,the set of maps SC (a1⊗ · · · ⊗ an, b) in the symmetric monoidal category is exactly theset of maps C (a1, . . . , an, b) in the multicategory category SC .

2.2. STRUCTURED FACTORIZATION ALGEBRAS 19

We can give an alternative definition of prefactorization algebra by working withthe symmetric monoidal category S FactM rather than the multicategory FactM.

2.1.3.1 Definition. Let S FactM denote the following symmetric monoidal category.

• The objects of S FactM consist of topological spaces U equipped with a map U → Mwhich, on each connected component of U, is an open embedding embedding.• A map from U → M to V → M is a commutative diagram

U i−→ V↓ M

where the map i is an embedding.• The symmetric monoidal structure on S FactM is given by disjoint union.

2.1.3.2 Lemma. S FactM is the universal symmetric monoidal category containing the mul-ticategory FactM.

The alternative definition of prefactorization algebra is as follows.

2.1.3.3 Definition. A prefactorization algebra with values in a symmetric monoidal categoryC is a symmetric monoidal functor

S FactM → C .

Remark: Although “algebra” appears in its name, a prefactorization algebra only al-lows one to ”multiply” elements that live on disjoint open sets. The category of pref-actorization algebras (taking values in some fixed target category) has a symmetricmonoidal product, so we can study commutative algebra objects in that category. Asan example, we will consider the observables for a classical field theory (chapter 4).

2.2. Structured factorization algebras

In this section we will define what it means to have a factorization algebra en-dowed with the structure of an algebra over an operad. Since, in this paper, we areprincipally concerned with factorization algebras taking values in the category of dif-ferentiable cochain complexes we will restrict attention to this case in the present sec-tion.


Not all operads work for this construction: only operads endowed with an extrastructure – that of a Hopf operad can be used.

2.2.0.4 Definition. A Hopf operad is an operad in the category of differential graded cocom-mutative coalgebras.

Any Hopf operad P is, in particular, a differential graded operad. In addition,the cochain complexes P(n) are endowed with the structure of differential gradedcommutative coalgebra. The operadic composition maps

i : P(n)⊗ P(m)→ P(n + m− 1)

are maps of coalgebras, as are the maps arising from the symmetric group action onP(n).

If P is a Hopf operad, then the category of dg P-algebras becomes a symmetricmonoidal category. If A, B are P-algebras, the tensor product A ⊗C B is also a P-algebra. The structure map

PA⊗B : P(n)⊗ (A⊗ B)⊗n → A⊗ B

is defined to be the composition

P(n)⊗ (A⊗ B)⊗n c(n)−−→ P(n)⊗ P(n)⊗ A⊗n ⊗ B⊗n PA⊗PB−−−→ A⊗ B.

In this diagram, c(n) : P(n)→ P(n)⊗2 is the comultiplication on c(n).

Any dg operad which is the homology operad of an operad in topological spacesis a Hopf operad (because topological spaces are automatically cocommutative coal-gebras, with comultiplication defined by the diagonal map). For example, the com-mutative operad Com is a Hopf operad, with coproduct defined on the generator? ∈ Com(2) by

c(?) = ?⊗ ?.

With the comultiplication defined in this way, the tensor product of commutative al-gebras is the usual one. If A and B are commutative algebras, the product on A⊗ B isdefined by

(a⊗ b) ? (a′ ⊗ b′) = (−1)|a′||b|(a ? a′)⊗ (b ? b′).

2.2. STRUCTURED FACTORIZATION ALGEBRAS 21

The Poisson operad is also a Hopf operad, with coproduct defined (on the gener-ators ?, −,− by

c(?) = ?⊗ ?

c(−,−) = −,−⊗ ?+ ?⊗ −,−.

If A, B are Poisson algebras, then the tensor product A⊗ B is a Poisson algebra withproduct and bracket defined by

(a⊗ b) ? (a′ ⊗ b′) = (−1)|a′||b|(a ? a′)⊗ (b ? b′)

a⊗ b, a′ ⊗ b′ = (−1)|a′||b| (a, a′ ⊗ (b ? b′) + (a ? a′)⊗ b, b′

).

2.2.1. Structured factorization algebras. Let dgDiff denote the dg multicategoryof differentiable cochain complexes. If Fi, G are differentiable cochain complexes, thenan element of

Homr(F1, . . . , Fk | G)

is a smooth multilinear mapF1 × · · · × Fk → G

of cohomological degree r. The differential arises in the usual way from the differen-tials on Fi and on G; closed elements of degree 0 are multilinear cochain maps.

If P is a dg Hopf operad, then we can talk about P-algebras in the multicategoryof differentiable cochain complexes. If F is a differentiable cochain complex, then wecan define the endomorphism dg operad End(F) whose n’th component is

End(F)(n) = Hom(F, . . . , F | F).

Then, a P-structure on F is a map of dg operads

P→ End(F).

We call such an object a differentiable P-algebra.

Such P-algebras themselves form a multicategory, in a natural way. To see this,let S dgDiff denote the universal symmetric monoidal dg category containing the dgmulticategory dgDiff of differentiable cochain complexes. If F1, . . . , Fk are P-algebrasin DVS, then they are P-algebras in S DVS. Since P is a Hopf operad F1 ⊗ · · · ⊗ Fk isthen a P-algebra in S DVS. A morphism in the multicategory of P-algebras is then amap of P algebras

F1 ⊗ · · · ⊗ Fk → G.


Note that forgetting the the forgetful functor from the multicategory of differentiableP-algebras to that of differentiable cochain complexes is faithful. That is, the map ofsets

HomP(F1, . . . , Fk | G)→ HomdgDiff(F1, . . . , Fk | G)

is injective. Further, the image of this map lies in the space of closed degree 0 multi-morphisms; these are the same as multilinear cochain maps.

Thus, if Fi, G are differentiable P-algebras, and if

φ : F1 × · · · × Fk → G

is a smooth multilinear cochain map, we can ask whether φ is a map of P-algebras.

2.2.1.1 Definition. Let P be a differential graded Hopf operad. A prefactorization differen-tiable P-algebra is a prefactorization algebra with values in the multicategory of differentiableP-algebras. A factorization P-algebra is a prefactorization P-algebra, such that the underly-ing prefactorization algebra with values in differentiable cochain complexes is a factorizationalgebra.

We can unpack this definition as follows. Suppose that F is a factorization P-algebra. Then, F is a factorization algebra; and, in addition, for all U ⊂ M, F (U) is aP-algebra. The structure maps

F (U1)× · · · × F (Un)→ F (V)

(defined when U1, . . . , Un are disjoint open subsets of V) are required to be P-algebramaps in the sense defined above.

2.2.2. Commutative factorization algebras. One of the most important examplesis when P is the operad Com of commutative algebras. Then, we find that F (U) is acommutative algebra for each U. Further, if U1, . . . , Uk ⊂ V are as above, the productmap

m : F (U1)× · · · × F (Uk)→ F (V)

is compatible with the commutative algebra structures, in the following sense.

(1) If 1 ∈ F (Ui) is the unit for the commutative product on each F(Ui), then

m(1, . . . , 1) = 1.

2.3. THE P0 OPERAD 23

(2) If αi, βi ∈ F (Ui), then

m(α1β1, . . . , αkβk) = ±m(α1, . . . , αk)m(β1, . . . , βk)

where ± indicates the usual Koszul rule of signs.

Note that the axioms of a factorization algebra imply that F (∅) is the ground ring k(which we normally take to be R or C for classical theories and R[[h]] or C[[h]] forquantum field theories. The axioms above, in the case that k = 1 and U1 = ∅, implythat the map

F (∅)→ F (U)

is a map of unital commutative algebras.

If F is a commutative prefactorization algebra, then we can recover F uniquelyfrom the underlying cosheaf of commutative algebras. Indeed, the maps

F (U1)× · · · × F (Uk)→ F (V)

can be described in terms of the commutative product on F (V) and the maps

F (Ui)→ F (V).

2.3. The P0 operad

Recall that the collection of observables in quantum mechanics form an associativealgebra. The observables of a classical mechanical system form a Poisson algebra.In the deformation quantization approach to quantum mechanics, one starts with aPoisson algebra Acl , and attempts to construct an associative algebra Aq, which is analgebra flat over the ring C[[h]], together with an isomorphism of associative algebrasAq/h ∼= Acl . In addition, if a, b ∈ Acl , and a, b are any lifts of a, b to Aq, then

limh→0

1h[a, b] = a, b ∈ Acl .

This book concerns the analog, in quantum field theory, of the deformation quan-tization picture in quantum mechanics. We have seen that the sheaf of solutions tothe Euler-Lagrange equation of a classical field theory can be encoded by a commu-tative factorization algebra. A commutative factorization algebra is the analog, in oursetting, of the commutative algebra appearing in deformation quantization. We have


argued (section 1.6) that the observables of a quantum field theory should form a fac-torization algebra . This factorization algebra is the analog of the associative algebraappearing in deformation quantization.

In deformation quantization, the commutative algebra of classical observables hasan extra structure – a Poisson bracket – which makes it “want” to deform into an asso-ciative algebra. In this section we will explain the analogous structure on a commuta-tive factorization algebra which makes it want to deform into a factorization algebra.Later (section 4.13) we will see that the commutative factorization algebra associatedto a classical field theory has this extra structure.

2.3.1. The E0 operad.

2.3.1.1 Definition. Let E0 be the operad defined by

E0(n) =

0 if n > 0

R if n = 0

Thus, an E0 algebra in the category of real vector spaces is a real vector space witha distinguished element in it. More generally, an E0 algebra in a symmetric monoidalcategory C is the same thing as an object A of C together with a map 1C → A

The reason for the terminology E0 is that this operad can be interpreted as theoperad of little 0-discs.

The inclusion of the empty set into every open set implies that, for any factoriza-tion algebra F , there is a unique map from the unit factorization algebra R→ F .

2.3.2. The P0 operad. The Poisson operad is an object interpolating between thecommutative operad and the associative (or E1) operad. We would like to find ananalog of the Poisson operad which interpolates between the commutative operadand the E0 operad.

Let us define the Pk operad to be the operad whose algebras are commutativealgebras equipped with a Poisson bracket of degree 1 − k. With this notation, theusual Poisson operad is the P1 operad.

2.3. THE P0 OPERAD 25

Recall that the homology of the En operad is the Pn operad, for n > 1. Thus, justas the semi-classical version of an algebra over the E1 operad is a Poisson algebra inthe usual sense (that is, a P1 algebra), the semi-classical version of an En algebra is aPn algebra.

Thus, we have the following table:

Classical Quantum? E0 operad

P1 operad E1 operadP2 operad E2 operad

......

This immediately suggests that the P0 operad is the semi-classical version of the E0

operad.

Note that the P0 operad is a Hopf operad: the coproduct is defined by

c(?) = ?⊗ ?

c(−,−) = −,−⊗ ?+ ?⊗ −,−.

In concrete terms, this means that if A and B are P0 algebras, their tensor productA⊗ B is again a P0 algebra, with product and bracket defined by

(a⊗ b) ? (a′ ⊗ b′) = (−1)|a′||b|(a ? a′)⊗ (b ? b′)

a⊗ b, a′ ⊗ b′ = (−1)|a′||b| (a, a′ ⊗ (b ? b′) + (a ? a′)⊗ b, b′

).

2.3.3. P0 factorization algebras. Since the P0 operad is a Hopf operad, it makessense to talk about P0 factorization algebras. We can give an explicit description ofthis structure. A P0 factorization algebra is a commutative factorization algebra F , to-gether with a Poisson bracket of cohomological degree 1 on each commutative algebraF (U), with the following additional properties. Firstly, if U ⊂ V, the map

F (U)→ F (V)

must be a homomorphism of P0 algebras.

The second condition is that observables coming from disjoint sets must Poissoncommute. More precisely, let U1, U2 be disjoint subsets f V. . Let ji : F (Ui) → F (V)


be the natural maps. Let αi ∈ F (Ui), and ji(αi) ∈ F (V). Then, we require that

j1(α1), j2(α2) = 0 ∈ F (V)

where −,− is the Poisson bracket on F (V).

2.3.4. Quantization of P0 algebras. We know what it means to quantize an Pois-son algebra in the ordinary sense (that is, a P1 algebra) into an E1 algebra.

There is a similar notion of quantization for P0 algebras. A quantization is simplyan E0 algebra over R[[h]] which, modulo h, is the original P0 algebra, and for whichthere is a certain compatibility between the Poisson bracket on the P0 algebra and thequantized E0 algebra.

Let A be a commutative algebra in the category of cochain complexes. Let A1 bean E0 algebra flat over R[[h]]/h2, and suppose that we have an isomorphism of chaincomplexes

A1 ⊗R[[h]]/h2 R ∼= A.

In this situation, we can define a bracket on A of degree 1, as follows.

We have an exact sequence

0→ hA→ A1 → A→ 0.

The boundary map of this exact sequence is a cochain map

D : A→ A

(well-defined up to homotopy).

Let us define a bracket on A by the formula

a, b = D(ab)− (−1)|a|aDb− (Da)b.

Because D is well-defined up to homotopy, so is this bracket. However, unless D is anorder two differential operator, this bracket is simply a cochain map A⊗ A→ A, andnot a Poisson bracket of degree 1.

In particular, this bracket induces one on the cohomology H∗(A) of A. The coho-mological bracket is independent of any choices.

2.4. THE BEILINSON-DRINFELD OPERAD 27

2.3.4.1 Definition. Let A be a P0 algebra in the category of cochain complexes. Then a quan-tization of A is an E0 algebra A over R[[h]], together with a quasi-isomorphism of E0 algebras

A⊗R[[h]] R ∼= A,

which satisfies the following correspondence principle: the bracket on H∗(A) induced by Amust coincide with that given by the P0 structure on A.

In the next section we will consider a more sophisticated, operadic notion of quan-tization, which is strictly stronger than this one. To distinguish between the two no-tions, one could call the definition of quantization presented here a weak quantization,while the definition introduced later will be called a strong quantization.

2.4. The Beilinson-Drinfeld operad

Beilinson and Drinfeld [BD04] constructed an operad over the formal disc whichgenerically is equivalent to the E0 operad, but which at 0 is equivalent to the P0 operad.We call this operad the Beilinson-Drinfeld operad.

The operad P0 is generated by a commutative associative product− ?−, of degree0; and a Poisson bracket −,− of degree +1.

2.4.0.2 Definition. The Beilinson-Drinfeld (or BD) operad is the differential graded operadover the ring R[[h]] which, as a graded operad, is simply

BD = P0 ⊗R[[h]];

but with differential defined by

d(− ?−) = h−,−.

If M is a flat differential graded R[[h]] module, then giving M the structure of aBD algebra amounts to giving M a commutative associative product, of degree 0, anda Poisson bracket of degree 1, such that the differential on M is a derivation of thePoisson bracket, and the following identity is satisfied:

d(m ? n) = (dm) ? n + (−1)|m|m ? (dn) + hm, n

2.4.0.3 Lemma. There is an isomorphism of operads,

BD⊗R[[h]] R ∼= P0,


and a quasi-isomorphism of operads over R((h)),

BD⊗R[[h]] R((h)) ' E0 ⊗R((h)).

Thus, the operad BD interpolates between the P0 operad and the E0 operad.

BD is an operad in the category of differential graded R[[h]] modules. Thus, wecan talk about BD algebras in this category, or in any symmetric monoidal categoryenriched over the category of differential graded R[[h]] modules.

The BD algebra is, in addition, a Hopf operad, with coproduct defined in the sameway as in the P0 operad. Thus, one can talk about BD factorization algebras.

2.4.1. BD quantization of P0 algebras.

2.4.1.1 Definition. Let A be a P0 algebra (in the category of cochain complexes). A BDquantization of A is a flat R[[h]] module Aq, flat over R[[h]], which is equipped with thestructure of a BD algebra, and with an isomorphism of P0 algebras

Aq ⊗R[[h]] R ∼= A.

Similarly, an order k BD quantization of A is a differential graded R[[h]]/hk+1 moduleAq, flat over R[[h]]/hk+1, which is equipped with the structure of an algebra over the operad

BD⊗R[[h]] R[[h]]/hk+1,

and with an isomorphism of P0 algebras

Aq ⊗R[[h]]/hk+1 R ∼= A.

This definition applies without any change in the world of factorization algebras.

2.4.1.2 Definition. Let F be a P0 factorization algebra on M. Then a BD quantization of Fis a BD factorization algebra F equipped with a quasi-isomorphism

F ⊗R[[h]] R ' F

of P0 factorization algebras on M.

2.5. THE FACTORIZATION ALGEBRAS ASSOCIATED TO FREE SCALAR FIELD THEORIES 29

2.4.2. Operadic description of ordinary deformation quantization. We will fin-ish this section by explaining how the ordinary deformation quantization picture canbe phrased in similar operadic terms.

Consider the following operad BD1 over R[[h]]. BD1 is generated by two binaryoperations, a product ∗ and a bracket [−,−]. The relations are that the product isassociative; the bracket is antisymmetric and satisfies the Jacobi identity; the bracketand the product satisfy a certain Leibniz relation, expressed in the identity

[ab, c] = a[b, c]± [b, c]a

(where ± indicates the Koszul sign rule); and finally the relation

a ∗ b∓ b ∗ a = h[a, b]

holds. This operad was introduced by Ed Segal [Seg10].

Note that, modulo h, BD1 is the ordinary Poisson operad P1. If we set h = 1, wefind that BD1 is the operad E1 of associative algebras. Thus, BD1 interpolates betweenP1 and E1 in the same way that BD0 interpolates between P0 and E0.

Let A be a P1 algebra. Let us consider possible lifts of A to a BD1 algebra.

2.4.2.1 Lemma. A lift of A to a BD1 algebra, flat over R[[h]], is the same as a deformationquantization of A in the usual sense.

PROOF. We need to describe BD1 structures on A[[h]] compatible with the givenPoisson structure. To give such a BD1 structure is the same as to give an associativeproduct on A[[h]], linear over R[[h]], and which modulo h is the given commutativeproduct on A. Further, the relations in the BD1 operad imply that the Poisson bracketon A is related to the associative product on A[[h]] by the formula

h−1 (a ∗ b∓ b ∗ a) = a, b mod h.

2.5. The factorization algebras associated to free scalar field theories

In this section, we will construct the factorization algebra associated to the freescalar field theory on a Riemannian manifold. We will show that, for one-dimensional


manifolds, this factorization algebra recovers the familiar Weyl algebra, the algebraobservables for quantum mechanics.

2.5.1. Before we describe these basic examples, we need to introduce some nota-tion.

Let E be a graded vector bundle on a manifold M, and let U be an open subsetof M. We let E (U) denote the cochain complex of smooth sections of E on U, andwe let Ec(U) denote the cochain complex of compactly supported sections of E on U.Throughout this book, we will often use the notation E (U) to denote the distributionalsections on U, defined by

E (U) = E (U)⊗C∞(U) D(U),

where D(U) is the space of distributions on U. Similarly, let E c(U) denote the com-pactly supported distributional sections of E on U. There are natural inclusions

Ec(U) → E c(U) → E (U),

Ec(U) → E (U) → E (U),

by viewing smooth functions as distributions.

If, as above, E is a graded vector bundle on M, let E! = E∨ ⊗ DensM. We giveE ! a differential that is the formal adjoint to Q on E. Let E !(U), E !

c (U) denote thecochain complexes of smooth and compactly supported sections of E!, and let E

!(U)

and E!c(U) denote the cochain complexes of distributional and compactly-supported

distributional sections of E!.

Note that E c(U) is the continuous dual to E !(U), and that Ec(U) is the continuousdual to E

!(U).

The prefactorization algebras we will discuss are constructed from symmetric al-gebras on the vector spaces Ec(U) and E c(U). This symmetric algebra can be definedin two ways: either using the completed projective tensor product of topological vec-tor spaces, or in terms of sections of bundles on Un. For concreteness, we will discussthe latter construction.

Thus, let us define (Ec(U))⊗n to be the tensor power defined using the completedprojective tensor product on the topological vector space Ec(U). Concretely, if En


denotes the vector bundle on Mn obtained as the external tensor product, then

(Ec(U))⊗n = Γc(Un, En)

is the compactly supported smooth sections of En on Un. Similarly, we have

(E c(U))⊗n = Γc(Un, En)

is the compactly supported distributional sections of En on Mn.

Symmetric (or exterior) powers of Ec(U) and E c(U) are defined by taking coin-variants of Ec(U)⊗n with respect to the action of the symmetric group Sn. Thus, wecan define, for example, the symmetric algebra

Sym∗ E !c (U) =

⊕n

Symn E !c (U)

Sym∗ E!c(U) =

⊕n

Symn E!c(U)

Note that, since E!c(U) is dual to E (U), we can view Sym E

!c(U) as the algebra of

polynomial functions on E (U). Thus, we often write

O(E (U)) = Sym E!c(U)

for this algebra of functions. Similarly, Sym E !c (U) is the algebra of polynomial func-

tions on E (U).

2.5.2. Defining the prefactorization algebra. After this preliminary discussion,we can introduce the the prefactorization algebras associated to free field theories. LetM be a Riemannian manifold, and so M is equipped with a natural density, arisingfrom the metric. We will use this natural density to integrate functions, and also toprovide an isomorphism between functions and densities that we use implicitly fromhereon. The field theory we will discuss has as fields φ ∈ C∞(M) and has as actionfunctional

S(φ) =∫

Mφ4φ,

where 4 is the Laplacian on M. (Normally we will reserve the symbol 4 for theBatalin-Vilkovisky Laplacian, but that’s not necessary in this section.)

If U ⊂ M is an open subset, then the space of solutions to the equation of motionon U is the space of harmonic functions on U.


In this book, we will always consider the derived space of solutions of the equationof motion. For more details about the derived philosophy, the reader should consultChapter 4. In this simple situation, the derived space of solutions to the free fieldequations, on an open subset U ⊂ M, is the two-term complex

E (U) =

(C∞(U)0 4−→ C∞(U)1

),

where the superscripts indicate the cohomological degree.

The classical observables of a field theory on an open subset U ⊂ M are functionson the derived space of solutions to the equations of motion on U. We will take ourfunctions to be polynomial functions, and thus the symmetric algebra of the dual. Thedual to the two-term complex E (U) above is the complex

E (U)∨ =

(Dc(U)−1 4−→ Dc(U)0

),

where Dc(U) indicates the space of compactly supported distributions on U.

Thus, as a first pass, one would want to define the classical observables as thesymmetric algebra on E (U)∨. (When we say symmetric algebra, we use the completedprojective tensor product of the topological vector space Dc(U).)

However, this will lead to difficulties defining the quantum observables. When wework with an interacting theory, these difficulties can only be surmounted using thetechniques of renormalization. For a free field theory, though, there is a much simplersolution.

We haveE !

c (U) ∼=(

C∞c (U)−1 → C∞

c (U)0)

,

using our identification between densities and functions. Note that there is a naturalmap of cochain complexes E !

c (U)→ E (U)∨, given by viewing a compactly supportedfunction as a distribution.

2.5.2.1 Lemma. The map E !c (U)→ E (U)∨ is a continuous homotopy equivalence.

PROOF. This is a special case of a general result proved in the appendix A.9. Notethat by continuous homotopy equivalence we mean that there is a continuous inversemap E (U)∨ → E !

c (U) and continuous cochain homotopies between the two composedmaps and the identity maps.


This lemma says that at the level of cohomology, we can replace a distributionallinear observable by a smooth linear observable. In more physical language, it saysthat it suffices to work with “smeared observables.”

Thus, we will define our classical observables to be

Obscl(U) = Sym∗(E !c (U)) = Sym∗(C∞

c (U)−1 4−→ C∞c (U)0),

the symmetric algebra on E !c (U), defined using the completed projective tensor prod-

uct of nuclear spaces. In section A.7 we describe some generalities about symmetricalgebras on spaces of sections of a vector bundle. In this case, the symmetric algebrais quite simple.

For example,

(2.5.2.1) Sym2(

C∞c (U)−1 4−→ C∞

c (U)0)

=(C∞

c (U2))−2

Alt2→ C∞

c (U2)−1 →(C∞

c (U2)S2

).

Here C∞c (U2)Alt2 indicates the coinvariants for the alternating action of the symmetric

group S2 on C∞c (U2).

2.5.3. The one-dimensional case, in detail. This space is particularly simple indimension 1.

2.5.3.1 Lemma. If U = (a, b) ⊂ R is an interval in R, then the algebra of classical observ-ables for the free field has cohomology

H∗(Obscl((a, b))) = R[p, q],

the polynomial algebra in two variables.

PROOF. We will show that the complex

E !c ((a, b)) =

(C∞

c ((a, b))−1 4−→ C∞c ((a, b))0

)is continuously homotopy equivalent to the complex R2 situated in degree 0.

Let us view the complex E !c ((a, b)) as a subspace of the dual of the complex of

fields

E ((a, b)) =(

C∞((a, b))0 4−→ C∞((a, b))1)

.


We denote an arbitrary field of degree 0 by φ ∈ C∞((a, b))0 and an arbitrary field ofdegree 1 by ψ ∈ C∞((a, b))1.

Choose a function f ∈ C∞c (a, b) with the property that

∫R

f (x)dx = 1. Then, let usdefine momentum and position observables p, q ∈ E !

c ((a, b)) by

q(φ, ψ) =∫

φ(x) f (x)dx,

p(φ, ψ) = −∫

φ(x)d fdx

dx.

These are the natural distributions associated to f and f ′ respectively. Note that p, qare in degree 0. We need to show that the cochain map

I : Rp, q → E !c ((a, b))

is a continuous homotopy equivalence (where Rp, q refers to the vector space spannedby p and q).

To define the inverse map, it suffices to write down a closed subspace K ⊂ C∞c ((a, b))

such that the quotient map

π : C∞c ((a, b))→ C∞

c ((a, b))/K

has dimension 2, with basis the images of f and f ′. For K, we take the space of func-tions g with the following properties.

(1)∫ b

a g(x)dx = 0.(2) The first property implies that g has a compactly supported antiderivative G.

We then require that∫ b

a G(x)dx = 0.

It is easy to verify that C∞c ((a, b)) is spanned by functions g ∈ K and by f and f ′.

Indeed, for h ∈ C∞c ((a, b)), let

c1 =∫ b

ah(x)dx

and set h1 = h − c1 f . Thus∫ b

a h1(x)dx = 0 and hence has a compactly supportedantiderivative H. Repeating this construction, we find the constant c2 such that H −c2 f satisfies

∫ ba H(x)− c2 f (x)dx = 0. Hence,

h(x) = h2(x) + c1 f (x) + c2 f ′(x),

where h2 lives in K.


Next, we need to define a homotopy between the identity and the endomorphismi π of E !

c ((a, b)). Equivalently, we need a homotopy that contracts the two-termcomplex

C∞c ((a, b))−1 4−→ K0.

The homotopy S is defined by sending g ∈ K to

S(g)(x) =∫ x

y=a

∫ y

u=ag(u)dudy

The result of performing this iterated integral has compact support, by the assumptionthat g ∈ K.

Thus we have constructed the desired continuous homotopy equivalence.

It is clear that classical observables form a prefactorization algebra. Indeed, Obscl(U)

is a commutative differential graded algebra. If U ⊂ V, there is a natural algebra ho-momorphism

iUV : Obscl(U)→ Obscl(V),

which on generators is just the natural map C∞c (U)→ C∞

c (V), extension by zero.

If U1, . . . , Un ⊂ V are disjoint open subsets, the prefactorization structure map isthe continuous multilinear map

Obscl(U1)× · · · ×Obscl(Un) → Obscl(V)

(α1, . . . , αn) 7→ ∏ni=1 iUi

V αi.

2.5.4. The Poisson bracket. We now return to the general case.

2.5.4.1 Lemma. There is a unique continuous Poisson bracket on Obscl(U) of cohomologicaldegree 1, with the property that for α ∈ C∞

c (U)−1 and β ∈ C∞c (U)−1, we have

α, β =∫

Uαβ.

PROOF. Uniqueness is immediate, since the dg algebra Obscl(U) is topologicallygenerated by α ∈ C∞

c (U)−1 and β ∈ C∞c (U)−1

Existence follows from the fact that the multilinear map

C∞c (Un)× C∞

c (U)× C∞c (U) → C∞

c (Un)

( f , α, β) 7→ f∫

U αβ


is continuous, and so extends to a map from the completed projective tensor product

C∞c (Un+2) = C∞

c (Un)⊗ C∞c (U)⊗ C∞

c (U)→ C∞c (Un).

Note that for U1, U2 disjoint open subsets of V and for observables αi ∈ Obscl(Ui),we have

iU1V α1, iU2

V α2 = 0.

That is, observables coming from disjoint open subsets “commute” with respect to thePoisson bracket. This means that Obscl(U) defines a P0 prefactorization algebra. (Wewill see later in section 3.2 that this prefactorization algebra is actually a factorizationalgebra.)

2.5.5. Quantum observables. As we explained in section 1.8, our philosophy isthat we should take a P0 factorization algebra and deform it into a BD factorizationalgebra. In the situation we are considering in this section, we will construct a fac-torization algebra of quantum observables Obsq with the property that, as a vectorspace,

Obsq(U) = Obscl(U)[h],

but with a differential d such that

(1) modulo h, d coincides with the differential on Obscl(U), and(2) the equation

d(a · b) = (da) · b + (−1)|a|a · b + ha, b

holds.

Here, · indicates the commutative product on Obscl(U). These properties imply thatObsq defines a BD factorization algebra quantizing the P0 factorization algebra Obscl .

Our construction starts with a certain graded Heisenberg Lie algebra. Let

H(U) =

(C∞

c (U)0 4−→ C∞c (U)1

)⊕ (Rh)1


where Rh is situated in degree 1. We give H(U) a Lie bracket by saying that, forα ∈ C∞

c (U)0 and β ∈ C∞c (U)1, the bracket is

[α, β] = h∫

Uαβ.

Thus, H(U) is a graded version of a Heisenberg algebra, centrally extending theabelian dg Lie algebra C∞

c (U)0 → C∞c (U)1.

Let

Obsq(U) = C∗(H(U)),

where C∗ denotes the Chevalley-Eilenberg complex for the Lie algebra homology ofH(U), defined using the completed projective tensor product. Thus,

Obsq(U) = (Sym∗H(U)[1], d)

=(

Obscl(U)[h], d)

where the differential arises from the Lie bracket on H(U). (Note that we alwayswork with cochain complexes, so our grading convention of C∗ is the negative of onepopular convention.)

Remark: Those readers who are operadically inclined might notice that the Lie algebrachain complex of a Lie algebra g is the E0 version of the universal enveloping algebraof a Lie algebra. Thus, our construction is an E0 version of the familiar construction ofthe Weyl algebra as a universal enveloping algebra of a Heisenberg algebra.

Next, we need to give Obsq(U) the structure of a prefactorization algebra. Observethat if U ⊂ V, there is a natural map of dg Lie algebras H(U) → H(V) arising fromthe natural map C∞

c (U) → C∞c (V), extended to the identity on the central extension.

Similarly, if U1, . . . , Un ⊂ V are disjoint, there is a natural map of dg Lie algebras

H(U1)⊕ · · · ⊕H(Un)→ H(V).

Applying Chevalley-Eilenberg chains to this map yields the factorization algebra struc-ture maps

Obsq(U1)× · · · ×Obsq(Un)→ Obsq(V).

It is easy to verify that Obsq satisfies the axioms of a prefactorization algebra in BDalgebras. It follows from the fact that Obscl is a factorization algebra (which we provein section 3.2) that Obsq is a factorization algebra over R[h].


2.5.6. Quantum mechanics and the Weyl algebra. Next, we will calculate the co-homology of the prefactorization algebra of the free scalar field theory in dimension1, and we will show that it recovers the familiar Weyl algebra, which is the algebra ofobservables in quantum mechanics.

First, we need to explain why one can recover an associative algebra from a factor-ization algebra on the real line.

2.5.6.1 Definition. Let F be a prefactorization algebra on R taking values in the category ofvector spaces (without any grading). We sayF is locally constant if the mapF (U)→ F (V)

is an isomorphism whenever the inclusion of opens U ⊂ V is a homotopy equivalence.

2.5.6.2 Lemma. Let F be a locally constant, unital prefactorization algebra on R taking val-ues in vector spaces. Let A = F (R). Then A has a natural structure of an associative algebra.

Remark: Recall thatF being unital means that the commutative algebraF (∅) is equippedwith a unit. We will find that A is an associative algebra over F (∅).

PROOF. For any interval (a, b) ⊂ R, the map

F ((a, b))→ F (R) = A

is an isomorphism. Thus, we have a canonical isomorphism

A = F ((a, b))

for all intervals (a, b).

Notice that if (a, b) ⊂ (c, d) then the diagram

A∼= //

Id

F ((a, b))

i(a,b)(c,d)

A∼= // F ((c, d))

commutes.

The product map m : A⊗ A → A is defined as follows. Let a < b < c < d. Then,the prefactorization structure on F gives a map

F ((a, b))⊗F ((c, d))→ F ((a, d)),


and so, after identifying F ((a, b)), F ((c, d)) and F ((a, d)) with A, we get a map

A⊗ A→ A.

This is the multiplication in our algebra.

It remains to check the following.

(1) This multiplication doesn’t depend on the intervals (a, b)q (c, d) ⊂ (a, d) wechose, as long as (a, b) < (c, d).

(2) This multiplication is associative and unital.

This is an easy (and instructive) exercise.

Finally, we can prove that our construction of the factorization algebra for a freefield theory, when restricted to dimension 1, reconstructs the Weyl algebra associatedto quantum mechanics.

2.5.6.3 Proposition. For Obsq denote the factorization algebra on R constructed from the freefield theory, as above, we have

(1) the cohomology H∗(Obsq) is locally constant, and(2) the corresponding associative algebra is the Weyl algebra, the associative algebra gen-

erated by p, q, h with the relation [p, q] = h and all other commutators being zero.

PROOF. First we will show that the cohomology H∗(Obsq) is locally constant. Re-call that Obsq(U) is the Lie algebra chains on the Heisenberg Lie algebraH(U). Let usfilter Obsq(U) by saying that

F≤i Obsq(U) = Sym≤i(H(U)[1]).

The associated graded for this filtration is Obscl(U)[h]. Thus, to show that H∗Obsq islocally constant, it suffices (by considering the spectral sequence associated to this fil-tration) to show that H∗Obscl is locally constant. We have already seen that H∗(Obscl(a, b)) =R[p, q] for any interval (a, b), and that the inclusion maps (a, b)→ (a′, b′) induces iso-morphisms. Thus, the cohomology of Obscl is locally constant.


Next, we will show that the associative algebra corresponding to H∗Obsq is theWeyl algebra. Recall that

Obsq((a, b)) = Sym∗(

C∞c ((a, b))−1 ⊕ C∞

c ((a, b))0)[h]

with a certain differential. We view Obsq((a, b)) as a space of functionals on the space

C∞((a, b))0 4−→ C∞((a, b))1 of fields. We will denote an arbitrary field of cohomologicaldegree 0 by φ ∈ C∞((a, b))0 and an arbitrary field of cohomological degree 1 by ψ ∈C∞((a, b))1.

Choose a function f0 ∈ C∞c ((− 1

2 , 12 )) with the property that

∫ ∞−∞ f0(x)dx = 1. Let

ft ∈ C∞c ((t− 1

2 , t + 12 )) be ft(x) = f (x− t). We define observables Pt, Qt by

Qt(φ, ψ) =∫

Rφ(x) ft(x)dx

Pt(φ, ψ) =∫

Rφ′(x) ft(x)dx.

Note that Qt and Pt represent measurements of positions and momenta of the field φ

in a neighborhood of t.

Because the cohomology classes [P0], [Q0] generate the commutative algebra H∗(Obscl(R)),it is automatic that they still generate the associative algebra H0 Obsq(R). We thusneed to show that they satisfy the Heisenberg commutation relation

[[P0], [Q0]] = h

for the associative product on H0 Obsq(R), which is an associative algebra by virtueof the fact that H∗Obsq is locally constant.

The first thing we need to show is that the cohomology class [Pt] ∈ H0 Obsq(R) isindependent of t. (This is conservation of momentum.)

To see this, let

hs,t(x) =∫ x

−∞( fs(u)− ft(u))du.

Let us define an observable

Hs,t(φ, ψ) =∫

Rhs,t(x)ψ(x)dx.


Thus Hs,t is an element of Sym2 of cohomological degree −1. Let d denote the differ-ential on Obsq(R). We have

(dHs,t) (φ, ψ) =∫

Rhs,t(x)φ′′(x)dx

= −∫

Rh′s,t(x)φ′(x)dx

= Pt(φ)− Ps(φ).

Thus, [Pt] = [Ps].

Note that if |t| > 1, the observables Pt and Q0 have disjoint support. This meansthat we can use the factorization structure map

Obsq((− 12 , 1

2 ))⊗Obsq((t− 12 , t + 1

2 ))→ Obsq(R)

to define a product observable

Q0 · Pt ∈ Obsq(R).

Concretely,

(Q0 · Pt) (φ, ψ) = Q0(φ, ψ)Pt(φ, ψ).

We will let ? denote the associative multiplication on H0 Obsq(R). We defined thismultiplication by

[Q0] ? [P0] = [Q0 · Pt] if t > 1

[P0] ? [Q0] = [Q0 · Pt] if t < −1.

Thus, it remains to show that, if t > 1,

[Q0 · Pt]− [Q0 · P−t] = h.

We will construct an observable whose differential is the difference between theleft and right hand sides. Consider the observable

S(φ, ψ) = Q0(φ, ψ)H−t,t(φ, ψ)

=

(∫R

f0(x)φ(x)dx)(∫

Rh−t,t(y)ψ(y)dy

),

where the functions f0 and h−t,t were defined above.


Recall that the differential on Obsq(R) has two terms: one coming from the Lapla-cian4mapping C∞

c (R)−1 to C∞c (R)0, and one arising from the bracket of the Heisen-

berg Lie algebra. The second term maps

Sym2(

C∞c (R)−1 ⊕ C∞

c (R)0)→ Rh.

Applying this differential to the observable S, we find that

(dS)(φ, ψ) =

(∫R

f0(x)φ(x)dx)(∫

Rh−t,t(y)φ′′(y)dy

)+ h

∫R

h−t,t(x) f0(x)dx

=Q0(φ, ψ) (Pt(φ, ψ)− P−t(φ, ψ)) + h.

Thus, [dS] = 0 so [Q0P−t]− [Q0Pt] = h, as desired.

2.6. Standard quantum mechanics as a prefactorization algebra

This section is a brief digression from the central theme of the book. Throughoutthis book we take the path integral formalism as fundamental, and hence we do notfocus on the Hamiltonian, or operator, approach to quantum physics. In this section,however, we will explain how to express the standard formalism of quantum mechan-ics in the language of prefactorization algebras.

2.6.1. Quantum mechanics.

Remark: Our goal here is to sketch the formal aspects of quantum mechanics, so weavoid technical issues (such as boundedness of operators or whether they are trace-class) by describing the simpler finite-dimensional setting.

Let V denote a finite-dimensional Hilbert space, A the continuous endomorphisms,and H ∈ A the Hamiltonian operator. We view V as a state space for our system, Aas where the observables live, and H as determining the time evolution of our system.We seek to describe the following experimental situation, which one might view as ascattering experiment:

• at time t = 0, we prepare our system in the initial state v0 ∈ V;

2.6. STANDARD QUANTUM MECHANICS AS A PREFACTORIZATION ALGEBRA 43

• we modify the governing Hamiltonian over some finite time interval (i.e.,apply an operator, i.e., an observable);• at time t = T, we measure whether our system is in the final state v1 ∈ V. If

we run this experiment many times, with the same initial and final states andthe same operator, we should find a statistical pattern in our data. To put thisin the usual Dirac notation, if we denote the operator by O and idealize it ashappening at a fixed moment t0 in time, then we are trying to compute thenumber

〈v1|ei(T−t0)HOeit0 H |v0〉.

Remark: A state is actually a ray, or line, in V. We address this issue in the next subsec-tion.

We want to describe this situation using a prefactorization algebra F on the interval[0, T]. Before jumping into details, here’s the guiding idea. Interior open intervalsdescribe moments when operators can act on our system. An interval that contains 0(but not the other end) should describe a state of the system (and dually for intervalscontaining the other endpoint). But not only do we assign V to a connected intervalcontaining exactly one endpoint and A to a connected interior interval; we also havea distinguished vector in each of these vector spaces. It is important to recall that aunital prefactorization algebra F always assigns a “pointed” vector space to an openset: F(∅) = C and so the inclusion of the empty set into an open U always gives a mapC→ F(U). Since the empty set has empty intersection with itself, we see that we havea distinguished map F(∅)⊗ F(∅) → F(∅), namely multiplication. Hence we assignC with multiplication to the empty set, and so we have a distinguished element in F(∅),namely 1, which picks out a distinguished vector in every F(U). Thus, a factorizationalgebra assigns a pair (F(U), u ∈ F(U)) to each open set.

Returning to quantum mechanics, we fix a vector v0 ∈ V and a vector v1 ∈ V. Toopen subintervals, our functor assigns the following vector spaces:

• [0, t) 7→ (V, eitHv0)

• (s, t) 7→ (A, ei(t−s)H)

• (t, T] 7→ (V, e−i(T−t)Hv1).

The first type of interval describes how the initial condition has evolved up to timet. The interior interval describes the possible operators that can act during that time


interval, and the evolution operator is the natural distinguished operator. The finaltype of interval describes the state that will evolve to the final condition. We deferdescribing F([0, T].

We must now describe the maps coming from inclusion of intervals. Hopefully,we give enough examples to pin down the idea.

First, we describe inclusions of connected intervals.

• For [0, s) ⊂ [0, t), we use the map v 7→ ei(t−s)Hv. This is an automorphism ofV sending eisHv0 to eitHv0, and so it respects our marked points.• For (s, t) ⊂ (s′, t′), we use O 7→ ei(t′−t)HOei(s−s′)H.• For (s, t) ⊂ [0, t′), we use O 7→ ei(t′−t)HOeisHv0. This sends an operator in A

to how it acts on the input state v0.• For (s, t) ⊂ (s′, T], we use O 7→ e−i(s−s′)HO†e−i(T−t)Hv1.

Next we describe the three simplest types of “multiplication,” namely a pair ofdisjoint intervals maps into a bigger, connected interval.

• For [0, s) ∪ (s′, t) ⊂ [0, t), we use v ⊗O 7→ Oei(s′−s)Hv. This map describeshow an operator acts on a state.• For (s, s′) ∪ (t, t′) ⊂ (s, t′), we use O ⊗P 7→ Pei(t−s′)HO.• For (s, s′) ∪ (t, T] ⊂ (s, T], we use O ⊗ v 7→ O†e−i(t−s′)Hv.

Finally, we describe what our functor assigns to the whole interval [0, T]. By thegluing axiom (described in the chapter on factorization algebras), we see that it mustassign the vector space V ⊗A V, where the left-hand V corresponds to the incomingstates (and A acts by multiplication) and the right-hand V corresponds to the outgoingstate (and A acts by multiplication of the adjoint). Hence we can view this as V⊗End(V)

V∨, which is isomorphic to the ground field C: we have the map

V ⊗C A⊗C V → V ⊗A V

v⊗O ⊗ w 7→ 〈w|Ov〉,

which is precisely the usual inner product.

The maps in this factorization algebra thus allow us to compute scattering as fol-lows. Suppose we apply some operator during the interval (s, t). The inclusion of

2.7. TRANSLATION-INVARIANT AND HOLOMORPHIC PREFACTORIZATION ALGEBRAS 45

intervals (s, t) → [0, T] yields a map A → V ⊗A V ∼= C. If we pick an operator Oacting during the interval (s, t), then its image in C is 〈v1|ei(T−t)HOeisH |v0〉.

2.6.2. Some subtleties. Our construction above captures much of the standardformalism of quantum mechanics, but there are a few loose ends we need to address.

First, in standard quantum mechanics, a state is not a vector in V but a line. Above,however, we fixed vectors v0 and v1, so there seems to be a discrepancy. The obser-vation that rescues us is a natural one, from the mathematical viewpoint. Considerscaling v0 and v1 by elements of C×. This defines a new factorization algebra, but it isisomorphic to what we described above, and the expectation value “〈v0|Ov1〉” of anoperator depends linearly in the rescaling of the input and output vectors. More pre-cisely, there is a natural equivalence relation we can place on the factorization algebrasdescribed above that corresponds to the usual notion of state in quantum mechanics.

Another issue that might bother the reader is that our formalism only matchesnicely with experiments that resemble scattering experiments. It does not seem well-suited to descriptions of systems like bound states (e.g., an atom sitting quietly, mind-ing its own business). For such systems, we might consider running over the wholespace of states (as described in the previous paragraph). Alternatively, we might dropthe endpoints and simply work with the factorization algebra on the open interval,which focuses on the algebra of operators A.

2.7. Translation-invariant and holomorphic prefactorization algebras

In this section we will analyze in detail the notion of translation-invariant prefactor-ization algebras on Rn or Cn. On Cn we can ask for a translation-invariant prefactor-ization algebra to have a holomorphic structure; this implies that all structure mapsof the prefactorization algebra are (in a sense we will explain shortly) holomorphic.There are many natural field theories where the corresponding prefactorization alge-bra is holomorphic: for instance, chiral conformal field theories in complex dimension1, and minimal twists [Cos11c] of supersymmetric field theories in complex dimension2.


2.7.1. We now turn to the definition of a translation-invariant prefactorizationalgebra. If U ⊂ Rn and x ∈ Rn, let

Tx(U) := y : y− x ∈ U

denote the translate of U by x.

2.7.1.1 Definition. A prefactorization algebra F on Rn is discretely translation invariantif we have isomorphisms

Tx : F (U) ∼= F (Tx(U))

for all x ∈ Rn and all open subsets U ⊂ Rn. These isomorphisms must satisfy a few condi-tions. First, we require that Tx Ty = Tx+y for every x, y ∈ Rn. Second, for all disjoint opensubsets U1, . . . , Uk in V, the diagram

F (U1)⊗ · · · ⊗ F (Uk)Tx //

F (TxU1)⊗ · · · ⊗ F (TxUk)

F (V)

Tx // F (TxV)

commutes. (Here the vertical arrows are the structure maps of the prefactorization algebra.)

We are interested in a refined version of this notion, where the structure maps ofthe prefactorization algebra depend smoothly on the position of the open sets. It is abit subtle to talk about “smoothly varying an open set,” and in order to do this, weintroduce some notation.

Firstly, we need to introduce the notion of a derivation of a prefactorization algebraon a manifold M. We will construct a differential graded Lie algebra of derivations ofany prefactorization algebra.

2.7.1.2 Definition. A degree k derivation of a prefactorization algebra F is a collection ofmaps DU : F (U) → F (U) of cohomological degree k for each open subset U ⊂ M, with theproperty that, if U1, . . . , Un ⊂ V are disjoint, and αi ∈ F (Ui), then

DVmU1,...,UnV (α1, . . . , αn) = ±∑ mU1,...,Un

V (α1, . . . , DUi αi, . . . , αn),

where ± indicates the usual Koszul rule of signs.

Let Derk(F ) denote the derivations of degree k; it is easy to verify that Der∗(F )forms a differential graded Lie algebra. The differential is defined by (dD)U = [dU , DU ],


where dU is the differential on F (U). The Lie bracket is defined by

[D, D′]U = [DU , D′U ].

The concept of derivation allows us to talk about the action of a dg Lie algebra ona prefactorization algebra F . Such an action is simply a homomorphism of differentialgraded Lie algebras

g→ Der∗(F ).

Next, let us introduce some notation which will help us describe the smoothnessconditions for a discretely translation-invariant prefactorization algebra.

Let U1, . . . , Uk ⊂ V be disjoint open subsets. Let W ⊂ (Rn)k be the set of thosex1, . . . , xk such that the sets Tx1(U1), . . . , Txk(Uk) are all disjoint and contained in V. Itparametrizes the way we can move the open sets without causing overlaps. Let usassume that W has non-empty interior, which happens when the closure of the Ui aredisjoint and contained in V.

Let F be any discretely translation-invariant prefactorization algebra. Then, foreach (x1, . . . , xk) ∈W, we have a multilinear map obtained as a composition

mx1,...,xk : F (U1)× · · · × F (Uk)→ F (Tx1U1)× · · · × F (Txk Uk)→ F (V),

where the second map arises from the inclusion

Tx1U1 q · · · q Txk Uk → V.

2.7.1.3 Definition. A discretely translation invariant prefactorization algebraF is smoothlytranslation invariant if the following conditions hold.

(1) The map mx1,...,xk above depends smoothly on (x1, . . . , xk) ∈W.(2) The prefactorization algebra F is equipped with an action of the Abelian Lie algebra

Rn of translations. If v ∈ Rn, we will denote the corresponding action maps by

ddv

: F (U)→ F (U).

We view this Lie algebra action as an infinitesimal version of the global translationinvariance.


(3) The infinitesimal action is compatible with the global translation invariance in thefollowing sense. If v ∈ Rn, let vi ∈ (Rn)k denote the vector with v placed in the ithposition and 0 in the other k− 1 slots. If αi ∈ F (Ui), then we require that

ddvi

mx1,...,xk(α1, . . . , αk) = mx1,...,xk

(α1, . . . ,

ddv

αi, . . . , αk

).

When we refer to a translation invariant prefactorization algebra without furtherqualification, we will always mean a smoothly translation invariant prefactorizationalgebra.

Remark: As always, we work with prefactorization algebras taking values in the cat-egory of differentiable cochain complexes. Generalities about differentiable cochaincomplexes are developed in appendix A. There we explain what it means for a smoothmultilinear map between differentiable cochain complexes to depend smoothly onsome parameters.

2.7.2. Next, we will explain how to think of the structure of a translation-invariantprefactorization algebra on Rn in more operadic terms. This description has a lot incommon with the En algebras familiar from topology.

Let r1, . . . , rk, s ∈ R>0. Let

Discsn(r1, . . . , rk | s) ⊂ (Rn)k

be the (possibly empty) open subset consisting of x1, . . . , xk ∈ Rn with the propertythat the closures of the balls Bri(xi) are all disjoint and contained in Bs(0) (where Br(x)denotes the open ball of radius r around x).

2.7.2.1 Definition. Let Discsn be the R>0-colored operad in the category of smooth mani-folds whose space of k-ary morphisms is the space Discsn(r1, . . . , rk | s) between ri, s ∈ R>0

described above.

Note that a colored operad is the same thing as a multicategory (recall remark2.1.2). An R>0-colored operad is thus a multicategory whose set of objects is R>0.


The essential data of the colored operad structure on the spaces Discsn(r1, . . . , rk |s) is the following. We have maps

i : Discsn(r1, . . . , rk | ti)×Discsn(t1, . . . , tm | s)

→ Discsn(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tm | s).

This map is defined by inserting the outgoing ball (of radius ti) of a configurationx ∈ Discsn(r1, . . . , rk | ti) into the ith incoming ball of a point y ∈ Discsn(t1, . . . , tk | s).

These maps satisfy the natural associativity and commutativity properties of amulticategory.

2.7.3. Next, let F be a translation-invariant prefactorization algebra on Rn. Let

Fr = F (Br(0))

denote the cochain complexF that assigns to a ball of radius r. This notation is reason-able because translation invariance gives us an isomorphism between F (Br(0)) andF (Br(x)) for any x ∈ Rn.

The structure maps for a translation invariant prefactorization algebra yield, foreach p ∈ Discsn(r1, . . . , rk | s), multiplication operations

m[p] : Fr1 × · · · × Frk → Fs.

The map m[p] is a smooth multilinear map of differentiable spaces; and furthermore,this map depends smoothly on p.

These operations make the complexes Fr into an algebra over the R>0-colored op-erad Discsn(r1, . . . , rk | s), valued in the multicategory of differentiable cochain com-plexes. In addition, the complexes Fr are endowed with an action of the Abelian Liealgebra Rn. This action is by derivations of the Discsn-algebra F compatible with theaction of translation on Discsn, as described above.

2.7.4. Now, let us unravel explicitly what it means to be such a Discsn algebra.

The first property is that, for each p ∈ Discsn(r1, . . . , rk | s), the map m[p] is amultilinear map, of cohomological degree 0, compatible with differentials.

Second, let N be a manifold and let fi : N → F diri be smooth maps into the space

F diri of elements of degree di. The smoothness properties of the map m[p] mean that


the map

N ×Discsn(r1, . . . , rk | s)→ Fs

(x, p) 7→ m[p]( f1(x), . . . , fk(x))

is smooth.

Next, note that a permutation σ ∈ Sk gives an isomorphism

σ : Discsn(r1, . . . , rk | s)→ Discsn(rσ(1), . . . , rσ(k) | s).

We require that, for each p ∈ Discsn(r1, . . . , rk | s) and each αi ∈ Fri ,

m[σ(p)](ασ(1), . . . , ασ(k)) = m[p](α1, . . . , αk).

Finally, we require that the maps m[p] are compatible with composition, in thefollowing sense. For p ∈ Discsn(r1, . . . , rk | ti), q ∈ Discsn(t1, . . . , tl | s), αi ∈ Fri , andβ j ∈ Ftj , we require that

m[q] (β1, . . . , βi−1, m[p](α1, . . . , αk), βi+1, . . . , βl)

= m[q i p](β1, . . . , βi−1, α1, . . . , αk, βi+1, . . . , βl).

In addition, the action of Rn on each F r is compatible with these multiplication maps,in the way described above.

2.7.5. Let us give one more equivalent way of rewriting these axioms, which willbe useful when we discuss the holomorphic context. These alternative axioms will saythat the spaces C∞(Discsn(r1, . . . , rk | s)) form an R>0-colored co-operad when we usethe appropriate completed tensor product. Since we know how to tensor a differen-tiable vector space with the space of smooth functions on a manifold, it makes senseto talk about an algebra over this colored co-operad in the category of differentiablecochain complexes.

The smoothness axiom for the product map

m[p] : Fr1 ⊗ · · · ⊗ Frk → Fs,

wherep ∈ Discsn(r1, . . . , rk | s), can be rephrased as follows. For any differentiablevector space V and smooth manifold M, we use the notation V⊗C∞(M) interchange-ably with the notation C∞(M, V); both indicate the differentiable vector space of smooth


maps M → V. The smoothness axiom states that the map above extends to a smoothmap of differentiable spaces

µ(r1, . . . , rk | s) : Fr1 × · · · × Frk → Fs ⊗ C∞(Discsn(r1, . . . , rk | s)).

In general, if V1, . . . , Vk, W are differentiable vector spaces and if X is a smoothmanifold, let

C∞(X, Hom(V1, . . . , Vk |W))

denote the space of smooth multilinear maps

V1 × · · · ×Vk → C∞(X, W).

Note that there is a natural gluing map

i : C∞(X, Hom(V1, . . . , Vk |Wi))× C∞(Y, Hom(W1, . . . , Wl | T))

→ C∞(XxY, Hom(W1, . . . , Wi−1, V1, . . . , Vk, Wi+1, . . . , Wl | T)).

With this notation in hand, there are elements

µ(r1, . . . , rk | s) ∈ C∞ (Discs(r1, . . . rk | s), Hom(Fr1 , . . . ,Frk | Fs))

with the following properties.

(1) µ(r1, . . . , rk | s) is closed under the natural differential, arising from the dif-ferentials on the cochain complexes Fri .

(2) If σ ∈ Sk, then

σ∗µ(r1, . . . , rk | s) = µ(rσ(1), . . . , rσ(k) | s)

where

σ∗ : C∞(Discs(r1, . . . rk | s), Hom(Fr1 , . . . ,Frk | Fs))

→ C∞(Discs(rσ(1), . . . rσ(k) | s), Hom(Frσ(1) , . . . ,Frσ(k) | Fs))

is the natural isomorphism.(3) As before, let

i : Discsn(r1, . . . , rk | ti)×Discsn(t1, . . . , tm | s)

→ Discsn(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tm | s).

denote the gluing map. Then, we require that

∗i µ(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tl) = µ(r1, . . . , rk | ti) i µ(t1, . . . , tl | s).


These elements equip the Fr with the structure of an algebra over the colored co-operad, as stated earlier.

2.7.6. Our next focus explain what it means for a (smoothly) translation-invariantprefactorization algebra F on Cn to be holomorphically translation invariant. For thisdefinition to make sense, we require that F is defined over C: that is, the vector spacesF (U) are complex vector spaces and all structure maps are complex linear.

Recall that such a factorization algebra has, as part of its structure, an action of thereal Lie algebra R2n = Cn by derivations. This action is as a real Lie algebra; since Fis defined over C, the action extends to an action of the complexified translation Liealgebra R2n ⊗R C. We will denote the action maps by

ddzi

,d

dzj: F (U)→ F (U).

2.7.6.1 Definition. A translation-invariant prefactorization algebra F on Cn is holomor-phically translation invariant if it is equipped with derivations ηi : F → F of cohomologicaldegree −1, for i = 1 . . . n, with the property that

dηi =d

dzi∈ Der(F ).

Here, d refers to the differential on the dg Lie algebra Der(F ).

We should understand this definition as saying that the vector fields ddzi

act homo-topically trivially on F .

2.7.7. Now we will interpret holomorphically translation invariant prefactoriza-tion algebras in the language of R>0-colored operads. When we work in complexgeometry, it is better to use polydiscs instead of balls, as is standard in complex anal-ysis.

Thus, if z ∈ Cn, let

PDr(z) = w ∈ Cn | |wi − zi| < r

be the polydisc of radius r around z. Let

PDiscsn(r1, . . . , rk | s) ⊂ (Cn)k


be the space of z1, . . . , zk ∈ Cn with the property that the closures of the polydiscsPDri(zi) are disjoint and contained in the polydisc PDs(0).

It is clear that the spaces PDiscsn(r1, . . . , rk | s) form a R>0-colored operad in thecategory of complex manifolds.

Now, let F be a holomorphically translation invariant prefactorization algebra onCn. Let Fr denote the differentiable cochain complex F (PDr(0)) associated to thepolydisc of radius r.

Then, as above, for each p ∈ PDiscsn(r1, . . . , rk | s) we have a map

m[p] : Fr1 × · · · × Frk → Fs.

This map is smooth, multilinear, and compatible with the differential. Further, thismap varies smoothly with p.

The fact that F is a holomorphically translation invariant prefactorization algebrameans that these maps are equipped with extra structure. We have derivations ηj ofF which make the derivations d

dzjhomotopically trivial.

For i = 1, . . . , k and j = 1, . . . , n, let zij, zij refer to coordinates on (Cn)k, and so onthe open subset

PDiscsn(r1, . . . , rk | s) ⊂ (Cn)k.

Thus, we have operations

dpdzij

m[p] : Fr1 × · · · × Frk → Fs.

Let m[p] i ηj denote the operation

m[p] i ηj : Fr1 × · · · × Frk → Fs

α1 × · · · × αk 7→ ±m[p](α1, . . . , ηjαi, . . . , αk)

(where ± indicates the usual Koszul rule of signs).

Then, the identity [d, m[p] i ηj

]=

dpdzij

m[p]

holds, describing the fact that the product map m[p] is holomorphic in p, up to ahomotopy given by ηi.


2.7.8. In the smooth case, we saw that we could describe the structure as that ofan algebra over a R>0-colored co-operad built from smooth functions on the spacesDiscsn(r1, . . . , rk | s). In this section we will see that there is an analogous story in thecomplex world, where we use the Dolbeault complex of the spaces PDiscsn(r1, . . . , rk |s).

Let us first introduce some notation. For any complex manifold X, and any collec-tion V1, . . . , Vk, W of differentiable cochain complexes over C, let

Ω0,∗(X, Hom(V1, . . . , Vk |Wi))

denote the cochain complex of smooth multilinear maps

V1 × · · · ×Vk → Ω0,∗(X, W).

Recall that

Ω0,∗(X, W) = C∞(X, W)⊗C∞(X) Ω0,∗(W);

as we see in the appendix, a differentiable vector space W has enough structure to de-fine the ∂ operator on Ω0,∗(X, W). The differential on Ω0,∗(X, Hom(V1, . . . , Vk |Wi)) isa combination of the Dolbeault differential on X with the differentials on the differen-tiable cochain complexes Vi, W.

Let

i : PDiscsn(r1, . . . , rk | ti)× PDiscsn(t1, . . . , tm | s)

→ PDiscsn(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tm | s).

be the composition map, which is a holomorphic map of complex manifolds. We let

∗i : Ω0,∗(PDiscsn(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tm | s))

→ Ω0,∗(PDiscsn(r1, . . . , rk | ti)× PDiscsn(t1, . . . , tm | s)

be the corresponding pullback map on Dolbeault complexes.

2.7.8.1 Proposition. Let F be a holomorphically translation invariant factorization algebraon Cn. Then, the product maps

m[p] : Fr1 × · · · × Frk → Fs

for p ∈ PDiscsn(r1, . . . , rk | s) lift to closed elements

µ∂(r1, . . . , rk | s) ∈ Ω0,∗(PDiscsn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | Fs))


satisfying the following properties.

(1) µ∂(r1, . . . , rk | s) is closed under the natural differential on

Ω0,∗(PDiscsn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | Fs))

which incorporates the Dolbeault differential as well as the internal differentials onthe the complexes Fri ,Fs.

(2) Let σ ∈ Sk. Then, as in the smooth case,

σ∗µ∂(r1, . . . , rk | s) = µ∂(rσ(1), . . . , rσ(k) | s)

where

σ∗ : Ω0,∗(PDiscsn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | Fs))

→ Ω0,∗(PDiscsn(rσ(1), . . . , rσ(k) | s), Hom(Frσ(1) , . . . ,Frσ(k) | Fs))

is the natural isomorphism.(3) For all 1 ≤ i ≤ m,

∗i µ∂(t1, . . . , ti−1, r1, . . . , rk, ti+1, . . . , tm | s) = µ∂(t1, . . . , tm | s) i µ∂(r1, . . . , rk | ti)

∈ Ω0,∗ (PDiscsn(r1, . . . , rk | ti)× PDiscsn(t1, . . . , tm | s),

Hom(Ft1 × · · · × Fti−1 ×Fr1 × . . .Frk ×Fti+1 × · · · × Ftm | Fs))

.

Essentially, this proposition asserts that we can construct a co-operad from thecomplexes Ω0,∗(PDiscsn(r1, . . . , rk | s)) and that this co-operad acts on the differen-tiable cochain complexes Fri .

PROOF. We will produce this action starting from the operations

µ0(r1, . . . , rk | s) ∈ C∞(PDiscs(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | s))

that we already have because F is a smoothly translation-invariant factorization alge-bra.

First, we need to introduce some notation. Recall that

PDiscsn(r1, . . . , rk | s) ⊂ (Cn)k

is an open (possibly empty) subset. Thus,

Ω0,∗ (PDiscsn(r1, . . . , rk | s)) = Ω0,0 (PDiscsn(r1, . . . , rk | s))⊗C[dzij]


where the dzij are commuting variables of cohomological degree 1, with i = 1, . . . , kand j = 1, . . . , n. We let d

d(dzij)denote the graded derivation which removes dzij.

As before, let ηj : Fr → Fr denote the derivation which cobounds the derivationd

dzj. We can compose any element

α ∈ Ω0,∗(PDiscn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | s))

with ηj acting on Fri , to get

α i ηj ∈ Ω0,∗(PDiscn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | s)).

Then, the cochains µ∂(r1, . . . , rk | s) are characterized by the following properties.

(1) When restricted to Ω0,0(PDiscsn(r1, . . . , rk | s)) they are the elements con-structed from the fact that F is a smoothly translation-invariant factorizationalgebra.

(2) The identity

dd(dzij)

µ∂(r1, . . . , rk | s) = µ∂(r1, . . . , rk | s) i ηj

holds for all i and j.

It is easy to verify that there is a unique µ∂ satisfying these properties.

Next, we need to verify that dµ∂(r1, . . . , rk | s) = 0, where d is the differential on

Ω0,∗(PDiscn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | s)).

To see this, note that, for all

α ∈ Ω0,∗(PDiscn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | s)),

dd

d(dzij)α +

dd(dzij)

dα =d

dzijα

d(α i ηj)− (dα) i ηj = (−1)|α|α id

dzj,

where on the right hand side of the second equation ddzj

indicates the action of thisvector field on Fri .


Now, we know that

µ0(r1, . . . , rk | s) ∈ Ω0,0(PDiscn(r1, . . . , rk | s), Hom(Fr1 , . . . ,Frk | s))

satisfies

dµ0(r1, . . . , rk | s) = 0

ddzij

µ0(r1, . . . , rk | s) = µ0(r1, . . . , rk | s) id

dzj.

It follows from these identities that µ∂(r1, . . . , rk | s) is closed.

It is straightforward to verify that the elements µ∂ are compatible with compositionand with the symmetric group actions.

CHAPTER 3

Factorization algebras: definition and formal aspects

Our definition of a prefactorization algebra is closely related to that of a precosheafor of a presheaf. Mathematicians have found it useful to refine the axioms of a presheafto that of a sheaf: a sheaf is a presheaf whose value on a large open set is determined,in a precise way, by values on arbitrarily small subsets. In this chapter we describea similar “descent” axiom for prefactorization algebras. We call a prefactorizationalgebra satisfying this axiom a factorization algebra.

After defining this axiom, our next task is to verify that the examples we haveconstructed so far (such as the free field) satisfy it. This we do in section 3.2.

The rest of this chapter is devoted to exploring formal aspects of the theory of fac-torization algebras: we show how factorization algebras form a (simplicially-enriched)multicategory, and construct push-forward and pull-back functors in certain cases.

Philosophically, our descent axiom for factorization algebras is important: a pref-actorization algebra satisfying descent (i.e. a factorization algebra) is built from localdata, in a way that a general prefactorization algebra is not. However, for practicalpurposes, this axiom is often not essential.

Thus, a reader with little taste for formal mathematics could skip this chapter andstill be able to follow the rest of this book.

3.1. Factorization algebras

A factorization algebra is a prefactorization algebra that satisfies the local-to-globalaxiom. This axiom is the analog of the gluing axiom for sheaves; it expresses how thevalues on big open sets are determined by the values on small open sets. For sheaves,the gluing axiom says that for any open set U and any cover of that open set, wecan determine the value of the sheaf on U from the values on the open cover. For

59

60 3. FACTORIZATION ALGEBRAS: DEFINITION AND FORMAL ASPECTS

factorization algebras, we require our covers to be fine enough that they capture allthe “multiplicative structure.”

We will describe the local-to-global axiom for factorization algebras taking valuesin vector spaces or chain complexes, but the generalization to an arbitrary symmetricmonoidal category is straightforward.

3.1.0.2 Definition. Let U be an open set and U := Ui | i ∈ I a cover of U by open sets. Thecover U is factorizing if for any finite collection of points x1, . . . , xk in U, there is a finitecollection of pairwise disjoint opens Ui1 , . . . , Uin from the cover such that x1, . . . , xk ⊂Ui1 ∪ · · · ∪Uin .

Remark: Every Hausdorff space admits a nontrivial factorizing cover (i.e., a cover notcontaining the whole space as an element).

Remark: For a smooth n-manifold M, there is a simple way to construct a factorizingbasis for M. Namely, fix a Riemannian metric on M, and consider

Br(x) : ∀x ∈ M, with 0 < r < InjRad(x),

the collection of open balls, running over each point x ∈ M, whose radii are less thanthe injectivity radius at x. Another construction is simply to take the collection of opensets in M diffeomorphic to the open n-ball.

3.1.1. Strict factorization algebras. The value of a factorization algebra on U isdetermined by its behavior on a factorizing cover, just as the value of a cosheaf on anopen set U is determined by its value on any cover of U.

In order to motivate our definition of factorization algebra, let us write brieflyrecall the cosheaf axiom. A precosheaf Φ on M is a cosheaf if, for every open coverUi | i ∈ I of an open set U ⊂ M, the sequence

⊕i,jΦ(Ui ∩Uj)→ ⊕kΦ(Uk)→ Φ(U)

is exact on the right. (Alternatively, one can say the map⊕Φ(Uk)→ Φ(U) coequalizesthe pair of maps ⊕Φ(Ui ∩Uj) ⇒ ⊕Φ(Uk).)

We will define the notion of factorization algebra in a similar way, except thatinstead of considering elements Ui of the cover, one considers finite collections of dis-joint elements of the cover.

3.1. FACTORIZATION ALGEBRAS 61

In order to make this precise, we need to introduce some notation. Let PI denotethe set of finite subsets α ⊂ I, with the property that if j, j′ ∈ α, Uj ∩Uj′ = ∅. Theseare the tuples of open subsets that appear in the structure maps of a prefactorizationalgebra.

If α ∈ PI, let us define F (α) by

F (α) = F (qj∈αUj).

Similarly, if α1, . . . , αk ∈ PI, we will let

F (α1, . . . , αk) = F(qj1∈α1,...,jk∈αk Uj1 ∩ · · · ∩Ujk

).

Note that there are natural maps

pi : F (α1, . . . , αk)→ F (α1, . . . , αi, . . . , αk)

for each 1 ≤ i ≤ k.

3.1.1.1 Definition. A prefactorization algebra is a lax factorization algebra if it has theproperty: For every open subset U ⊂ M and every factorizing cover Ui | i ∈ I of U, thesequence

⊕α1,α2∈PIF (α1, α2)p1−p2−−−→ ⊕β∈PIF (β)→ F (U)

is exact on the right.

A lax factorization algebra is a strict factorization algebra if, in addition, for every pairof disjoint open sets U, V ∈ M, the natural map

F (U)⊗F (V)→ F (U qV)

is an isomorphism.

3.1.2. The Cech complex and homotopy factorization algebras. Now supposewe have a prefactorization algebraF , taking values in complexes. We will define whatit means for F to be a homotopy factorization algebra. This will happen when F (U) isquasi-isomorphic to a certain Cech complex constructed from any factorizing cover.

To motivate the definition, let us first recall the definition of a homotopy cosheaf.Let Φ be a pre-cosheaf on M, and let U = Ui | i ∈ I be a cover of some open subsetU of M. The Cech complex of U with coefficients in Φ is is defined in the usual way, as

⊕k ⊕j1,...,jk∈I Φ(Uj1 ∩ · · · ∩Ujk)[k− 1]


where the differential is defined in the usual way. We say that Φ is a homotopy cosheafif the natural map from the Cech complex to Φ(U) is a quasi-isomorphism, for everyopen U ⊂ M and every open cover of U.

Now let F be a prefactorization algebra on M, and let U = Ui | i ∈ I be afactorizing cover of an open subset U ⊂ M. The Cech complex of U with coefficientsin F is defined by

C(U,F ) = ⊕k≥0 ⊕α1,...,αk∈PI F (α1, . . . , αk)[k− 1]

with differential defined as the alternating sum of the restriction mapsF (α1, . . . , αk)→F (α1, . . . , αi, . . . , αk), as in the Cech complex for a homotopy cosheaf.

3.1.2.1 Definition. A lax homotopy factorization algebra on X is a prefactorization algebraF valued in cochain complexes, with the property that for every open set U ⊂ X, factorizingcover U of U, the natural map

C(U,F )→ F (U)

is a quasi-isomorphism.

A lax homotopy factorization algebra is a homotopy factorization algebra if, in addition,for every pair U, V of disjoint open subsets of X, the natural map

F (U)⊗F (V)→ F (U qV)


Remark: The notion of strict factorization algebra is not appropriate for the world ofcochain complexes. Whenever we refer to a factorization algebra in cochain com-plexes, we will mean a homotopy factorization algebra.

3.1.3. Factorization algebras valued in a multicategory. The factorization alge-bras of ultimate interest to us take values, not in the symmetric monoidal categoryof cochain complexes, but in the multicategory of differentiable cochain complexes(section A).

Thus, we need to define what it means to be a factorization algebra in a multicate-gory C. We will assume that C is equipped with a realization functor from the categoryC4 of simplicial objects of C to the original category C. This will allow us to define theCech complex of an object of C. (In the category of differentiable cochain complexes,

3.1. FACTORIZATION ALGEBRAS 63

the geometric realization of a simplicial object is defined in the same way as it is in thecategory of ordinary cochain complexes).

We will also assume that C is equipped with some notion of weak equivalence(weak equivalences in differentiable cochain complexes are defined in section A).

As before, let Op(X) be the multicategory whose objects are open sets in X andwhose multi-morphisms are defined by

Hom(U1, . . . , Un; V) =

∗ if U1 q . . . Un ⊂ V

∅ otherwise.

3.1.3.1 Definition. Let C be a multicategory with the structures listed above. A factorizationalgebra F with values in C is a functor Op(X) → C with the property that, for all opensubsets U ⊂ X, and all factorizing open covers U of U, the map

C(U,F )→ F (U)

is a weak equivalence.

Remark: Suppose that C is the multicategory underlying a symmetric monoidal cate-gory C⊗. Then a factorization algebra valued in C, considered as a multicategory, is thesame as a lax factorization algebra valued in C⊗, considered as a symmetric monoidalcategory.

3.1.3.2 Definition. A differentiable factorization algebra is a factorization algebra valued inthe multicategory of differentiable cochain complexes.

3.1.4. Factorization algebras in quantum field theory. We have seen (section 1.4)how prefactorization algebras appear naturally when one thinks about the structure ofobservables of a quantum field theory. It is natural to ask whether the local-to-globalaxiom which distinguishes factorization algebras from prefactorization algebras alsohas a quantum-field theoretic interpretation.

The local-to-global axiom we posit states, roughly speaking, that all observableson an open set U ⊂ M can be built up as sums of observables supported on arbitrarilysmall open subsets of M. To be concrete, let us consider a factorizing cover Uε of M,


consisting of all balls in M of radius < ε. Applied to this factorizing cover, our local-to-global axiom states that any observable O ∈ Obs(U) can be written as a sum ofobservables of the form O1O2 · · ·Ok, where Oi ∈ Obs(Bδi(xi)) and x1, . . . , xk ∈ M.

By taking ε to be very small, we see that our local-to-global axiom implies that allobservables can be written as sums of products of observables which are supported asclose as we like to points in U.

This is a physically reasonable assumption: most of the observables (or operators)which are considered in quantum field theory textbooks are supported at points, so itmight make sense to restrict attention to observables built from these.

However, more global observables are also considered in the physics literature.For example, in a gauge theory, one might consider the observable which measuresthe monodromy of a connection around some loop in the space-time manifold. Howwould such observables fit into the factorization algebra picture?

The answer reveals a key limitation of our axioms: the concept of factorization algebrais only appropriate for perturbative quantum field theories. Indeed, in a perturbative gaugetheory, the gauge field (i.e., the connection) is taken to be an infinitesimally smallperturbation A0 + δA of a fixed connection A0, which is a solution to the equationsof motion. There is a well-known formula (the time-ordered exponential) expressingthe holonomy of A0 + δA as a power series in δA, where the coefficients of the powerseries are given as integrals over Lk, where L is the loop which we are considering.

This expression shows that the holonomy of A0 + δA can be built up from observ-ables supported at points (which happen to lie on the loop L). This, the holonomyobservable will form part of our factorization algebra.

However, if we are not working in a perturbative setting, this formula does notapply, and we would not expect (in general) that the prefactorization algebra of ob-servables satisfies the local-to-global axiom.

3.2. FACTORIZATION ALGEBRAS FROM COSHEAVES 65

3.2. Factorization algebras from cosheaves

The goal of this section is to describe a natural class of factorization algebras. Thefactorization algebras which we construct from classical and quantum field theory willbe closely related to the factorization algebras discussed here.

The main result of this section is that, given a nice cosheaf of vector spaces orcochain complexes F on a manifold M, the functor Sym∗ F : U 7→ Sym(F(U)) is afactorization algebra. It is clear how this functor is a prefactorization algebra; thehard part is verifying when it satisfies the local-to-global axiom. The examples weare ultimately interested in arise from cosheaves F which are compactly supportedsections of a vector bundle, so we will focus on cosheaves like this.

We begin by providing the definitions necessary to state the main result of thissection. We then state the main result and explain its role for the rest of the book.Finally, we prove the lemmas that culminate in the proof of the main result.

3.2.1. Preliminary definitions.

3.2.1.1 Definition. A local cochain complex on M is a graded vector bundle E on M, whosesections will be denoted by E , equipped with a differential operator d : E → E of cohomologicaldegree 1 satisfying d2 = 0.

Let E be a local cochain complex on M, and let U be an open subset of M. Welet E (U) denote the cochain complex of smooth sections of E on U, and Ec(U) denotethe cochain complex of compactly supported sections of E on U. Similarly, let E (U)

denote the distributional sections on U, defined by

E (U) = E (U)⊗C∞(U) D(U)

where D(U) is the space of distributions on U. Let E c(U) denote the compactly sup-ported distributional sections of E on U. In the appendix (A) it is shown that thesefour cochain complexes are differentiable cochain complexes in a natural way.

If, as above, E is a graded vector bundle on M, let E! = E⊗DensM. Let us giveE! a differential which is the formal adjoint to that on E. Let E !(U), E !

c (U) denote thecochain complexes of smooth and compactly supported sections of E!, and let E

!(U)


and E!c(U) denote the cochain complexes of distributional and compactly-supported

distributional sections of E!.

Note that E c(U) is the continuous dual to E !(U), and that Ec(U) is the continuousdual to E

!(U).

The factorization algebras we will discuss are constructed from the symmetric al-gebra on the vector spaces Ec(U) and E c(U). This symmetric algebra can be defined intwo ways: either using the completed projective tensor product of topological vectorspaces, or in terms of sections of bundles on Un. For concreteness, we will discuss thelatter construction.

Thus, let us define (Ec(U))n to be the tensor power defined using the completedprojective tensor product on the topological vector space Ec(U). Concretely, if En

denotes the vector bundle on Mn obtained as the external tensor product, then

(Ec(U))n = Γc(Un, En)

is the compactly supported smooth sections of En on Un. Similarly, we have

(E c(U))n = Γc(Un, En)

is the compactly supported distributional sections of En.

The symmetric powers Symn Ec(U) and Symn E c(U) are defined as the coinvari-ants of the symmetric group action on the tensor powers. Finally, we can define thesymmetric and completed symmetric algebra of Ec(U) and E c(U) as

Sym∗ Ec(U) = ⊕n Symn Ec(U)

Sym∗ E c(U) = ⊕n Symn E c(U)

Sym∗Ec(U) = ∏

nSymn Ec(U)

Sym∗E c(U) = ∏

nSymn E c(U).

Note that these are all commutative algebras in the multi-category of differentiablecochain complexes (section A), and the completed symmetric algebras are commu-tative algebras in the multi-category of differentiable pro-cochain complexes (sectionA).

3.2. FACTORIZATION ALGEBRAS FROM COSHEAVES 67

Note that, since E!c(U) is dual to E (U), we can view SymE

!c(U) as the algebra of

formal power series on E (U). Thus, we often write

SymE!c(U) = O(E (U)).

3.2.2. Note that if U → V is an inclusion of open sets in M, then there are naturalmaps of commutative dg algebras

Sym∗ Ec(U)→ Sym∗ Ec(V)

Sym∗Ec(U)→ Sym

∗Ec(V)

Sym∗ E c(U)→ Sym∗ E c(V)

Sym∗E c(U)→ Sym

∗brEc(V).

Thus, each of these symmetric algebras forms a precosheaf of commutative algebras,and thus a prefactorization algebra. We denote these prefactorization algebras bySym∗ Ec, etc.

3.2.3. The main result of this section is the following.

3.2.3.1 Theorem. (1) Let E be a vector bundle on M. Then,(a) Sym∗ Ec and Sym∗ E c are strict (non-homotopical) factorization algebras val-

ued in the category of differentiable vector spaces.(b) Sym

∗Ec and Sym

∗E c are strict (non-homotopical) factorization algebras valued

in the category of differentiable pro-vector spaces.(2) Let E be a local cochain complex on M. Then,

(a) Sym∗ Ec and Sym∗ Ec are homotopy factorization algebras valued in the cate-gory of differentiable cochain complexes.

(b) Sym∗Ec and Sym

∗Ec are homotopy factorization algebras valued in the category

of differentiable pro-cochain complexes.

PROOF. Let us first prove the strict (non-homotopy) version of the result. To startwith, consider the case of Sym∗ Ec. We need to verify the local-to-global axiom (section3.1).

Let U be an open set in M and U = Ui | i ∈ I be a factorizing cover of U. Asbefore, let PI denote the set of finite subsets of I, where for each α ∈ PI, and every


i, j ∈ α, Ui ∩Uj = ∅. If α ∈ PI let Uα = qi∈αUi. More generally, let

Uα1,...,αn = qi1∈α1,...,in∈αn (Ui1 ∩ · · · ∩Uin) .

We need to prove that Sym∗ Ec(U) is the cokernel of the map

⊕α,β∈PI Sym∗(Ec(Uα,β))→ ⊕γ∈PI Syma st(Ec(Uγ))

This map compatible with the decomposition of Sym∗ Ec(U) into symmetric powers.Thus, it suffices to show that, for all m,

Symm Ec(U) = coker(⊕α,β∈PI Symm(Ec(Uα ∩Uβ)→ ⊕γ∈PI Symm Ec(Uγ)

).

Now, observe that

Ec(U)⊗m = E mc (Um)

where E mc is the cosheaf on Um obtained as the external product of Ec with itself m

times.

Thus it is enough to show that

E mc (Um) = coker

(⊕α,β∈PIE

mc((Uα ∩Uβ)

m)→ ⊕γ∈PIEmc(Um

γ

)).

Our cover U is a factorizing cover. This means that, for every finite set of pointsx1, . . . , xk ∈ M we can find disjoint open subsets Ui1 , . . . , Uik in the cover U withxi ∈ Uii . This implies that the subsets of Um of the form (Uα)m, where α ∈ PI, coverUm. Further,

(Uα)m ∩ (Uβ)

m = (Uα ∩Uβ)m.

The desired isomorphism now follows from the fact that E mc is a cosheaf on Mm.

The same argument applies to show that Sym∗ E !c is a factorization algebra. In

the completed case, essentially the same argument applies, with the subtlety (A) thatwhen working with pro-cochain complexes the direct sum is completed.

For the homotopy case, the argument is similar. Let U be a factorizing cover of anopen subset V of M, indexed by a set I. Let PI denote the set of finite subsets α ⊂ Iwith the property that, for i, j ∈ α, Ui ∩Uj = ∅. Let Uα = qi∈αUi. Let Uα1,...,αn denotethe intersection Uα1 ∩ · · · ∩Uαn .

Let F = Sym∗ Ec denote the prefactorization algebras we are considering (theargument will below will apply when we use the completed symmetric product or

3.3. LOCALLY CONSTANT FACTORIZATION ALGEBRAS 69

use E c instead of Ec). We need to show that map

C(U,F ) = ⊕α1,...,αnF (Uα1,...,αn)[n− 1]→ F (U)

is an equivalence (where the left hand side is equipped with the standard Cech differ-ential).

Let Fm(U) = Symm Ec. Both sides of the displayed equation above split as a directsum over m, and the map is compatible with this splitting. (If we use the completedsymmetric product, this decomposition is of course as a product).

We thus need to show that the map

⊕α1,...,αn Symm (Ec(Uα1,...,αn)) [n− 1]→ Symm (Ec(U))


For α ∈ PI, we get an open subset Umα ⊂ Um. Since U is a factorizing cover of V,

these open subsets form a cover of Vm. Note that

(Uα1)m ∩ · · · ∩ (Uαn)

m = (Uα1,...,αn)m .

Note that Ec(U)⊗m can be naturally identified with Γc(Um, Em) (where the tensorproduct is the completed projective tensor product).

Thus, to show that the Cech descent axiom holds, we need to verify that the map

⊕α1,...,αn∈PIΓc(Um

α1∩ · · · ∩Um

αn, Em) [n− 1]→ Γc(Vm, Em)

is a quasi-isomorphism. The left hand side above is the Cech complex for the cosheafof compactly supported sections of Em on Vm. Standard partition-of-unity argumentsshow that this map is a weak equivalence.

3.3. Locally constant factorization algebras

If M is an n-dimensional manifold, then prefactorization algebras locally bear aresemblance to En algebras. After all, a prefactorization algebra prescribes a way tocombine the elements associated to k distinct balls into an element associated to a bigball containing all k balls. In fact, En algebras form a full subcategory of factorizationalgebras on Rn.


3.3.0.2 Definition. A factorization algebra F on an n-manifold M is locally constant ifwhenever U, U′ are open subsets with U ⊂ U′, such that U is a deformation retraction of U′,then the map F (U)→ F (U′) is a quasi-isomorphism of cochain complexes.

Lurie [] has shown the following.

3.3.0.3 Theorem. There is an equivalence of (∞, 1)-categories between locally constant fac-torization algebras on Rn, and En algebras.

3.3.1. Many examples of En algebras arise naturally from topology, such as la-belled configuration spaces (as discussed in the work of Segal, McDuff, Bodigheimer,Salvatore, and Lurie). We will discuss an important example, that of mapping spaces.

Recall that (in the appropriate category of spaces) Maps(UtV, X) ∼= Maps(U, X)×Maps(V, X). This fact suggests that we might fix a target space X and define a factor-ization algebra by sending an open set U to Maps(U, X). This construction almostworks, but it is not clear how to “extend” a map f : U → X from U to a larger openset V 3 U. By working with “compactly-supported” maps, we solve this issue.

Fix (X, p) a pointed space. Let F denote the prefactorization algebra on M sendingan open set U to the space of compactly-supported maps from U to (X, p). (Here, “ fis compactly-supported” means that the closure of f−1(X − p) is compact.) Then Fis a prefactorization algebra in the category of pointed spaces. (Composing with thesingular chains functor gives a prefactorization algebra in abelian groups, but we willwork at the level of spaces.)

Note that this prefactorization algebra is locally constant: if U → U′ is an inclusionof open subsets where U is a deformation retraction of U′, then the map F(U)→ F(U′)is a weak homotopy equivalence.

Note also that there is a natural isomorphism

F(U1 qU2) = F(U1)× F(U2)

if U1, U2 are disjoint.

Let’s consider for a moment the case when M = Rn. Then, if D ⊂ Rn is a ball,there is a equivalence

F(D) ' ΩnX

3.4. THE CATEGORY OF FACTORIZATION ALGEBRAS 71

between the space of compactly supported maps D → X and the n-fold based loopspace of X. (We are using the topologists notation ΩnX for the n-fold loop space: weapologize with the conflict of notation with the space of n-forms).

To see this, note that a compactly supported map D → X extends uniquely to amap from the closed ball D, sending the boundary ∂D to the base point p of X. SinceΩn(X) is defined to be the space of maps of pairs

(D, ∂D)→ (X, p)

we have constructed the desired map from F(D) to Ωn(X). It is easily verified thatthis map is a homotopy equivalence.

If D1, D2 are disjoint discs contained in a disc D3, the prefactorization structuregives us a map

F(D1)× F(D2)→ F(D3).

These maps correspond to the standard En structure on the n-fold loop space.

For a particularly nice example, let M = R. Then, these maps describe the stan-dard product on the space ΩX of based loops in X. This is the product which, oncomponents, is the standard product on π0ΩX = π1(X, x).

This prefactorization algebra does not always satisfy the gluing axiom. However,Salvatore [Sal99] and Lurie [Lur09a] has shown that if X is sufficiently connected, thisprefactorization algebra is in fact a factorization algebra. (Technically, Salvatore andLurie use a slightly different descent axiom than we do here, but in the locally constantcase we believe our descent axioms our equivalent).

3.4. The category of factorization algebras

In this section, we explain how prefactorization algebras and factorization alge-bras form categories. In fact, they naturally form multicategories (or colored oper-ads). We also explain how these multicategories are enriched in simplicial sets whenthe (pre)factorization algebras take values in cochain complexes.

3.4.1. Morphisms and the category structure.


3.4.1.1 Definition. A morphism of prefactorization algebras φ : F → G consists of a mapφU : F(U) → G(U) for each open U ⊂ M, compatible with the structure maps. That is, forany open V and any finite collection U1, . . . , Uk of pairwise disjoint open sets, each containedin V, the following diagram commutes:

F(U1)⊗ · · · ⊗ F(Uk)φU1⊗···⊗φUk−→ G(U1)⊗ · · · ⊗ G(Uk)

↓ ↓F(V)

φV−→ G(V)

Likewise, all the obvious associativity relations are respected.

Remark: When our prefactorization algebras take values in cochain complexes, we re-quire the φU to be cochain maps, i.e., they each have degree 0 and commute with thedifferentials. When our prefactorization algebras take values in differentiable cochaincomplexes, we require in addition that the maps φU are smooth.

3.4.1.2 Definition. On a space X, we denote the category of prefactorization algebras on Xtaking values in the multicategory C by PreFA(X, C). The category of factorization algebras,FA(X, C), is the full subcategory whose objects are the factorization algebras.

In practise, C will normally be the multicategory of differentiable cochain com-plexes.

3.4.2. The multicategory structure. Let SC denote the universal symmetric monoidalcategory containing the multicategory C. Any prefactorization algebra valued in Cgives rise to one valued in SC.

There is a natural tensor product on PreFA(X, SC), as follows. Let F, G be prefac-torization algebras. We define F⊗ G by

F⊗ G(U) := F(U)⊗ G(U),

and we simply define the structure maps as the tensor product of the structure maps.For instance, if U ⊂ V, then the structure map is

F(U ⊂ V)⊗ G(U ⊂ V) : F⊗ G(U) = F(U)⊗ G(U)→ F(V)⊗ G(V) = F⊗ G(V).

3.4.2.1 Definition. Let PreFAmc(X, C) denote the multicategory arising from the symmetricmonoidal product on PreFA(X, SC). That is, if Fi, G are prefactorization algebras valued in C,

3.4. THE CATEGORY OF FACTORIZATION ALGEBRAS 73

we define the set of multi-morphisms by

PreFAmc(F1, · · · , Fn | G)

to be the set of maps of SC-valued prefactorization algebras

F1 ⊗ · · · ⊗ Fn → G.

Factorization algebras inherit this multicategory structure.

3.4.3. Enrichment over simplicial sets. In this subsection we will explain how themulticategory of factorization algebras with values in an appropriate multicategory issimplicially enriched, in a natural way. In order to define this simplicial enrichment,we need to introduce some notation.

The first thing to define is the algebra Ω∗(X) of smooth forms on a simplicial setX. We define an element ω ∈ Ωi(X) is, for every n-simplex f : 4n → X, an i-formf ∗Ωi(4n). If σ : 4m → 4n is a face or degeneracy map, we require that σ∗ f ∗ω =

( f σ)∗ω.

If V is a differentiable cochain complex, we can define a complex

Ω∗(X, V) = Ω∗(X)⊗C∞(X) C∞(X, V)

where C∞(X, V) refers to the cochain complex of smooth maps from X to V. In thisway, we see that the multicategory differentiable cochain complexes is tensored overthe opposite category SSetop to the category of simplicial sets.

This allows us to lift the multicategory DVS of differentiable cochain complexes toa simplicially enriched multicategory, where we define the n-simplices in the simpli-cial set of multimorphisms by

Hom(V1, . . . , Vn |W)[n] = Hom(V1, . . . , Vn | Ω∗(4n, W))

where on the right hand side, Hom denotes smooth multilinear cochain maps

V1 × · · · ×Vn → Ω∗(4n, W)

which are compatible with differentials.

Now, in general, suppose we have a multicategory C which is tensored over SSetop.Then the multicategory PreFA(X, C) is also tensored over SSetop: the tensor product


F X of a prefactorization algebra F with a simplicial set X is defined by

(F X)(U) = F(U) X.

Then, we can lift our multicategory of C-valued prefactorization algebras to a simpli-cially enriched multicategory, by defining

PreFA4mc(F1, . . . , Fn | G) = PreFA4mc(F1, . . . , Fn | G X).

In particular, we see that the multicategory of prefactorization algebras valued in dif-ferentiable cochain complexes is simplicially enriched.

3.4.4. Equivalences. In the appendix A, we define a notion of weak equivalence orquasi-isomorphism of differentiable cochain complexes.

3.4.4.1 Definition. Let F, G be factorization algebras valued in cochain complexes. Let φ :F → G be a map. We say that φ is a weak equivalence if, for all open subsets U ⊂ M, the mapF(U)→ G(U) is a quasi-isomorphism of cochain complexes.

Similarly, let F, G be factorization algebras valued in differentiable cochain complexes. Wesay a map φ : F → G is a weak equivalence if, for all U, the map F(U) → G(U) is a weakequivalence of differentiable cochain complexes.

3.4.4.2 Lemma. A map F → G between differentiable factorization algebras is a weak equiv-alence if and only if, for all factorizing bases U of X, and all U1, . . . , Uk disjoint elements of U ,the maps

F(U1 q · · · qUk)→ G(U1 q · · · qUk)

are weak equivalences.

PROOF. For any open subset V ⊂ X, let UV denote the factorizing cover of Vconsisting of open subsets in U which lie in V. By the descent axiom the map

C(UV , F)→ F(V)

is a weak equivalence, and similarly for G. Thus, it suffices to check that the map

C(UV , F)→ C(UV , G)

is a weak equivalence. This follows from a spectral sequence argument and the factthat the maps

F(U1 q · · · qUk)→ G(U1 q · · · qUk)

are weak equivalences.

3.6. EXTENSION FROM A BASIS 75

3.5. Pushing forward factorization algebras

A crucial feature of factorization algebras is that they push forward nicely. LetM and N be topological spaces admitting factorizing covers and let f : M → Nbe a continuous map. Given a factorizing cover U = Uα of an open U ⊂ N, letf−1U = f−1Uα denote the preimage cover of f−1U ⊂ M. Observe that f−1U isfactorizing: given a finite collection of points x1, . . . , xn in f−1U, the image points f (x1), . . . , f (xn) can be covered by a disjoint collection of opens Uα1 , . . . , Uαk in U

and hence f−1Uα1 , . . . , f−1Uαk is a disjoint collection of opens in f−1U covering the xj.

3.5.0.3 Definition. Given a factorization algebra F on a space M and a continuous mapf : M→ N, the pushforward factorization algebra f∗F on N is defined by

f∗F (U) := F ( f−1(U)).

Note that for the map to a point f : M→ pt, the pushforward factorization algebraf∗F is simply the global sections of F . We also call this the factorization homology of Fon M. We sometimes denote this FH(M,F ).

3.6. Extension from a basis

3.6.1. Factorization algebras defined on a factorizing basis. Let X be a topologi-cal space, and let U be a basis for X, which is closed under taking finite intersections.It is well-known that there is an equivalence of categories between sheaves on X andsheaves which are only defined for open sets in the basis U. In this section we willprove a similar statement for factorization algebras. This will allow us to performseveral useful formal constructions with factorization algebras, such as gluing.

3.6.1.1 Definition. A factorizing basis for X is a basis U of open sets of X which is closedunder finite intersections, and which is also a factorizing cover.

Let U be a factorizing basis.

3.6.1.2 Definition. A U-prefactorization algebra F is like a factorization algebra, except thatF (U) is only defined for sets U of the form U = U1 q . . . Uk with Ui disjoint sets in U. AU-factorization algebra is a U-prefactorization algebra with the property that, for all U which


is a disjoint union of sets in U and all factorizing covers V of U consisting of open sets in U,

C(V,F ) ' F (U),

where C(V,F ) denotes the Cech complex described earlier (section 3.1).

In this section we will show that any U-factorization algebra on X extends to afactorization algebra on X. This extension is unique up to quasi-isomorphism.

Let F be a U-factorization algebra. Let us define a prefactorization algebra iU∗F onX by

iU∗ (F )(V) = C(UV ,F ),

- for each V ⊂ X open. Here UV is the cover of V consisting of those open subsets inthe cover U which are contained in V.

3.6.1.3 Proposition. With this definition, iU∗ (F ) is a factorization algebra whose restrictionto open sets in the cover U is quasi-isomorphic to F .

The proof is rather technical, so we put it in the appendix B. The statement holds ifF is a factorization algebra valued in differentiable cochain complexes or in ordinarycochain complexes.

3.7. Pulling back factorization algebras

Let F be a factorization algebra on M. Let U ⊂ M be an open subset. Then we canrestrict F to a factorization algebra F |U on U, whose value on an open subset V ⊂ Uis simply F (V).

In this section we will discuss a generalization of this construction. We will not tryto define pull-backs for arbitrary maps, but only for open immersions.

Let f : N → M be an open immersion. Let U f be the cover of N consisting of thoseopen subsets U ⊂ N with the property that

f |U : U → f (U)

is a homeomorphism. (To say that f is an open immersion means that sets of this formcover N).

3.8. DESCENT AND GLUING 77

Now, U f is a factorizing basis for N. Let us define a U f -prefactorization algebraf ∗F by

f ∗F (U) = F ( f (U))

if U ∈ U f .

3.7.0.4 Lemma. f ∗F is a U f -factorization algebra.

PROOF. We need to verify that if U ∈ U f , and V is a factorizing cover of U byelements of U f , that

C(V, f ∗F ) ' f ∗F (U) = F ( f (U)).

Now, f (V) is a factorizing cover of f (U), and

C(V, f ∗F ) = C( f (V,F ).

The result follows from the fact that F is a factorization algebra on M.

So far we have defined f ∗F as a U f -factorization algebra. We can extend (section3.6) f ∗F to an actual factorization algebra, which we will continue to call f ∗F .

3.8. Descent and gluing

3.8.1. Gluing. Suppose we have open cover Ui of X and a factorization alge-bra Fi on each Ui, with gluing data on double intersections, coherence data on tripleintersections, and so on. In this section we will show that we can construct a uniquefactorization algebra F on X from this data. In fact, this is an easy consequence of thefact that we can extend factorization algebras from a factorizing basis. This construc-tion only works for factorization algebras valued in a symmetric monoidal category.

3.8.2. Covers by two open sets. Let us start with the simple case when we cover Xby two open sets U, V. LetFU be a factorization algebra on U andFV be a factorizationalgebra on V, both valued in a symmetric monoidal category C.


Let FU∩V be a factorization algebra on U ∩V equipped with weak equivalences offactorization algebras on U ∩V,

FU∩V → i∗UFU

FU∩V → i∗VFV .

This is one natural way of saying what it means to have “gluing data”.

Let us define a factorizing basis W of X by saying that a set is in W if it lies in U orin V.

We want to define an W prefactorization algebra; to do this, we need to defineF (W) for all sets W which are disjoint unions of sets in W. Thus, given such a W, letus write it as

W = U1 q · · · qUk qV1 q · · · qVl

where the Ui ⊂ U and Vi ⊂ V are open subsets. We then define

F (W) = ⊗FU(Ui)⊗FV(Vi).

It is clear that F is a W-prefactorization algebra (the structure maps use the weakequivalences FU∩V(W)→ FU(W) if W ⊂ U ∩V).

3.8.2.1 Lemma. F is a W-factorization algebra.

PROOF. Let W ∈W, and let W′ be a factorizing cover of W consisting of sets in W.We need to check that

C(W′,F ) ' F (W).

There are three cases: either W ⊂ U ∩ V, or W ⊂ U and W 6⊂ V, or W ⊂ V andW 6⊂ U. The first case is immediate from the fact that FU∩V is a factorization algebraon U ∩V.

The second two cases are the same; so let us assume that W ⊂ U and W 6⊂ V.

We need to verify thatC(W′,F ) ' F (W).

Now, there are two prefactorization algebras on U, F and FU . FU is given to us, andF is defined by

F (W ′) =

FU(W ′) if W ′ 6⊂ V

FU∩V(W ′) if W ′ ⊂ V.


What we’re checking is that F is a factorization algebra on U.

If W ′ ∈W′, and W ′ ⊂ U ∩V, then there is a weak equivalence

FU∩V(W ′)→ FU(W ′).

This induces a weak equivalence map of prefactorization algebras on U,

F → FU .

Since FU is a factorization algebra, and this map is a weak equivalence, F is also afactorization algebra.

3.8.3. The general case. Now let us consider a general open cover U of X. Wewant to show how to construct, from a factorization algebra on every set in U withcertain gluing data, a factorization algebra on X.

The way we will encode the factorization algebra on the sets in the cover U, to-gether with gluing data, is as follows. We have, for each finite subset I ⊂ U, a factor-ization algebra FI on

UI = ∩i∈IUi.

Further, we have weak equivalences

FI → r∗i FI\i

of factorization algebras on UI , for each i ∈ I.

ri : UI → UI\i

is the natural inclusion.

Finally, we require that, for every I and every i, j ∈ I, the following diagram ofcommutes:

FI → FI\i↓ ↓FI\j → FI\i,j

We can think of this data as defining a factorization algebra Fi for each i ∈ U,together with weak equivalences on double intersections, provided by Fi,j; and withcoherences provided by the factorization algebras FI where #I ≥ 3.


In this situation, we can construct a factorization algebra on X, as follows. We willlet W be the factorizing basis of X consisting of open sets subordinate to the cover U.We will define a W-factorization algebra F by saying that, if W ∈W,

F (W) = FI(W)

where I is the largest subset of U such that

W ⊂ UI .

If W = W1 q · · · qWk where Wi ∈W, we set

F (W) = ⊗ki=1F (Wi).

It is straightforward to check that F is a W-factorization algebra, and so extends tofactorization algebra on X.

3.8.4. Descent. Let G be a discrete group acting on a space X.

3.8.4.1 Definition. A G-equivariant factorization algebra on X is a factorization algebra Fon X together with isomorphisms

ρg : g∗F ∼= F ,

for each g ∈ G, such that

ρId = Id

ρgh = ρh h∗(ρg) : h∗g∗F → F .

3.8.4.2 Proposition. Let G be a discrete group acting properly discontinuously on X, so thatX → X/G is a principal G-bundle. Then, there is an equivalence of categories between G-equivariant factorization algebras on X and factorization algebras on X/G.

This result holds for factorization algebras valued in multicategories as well assymmetric monoidal categories.

PROOF. If F is a factorization algebra on X/G, then f ∗F is a G-equivariant factor-ization algebra on X.

Conversely, let F be a G-equivariant factorization algebra on F . Let U be the opencover of X/G consisting of those connected sets where the G-bundle X → X/G admitsa section. Note that U is a factorizing basis for X/G, and that U is closed under taking


both finite intersection and disjoint union. We will define a U-factorization algebra FG

by definingFG(U) = F (σ(U))

where σ is any section of the G-bundle π−1(U)→ U.

BecauseF is G-equivariant, F (σ(U)) is independent of the section σ chosen. SinceU is a factorizing basis, FG extends canonically to a factorization algebra on X/G.

CHAPTER 4

Classical field theory

4.1. Introduction

Our goal here is to describe how the observables of a classical field theory natu-rally form a factorization algebra (section 3.1). More accurately, we are interested inwhat might be called classical perturbative field theory. “Classical” means that themain object of interest is the sheaf of solutions to the Euler-Lagrange equations forsome local action functional. “Perturbative” means that we will only consider thosesolutions which are infinitesimally close to a given solution. Much of the chapter isdevoted to providing a precise mathematical definition of these ideas, with inspirationtaken from deformation theory and derived geometry.

4.1.1. The Euler-Lagrange equations. The fundamental objects of a physical the-ory are the observables of a theory, that is, the measurements one can make in thattheory. In a classical field theory, our fields are constrained to be solutions to theEuler-Lagrange equations. Thus, the measurements one can make are the functionson the space of solutions to the Euler-Lagrange equations.

However, it is essential that we do not take the naive moduli space of solutions.Instead, we consider the derived moduli space of solutions. Since we are workingperturbatively – that is, infinitesimally close to a given solution – this derived modulispace will be a “formal moduli problem” [Lur10, Lur11]. In the physics literature, theprocedure of taking the derived critical locus is implemented by the BV formalism.Thus, the first step (section 4.3.3) in our treatment of classical field theory is to developa language to treat formal moduli problems cut out by systems of partial differentialequations on a manifold M. Since it is essential that the differential equations weconsider are elliptic, we call such an object a formal elliptic moduli problem.

83

84 4. CLASSICAL FIELD THEORY

Since one can consider the solutions to a differential equation on any open subsetU ⊂ M, a formal elliptic moduli problem F yields, in particular, a sheaf of formalmoduli problems on M; which sends U to the formal moduli space F (U) of solutionson U.

We will use the notation EL to denote the formal elliptic moduli problem of so-lutions to the Euler-Lagrange equation on M; so that EL(U) will denote the space ofsolutions on an open subset U ⊂ M.

4.1.2. Observables. In a field theory, we tend to focus on measurements that arelocalized in spacetime. Hence, we want a method that associates a set of observablesto each region in M. If U ⊂ M is an open subset, the observables on U are

Obscl(U) = O(EL(U)),

our notation for the algebra of functions on the formal moduli space EL(U) of so-lutions to the Euler-Lagrange equations on U. (We will be more precise about whichclass of functions we are using later). As we are working in the derived world, Obscl(U)

is a differential-graded commutative algebra. Using these functions, we can answerany question we might ask about the behavior of our system in the region U.

The factorization algebra structure arises naturally on the observables in a classicalfield theory. Let U be an open set in M, and V1, . . . , Vk a disjoint collection of opensubsets of U. Then restriction of solutions from U to each Vi induces a natural map

EL(U)→ EL(V1)× · · · × EL(Vk).

Since functions pullback under maps of spaces, we get a natural map

Obscl(V1)⊗ · · · ⊗Obscl(Vk)→ Obscl(U)

so that Obscl forms a prefactorization algebra. To see that Obscl is indeed a factorizationalgebra, it suffices to observe that the functor EL is a sheaf.

Since the space Obscl(U) of observables on a subset U ⊂ M is a commutativealgebra, and not just a vector space, we see that the observables of a classical fieldtheory form a commutative factorization algebra (section 2.2).

4.1.3. The symplectic structure. Above, we outlined a way to construct, from theelliptic moduli problem associated to the Euler-Lagrange equations, a commutative

4.1. INTRODUCTION 85

factorization algebra. However, this construction would apply equally well to anysystem of differential equations. The Euler-Lagrange equations, of course, have thespecial property that they arise as the critical points of a functional.

In finite dimensions, a formal moduli problem which arises as the derived criti-cal locus (section 4.8) of a function is equipped with an extra structure: a symplecticform of cohomological degree −1. For us, this symplectic form is an intrinsic wayof indicating that a formal moduli problem arises as the critical locus of a functional.Indeed, any formal moduli problem with such a symplectic form can be expressed(non-uniquely) in this way.

We give (section 4.8.2) a definition of symplectic form on an elliptic moduli prob-lem. We then simply define a classical field theory to be a formal elliptic moduli prob-lem equipped with a symplectic form of cohomological degree −1.

Given a local action functional satisfying certain non-degeneracy properties, weconstruct (section 4.9.1) an elliptic moduli problem describing the corresponding Euler-Lagrange equations, and show that this elliptic moduli problem has a symplectic formof degree −1.

In ordinary symplectic geometry, the simplest construction of a symplectic mani-fold is as a cotangent bundle. In our setting, there is a similar construction: given anyelliptic moduli problem F , we construct (section 4.12) a new elliptic moduli problemT∗[−1]F which has a symplectic form of degree −1. It turns out that many examplesof field theories of interest in mathematics and physics arise in this way.

4.1.4. The P0 structure. In finite dimensions, if X is a formal moduli problem witha symplectic form of degree−1, then the dga O(X) of functions on X is equipped witha Poisson bracket of degree 1. In other words, O(X) is a P0 algebra (section 2.3).

In infinite dimensions, we show that something similar happens. If F is a classicalfield theory, then we show that the commutative algebra O(F (U)) = Obscl(U) has aP0 structure; and that the commutative factorization algebra Obscl forms a P0 factor-ization algebra. This is not quite trivial; it is at this point that we need the assumptionthat our Euler-Lagrange equations are elliptic.


4.2. Elliptic moduli problems and local Lie algebras

The essential data of a classical field theory is the moduli space of solutions tothe equations of motion of the field theory. For us, it is essential that we take not thenaive moduli space of solutions, but rather the derived moduli space of solutions. Inthe physics literature, the procedure of taking the derived moduli of solutions to theEuler-Lagrange equations is known as the Batalin-Vilkovisky formalism.

The derived moduli space of solutions to the equations of motion of a field theoryon X is a sheaf on X. In this section we will introduce a general language for discussingsheaves of “derived spaces” on X which are cut out by differential equations.

4.3. Formal moduli problems and Lie algebras

Before we discuss the concepts specific to classical field theory, we will explainsome general techniques from deformation theory.

In ordinary algebraic geometry, the fundamental objects are commutative rings. Inderived algebraic geometry, commutative rings are replaced by commutative differ-ential graded rings concentrated in non-positive degrees (or, if one prefers, simplicialcommutative rings; over Q, there is no difference).

We are interested in formal derived geometry, which is described by nilpotentcommutative dgas.

4.3.0.1 Definition. An Artinian dga over a field K of characteristic zero is a differential gradedK-algebra R, concentrated in degrees ≤ 0, such that

(1) Each graded component Ri is finite dimensional, and Ri = 0 for i << 0.(2) R has a unique maximal differential ideal m such that R/m = K, and such that

mN = 0 for N >> 0.

Given the first condition, the second condition is equivalent to the statement thatH0(R) is Artinian in the classical sense.

The category of Artinian dg rings is a simplicially enriched category. A map R→ Sis simply a map of dg rings taking the maximal ideal mR to that of mS. Equivalently,

4.3. FORMAL MODULI PROBLEMS AND LIE ALGEBRAS 87

such a map is a map of non-unital dg rings mR → mS. An n-simplex in the spaceHom(R, S) of maps from R to S is defined to be a map of non-unital dg rings

mR → mS ⊗Ω∗(4n)

where Ω∗(4n) is some commutative algebra model for the cochains on the n-simplex.(Normally, we will work over R, and Ω∗(4n) will be the usual de Rham complex).

We will (temporarily) let Art denote the simplicially enriched category of Artiniandg rings over K.

4.3.0.2 Definition. A formal moduli problem over a field k is a functor (of simplicially en-riched categories)

F : Artk → sSets

from Artk to the category sSets of simplicial sets, with the following additional properties.

(1) F(k) is contractible.(2) F takes surjective maps of dg Artinian rings to fibrations of simplicial sets.(3) Suppose that A, B, C are dg Artinian rings, and that B → A, C → A are surjective

maps. Then we can form the fiber product B×A C. We require that the natural map

F(B×A C)→ F(B)×F(A) F(C)

is a weak homotopy equivalence.

Note that, in light of the second property, the fiber product F(B)×F(A) F(C) coin-cides with the homotopy fiber product.

The category of formal moduli problems is itself simplicially enriched, in an evi-dent way. If F, G are formal moduli problems, and φ : F → G is a map, we say that φ

is a weak equivalence if for all dg Artinian rings R, the map

φ(R) : F(R)→ G(R)

is a weak homotopy equivalence of simplicial sets.

4.3.1. Formal moduli problems and L∞ algebras. One very important way inwhich formal moduli problems arise is as the solutions to the Maurer-Cartan equa-tion in an L∞ algebra. As we will see later, all formal moduli problems are equivalentto formal moduli problems of this form.


If g is an L∞ algebra, and (R, m) is a dg Artinian ring, we will let

MC(g⊗m)

denote the simplicial set of solutions to the Maurer-Cartan equation in g⊗ m. Thus,an n-simplex in this simplicial set is an element

α ∈ g⊗m⊗Ω∗(4n)

of cohomological degree 1, which satisfies the Maurer-Cartan equation

dα + ∑n≥2

1n! ln(α, . . . , α) = 0.

It is a standard lemma that sending R to MC(g⊗m) defines a formal moduli problem.We will often use the notation Bg to denote this formal moduli problem.

If g is finite dimensional, then a Maurer-Cartan element of g⊗m is the same thingas a map of commutative dgas

C∗(g)→ R

which takes the maximal ideal of C∗(g) to that of R.

Thus, we can think of the Chevalley-Eilenberg cochain complex C∗(g) as the alge-bra of functions on Bg.

Under the dictionary between formal moduli problems and L∞ algebras, a dg vec-tor bundle on Bg is the same thing as a dg module over g. The cotangent complexto Bg corresponds to the g-module g∨[−1]. The tangent complex corresponds to theg-module g[1].

If M is a g-module, then sections of the corresponding vector bundle on Bg is theChevalley cochains with coefficients in M. Thus, we can define Ω1(Bg) to be

Ω1(Bg) = C∗(g, g∨[−1]).

Similarly, the complex of vector fields on Bg is

Vect(Bg) = C∗(g, g[1]).

Note that, if g is finite dimensional, this is the same as the cochain complex of deriva-tions of C∗(g). Even if g is not finite dimensional, the complex Vect(Bg) is, up to a shiftof one, the Lie algebra controlling deformations of the L∞ structure on g.

4.3. FORMAL MODULI PROBLEMS AND LIE ALGEBRAS 89

4.3.2. The fundamental theorem of deformation theory. The following statementis at the heart of the philosophy of deformation theory:

There is an equivalence of (∞, 1) categories between the category ofdifferential graded Lie algebras, and the category of formal pointedderived moduli problems.

In a different guise, this statement goes back to Quillen’s work [Qui69] on rationalhomotopy theory. A precise formulation of this theorem has been proved by Hinich[Hin01]; more general theorems of this nature are considered in [Lur11] and in [Lur10],which is also an excellent survey of these ideas.

It would take us too far afield to describe the language in which this statementcan be made precise. We will simply use this statement as motivation: we will onlyconsider formal moduli problems described by L∞ algebras, and this statement assertsthat we lose no information in doing so.

4.3.3. Elliptic moduli problems. We are interested in formal moduli problemswhich describe solutions to differential equations on a manifold M. Since we candiscuss solutions to a differential equation on any open subset of M, such an objectwill give a (homotopy) sheaf of derived moduli problems on M. Let us give a formaldefinition of such a sheaf.

4.3.3.1 Definition. Let M be a manifold. A simplicial presheaf on M is a homotopy sheaf ifit satisfies Cech descent. A homotopy sheaf of formal moduli problems on M is a presheaf F offormal moduli problems, with the property that for all Artinian dgas R, the simplicial presheafF(R) is a homotopy sheaf.

We will often refer to a homotopy sheaf as just a sheaf.

4.3.4. Elliptic moduli problems. We are interested in elliptic derived moduli prob-lems: that is, derived moduli problems described by a system of elliptic partial differ-ential equations on a manifold M. We will define a formal pointed elliptic moduliproblem on a manifold M to be a sheaf of formal moduli problems represented by asheaf of L∞ algebras on M of a certain kind.


4.3.4.1 Definition. Let M be a manifold. A local L∞ algebra on M consists of the followingdata.

(1) A graded vector bundle L on M, whose space of smooth sections will be denoted L.(2) A differential operator d : L → L, of cohomological degree 1 and square 0.(3) A collection of poly-differential operators

ln : L⊗n → L

which are alternating, of cohomological degree 2− n, and which endow L with thestructure of L∞ algebra.

4.3.4.2 Definition. An elliptic L∞ algebra is a local L∞ algebra L as above with the propertythat (L, d) is an elliptic complex.

If L is a local L∞ algebra on a manifold M, then it yields a presheaf BL of formalmoduli problems on M. This presheaf sends a dg Artinian ring (R, m), and an opensubset U ⊂ M, to the simplicial set then we can consider the simplicial set

BL(U)(R) = MC(L(U)⊗m)

of Maurer-Cartan elements of the L∞ algebra L(U) ⊗ m (where L(U) refers to thesections of L on U). We will think of this as the R-points of the formal pointed moduliproblem associated to L(U).

4.3.4.3 Definition. A formal pointed elliptic moduli problem (or simply elliptic moduli prob-lem) is a sheaf of formal moduli problems on M, which is represented by an elliptic L∞ algebra.

4.4. Examples of elliptic moduli problems related to gauge theories

4.4.1. Flat bundles. Next, let us discuss a more geometric example of an ellipticmoduli problem: that describing flat bundles on a manifold M. In this case, becauseflat bundles have automorphisms, it is more difficult to give a direct definition of theformal moduli problem.

Thus, let G be a Lie group, and let P→ M be a principal G-bundle equipped with aflat connection. Let gP be the adjoint bundle (associated to P by the adjoint action of Gon its Lie algebra g). Thus, gP is a bundle of Lie algebras on M, with a flat connection.

4.4. EXAMPLES OF ELLIPTIC MODULI PROBLEMS RELATED TO GAUGE THEORIES 91

Let R be an Artinian dg ring. We want to define the simplicial set DefP(R) ofR-families of flat G-bundles on M which deform P.

As the underlying topological bundle of P is rigid, we can only deform the flatconnection on P. A deformation of the connection on P is given by an element

A ∈ Ω1(M, gP)⊗m

of cohomological degree 0.

We would like to ask that A is flat up to homotopy. The curvature F(A) is

F(A) = dA + 12 [A, A] ∈ Ω2(M, gP)⊗m.

Note that, by the Bianchi identity, dF(A) + [A, F(A)] = 0.

For A to be flat up to homotopy, we should ask that F(A) is exact in the cochaincomplex Ω2(M, gP)⊗m of two-forms on M. However, we should also ask that F(A)

be made exact in a way compatible with the Bianchi identity.

Thus, as a first approximation, we will define the zero-simplices of the deforma-tion functor by

DefprelimP (R)[0] =

A ∈ Ω1(M, gP)⊗m, B ∈ Ω2(M, gP)⊗m | F(A) = dRB, ddRB + [A, B] = 0.

Here, A is of cohomological degree 0 and B is of cohomological degree −1.

Note that if m is of square zero, then the Bianchi constraint on B just says thatddRB = 0. This leads to a problem: the sheaf of closed 2-forms on M is not fine: it hashigher cohomology groups. Thus, we cannot hope to construct a deformation functorwith values in homotopy sheaves of simplicial sets on M in this way.

Instead, we will ask that B satisfy the Bianchi constraint up a sequence of higherhomotopies. Thus, the zero simplices of our simplicial set of deformations are definedby

DefP(R)[0] = A ∈ Ω1(M, gP)⊗m, B ∈ Ω≥2(M, gP)⊗m

| F(A) + ddRB + [A, B] + 12 [B, B] = 0..

Here, d refers to the total differential on the tensor product cochain complex Ω≥2(M, gP)⊗m. As before, A is of cohomological degree 0 and B is of cohomological degree −1.


If we let Bi ∈ Ωi(M, gP)⊗m, then the first few constraints on the Bi can be writtenas

ddRB2 + [A, B2] + dRB3 = 0

ddRB3 + [A, B3] +12 [B2, B2] + dRB4 = 0.

Thus, B2 satisfies the Bianchi constraint up to a homotopy defined by B3, and so on.

The higher simplices of this simplicial set must relate gauge-equivalent flat con-nections. If the dg ring R is concentrated in degree 0 (and so has zero differential), thenwe can define the simplicial set DefP(R) to be the homotopy quotient of DefP(R)[0] bythe nilpotent group associated to the nilpotent Lie algebra Ω0(M, gP)⊗m, which actson DefP(R)[0] in a standard way.

If R is not concentrated in degree 0, however, then the higher simplices of DefP(R)must also involve elements of R of negative cohomological degree. Indeed, degree−1elements of R should be thought of as homotopies between degree 0 elements of R,and so should contribute 1-simplices to our simplicial set.

A slick way to define a simplicial set with both desiderata is to set

DefP(R)[n] = A ∈ Ω∗(M, gP)⊗m⊗Ω∗(4n) | ddR A + dR A + 12 [A, A] = 0.

Suppose that R is concentrated in degree 0 (so that the differential on R is zero).Then, the higher forms on M don’t play any role, and

DefP(R)[0] = A ∈ Ω!(M, gP)⊗m | ddR A + 12 [A, A] = 0.

One can show [Get04] that the simplicial set DefP(R) is weakly homotopy equivalentto the homotopy quotient of DefP(R)[0] by the nilpotent group associated to the nilpo-tent Lie algebra Ω0(M, gP)⊗ m. Indeed, a one-simplex in the simplicial set DefP(R)is given by a family of the form A0(t) + A1(t)dt, where A0(t) is a smooth family ofelements of Ω1(M, gP)⊗ m depending on t ∈ [0, 1], and A1(t) is a smooth family ofelements of Ω0(M, gP)⊗m. The Maurer-Cartan equation in this context says that

ddR A0(t) + 12 [A0(t), A0(t)] = 0

ddt A0(t) + [A1(t), A0(t)] = 0.

The first equation says that A0(t) defines a family of flat connections. The secondequation says that the gauge equivalence class of A0(t) is independent of t. In thisway, gauge equivalences are represented by one-simplices in DefP(R).

4.5. EXAMPLES OF ELLIPTIC MODULI PROBLEMS RELATED TO SCALAR FIELD THEORIES 93

It is immediate that the formal moduli problem DefP(R) is represented by theelliptic dg Lie algebra

L = Ω∗(M, g).

The differential on L is the de Rham differential on M coupled to the flat connectionon g.

4.4.2. Self-dual bundles. Next, we will discuss the formal moduli problem asso-ciated to the self-duality equations on a 4-manifold. We won’t go into as much detailas we did for flat connections; instead, we will simply write down the elliptic L∞ al-gebra representing this formal moduli problem.

Let M be an oriented 4-manifold. Let G be a Lie group, and let P → M be aprincipal G-bundle, and let gP be the adjoint bundle of Lie algebras. Suppose we havea connection A on P with anti-self-dual curvature:

F(A)+ = 0 ∈ Ω2+(M, gP)

(here Ω2+(M) denotes the space of self-dual two-forms).

Then, the elliptic Lie algebra controlling deformations of (P, A) is described by thediagram

Ω0(M, gP)d−→ Ω1(M, gP)

d+−→ Ω2+(M, gP).

Here d+ is the composition of the de Rham differential (coupled to the connection ongP) with the projection onto Ω2

+(M, gP).

Note that this elliptic Lie algebra is a quotient of that describing the moduli of flatG-bundles on M.

4.4.3. Holomorphic bundles. In a similar way, if M is a complex manifold and ifP→ M is a holomorphic principal G-bundle, then the elliptic dg Lie algebra Ω0,∗(M, gP),with differential ∂, describes the formal moduli space of holomorphic G-bundles onM.

4.5. Examples of elliptic moduli problems related to scalar field theories

4.5.1. The free scalar field theory. Let us start with the most basic example of anelliptic moduli problem, that of harmonic functions. Let M be a Riemannian manifold.


We want to consider the formal moduli problem describing functions φ on M whichare harmonic, that is, satisfy D φ = 0 where D is the Laplacian. The base point of thisformal moduli problem is the zero function.

The elliptic L∞ algebra describing this formal moduli problem is defined by

L = C∞(M)D−→ C∞(M).

This is situated in degrees 1 and 2. The products ln in this L∞ algebra are all zero forn ≥ 2.

In order to justify this definition, let us analyze the Maurer-Cartan functor of thisL∞ algebra. Let R be an ordinary (not dg) Artinian ring, and let m be the maximalideal of R. The set of 0-simplices of the simplicial set MCL (R) is the set

φ ∈ C∞(M)⊗m | D φ = 0.

Indeed, because the L∞ algebra L is Abelian, the set of solutions to the Maurer-Cartanequation is simply the set of closed degree 1 elements of the cochain complex L ⊗m.All higher simplices in the simplicial set MCL (R) are constant. Indeed, if φ ∈ L ⊗m⊗Ω∗(4n) is a closed element in degree 1, then φ must be in C∞(M)⊗m⊗Ω0(4n).The fact that φ is closed amounts to the statement that D φ = 0 and that ddRφ = 0,where ddR is the de Rham differential on Ω∗(4n).

Let us now consider the Maurer-Cartan simplicial set associated to a differentialgraded Artinian ring (R, m) with differential dR. The the set of 0-simplices of MCL (R)is the set

φ ∈ C∞(M)⊗m0, ψ ∈ C∞(M)⊗m−1 | D φ = dRψ.

(The superscripts on m indicate the cohomological degree). Thus, the zero-simplicesof our simplicial set can be identified with the set R-valued smooth functions φ onM which are harmonic up to a homotopy given by ψ, and which vanish modulo themaximal ideal m.

Next, let us identify the set of 1-simplices of the Maurer-Cartan simplicial setMCL (R). This is the set of closed degree 0 elements of L ⊗ m ⊗ Ω∗([0, 1]). Such

4.5. EXAMPLES OF ELLIPTIC MODULI PROBLEMS RELATED TO SCALAR FIELD THEORIES 95

a closed degree 0 element has four terms:

φ0(t) ∈ C∞(M)⊗m0 ⊗Ω0([0, 1])

φ1(t)dt ∈ C∞(M)⊗m−1 ⊗Ω1([0, 1])

ψ0(t) ∈ C∞(M)⊗m−1 ⊗Ω0([0, 1])

ψ1(t)dt ∈ C∞(M)⊗m−2 ⊗Ω1([0, 1]).

Being closed amounts to the sequence of equations

D φ0(t) = dRψ0(t)

ddt

φ0(t) = dRφ1(t)

D φ1(t) +ddt

ψ0(t) = dRψ1(t).

These equations can be interpreted as follows. We think of φ0(t) as providing a familyof R-valued smooth functions on M, which are harmonic up to a homotopy specifiedby ψ0(t). Further, φ0(t) is independent of t, up to a homotopy specified by φ1(t).Finally, we have a coherence condition among our two homotopies.

The higher simplices of the simplicial set have a similar interpretation.

4.5.2. Interacting scalar field theories. Next, we will consider an elliptic moduliproblem which arises as the Euler-Lagrange equation for an interacting scalar fieldtheory. The Euler-Lagrange equation for the action functional 1

2

∫φ D φ + 1

4! φ4 (where

φ ∈ C∞(M) is a smooth function) is the equation

D φ + 13! φ

3 = 0.

The formal moduli problem of solutions to this equation can be described as thesolutions to the Maurer-Cartan equation in a certain elliptic L∞ algebra which (as al-ways) we call L. As a cochain complex, L is

L = C∞(M)[−1] D−→ C∞(M)[−2].

Thus, C∞(M) is situated in degrees 1 and 2, and the differential is the Laplacian.


The L∞ brackets ln are all zero except for l3. The cubic bracket l3 is the map

l3 : C∞(M)⊗3 → C∞(M)

φ1 ⊗ φ2 ⊗ φ3 7→ φ1φ2φ3.

Here, the copy of C∞(M) appearing in the source of l3 is the one situated in degree 1,whereas that appearing in the target is the one situated in degree 2.

If R is an ordinary (not dg) Artinian ring, then the Maurer-Cartan simplicial setMCL (R) associated to R has for 0-simplices the set φ ∈ C∞(M)⊗m such that D φ +13! φ

3 = 0. The higher simplices of this simplicial set are constant.

If R is a dg Artinian ring, then the simplicial set MCL (R) has for zero simplicesthe set of pairs φ ∈ C∞(M)⊗m0 and ψ ∈ C∞(M)⊗m−1 such that

D φ + 13! φ

3 = dRψ.

We should interpret this as saying that φ satisfies the Euler-Lagrange equations up toa homotopy given by ψ.

The higher simplices of this simplicial set have an interpretation similar to thatdescribed for the free theory.

4.6. Cochains of a local Lie algebra

Let L be a local L∞ algebra on M. If U ⊂ M is an open subset, then L (U) denotesthe L∞ algebra of sections of L on U. Let Lc(U) ⊂ L(U) denote the sub-L∞ algebra ofcompactly supported sections.

In the appendix (section A.7) we defined the algebra of functions on the space ofsections on a vector bundle on a manifold. We are interested in the algebra

O(L(U)[1]) = ∏n≥0

Hom((L(U)[1])⊗n, R

)Sn

where the tensor product is the completed projective tensor product, and Hom de-notes the space of continuous linear maps.

This space is naturally a graded differentiable vector space. However, it is im-portant that we treat this object as a differentiable pro-vector space. Basic facts about

4.6. COCHAINS OF A LOCAL LIE ALGEBRA 97

differentiable pro-vector spaces are developed in the Appendix A. The pro-structurecomes the filtration

FiO(L(U)[1]) = ∏n≥i

Hom((L(U)[1])⊗n, R

)Sn

.

The L∞ algebra structure on L(U) gives, as usual, a differential on O(L(U)[1]),making O(L(U)[1]) into a differentiable pro-cochain complex.

4.6.0.1 Definition. Define the Lie algebra cochain complex C∗(L(U)) to be

C∗(L(U)) = O(L(U)[1])

equipped with the usual Chevalley-Eilenberg differential. Similarly, define

C∗red(L(U)) ⊂ C∗(L(U))

to be the reduced Chevalley-Eilenberg complex, that is, the kernel of the natural augmentationmap C∗(L(U)→ R. These are both differentiable pro-cochain complexes.

Of course, one can define C∗(Lc(U) in the same way.

We will think of C∗(L(U)) as the algebra of functions on the formal moduli prob-lem BL(U) associated to the L∞ algebra L(U).

4.6.1. Cochains with coefficients in a module. Let L be a local L∞ algebra on M,and let E be a graded vector bundle on M, equipped with a differential which is adifferential operator. As usual, we will let L and E denote the global sections of Land E, respectively.

4.6.1.1 Definition. A local action of L on E is an action of L on E with the property that thestructure maps

L ⊗n ⊗ E → E

are all polydifferential operators.

Note that L! = L∨ ⊗C∞M

DensM has the structure of a local module over L .

If E is a local module over L, then, for each U ⊂ M, we can define the Chevalley-Eilenberg cochains

C∗(L(U), E (U))


of L(U) with coefficients in E (U). As above, one needs to take account of the topolo-gies on the vector spaces L(U) and E (U) when defining this Chevalley-Eilenbergcochain complex. Thus, as a graded vector space,

C∗(L(U), E (U)) = ∏n≥0

Hom((L(U)[1])⊗n, E (U))Sn

where the tensor product is the completed projective tensor product, and Hom de-notes the space of continuous linear maps. Again, we treat this object as a differen-tiable pro-cochain complex.

As explained in the section on formal moduli problems (section 4.3), we shouldthink of a local module E over L as providing, on each open subset U ⊂ M, a vectorbundle on the formal moduli problem BL(U) associated to L(U). Then the Chevalley-Eilenberg cochain complex C∗(L(U), E (U)) should be thought of as the space of sec-tions of this vector bundle.

4.7. D-modules and local Lie algebras

Our definition of local L∞ algebra is designed to encode derived moduli spaces ofsolutions to non-linear differential equations. An alternative language for describingdifferential equations is the theory of D-modules. In this section we will show howour local L∞ algebras can also be viewed as L∞ algebras in the symmetric monoidalcategory of D-modules.

The main motivation for this extra layer of formalism is that local action function-als – which play a central role in classical field theory – are elegantly described usingthe language of D-modules.

Let C∞M denote the sheaf of smooth functions on the manifold M, DensM the sheaf

of densities, and DM the sheaf of differential operators. The infinite jet bundle Jet(E)of our vector bundle E is the vector bundle whose fiber at a point x ∈ M is the spaceof formal germs at x of sections of E. The sheaf of sections of Jet(E), denoted J(E),is equipped with a canonical DM-module structure, i.e., the natural flat connectionsometimes known as the Cartan distribution. (For motivation, observe that a field φ

(a section of E) gives a section of Jet(E) that encodes all the local information about φ.)

4.7. D-MODULES AND LOCAL LIE ALGEBRAS 99

The category of DM modules has a symmetric monoidal structure, given by ten-soring over C∞

M. The following lemma allows us to translate our definition of local L∞

algebra into the world of D-modules.

4.7.0.2 Lemma. Let E1, . . . , En, F be vector bundles on M, and let Ei, F denote their spacesof global sections. Then, there is a natural bijection

PolyDiff(E1 × · · · × En, F ) ∼= HomDM(J(E1)⊗ · · · ⊗ J(En), J(F))

where PolyDiff refers to the space of polydifferential operators. Further, this bijection is com-patible with composition.

A more formal statement of this lemma is that the multi-category of vector bun-dles on M, with morphisms given by polydifferential operators, is a full subcategory ofthe symmetric monoidal category of DM modules; with the embedding given by tak-ing jets. The proof of this lemma (which is straightforward) is presented in [Cos11c],Chapter 5.

4.7.0.3 Corollary. Let L be a local L∞ algebra on M. Then, J(L) has the structure of L∞

algebra in the category of DM modules.

Indeed, the lemma implies that to give a local L∞ algebra on M is the same as togive a graded vector bundle L on M together with an L∞ structure on the DM moduleJ(L).

We are interested in the Chevalley cochains of J(L), taken in the category of DM

modules. Because J(L) is an inverse limit of the sheaves of finite-order jets, some careneeds to be taken when defining this Chevalley cochain complex.

In general, if E is a vector bundle, let J(E)∨ denote the sheaf HomC∞M(J(E), C∞

M),where HomC∞

Mdenotes continuous linear maps of C∞

M-modules. This sheaf is naturallya DM-module. We can form the completed symmetric algebra

Ored(J(E)) = ∏n>0

Symn J(E)∨,

which is a DM-algebra.


We should think of an element of Ored(J(E)) as a Lagrangian on the space E ofsections of E. Indeed, a section

Fn ∈ HomC∞M(J(E)⊗n, C∞

M)

is something which takes n elements φ1, . . . , φn ∈ E and yields a smooth functionFn(φ1, . . . , φn) ∈ C∞(M), with the property that Fn(φ1, . . . , φn)(x) only depends on thejet of φi at x.

Let F be a section of Ored(E), and let us write F as a sum F = ∑ Fn, where

Fn ∈ HomC∞M(J(E)⊗n, C∞

M)Sn .

Then, we can interpret F as something which takes a section φ ∈ E and yields a smoothfunction

∑ Fn(φ, . . . , φ) ∈ C∞(M),

with the property that F(φ)(x) only depends on the jet of φ at x.

Of course, the functional F is a formal power series in the variable φ. A formal wayto say what such a power series is is to use the functor of points: if R is an auxiliarygraded Artin ring with maximal ideal m, and if φ ∈ E ⊗m, then F(φ) is an element ofC∞(M)⊗m. This assignment is functorial with respect to maps of graded Artin rings.

4.7.1. Local functionals. We have seen that we can interpret Ored(J(E)) as thesheaf of Lagrangians on a graded vector bundle E on M. Thus, the sheaf

DensM⊗C∞MOred(J(E))

is the sheaf of Lagrangian densities on M. A section F of this sheaf is something whichtakes a section φ ∈ E of E , and yields a density F(φ) on M, in such a way that F(φ)(x)only depends on the jet of φ at x. (As before, F is of course a formal power series inthe variable φ).

The sheaf of local action functionals is the sheaf of Lagrangians, modulo totalderivatives. The formal definition is as follows.

4.7.1.1 Definition. Let E be a graded vector bundle on M, whose space of global sections isE . Then, the space of local action functionals on E is

Oloc(E ) = DensM⊗DMOred(J(E)).

4.7. D-MODULES AND LOCAL LIE ALGEBRAS 101

Here, DensM is the right DM-module of densities on M.

Let Ored(Ec) denote the algebra of functionals modulo constants on the space Ec ofcompactly supported sections of E. Integration over M provides a natural inclusion

Oloc(E )→ Ored(Ec).

4.7.2. Local Chevalley complex of a local Lie algebra. Let L be a local Lie algebra.Then, we can form, as above, the reduced Chevalley cochain complex C∗red(J(L)) ofL. This is the DM-algebra Ored(J(L)[1]) equipped with a differential encoding the L∞

structure on L.

In general, if g is an L∞ algebra, we will think of the Lie algebra cochain complexC∗(g) as being the algebra of functions on Bg. Thus, we will let

Oloc(BL ) = DensM⊗DM C∗red(J(L))

denote the space of local action functionals on J(L)[1], equipped with the Chevalley-Eilenberg differential. This is the local Chevalley cochain complex. We could also usethe notation C∗red,loc(L) for this complex.

Note that there’s a natural inclusion of cochain complexes

Oloc(BL )→ C∗red(Lc)

where Lc denotes the L∞ algebra of compactly supported sections of L.

4.7.3. Modules over local L∞ algebras. Let L be a local L∞ algebra on M, and letE be a local module for L. Then, J(E) has an action of the L∞ algebra J(L), in a waycompatible with the DM-module on both J(E) and J(L).

4.7.3.1 Definition. Suppose that E has a local action of L. Then the local cochains C∗loc(L , E )

of L with coefficients in E is defined to be the flat sections of the DM-module of cochains ofJ(L) with coefficients in J(E).

More explicitly, the DM-module C∗(J(L), J(E) is

∏n≥0

HomC∞M

((J(L)[1])⊗n, J(E)

)Sn


equipped with the usual Chevalley-Eilenberg differential. The sheaf of flat sections ofthis DM module is the subsheaf

∏n≥0

HomDM

((J(L)[1])⊗n, J(E)

)Sn

where the maps must be DM-linear. In light of the fact that

HomDM

(J(L)⊗n, J(E)

)= PolyDiff(L ⊗n, E )

we see that C∗loc(L , E ) is precisely the subcomplex of the Chevalley-Eilenberg cochaincomplex

C∗(L , E ) = ∏n≥0

HomR((L [1])⊗n, E )Sn

consisting of those cochains built up from polydifferential operators.

4.8. The classical BV formalism in finite dimensions

Before we discuss the Batalin-Vilkovisky formalism for classical field theory, wewill discuss a finite-dimensional toy model (which we can think of as a 0-dimensionalclassical field theory). Our model for the space of fields is a finite-dimensional smoothmanifold manifold M. The “action functional” is given by a smooth function S ∈C∞(M). Classical field theory is concerned with solutions to the equations of motion.In our setting, the equations of motion are given by the subspace Crit(S) ⊂ M. Ourtoy model will not change if M is a smooth algebraic variety or a complex manifold, orindeed a smooth formal scheme. Thus we will write O(M) to indicate whatever classof functions (smooth, polynomial, holomorphic, power series) we are considering onM.

If S is not a nice function, then this critical set can by highly singular. The classicalBatalin-Vilkovisky formalism tells us to take, instead the derived critical locus of S. (Ofcourse, this is exactly what a derived algebraic geometer ([Lur09b], [Toe06]) wouldtell us to do as well).

The critical locus of S is the intersection of the graph

Γ(dS) ⊂ T∗M

4.8. THE CLASSICAL BV FORMALISM IN FINITE DIMENSIONS 103

with the zero-section of the cotangent bundle of M. Algebraically, this means that wecan write the algebra O(Crit(S)) of functions on Crit(S) as a tensor product

O(Crit(S)) = O(Γ(dS))⊗O(T∗M) O(M).

Derived algebraic geometry tells us that the derived critical locus is obtained by re-placing this tensor product with a derived tensor product. Thus, the derived criticallocus of S (which we denote Crith(S) is an object such that

O(Crith(S)) = O(Γ(dS))⊗LO(T∗M) O(M).

In derived algebraic geometry, as in ordinary geometry, spaces are determined by theiralgebras of functions. In derived geometry, however, one allows differential-gradedalgebras as algebras of functions (normally one restricts attention to differential-gradedalgebras concentrated in non-positive cohomological degrees).

We will take this derived tensor product as a definition of O(Crith(S)).

4.8.1. An explicit model. It is convenient to consider an explicit model for thederived tensor product. By taking a standard Koszul resolution of O(M) as a moduleover O(T∗M), one sees that O(Crith(S)) can be realized as the complex

O(Crith(S)) ' . . . ∨dS−−→ Γ(M,∧2TM)∨dS−−→ Γ(M, TM)

∨dS−−→ O(M).

In other words, we can identify O(Crith(S)) with functions on the graded manifoldT∗[−1]M, equipped with the differential given by contracting with dS.

Note that

O(T∗[−1]M) = Γ(M,∧∗TM)

has a Poisson bracket of cohomological degree 1, called the Schouten-Nijenhuis bracket.This Poisson bracket is characterized by the fact that if f , g ∈ O(M) and X, Y ∈Γ(M, TM), then

X, Y = [X, Y]

X, f = X f

f , g = 0

(the Poisson bracket between other elements of O(T∗[−1]M) is inferred from the Leib-niz rule).


The differential on O(T∗[−1]M) corresponding to that on O(Crith(S)) is given by

dφ = S, φ

for φ ∈ O(T∗[−1]M).

The derived critical locus of any function is a dg manifold equipped with a sym-plectic form of cohomological degree −1. In the Batalin-Vilkovisky formalism, thespace of fields always has such a symplectic structure. However, one does not requirethat the space of fields arises as the derived critical locus of a function.

4.8.2. The classical BV formalism in infinite dimensions. We would like to con-sider classical field theories in the BV formalism. For us, such a classical field theorywill be specified by an elliptic moduli problem equipped with a symplectic form ofcohomological degree −1.

We defined the notion of formal elliptic moduli problem on a manifold M usingthe language of L∞ algebras. Thus, in order to give the definition of a classical fieldtheory, we need to understand the following question: what extra structure on an L∞

algebra g endows the corresponding formal moduli problem with a symplectic form?

The answer to this question was given by Kontsevich [Kon93]. Given a pointedformal moduli problemM, the associated L∞ algebra gM has the property that

gM = TpM[−1].

Further, we can identify geometric objects onM in terms of gM as follows.

C∗(gM) The algebra O(M) of functions onMgM-modules V OM-modules

C∗(gM, V) the OM module corresponding to VThe gM-module gM[1] TM

Following this logic, we see that the complex of two-forms on M can be identifiedwith C∗(gM,∧2(g∨M[−1])).

However, on a symplectic formal manifold, one can always choose Darboux co-ordinates. Changes of coordinates onM correspond to L∞ isomorphisms on gM. InDarboux coordinates, the symplectic form has constant coefficients, and thus can beviewed as a gM-invariant element of ∧2(g∨M[−1]).

4.8. THE CLASSICAL BV FORMALISM IN FINITE DIMENSIONS 105

Note that the usual Koszul rules of signs imply that

∧2(g∨M[−1]) = Sym2(g∨M)[−2].

To give a gM-invariant element of Sym2(g∨M) is the same as to give an invariant sym-metric bilinear form on gM.

Thus, we arrive at the following principle:

To give a formal pointed derived moduli problem with a symplecticform of cohomological degree k is the same as to give an L∞ alge-bra with an invariant and non-degenerate pairing of cohomologicaldegree k− 2.

We will define a classical field theory to be an elliptic L∞ algebra equipped with anon-degenerate invariant pairing of cohomological degree−3. Let us first define whatit means to have an invariant pairing on an elliptic L∞ algebra.

4.8.2.1 Definition. Let M be a manifold, and let E be an elliptic L∞ algebra on M. Aninvariant pairing on E of cohomological degree k is a symmetric vector bundle map

〈−,−〉E : E⊗ E Dens(M)[k]

satisfying some additional conditions:

(1) Non-degeneracy: we require that this pairing induces a vector bundle isomorphism

E→ E∨ ⊗Dens(M)[−3].

(2) Invariance: let Ec denotes the space of compactly supported sections of E. The pairingon E induces an inner product on Ec, defined by

〈−,−〉 : Ec ⊗ Ec → R

α⊗ β→∫

M〈α, β〉 .

We require that this is an invariant pairing on the L∞ algebra Ec.


Recall that a symmetric pairing on an L∞ algebra g is called invariant if, for all n,the linear map

g⊗n+1 → R

α1 ⊗ · · · ⊗ αn+1 7→ 〈ln(α1, . . . , αn), αn+1〉

is graded anti-symmetric in the αi.

4.8.2.2 Definition. A formal pointed elliptic moduli problem on with a symplectic form ofcohomological degree k on a manifold M is an elliptic L∞ algebra on M with an invariantpairing of cohomological degree k− 2.

4.8.2.3 Definition. A (perturbative) classical field theory on M in the BV formalism is aformal pointed elliptic moduli problem on M with a symplectic form of cohomological degree−1.

4.9. The exterior derivative of a local action functional

The critical locus of a function f is, of course, the zero locus of the one-form d f .We are interested in constructing the derived critical locus of a local functional S ∈Oloc(BL) on the formal moduli problem associated to a local L∞ algebra on a manifoldM. Thus, we need to understand what kind of object the exterior derivative dS of suchan S is.

If g is an L∞ algebra, then we should think of C∗red(g) as the algebra of functionson the formal moduli problem Bg associated to g, which vanish at the base point.Similarly, C∗(g, g∨[−1]) should be the thought of as the space of one-forms on Bg. Theexterior derivative is thus a map

C∗red(g)→ C∗(g, g∨[−1]).

We will define a similar exterior derivative for a local Lie algebra L on M. Theanalog of g∨ is the L-module L!. Thus, our exterior derivative will be a map

Oloc(BL)→ C∗loc(L,L![−1]).

4.9. THE EXTERIOR DERIVATIVE OF A LOCAL ACTION FUNCTIONAL 107

Recall that Oloc(BL) is the subcomplex of the complex C∗red(Lc(M)) consisting oflocal functionals. The exterior derivative for the L∞ algebra Lc(M) is a map

d : C∗red(Lc(M))→ C∗(Lc(M),Lc(M)∨[−1]).

Note that the dual Lc(M)∨ of Lc(M) is the space L!(M) of distributional sections of

the bundle L! on M. Thus, the exterior derivative can be viewed as a map

d : C∗red(Lc(M))→ C∗(Lc(M),L!(M)[−1]).

Note that

C∗loc(L,L![−1]) ⊂ C∗(Lc(M),L!(M)) ⊂ C∗(Lc(M),L!(M)).

4.9.0.4 Lemma. The exterior derivative takes the subspace Oloc(BL) of C∗red(Lc(M)) to the

subspace C∗loc(L,L![−1]) of C∗(Lc(M),L!(M)).

PROOF. The content of this lemma is the familiar statement that the Euler-Lagrangeequations associated to a local action functional are differential equations. We will givea formal proof, but the reader will see that all that is used is integration by parts.

Any functional

F ∈ Oloc(BL)

can be written as a sum F = ∑ Fn where

Fn ∈ DensM⊗DM HomC∞M

(J(L)⊗n, C∞

M)

Sn.

Any such Fn can be written as a finite sum

Fn = ∑i

ωDi1 . . . Di

n

where ω is a section of DensM and Dij are differential operators from L → C∞

M.

If we view F ∈ O(Lc(M)), then the nth Taylor component of F is the linear map

Lc(M)⊗n → R

defined by

φ1 ⊗ · · · ⊗ φn →∑i

∫M

ω(Di1φ1) . . . (Di

nφn).


Thus, the (n− 1)th Taylor component of dF is given by the linear map

dFn : Lc(M)⊗n−1 → L!(M) = Lc(M)∨

φ1 ⊗ · · · ⊗ φn−1 ∑i7→ ω(Di

1φ1) . . . (Din−1φn−1)Di

n(−) + symmetric terms

where the right hand side is viewed as a linear map from Lc(M) to R. Now, by inte-gration by parts, we see that

(dFn)(φ1, . . . , φn−1)

is in the subspace L!(M) ⊂ L!(M) of smooth sections of the bundle L!(M), inside the

space of distributional sections.

It is clear from the explicit expressions that the map

dFn : Lc(M)⊗n−1 → L!(M)

is a polydifferential operator, and so defines an element of C∗loc(L,L![−1]) as desired.

4.9.1. Field theories from action functionals. Physicists normally think of a clas-sical field theory as being associated to an action functional. In this section we willshow how to construct a classical field theory in our sense from an action functional.

We will work in a quite general setting. Recall (section 4.3.3) that we defined alocal L∞ algebra on a manifold M to be a sheaf of L∞ algebras where the structuremaps are given by differential operators. We will think of a local L∞ algebra L on Mas defining a formal moduli problem cut out by some elliptic equations. We will usethe notation BL to denote this formal moduli problem.

We want to take the derived critical locus of a local action functional

S ∈ Oloc(BL )

of cohomological degree 0. (We also need to assume that S is at least quadratic: thismeans that the base-point of our formal moduli problem BL is a critical point ofS). We have seen (section 4.9) how to apply the exterior derivative to a local actionfunctional S yields an element

dS ∈ C∗loc(L,L![−1])

which we think of as being a local one-form on BL .


The critical locus of S is the zero locus of dS. We thus need to explain how toconstruct a new local L∞ algebra which we interpret as being the zero locus of dS.

4.9.2. Finite dimensional model. We will first describe the analogous construc-tion in finite dimensions. Let g be an L∞ algebra, M be a g-module of finite totaldimension, and α be a closed degree zero element of C∗red(g, M). The subscript red in-dicates that we are taking the reduced cochain complex, so that α is in the kernel ofthe augmentation map C∗(g, M)→ M.

We should think of M as a dg vector bundle on the formal derived moduli problemBg, and α as a section of this vector bundle. The condition that α is in the reducedcochain complex means translates into the statement that α vanishes at the basepointof Bg. We are interested in constructing the L∞ algebra representing the zero locus ofα.

The commutative dga representing this zero locus is given by the total complex ofthe double complex

· · · → C∗(g,∧2M∨) α∨−→ C∗(g, M∨) α∨−→ C∗(g).

This commutative dga is the symmetric algebra on the dual of g[1]⊕M[−1]. It followsthat this commutative dga is the Chevalley-Eilenberg cochain complex of g⊕M[−2],equipped with an L∞ structure arising from the differential on this complex.

Note that g⊕M[−2] has a natural semi-direct product L∞ structure, arising fromthe L∞ structure on g and the g action on M[−2]. This L∞ structure corresponds to thecase α = 0.

4.9.2.1 Lemma. The L∞ structure on g⊕M[−2] describing the zero locus of α is a deforma-tion of the semidirect product L∞ structure, obtained by adding to the structure maps ln themaps

Dnα : g⊗n → M

X1 ⊗ · · · ⊗ Xn 7→∂

∂X1. . .

∂

∂Xnα.

PROOF. The proof is a straightforward computation.


Note that the maps Dnα in the statement of the lemma are simply the homoge-neous components of the cochain α.

We will let Z(α) denote g⊕M[−2], equipped with this L∞ structure.

Recall that the formal moduli problem Bg is the functor from dg Artin rings (R, m)

to simplicial sets, sending (R, m) to the simplicial set of Maurer-Cartan elements ofg⊗m. In order to check that we have constructed the correct derived zero locus for α,we should describe the formal moduli problem associated Z(α).

Thus, let (R, m) be a dg Artin ring, and x ∈ g ⊗ m be an element of degree 1,and y ∈ M⊕m be an element of degree −1. Then, (x, y) satisfies the Maurer-Cartanequation in Z(α) if and only if:

(1) x satisfies the Maurer-Cartan equation in g⊗m.(2) α(x) = dxy ∈ M, where dx : M → M is the differential obtained by deform-

ing the original differential by that arising from the Maurer-Cartan elementx.

In other words, we see that an R-point of BZ(α) is an R-point x of Bg, and a ho-motopy between α(x) and 0, in the fiber Mx of the bundle M at x ∈ Bg.

4.9.3. The derived critical locus of a local functional. Let us now return to thesituation where L is a local L∞ algebra on a manifold M, and S ∈ O(BL) is a localfunctional which is at least quadratic. We let

dS ∈ C∗loc(L,L![−1])

denote the exterior derivative of S. Note that dS is in the reduced cochain complex(that is, the kernel of the augmentation map C∗loc(L,L![−1])→ L![−1]).

LetdnS : L⊗n → L!

be the nth Taylor component of dS. The fact that dS is a local cochain means that dnSis a polydifferential operator.

4.9.3.1 Definition. The derived critical locus of S is the local L∞ algebra obtained by addingto the structure maps ln of the semi-direct product L∞ algebra L⊕L![−3] the maps

dnS : L⊗n → L!.


Let us denote this local L∞ algebra by Crit(S). If (R, m) is an auxiliary Artiniandg ring, then a solution to the Maurer-Cartan equation in Crit(S)⊗ m consists of thefollowing data.

(1) A Maurer-Cartan element x ∈ L⊗m.(2) An element y ∈ L! ⊗m, such that

(dS)(x) = dxy.

Here, dxy is the differential on L!⊗m induced by the Maurer-Cartan element x. Thesetwo equations say that x is an R-point of BL which satisfies the Euler-Lagrange equa-tions up to a homotopy specified by y.

4.9.4. Symplectic structure on the derived critical locus. Recall that a classicalfield theory is given by a local L∞ algebra which is elliptic, and which also has aninvariant pairing of degree−3. The pairing on the local L∞ algebra Crit(S) constructedabove is evident: it is given by the natural bundle isomorphism

(L⊕ L![−3])![−3] ∼= L⊕ L![−3].

In other words, the pairing arises from the bundle map

L⊗ L! → DensM .

4.9.4.1 Lemma. This pairing is invariant under the L∞ structure constructed from dS.

PROOF. The original L∞ structure on L ⊕ L![−3] (that is, the L∞ structure not in-volving S) is easily seen to be invariant. We will verify that the deformation of thisstructure coming from S is also invariant.

We need to show that, if

α1, . . . , αn+1 ∈ Lc ⊕L!c[−3]

are compactly supported sections of L⊕ L1[−3], that

〈ln(α1, . . . , αn), αn+1〉

is totally antisymmetric in the variables αi. Now, the part of this expression whichcomes from S is just (

∂

∂α1. . .

∂

∂αn+1

)S(0).


The fact that partial derivatives commute, and the shift in grading coming from thefact that C∗(Lc) = O(Lc[1]), immediately implies that this is totally antisymmetric.

Note that, although the local L∞ algebra Crit(S) has a symplectic form, it does notalways define a classical field theory. It only does so under the additional assumptionthat the local L∞ algebra Crit(S) is elliptic.

4.10. A succinct definition of a classical field theory

We defined a classical field theory to be a formal elliptic moduli problem equippedwith a symplectic form of degree −1. In this section we will rewrite this definition ina more concise (but less conceptual) way. This is included largely for consistency with[Cos11c], and for ease of reference when we discuss the quantum theory.

4.10.0.2 Definition. Let E be a graded vector bundle on a manifold M. Then, a degree −1symplectic structure on E is an isomorphism of graded vector bundles

φ : E ∼= E![−1]

which is anti-symmetric, in the sense that φ∗ = −φ where φ∗ is the formal adjoint of φ.

Note that if L is an elliptic L∞ algebra on M with an invariant pairing of degree−3, then the graded vector bundle L[1] on M has a −1 symplectic form. Indeed, bydefinition, L is equipped with a symmetric isomorphism L ∼= L![−3], which becomesan antisymmetric isomorphism L[1] ∼= (L[1])![−1].

Note also that the tangent space at the basepoint to the formal moduli problemBL associated to L is L[1] (equipped with the differential induced from that on L).Thus, the algebra C∗(L) of cochains of L is isomorphic, as a graded algebra withoutthe differential, to the algebra O(L[1]) of functionals on L[1].

Now suppose that E is a graded vector bundle equipped with a −1 symplecticform. Let Oloc(E ) denote the space of local functionals on E .

4.10.0.3 Proposition. (1) The symplectic form on E induces a Poisson bracket on Oloc(E ),of degree +1.

4.10. A SUCCINCT DEFINITION OF A CLASSICAL FIELD THEORY 113

(2) To give a local L∞ algebra structure on E[1], compatible with the given pairing onE[1], is the same as to give an element S ∈ Oloc(E ) which is of cohomological degree0, at least quadratic, and satisfies the classical master equation

S, S = 0.

PROOF. Let L = E[−1]. Note that L is a local L∞ algebra, with 0 differential andbracket. We have seen that the exterior derivative (section 4.9) gives a map

d : Oloc(E ) = Oloc(BL)→ C∗loc(L, L![−1]).

Note that the isomorphism

L ∼= L![−3]

gives an isomorphism

C∗loc(L, L![−1]) ∼= C∗loc(L, L[2]).

Finally, C∗loc(L, L[2]) is the Lie algebra controlling the deformations of L as a local L∞

algebra. It thus remains to verify that Oloc(BL) ⊂ C∗loc(L, L[2]) is a sub-Lie algebra;but this is straightforward.

Note that the finite-dimensional analog of this statement is simply the fact that,on a formal symplectic manifold, all symplectic derivations (which correspond, aftera shift, to deformations of the formal symplectic manifold) are given by Hamiltonianfunctions, defined up to the addition of an additive constant. The additive constant isnot mentioned in our formulation because Oloc(E ) by definition consists of functionalswithout a constant term.

Thus, we can make a concise definition of a field theory.

4.10.0.4 Definition. A pre-classical field theory on a manifold M consists of a graded vectorbundle E on M, equipped with a symplectic pairing of degree −1, and a local functional

S ∈ Oloc(Ec(M))

of cohomological degree 0, satisfying the following properties.

(1) S satisfies the classical master equation S, S = 0.(2) S is at least quadratic (so that 0 ∈ Ec(M) is a critical point of S).


In this situation, we can write S as a sum (in a unique way)

S(e) = 〈e, Qe〉+ I(e)

where Q : E → E is a skew self-adjoint differential operator of cohomological degree1 and square zero.

4.10.0.5 Definition. A pre-classical field is a classical field theory if the complex (E , Q) iselliptic.

There is one more property we need of a classical field theories in order to be applythe quantization machinery of [Cos11c].

4.10.0.6 Definition. A gauge fixing operator is a map

QGF : E (M)→ E (M)

which is a differential operator of cohomological degree −1 and square zero, such that

[Q, QGF] : E (M)→ E (M)

is a generalized Laplacian in the sense of [BGV92].

The only classical field theories we can try to quantize are those which admit agauge fixing operator. We will only consider classical field theories which have agauge fixing operator.

4.11. Examples of field theories from action functionals

Let us now give some basic examples of field theories arising as the derived criticallocus of an action functional. We will only discuss scalar field theories in this section.

Let M be a Riemannian manifold. Let E = R be the trivial vector bundle on M,and let

S(φ) = 12

∫M

φ D φ

denote the action functional for the free massless field theory on M. Here D is theLaplacian on M, viewed as a differential operator from C∞(M) to Dens(M).

The derived critical locus of S is described by the elliptic L∞ algebra

L = C∞(M)[−1] D−→ Dens(M)[−2]

4.12. COTANGENT FIELD THEORIES 115

where Dens(M) is the global sections of the bundle of densities on M. Thus, C∞(M)

is situated in degree 1, and the space Dens(M) is situated in degree 2. The pairingbetween Dens(M) and C∞(M) gives the invariant pairing on L, which is symmetricof degree −3 as desired.

4.11.1. Interacting scalar field theories. Next, let us write down the derived crit-ical locus for a basic interacting scalar field theory, given by the action functional

S(φ) = 12

∫M

φ D φ + 14!

∫M

φ4.

The cochain complex underlying our elliptic L∞ algebra is, as before,

L = C∞(M)[−1] D−→ Dens(M)[−2].

The interacting term 14!

∫M φ4 gives rise to a higher bracket l3 on L, defined by the map

C∞(M)⊗3 → Dens(M)

φ1 ⊗ φ2 ⊗ φ3 7→ φ1φ2φ3dVol.

Let (R, m) be a nilpotent Artinian ring, concentrated in degree 0. Then, a sectionof φ ∈ C∞(M)⊗m satisfies the Maurer-Cartan equation in this L∞ algebra if and onlyif

D φ + 13! φ

3dVol = 0.

Note that this is precisely the Euler-Lagrange equation for S. Thus, the formal moduliproblem associated to L is, as desired, the derived version of the moduli of solutionsto the Euler-Lagrange equations for S.

4.12. Cotangent field theories

We have defined a field theory to be a formal elliptic moduli problem equippedwith a symplectic form of degree −1. The basic way symplectic manifolds arise ingeometry is, of course, as cotangent bundles. We can apply this in our setting: givenany elliptic moduli problem, we will construct a new elliptic moduli problem – itsshifted cotangent bundle – which has a symplectic form of degree −1. We will callfield theories which arise by this construction cotangent field theories. It turns out that asurprising number of field theories of interested in mathematics and physics (includ-ing, for example, both the A- and the B-models of mirror symmetry, as well as theirhalf-twisted versions) arise as cotangent theories.


We should regard cotangent field theories as the simplest and most basic class ofnon-linear field theories, just as cotangent bundles are the simplest class of symplecticvector spaces. One can show, for example, that the phase space of a cotangent fieldtheory is always an (infinite-dimensional) cotangent bundle, whose classical Hamil-tonian function is linear on the cotangent fibers.

4.12.1. The cotangent bundle to an elliptic moduli problem. Let L be an ellipticL∞ algebra on a manifold X; and letML be the associated elliptic moduli problem.

Let L! be the bundle L∨ ⊗Dens(X). Note that there is a natural pairing betweencompactly supported sections of L and compactly supported sections of L!.

Recall that we use the notation L to denote the space of sections of L; we will letL ! denote the space of sections of L!.

4.12.1.1 Definition. Let us define T∗[k]ML to be the elliptic moduli problem associated tothe elliptic L∞ algebra L⊕ L![k− 2].

This elliptic L∞ algebra has a pairing of cohomological degree k− 2.

The L∞ structure on the space L ⊕L ![k− 2] of sections of the direct sum bundleL⊕ L![k− 2] arises from the natural L -module structure on L !.

4.12.1.2 Definition. LetM be an elliptic moduli problem. Then, the cotangent field theoryassociated toM is the −1-symplectic elliptic moduli problem T∗[−1]M.

4.12.2. Examples. In this section we will list some basic examples of cotangenttheories.

In order to make the discussion more transparent, we will normally not explicitlydescribe the elliptic L∞ algebra related to an elliptic moduli problem; instead, we willsimply define the elliptic moduli problem in terms of the geometric objects it classifies.In all examples, it is straightforward using the techniques we have discussed so far toright down the elliptic L∞ algebra describing the formal neighborhood of a point inany of the elliptic moduli problems we will consider.

4.12. COTANGENT FIELD THEORIES 117

4.12.3. Self-dual Yang-Mills theory. Let X be an oriented 4-manifold equippedwith a conformal class of a metric. Let G be a compact Lie group. LetM(X, G) denotethe elliptic moduli problem parametrizing principal G-bundles on X with a connectionwhose curvature is self-dual.

Then, we can consider the cotangent theory T∗[−1]M(X, G). This theory is knownin the physics literature as self-dual Yang-Mills theory.

Let us describe the L∞ algebra of this theory explicitly. Observe that the elliptic L∞

algebra describing the completion ofM(X, G) near a point (P,∇) is

Ω0(X, gP)d−→ Ω1(X, gP)

d−−→ Ω2−(X, gP)

where gP is the adjoint bundle of Lie algebras associated to the principal G-bundle P.

Thus, the elliptic L∞ algebra describing T∗[−1]M is given by the diagram

Ω0(X, gP)d−→Ω1(X, gP)

d−−→ Ω2−(X, gP)

⊕ ⊕

Ω2−(X, gP)

d−→ Ω3(X, gP)d−→Ω4(X, gP)

This is a standard presentation of the fields of self-dual Yang-Mills theory in the BVformalism.

Ordinary Yang-Mills theory arises as a deformation of the self-dual theory. The de-formation is given by simply deforming the differential in the dg Lie algebra presentedin the diagram above by including a term in the differential which is the identity map-ping Ω2

−(X, gP) in degree 1 to the copy of Ω2−(X, gP) situated in degree 2.

4.12.4. The holomorphic σ-model. Let E be an elliptic curve and let X be a com-plex manifold. LetM(E, X) denote the elliptic moduli problem parametrizing holo-morphic maps from E → X. As before, there is an associated cotangent field theoryT∗[−1]M(E, X). (In [Cos11a] it is explained how to describe the formal neighborhoodof any point in this mapping space in terms of an elliptic L∞ algebra on E).

In [Cos10], this field theory was called a holomorphic Chern-Simons theory. In thephysics literature ([Wit05], [Kap05]) this theory is known as the (0, 2) supersymmetricsigma model.


This theory has an interesting role in both mathematics and physics. For instance,it was shown in [Cos10, Cos11a] that the partition function of this theory (at least, thepart which discards the contributions of non-constant maps to X) is the Witten genusof X.

4.12.5. Twisted supersymmetric gauge theories. Of course, there are a great manymore examples of cotangent theories, as there are very many elliptic moduli problems.In [Cos11b], it is shown how twisted versions of supersymmetric gauge theories canbe written as cotangent theories.

The most basic example is the twisted N = 1 field theory. If X is a complexsurface, and G is a complex Lie group, then the N = 1 twisted theory is simply thecotangent theory to the elliptic moduli problem of holomorphic principal G-bundleson X. If we fix one such principal G-bundle P → X, then the elliptic L∞ algebradescribing this formal moduli problem near P is

Ω0,∗(X, gP)

where gP is the adjoint bundle of Lie algebras associated to P.

The cotangent theory to this elliptic moduli problem is thus described by the ellip-tic L∞ algebra

Ω0,∗(X, gP ⊕ g∨P ⊗ KX[−1].).

4.12.6. The twisted N = 2 theory. Twisted versions of gauge theories with moresupersymmetry have similar descriptions, as is explained in [Cos11b]. The N =

2 theory is the cotangent theory to the elliptic moduli problem for holomorphic G-bundles P → X together with a holomorphic section of the adjoint bundle gP. Theelliptic L∞ algebra describing this moduli problem is

Ω0,∗(X, gP + gP[−1]).

Thus, the elliptic L∞ algebra for the cotangent theory is

Ω0,∗(X, gP + gP[−1]⊕ g∨P ⊗ KX ⊕ g∨P ⊗ KX[−1]).

4.13. THE GRADED POISSON STRUCTURE ON CLASSICAL OBSERVABLES 119

4.12.7. The twisted N = 4 theory. Finally we will describe the twisted N =

4 theory. There are two versions of this twisted theory: one used in the work ofVafa-Witten [VW94] on S-duality, and another considered more recently by Kapustin-Witten [KW06] in their work on geometric Langlands. Here we will describe only thelatter.

Let X again be a complex surface, and G a complex Lie group. Then, the twistedN = 4 theory is the cotangent theory to the elliptic moduli problem describing prin-cipal G-bundles P → X, together with a holomorphic section φ ∈ H0(X, T∗X ⊗ gP),satisfying

[φ, φ] = 0 ∈ H0(X, KX ⊗ gP).

Here T∗X is the holomorphic cotangent bundle of X.

The elliptic L∞ algebra describing this is

Ω0,∗(X, gP ⊕ T∗X⊗ gP[−1]⊕ KX ⊗ gP[−2]).

Of course, this elliptic L∞ algebra can be rewritten as

(Ω∗,∗(X, gP), ∂)

(so that the differential is just ∂ and does not involve ∂). The Lie bracket arises fromthe commutative algebra structure on the algebra Ω∗,∗(X) of forms on X, and the Liebracket on gP.

Thus, the elliptic Lie algebra describing the corresponding cotangent theory is

Ω∗,∗(X, gP)⊕Ω∗,∗(X, gP)[1].

4.13. The graded Poisson structure on classical observables

Recall the following definition.

4.13.0.1 Definition. A P0 algebra (in the category of cochain complexes) is a commutativedifferential graded algebra together with a Poisson bracket −,− of cohomological degree 1,which satisfies the Jacobi identity and the Leibniz rule.

The main result of this section is the following.


4.13.0.2 Theorem. For any classical field theory (section 4.10) on M, there is a P0 factoriza-

tion algebra Obscl

, together with a weak equivalence of commutative factorization algebras.

Obscl ∼= Obscl .

Concretely, Obscl(U) is built from functionals on the space of solutions to the

Euler-Lagrange equations which have more regularity than the functionals in Obscl(U).

The idea of the definition of the P0 structure is very simple. Let us start with a finitedimensional model. Let g be an L∞ algebra equipped with an invariant antisymmetricelement P ∈ g⊗ g, of cohomological degree 3. This can be viewed (according to thecorrespondence between formal moduli problems and Lie algebras (section 4.3)) asa bivector on Bg, and so defines a Poisson bracket on O(Bg) = C∗(g). Concretely,this Poisson bracket is defined, on the generators g∨[−1] of C∗(g), to be the tensor Pviewed as a map

g∨ ⊗ g∨ → R.

Now, let L be an elliptic L∞ algebra describing a classical field theory. Then, thekernel for the isomorphism L(U) ∼= L!(U)[−3] is an element P ∈ L(U) ⊗ L(U),which is symmetric, invariant, and of degree 3.

We would like to use this idea to define the Poisson bracket on

Obscl(U) = C∗(L(U)).

As in the finite dimensional case, in order to define such a Poisson bracket we wouldneed an invariant tensor in L(U)⊗2. The tensor representing our pairing is instead inL(U)⊗2, which contains L(U)⊗2 as a dense subspace.

We solve this problem by finding a subcomplex

Obscl(U) ⊂ Obscl(U)

on which the Poisson bracket is well-defined and such that the inclusion is a weakequivalence.

4.13.1. Overview. Here is an overview of this section.

(1) The Poisson structure for free field theories (section 4.14) In this section, wegive a simple direct construction of the P0 structure on the observables for a

4.14. THE POISSON STRUCTURE FOR FREE FIELD THEORIES 121

free field theory. For this class of field theories, we can find a simple modelfor the classical observables which has a P0 structure.

(2) The Poisson structure for a general classical field theory (section 4.15). In thissection we construct the Poisson structure in general: this construction is alittle more involved.

4.14. The Poisson structure for free field theories

In this section, we will construct a P0 structure on the factorization algebra of ob-servables of a free field theory. More precisely, in each case, we will construct a sub-complex

Obscl(U) ⊂ Obscl(U)

of the complex of classical observables for each open subset U, such that

(1) Obscl

forms a sub-commutative factorization algebra of Obscl ;

(2) The inclusion Obscl(U) ⊂ Obscl(U) is a weak equivalence of differentiable

pro-cochain complexes, for each U;

(3) Obscl

has the structure of P0 factorization algebra.

Recall that a free field theory is a classical field theory associated to an elliptic L∞

algebra L which is abelian, that is, where all the brackets ln | n ≥ 2 vanish.

Thus, let L be the graded vector bundle associated to an abelian elliptic L∞ algebra,and let L(U) be the elliptic complex of sections of L on U. To say that L defines a fieldtheory means we have a symmetric isomorphism L ∼= L![−3].

Recall (section A.2) that we use the notation L(U) to denote the space of distri-butional sections of L on U. A lemma of Atiyah-Bott (section A.9) shows that theinclusion

L(U) → L(U)

is a homotopy equivalence of topological cochain complexes.

It follows that the natural map

C∗(L(U)) → C∗(L(U))


is a cochain homotopy equivalence. Indeed, because we are dealing with an abelianL∞ algebra,

C∗(L(U)) = Sym(L(U)∨[−1])

C∗(L(U)) = Sym(L(U)∨[−1]),

where, as always, the symmetric algebra is defined using the completed tensor prod-uct.

Note that

L(U)∨ = L!c(U) = Lc(U)[3].

Thus,

C∗(L(U)) = Sym(Lc(U)[2])

C∗(L(U)) = Sym(Lc(U)[2]).

We can define a Poisson bracket of degree 1 on C∗(L(U) as follows. On the gener-ators Lc(U)[2], it is defined to be the given pairing

〈−,−〉 : Lc(U)×Lc(U)→ R.

This extends uniquely (by the Leibniz rule) to continuous bilinear map

C∗(L(U)× C∗(L(U)→ C∗(L(U).

In particular, we see that C∗(L(U)) has the structure of a P0 algebra in the multicate-gory of differentiable cochain complexes.

Let us define the modified observables in this theory by

Obscl(U) = C∗(L(U)).

We have seen that Obscl(U) is homotopy equivalent to Obscl(U), and that Obs

cl(U)

has a P0 structure.

4.14.0.1 Lemma. Obscl(U) has the structure of a P0 factorization algebra.

PROOF. It remains to verify that, if U1, . . . , Un are disjoint open subsets of M, eachcontained in an open subset W, then the map

Obscl(U1)× · · · × Obs

cl(Un)→ Obs

cl(W)

4.15. THE POISSON STRUCTURE FOR A GENERAL CLASSICAL FIELD THEORY 123

is compatible with the P0 structures. This map is automatically respects the commu-

tative structure; so it suffices to verify that α ∈ Obscl(Ui), and β ∈ Obs

cl(Uj), where

i 6= j, then

α, β = 0 ∈ Obscl(W).

This follows immediately from the fact that if φ, ψ ∈ Lc(W) have disjoint support,then

〈φ, ψ〉 = 0.

4.15. The Poisson structure for a general classical field theory

In this section we will prove the following.

4.15.0.2 Theorem. For any classical field theory (section 4.10) on M, there is a P0 factoriza-

tion algebra Obscl

, together with a quasi-isomorphism of factorization algebras

Obscl ∼= Obscl .

4.15.1. Functionals with smooth first derivative. As in our discussion of the P0

algebra for free field theories (section 4.14), we will define a subalgebra Obscl(U) ⊂

Obscl(U) consisting of functionals which have some additional smoothness proper-ties.

Let L be an elliptic L∞ algebra on M, which defines a classical field theory. Recallthat the cochain complex of observables is

Obscl(U) = C∗(L(U)),

where L(U) is the L∞ algebra of sections of L on U.

Recall that, as a graded vector space, C∗(L(U)) is the algebra of functionals O(L(U)[1])on the graded vector space L(U)[1]. In the appendix (section A.7), given any gradedvector bundle E on M, we define a subspace

O sm(E (U)) ⊂ O(E (U))


of functionals which have “smooth first derivative”. Precisely, a function Φ ∈ O(E (U))

is in O sm(E (U)) if

dΦ ∈ O(E (U))⊗ E !c (U).

(The exterior derivative of a general function in O(E (U)) will lie in the larger spaceO(E (U))⊗ E

!c(U)). The space O sm(E (U)) is a differentiable pro-vector space.

Recall that, if g is an L∞ algebra, the exterior derivative maps C∗(g) to C∗(g, g∨[−1]).The complex C∗sm(L(U)) of cochains with smooth first derivative is thus defined to bethe subcomplex of C∗(L(U)) consisting of this cochains whose first derivative lies inC∗(L(U),L!

c(U)[−1]), which is a subcomplex of C∗(L(U),⊗L(U)∨[−1]).

In other words, C∗sm(L(U)) is defined by the fiber diagram

C∗sm(L(U))d−→ C∗(L(U),L!

c(U)[−1])↓ ↓

C∗(L(U))d−→ C∗(L(U),Lc

!(U)[−1]).

(Note that differentiable pro-cochain complexes are closed under taking limits, so thatthis fiber product is again a differentiable pro-cochain complex; more details are pro-vided in the appendix A.7).

Note that

C∗sm(L(U)) ⊂ C∗(L(U))

is a sub-commutative dga, and further, as U varies, C∗sm(L(U)) defines a sub-commutativeprefactorization algebra of that defined by C∗(L(U)).

We will let

Obscl(U) = C∗sm(L(U)) ⊂ C∗(L(U)) = Obscl(U).

4.15.2. The Poisson bracket. Because the elliptic L∞ algebra L defines a classicalfield theory, it is equipped with an isomorphism L ∼= L![−3]. Thus, we have an iso-morphism

Φ : C∗(L(U),L!c(U)[−1]) ∼= C∗(L(U),Lc(U)[2]).


In the appendix (section A.8) we show that C∗(L(U),L(U)[1]) (which we think ofas vector fields on the formal manifold BL(U)) has a natural structure of dg Lie al-gebra, in the multicategory category of differentiable pro-cochain complexes. Fur-ther, C∗(L(U),L(U)[1]) acts on C∗(L(U)) by derivations. This action is in the multi-category of differentiable pro-cochain complexes: the map

C∗(L(U),L(U)[1])× C∗(L(U))→ C∗(L(U))

is a smooth bilinear cochain map. We will write Der(C∗(L(U)) for the Lie algebraC∗(L(U),L(U)[1]).

Thus, composing the map Φ above with the exterior derivative d, and with theinclusion Lc(U) → L(U), we find a cochain map

C∗sm(L(U))→ C∗(L(U),Lc(U)[2])→ Der(C∗(L(U))[1].

If f ∈ C∗sm(L(U)) we will let X f ∈ Der(C∗(L(U)) be the corresponding derivation. Iff has cohomological degree k, then X f has cohomological degree k + 1.

If f , g ∈ C∗sm(L(U)) = Obscl(U), we define

f , g = X f g ∈ Obscl(U).

This bracket defines a bilinear map

Obscl(U)× Obs

cl(U)→ Obs

cl(U).

4.15.2.1 Lemma. This map is smooth, that is, is a bilinear map in the multicategory of differ-entiable pro-cochain complexes.

PROOF. This follows from the fact that the map

d : Obscl(U)→ Der(C∗(L(U))[1]

is smooth (which is immediate from the definitions), and from the fact that the map

Der(C∗(L(U))× C∗(L(U))→ C∗(L(U))

is smooth (which is proved in the appendix A.8).

4.15.2.2 Lemma. This bracket satisfies the Jacobi rule and the Leibniz rule. Further, if U, V

are disjoint subsets of M, both contained in W, and if f ∈ Obscl(U), g ∈ Obs

cl(V), then

f , g = 0 ∈ Obscl(W).


PROOF. The proof is straightforward.

4.15.2.3 Corollary. Obscl

defines a P0 factorization algebra in the valued in the multicategoryof differentiable pro-cochain complexes.

The final thing we need to verify is the following.

4.15.2.4 Proposition. For all open subset U ⊂ M, the map

Obscl(U)→ Obscl(U)


PROOF. It suffices to show that it is a weak equivalence on the associated graded

for the natural filtration on both sides. Now, Grn Obscl(U) fits into a fiber diagram

Grn Obscl(U) //

Symn(L!c(U)[−1])⊗L!

c(U)

Grn Obscl(U) // Symn(L!c(U)[−1])⊗L!

c(U).

Note also that

Grn Obscl(U) = Symn L!c(U).

The Atiyah-Bott lemma A.9 shows that the inclusion

L!c(U) → L!

c(U)

is a continuous cochain homotopy equivalence. We can thus choose a homotopy in-verse PL!

c(U) → L!c(U), and a homotopy H : L!

c(U) → L!c(U) with [d, H] = P− Id;

with the property that H preserves the subspace L!c(U).

Now,

Symn L!c(U) ⊂ Grn Obs

cl(U) ⊂ Symn L!

c(U).

From the projector P and the homotopy H one can construct a projector

Pn = P⊗n : L!c(U)⊗n → L!

c(U)⊗n

and a homotopy

Hn : L!c(U)⊗n → L!

c(U)⊗n.


The homotopy Hn is defined inductively by the formula

Hn = H ⊗ Pn−1 + 1⊗ Hn−1.

This defines a homotopy, because

[d, Hn] = P⊗ Pn−1 − 1⊗ Pn−1 + 1⊗ Pn−1 − 1⊗ 1.

Notice that the homotopy Hn preserves all the subspaces of the form

L!c(U)⊗k ⊗L!

c(U)⊗ L!c(U)⊗n−k−1.

This will be important momentarily. Next, let

π : L!c(U)⊗n[−n]→ Symn(L!

c(U)[−1])

be the projection, and let

Γn = π−1 Grn Obscl(U).

Then, Γn is acted on by the symmetric group Sn, and the Sn invariants are Obscl(U).

Thus, it suffices to show that the inclusion

Γn → Lc(U)⊗n

is a weak equivalence of differentiable spaces; we will show that it is continuous ho-motopy equivalence.

The definition of Obscl(U) allows one can identify

Γn = ∩n−1k=0 L!

c(U)⊗k ⊗L!c(U)⊗ L!

c(U)⊗n−k−1.

The homotopy Hn preserves Γn, and the projector Pn maps

L!c(U)⊗n → Lc(U)⊗n ⊂ Γn.

Thus, Pn and Hn provide a continuous homotopy equivalence between L!c(U)⊗n and

Γn, as desired.


4.16. Symmetries and deformations of a classical field theory

In this section we will give the definition of an action of an L∞ algebra on a classicalfield theory. We will start by saying what it means for an L∞ algebra to act on an ellipticmoduli problem.

Recall that in homotopy theory, to give an action of a group G on an object is thesame as to give a family of objects over the classifying space BG. There is a similarpicture in homotopical algebra: to given an action of an L∞ algebra g on some objectis the same as to give a family of such objects over C∗(g). We will take this as ourdefinition of action of an L∞ algebra g on an elliptic L∞ algebra. Thus, we need todefine what it means to have a family of elliptic L∞ algebras over some differentialgraded base ring.

Suppose that R is a differential graded algebra. Let R] refer to R without the dif-ferential.

4.16.0.5 Definition. An R-family of elliptic L∞ algebras on X consists of graded bundle Lof R]-modules on X, whose sheaf of sections will be denoted L; together with an R]-lineardifferential operator

d : L → Lwhich makes L into a sheaf of dg R-modules; and, collection of R-linear polydifferential opera-tors

ln : L⊗n → Lmaking L into a sheaf of L∞ algebras on X over R.

Remark: Note that in this definition, R can be a nuclear Frchet dg algebra. In that case,the tensor products should be the completed projective tensor product.

4.16.0.6 Definition. If g is an L∞ algebra, and L is an elliptic L∞ algebra on a space X,a g-action on L is a family of elliptic moduli problems L g on X, over the base ring C∗(g),which specialize to L modulo the maximal ideal C>0(g) of C∗(g).

Remark: The Chevalley-Eilenberg cochain complex C∗(g) is the completed pro-nilpotentdg algebra, which is an inverse limit

C∗(g) = lim←−C∗(g)/In

4.16. SYMMETRIES AND DEFORMATIONS OF A CLASSICAL FIELD THEORY 129

where I is the maximal ideal C>0(g).

Let R be a differential graded algebra, and let L be an R-family of elliptic L∞

algebras. Recall that this means that we have a graded bundle L of R]-modules on X,whose sheaf L of sections is equipped with a differential making it into a sheaf of dgR-modules, and with an R-linear L∞ structure. We will let

L! = L∨ ⊗DensX

where L∨ is the R]-linear dual of L. We will let L ! denote the sheaf of sections of L!.This has a natural structure of sheaf of dg modules over R, with an L∞ action of L .

4.16.0.7 Definition. An invariant pairing of degree k on an R-family of elliptic L∞ algebrasL is an R-linear isomorphism

L ∼= L ![k]

of sheaves of L -modules, which is symmetric as before.

4.16.0.8 Definition. Let g be an L∞ algebra, and let L be a classical field theory on a spaceX. Thus L is an elliptic L∞ algebra on X with an invariant pairing L ∼= L ![−3] of degree−3. Then a g-action on L is a family of elliptic moduli problems L g on X, flat over the basering C∗(g), equipped with an invariant pairing of degree −3, which specializes to L modulothe maximal ideal C>0(g) of C∗(g).

If L is an elliptic L∞ algebra on X with an action of g, then the cotangent fieldtheory T∗[−1]L also has a natural action of g, compatible with the invariant pairing.

4.16.1. Symmetries and local functionals. Let L be a classical field theory, so thatL is a local L∞ algebra on M equipped with a non-degenerate symmetric isomorphismL ∼= L![−3]. Recall (section 4.10) that the L∞ structure on L can be described by a localfunctional S ∈ Oloc(BL), satisfying the classical master equation S, S = 0.

We will use this presentation to show that the dg Lie algebra controlling symme-tries and deformations of the classical field theory L is Oloc(BL)[−1], with the bracket−,− and differential S,−.

More precisely, we will verify the following.

4.16.1.1 Proposition.


//Let g be an L∞ algebra. Then every g action on L (in the sense described above)arises from an L∞ map g→ Oloc(BL)[−1]. This L∞ map is unique up to homotopy. //

PROOF. An L∞ map g→ Oloc(BL)[−1] is the same as an element

α ∈ C∗(g)⊗Oloc(BL)[−1]

which

Satisfies the Maurer-Cartan equation.

Vanishes modulo the maximal ideal C>0(g) of C∗(g).

The dg Lie algebra structure on C∗(g)⊗Oloc(BL)[−1] arises from the dg commu-tative algebra structure on C∗(g) and the dg Lie algebra structure on Oloc(BL)[−1].

The Maurer-Cartan equation is, of course,

dgα + S, α+ 12 α, α = 0

where dg refers to the differential on C∗red(g).

Given such an α, it is immediate that α+ S gives C∗(g)⊗ L the structure of a familyof classical field theories over C∗(g), as desired.

Conversely, suppose that we have a family of classical field theories over C∗(g),whose space of fields is the sections of some bundle L′ of C∗(g)-modules L′ on M.Let us suppose that this family reduces modulo the maximal ideal C>0(g) of C∗(g)to the given classical field theory, with underlying vector bundle L on M and actionfunctional S ∈ Oloc(BL). Then, it is easy to verify that we can choose an isomorphismof graded C](g)-modules L′ ∼= C](g) ⊗ L. The structure of family of classical fieldtheories on sections of L′ is encoded in some φ ∈ C∗(g) ⊗ Oloc(BL)[−1]. Modulothe maximal ideal of C∗(g), φ must reduce to the given action functional S, so thatφ = S + α where α is as above.

The final thing we need to verify is the statement that the L∞ map g→ Oloc(BL)[−1]corresponding to a given family of classical field theories over C∗(g) is unique up tohomotopy. The L∞ map is associated to the family of classical field theories togetherwith the isomorphism of graded C](g)-modules

Γ : L′ ∼= C](g)⊗ L

4.17. CONSERVED QUANTITIES AND NOETHER’S THEOREM 131

(of course, this isomorphism must be the given one modulo the maximal ideal ofC∗(g)). The space of such isomorphisms is contractible. Thus, it suffices to verifythat if we have a family of such isomorphisms, parametrized by the n-simplex, we geta family of L∞ maps over the base ring Ω∗(4n). If

Γ : L′ ⊗ C∞(4n) ∼= C](g)⊗ L⊗ C∞(4n)

is such a family, it extends by linearity to an isomorphism of graded C](g)⊗Ω](4n)-modules

L′ ⊗Ω](4n) ∼= C](g)⊗ L⊗Ω](4n).

The left hand side is equipped with the structure of a family of classical field theoriesover C∗(g) ⊗ Ω∗(4n) (this family is constant in the 4n-directions). As above, thisstructure can be interpreted as giving a Maurer-Cartan element α of C∗(g)⊗Ω∗(4n)⊗Oloc(BL)[−1], which vanishes modulo the maximal ideal of C∗(g). Such a Maurer-Cartan element is precisely the same as a family of L∞ maps g → Oloc(BL)[−1] overthe base ring Ω∗(4n).

4.17. Conserved quantities and Noether’s theorem

There is a famous theorem of E. Noether that says, in essence, that to every infini-tesimal local symmetry of a classical field theory corresponds a ”conserved current.”We will now describe that statement in our formalism. For simplicity, we will assumethat our space-time manifold M is oriented. It is straightforward to modify our con-structions to deal with the general case.

In the usual framework, a current J is something which associates to a field φ ann − 1 form J(φ) on space-time, defined up to the addition of an exact n − 1 form.Thus, we can integrate J(φ) over any oriented codimension 1 hypersurface in space-time. The association of J(φ) to φ must be local in nature: the value of J(φ) at a pointp ∈ M must only depend on the values of the derivatives of φ at p.

In our context, the space of fields is described by an elliptic L∞ algebra L on space-time M. The space of n− 1 forms is described by the Abelian L∞ algebra Ωn−1

M [−1],concentrated in degree 1. The space of n− 1 forms modulo exact n− 1 forms is bestdescribed by the truncated de Rham complex Ω≤n−1

M [n− 2], shifted so that Ωn−1M is in

degree 1. We will view Ω≤n−1M [n− 2] as an Abelian local L∞ algebra.


4.17.0.2 Definition. A current on the classical field theory described by L is a map of localL∞ algebras

J : L → Ω≤n−1M [n− 2].

A map of local L∞ algebras is a map of L∞ algebras L(M) → Ω≤n−1M [n− 2], with

the property that the constituent maps

L⊗n → Ω≤n−1M [n− 2]

are polydifferential operators.

A map of local L∞ algebras like this induces a map of sheaves of formal moduliproblems. We will let EL(U) be the formal moduli problem associated to the L∞ alge-bra L(U); so that if (R, m) is a dg Artinian algebra, EL(U)(R) is the simplicial set ofMaurer-Cartan elements of L(U)⊗m.

The formal moduli problem associated to the Abelian L∞ algebra Ω≤n−1(U)[n− 2]sends a dg Artinian algebra R to the Dold-Kan simplicial set of the cochain complexΩ≤n−1(U)[n− 1]⊗m. Recall that π0 of this Dold-Kan simplicial set is H0(Ω≤n−1(U)[n−1]⊗m). Since R is concentrated in degrees ≤ 0, we see that π0 of this simplicial set is(

Ωn−1(U)/dΩn−2(U))⊗ H0(m).

It follows that if J is a current, then for every dg Artinian ring (R, m), and for everyopen set U ⊂ M, we get a map

π0 J : π0(EL(U)(m))→(

Ωn−1(U)/dΩn−2(U))⊗ H0(m).

This map takes a homotopy class of solution to the Euler-Lagrange equations on U,and yields an n− 1 form modulo an exact n− 1 form.

4.17.1. Conserved currents. Suppose that N ⊂ M is a compact oriented submani-fold of codimension one. We will see how we can integrate J a current over N to yielda function on the formal moduli problem EL(M).

Let A1 denote the formal moduli problem which sends a dg Artinian ring (R, m)

to the Dold-Kan simplicial set for the cochain complex m. The reason for this notationis that A1 is represented by the pro-Artinian algebra R[[t]]. Thus, given any formal


moduli problem F, a map of formal moduli problems F → A1 should be regarded asa function on F.

Given a compact oriented codimension 1 submanifold N of M, a current J as aboveyields a map of formal moduli problems∫

NJ : EL(M)→ A1.

This sends an n-simplex φ ∈ EL(M)(R)[n] to∫

N J(φ), which is a closed degree 0element of m⊗Ω∗(4n).

Note that∫

N J(φ) only depends on the behavior of the field φ on a neighborhoodof N in M. Thus, we will let EL(N) denote the formal moduli problem of germs ofsolutions to the Euler-Lagrange equation near N:

EL(N)(R) = lim←−N⊂U⊂M

EL(U)(R).

The integral∫

N J is a function on the formal moduli problem EL(N).

In the usual treatment, a conserved current is a current with the property that,if we let a field evolve from one space-like hypersurface to another, the value of thecurrent does not change.

In order to explain this concept more precisely, let us assume that our space-timemanifold is of the form M = N ×R, where N is compact. We will let Nt denote thehypersurface N × t. Then, for each t ∈ R we have a map∫

Nt

J : EL(Nt)→ A1.

The condition for∫

NtJ to be a conserved quantity is that, if φt ∈ EL(Nt)(R) is a family

of fields which evolve according to the equations of motion, then

ddt

∫Nt

J(φt) = 0.

To say that a family of fields φt evolves according to the equation of motion meansprecisely that this family arises from a global solution φ ∈ EL(M)(R).

Thus, the statement that a current is conserved can be recast as saying that, for allsolutions φ ∈ EL(M)(R) of the equations of motion,

∫Nt

J(φ) is independent of t. Ofcourse, this will happen when the n− 1 form J(φ) is closed.


A conserved current should thus be a map of local L∞ algebras from L to the localL∞ algebra describing closed n − 1 forms, modulo total derivative. The latter is bestdescribed by the Abelian L∞ algebra Ω∗M[n− 2] of all forms, shifted so that Ωn−1

M is indegree 1.

4.17.1.1 Definition. A conserved current is a map of local L∞ algebras

L → Ω∗M[n− 2].

4.17.2. Noether’s theorem. Now we are ready to state Noether’s theorem, relat-ing conserved currents and symmetries. We have seen that the dg Lie algebra control-ling deformations and symmetries of the classical field theory L is Oloc(BL), the localfunctionals on L.

In our framework, Noether’s theorem is a statement about the simplicial set ofconserved currents. An n-simplex in this simplicial set is simply a family of conservedcurrents over the base ring Ω∗(4n).

4.17.2.1 Theorem. There is a natural weak homotopy equivalence between the simplicial set ofconserved currents and the Dold-Kan simplicial set for the cochain complex Oloc(BL)[−1]. Inparticular, homotopy classes of conserved currents are in bijection with the group H−1(Oloc(BL)),which is the group of homotopy classes of infinitesimal symmetries of the classical field theoryL.

PROOF. To give a map L → Ω∗M[n− 2] of local L∞ algebras is the same as to givea map J(L)→ J(Ω∗M)[n− 2] of DM L∞ algebras.

In general, if g and h are L∞ algebras, the L∞ algebra controlling maps from g to h

is C∗red(g)⊗ h, the tensor product of the reduced cochain complex of g with h.

It follows that to give a map of DM L∞ algebras J(L) → J(Ω∗M)[n− 2] is the sameas to give a flat section of

C∗red(J(L))⊗C∞M

J(Ω∗M)[n− 1].

The sheaf of flat sections of this can be identified with

Ω∗(M, C∗red(J(L)))[n− 1],

the de Rham complex of M with coefficients in the DM-module C∗red(J(L)), with a shiftof n− 1.


More generally, the simplicial set of maps of local L∞ algebras L → Ω∗M[n− 2] isthe Dold-Kan simplicial set associated to the cochain complex Ω∗(M, C∗red(J(L)))[n−1].

Since M is oriented, and since C∗red(J(L)) is a flat DM-module, this cochain coin-cides with

ωM ⊗DM C∗red(J(L))[−1] = Oloc(BL)[−1].

Thus, the simplicial set of conserved currents is the Dold-Kan simplicial set for Oloc(BL)[−1],as desired.

Note that is not difficult to modify these constructions to relate “higher conservedcurrents” (which associate to a field a closed n− k-form) and higher infinitesimal sym-metries of a classical field theory, corresponding to the groups Hn−k(Oloc(BL)).

4.17.3. A simple example. Consider the free particle moving in the Euclidean vec-tor space V ∼= Rn. We will show how to recover the usual conserved quantity of linearmomentum using the formalism described above. We encode this physical system asa free theory on the real line R. Let V also denote the trivial vector bundle over R1

with fiber V.

A section φ ∈ Γ(R1, V) satisfies the equation of motion if d2

dt2 φ = 0. Thus, theelliptic L∞ algebra describing this classical field theory is

L = Γ(R1, V)[−1]d2

dt2−−→ Γ(R1, V[−2]).

This L∞ is Abelian. The pairing between L1 = Γ(R1, V) and L2 = Γ(R1, V) is

〈φ, ψ〉 =∫

R1〈φ(t), ψ(t)〉V dt,

where 〈−,−〉V is the Euclidean inner product on V.

We now describe the obstruction-deformation complex Oloc(BL) for this free the-ory. Let J denote the infinite jet bundle of smooth functions on R1. Then the jetbundle J(V) for sections of V is isomorphic to J ⊗ V. Likewise, recall that J∨ :=HomC∞(J, C∞) = DR, the ring of differential operators on R1, and so J(V)∨ ∼= D⊗V∨.Hence the DR L∞ algebra associated to L is

J(L) = J ⊗ (V[−1]⊕V[−2]).


This system is invariant under the translation action of the vector space V. We willshow how to construct the conserved current associated to translation by each v ∈ V.This conserved current plays the role of momentum.

The Lie algebra of symmetries and deformations of our theory is

Oloc(BL)[−1] = ωR1 ⊗DR1 C∗red(J(L))[−1],

which we can identify withΩ∗(R1, C∗red(J(L))).

Since we are dealing with a free field theory, the differential on the Lie algebracochain complex C∗red(J(L)) preserves the grading into homogeneous components.Since momentum is a linear function on the vector space V, we are interested in sym-metries coming from J(L)∨[−2].

The de Rham complex of the D module J(L)∨[−2] is concentrated in cohomologi-cal degrees −1, 0, and 1:

0→ Ω0⊗DR1 ⊗V∨ → (Ω0⊗DR1 ⊗V∨0 )⊕ (Ω1⊗DR1 ⊗V∨1 )→ Ω1⊗DR1 ⊗V∨0 → 0,

and the differential is∇+ Q, where∇ denotes the flat connection on the bundle DR1 ,and Q is the differential on J(L)∨[−1].

An element of degree 0 is a sum of terms

f ⊗ ∂k ⊗ v,

with f a smooth function, ∂ = ∂/∂x, and v ∈ V∨, and of terms

f dx⊗ ∂k ⊗ v.

We compute the differential as follows:

f ⊗ ∂k ⊗ v + g dx⊗ ∂l ⊗ w 7→ (∂ f )dx⊗ ∂k ⊗ v + f dx⊗ ∂k+1 ⊗ v + g dx⊗ ∂l+2 ⊗ w,

since we apply the connection on DR1 to terms in de Rham degree 0 and the differentialon J(L)∨ to terms in de Rham degree 1.

To each v ∈ V we have a translation symmetry, given by the closed element ofdegree 0

Xv = 1⊗ ∂⊗ v∨ − dx⊗ 1⊗ v∨,

where v∨ ∈ V∨ is the linear functional given by inner product with v.


The associated current J(Xv) simply picks out the 0-form component of Xv. Thus,the current is

J(Xv) = 1⊗ ∂⊗ v.

Applied to a field φ ∈ C∞(R1)⊗V, the current J(Xv) yields

〈v, ∂φ〉V ∈ C∞(R1).

Evaluated at a point t ∈ R1, we find 〈v, ∂φ〉V (t), which is the momentum of φ in thev-direction at t.

CHAPTER 5

Quantum field theory

We have already explained in 1.7 the philosophy of this book: a classical field the-ory gives a P0 factorization algebra, and a quantum field theory gives a quantizationof this P0 factorization algebra. In this section we will prove the following theorem.

5.0.3.1 Theorem. Any quantum field theory on a manifold M, in the sense of [Cos11c], givesrise to a factorization algebra Obsq on M of quantum observables. This is a factorizationalgebra over C[[h]], valued in differentiable pro-cochain complexes, and it quantizes (in thesense of 1.7) the P0 factorization algebra of classical observables of the corresponding classicalfield theory.

The machinery of [Cos11c] allows one to construct many examples of quantumfield theories. Given a classical field theory, the results of [Cos11c] allow one to de-scribe possible quantizations in terms of certain cohomology groups of local function-als. By calculating these cohomology groups, one can construct many examples ofquantum field theory. For example, in [Cos11c], the quantum Yang-Mills gauge the-ory is constructed.

Thus, this theorem, together with the results of [Cos11c], produces many interest-ing examples of factorization algebras.

Here is an overview of this chapter.

(1) In section 5.1 we recall the definition of a free theory in the BV formalism andconstruct the factorization algebra of quantum observables of a general freetheory.

(2) In sections 5.2 to 5.6 we give an overview of the definition of QFT developedin [Cos11c].

139

140 5. QUANTUM FIELD THEORY

(3) In section 5.7 we show how the definition of a QFT leads immediately toa construction of a BD algebra of “global observables” on the manifold M,which we denote Obsq

P(M).(4) In section 5.8 we start the construction of the factorization algebra associ-

ated to a QFT. We construct a cochain complex Obsq(M) of global observ-ables, which is quasi-isomorphic to (but much smaller than) the BD algebraObsq

P(M).(5) In section 5.10 we construct, for every open subset U ⊂ M, the subspace

Obsq(U) ⊂ Obsq(M) of observables supported on U.(6) Section 5.11 accomplishes the primary aim of the chapter. In it, we prove that

the cochain complexes Obsq(U) form a factorization algebra. The proof ofthis result is the most technical part of the chapter.

(7) In section 5.12 we show that translation-invariant theories have translation-invariant factorization algebras of observables, and we treat the holomorphicsituation as well.

(8) In section 5.13 we explain how to interpret our definition of a QFT in thespecial case of a cotangent theory: roughly speaking, a quantization of thecotangent theory to an elliptic moduli problem yields a locally-defined vol-ume form on the moduli problem we start with.

Remark: We forewarn the reader that our definitions and constructions involve a heavyuse of functional analysis and (perhaps more surprisingly) simplicial sets. Makinga quantum field theory typically requires many choices, and as mathematicians, wewish to pin down precisely how the quantum field theory depends on these choices.The machinery we use gives us very precise statements, but statements that can beforbidding at first pass. We encourage the reader, on a first pass through this material,to simply make all necessary choices (such as a parametrix) and focus on the outputof our machine, namely the factorization algebra of quantum observables. Keepingtrack of the dependence on choices requires careful bookkeeping (aided by the ma-chinery of simplicial sets, etc) but is straightforward once the primary construction isunderstood.

5.1. FREE FIELDS 141

5.1. Free fields

Before we give our general construction of the factorization algebra associated toa quantum field theory, we will give the much easier construction of the factorizationalgebra for a free field theory.

Let us recall the definition of a free BV theory.

5.1.0.2 Definition. A free BV theory on a manifold M consists of the following data:

(1) a Z-graded super vector bundle π : E→ M that has finite rank;(2) an antisymmetric map of vector bundles 〈−,−〉loc : E⊗ E → Dens(M) of degree−1 that is fiberwise nondegenerate. It induces a symplectic pairing on compactlysupported smooth sections Ec of E:

〈φ, ψ〉 =∫

x∈M〈φ(x), ψ(x)〉loc;

(3) a square-zero differential operator Q : E → E of cohomological degree 1 that is skewself adjoint for the symplectic pairing.

Remark: When we consider deforming free theories into interacting theories, we willneed to assume the existence of a “gauge fixing operator”: this is a degree−1 operatorQGF : E → E such that [Q, QGF] is a generalized Laplacian in the sense of [BGV92].

On any open set U ⊂ M, the commutative dg algebra of classical observablessupported in U is

Obscl(U) = (Sym(E ∨(U)), Q),

where

E ∨(U) = E!c(U)

denotes the distributions dual to E with compact support in U and Q is the derivationgiven by extending the natural action of Q on the distributions.

In section 4.14 we constructed a sub-factorization algebra

Obscl(U) = (Sym(E !

c (U)), Q)

defined as the symmetric algebra on the compactly-supported smooth (rather than

distributional) sections of the bundle E!. We showed that the inclusion Obscl(U) →


Obscl(U) is a weak equivalence of factorization algebras. Further, Obscl(U) has a Pois-

son bracket of cohomological degree 1, defined on the generators by the natural pair-ing

E !c (U)⊗ E !

c (U)→ R,

which arises from the dual pairing on Ec(U). In this section we will show how to

construct a quantization of the P0 factorization algebra Obscl

.

5.1.1. The Heisenberg algebra construction. Our quantum observables on an openset U will be built from a certain Heisenberg Lie algebra.

Recall the usual construction of a Heisenberg algebra. If V is a symplectic vectorspace, viewed as an abelian Lie algebra, then the Heisenberg algebra Heis(V) is thecentral extension

0→ C · h→ Heis(V)→ V

whose bracket is [x, y] = h〈x, y〉.

Since the element h ∈ Heis(V) is central, the algebra U(Heis(V)) is an algebraover C[[h]], the completed universal enveloping algebra of the Abelian Lie algebraC · h.

In quantum mechanics, this Heisenberg construction typically appears in the studyof systems with quadratic Hamiltonians. In this context, the space V can be viewedin two ways. Either it is the space of solutions to the equations of motion, which is alinear space because we are dealing with a free field theory; or it is the space of linearobservables dual to the space of solutions to the equations of motion. The natural sym-plectic pairing on V gives an isomorphism between these descriptions. The algebraU(Heis(V)) is then the algebra of non-linear observables.

Our construction of the quantum observables of a free field theory will be formallyvery similar. We will start with a space of linear observables, which (after a shift) isa cochain complex with a symplectic pairing of cohomological degree 1. Then, in-stead of applying the usual universal enveloping algebra construction, we will takeChevalley-Eilenberg chain complex, whose cohomology is the Lie algebra homology.1

This fits with our operadic philosophy: Chevalley-Eilenberg chains are the E0 analogof the universal enveloping algebra.

1As usual, we always use gradings such that the differential has degree +1.

5.1. FREE FIELDS 143

5.1.2. The basic homological construction. Let us start with a 0-dimensional freefield theory. Thus, let V be a cochain complex equipped with a symplectic pairing ofcohomological degree −1. We will think of V as the space of fields of our theory. Thespace of linear observables of our theory is V∨; the Poisson bracket on O(V) inducesa symmetric pairing of degree 1 on V∨. We will construct the space of all observablesfrom a Heisenberg Lie algebra built on V∨[−1], which has a symplectic pairing 〈−,−〉of degree −1. Note that there is an isomorphism V ∼= V∨[−1] compatible with thepairings on both sides.

5.1.2.1 Definition. The Heisenberg algebra Heis(V) is the Lie algebra central extension

0→ C · h[−1]→ Heis(V)→ V∨[−1]→ 0

whose bracket is[v + ha, w + hb] = h 〈v, w〉

The element h labels the basis element of the center C[−1].

Putting the center in degree 1 may look strange, but it is necessary to do this inorder to get a Lie bracket of cohomological degree 0.

Let C∗(Heis(V)) denote the completion2 of the Lie algebra chain complex of Heis(V),defined by the product of the spaces Symn Heis(V), instead of their sum.

In this zero-dimensional toy model, the classical observables are

Obscl = O(V) = ∏n

Symn(V∨).

This is a commutative dg algebra equipped with the Poisson bracket of degree 1 aris-ing from the pairing on V. Thus, O(V) is a P0 algebra.

5.1.2.2 Lemma. The completed Chevalley-Eilenberg chain complex C∗(Heis(V)) is a BDalgebra (section 2.4) which is a quantization of the P0 algebra O(V).

PROOF. The completed Chevalley-Eilenberg complex for Heis(V) has the com-pleted symmetric algebra Sym(Heis(V)[1]) as its underlying graded vector space.Note that

Sym(Heis(V)[1]) = Sym(V∨ ⊕C · h) = Sym(V∨)[[h]],

2One doesn’t need to take the completed Lie algebra chain complex. We do this to be consistent withour discussion of the observables of interacting field theories, where it is essential to complete.


so that C∗(Heis(V)) is a flat C[[h]] module which reduces to Sym(V∨) modulo h. TheChevalley-Eilenberg chain complex C∗(Heis(V)) inherits a product, correspondingto the natural product on the symmetric algebra Sym(Heis(V)[1]). Further, it has anatural Poisson bracket of cohomological degree 1 arising from the Lie bracket onHeis(V), extended to be a derivation of C∗(Heis(V)). Note that, since C · h[−1] is cen-tral in Heis(V), this Poisson bracket reduces to the given Poisson bracket on Sym(V∨)modulo h.

In order to prove that we have a BD quantization, it remains to verify that, al-though the commutative product on C∗(Heis(V)) is not compatible with the product,it satisfies the BD axiom:

d(a · b) = (da) · b + (−1)|a|a · (db) + ha, b.

This follows by definition.

5.1.3. Cosheaves of Heisenberg algebras. Next, let us give the analog of this con-struction for a general free BV theory E on a manifold M. As above, our classicalobservables are defined by

Obscl(U) = Sym E !

c (U)

which has a Poisson bracket arising from the pairing on E !c (U). Recall that this is a

factorization algebra.

To construct the quantum theory, we define, as above, a Heisenberg algebra Heis(U)

as a central extension

0→ C[−1] · h→ Heis(U)→ E !c (U)[−1]→ 0.

Note that Heis(U) is a pre-cosheaf of Lie algebras. The bracket in this Heisenbergalgebra arises from the pairing on E !

c (U).

We then define the quantum observables by

Obsq(U) = C∗(Heis(U)).

The underlying cochain complex is, as before,

Sym(Heis(U)[1])

where the completed symmetric algebra is defined (as always) using the completedtensor product.

5.2. OVERVIEW OF PERTURBATIVE QUANTUM FIELD THEORY 145

5.1.3.1 Proposition. Sending U to Obsq(U) defines a BD factorization algebra in the cate-gory of differentiable pro-cochain complexes over R[[h]], which quantizes Obscl(U).

PROOF. First, we need to define the filtration on Obsq(U) making it into a differ-entiable pro-cochain complex. The filtration is defined, in the identification

Obsq(U) = Sym E !c (U)[[h]]

by sayingFn Obsq(U) = ∏

khk Sym≥n−2k E !

c (U).

This filtration is engineered so that the Fn Obsq(U) is a subcomplex of Obsq(U).

It is immediate that Obsq is a BD pre-factorization algebra quantizing Obscl(U).The fact that it is a factorization algebra follows from the fact that Obscl(U) is a factor-ization algebra, and then a simple spectral sequence argument. (A more sophisticatedversion of this spectral sequence argument, for interacting theories, is given in section5.11.)

5.2. Overview of perturbative quantum field theory

In sections 5.2 to 5.6, we will give a summary of the definition of a QFT as devel-oped in [Cos11c]. We will emphasize the aspects which we will use in our constructionof the factorization algebra associated to a QFT. This means that important aspects ofthe story there — such as the concept of renormalizability — will not be mentioned.

The introductory chapter of [Cos11c] is a leisurely exposition of the main physicaland mathematical ideas, and we encourage the reader to examine it before delving intothis book. The approach is perturbative and hence has the flavor of formal geometry(that is, geometry with formal manifolds).

A perturbative field theory is defined to be a family of effective field theoriesparametrized by some notion of “scale”. The notion of scale can be quite flexible:the simplest version is where the scale is a positive real number, the length. Then, theeffective theory at a length scale L is obtained from the effective theory at scale ε byintegrating out over fields with length scale between ε and L. In order to construct fac-torization algebras, we need a more refined notion of “scale”, where there is a scale forevery parametrix Φ of a certain elliptic operator. We denote such a family of effective


field theories by I[Φ], where I[Φ] is the “interaction term” in the action functionalS[Φ] at “scale” Φ. We always study families with respect to a fixed free theory.

A local action functional (see section 5.3) S is a real-valued function on the space offields such tat S(φ) is given by integrating some function of the field and its derivativesover the base manifold (the “spacetime”). The main result of [Cos11c] states that thespace of perturbative QFTs is the “same size” as the space of local action functionals.More precisely, the space of perturbative QFTs defined modulo hn+1 is a torsor overthe space of QFTs defined modulo hn for the abelian group of local action functionals.

The starting point for many physical constructions – such as the path integral – isa local action functional. However, a naive application of these constructions to suchan action functional yields a nonsensical answer. Many of these constructions do workif, instead of applying them to a local action functional, they are applied to a familyI[Φ] of effective action functionals. Thus, one can view the family of effective actionfunctionals I[Φ] as a quantum version of the local action functional defining classi-cal field theory. The results of [Cos11c] allow one to construct such families of actionfunctionals. Many formal manipulations with path integrals in the physics literatureapply rigorously to families I[Φ] of effective actions. Our strategy for constructingthe factorization algebra of observables is to mimic path-integral definitions of observ-ables one can find in the physics literature, but replacing local functionals by familiesof effective actions.

5.3. Local action functionals

In studying field theory, there is a special class of functions on the fields, known aslocal action functionals, that parametrize the possible classical physical systems. LetM be a smooth manifold. Let E = C∞(M, E) denote the smooth sections of a Z-gradedsuper vector bundle E on M, which has finite rank when all the graded componentsare included. We call E the fields.

Spaces of functions on the space of fields are defined in the appendix A.7.

5.3.0.2 Definition. A functional F is an element of

O(E ) =∞

∏n=0

HomDVS(E⊗n, R)Sn ,

5.4. THE DEFINITION OF A QUANTUM FIELD THEORY 147

namely the completed symmetric algebra of E ∨, the continuous dual space to the fields.

Let Ored(E ) = O(E )/C be the space of functionals on E modulo constants.

The local functionals depend only on the local behavior of a field, so that at eachpoint of M, a local functional should only depend on the jet of the field at that point.In the Lagrangian formalism for field theory, their role is to describe the permittedactions, so we call them local action functionals. A local action functional is the essentialdatum of a classical field theory.

5.3.0.3 Definition. A functional F is local if each homogeneous component Fn is a finite sumof terms of the form

Fn(φ) =∫

M(D1φ) · · · (Dnφ) dµ,

where each Di is a differential operator from E to C∞(M) and dµ is a density on M.

We let

Oloc(E ) ⊂ Ored(E )

denote denote the space of local action functionals modulo constants.

dg algebra

5.4. The definition of a quantum field theory

In this section, we will give the formal definition of a quantum field theory. Thedefinition is a little long and somewhat technical. The reader should consult the firstchapter of [Cos11c] for physical motivations for this definition. We will provide somejustification for the definition from the point of view of homological algebra shortly(section 5.7).

5.4.1. In section 5.1 we gave a definition of a free BV theory.

5.4.1.1 Definition. A free BV theory on a manifold M consists of the following data:

(1) a Z-graded super vector bundle π : E→ M that is of finite rank;


(2) an antisymmetric map of vector bundles 〈−,−〉loc : E⊗ E → Dens(M) of degree−1 that is fiberwise nondegenerate. It induces a symplectic pairing on compactlysupported smooth sections Ec of E:

〈φ, ψ〉 =∫

x∈M〈φ(x), ψ(x)〉loc;

(3) a square-zero differential operator Q : E → E of cohomological degree 1 that is skewself adjoint for the symplectic pairing.

In our constructions, we require the existence of a gauge-fixing operator QGF : E →E with the following properties:

(1) it is a square-zero differential operator of cohomological degree −1 ;(2) it is self adjoint for the symplectic pairing;(3) D = [Q, QGF] is a generalized Laplacian on M, in the sense of [BGV92]. This

means that D is an order 2 differential operator whose symbol σ(D), whichis an endomorphism of the pullback bundle p∗E on the cotangent bundle p :T∗M→ M, is

σ(D) = g Idp∗E

where g is some Riemannian metric on M, viewed as a function on T∗M.

All our constructions vary homotopically with the choice of gauge fixing operator.In practice, there is a natural contractible space of gauge fixing operators, so that ourconstructions are independent (up to contractible choice) of the choice of gauge fixingoperator.

5.4.2. Operators and kernels. Let us recall the relationship between kernels andoperators on E . Any continuous linear map F : Ec → E can be represented by a kernel

KF ∈ E ⊗ E!= HomDVS(Ec, E ).

The symplectic pairing on E gives an isomorphism between E and E![−1]. This allows

us to view the kernel for any continuous linear map F as an element KF ∈ E ⊗ E . If Fis of cohomological degree k, then the kernel KF is of cohomological degree k + 1.

If the map F : Ec → E has image in E c and extends to a continuous linear mapE → E c, then the kernel KF has compact support. If F has image in E and extends toa continuous linear map E c → E , then the kernel KF is smooth.


Our conventions are such that the following hold.

(1) K[Q,F] = QKF, where Q is the total differential on E ⊗ E .(2) Suppose that F : Ec → Ec is skew-symmetric with respect to the degree −1

pairing on Ec. Then KF is symmetric. Similarly, if F is symmetric, then KF isanti-symmetric.

5.4.3. The heat kernel. In this section we will discuss heat kernels associated tothe generalized Laplacian D = [Q, QGF]. These generalized heat kernels will not beessential to our story; most of our constructions will work with a general parametrixfor the operator D, and the heat kernel simply provides a convenient example.

Suppose that we have a free BV theory with a gauge fixing operator QGF. Asabove, let D = [Q, QGF]. If our manifold M is compact, then this leads to a heatoperator e−tD acting on sections E . The heat kernel Kt is the corresponding kernel,which is an element of E ⊗ E ⊗ C∞(R≥0). Further, if t > 0, the operator e−tD is asmoothing operator, so that the kernel Kt is in E ⊗ E . Since the operator e−tD is skewsymmetric for the symplectic pairing on E , the kernel Kt is symmetric.

The kernel Kt is uniquely characterized by the following properties:

(1) The heat equation:

ddt

Kt + (D⊗ 1)Kt = 0.

(2) The initial condition that K0 ∈ E ⊗ E is the kernel for the identity operator.

On a non-compact manifold M, there is more than one heat kernel satisfying theseproperties. We will only consider heat kernels on a compact manifold.

5.4.4. Parametrices. In [Cos11c], two equivalent definitions of a field theory aregiven: one based on the heat kernel, and one based on a general parametrix. We willuse exclusively the parametrix version in this book.

Before we define the notion of parametrix, we need a technical definition.

5.4.4.1 Definition. If M is a manifold, a subset V ⊂ Mn is proper if all of the projectionmaps π1, . . . , πn : V → M are proper. We say that a function, distribution, etc. on Mn hasproper support if its support is a proper subset of Mn.


5.4.4.2 Definition. A parametrix Φ is a distributional section

Φ ∈ E (M)⊗ E (M)

of the bundle E E on M×M with the following properties.

(1) Φ is symmetric under the natural Z/2 action on E (M)⊗ E (M).(2) Φ is of cohomological degree 1.(3) Φ is closed under the natural differential Q⊗ 1 + 1⊗Q on E (M)⊗ E (M).(4) Φ has proper support.(5) Let QGF : E → E be the gauge fixing operator. We require that

([Q, QGF]⊗ 1)Φ− KId

is a smooth section of E E on M×M. Thus,

([Q, QGF]⊗ 1)Φ− KId ∈ E (M)⊗ E (M).

(Here KId is the kernel corresponding to the identity operator).

Remark: For clarity’s sake, note that our definition depends on a choice of QGF. Thus,we are defining here parametrices for the generalized Laplacian [Q, QGF], not generalparametrices for the elliptic complex E .

Note that the parametrix Φ can be viewed (using the correspondence betweenkernels and operators described above) as a linear map AΦ : E → E . This operator isof cohomological degree 0, and has the property that

AΦ[Q, QGF] = Id+ a smoothing operator

[Q, QGF]AΦ = Id+ a smoothing operator.

This property – being both a left and right inverse to the operator [Q, QGF], up to asmoothing operator – is the standard definition of a parametrix.

An example of a parametrix is the following. For M compact, let Kt ∈ E ⊗ E bethe heat kernel. Then, the kernel

∫ L0 Ktdt is a parametrix, for any L > 0.

It is a standard result in the theory of pseudo-differential operators (see e.g. [Tar87])that every elliptic operator admits a parametrix. Normally a parametrix is not as-sumed to have proper support; however, if Φ is a parametrix satisfying all conditionsexcept that of proper support, and if f ∈ C∞(M×M) is a smooth function with proper


support which is 1 in a neighborhood of the diagonal, then f Φ is a parametrix withproper support. This shows that parametrices with proper support always exist.

Here is a key property of parametrices that we will use repeatedly.

5.4.4.3 Lemma. If Φ, Ψ are parametrices, then Φ−Ψ ∈ E ⊗ E is smooth.

PROOF. Indeed, Φ−Ψ is annihilated by the elliptic operator ([Q, QGF])⊗ 1 + 1⊗([Q, QGF]) on E (M)⊗ E (M). Hence it is a smooth section.

If Φ, Ψ are parametrices, we say that Φ < Ψ if the support of Φ is contained in thesupport of Ψ. In this way, parametrices acquire a partial order.

5.4.5. The propagator for a parametrix. If Φ is a parametrix, we let

P(Φ) = (QGF ⊗ 1)Φ ∈ E ⊗ E .

This is the propagator associated to Φ. We let

KΦ = KId − ([Q, QGF]⊗ 1)Φ.

To relate to section 5.4.3 and [Cos11c], we note that if M is a compact manifold and if

Φ =∫ L

0Ktdt

is the parametrix associated to the heat kernel, then

P(Φ) = P(0, L) =∫ L

0(QGF ⊗ 1)Ktdt

and

KΦ = KL.

5.4.6. Classes of functionals. In the appendix A.7 we define various classes offunctions on the space Ec of compactly-supported fields. Here we give an overview ofthose classes. Many of the conditions seem somewhat technical at first, but they arisenaturally as one attempts both to discuss the support of an observable and to extendthe algebraic ideas of the BV formalism in this infinite-dimensional setting.


We are interested, firstly, in functions modulo constants, which we call Ored(Ec).Every functional F ∈ Ored(Ec) has a Taylor expansion in terms of symmetric continu-ous linear maps

Fk : E ⊗kc → C

(for k > 0). We say that F has proper support if the support of each Fk (as defined above)is a proper subset of Mk. The space of functionals with proper support is denotedOP(Ec) (as always in this section, we work with functionals modulo constants). Thiscondition equivalently means that, when we think of Fk as an operator

E ⊗k−1c → E

!,

it extends to a continuous linear map

Fk : E ⊗k−1 → E!.

At various points in this book, we will need to consider functionals with smoothfirst derivative, which are functionals satisfying a certain technical regularity constraint.Functionals with smooth first derivative are needed in two places in the text: when wedefine the Poisson bracket on classical observables, and when we give the definitionof a quantum field theory. In terms of the Taylor components Fk, viewed as multilinearoperators E ⊗k−1

c → E!, this condition means that the Fk has image in E !. (For more

detail, see Appendix A, section A.7.)

We are interested in the functionals with smooth first derivative and with propersupport. We denote this space by OP,sm(E ). These are the functionals with the prop-erty that the Taylor components Fk, when viewed as operators, give continuous linearmaps

E ⊗k−1 → E !.

5.4.7. The renormalization group flow. Suppose that Φ, Ψ are parametrices. ThenP(Φ)− P(Ψ) is a smooth kernel with proper support.

Given any element α ∈ E ⊗ E of cohomological degree 0, we define an operator

∂α : O(E )→ O(E ).

This map is an order 2 differential operator, which, on components, is the map givenby contraction with α:

α ∨− : Symn E ∨ → Symn−2 E ∨.


The operator ∂α is the unique order 2 differential operator which is given by pairingwith α on Sym2 E ∨ and which is zero on Sym≤1 E ∨.

We define a map

W (α,−) : O+(E )[[h]]→ O+(E )[[h]]

F 7→ h log(

eh∂α eF/h)

,

known as the RG flow with respect to α. (When α = P(Φ) − P(Ψ), we call it theRG flow from Ψ to Φ.) This formula is a succinct way of summarizing a Feynmandiagram expansion. In particular, W (α, F) can be written as a sum over Feynman dia-grams with the Taylor components Fk of F labelling vertices of valence k, and with α aspropagator. (All of this, and indeed everything else in this section, is explained in fargreater detail in chapter 2 of [Cos11c].) For this map to be well-defined, the functionalF must have only cubic and higher terms modulo h. The notation O+(E )[[h]] denotesthis restricted class of functionals.

If α ∈ E ⊗ E has proper support, then the operator W (α,−) extends (uniquely, ofcourse) to a continuous operator

W (α,−) : O+P,sm(Ec)[[h]]→ O+

P,sm(Ec)[[h]].

Our philosophy is that a parametrix Φ is like a choice of “scale” for our field theory.The renormalization group flow relating the scale given by Φ and that given by Ψ isW (P(Φ)− P(Ψ),−).

Because P(Φ) is not a smooth kernel, the operator W (P(Φ),−) is not well-defined.This is just because the definition of W (P(Φ),−) involves multiplying distributions.In physics terms, the singularities that appear when one tries to define W (P(Φ),−)are called ultraviolet divergences.

However, if I ∈ O+P,sm(E ), the tree level part

W0 (P(Φ), I) = W ((P(Φ), I) mod h

is a well-defined element of O+P,sm(E ). The h → 0 limit of W (P(Φ), I) is called the

tree-level part because, whereas the whole object W (P(Φ), I) is defined as a sum overgraphs, the h → 0 limit W0 (P(Φ), I) is defined as a sum over trees. It is straightfor-ward to see that W0 (P(Φ), I) only involves multiplication of distributions with trans-verse singular support, and so is well defined.


5.4.8. The BD algebra structure associated to a parametrix. A parametrix alsoleads to a BV operator

4Φ = ∂KΦ : O(E )→ O(E ).

Again, this operator preserves the subspace OP,sm(E ) of functions with proper sup-port and smooth first derivative. The operator 4Φ commutes with Q, and it satisfies(4Φ)

2 = 0. In a standard way, we can use the BV operator4Φ to define a bracket onthe space O(E ), by

I, JΦ = 4Φ(I J)− (4Φ I)J − (−1)|I| I4Φ J.

This bracket is a Poisson bracket of cohomological degree 1. If we give the graded-commutative algebra O(E )[[h]] the standard product, the Poisson bracket −,−Φ,and the differential Q + h4Φ, then it becomes a BD algebra.

The bracket −,−Φ extends uniquely to a continuous linear map

OP(E )×O(E )→ O(E ).

Further, the space OP,sm(E ) is closed under this bracket. (Note, however, that OP,sm(E )

is not a commutative algebra if M is not compact: the product of two functionals withproper support no longer has proper support.)

A functional F ∈ O(E )[[h]] is said to satisfy the Φ-quantum master equation if

QF + h4ΦF + 12F, FΦ = 0.

It is shown in [Cos11c] that if F satisfies the Φ-QME, and if Ψ is another parametrix,then W (P(Ψ)− P(Φ), F) satisfies the Ψ-QME. This follows from the identity

[Q, ∂P(Φ) − ∂P(Ψ)] = 4Ψ −4Φ

of order 2 differential operators on O(E ). This relationship between the renormaliza-tion group flow and the quantum master equation is a key part of the approach to QFTof [Cos11c].

5.4.9. The definition of a field theory. Our definition of a field theory is as fol-lows.

5.4.9.1 Definition. Let (E , Q, 〈−,−〉) be a free BV theory. Fix a gauge fixing condition QGF.Then a quantum field theory (with this space of fields) consists of the following data.


(1) For all parametrices Φ, a functional

I[Φ] ∈ O+P,sm(Ec)[[h]]

which we call the scale Φ effective interaction. As we explained above, the subscriptsindicate that I[Φ] must have smooth first derivative and proper support. The super-script + indicates that, modulo h, I[Φ] must be at least cubic. Note that we workwith functions modulo constants.

(2) For two parametrices Φ, Ψ, I[Φ] must be related by the renormalization group flow:

I[Φ] = W (P(Φ)− P(Ψ), I[Ψ]) .

(3) Each I[Φ] must satisfy the Φ-quantum master equation

(Q + h4Φ)eI[Φ]/h = 0.

Equivalently,

QI[Φ] + h4Φ I[Φ] + 12I[Φ], I[Φ]Φ.

(4) Finally, we require that I[Φ] satisfies a locality axiom. Let

Ii,k[Φ] : E ⊗kc → C

be the kth Taylor component of the coefficient of hi in I[Φ].Our locality axiom says that, as Φ tends to zero, the support of

Ii,k[Φ]

becomes closer and closer to the small diagonal in Mk.For the constructions in this book, it turns out to be useful to have precise bounds

on the support of Ii,k[Φ]. To give these bounds, we need some notation. Let Supp(Φ) ⊂M2 be the support of the parametrix Φ, and let Supp(Φ)n ⊂ M2 be the subset ob-tained by convolving Supp(Φ) with itself n times. (Thus, (x, y) ∈ Supp(Φ)n ifthere exists a sequence x = x0, x1, . . . , xn = y such that (xi, xi+1) ∈ Supp(Φ).)

Our support condition is that, if ej ∈ Ec, then

Ii,k(e1, . . . , ek) = 0

unless, for all 1 ≤ r < s ≤ k,

Supp(er)× Supp(es) ⊂ Supp(Φ)3i+k.


Remark: (1) The locality axiom condition as presented here is a little unappealing.An equivalent axiom is that for all open subsets U ⊂ Mk containing the smalldiagonal M ⊂ Mk, there exists a parametrix ΦU such that

Supp Ii,k[Φ] ⊂ U for all Φ < ΦU .

In other words, by choosing a small parametrix Φ, we can make the supportof Ii,k[Φ] as close as we like to the small diagonal on Mk.

We present the definition with a precise bound on the size of the supportof Ii,k[Φ] because this bound will be important later in the construction of thefactorization algebra. Note, however, that the precise exponent 3i + k whichappears in the definition (in Supp(Φ)3i+k) is not important. What is importantis that we have some bound of this form.

(2) It is important to emphasize that the notion of quantum field theory is onlydefined once we have chosen a gauge fixing operator. Later, we will explain indetail how to understand the dependence on this choice. More precisely, wewill construct a simplicial set of QFTs and show how this simplicial set onlydepends on the homotopy class of gauge fixing operator (in most examples,the space of natural gauge fixing operators is contractible).

Suppose that I0 ∈ Oloc(E) is a local functional (defined modulo constants) whichsatisfies the classical master equation

QI0 +12I0, I0 = 0.

Let us suppose that I0 is at least cubic.

Then, as we have seen above, we can define a family of functionals

I0[Φ] = W0 (P(Φ), I0) ∈ OP,sm(E )

as the tree-level part of the renormalization group flow operator from scale 0 to thescale given by the parametrix Φ. The compatibility between this classical renormal-ization group flow and the classical master equation tells us that I0[Φ] satisfies theΦ-classical master equation

QI0[Φ] + 12I0[Φ], I0[Φ]Φ = 0.

5.4.9.2 Definition. Let I[Φ] ∈ O+P,sm(E )[[h]] be the collection of effective interactions defin-

ing a quantum field theory. Let I0 ∈ Oloc(E ) be a local functional satisfying the classical


master equation, and so defining a classical field theory. We say that the quantum field theoryI[Φ] is a quantization of the classical field theory defined by I0 if

I[Φ] = I0[Φ] mod h,

or, equivalently, if

limΦ→0

I[Φ]− I0 mod h = 0.

5.4.10. Allowing families. Before discussing the interpretation of these axiomsand also explaining the results of [Cos11c] that allow one to construct such quantumfield theories, we will explain how to define families of quantum field theories oversome base dg algebra. The fact that we can work in families in this way means thatthe moduli space of quantum field theories is something like a derived stack. Forinstance, by considering families over the base dg algebra of forms on the n-simplex,we see that the set of quantizations of a given classical field theory is a simplicial set.

One particularly important use of the families version of the theory is that it allowsus to show that our constructions and results are independent, up to homotopy, ofthe choice of gauge fixing condition (provided one has a contractible — or at leastconnected — space of gauge fixing conditions, which happens in most examples).

In later sections, we will work implicitly over some base dg ring in the sense de-scribed here, although we will normally not mention this base ring explicitly.

5.4.10.1 Definition. A nilpotent dg manifold is a manifold X (possibly with corners),equipped with a sheaf A of commutative differential graded algebras over the sheaf Ω∗X, withthe following properties.

(1) A is concentrated in finitely many degrees.(2) Each A i is a locally free sheaf of Ω0

X-modules of finite rank. This means that A i isthe sheaf of sections of some finite rank vector bundle Ai on X.

(3) We are given a map of dg Ω∗X-algebras A → C∞X .

We will let I ⊂ A be the ideal which is the kernel of the map A → C∞X :

we require that I , its powers I k, and each A /I k are locally free sheaves of C∞X -

modules. Also, we require that I k = 0 for k sufficiently large.

Note that the differential d on A is necessary a differential operator.


We will use the notation A] to refer to the bundle of graded algebras on X whose smoothsections are A ], the graded algebra underlying the dg algebra A .

If (X, A ) and (Y, B) are nilpotent dg manifolds, a map (Y, B) → (X, A ) is a smoothmap f : Y → X together with a map of dg Ω∗(X)-algebras A → B.

Here are some basic examples.

(1) A = C∞(X) and I = 0. This describes the smooth manifold X.(2) A = Ω∗(X) and I = Ω>0(X). This equips X with its de Rham complex as a

structure sheaf. (Informally, we can say that “constant functions are the onlyfunctions on a small open” so that this dg manifold is sensitive to topologicalrather than smooth structure.)

(3) If R is a dg Artinian C-algebra with maximal ideal m, then R can be viewedas giving the structure of nilpotent graded manifold on a point.

(4) If again R is a dg Artinian algebra, then for any manifold (X, R⊗Ω∗(X)) is anilpotent dg manifold.

(5) If X is a complex manifold, then A = (Ω0,∗(X), ∂) is a nilpotent dg manifold.

Remark: We study field theories in families over nilpotent dg manifolds for both prac-tical and structural reasons. First, we certainly wish to discuss familes of field theoriesover smooth manifolds. Nilpotent dg manifolds provide a version of derived geome-try that is a modest enlargement of smooth manifolds. In particular, by only allowing(derived) Artinian directions, we avoid many subtleties of “global” derived geome-try and can rely on the ideas and techniques of deformation theory. Second, from apractical point of view, our arguments are tractable when working over nilpotent dgmanifolds. Families over more sophisticated spaces might very well work, but nilpo-tent dg manifolds are well-suited to our methods. We find it convenient, for example,that the algebra of functions on a nilpotent dg manifold is a nice Frechet space.

We can now give a precise notion of “family of field theories.”

5.4.10.2 Definition. Let M be a manifold. A family of free BV theories on M, parameterizedby a nilpotent dg manifold (X, A ), is the following data.

(1) A graded bundle E on M × X of locally free A]-modules. We will refer to globalsections of E as E . The space of those sections s ∈ Γ(M× X, E) with the property


that the map Supp s → X is proper will be denoted Ec. Similarly, we let E denotethe space of sections which are distributional on M and smooth on X, that is,

E = E ⊗C∞(M×X) (D(M)⊗ C∞(X)) .

(This is just the algebraic tensor product, which is reasonable as E is a finitely gener-ated projective C∞(M× X)-module).

As above, we let

E! = HomA](E, A])⊗DensM

denote the “dual” bundle. There is a natural A ]-valued pairing between E and E !c .

(2) A differential operator Q : E → E , of cohomological degree 1 and square-zero,making E into a dg module over the dg algebra A .

(3) A map

E⊗A] E→ DensM⊗A]

which is of degree −1, anti-symmetric, and leads to an isomorphism

HomA](E, A])⊗DensM → E

of sheaves of A]-modules on M× X.This pairing leads to a degree −1 anti-symmetric A -linear pairing

〈−,−〉 : Ec ⊗ Ec → A .

We require it to be a cochain map. In other words, if e, e′ ∈ Ec,

dA

⟨e, e′⟩=⟨

Qe, e′⟩+ (−1)|e|

⟨e, Qe′

⟩.

5.4.10.3 Definition. Let (E, Q, 〈−,−〉) be a family of free BV theories on M parameterizedby A . A gauge fixing condition on E is an A -linear differential operator

QGF : E → E

such that

D = [Q, QGF] : E → E

is a generalized Laplacian, in the following sense.

Note that D is an A -linear cochain map. Thus, we can form

D0 : E ⊗A C∞(X)→ E ⊗A C∞(X)

by reducing modulo the maximal ideal I of A .


Let E0 = E/I be the bundle on M × X obtained by reducing modulo the ideal I in thebundle of algebras A. Let

σ(D0) : π∗E0 → π∗E0

be the symbol of the C∞(X)-linear operator D0. Thus, σ(D0) is an endomorphism of the bundleof π∗E0 on (T∗M)× X.

We require that σ(D0) is the product of the identity on E0 with a smooth family of metricson M parameterized by X.

Throughout this section, we will fix a family of free theories on M, parameterizedby A . We will take A to be our base ring throughout; thus, tensor products, etc. willbe taken over A . All of our constructions will be functorial with respect to change ofthe base ring A .

5.4.11. Now that we have defined free theories over a base ring A , the definitionof an interacting theory over A is very similar to the definition given when A = C.First, one defines a parametrix to be an element

Φ ∈ E ⊗A E

with the same properties as before, but where now we take all tensor products (andso on) over A . More precisely,

(1) Φ is symmetric under the natural Z/2 action on E ⊗ E .(2) Φ is of cohomological degree 1.(3) Φ is closed under the differential on E ⊗ E .(4) Φ has proper support: this means that the map Supp Φ→ M× X is proper.(5) Let QGF : E → E be the gauge fixing operator. We require that

([Q, QGF]⊗ 1)Φ− KId

is an element of E ⊗ E (where, as before, KId ∈ E ⊗ E is the kernel for theidentity map).

An interacting field theory is then defined to be a family of A -linear functionals

I[Φ] ∈ Ored(E )[[h]] = ∏n≥1

HomA (E ⊗A n, A )Sn [[h]]


satisfying the renormalization group flow equation, quantum master equation, andlocality condition, just as before. In order for the RG flow to make sense, we requirethat each I[Φ] has proper support and smooth first derivative. In this context, thismeans the following. Let Ii,k[Φ] : E ⊗k → A be the kth Taylor component of thecoefficient of hi in Ii,k[Φ]. Proper support means that any projection map

Supp Ii,k[Φ] ⊂ Mk × X → M× X

is proper. Smooth first derivative means, as usual, that when we think of Ii,k[Φ] as anoperator E ⊗k−1 → E , the image lies in E .

If we have a family of theories over (X, A ), and a map

f : (Y, B)→ (X, A )

of dg manifolds, then we can base change to get a family over (Y, B). The space offields for this family is

f ∗E = E ⊗A B.

The gauge fixing operator

QGF : E ⊗A B → E ⊗A B

is the B-linear extension of the gauge fixing condition for the family of theories overA .

If

Φ ∈ E ⊗A E ⊂ f ∗E ⊗B f ∗E

is a parametrix for the family of free theories E over A , then it defines a parametrixf ∗Φ for the family of free theories f ∗E over B. For parametrices of this form, theeffective action functionals

f ∗ I[ f ∗Φ] ∈ O+sm,P( f ∗E )[[h]] = O+

sm,P(E )[[h]]⊗A B

is simply the image of the original effective action functional

I[Φ] ∈ O+sm,P(E )[[h]] ⊂ O+

sm,P( f ∗E )[[h]].

For a general parametrix Ψ for f ∗E , the effective action functional is defined by therenormalization group equation

f ∗ I[Ψ] = W (P(Ψ)− P( f ∗Φ), f ∗ I[ f ∗Φ]) .


This is well-defined because

P(Ψ)− P( f ∗Φ) ∈ f ∗E ⊗B f ∗E

has no singularities.

The compatibility between the renormalization group equation and the quantummaster equation guarantees that the effective action functionals f ∗ I[Ψ] satisfy theQME for every parametrix Ψ. The locality axiom for the original family of effectiveaction functionals I[Φ] guarantees that the pulled-back family f ∗ I[Ψ] satisfy the local-ity axiom necessary to define a family of theories over B.

5.5. The simplicial set of theories

One of the main reasons for introducing theories over a nilpotent dg manifold(X, A ) is that this allows us to talk about the simplicial set of theories. This is essential,because the main result we will use from [Cos11c] is homotopical in nature: it relatesthe simplicial set of theories to the simplicial set of local functionals.

Let us introduce some notation. Let us fix a family of classical field theories ona manifold M over a nilpotent dg manifold (X, A ). As above, the fields of such atheory are a dg A -module E equipped with an A -linear local functional I ∈ Oloc(E )

satisfying the classical master equation QI + 12I, I = 0.

By pulling back along the projection map

(X×4n, A ⊗ C∞(4n))→ (X, A ),

we get a new family of classical theories over the dg base ring A ⊗ C∞(4n), whosefields are E ⊗ C∞(4n). We can then ask for a gauge fixing operator

QGF : E ⊗ C∞(4n)→ E ⊗ C∞(4n).

for this family of theories. This is the same thing as a smooth family of gauge fixingoperators for the original theory depending on a point in the n-simplex.

5.5.0.1 Definition. Let (E , I) denote the classical theory we start with over A . Let G F (E , I)denote the simplicial set whose n-simplices are such families of gauge fixing operators overA ⊗ C∞(4n). If there is no ambiguity as to what classical theory we are considering, we willdenote this simplicial set by G F .

5.5. THE SIMPLICIAL SET OF THEORIES 163

Any such gauge fixing operator extends, by Ω∗(4n)-linearity, to a linear map E ⊗Ω∗(4n) → E ⊗Ω∗(4n), which thus defines a gauge fixing operator for the family oftheories over A ⊗Ω∗(4n) pulled back via the projection

(X×4n, A ⊗Ω∗(4n))→ (X, A ).

(Note that Ω∗(4n) is equipped with the de Rham differential.)

Example: Suppose that A = C, and the classical theory we are considering is Chern-Simons theory on a 3-manifold M, where we perturb around the trivial bundle. Then,the space of fields is E = Ω∗(M)⊗ g[1] and Q = ddR. For every Riemannian metricon M, we find a gauge fixing operator QGF = d∗. More generally, if we have a smoothfamily

gσ | σ ∈ 4n

of Riemannian metrics on M, depending on the point σ in the n-simplex, we get ann-simplex of the simplicial set G F of gauge fixing operators.

Thus, if Met(M) denotes the simplicial set whose n-simplices are the set of Rie-mannian metrics on the fibers of the submersion M×4n → 4n, then we have a mapof simplicial sets

Met(M)→ G F .

Note that the simplicial set Met(M) is (weakly) contractible (which follows from thefamiliar fact that, as a topological space, the space of metrics on M is contractible).

A similar remark holds for almost all theories we consider. For example, supposewe have a theory where the space of fields

E = Ω0,∗(M, V)

is the Dolbeault complex on some complex manifold M with coefficients in some holo-morphic vector bundle V. Suppose that the linear operator Q : E → E is the ∂-operator. The natural gauge fixing operators are of the form ∂

∗. Thus, we get a gauge

fixing operator for each choice of Hermitian metric on M together with a Hermitianmetric on the fibers of V. This simplicial set is again contractible.

It is in this sense that we mean that, in most examples, there is a natural con-tractible space of gauge fixing operators.


5.5.1. We will use the shorthand notation (E , I) to denote the classical field theoryover A that we start with; and we will use the notation (E4n , I4n) to refer to thefamily of classical field theories over A ⊗Ω∗(4n) obtained by base-change along theprojection (X×4n, A ⊗Ω∗(4n))→ (X, A ).

5.5.1.1 Definition. We let T (n) denote the simplicial set whose k-simplices consist of thefollowing data.

(1) A k-simplex QGF4k ∈ G F [k], defining a gauge-fixing operator for the family of theo-

ries (E4k , I4k) over A ⊗Ω∗(4k).(2) A quantization of the family of classical theories with gauge fixing operator (E4k , I4k , QGF

4k ),

defined modulo hn+1.

We let T (∞) denote the corresponding simplicial set where the quantizations are defined to allorders in h.

Note that there are natural maps of simplicial sets T (n) → T (m), and that T (∞) =

lim←−T (n). Further, there are natural maps T (n) → G F .

Note further that T (0) = G F .

This definition describes the most sophisticated version of the set of theories wewill consider. Let us briefly explain how to interpret this simplicial set of theories.

Suppose for simplicity that our base ring A is just C. Then, a 0-simplex of T (0)

is simply a gauge-fixing operator for our theory. A 0-simplex of T (n) is a gauge fix-ing operator, together with a quantization (defined with respect to that gauge-fixingoperator) to order n in h.

A 1-simplex of T (0) is a homotopy between two gauge fixing operators. Supposethat we fix a 0-simplex of T (0), and consider a 1-simplex of T (∞) in the fiber over this0-simplex. Such a 1-simplex is given by a collection of effective action functionals

I[Φ] ∈ O+P,sm(E )⊗Ω∗([0, 1])[[h]]

one for each parametrix Φ, which satisfy a version of the QME and the RG flow, asexplained above.

5.5. THE SIMPLICIAL SET OF THEORIES 165

We explain in some more detail how one should interpret such a 1-simplex in thespace of theories. Let us fix a parametrix Φ on E and extend it to a parametrix for thefamily of theories over Ω∗([0, 1]). We can then expand our effective interaction I[Φ] as

I[Φ] = J[Φ](t) + J′[Φ](t)dt

where J[Φ](t), J′[Φ](t) are elements

J[Φ](t), J′[Φ](t) ∈ O+P,sm(E )⊗ C∞([0, 1])[[h]].

Here t is the coordinate on the interval [0, 1].

The quantum master equation implies that the following two equations hold, foreach value of t ∈ [0, 1],

QJ[Φ](t) + 12J[Φ](t), J[Φ](t)Φ + h4Φ J[Φ](t) = 0,

∂

∂tJ[Φ](t) + QJ′[Φ](t) + J[Φ](t), J′[Φ](t)Φ + h4Φ J′[Φ](t) = 0.

The first equation tells us that for each value of t, J[Φ](t) is a solution of the quan-tum master equation. The second equation tells us that the t-derivative of J[Φ](t) ishomotopically trivial as a deformation of the solution to the QME J[Φ](t).

In general, if I is a solution to some quantum master equation, a transformation ofthe form

I 7→ I + εJ = I + εQI′ + I, I′+ h4I′

is often called a “BV canonical transformation” in the physics literature. In the physicsliterature, solutions of the QME related by a canonical transformation are regarded asequivalent: the canonical transformation can be viewed as a change of coordinates onthe space of fields.

For us, this interpretation is not so important. If we have a family of theoriesover Ω∗([0, 1]), given by a 1-simplex in T (∞), then the factorization algebra we willconstruct from this family of theories will be defined over the dg base ring Ω∗([0, 1]).This implies that the factorization algebras obtained by restricting to 0 and 1 are quasi-isomorphic.

5.5.2. Generalizations. We will shortly state the theorem which allows us to con-struct such quantum field theories. Let us first, however, briefly introduce a slightlymore general notion of “theory.”


We work over a nilpotent dg manifold (X, A ). Recall that part of the data ofsuch a manifold is a differential ideal I ⊂ A whose quotient is C∞(X). In the abovediscussion, we assumed that our classical action functional S was at least quadratic;we then split S as

S = 〈e, Qe〉+ I(e)

into kinetic and interacting terms.

We can generalize this to the situation where S contains linear terms, as long asthey are accompanied by elements of the ideal I ⊂ A . In this situation, we also havesome freedom in the splitting of S into kinetic and interacting terms; we require onlythat linear and quadratic terms in the interaction I are weighted by elements of thenilpotent ideal I .

In this more general situation, the classical master equation S, S = 0 does notimply that Q2 = 0, only that Q2 = 0 modulo the ideal I . However, this does not leadto any problems; the definition of quantum theory given above can be easily modifiedto deal with this more general situation.

In the L∞-language used in Chapter 4, this more general situation describes a fam-ily of curved L∞ algebras over the base dg ring A with the property that the curvingvanishes modulo the nilpotent ideal I .

Recall that ordinary (not curved) L∞ algebras correspond to formal pointed mod-uli problems. These curved L∞ algebras correspond to families of formal moduli prob-lems over A which are pointed modulo I .

5.6. The theorem on quantization

Let M be a manifold, and suppose we have a family of classical BV theories onM over a nilpotent dg manifold (X, A ). Suppose that the space of fields on M is theA -module E . Let Oloc(E ) be the dg A -module of local functionals with differentialQ + I,−.

Given a cochain complex C, we denote the Dold-Kan simplicial set associated to Cby DK(C). Its n-simplices are the closed, degree 0 elements of C⊗Ω∗(4n).

5.6. THE THEOREM ON QUANTIZATION 167

5.6.0.1 Theorem. All of the simplicial sets T (n)(E , I) are Kan complexes and T (∞)(E , I).The maps p : T (n+1)(E , I)→ T (n)(E , I) are Kan fibrations.

Further, there is a homotopy fiber diagram of simplicial sets

T (n+1)(E , I)

p

// 0

T (n)(E , I) O // DK(Oloc(E )[1], Q + I,−)

where O is the “obstruction map.”

In more prosaic terms, the second part of the theorem says the following. Ifα ∈ T (n)(E , I)[0] is a zero-simplex of T (n)(E , I), then there is an obstruction O(α) ∈Oloc(E ). This obstruction is a closed degree 1 element. The simplicial set p−1(α) ∈T (n+1)(E , I) of extensions of α to the next order in h is homotopy equivalent to thesimplicial set of ways of making O(α) exact. In particular, if the cohomology class[O(α)] ∈ H1(Oloc(E), Q + I,−) is non-zero, then α does not admit a lift to the nextorder in h. If this cohomology class is zero, then the simplicial set of possible lifts is atorsor for the simplicial Abelian group DK(Oloc(E ))[1].

Note also that a first order deformation of the classical field theory (E , Q, I) isgiven by a closed degree 0 element of Oloc(E ). Further, two such first order defor-mations are equivalent if they are cohomologous. Thus, this theorem tells us that themoduli space of QFTs is “the same size” as the moduli space of classical field theories:at each order in h, the data needed to describe a QFT is a local action functional.

The first part of the theorem says can be interpreted as follows. A Kan simplicialset can be thought of as an “infinity-groupoid.” Since we can consider families oftheories over arbitrary nilpotent dg manifolds, we can consider T ∞(E , I) as a functorfrom the category of nilpotent dg manifolds to that of Kan complexes, or infinity-groupoids. Thus, the space of theories forms something like a “derived stack” [Toe06,Lur11].

This theorem also tells us in what sense the notion of “theory” is independent ofthe choice of gauge fixing operator. The simplicial set T (0)(E , I) is the simplicial setG F of gauge fixing operators. Since the map

T (∞)(E , I)→ T (0)(E , I) = G F


is a fibration, a path between two gauge fixing conditions QGF0 and QGF

1 leads to ahomotopy between the corresponding fibers, and thus to an equivalence between the∞-groupoids of theories defined using QGF

0 and QGF1 .

As we mentioned several times, there is often a natural contractible simplicial setmapping to the simplicial set G F of gauge fixing operators. Thus, G F often has acanonical “homotopy point”. From the homotopical point of view, having a homotopypoint is just as good as having an actual point: if S→ G F is a map out of a contractiblesimplicial set, then the fibers in T (∞) above any point in S are canonically homotopyequivalent.

5.7. The BD algebra of global observables

In this section, we will try to motivate our definition of a quantum field theoryfrom the point of view of homological algebra. All of the constructions we will explainwill work over an arbitrary nilpotent dg manifold (X, A ), but to keep the notationsimple we will not normally mention the base ring A .

Thus, suppose that (E , I, Q, 〈−,−〉) is a classical field theory on a manifold M. Wehave seen (Chapter 4, section 4.13) how such a classical field theory gives immediatelya commutative factorization algebra whose value on an open subset is

Obscl(U) = (O(E (U)), Q + I,−) .

Further, we saw that there is a P0 sub-factorization algebra

Obscl(U) = (Osm(E (U)), Q + I,−) .

In particular, we have a P0 algebra Obscl(M) of global sections of this P0 algebra. We

can think of Obscl(M) as the algebra of functions on the derived space of solutions to

the Euler-Lagrange equations.

In this section we will explain how a quantization of this classical field theory will

give a quantization (in a homotopical sense) of the P0 algebra Obscl(M) into a BD

algebra Obsq(M) of global observables. This BD algebra has some locality properties,which we will exploit later to show that Obsq(M) is indeed the global sections of afactorization algebra of quantum observables.

5.7. THE BD ALGEBRA OF GLOBAL OBSERVABLES 169

In the case when the classical theory is the cotangent theory to some formal el-liptic moduli problem BL on M (encoded in an elliptic L∞ algebra L on M), there is aparticularly nice class of quantizations, which we call cotangent quantizations. Cotan-gent quantizations have a very clear geometric interpretation: they are locally-definedvolume forms on the sheaf of formal moduli problems defined by L.

5.7.1. The BD algebra associated to a parametrix. Suppose we have a quantiza-tion of our classical field theory (defined with respect to some gauge fixing condition,or family of gauge fixing conditions). Then, for every parametrix Φ, we have seenhow to construct a cohomological degree 1 operator

4Φ : O(E )→ O(E )

and a Poisson bracket

−,−Φ : O(E )×O(E )→ O(E )

such that O(E )[[h]], with the usual product, with bracket −,−Φ and with differen-tial Q + h4Φ, forms a BD algebra.

Further, since the effective interaction I[Φ] satisfies the quantum master equation,we can form a new BD algebra by adding I[Φ],−Φ to the differential of O(E )[[h]].

5.7.1.1 Definition. Let ObsqΦ(M) denote the BD algebra

ObsqΦ(M) = (O(E )[[h]], Q + h4Φ + I[Φ],−Φ) ,

with bracket −,−Φ and the usual product.

Remark: Note that I[Φ] is not in O(E )[[h]], but rather in O+P,sm(E )[[h]]. However, as we

remarked earlier in 5.4.8, the bracket

I[Φ],−Φ : O(E )[[h]]→ O(E )[[h]]

is well-defined.

Remark: Note that we consider ObsqΦ(M) as a BD algebra valued in the multicategory

of differentiable pro-cochain complexes (see Appendix A). This structure includes afiltration on Obsq

Φ(M) = O(E )[[h]]. The filtration is defined by saying that

FnO(E )[[h]] = ∏i

hi Sym≥(n−2i)(E ∨);

it is easily seen that the differential Q + h4Φ + I[Φ],−Φ preserves this filtration.


We will show that for varying Φ, the BD algebras ObsqΦ(M) are canonically weakly

equivalent. Moreover, we will show that there is a canonical weak equivalence of P0

algebras

ObsqΦ(M)⊗C[[h]] C ' Obs

cl(M).

To show this, we will construct a family of BD algebras over the dg base ring of formson a certain contractible simplicial set of parametrices that restricts to Obsq

Φ(M) ateach vertex.

Before we get into the details of the construction, however, let us say somethingabout how this result allows us to interpret the definition of a quantum field theory.

A quantum field theory gives a BD algebra for each parametrix. These BD algebrasare all canonically equivalent. Thus, at first glance, one might think that the data of aQFT is entirely encoded in the BD algebra for a single parametrix. However, this doesnot take account of a key part of our definition of a field theory, that of locality.

The BD algebra associated to a parametrix Φ has underlying commutative algebraO(E )[[h]], equipped with a differential which we temporarily denote

dΦ = Q + h4Φ + I[Φ],−Φ.

If K ⊂ M is a closed subset, we have a restriction map

E = E (M)→ E (K),

where E (K) denotes germs of smooth sections of the bundle E on K. There is a dualmap on functionals O(E (K)) → O(E ). We say a functional f ∈ O(E )[[h]] is supportedon K if it is in the image of this map.

As Φ → 0, the effective interaction I[Φ] and the BV Laplacian 4Φ become moreand more local (i.e., their support gets closer to the small diagonal). This tells usthat, for very small Φ, the operator dΦ only increases the support of a functional inO(E )[[h]] by a small amount. Further, by choosing Φ to be small enough, we canincrease the support by an arbitrarily small amount.

Thus, a quantum field theory is

(1) A family of BD algebra structures on O(E )[[h]], one for each parametrix,which are all homotopic (and which all have the same underlying gradedcommutative algebra).


(2) The differential dΦ defining the BD structure for a parametrix Φ increasessupport by a small amount if Φ is small.

This property of dΦ for small Φ is what will allow us to construct a factorizationalgebra of quantum observables. If dΦ did not increase the support of a functionalf ∈ O(E )[[h]] at all, the factorization algebra would be easy to define: we would justset Obsq(U) = O(E (U))[[h]], with differential dΦ. However, because dΦ does increasesupport by some amount (which we can take to be arbitrarily small), it takes a littlework to push this idea through.

Remark: The precise meaning of the statement that dΦ increases support by an ar-bitrarily small amount is a little delicate. Let us explain what we mean. A func-tional f ∈ O(E )[[h]] has an infinite Taylor expansion of the form f = ∑ hi fi,k, wherefi,k : E ⊗k → C is a symmetric linear map. We let Supp≤(i,k) f be the unions of the sup-ports of fr,s where (r, s) ≤ (i, k) in the lexicographical ordering. If K ⊂ M is a subset,let Φn(K) denote the subset obtained by convolving n times with Supp Φ ⊂ M2. Thedifferential dΦ has the following property: there are constants ci,k ∈ Z>0 of a purelycombinatorial nature (independent of the theory we are considering) such that, for allf ∈ O(E )[[h]],

Supp≤(i,k) dΦ f ⊂ Φci,k(Supp≤(i,k) f ).

Thus, we could say that dΦ increase support by an amount linear in Supp Φ. We willuse this concept in the main theorem of this chapter.

5.7.2. Let us now turn to the construction of the equivalences between ObsqΦ(M)

for varying parametrices Φ. The first step is to construct the simplicial set P of para-metrices; we will then construct a BD algebra Obsq

P(M) over the base dg ring Ω∗(P),which we define below.

LetV ⊂ E ⊗ E

denote the subspace of those elements which are cohomologically closed and of degree1, symmetric, and have proper support.

Note that the set of parametrices has the structure of an affine space for V: if Φ, Ψare parametrices, then

Φ−Ψ ∈ V ⊂ E ⊗ E ,

and, conversely, if Φ is a parametrix and A ∈ V, then Φ + A is a new parametrix.


Let P denote the simplicial set whose n-simplices are affine-linear maps from4n

to the affine space of parametrices. It is clear that P is contractible.

For any vector space V, let V4 denote the simplicial set whose k-simplices areaffine linear maps 4k → V. For any convex subset U ⊂ V, there is a sub-simplicialset U4 ⊂ V4 whose k-simplices are affine linear maps 4k → U. Note that P is a

sub-simplicial set of E⊗24 , corresponding to the convex subset of parametrices inside

E⊗2

.

Let C P [0] ⊂ E⊗2

denote the cone on the affine subspace of parametrices, withvertex the origin 0. An element of C P [0] is an element of E

⊗2of the form tΦ, where

Φ is a parametrix and t ∈ [0, 1]. Let C P denote the simplicial set whose k-simplicesare affine linear maps to C P [0].

Recall that the simplicial de Rham algebra Ω∗4(S) of a simplicial set S is defined asfollows. Any element ω ∈ Ωi

4(S) consists of an i-form

ω(φ) ∈ Ωi(4k)

for each k-simplex φ : 4k → S. If f : 4k → 4l is a face or degeneracy map, then werequire that

f ∗ω(φ) = ω(φ f ).

The main results of this section are as follows.

5.7.2.1 Theorem. There is a BD algebra ObsqP(M) over Ω∗(P) which, at each 0-simplex

Φ, is the BD algebra ObsqΦ(M) discussed above.

The underlying graded commutative algebra of ObsqP(M) is O(E )⊗Ω∗(P)[[h]].

For every open subset U ⊂ M×M, let PU denote the parametrices whose support is inU. Let Obsq

PU(M) denote the restriction of Obsq

P(M) to U. The differential on ObsqPU

(M)

increases support by an amount linear in U (in the sense explained precisely in the remarkabove).

The bracket −,−PU on ObsqPU

(M) is also approximately local, in the following sense.If O1, O2 ∈ Obsq

PU(M) have the property that

Supp O1 × Supp O2 ∩U = ∅ ∈ M×M,

then O1, O2PU = 0.


Further, there is a P0 algebra ObsclC P(M) over Ω∗(C P) equipped with a quasi-isomorphism

of P0 algebras over Ω∗(P),

ObsclC P(M)

∣∣∣P' Obsq

P(M) modulo h,

and with an isomorphism of P0 algebras,

ObsclC P(M)

∣∣∣0∼= Obs

cl(M),

where Obscl(M) is the P0 algebra constructed in Chapter 4.

The underlying commutative algebra of ObsclC P(M) is Obs

cl(M) ⊗Ω∗(C P), the dif-

ferential on ObsclC P(M) increases support by an arbitrarily small amount, and the Poisson

bracket on ObsclC P(M) is approximately local in the same sense as above.

PROOF. We need to construct, for each k-simplex φ : 4k → P , a BD algebraObsq

φ(M) over Ω∗(4k). We view the k-simplex as a subset of Rk+1 by

4k :=

(λ0, . . . , λk) ⊂ [0, 1]k+1 : ∑

iλi = 1

.

Since simplices in P are affine linear maps to the space of parametrices, the simplexφ is determined by k + 1 parametrices Φ0, . . . , Φk, with

φ(λ0, . . . , λk) = ∑i

λiΦi

for λi ∈ [0, 1] and ∑ λi = 1.

The graded vector space underlying our BD algebra is

Obsqφ(M) = O(E )[[h]]⊗Ω∗(4k).

The structure as a BD algebra will be encoded by an order two, Ω∗(4k)-linear differ-ential operator

4φ : Obsqφ(M)→ Obsq

φ(M).

We need to recall some notation in order to define this operator. Each parametrixΦ provides an order two differential operator 4Φ on O(E ), the BV Laplacian corre-sponding to Φ. Further, if Φ, Ψ are two parametrices, then the difference between thepropagators P(Φ) − P(Ψ) is an element of E ⊗ E , so that contracting with P(Φ) −P(Ψ) defines an order two differential operator ∂P(Φ) − ∂P(Ψ) on O(E ). (This operator


defines the infinitesimal version of the renormalization group flow from Ψ to Φ.) Wehave the equation

[Q, ∂P(Φ) − ∂P(Ψ)] = −4Φ +4Ψ.

Note that although the operator ∂P(Φ) is only defined on the smaller subspace O(E ),because P(Φ) ∈ E ⊗ E , the difference ∂P(Φ) and ∂P(Ψ) is nonetheless well-defined onO(E ) because P(Φ)− P(Ψ) ∈ E ⊗ E .

The BV Laplacian 4φ associated to the k-simplex φ : 4k → P is defined by theformula

4φ =k

∑i=0

λi4Φi −k

∑i=0

dλi∂P(Φi),

where the λi ∈ [0, 1] are the coordinates on the simplex 4k and, as above, the Φi arethe parametrices associated to the vertices of the simplex φ.

It is not entirely obvious that this operator makes sense as a linear map O(E ) →O(E )⊗Ω∗(4k), because the operators ∂P(Φ) are only defined on the smaller subspaceO(E ). However, since ∑ dλi = 0, we have

∑ dλi∂P(Φi) = ∑ dλi(∂P(Φi) − ∂P(Φ0)),

and the right hand side is well defined.

It is immediate that 42φ = 0. If we denote the differential on the classical observ-

ables O(E )⊗Ω∗(4n) by Q + ddR, we have

[Q + ddR,4φ] = 0.

To see this, note that

[Q + ddR,4φ] = ∑ dλi4Φi + ∑ dλi[Q, ∂Φi − ∂Φ0 ]

= ∑ dλi4Φi −∑ dλi(4Φi −4Φ0)

= ∑ dλi4Φ0

= 0,

where we use various identities from earlier.

The operator 4φ defines, in the usual way, an Ω∗(4k)-linear Poisson bracket−,−φ on O(E )⊗Ω∗(4k).


We have effective action functionals I[Ψ] ∈ O+sm,P(E )[[h]] for each parametrix Ψ.

Let

I[φ] = I[∑ λiΦi] ∈ O+sm,P(E )[[h]]⊗ C∞(4k).

The renormalization group equation tells us that I[∑ λiΦi] is smooth (actually polyno-mial) in the λi.

We define the structure of BD algebra on the graded vector space

Obsqφ(M) = O(E )[[h]]⊗Ω∗(4k)

as follows. The product is the usual one; the bracket is −,−φ, as above; and thedifferential is

Q + ddR + h4φ + I[φ],−φ.

We need to check that this differential squares to zero. This is equivalent to the quan-tum master equation

(Q + ddR + h4φ)eI[φ]/h = 0.

This holds as a consequence of the quantum master equation and renormalizationgroup equation satisfied by I[φ]. Indeed, the renormalization group equation tells usthat

eI[φ]/h = exp(

h ∑ λi

(∂PΦi) − ∂P(Φ0)

))eI[Φ0]/h.

Thus,

ddReI[φ]/h = h ∑ dλi∂P(Φi)eI[φ]/h

The QME for each I[∑ λiΦi] tells us that

(Q + h ∑ λi4Φi)eI[φ]/h = 0.

Putting these equations together with the definition of4φ shows that I[φ] satisfies theQME.

Thus, we have constructed a BD algebra Obsqφ(M) over Ω∗(4k) for every simplex

φ : 4k →P . It is evident that these BD algebras are compatible with face and degen-eracy maps, and so glue together to define a BD algebra over the simplicial de Rhamcomplex Ω∗4(P) of P .

Let φ be a k-simplex of P , and let

Supp(φ) = ∪λ∈4k Supp(∑ λiΦi).


We need to check that the bracket O1, O2φ vanishes for observables O1, O2 such that(Supp O1 × Supp)O2 ∩ Supp φ = ∅. This is immediate, because the bracket is definedby contracting with tensors in E ⊗ E whose supports sit inside Supp φ.

Next, we need to verify that, on a k-simplex φ of P , the differential Q + I[φ],−φ

increases support by an amount linear in Supp(φ). This follows from the supportproperties satisfied by I[Φ] (which are detailed in the definition of a quantum fieldtheory, definition 5.4.9.1).

It remains to construct the P0 algebra over Ω∗(C P). The construction is almostidentical, so we will not give all details. A zero-simplex of C P is an element of E ⊗ E

of the form Ψ = tΦ, where Φ is a parametrix. We can use the same formulae we usedfor parametrices to construct a propagator P(Ψ) and Poisson bracket −,−Ψ for eachΨ ∈ C P . The kernel defining the Poisson bracket −,−Ψ need not be smooth. Thismeans that the bracket −,−Ψ is only defined on the subspace Osm(E ) of functionalswith smooth first derivative. In particular, if Ψ = 0 is the vertex of the cone C P , then

−,−0 is the Poisson bracket defined in Chapter 4 on Obscl(M) = Osm(E ).

For each Ψ ∈ C P , we can form a tree-level effective interaction

I0[Ψ] = W0 (P(Ψ), I) ∈ Osm,P(E ),

where I ∈ Oloc(E ) is the classical action functional we start with. There are no diffi-culties defining this expression because we are working at tree-level and using func-tionals with smooth first derivative. If Ψ = 0, then I0[0] = I.

The P0 algebra over Ω∗(C P) is defined in almost exactly the same way as wedefined the BD algebra over Ω∗P . The underlying commutative algebra is Osm(E )⊗Ω∗(C P). On a k-simplex ψ with vertices Ψ0, . . . , Ψk, the Poisson bracket is

−,−ψ = ∑ λi−,−Ψi + ∑ dλi−,−P(Ψi),

where −,−P(Ψi) is the Poisson bracket of cohomological degree 0 defined using thepropagator P(Ψi) ∈ E ⊗ E as a kernel. If we let I0[ψ] = I0[∑ λiΨi], then the differentialis

dψ = Q + I0[ψ],−ψ.

The renormalization group equation and classical master equation satisfied by the

I0[Ψ] imply that d2ψ = 0. If Ψ = 0, this P0 algebra is clearly the P0 algebra Obs

cl(M)

constructed in Chapter 4. When restricted to P ⊂ C P , this P0 algebra is the sub P0

5.8. GLOBAL OBSERVABLES 177

algebra of ObsqP(M)/h obtained by restricting to functionals with smooth first deriv-

ative; the inclusion

ObsclC P(M) |P → Obsq

P(M)/h

is thus a quasi-isomorphism, using proposition 4.15.2.4 of Chapter 4.

5.8. Global observables

In the next few sections, we will prove the first version (section 1.7) of our quanti-zation theorem. Our proof is by construction, associating a factorization algebra on Mto a quantum field theory on M, in the sense of [Cos11c]. This is a quantization (in theweak sense) of the P0 factorization algebra associated to the corresponding classicalfield theory.

More precisely, we will show the following.

5.8.0.2 Theorem. For any quantum field theory on a manifold M over a nilpotent dg manifold(X, A ), there is a factorization algebra Obsq on M, valued in the multicategory of differen-tiable pro-cochain complexes flat over A [[h]].

There is an isomorphism of factorization algebras

Obsq⊗A [[h]]A∼= Obscl

between Obsq modulo h and the commutative factorization algebra Obscl .

Further, Obsq is a weak quantization (in the sense of Chapter 1, section 1.7) of the P0

factorization algebra Obscl of classical observables.

5.8.1. So far we have constructed a BD algebra ObsqΦ(M) for each parametrix

Φ; these BD algebras are all weakly equivalent to each other. In this section we willdefine a cochain complex Obsq(M) of global observables which is independent of thechoice of parametrix. For every open subset U ⊂ M, we will construct a subcomplexObsq(U) ⊂ Obsq(M) of observables supported on U. The complexes Obsq(U) willform our factorization algebra.

Thus, suppose we have a quantum field theory on M, with space of fields E and ef-fective action functionals I[Φ], one for each parametrix (as explained in section 5.2).An observable for a quantum field theory (that is, an element of the cochain complex


Obsq(M)) is simply a first-order deformation I[Φ] + δO[Φ] of the family of effectiveaction functionals I[Φ], which satisfies a renormalization group equation but does notnecessarily satisfy the locality axiom in the definition of a quantum field theory. Defi-nition 5.8.1.3 makes this idea precise.

Remark: This definition is motivated by a formal argument with the path integral. LetS(φ) be the action functional for a field φ, and let O(φ) be another function of the field,describing a measurement that one could make. Heuristically, the expectation valueof the observable is

〈O〉 = 1ZS

∫O(φ)e−S(φ)/h Dφ,

where ZS denotes the partition function, simply the integral without O. A formalmanipulation shows that

〈O〉 = ddδ

1ZS

∫e(−S(φ)+hδO(φ))/h Dφ.

In other words, we can view O as a first-order deformation of the action functional Sand compute the expectation value as the change in the partition function. Becausethe book [Cos11c] gives an approach to the path integral that incorporates the BV for-malism, we can define and compute expectation values of observables by exploitingthe second description of 〈O〉 given above.

Earlier we defined cochain complexes ObsqΦ(M) for each parametrix. The under-

lying graded vector space of ObsqΦ(M) is O(E )[[h]]; the differential on Obsq

Φ(M) is

QΦ = Q + I[Φ],−Φ + h4Φ.

5.8.1.1 Definition. Define a linear map

WΦΨ : O(E )[[h]]→ O(E )[[h]]

by requiring that, for an element f ∈ O(E )[[h]] of cohomological degree i,

I[Φ] + δWΦΨ ( f ) = W (P(Φ)− P(Ψ), I[Ψ] + δ f )

where δ is a square-zero parameter of cohomological degree −i.

5.8.1.2 Lemma. The linear map

WΦΨ : Obsq

Ψ(M)→ ObsqΦ(M)

is an isomorphism of differentiable pro-cochain complexes.

5.9. LOCAL OBSERVABLES 179

PROOF. The fact that WΦΨ intertwines the differentials QΦ and QΨ follows from the

compatibility between the quantum master equation and the renormalization groupequation described in [Cos11c], Chapter 5 and summarized in section 5.2. It is nothard to verify that WΦ

Ψ is a map of differentiable pro-cochain complexes. The inverseto WΦ

Ψ is WΨΦ .

5.8.1.3 Definition. A global observable O of cohomological degree i is an assignment to everyparametrix Φ of an element

O[Φ] ∈ ObsqΦ(M) = O(E)[[h]]

of cohomological degree i such that

WΦΨ O[Ψ] = O[Φ].

If O is an observable of cohomological degree i, we let QO be defined by

Q(O)[Φ] = QΦ(O[Φ]) = QO[Φ] + I[Φ], O[Φ]Φ + h4ΦO[Φ].

This makes the space of observables into a differentiable pro-cochain complex, which we callObsq(M).

Thus, if O ∈ Obsq(M) is an observable of cohomological degree i, and if δ is asquare-zero parameter of cohomological degree −i, then the collection of effectiveinteractions I[Φ] + δO[Φ] satisfy most of the axioms needed to define a family ofquantum field theories over the base ring C[δ]/δ2. The only axiom which is not satis-fied is the locality axiom: we have not imposed any constraints on the behavior of theO[Φ] as Φ→ 0.

5.9. Local observables

So far, we have defined a cochain complex Obsq(M) of global observables on thewhole manifold M. If U ⊂ M is an open subset of M, we would like to isolate thoseobservables which are “supported on U”.

The idea is to say that an observable O ∈ Obsq(M) is supported on U if, for suf-ficiently small parametrices, O[Φ] is supported on U. The precise definition is as fol-lows.


5.9.0.4 Definition. An observable O ∈ Obsq(M) is supported on U if, for each (i, k) ∈Z≥0 ×Z≥0, there exists a compact subset K ⊂ Uk and a parametrix Φ, such that for allparametrices Ψ ≤ Φ

Supp Oi,k[Ψ] ⊂ K.

Remark: Recall that Oi,k[Φ] : E ⊗k → C is the kth term in the Taylor expansion of thecoefficient of hi of the functional O[Φ] ∈ O(E )[[h]].

Remark: As always, the definition works over an arbitrary nilpotent dg manifold (X, A ),even though we suppress this from the notation. In this generality, instead of a com-pact subset K ⊂ Uk we require K ⊂ Uk × X to be a set such that the map K → X isproper.

We let Obsq(U) ⊂ Obsq(M) be the sub-graded vector space of observables sup-ported on U.

5.9.0.5 Lemma. Obsq(U) is a sub-cochain complex of Obsq(M). In other words, if O ∈Obsq(U), then so is QO.

PROOF. The only thing that needs to be checked is the support condition. Weneed to check that, for each (i, k), there exists a compact subset K of Uk such that, forall sufficiently small Φ, QOi,k[Φ] is supported on K.

Note that we can write

QOi,k[Φ] = QOi,k[Φ] + ∑a+b=i

r+s=k+2

Ia,r[Φ], Ob,s[Φ]Φ + ∆ΦOi−1,k+2[Φ].

We now find a compact subset K for QOi,k[Φ]. We know that, for each (i, k) and for allsufficiently small Φ, Oi,k[Φ] is supported on K, where K is some compact subset of Uk.It follows that QOi,k[Φ] is supported on K.

By making K bigger, we can assume that for sufficiently small Φ, Oi−1,k+2[Φ] issupported on L, where L is a compact subset of Uk+2 whose image in Uk, under everyprojection map, is in K. This implies that ∆ΦOi−1,k+2[Φ] is supported on K.

The locality condition for the effective actions I[Φ] implies that, by choosing Φ tobe sufficiently small, we can make Ii,k[Φ] supported as close as we like to the smalldiagonal in Mk. It follows that, by choosing Φ to be sufficiently small, the support of

5.9. LOCAL OBSERVABLES 181

Ia,r[Φ], Ob,s[Φ]Φ can be taken to be a compact subset of Uk. Since there are only afinite number of terms like Ia,r[Φ], Ob,s[Φ]Φ in the expression for (QO)i,k[Φ], we seethat for Φ sufficiently small, (QO)i,k[Φ] is supported on a compact subset K of Uk, asdesired.

5.9.0.6 Lemma. Obsq(U) has a natural structure of differentiable pro-cochain complex space.

PROOF. Our general strategy for showing that something is a differentiable vec-tor space is to ensure that everything works in families over an arbitrary nilpotent dgmanifold (X, A ). Thus, suppose that the theory we are working with is defined over(X, A ). If Y is a smooth manifold, we say a smooth map Y → Obsq(U) is an ob-servable for the family of theories over (X×Y, A ⊗ C∞(Y)) obtained by base-changealong the map X×Y → X (so this family of theories is constant over Y).

The filtration on Obsq(U) (giving it the structure of pro-differentiable vector space)is inherited from that on Obsq(M). Precisely, an observable O ∈ Obsq(U) is in Fk Obsq(U)

if, for all parametrices Φ,

O[Φ] ∈∏ hi Sym≥(2k−i) E ∨.

The renormalization group flow WΨΦ preserves this filtration.

So far we have verified that Obsq(U) is a pro-object in the category of pre-differentiablecochain complexes. The remaining structure we need is a flat connection

∇ : C∞(Y, Obsq(U))→ Ω1(Y, Obsq(U))

for each manifold Y, where C∞(Y, Obsq(U)) is the space of smooth maps Y → Obsq(U).

This flat connection is equivalent to giving a differential on

Ω∗(Y, Obsq(U)) = C∞(Y, Obsq(U))⊗C∞(Y) Ω∗(Y)

making it into a dg module for the dg algebra Ω∗(Y). Such a differential is providedby considering observables for the family of theories over the nilpotent dg manifold(X×Y, A ⊗Ω∗(Y)), pulled back via the projection map X×Y → Y.


5.10. Local observables form a prefactorization algebra

At this point, we have constructed the cochain complex Obsq(M) of global observ-ables of our factorization algebra. We have also constructed, for every open subsetU ⊂ M, a sub-cochain complex Obsq(U) of observables supported on U.

In this section we will see that the local quantum observables Obsq(U) of a quan-tum field on a manifold M form a prefactorization algebra.

The definition of local observables makes it clear that they form a pre-cosheaf:there are natural injective maps of cochain complexes

Obsq(U)→ Obsq(U′)

if U ⊂ U′ is an open subset.

Let U, V be disjoint open subsets of M. The structure of prefactorization algebraon the local observables is specified by the pre-cosheaf structure mentioned above,and a bilinear cochain map

Obsq(U)×Obsq(V)→ Obsq(U qV).

These product maps need to be a map of cochain complexes which is compatible withthe pre-cosheaf structure and with reordering of the disjoint opens.

5.10.1. Defining the product map. Suppose that O ∈ Obsq(U) and O′ ∈ Obsq(V)

are observables on U and V respectively. Note that O[Φ] and O′[Φ] are elements ofthe cochain complex

ObsqΦ(M) =

(O(E )[[h]], QΦ

)which is a BD algebra and so a commutative algebra (ignoring the differential, ofcourse). (The commutative product is simply the usual product of functions on E .)In the definition of the prefactorization product, we will use the product of O[Φ]

and O′[Φ] taken in the commutative algebra O(E). This product will be denotedO[Φ] ∗O′[Φ] ∈ O(E).

Recall (see definition 5.8.1.1) that we defined a linear renormalization group flowoperator WΨ

Φ , which is an isomorphism between the cochain complexes ObsqΦ(M) and

ObsqΨ(M).

The main result of this section is the following.

5.10. LOCAL OBSERVABLES FORM A PREFACTORIZATION ALGEBRA 183

5.10.1.1 Theorem. For all observables O ∈ Obsq(U), O′ ∈ Obsq(V), where U and V aredisjoint, the limit

limΨ→0

WΦΨ(O[Ψ] ∗O′[Ψ]

)∈ O(E )[[h]]

exists. Further, this limit satisfies the renormalization group equation, so that we can definean observable m(O, O′) by

m(O, O′)[Φ] = limΨ→0

WΦΨ(O[Ψ] ∗O′[Ψ]

).

The map

Obsq(U)×Obsq(V) 7→ Obsq(U qV)

O×O′ 7→ m(O, O′)

is a smooth bilinear cochain map, and it makes Obsq into a prefactorization algebra in themulticategory of differentiable pro-cochain complexes.

PROOF. We will show that, for each i, k, the Taylor term

WΨΦ (O[Φ] ∗O′[Φ])i,k : E ⊗k → C

is independent of Ψ for Ψ sufficiently small.

Note thatWΨ

Γ

(WΓ

Φ(O[Φ] ∗O′[Φ]

))= WΨ

Φ(O[Φ] ∗O′[Φ]

).

Thus, to show that the limit limΦ→0 WΨΦ (O[Φ] ∗O′[Φ]) is eventually constant, it suf-

fices to show that, for all sufficiently small Φ, Γ satisfying Φ < Γ,

WΓΦ(O[Φ] ∗O′[Φ])i,k = (O[Γ] ∗O′[Γ])i,k.

This turns out to be an exercise in the manipulation of Feynman diagrams. In or-der to prove this, we need to recall a little about the Feynman diagram expansionof WΓ

Φ(O[Φ]). (Feynman diagram expansions of the renormalization group flow arediscussed extensively in [Cos11c].)

We have a sum of the form

WΓΦ(O[Φ])i,k = ∑

G

1|Aut(G)|wG

(O[Φ] ∗O′[Φ]; I[Φ]; P(Γ)− P(Φ)

).

The sum is over all connected graphs G with the following decorations and properties.

(1) The vertices v of G are labelled by an integer g(v) ∈ Z≥0, which we call thegenus of the vertex.


(2) The first Betti number of G, plus the sum of over all vertices of the genus g(v),must be i (the “total genus”).

(3) G has one special vertex.(4) G has k tails (or external edges).

The weight wG (O[Φ]; I[Φ]; P(Γ)− P(Φ)) is computed by the contraction of a collec-tion of symmetric tensors. One places O[Φ]r,s at the special vertex, when that vertexhas genus r and valency s; places I[Φ]g,v at every other vertex of genus g and valencyv; and puts the propagator P(Γ)− P(Φ) on each edge.

Let us now consider WΓΦ(O[Φ] ∗ O′[Φ]). Here, we a sum over graphs with one

special vertex, labelled by O[Φ] ∗O′[Φ]. This is the same as having two special ver-tices, one of which is labelled by O[Φ] and the other by O′[Φ]. Diagrammatically, itlooks like we have split the special vertex into two pieces. When we make this maneu-ver, we introduce possibly disconnected graphs; however, each connected componentmust contain at least one of the two special vertices.

Let us now compare this to the graphical expansion of

O[Γ] ∗O′[Γ] = WΓΦ(O[Φ]) ∗WΓ

Φ(O′[Φ]).

The Feynman diagram expansion of the right hand side of this expression consistsof graphs with two special vertices, labelled by O[Φ] and O′[Φ] respectively (and, ofcourse, any number of other vertices, labelled by I[Φ], and the propagator P(Γ) −P(Φ) labelling each edge). Further, the relevant graphs have precisely two connectedcomponents, each of which contains one of the special vertices.

Thus, we see that

WΓΦ(O[Φ] ∗O′[Φ])−WΓ

Φ(O[Φ]) ∗WΓΦ(O

′[Φ]).

is a sum over connected graphs, with two special vertices, one labelled by O[Φ] andthe other by O′[Φ]. We need to show that the weight of such graphs vanish for Φ, Γsufficiently small, with Φ < Γ.

Graphs with one connected component must have a chain of edges connecting thetwo special vertices. (A chain is a path in the graph with no repeated vertices or edges.)For a graph G with “total genus” i and k tails, the length of any such chain is boundedby an expression involving only i and k. (It is important to note here that we requirea vertex of genus zero to have valence at least three and a vertex of genus one to have

5.10. LOCAL OBSERVABLES FORM A PREFACTORIZATION ALGEBRA 185

valence at least one. See [Cos11c] for more discussion.) Each step along such a chaininvolves a tensor with some support that depends on the choice of parametrix Φ. Aswe move from the special vertex O toward the other O′, we extend the support, andour aim is to show that we can choose Φ small enough that the support of the chain,excluding O′[Φ], is disjoint from the support of O′[Φ]. The contraction of a distributionand function with disjoint supports is zero, so that the weight will vanish. We nowmake this idea precise.

Let us choose arbitrarily a metric on M. By taking Φ and Γ to be sufficiently small,we can assume that the support of the propagator on each edge is within ε of thediagonal in this metric. By choosing Γ to be sufficiently small, we can take ε as smallas we like. Similarly, the support of the Ir,s[Γ] labelling a vertex of genus r and valencys can be taken to be within cr,sε of the diagonal, where cr,s is a combinatorial constant.In addition, by choosing Φ to be small enough we can ensure that the supports of O[Φ]

and O′[Φ] are disjoint.

Now let G′ denote the graph G with the special vertex for O′ removed. This graphcorresponds to a symmetric tensor whose support is within some distance CGε of thesmall diagonal, where CG is a combinatorial constant depending on the graph G′. Asthe supports K and K′ (of O and O′, respectively) have a finite distance d betweenthem, we can choose ε small enough that CGε < d. It follows that, by choosing Φ andΓ to be sufficiently small, the weight of any connected graph is obtained by contractinga distribution and a function which have disjoint support. The graph hence has weightzero.

As there are finitely many such graphs with total genus i and k tails, we see that wecan choose Γ small enough that for any Φ < Γ, the weight of all such graphs vanishes.

Thus we have proved the first part of the theorem and have produced a bilinearmap

Obsq(U)×Obsq(V)→ Obsq(U qV).

It is a straightforward to show that this is a cochain map and satisfies the associativityand commutativity properties necessary to define a prefactorization algebra. The factthat this is a smooth map of differentiable pro-vector spaces follows from the fact thatthis construction works for families of theories over an arbitrary nilpotent dg manifold(X, A ).


5.11. Local observables form a factorization algebra

We have seen how to define a prefactorization algebra Obsq of observables for ourquantum field theory. In this section we will show that this prefactorization algebra isin fact a factorization algebra. In the course of the proof, we show that modulo h, thisfactorization algebra is isomorphic to Obscl .

5.11.0.2 Theorem. (1) The prefactorization algebra Obsq of quantum observables is, infact, a factorization algebra.

(2) Further, there is an isomorphism

Obsq⊗C[[h]]C∼= Obscl

between the reduction of the factorization algebra of quantum observables modulo h,and the factorization algebra of classical observables.

5.11.1. Proof of the theorem. This theorem will be a corollary of a more technicalproposition.

5.11.1.1 Proposition. For any open subset U ⊂ M, filter Obsq(U) by saying that the k-thfiltered piece Gk Obsq(U) is the sub C[[h]]-module consisting of those observables which arezero modulo hk. Note that this is a filtration by sub prefactorization algebras over the ringC[[h]].

Then, there is an isomorphism of prefactorization algebras (in differentiable pro-cochaincomplexes)

Gr Obsq ' Obscl ⊗CC[[h]].

This isomorphism makes Gr Obsq into a factorization algebra.

Remark: We can give Gk Obsq(U) the structure of a pro-differentiable cochain complex,as follows. The filtration on Gk Obsq(U) that defines the pro-structure is obtainedby intersecting Gk Obsq(U) with the filtration on Obsq(U) defining the pro-structure.Then the inclusion Gk Obsq(U) → Obsq(U) is a cofibration of differentiable pro-vectorspaces (see definition A.5.0.7).

PROOF OF THE THEOREM, ASSUMING THE PROPOSITION. We need to show that forevery open U and for every factorizing cover U, the natural map

(†) C(U, Obsq)→ Obsq(U)

5.11. LOCAL OBSERVABLES FORM A FACTORIZATION ALGEBRA 187

is a quasi-isomorphism of differentiable pro-cochain complexes.

The basic idea is that the filtration induces a spectral sequence for both C(U, Obsq)

and Obsq(U), and we will show that the induced map of spectral sequences is an iso-morphism on the first page. Because we are working with differentiable pro-cochaincomplexes, this is a little subtle. The relevant statements about spectral sequences inthis context are developed in this context in Appendix A.

Note that C(U , Obsq) is filtered by C(U, Gk Obsq). The map (†) preserves the filtra-tions. Thus, we have a maps of inverse systems

C(U, Obsq /Gk Obsq)→ Obsq(U)/Gk Obsq(U).

These inverse systems satisfy the properties of Appedix A, lemma A.5.0.11. Further, itis clear that

Obsq(U) = lim←−Obsq(U)/Gk Obsq(U).

We also have

C(U, Obsq) = lim←− C(U, Obsq /Gk Obsq).

This equality is less obvious, and uses the fact that the Cech complex is defined usingthe completed direct sum as described in Appendix A, section A.5.

Using lemma A.5.0.11, we need to verify that the map

C(U, Gr Obsq)→ Gr Obsq(U)

is an equivalence. This follows from the proposition because Gr Obsq is a factorizationalgebra.

PROOF OF THE PROPOSITION. The first step in the proof of the proposition is thefollowing lemma.

5.11.1.2 Lemma. Let Obsq(0) denote the prefactorization algebra of observables which are only

defined modulo h. There is an isomorphism of prefactorization algebras

Obsq(0) ' Obscl

of differential graded prefactorization algebras.

PROOF OF LEMMA. Let O ∈ Obscl(U) be a classical observable. Thus, O is anelement of the cochain complex O(E (U)) of functionals on the space of fields on U.


We need to produce an element of Obsq(0) from O. An element of Obsq

(0) is a collectionof functionals O[Φ] ∈ O(E ), one for every parametrix Φ, satisfying a classical versionof the renormalization group equation and an axiom saying that O[Φ] is supported onU for sufficiently small Φ.

Given an elementO ∈ Obscl(U) = O(E (U)),

we define an elementO[Φ] ∈ Obsq

(0)

by the formulaO[Φ] = lim

Γ→0WΦ

Γ (O) modulo h.

The Feynman diagram expansion of the right hand side only involves trees, sincewe are working modulo h. As we are only using trees, the limit exists. The limit isdefined by a sum over trees with one special vertex, where each edge is labelled bythe propagator P(Φ), the special vertex is labelled by O, and every other vertex islabelled by the classical interaction I0 ∈ Oloc(E ) of our theory.

The mapObscl(U)→ Obsq

(0)(U)

we have constructed is easily seen to be a map of cochain complexes, compatible withthe structure of prefactorization algebra present on both sides. (The proof is a varia-tion on the argument in section 11, chapter 5 of [Cos11c], about the scale 0 limit of adeformation of I modulo h.)

A simple inductive argument on the degree shows this map is an isomorphism.

Because the construction works over an arbitrary nilpotent dg manifold, it is clearthat these maps are maps of differentiable cochain complexes.

The next (and most difficult) step in the proof of the proposition is the followinglemma. We use it to work inductively with the filtration of quantum observables.

Let Obsq(k) denote the prefactorization algebra of observables defined modulo hk+1.

5.11.1.3 Lemma. For all open subsets U ⊂ M, the natural quotient map of differentiablepro-cochain complexes

Obsq(k+1)(U)→ Obsq

(k)(U)


is a fibration of differentiable pro-cochain complexes (see Appendix A, Definition A.5.0.7 forthe definition of a fibration). The fiber is isomorphic to Obscl(U).

PROOF OF LEMMA. We give the set (i, k) ∈ Z≥0 ×Z≥0 the lexicographical order-ing, so that (i, k) > (r, s) if i > r or if i = r and k > s.

We will let Obsq≤(i,k)(U) be the quotient of Obsq

(i) consisting of functionals

O[Φ] = ∑(r,s)≤(i,k)

hrO(r,s)[Φ]

satisfying the renormalization group equation and locality axiom as before, but whereO(r,s)[Φ] is only defined for (r, s) ≤ (i, k). Similarly, we will let Obsq

<(i,k)(U) be thequotient where the O(r,s)[Φ] are only defined for (r, s) < (i, k).

We will show that the quotient map

q : Obsq≤(i,k)(U)→ Obsq

<(i,k)(U)

is a fibration. The result will follow.

Recall what it means for a map f : V → W of differentiable cochain complexes tobe a fibration. For X a manifold, let C∞

X (V) denote the sheaf of cochain complexes on Xof smooth maps to V. We say f is a fibration if for every manifold X, the induced mapof sheaves C∞

X (V) → C∞X (W) is surjective in each degree. Equivalently, we require

that for all smooth manifolds X, every smooth map X →W lifts locally on X to a mapto V.

Now, by definition, a smooth map from X to Obsq(U) is an observable for theconstant family of theories over the nilpotent dg manifold (X, C∞(X)). Thus, in orderto show q is a fibration, it suffices to show the following. For any family of theoriesover the nilpotent dg manifold (X, A ), any open subset U ⊂ M, and any observableα in the A -module Obsq

<(i,k)(U), we can lift α to an element of Obsq≤(i,k)(U) locally on

X.

To prove this, we will first define, for every parametrix Φ, a map

LΦ : Obsq<(i,k)(U)→ Obsq

≤(i,k)(M)

with the property that the composed map

Obsq<(i,k)(U)

LΦ−→ Obsq≤(i,k)(M)→ Obsq

<(i,k)(M)


is the natural inclusion map. Then, for every observable O ∈ Obsq<(i,k)(U), we will

show that LΦ(O) is supported on U, for sufficiently small parametrices Φ, so thatLΦ(O) provides the desired lift.

ForO ∈ Obsq

<(i,k)(U),

we defineLΦ(O) ∈ Obsq

≤(i,k)(M)

by

LΦ(O)r,s[Φ] =

Or,s[Φ] if (r, s) < (i, k)

0 if (r, s) = (i, k).

For Ψ 6= Φ, we obtain LΦ(O)r,s[Ψ] by the renormalization group flow from LΦ(O)r,s[Φ].If (r, s) < (i, k), then

LΦ(O)r,s[Ψ] = Or,s[Ψ].

ButIi,k[Ψ] + δ (LΦ(O)i,k[Ψ]) = Wi,k (P(Ψ)− P(Φ), I[Φ] + δO[Φ])

for δ a square-zero parameter of cohomological degree opposite to that of O. HenceLΦ(O)r,s satisfies the relevant RGE.

To complete the proof of this lemma, we prove the required local lifting propertyin the sublemma below.

5.11.1.4 Sub-lemma. For each O ∈ Obsq<(i,k)(U), we can find a parametrix Φ — locally

over the parametrizing manifold X — so that LΦO lies in Obsq≤(i,k)(U) ⊂ Obsq

≤(i,k)(M).

PROOF. Although the observables Obsq form a factorization algebra on the man-ifold M, they also form a sheaf on the parametrizing base manifold X. That is, forevery open subset V ⊂ X, let Obsq(U) |V denote the observables for our family oftheories restricted to V. In other words, Obsq(U) |V denotes the sections of this sheafObsq(U) on V.

The map LΦ constructed above is then a map of sheaves on X.

For every observable O ∈ Obsq<(i,k)(U), we need to find an open cover

X =⋃α

Yα


of X, and on each Yα a parametrix Φα (for the restriction of the family of theories to Yα)such that

LΦα(O |Yα) ∈ Obsq

≤(i,k)(U) |Yα.

More informally, we need to show that locally in X, we can find a parametrix Φ suchthat for all sufficiently small Ψ, the support of LΦ(O)(i,k)[Ψ] is in a subset of Uk × Xwhich maps properly to X.

This argument resembles previous support arguments (e.g., the product lemmafrom section 5.10). The proof involves an analysis of the Feynman diagrams appearingin the expression

(?) LΦ(O)i,k[Ψ] = ∑γ

1|Aut(γ)|wγ (O[Φ]; I[Φ]; P(Ψ)− P(Φ)) .

The sum is over all connected Feynman diagrams of genus i with k tails. The edgesare labelled by P(Ψ)− P(Φ). Each graph has one special vertex, where O[Φ] appears.More explicitly, if this vertex is of genus r and valency s, it is labelled by Or,s[Φ]. Eachnon-special vertex is labelled by Ia,b[Φ], where a is the genus and b the valency of thevertex. Note that only a finite number of graphs appear in this sum.

By assumption, O is supported on U. This means that there exists some parametrixΦ0 and a subset K ⊂ U × X mapping properly to X such that for all Φ < Φ0, Or,s[Φ]

is supported on Kr. (Here by Ks ⊂ Us × X we mean the fibre product over X.)

Further, each Ia,b[Φ] is supported as close as we like to the small diagonal M×X inMk × X. We can find precise bounds on the support of Ia,b[Φ], as explained in section5.4. To describe these bounds, let us choose metrics for X and M. For a parametrixΦ supported within ε of the diagonal M× X in M×M× X, the effective interactionIa,b[Φ] is supported within (2a + b)ε of the diagonal.

(In general, if A ⊂ Mn×X, the ball of radius ε around A is defiend to be the unionof the balls of radius ε around each fibre Ax of A→ X. It is in this sense that we meanthat Ia,b[Φ] is supported within (2a + b)ε of the diagonal.)

Similarly, for every parametrix Ψ with Ψ < Φ, the propagator P(Ψ) − P(Φ) issupported within ε of the diagonal.

In sum, there exists a set K ⊂ U × X, mapping properly to X, such that for allε > 0, there exists a parametrix Φε, such that


(1) O[Φε]r,s is supported on Ks for all (r, s) < (i, k).(2) Ia,b[Φε] is supported within (2a + b)ε of the small diagonal.(3) For all Ψ < Φε, P(Ψ)− P(Φε) is supported within ε of the small diagonal.

The weight wγ in the graphical expansion (?) above (using the parametrices Φε andany Ψ < Φε) is thus supported in the ball of radius cε around Kk (where c is somecombinatorial constant, depending on the number of edges and vertices in γ). Thereare a finite number of such graphs in the sum, so we can choose the combinatorialconstant c uniformly over the graphs.

Since K ⊂ U × X maps properly to X, locally on X, we can find an ε so that theclosed ball of radius cε is still inside Uk × X. This completes the proof.

5.12. Translation-invariant factorization algebras from translation-invariantquantum field theories

In this section, we will show that a translation-invariant quantum field theory onRn gives rise to a smoothly translation-invariant factorization algebra on Rn (see sec-tion 2.7). We will also show that a holomorphically translation-invariant field theoryon Cn gives rise to a holomorphically translation-invariant factorization algebra.

5.12.1. First, we need to define what it means for a field theory to be translation-invariant. Let us consider a classical field theory on Rn. Recall that this is given by

(1) A graded vector bundle E whose sections are E ;(2) An antisymmetric pairing E⊗ E→ DensRn ;(3) A differential operator Q : E → E making E into an elliptic complex, which

is skew-self adjoint;(4) A local action functional I ∈ Oloc(E ) satisfying the classical master equation.

A classical field theory is translation-invariant if

(1) The graded bundle E is translation-invariant, so that we are given an isomor-phism between E and the trivial bundle with fibre E0.

5.12. TRANSLATION INVARIANCE 193

(2) The pairing, differential Q, and local functional I are all translation-invariant.

It takes a little more work to say what it means for a quantum field theory to betranslation-invariant. Suppose we have a translation-invariant classical field theory,equipped with a translation-invariant gauge fixing operator QGF. As before, a quanti-zation of such a field theory is given by a family of interactions I[Φ] ∈ Osm,P(E ), onefor each parametrix Φ.

5.12.1.1 Definition. A translation-invariant quantization of a translation-invariant classicalfield theory is a quantization with the property that, for all translation-invariant parametricesΦ, I[Φ] is translation-invariant.

Remark: In general, in order to give a quantum field theory on a manifold M, we donot need to give an effective interaction I[Φ] for all parametrices. We only need tospecify I[Φ] for a collection of parametrices such that the intersection of the supportsof Φ is the small diagonal M ⊂ M2. The functional I[Ψ] for all other parametricesΨ is defined by the renormalization group flow. It is easy to construct a collection oftranslation-invariant parametrices satisfying this condition.

5.12.1.2 Proposition. The factorization algebra associated to a translation-invariant quan-tum field theory is smoothly translation-invariant (section 2.7).

PROOF. Let Obsq denote the factorization algebra of quantum observables for ourtranslation-invariant theory. An observable supported on U ⊂ Rn is defined by afamily O[Φ] ∈ O(E )[[h]], one for each translation-invariant parametrix, which satis-fiese the RG flow and (in the sense we explained in section 5.9) is supported on U forsufficiently small parametrices. The renormalization group flow

WΦΨ : O(E )[[h]]→ O(E )[[h]]

for translation-invariant parametrices Ψ, Φ commutes with the action of Rn on O(E )

by translation, and therefore acts on Obsq(Rn). For x ∈ Rn and U ⊂ Rn, let TxUdenote the x-translate of U. It is immediate that the action of x ∈ Rn on Obsq(Rn)

takes Obsq(U) ⊂ Obsq(Rn) to Obsq(TxU). It is not difficult to verify that the resultingmap

Obsq(U)→ Obsq(TxU)

is an isomorphism of differentiable pro-cochain complexes and that it is compatiblewith the structure of a factorization algebra.


We need to verify the smoothness hypothesis of a smoothly translation-invariantfactorization algebra. This is the following. Suppose that U1, . . . , Uk are disjoint opensubsets of Rn, all contained in an open subset V. Let A′ ⊂ Rnk be the subset consistingof those x1, . . . , xk such that the closures of Txi Ui remain disjoint and in V. Let A bethe connected component of 0 in A′. We need only examine the case where A is non-empty.

We need to show that the composed map

mx1,...,xk : Obsq(U1)× · · · ×Obsq(Uk)→

Obsq(Tx1U1)× · · · ×Obsq(Txk Uk)→ Obsq(V)

varies smoothly with (x1, . . . , xk) ∈ A. In this diagram, the first map is the productof the translation isomorphisms Obsq(Ui) → Obsq(Txi Ui), and the second map is theproduct map of the factorization algebra.

The smoothness property we need to check says that the map mx1,...,xk lifts to amultilinear map of differentiable pro-cochain complexes

Obsq(U1)× · · · ×Obsq(Uk)→ C∞(A, Obsq(V)),

where on the right hand side the notation C∞(A, Obsq(V)) refers to the smooth mapsfrom A to Obsq(V).

This property is local on A, so we can replace A by a smaller open subset if neces-sary.

Let us assume (replacing A by a smaller subset if necessary) that there exist opensubsets U′i containing Ui, which are disjoint and contained in V and which have theproperty that for each (x1, . . . , xk) ∈ A, Txi Ui ⊂ U′i .

Then, we can factor the map mx1,...,xk as a composition

(†) Obsq(U1)× · · · ×Obsq(Uk)ix1×···×ixk−−−−−−→ Obsq(U′1)× · · · ×Obsq(U′k)→ Obsq(V).

Here, the map ixi : Obsq(Ui)→ Obsq(U′i ) is the composition

Obsq(Ui)→ Obsq(Txi Ui)→ Obsq(U′i )


of the translation isomorphism with the natural inclusion map Obsq(Txi Ui)→ Obsq(U′i ).The second map in equation (†) is the product map associated to the disjoint subsetsU′1, . . . , U′k ⊂ V.

By possibly replacing A by a smaller open subset, let us assume that A = A1 ×· · · × Ak, where the Ai are open subsets of Rn containing the origin. It remains to showthat the map

ixi : Obsq(Ui)→ Obsq(U′i )

is smooth in xi, that is, extends to a smooth map

Obsq(Ui)→ C∞(Ai, Obsq(U′i )).

Indeed, the fact that the product map

m : Obsq(U′1)× · · · ×Obsq(U′k)→ Obsq(V)

is a smooth multilinear map implies that, for every collection of smooth maps αi :Yi → Obsq(U′i ) from smooth manifolds Yi, the resulting map

Y1 × · · · ×Yk → Obsq(V)

(y1, . . . yk) 7→ m(α1(y), . . . , αk(y))

is smooth.

Thus, we have reduced the result to the following statement: for all open subsetsA ⊂ Rn and for all U ⊂ U′ such that TxU ⊂ U′ for all x ∈ A, the map ix : Obsq(U)→Obsq(U′) is smooth in x ∈ A.

But this statement is tractable. Let

O ∈ Obsq(U) ⊂ Obsq(U′) ⊂ Obsq(Rn)

be an observable. It is obvious that the family of observables TxO, when viewed aselements of Obsq(Rn), depends smoothly on x. We need to verify that it dependssmoothly on x when viewed as an element of Obsq(U′).

This amounts to showing that the support conditions which ensure an observableis in Obsq(U′) hold uniformly on x.

The fact that O is in Obsq(U) means the following. For each (i, k), there exists acompact subset K ⊂ U and ε > 0 such that for all translation-invariant parametrices


Φ supported within ε of the diagonal and for all (r, s) ≤ (i, k) in the lexicographicalordering, the Taylor coefficient Or,s[Φ] is supported on Ks.

We need to enlarge K to a subset L ⊂ U′ × A, mapping properly to A, such thatTxO is supported on L in this sense (again, for (r, s) ≤ (i, k)). Taking L = K × A,embedded in U′ × A by

(k, x) 7→ (Txk, x)

suffices.

Remark: Essentially the same proof will give us the somewhat stronger result that forany manifold M with a smooth action of a Lie group G, the factorization algebra cor-responding to a G-equivariant field theory on M is smoothly G-equivariant.

5.12.2. Similarly, we can talk about holomorphically translation-invariant classi-cal and quantum field theories on Cn. In this context, we will take our space of fieldsto be Ω0,∗(Cn, V), where V is some translation-invariant holomorphic vector bundleon Cn. The pairing must arise from a translation-invariant map of holomorphic vectorbundles

V ⊗V → KCn

of cohomological degree n− 1, where KCn denotes the canonical bundle. This meansthat the composed map

Ω0,∗c (Cn, V)⊗2 → Ω0,∗

c (Cn, KCn)

∫−→ C

is of cohomological degree −1.

Let

ηi =∂

∂zi∨− : Ω0,k(Cn, V)→ Ω0,k−1(Cn, V)

be the contraction operator. The cohomological differential operator Q on Ω0,∗(V)

must be of the form

Q = ∂ + Q0

where Q0 is translation-invariant and

[Q0, ηi] = 0,

for i = 1, . . . , n. Of course, Q must be skew self-adjoint with respect to the pairing onΩ0,∗

c (Cn, V).


The other piece of data of a classical field theory is the local action functional I ∈Oloc(Ω0,∗(Cn, V)). We assume that I is translation-invariant, of course, but also that

ηi I = 0,

where the linear map ηi on Ω0,∗(Cn, V) is extended in the natural way to a derivationof the algebra O(Ω0,∗

c (Cn, V)) preserving the subspace of local functionals.

We then take our gauge fixing operator to be

∂∗= ∑ ηi

∂

∂zi.

Since [ηi, Q0] = 0, we see that [Q, ∂∗] = [∂, ∂

∗] is the Laplacian. (More generally, we can

consider a family of gauge fixing operators coming from the ∂∗

operator for a familyof flat Hermitian metrics on Cn. Since the space of such metrics is GL(n, C)/U(n) andthus contractible, we see that everything is independent up to homotopy of the choiceof gauge fixing operator.)

We say a translation-invariant parametrix

Φ ∈ Ω0,∗(Cn, V)⊗2

is holomorphically translation-invariant if

(ηi ⊗ 1 + 1⊗ ηi)Φ = 0

for i = 1, . . . , n. For example, if f ∈ C∞(Cn × Cn) is a smooth function with propersupport which is 1 near the diagonal, then

Φ = f∫ L

0(∂∗ ⊗ 1)Ktdt

defines such a parametrix. Clearly, we can find holomorphically translation-invariantparametrices which are supported arbitrarily close to the diagonal. This means thatwe can define a field theory by only considering I[Φ] for holomorphically translation-invariant parametrices Φ.

5.12.2.1 Definition. A holomorphically translation-invariant quantization of a holomorphi-cally translation-invariant classical field theory as above is a translation-invariant quanti-zation such that for each holomorphically translation-invariant parametrix Φ, the effectiveinteraction I[Φ] satisfies

ηi I[Φ] = 0


for i = 1, . . . , n. Here ηi abusively denotes the natural extension of the contraction ηi to aderivation on O(Ω0,∗

c (Cn, V)).

5.12.2.2 Proposition. A holomorphically translation-invariant quantum field theory on Cn

leads to a holomorphically translation-invariant factorization algebra.

PROOF. This follows immediately from proposition 5.12.1.2. Indeed, quantum ob-servables form a smoothly translation-invariant factorization algebra. Such an observ-able O is specified by a family O[Φ] ∈ O(Ω0,∗(Cn, V)) of functionals defined for eachholomorphically translation-invariant parametrix Φ. The operators ∂

∂zi, ∂

∂zi, ηi act in a

natural way on O(Ω0,∗(Cn, V)) by derivations, and each commutes with the renormal-ization group flow WΦ

Ψ for holomorphically translation-invariant parametrices Ψ, Φ.Thus, ∂

∂zi, ∂

∂ziand ηi define derivations of the factorization algebra Obsq. Explicitly,

if O ∈ Obsq(U) is an observable, then for each holomorphically translation-invariantparametrix Φ, (

∂

∂ziO)[Φ] =

∂

∂zi(O[Φ]),

and similarly for ∂∂zi

and ηi.

By definition (Definition 2.7.6.1), a holomorphically translation-invariant factor-ization algebra is a translation-invariant factorization algebra where the derivationoperator ∂

∂zion observables is homotopically trivialized.

Note that, for a holomorphically translation-invariant parametrix Φ, [ηi,4Φ] = 0and ηi is a derivation for the Poisson bracket −,−Φ. It follows that

[Q + I[Φ],−Φ + h4Φ, ηi] = [Q, ηi]

as operators on O(Ω0,∗(Cn, V)). Since we wrote Q = ∂+Q0 and required that [Q0, ηi] =

0, we have

[Q, ηi] = [∂, ηi] =∂

∂zi.

Since the differential on Obsq(U) is defined by

(QO)[Φ] = QO[Φ] + I[Φ], O[Φ]Φ + h4ΦO[Φ],

we see that [Q, ηi] =∂

∂zi, as desired.

5.13. COTANGENT THEORIES AND VOLUME FORMS 199

5.13. Cotangent theories and volume forms

In this section we will examine the case of a cotangent theory, in which our def-inition of a quantization of a classical field theory acquires a particularly nice inter-pretation. Suppose that L is an elliptic L∞ algebra on a manifold M describing anelliptic moduli problem, which we denote by BL. As we explained in Chapter 4, sec-tion 4.12, we can construct a classical field theory from L, whose space of fields isE = L[1] ⊕ L![−2]. The main observation of this section is that a quantization ofthis classical field theory can be interpreted as a kind of “volume form” on the ellip-tic moduli problem BL. This point of view was developed in [Cos11b], and used in[Cos11a] to provide a geometric interpretation of the Witten genus.

5.13.1. A finite dimensional model. We first need to explain an algebraic inter-pretation of a volume form in finite dimensions. Let X be a manifold (or complexmanifold or smooth algebraic variety; nothing we will say will depend on which geo-metric category we work in). Let O(X) denote the smooth functions on X, and letVect(X) denote the vector fields on X.

If ω is a volume form on X, then it gives a divergence map

Divω : Vect(X)→ O(X)

defined via the Lie derivative:

Divω(V)ω = LVω

for V ∈ Vect(X). Note that the divergence operator Divω satisfies the equations

(†)Divω( f V) = f Divω V + V( f ).

Divω([V, W]) = V Divω W −W Divω V.

The volume form ω is determined up to a constant by the divergence operator Divω.

Conversely, to give an operator Div : Vect(X) → O(X) satisfying equations (†) isthe same as to give a flat connection on the canonical bundle KX of X, or, equivalently,to give a right D-module structure on the structure sheaf O(X).

5.13.1.1 Definition. A projective volume form on a space X is an operator Div : Vect(X)→O(X) satisfying equations (†).


The advantage of this definition is that it makes sense in many contexts wheremore standard definitions of a volume form are hard to define. For example, if A isa quasi-free differential graded commutative algebra, then we can define a projectivevolume form on the dg scheme Spec A to be a cochain map Der(A) → A satisfyingequations (†). Similarly, if g is a dg Lie or L∞ algebra, then a projective volume formon the formal moduli problem Bg is a cochain map C∗(g, g[1]) → C∗(g) satisfyingequations (†).

5.13.2. There is a generalization of this notion that we will use where, instead ofvector fields, we take any Lie algebroid.

5.13.2.1 Definition. Let A be a commutative differential graded algebra over a base ring k. ALie algebroid L over A is a dg A-module with the following extra data.

(1) A Lie bracket on L making it into a dg Lie algebra over k. This Lie bracket will betypically not A-linear.

(2) A homomorphism of dg Lie algebras α : L→ Der∗(A), called the anchor map.(3) These structures are related by the Leibniz rule

[l1, f l2] = (α(l1)( f )) l2 + (−1)|l1|| f | f [l1, l2]

for f ∈ A, li ∈ L.

In general, we should think of L as providing the derived version of a foliation:an ordinary foliation consists of an ordinary commutative algebra A with a projectiveA-module L and an injective anchor map.

5.13.2.2 Definition. If A is a commutative dg algebra and L is a Lie algebroid over A, thenan L-projective volume form on A is a cochain map

Div : L→ A

satisfying

Div(al) = a Div l + (−1)|l||a|α(l)a.

Div([l1, l2]) = l1 Div l2 − (−1)|l1||l2|l2 Div l1.


Of course, if the anchor map is an isomorphism, then this structure is the sameas a projective volume form on A. In the more general case, we should think of an L-projective volume form as giving a projective volume form on the leaves of the derivedfoliation.

5.13.3. Let us explain how this definition relates to the notion of quantization ofP0 algebras.

5.13.3.1 Definition. Give the operad P0 a C× action where the product has weight 0 and thePoisson bracket has weight 1. A graded P0 algebra is a C×-equivariant differential gradedalgebra over this dg operad.

Note that, if X is a manifold, O(T∗[−1]X) has the structure of graded P0 algebra,where the C× action on O(T∗[−1]X) is given by rescaling the cotangent fibers.

Similarly, if L is a Lie algebroid over a commutative dg algebra A, then Sym∗A L[1]is a C×-equivariant P0 algebra. The P0 bracket is defined by the bracket on L and theL-action on A; the C× action gives Symk L[1] weight −k.

5.13.3.2 Definition. Give the operad BD over C[[h]] a C× action, covering the C× action onC[[h]], where h has weight −1, the product has weight 0, and the Poisson bracket has weight1.

Note that this C× action respects the differential on the operad BD, which is de-fined on generators by

d(− ∗−) = h−,−.

Note also that by describing the operad BD as a C×-equivariant family of operadsover A1, we have presented BD as a filtered operad whose associated graded operadis P0.

5.13.3.3 Definition. A filtered BD algebra is a BD algebra A with a C× action compatiblewith the C× action on the ground ring C[[h]], where h has weight −1, and compatible withthe C× action on BD.

5.13.3.4 Lemma. If L is Lie algebroid over a dg commutative algebra A, then every L-projectivevolume form yields a filtered BD algebra structure on Sym∗A(L[1])[[h]], quantizing the gradedP0 algebra Sym∗A(L[1]).


PROOF. If Div : L → A is an L-projective volume form, then it extends uniquelyto an order two differential operator4 on Sym∗A(L[1]) which maps

SymiA(L[1])→ Symi−1

A (L[1]).

Then Sym∗A L[1][[h]], with differential d + h4, gives the desired filtered BD algebra.

5.13.4. Let BL be an elliptic moduli problem on a compact manifold M. Themain result of this section is that there exists a special kind of quantization of thecotangent field theory for BL that gives a projective volume on this formal moduliproblem BL. Projective volume forms arising in this way have a special “locality”property, reflecting the locality appearing in our definition of a field theory.

Thus, let L be an elliptic L∞ algebra on M. This gives rise to a classical field theorywhose space of fields is E = L[1] ⊕ L![−2], as described in Chapter 4, section 4.12.Let us give the space E a C×-action where L[1] has weight 0 and L![−1] has weight 1.This induces a C× action on all associated spaces, such as O(E ) and Oloc(E ).

This C× action preserves the differential Q + I,− on O(E ), as well as the com-mutative product. Recall (Chapter 4, section 4.13) that the subspace

Obscl(M) = Osm(E ) ⊂ O(E )

of functionals with smooth first derivative has a Poisson bracket of cohomologicaldegree 1, making it into a P0 algebra. This Poisson bracket is of weight 1 with respect

to the C× action on Obscl(M), so Obs

cl(M) is a graded P0 algebra.

We are interested in quantizations of our field theory where the BD algebra ObsqΦ(M)

of (global) quantum observables (defined using a parametrix Φ) is a filtered BD alge-bra.

5.13.4.1 Definition. A cotangent quantization of a cotangent theory is a quantization, givenby effective interaction functionals I[Φ] ∈ O+

sm,P(E )[[h]] for each parametrix Φ, such thatI[Φ] is of weight −1 under the C× action on the space O+

sm,P(E )[[h]] of functionals.

This C× action gives h weight −1. Thus, this condition means that if we expand

I[Φ] = ∑ hi Ii[Φ],


then the functionals Ii[Φ] are of weight i− 1.

Since the fields E = L[1]⊕ L![−2] decompose into spaces of weights 0 and 1 under theC× action, we see that I0[Φ] is linear as a function of L![−2], that I1[Φ] is a function only ofL[1], and that Ii[Φ] = 0 for i > 1.

Remark: (1) The quantization I[Φ] is a cotangent quantization if and only ifthe differential Q + I[Φ],−Φ + h4Φ preserves the C× action on the spaceO(E )[[h]] of functionals. Thus, I[Φ] is a cotangent quantization if and onlyif the BD algebra Obsq

Φ(M) is a filtered BD algebra for each parametrix Φ.(2) The condition that I0[Φ] is of weight −1 is automatic.(3) It is easy to see that the renormalization group flow

W (P(Φ)− P(Ψ),−)

commutes with the C× action on the space O+sm,P(E )[[h]].

5.13.5. Let us now explain the volume-form interpretation of cotangent quanti-zation. Let L be an elliptic L∞ algebra on M, and let O(BL) = C∗(L) be the Chevalley-Eilenberg cochain complex of M. The cochain complexes O(BL(U)) for open subsetsU ⊂ M define a commutative factorization algebra on M.

As we have seen in Chapter 4, section 4.2, we should interpret modules for an L∞

algebra g as sheaves on the formal moduli problem Bg. The g-module g[1] correspondsto the tangent bundle of Bg, and so vector fields on g correspond to the O(Bg)-moduleC∗(g, g[1]).

Thus, we use the notation

Vect(BL) = C∗(L,L[1]);

this is a dg Lie algebra and acts on C∗(L) by derivations (see Appendix A, section A.8,for details).

For any open subset U ⊂ M, theL(U)-moduleL(U)[1] has a sub-moduleLc(U)[1]given by compactly supported elements of L(U)[1]. Thus, we have a sub-O(BL(U))-module

Vectc(BL(U)) = C∗(L(U),Lc(U)[1]) ⊂ Vect(BL(U)).


This is in fact a sub-dg Lie algebra, and hence a Lie algebroid over the dg commutativealgebra O(BL(U)). Thus, we should view the subspace Lc(U)[1] ⊂ Lc(U)[1] as pro-viding a foliation of the formal moduli problem BL(U), where two points of BL(U)

are in the same leaf if they coincide outside a compact subset of U.

If U ⊂ V are open subsets of M, there is a restriction map of L∞ algebras L(V) →L(U). The natural extension map Lc(U)[1] → Lc(V)[1] is a map of L(V)-modules.Thus, by taking cochains, we find a map

Vectc(BL(U))→ Vectc(BL(V)).

Geometrically, we should think of this map as follows. If we have an R-point α ofBL(V) for some dg Artinian ring R, then any compactly-supported deformation ofthe restriction α |U of α to U extends to a compactly supported deformation of α.

We want to say that a cotangent quantization of L leads to a “local” projectivevolume form on the formal moduli problem BL(M) if M is compact. If M is compact,then Vectc(BL(M)) coincides with Vect(BL(M)). A local projective volume form onBL(M) should be something like a divergence operator

Div : Vect(BL(M))→ O(BL(M))

satisfying the equations (†), with the locality property that Div maps the subspace

Vectc(BL(U)) ⊂ Vect(BL(M))

to the subspace O(BL(U)) ⊂ O(BL(M)).

Note that a projective volume form for the Lie algebroid Vectc(BL(U)) over O(BL(U))

is a projective volume form on the leaves of the foliation of BL(U) given by Vectc(BL(U)).The leaf space for this foliation is described by the L∞ algebra

L∞(U) = L(U)/Lc(U) = colimK⊂U

L(U \ K).

(Here the colimit is taken over all compact subsets of U.) Consider the one-point com-pactification U∞ of U. Then the formal moduli problem L∞(U) describes the germs at∞ on U∞ of sections of the sheaf on U of formal moduli problems given by L.

Thus, the structure we’re looking for is a projective volume form on the fibers ofthe maps BL(U) → BL∞(U) for every open subset U ⊂ M, where the divergenceoperators describing these projective volume forms are all compatible in the sensedescribed above.


What we actually find is something a little weaker. To state the result, recall (sec-tion 5.7) that we use the notation P for the contractible simplicial set of parametrices,and C P for the cone on P . The vertex of the cone C P will denoted 0.

5.13.5.1 Theorem. A cotangent quantization of the cotangent theory associated to the ellipticL∞ algebra L leads to the following data.

(1) A commutative dg algebra OC P(BL) over Ω∗(C P). The underlying graded al-gebra of this commutative dg algebra is O(BL)⊗Ω∗(C P). The restriction of thiscommutative dg algebra to the vertex 0 of C P is the commutative dg algebra O(BL).

(2) A dg Lie algebroid VectC Pc (BL) over OC P(BL), whose underlying graded OC P(BL)-

module is Vectc(BL) ⊗ Ω∗(C P). At the vertex 0 of C P , the dg Lie algebroidVectC P

c (BL) coincides with the dg Lie algebroid Vectc(BL).(3) We let OP(BL) and VectPc (BL) be the restrictions of OC P(BL) and VectC P

c (BL)to P ⊂ C P . Then we have a divergence operator

DivP : OP(BL)→ VectPc (BL)

defining the structure of a VectPc (BL) projective volume form on OP(BL) andVectPc (BL).

Further, when restricted to the sub-simplicial set PU ⊂ P of parametrices with support ina small neighborhood of the diagonal U ⊂ M × M, all structures increase support by anarbitrarily small amount (more precisely, by an amount linear in U, in the sense explained insection 5.7).

PROOF. This follows almost immediately from theorem 5.7.2.1. Indeed, becausewe have a cotangent theory, we have a filtered BD algebra

ObsqP(M) =

(O(E )[[h]]⊗Ω∗(P), QP , −,−P

).

Let us consider the sub-BD algebra ObsqP(M), which, as a graded vector space, is

Osm(E )[[h]]⊗Ω∗(P) (as usual, Osm(E ) indicates the space of functionals with smoothfirst derivative).

Because we have a filtered BD algebra, there is a C×-action on this complex ObsqP(M).

We let

OP(BL) = ObsqP(M)0


be the weight 0 subspace. This is a commutative differential graded algebra overΩ∗(P), whose underlying graded algebra is O(BL); further, it extends (using againthe results of 5.7.2.1) to a commutative dg algebra OC P(BL) over Ω∗(C P), whichwhen restricted to the vertex is O(BL).

Next, consider the weight −1 subspace. As a graded vector space, this is

ObsqP(M)−1 = Vectc(BL)⊗Ω∗(P)⊕ hOP(BL).

We thus letVectPc (BL) = Obs

qP(M)−1/hOP(BL).

The Poisson bracket on ObsqP(M) is of weight 1, and it makes the space Obs

qP(M)−1

into a sub Lie algebra.

We have a natural decomposition of graded vector spaces

ObsqP(M)−1 = VectPc (BL)⊕ hOP(BL).

The dg Lie algebra structure on ObsqP(M)−1 gives us

(1) The structure of a dg Lie algebra on VectPc (BL) (as the quotient of ObsqP(M)−1

by the differential Lie algebra ideal hOP(BL)).(2) An action of VectPc (BL) on OP(BL) by derivations; this defines the anchor

map for the Lie algebroid structure on VectPc (BL).(3) A cochain map

VectPc (BL)→ hOP(BL).

This defines the divergence operator.

It is easy to verify from the construction of theorem 5.7.2.1 that all the desired proper-ties hold.

5.13.6. The general results about quantization of [Cos11c] thus apply to this situ-ation, to show that the following.

5.13.6.1 Theorem. Consider the cotangent theory E = L[1]⊕ L![−2] to an elliptic moduliproblem described by an elliptic L∞ algebra L on a manifold M.

The obstruction to constructing a cotangent quantization is an element in

H1(Oloc(E )C×) = H1(Oloc(BL)).


If this obstruction vanishes, then the simplicial set of cotangent quantizations is a torsor forthe simplicial Abelian group arising from the cochain complex Oloc(BL) by the Dold-Kancorrespondence.

As in Chapter 4, section 4.7, we are using the notation Oloc(BL) to refer to a “local”Chevalley-Eilenberg cochain for the elliptic L∞ algebra L. If L is the vector bundlewhose sections are L, then as we explained in [Cos11c], the jet bundle J(L) is a DM L∞

algebra andOloc(BL) = DensM⊗DM C∗red(J(L)).

There is a de Rham differential (see section 4.9) mapping Oloc(BL) to the complex oflocal 1-forms,

Ω1loc(BL) = C∗loc(L,L![−1]).

The de Rham differential maps Oloc(BL) isomorphically to the subcomplex of Ω1locBL)

of closed local one-forms.

Thus, we should view the obstruction to quantizing the cotangent theory, whichis a class in H1(Ω1

loc(BL)cl), as being the local version of the first Chern class of thecanonical bundle of L.

APPENDIX A

Homological algebra with differentiable vector spaces

The factorization algebras we consider take values in vector spaces of an analyt-ical nature, like the space of smooth functions on a manifold. We would thus like toperform homological algebra in this setting. The standard approach to working withobjects of this nature is to treat them as topological vector spaces. However, it is notcompletely obvious how one should set up homological algebra when using topolog-ical vector spaces. It is also not straightforward to construct the topology on vectorspaces which appear in our most important examples of factorization algebras: theobservables of a quantum field theory.

Thus we will work with a weaker and more flexible concept, that of differentaiblevector space. This Appendix develops homological algebra in the category of differ-entiable vector spaces. Related approaches to functional analysis are developed in[Pau10] and in [KM97].

A.1. Diffeological vector spaces

Let us remind the reader of the concept of diffeological space [?].

A.1.0.2 Definition. The site of smooth manifolds is the site whose objects are smooth mani-folds, morphisms are smooth maps, and where a map M → N is an open covering if it is asurjective local diffeomorphism.

A diffeological space X is a sheaf of sets on the site of smooth manifolds with the propertythat, for all smooth manifolds M, the map

X(M)→ HomSets(M, X(∗))

is injective. (On the right hand side, X(∗) is the value of X on a point.)

A map of diffeological spaces is a map of sheaves of sets on the smooth site.

209

210 A. HOMOLOGICAL ALGEBRA WITH DIFFERENTIABLE VECTOR SPACES

We will sometimes refer to maps of diffeological spaces as smooth maps, to distin-guish them from maps of the underlying sets.

We can rewrite the axioms of a diffeological space in more explicit terms, as fol-lows. A diffeological space is determined by a set X = X(∗), and for each smoothmanifold M, a subset X(M) ⊂ Maps(M, X) of smooth maps from M to X. These sub-sets must satisfy the following conditions. If f : M → X is a smooth map and ifg : N → M is a smooth map of ordinary manifolds, then f g : N → X is a smoothmap. Further, a map f : M → X is smooth if and only if it is smooth locally on M.Finally, all constant maps to X are smooth. We call this collection of smooth maps thediffeology of X.

Note that if X, Y are diffeological spaces, then so is X × Y: a map M → X × Y issmooth if the composition with both projection maps is smooth.

A.1.0.3 Definition. A diffeological vector space is a vector space V together with a diffeologycompatible with the vector space structure. Thus, the sum map V × V → V and the scalarmultiplication map R×V → V are maps of diffeological spaces (where R is given the standarddiffeology).

A diffeological vector space V has enough structure to talk about smooth mapsfrom a manifold M to V. We also want to be able to differentiate such maps. Thisrequires extra structure.

We use the notation C∞(M, V) to denote the C∞(M)-module of smooth maps fromM to V. Thus, C∞(M, V) is, as a vector space, just V(M), equipped with the naturalstructure of module over the (discrete) algebra C∞(M).

Similarly, we let

Ωk(M, V) = Ωk(M)⊗C∞(M) C∞(M, V)

denote the space of k-forms with values in V. This is just the algebraic tensor prod-uct. This is a reasonable thing to do because Ωk(M) is a finitely-generated projectiveC∞(M) module: it is a direct summand of a free finite rank C∞(M)-module. Thisimplies that Ωk(M, V) is a direct summand of C∞(M, V)⊕l for some l.

A.1. DIFFEOLOGICAL VECTOR SPACES 211

A.1.0.4 Definition. A differentiable vector space is a diffeological vector space together with,for each smooth manifold M, a flat connection

∇M,V : C∞(M, V)→ Ω1(M, V)

such that, for all smooth maps f : N → M,

f ∗∇M,V = ∇N,V .

To say that ∇M,V is a flat connection means, of course, that it satisfies the Leibnizrule,

∇M,V( f · s) = (d f )s + f∇M,Vs,

and that the curvature

F(∇M,V) = (∇M,V)2 : C∞(M, V)→ Ω2(M, V)

vanishes.

The flat connection ∇M,V allows us to differentiate smooth maps M → V. If f :M→ V is a smooth map and if X ∈ Vect(M) is a vector field on M, we define

X( f ) = 〈X,∇M,V f 〉 ∈ C∞(M, V),

where 〈−,−〉 indicates the C∞(M)-linear pairing

Vect(M)×Ω1(M, V)→ C∞(M, V).

Differentiable vector spaces form a category that we denote DVS. An object is adifferentiable vector space V. A morphism φ : V → W is a linear map such that forevery smooth map f : M→ V, the map φ f is smooth, and which is compatible withconnections in the sense that, for all smooth manifolds M,

φ ∇M,V = ∇M,V φ.

We will often refer to morphisms of differentiable vector spaces as smooth linear maps.

Differentiable vector spaces appear naturally in geometry. In section A.2 below,we show that for M a manifold and E is a vector bundle on M, the space C∞(N, E) ofsmooth sections of E has a natural structure of differentiable vector space. The sameholds for the space of compactly supported or distributional sections of E. Most of ourexamples of differentiable vector spaces arise in this way.


Below, we will examine various properties and examples of differentiable vectorspaces. Many of these constructions work equally well for diffeological vector spaces.

Remark: One of the main reasons we consider differentiable vector spaces instead oftopological vector spaces is that homological algebra for sheaves of vector spaces on asite is relatively standard, whereas homological algebra for topological vector spacesis trickier.

Another reason is that, when we consider our construction of factorization alge-bras in families, we will only use families where the base is a smooth manifold. Dif-ferentiable vector spaces have just enough structure to talk about such families.

A.1.1. Limits and colimits. Let V be a differentiable vector space, and let i : W ⊂V be a sub-vector space. The subspace diffeology on W is defined by saying that a mapf : M → W is smooth if the composed map i f : M → V is smooth. We say that thediffeological subspace W is a differentiable subspace if, for all smooth manifolds M, theconnection ∇M,V maps C∞(M, W) ⊂ C∞(M, V) to Ω1(M, W) ⊂ Ω1(M, V).

Differentiable subspaces have the usual universal property: if A is another differ-entiable vector space, a linear map A→W is smooth if and only if the composed mapA→ V is.

If W ⊂ V is a differentiable subspace, then we can form the quotient V/W. Amap from M to V/W is smooth if, locally on M, it lifts to a smooth map to V. Theconnection on V/W is uniquely determined by the requirement that the map V →V/W is compatible with connections (and so a map of differentiable spaces). We callV/W a differentiable quotient of V.

Again, this has the usual universal property: if A is another differentiable space, alinear map V/W → A is smooth if and only if the composed map V → A is smooth.

A.1.1.1 Lemma. The category of differentiable spaces admits all products and coproducts.

These can be described explicitly as follows. Let Vi | i ∈ I be some family of differentiablespaces indexed by a set I. The product ∏i Vi differentiable space has, as underlying vectorspace, the product vector space ∏ Vi. A map M→ ∏i Vi is smooth if and only if the composedmaps M→ Vi are smooth for all i. The connection map

∇M,∏ Vi : C∞(M, ∏ Vi) = ∏ C∞(M, Vi)→ Ω1(M, ∏ Vi) = ∏ Ω1(M, Vi)


is the product of the connections ∇M,Vi .

Similarly, the differentiable coproduct of the Vi has, as underlying vector space, the ordi-nary direct sum ⊕Vi. A map f : M → ⊕Vi is smooth if, locally on M, f can be written as afinite sum of smooth maps to some Vi1 , . . . , Vik . The connection

∇M,⊕Vi : C∞(M,⊕Vi)→ Ω1(M,⊕Vi)

is the unique connection which restricts to ∇M,Vi on the subspace C∞(M, Vi).

PROOF. We need to verify that the product and coproduct as described above havethe desired universal properties. For the product, this is immediate. Let’s verify it forthe coproduct. Let A be another differentiable vector space. Let f : ⊕Vi → A be alinear map. Suppose that the maps fi : Vi → A are all smooth. We need to show that fis smooth. Let φ : M→ ⊕Vi be a smooth map. To show that f φ is smooth, it sufficesto do so locally on M. Thus, we can assume that φ can be written as a finite sum ofsmooth maps φi : M → Vi. Then, f φ is a finite sum of f φi, and by assumption,f φi : M → A are smooth. It is straightforward to verify that the fact that the mapsfi are compatible with the connections on Vi and A imply that f is compatible withconnections.

A.1.1.2 Corollary. The category of differentiable vector spaces admits all limits (and so iscomplete).

PROOF. Arbitrary limits are obtained from products and kernels.Thus, we need toverify that the category of differentiable vector spaces admits kernels.

Let f : V →W be a map of differentiable vector spaces. Let us consider the kernelKer f ⊂ V, just as an ordinary vector space. We say that a map M→ Ker f is smooth ifand only if the composed map to V is smooth: this gives Ker f the subspace diffeology.Then, the sequence

0→ C∞(M, Ker f )→ C∞(M, V)→ C∞(M, W)

is exact.

We need to give Ker f a connection. Since the map C∞(M, V)→ C∞(M, W) is com-patible with connections, the connection ∇M,V on C∞(M, V) must map C∞(M, Ker f )to Ω1(M, Ker f ).


It is easy to verify that Ker f satisfies the universal property of a kernel.

Note that the forgetful functor DVS→ Vect preserves all limits.

A.1.2. Cokernels and exact sequences. The category of differentiable spaces doesnot admit all cokernels. Here is the prime example. Let W be a differentiable space,and let V ⊂W be an arbitrary linear subspace. We equip V with the initial diffeology,by saying that the space of smooth maps M → V is the algebraic tensor productC∞(M) ⊗alg V, rather than the subspace diffeology. Suppose these two diffeologiesdiffer. The quotient W/V has a natural diffeology, by saying that a map M→W/V issmooth if locally it lifts to a smooth map to W. The fact that the sequence

C∞(M)⊗alg V = C∞(M, V)→ C∞(M, W)→ C∞(M, W/V)→ 0

is not exact means that the connection on C∞(M, W) need not descend to one onC∞(M, V/W).

Happily, this example is the only way that things can go wrong.

A.1.2.1 Definition. A map f : V → W of differentiable spaces is admissible if, for allmanifolds M and all maps φ : M → Im f , the composed map M → W is smooth if and onlyif φ lifts locally to a smooth map to V.

In other words, f is admissible if the two natural pre-diffeologies on Im f (where we viewit as a quotient of V or a subspace of W) coincide.

A.1.2.2 Lemma. Cokernels of admissible maps exist.

PROOF. If f : V → W is an admissible map, we give Coker f the quotient diffe-ology: a map M → Coker f is smooth if locally it lifts to a smooth map to W. LetC∞

M(V) denote the sheaf on M which sends U ⊂ M to C∞(U, V). Then, the sequenceof sheaves

C∞M(V)→ C∞

M(W)→ C∞M(Coker f )→ 0

is exact. This implies that the connection on C∞(M, W) descends uniquely to one onC∞(M, Coker f ).

Another simple class of colimits that exist is the following.


A.1.2.3 Lemma. The category of differentiable vector spaces is closed under taking sequentialcolimits of injective maps.

By injective, we just mean that the map on the underlying vector space is injective.

PROOF. Let Vi for i ∈ Z≥0 be a sequence of differentiable vector spaces, and letfij : Vi → Vj be injective maps with f jk fij = fik. Let V denote the ordinary vector space

V = colim Vi = ∪Vi.

We say a map from a smooth manifold M to V is smooth if, locally on M, it comes froma smooth map to one of the Vi. Let C∞

M(V) denote the sheaf on M of smooth maps toV; then

C∞M(V) = colim C∞

M(Vi).

This identification uses the fact that the maps in our directed system are injective.Recall also that the colimit in the category of sheaves is defined to be the sheafificationof the colimit in the category of presheaves.

Now, we define the flat connection ∇M,V to be the map of sheaves

∇M,V : C∞M(V)→ Ω1

M(V)

which arises as the colimit of the maps of sheaves

C∞M(Vi)→ Ω1

M(Vi).

A.1.2.4 Definition. A sequence 0 → A → B → C → 0 of differentiable spaces is exact if itis exact as a sequence of ordinary vector spaces, A ⊂ B is a differentiable subspace, and C is adifferentiable quotient.

Equivalently, the sequence is exact if A is the kernel of the map B → C and C is thecokernel of the map A→ B.

Let V be a differentiable vector space. By evaluating V on open subsets of Rn, Vbecomes a sheaf on Rn. We can thus define the stalk

Stalkn(V) = colim0∈U⊂Rn V(U)


of V at the origin in Rn. The colimit above is taken over open subsets of Rn containingthe origin.

Note that the stalk of V at a point in any manifold can be defined in the same way,but the stalk at a point in a n-dimensional manifold is the same as the stalk at theorigin in Rn.

A.1.2.5 Lemma. A sequence of differentiable vector spaces 0→ A→ B→ C → 0 is exact ifand only if, for all n, the sequence

0→ Stalkn(A)→ Stalkn(B)→ Stalkn(C)→ 0

of vector spaces is exact.

PROOF. Suppose 0→ A → B → C → 0 is an exact sequence of differentiable vec-tor spaces. Then, for any manifold M, the sequence 0 → C∞(M, A) → C∞(M, B) →C∞(M, C) is exact. Further, a map M→ C is smooth if locally on M it lifts to a smoothmap to B. Thus, if C∞

M(B) denotes the sheaf on M of smooth maps to B, the sequence

(†) 0→ C∞M(A)→ C∞

M(B)→ C∞M(C)→ 0

of sheaves on M is exact. This implies that the corresponding sequence on stalks isexact.

Conversely, if the sequence of stalks is exact for all n, then the sequence (†) ofsheaves is exact for all manifolds M. This implies that the sequence

0→ Ω1M(A)→ Ω1

M(B)→ Ω1M(C)→ 0

is also exact, where we define

Ω1M(A) = Ω1

M ⊗C∞M

C∞M(A).

It follows that the connection ∇M,C on C is the unique connection which descendsfrom that on B, and that the connection on A is the restriction of the connection on Bto A. These are the conditions we imposed for A to be a differentiable subspace of Band for C to be a differentiable quotient.

A.1.3. The multicategory structure. We will consider the category of differen-tiable vector spaces as a multicategory, instead of a symmetric monoidal category.


A.1.3.1 Definition. If V1, . . . , Vk, W are differentiable vector spaces, then a smooth multilin-ear map

φ : V1 × · · · ×Vk →W

is a multilinear map with the following properties.

(1) It is smooth: if fi : M→ Vi are smooth maps from a manifold M, then

φ( f1, . . . , fk) : M→W

is a smooth map.(2) It is compatible with flat connections: if fi : M → V are smooth and if X is a vector

field on M, then

∇Xφ( f1, . . . , fk) = ∑i

φ( f1, . . . ,∇X fi, . . . , fk).

Differentiable vector spaces form a multicategory where the multi-morphisms Hom(V1, . . . Vk; W)

are smooth multilinear maps.

A.1.4. In general, we can not tensor differentiable vector spaces together. Nonethe-less, certain tensor products arise naturally and we will use them repeatedly.

First, we can tensor a differentiable vector space V with the algebra of smoothfunctions on a manifold: we use the notation

C∞(M)⊗V = C∞(M, V)

when V is a differentiable vector space.

Similarly, if E is a vector bundle on M, then C∞(M, E) is a projective module overC∞(M). Thus, it’s reasonable to form the algebraic tensor product

C∞(M, E)⊗C∞(M) C∞(M, V).

We interpret the output as a kind of completed tensor product of V with C∞(M, E).We will use the notation C∞(M, E ⊗ V) to denote this tensor product. This notationis natural: we can tensor a differentiable vector space with a finite-dimensional vectorspace, so that E⊗ V can be thought of as a bundle of differentiable vector spaces onM.


A.1.4.1 Lemma. Let E, F be vector bundles on M. Let D : C∞(M, E) → C∞(M, F) be adifferential operator. Let V be a differentiable vector space. Then, the map

D⊗ 1 : C∞(M, E)⊗alg V → C∞(M, F)⊗alg V

extends canonically to a map

C∞(M, E⊗V)→ C∞(M, F⊗V).

PROOF. Let’s start with the case when E and F are trivial of rank 1. Then we areasserting that differential operators on M act naturally on C∞(M, V). This action arisesin the standard way from the connection∇M,V : C∞(M, V)→ Ω1(M, V) which we aregiven as part of the structure of a differentiable vector space.

In the case when E and F are non-trivial, one constructs the desired map in localtrivializations and then verifies that it is independent of the choice of local trivializa-tions. This is standard.

Here is a formal way to describe these structures. Let C denote the following sym-metric monoidal category: the objects are smooth manifolds M equipped with a vectorbundle E; a morphism (M, E) → (N, F) is a smooth map f : M → N together with adifferential operator C∞(M, f ∗F)→ C∞(M, E); and the tensor product is defined by

(M, E)⊗ (N, F) = (M× N, E F).

Then, the category of differentiable vector spaces DVS is tensored over Cop.

A.2. Differentiable vector spaces from sections of a vector bundle

In this section, we will describe various classes of sections of a vector bundle on amanifold M and show that each is equipped with a natural diffeology and flat connec-tion. There are various ways to express this construction; we begin in a more geometriclanguage and then rephrase in the language of topological vector spaces.

The most basic example is as follows. Let M be a manifold, and let E be a vectorbundle on M. Then the space

E = Γ(M, E)

of smooth sections of E is a differentiable vector space. We let

Ec = Γc(M, E)

A.2. DIFFERENTIABLE VECTOR SPACES FROM SECTIONS OF A VECTOR BUNDLE 219

be the space of compactly supported smooth sections of E on M.

A.2.0.2 Definition. Equip E with the diffeology where, for N a smooth manifold, a smoothmap f : N → E is a section of the pullback bundle π∗ME on N×M arising from the projectionmap πM : N ×M→ M.

Similarly, give Ec a diffeology by saying that a smooth map N → Ec is a section s of π∗MEon N × M with the property that the map Supp(s) → N is proper (where Supp(s) is theclosure of the locus on which s is non-vanishing).

Note that the spaces E and Ec are complete locally-convex topological vector spaces,using the standard topologies for these spaces. As is explained in [KM97], for exam-ple, one has a notion of smooth map N → V for any manifold N and for any suchtopological vector space V. Thus, V defines a diffeological space. The diffeologiesdescribed above on E and Ec arise from the standard topologies on these spaces.

Notice as well that C∞(N, E ) is the vector space given by the completed projectivetensor product C∞(N)⊗ E , which is their natural tensor product as nuclear spaces.

Next, we explain the flat connections on the spaces E and Ec.

A.2.0.3 Definition. Let N be a smooth manifold. Equip the pullback bundle π∗ME on N×Mwith the natural flat connection along the fibers of the projection map πM : N×M→ M. Wethus obtain a map

∇N,E : Γ(N ×M, π∗ME)→ Γ(N ×M, T∗N E)

or, equivalently, a map∇N,E : C∞(N, E )→ Ω1(N, E ).

This defines the flat connection on C∞(N, E ) and so gives E the structure of a differentiablevector space.

This flat connection preserves the subspace C∞(N, Ec), giving Ec the structure of a differ-entiable vector space.

A.2.1. We are also interested in distributional sections of a vector bundle E. LetD(M) denote the space of distributions on M, that is, the continuous dual of the spaceC∞

c (M). Let Dc(M) denote the space of compactly supported distributions on M,which is the continuous dual of C∞(M).


We letE (M) = E (M)⊗C∞(M) D(M)

be the space of distributional sections of E. (These are sections whose coefficients aredistributions rather than functions.) We let

E c(M) = Ec(M)⊗C∞c (M) Dc(M)

be the space of compactly supported distributional sections of E.

A.2.1.1 Definition. Equip the space D(M) of distributions on M with the diffeology wherea smooth map N → D(M) is a continuous linear map C∞

c (M) → C∞(N). Similarly, giveDc(M) a diffeology by saying that a smooth map N → Dc(M) is a continuous linear mapC∞(M)→ C∞(N).

Equip E (M) with the diffeology in which the vector space of smooth maps N → E (M) is

C∞(N, E (M)) = C∞(N, E (M))⊗C∞(M×N) C∞(N,D(M))

(where we use the notation C∞(N, V) to indicate the space of smooth maps from N to a diffeo-logical vector space V).

Similarly, give E c(M) a diffeology by saying that

C∞(N, E c(M)) = C∞(N, Ec(M))⊗C∞(N,C∞c (M)) C∞(N,Dc(M)).

Remark: The diffeologies we have defined on these spaces of distributions again arisefrom the standard topology on these spaces. In particular, note that

C∞(N,D(M)) = C∞(N)⊗ D(M)

as vector spaces, where ⊗ denotes the completed projective tensor product. Comparethis to the analogous definition of C∞(N, C∞(M)) from earlier.

We need to equip these diffeological vector spaces with flat connections to makethem into differentiable vector spaces. There is a natural choice.

A.2.1.2 Definition. We extend the diffeological vector space D(M) to a differentiable vectorspace as follows. There is a natural inclusion

C∞(N,D(M)) → D(N ×M).

The Lie algebra Vect(N) of vector fields on N acts naturally on D(N ×M); this action pre-serves the subspace C∞(N,D(M)), giving this subspace the desired flat connection.

A.3. DIFFERENTIABLE COCHAIN COMPLEXES 221

The action of Vect(N) also preserves the smaller subspace

C∞(N,Dc(M)) ⊂ D(N ×M)

and so gives C∞(N,Dc(M)) the structure of differentiable vector space.

Similarly, the space D(N × M, π∗ME) of distributional sections on N × M of the vec-tor bundle π∗ME has a natural action of vector fields on M. This preserves the subspacesC∞(N, E(M)) and C∞(N, Ec(M)), and gives those the structure of differentiable vector spaces.

Let E! denote the vector bundle E⊗DensM, where DensM denotes the bundle ofdensities on M. Then, as above, we can define vector spaces

E ! = Γ(M, E!)

E !c = Γc(M, E!).

The vector spaces have natural diffeologies and topologies. There are natural identifi-cations

E (N) = Homcont(E!

c , C∞(N))

E c(N) = Homcont(E!, C∞(N))

where Homcont denotes the vector space of continuous linear maps.

A.3. Differentiable cochain complexes

A.3.0.3 Definition. A differentiable cochain complex is a cochain complex V where each Vi

is a differentiable vector space and each differential d : Vi → Vi+1 is a smooth map.

A map of differentiable cochain complexes is simply a cochain map V → W whose con-stituent maps Vi →W i are all smooth.

A cochain homotopy of such maps is a cochain homotopy whose constituent maps Vi →W i−1 are all smooth.

We want to perform standard constructions from homological algebra with differ-entiable cochain complexes. Of course, we need to make sure the definitions take intoaccount the diffeological structure.


A.3.0.4 Definition. (1) A sequence 0 → A → B → C → 0 of differentiable cochaincomplexes is exact if the component sequences 0→ Ai → Bi → Ci → 0 are exact.

(2) A map f : A→ B of differentiable cochain complexes is a cofibration if it fits into an

exact sequences 0→ Af→ B→ C → 0.

(3) Similarly, a map f : A → B is a fibration if it fits into an exact sequence 0 → C →A

f→ B→ 0.

Note that A→ B is a cofibration (respectively, fibration) if and only if, for all n, themap Stalkn(A) → Stalkn(B) is an injective (respectively, surjective) map of cochaincomplexes. Equivalently, the map A → B is a cofibration if in each cohomologicaldegree, Ai is a differentiable subspace of Bi. This means that Ai → Bi is injectiveand that a map M → Ai is smooth if and only if the composed map to Bi is smooth.Similarly, A → B is a fibration if and only if every Ai → Bi is surjective and a mapM→ Bi is smooth if and only if it locally lifts to a map to Ai.

We have seen above that one can take kernels and cokernels in the category ofdifferentiable vector spaces. Thus, we can define the cohomology groups Hi(A) ofany differentiable cochain complex. These are differentiable vector spaces.

A.3.0.5 Definition. A map A → B is a weak equivalence if and only if, for all n, the map ofcochain complexes Stalkn A→ Stalkn B is a quasi-isomorphism.

Standard constructions and lemmas from ordinary homological algebra hold inthis setting. For example, if f : V → W is a map of differentiable cochain complexes,we can form the cone Cone( f ), whose underlying graded differentiable space is V[1]⊕W, but equipped with differential (

dV[1] f0 dW

).

If f is a fibration, then the map

Ker f [1]→ Cone( f )

is an equivalence. If f is a cofibration, then the map

Cone( f )→ Coker( f )

is an equivalence.

A.3. DIFFERENTIABLE COCHAIN COMPLEXES 223

A.3.1. We will often use versions of spectral sequence arguments in the categoryof differentiable complexes.

Suppose that Vi is a directed system of differentiable cochain complexes indexedby i ∈ Z≥0. Thus, we have maps fi : Vi → Vi+1.

Let us suppose that the maps fi are cofibrations. Then, since the category of dif-ferentiable vector spaces is closed under colimits of cofibrant maps, we can form thedifferentiable cochain complex colimi V, which in cohomological degree k is colimi Vk

i .

A.3.1.1 Lemma. Let V∗, W∗ be sequential directed systems where the maps Vi → Vi+1, Wi →Wi+1 are cofibrations. Let V∗ →W∗ be a map of directed systems.

Suppose that the maps Vi/Vi−1 →Wi/Wi−1 are all weak equivalences.

Then the map

colim Vi → colim Wj


PROOF. We need to verify that the maps are equivalences at the level of stalks. Theforgetful functors

Stalkn : DVS→ Vect

commute with all colimits of cofibrations. It follows that, in the situation above,

colim Stalkn Vi = Stalkn colim Vi.

Stalkn(Vi/Vi−1) = Stalkn Vi/ Stalkn Vi−1.

Now, Stalkn Vi is a directed system of cochain complexes where the maps are injective,and likewise for Stalkn Wi. Therefore, by the usual spectral sequence argument, if themap

Stalkn Vi/ Stalkn Vi−1 → Stalkn Wi Stalkn Wi−1

is a quasi-isomorphism for all n, the map

colim Stalkn Vi → colim Stalkn Wi

is also a weak equivalence, giving the desired result.

Similarly, we have the following.


A.3.1.2 Lemma. Let V∗, W∗ be sequential directed systems of differentiable cochain com-plexes, where the maps Vi → Vi+1, Wi → Wi+1 are cofibrations. Let V∗ → W∗ be a mapof systems, such that the constituent maps Vi → Wi are quasi-isomorphisms. Then the mapcolim Vi → colim Wi is a quasi-isomorphism.

PROOF. The proof is almost identical to that of the previous lemma.

We have a similar statement for inverse systems, but only under some strongerhypothesis.

A.3.1.3 Lemma. Let V∗, W∗ be sequential inverse systems of differentiable cochain complexes.Thus, there are maps fi : Vi → Vi−1 and gi : Wi → Wi−1. Suppose that these maps arefibrations and that the systems Vi, Wi are eventually constant. In other words, the maps fi, gi

are isomorphisms for i sufficiently large.

Let V∗ →W∗ be a map of inverse systems, which induces a quasi-isomorphism

Ker fi → Ker gi

for each i.

Then the map lim←−V → lim←−W is a quasi-isomorphism.

PROOF. The functor of stalks commutes with finite limits of fibrations. The factthat the maps Vi → Vi−1 are isomorphisms for sufficiently large i ≥ N thus impliesthat

Stalkn lim←−Vi = lim←− Stalkn Vi.

Because the maps Vi → Vi−1 are fibrations, the maps Stalkn Vi → Stalkn Vj are sur-jective maps of cochain complexes. The spectral sequence argument implies that themap

lim←− Stalkn Vi → lim←− Stalkn Wi

is a quasi-isomorphism as desired.

A.3.2. With these definitions, we can define a factorization algebra valued in themulticategory of differentiable cochain complexes. Indeed, we have already given ageneral definition of factorization algebra valued in a multicategory. The definitionpresented here is just an exegesis of the general definition.

A.4. PRO-COCHAIN COMPLEXES 225

A.3.2.1 Definition. A prefactorization algebra on a manifold M valued in the multicategoryof differentiable cochain complexes is the assignment of a differentiable cochain complex F (U)

to every open subset U ⊂ M, together with smooth multilinear cochain maps

F (U1)× · · · × F (Un)→ F (V)

if U1, . . . , Un are disjoint open subsets of V, and satisfying the coherence axioms explainedearlier.

Given any such prefactorization algebra F and any factorizing cover

U = Ui | i ∈ I

of an open set V ⊂ M, we can form the Cech complex

C(U,F ).

As usual, this is a direct sum

C(U,F ) = ⊕α1,...,αk⊂PIF (Uα1 ∩ · · · ∩Uαn)[k− 1].

Here, PI indicates the set of finite subsets α ⊂ I with the property that the open setsUi for i ∈ α are disjoint. The set Uα indicates the disjoint union of the Ui for i ∈ α.

Since differentiable cochain complexes admit all coproducts, this Cech complex isagain a differentiable cochain complex.

A.3.2.2 Definition. A differentiable factorization algebra on M is a differentiable prefactor-ization algebra F on M with the property that, for every factorizing cover U of an open subsetV ⊂ M, the map

C(U,F )→ F (V)

is a weak equivalence of differentiable cochain complex (as defined above).

A.4. Pro-cochain complexes

This section involves ordinary vector spaces, not differentiable vector spaces, butit prepares us for an important notion we use throughout the book.

Most of the examples of factorization algebras we construct will take values not inthe category of ordinary cochain complexes but in the category of pro-cochain com-plexes or, equivalently, of complete filtered cochain complexes.


A.4.0.3 Definition. A complete filtered cochain complex is a cochain complex V equippedwith a decreasing filtration FiV ⊂ V by sub-cochain complexes indexed by i ∈ Z≥0, such thatF0V = V and

V = lim←−V/FiV.

A map of complete filtered cochain complexes is a map V →W which preserves the filtration.

Such a map is a weak equivalence if the map Gri V → Gri W is a quasi-isomorphism forall i. (Note that this implies that the map V →W is a quasi-isomorphism.)

Some care is needed when defining colimits of complete filtered cochain com-plexes.

A.4.0.4 Definition. Let Vα | α ∈ A be a collection of complete filtered cochain complexesindexed by some set A. Then the direct sum

⊕α∈A Vα is defined by

⊕α∈A

Vα = lim←−i∈Z≥0

(⊕α

Vα/FiVα

).

(On the right hand side of this equation,⊕

Vα/FiVα indicates the ordinary direct sum ofcochain complexes.)

The reason for making this definition is that the filtration on the naive direct sumof the Vα is not complete. It is easy to verify that the direct sum defined above is acoproduct in the category of complete filtered cochain complexes.

Similarly, the tensor product of complete filtered cochain complexes needs to becompleted.

A.4.0.5 Definition. Let V, W be complete filtered cochain complexes. The tensor productV ⊗W is defined as the limit

V ⊗W = lim←−i,j

(V/FiV)⊗ (V/FjW).

The filtration on V ⊗W is defined by

Fk(V ⊗W) = colimi+j≥k

FiV ⊗ FjW,

where the tensor product FiV ⊗ FjW is defined as the limit of FiV/FrV ⊗ FjW/FsV.

A.5. DIFFERENTIABLE PRO-COCHAIN COMPLEXES 227

Again, the reason for this definition is that the filtration on the naive tensor productof V ⊗W is not complete.

With this definition of completed direct sum and tensor product, it is straightfor-ward to modify our definition of factorization algebra to take values in the categoryof complete filtered cochain complexes.

A.4.0.6 Definition. A complete filtered factorization algebra F on X is a prefactorizationalgebra F taking values in the symmetric monoidal category of complete filtered cochain com-plexes, using the tensor product described above, such that, for every factorizing open cover Uof an open subset U of X, the map

C(U,F )→ F (U)

is an equivalence. The direct sums and tensor products appearing in the definition of the Cechcomplex are completed, as above.

A.5. Differentiable pro-cochain complexes

As a final elaboration on the concept of cochain complex, we will put together thetwo ideas described above.

A.5.0.7 Definition. A differentiable pro-cochain complex is a differentiable cochain complexV equipped with a decreasing filtration by differentiable subcomplexes FiV with the followingproperties.

(1) F0V = V.(2) The maps FiV → FjV if i > j are cofibrations. This means that they are injective

and that in each cohomological degree, the map FiVk → FjVk has the property thata map M→ FjVk is smooth if it lifts to a smooth map to FiVk.

This implies that we can form the quotient differentiable vector space V/FiV,and that the maps

V/FiV → V/FjV

are cofibrations.(3) We require that

V = lim←−V/FiV.


A map of differentiable pro-cochain complexes is a filtration-preserving map V → W.Such a map is a weak equivalence if the maps Gri V → Gri W are weak equivalences of differ-entiable cochain complexes. Note that this implies that the maps V/FiV →W/FiW are weakequivalences of differentiable cochain complexes.

A map V → W of differentiable pro-cochain complexes is a fibration (respectively, a cofi-bration) if the map V/FiV →W/FiW are fibrations (cofibrations) for all i.

As before, we need to define the completed direct sum and multilinear maps ofdifferentiable cochain complexes.

A.5.0.8 Definition. If Vα | α ∈ A is a collection of complete filtered differentiable cochaincomplexes, indexed by some set A, then the completed direct sum

⊕α∈AVα is defined to be the

inverse limit ⊕α∈A

Vα = lim←−i∈Z≥0

⊕α∈A

(Vα/FiVα

),

where on the right hand side we use the ordinary direct sum of differentiable spaces.

We can define the stalks of a differentiable pro-cochain complex Stalkn(V) as thecolimit

Stalkn(V) = colim0∈U⊂Rn

V(U),

where the colimit of the pro-cochain complexes V(U) is completed as above. Thus,Stalkn(V) is a pro-cochain complex, and

Stalkn(V)/Fi Stalkn(V) = Stalkn(V/FiV).

A.5.0.9 Lemma. A map V → W of differentiable pro-cochain complexes is an equivalence ifand only if the maps Stalkn(V)→ Stalkn(W) are weak equivalences of pro-cochain complexes.

PROOF. Immediate.

A.5.0.10 Lemma. Let V∗, W∗ be sequential directed systems of differentiable pro-cochain com-plexes, and let V∗ →W∗ be a map of directed systems.

Suppose that the maps Vi → Vj and Wi → Wj are all cofibrations and suppose that themaps Vi →Wi are all equivalences.

A.5. DIFFERENTIABLE PRO-COCHAIN COMPLEXES 229

Then the map

colim Vi → colim Wj

is an equivalence.

PROOF. The proof is almost identical to the proof of lemma A.3.1.1.

Similarly, we can have spectral sequences for inverse systems, but only undersome more restrictive hypotheses.

A.5.0.11 Lemma. Let V∗, W∗ be sequential inverse systems of differentiable pro-cochain com-plexes, and let V∗ →W∗ be a map of inverse systems. Let V = lim V∗ and W = lim W∗.

Suppose that

(1) The maps fi : Vi → Vi−1, gi : Wi → Wi−1 are fibrations of differentiable cochaincomplexes.

(2) For each k, the inverse systems V∗/FkV∗ and W∗/FkW∗ are eventually constant, asin lemma A.3.1.3.

(3) The maps Ker fi → Ker gi are quasi-isomorphisms of differentiable cochain com-plexes.

Then, the map V →W is a quasi-isomorphism of differentiable pro-cochain complexes.

PROOF. This follows immediately from lemma A.3.1.3.

A.5.1. Differentiable pro-cochain complexes form a multicategory, just like dif-ferentiable cochain complexes.

A.5.1.1 Definition. Let V1, . . . , Vk, W be differentiable pro-cochain complexes. In the mul-ticategory of differentiable pro-cochain complexes, an element of Hom(V1, . . . , Vk; W) is asmooth multilinear cochain map

Φ : V1 × · · · ×Vk →W = lim W/FiW

which preserves filtrations: if vi ∈ Fri(Vi), then

Φ(v1, . . . , vk) ∈ Fr1+···+rkW.


Factorization algebras valued in differentiable pro-cochain complexes are definedas before.

A.6. Differentiable cochain complexes over a differentiable dg ring

The category of differentiable cochain complexes is a differential graded multi-category. Thus, we can talk about commutative differentiable dg algebras R. This isjust a commutative dg algebra R, with the structure of a differentiable vector space,such that all the structure maps are smooth. Similarly, a commutative differentiablepro-algebra is a commutative dg algebra in the multi-category of differentiable pro-cochain complexes.

In either context, we can define an R-module M to be a differentiable (pro-)cochaincomplex equipped with an action of the commutative differentiable (pro-)algebra R,in the obvious way. We say a map M → M′ is a weak equivalence if it is a weakequivalence (as defined above) in the category of differentiable (pro-)cochain complex.

In either context, we say a sequence of R-modules 0 → M1 → M2 → M3 → 0is exact if it is exact in the category of differentiable (pro-)cochain complexes. A mapM1 → M2 is a cofibration if it can be extended to an exact sequence 0→ M1 → M2 →M3 → 0; it is a fibration if it can be extended to an exact sequence 0 → M3 → M1 →M2 → 0.

The category of modules over a differentiable (pro-)dg algebra R is, as above, mul-ticategory. In either case, the multi-maps

HomR(M1, . . . , Mn; N)

are the multi-maps in the category of differentiable (pro-)cochain complexes whoseunderlying multilinear map M1 × · · · ×Mn → N are R-multilinear.

A.7. Classes of functions on the space of sections of a vector bundle

Let M be a manifold and E a graded vector bundle on M. Let U ⊂ M be anopen subset. In this section we will introduce some notation for various classes offunctionals on sections E (U) of E on U. These spaces of functionals will all be gradeddifferentiable pro-vector spaces.

A.7. CLASSES OF FUNCTIONS ON THE SPACE OF SECTIONS OF A VECTOR BUNDLE 231

A.7.1. We are interested in symmetric algebras on vector spaces of the form E!c(U),

E !c (U), etc. These symmetric algebras can be defined in two ways: either using the

completed projective tensor product of topological vector spaces, or in terms of sec-tions of bundles on Un. We will explain both points of view.

Thus, let us first define (E (U))⊗n to be the tensor power defined using the com-pleted projective tensor product on the topological vector space E (U). Then a moreconcrete description of this space is as follows. Let En denotes the vector bundle onMn obtained as the external tensor product, so

(E (U))⊗n = Γ(Un, En)

is the space of smooth sections of En on Un. Similarly, we can identify

(Ec(U))⊗n = Γc(Un, En)

(E c(U))⊗n = Γc(Un, En)

(E (U))⊗n = Γ(Un, En)

where Γ indicates the space of distributional sections and the subscript c indicatescompactly supported distributional sections.

We have already seen how to equip the various kinds of spaces of sections of avector bundle with the structure of a differentiable vector space. Since the spaceslisted above are expressed as sections of various kinds of a vector bundle on Un, theyall have the structure of differentiable vector spaces.

Symmetric (or exterior) powers of the spaces Ec(U), E c(U), E (U), E (U) are de-fined by taking coinvariants of the tensor powers defined above with respect to theaction of the symmetric group. These symmetric powers inherit the structure of dif-ferentiable vector space.

Thus, we can define, for example, the completed symmetric algebra

Sym∗E !

c (U) = ∏n

Symn E !c (U)

Sym∗E

!c(U) = ∏

nSymn E

!c(U)


Note that since E!c(U) is dual to E (U), we can view Sym E

!c(U) as the algebra of formal

power series on E (U). Thus, we often write

Sym E!c(U) = O(E (U)).

Similarly, Sym E !c (U) is the algebra of formal power series on E (U).

In a similar way, we can construct

O(E (U)) = ∏n

Symn(E !c (U))

O(E c(U)) = ∏n

Symn(E !(U)).

These spaces of functionals are all products of the differentiable vector spaces ofsymmetric powers, and so they are themselves differentiable vector spaces. We willequip all of these spaces of functionals with the structure of a differentiable pro-vectorspace, induced by the filtration

FiO(E (U)) = ∏n≥i

Symi E!c(U)

(and similarly for O(Ec(U)), O(E (U)) and O(E c(U))).

The natural product O(E (U)) is compatible with the differentiable structure, mak-ing O(E (U)) into a commutative algebra in the multicategory of differentiable gradedpro-vector spaces. The same holds for the spaces of functionals O(Ec(U)), O(E (U))

and O(E c(U)).

A.7.2. One-forms. Recall that if V is a vector space, we can define the space ofone-forms on V (treated as formal scheme) as

Ω1(V) = O(V)⊗V∨.

Similarly, we can define

Ω1(E (U)) = O(E (U))⊗ E!c(U),

where ⊗ denotes the completed projective tensor product.

In concrete terms,

Ω1(E (U)) = ∏n

Symn(E!c(U))⊗ E

!c(U)


and we can identify the space

Symn(E!c(U))⊗ E

!c(U)) ⊂ E

!c(U)⊗n+1 = Γc(Un+1, (E!)n+1)

as the space of compactly supported distributional sections of (E!)n+1 that are sym-metric in the first n variables.

In this way, Ω1(E (U)) becomes a differentiable pro-cochain complex, where thefiltration is defined by

FiΩ1(E (U)) = ∏n≥i−1

Symn(E!c(U))⊗ E

!c(U).

Further, Ω1(E (U)) is a module for the commutative algebra O(E (U)), where the mod-ule structure is defined in the multicategory of differentiable pro-vector spaces.

If V is a finite-dimensional vector space, the exterior derivative map

d : O(V)→ O(V)⊗V∨

is, in components, just the composition

Symn+1 V∨ → (V∨)⊗n+1 → Symn(V∨)⊗V∨

where the maps are the inclusion followed by the natural projection (up to an overallcombinatorial constant).

We can, in a similar way, define the exterior derivative

d : O(E (U))→ Ω1(E (U))

by saying that on components it is given by the same formula as in the finite-dimensionalcase.

A.7.3. Other classes of sections of a vector bundle. Before we introduce our nextclass of functionals — those with proper support — we need to introduce some furthernotation concerning classes of sections of a vector bundle.

Let M be a manifold, and let f : M → N be a fibration. Let E be a vector bundleon M. We say a section s ∈ Γ(M, E) has relative compact support if the map

f : Supp(s)→ N

is proper. We let Γc/ f (M, E) denote the space of sections with relative compact sup-port. This is a differentiable vector space: if X is an auxiliary manifold, a smooth map


X → Γc/ f (M, E) is a section of the bundle π∗ME on X×M which has relative compactsupport relative to the map

M× X → N × X.

(It is straightforward to write down a flat connection on C∞(X, Γc/ f (M, E)), usingarguments of the type described in section A.2.)

Next, we need to consider spaces of the form E (M)⊗F (N), where M and N aremanifolds and E, F are vector bundles on M and N respectively. Of course, we cangive an abstract definition using the projective tensor product, but we want a moregeometric interpretation.

There are several ways to identify this space geometrically. We will view E (M)⊗F (N) as a subspace

E (M)⊗F (N) ⊂ E (M)⊗F (N).

It consists of those elements D with the property that, if φ ∈ E !c (M), then map

D(φ) : F !c(N) → R

ψ 7→ D(φ⊗ ψ)

comes from an element of F (N).

Alternatively, E (M)⊗ F (N) is the space of continuous linear maps from E !c (M)

to F (N).

We can similarly define E c(M)⊗F (N) as the subspace of those elements of E (M)⊗F (N) that have compact support relative to the projection M× N → N.

These spaces form differentiable vector spaces in a natural way: a smooth mapfrom an auxiliary manifold X to E (M) ⊗ F (N) is an element of E (N) ⊗ F (N) ⊗C∞(X). Similarly, a smooth map to E c(M)⊗F (N) is an element of E (M)⊗F (N)⊗C∞(X) whose support is compact relative to the map M× N × X → N × X.

A.7.4. Functions with proper support. Recall that

Ω1(Ec(U)) = O(Ec(U))⊗ E!(U).

We can thus define a subspace

O(E (U))⊗ E!(U) ⊂ Ω1(Ec(U)).


The Taylor components of elements of this subspace are in the space

Symn(E!c(U))⊗ E

!(U),

which in concrete terms is the Sn invariants of

E!c(U)⊗n ⊗ E

!(U).

A.7.4.1 Definition. A function Φ ∈ O(Ec(U)) has proper support if

dΦ ∈ O(E (U))⊗ E!(U) ⊂ O(Ec(U))⊗ E

!(U)∨.

The reason for the terminology is as follows. Let Φ ∈ O(Ec(U)) and let

Φn ∈ Hom(Ec(U)⊗n, R)

be the nth term in the Taylor expansion of Φ.

Then, Φ has proper support if and only if, for all n, the composition with a projec-tion map

Supp(Φn) ⊂ Un → Un−1

is proper.

We will let

OP(Ec(U)) ⊂ O(Ec(U))

be the subspace of functions with proper support. Note that functions with propersupport are not a subalgebra.

Because OP(Ec(U)) fits into a fiber square

OP(Ec(U)) → O(E (U))⊗ Ec(U)∨

↓ ↓O(Ec(U)) → O(Ec(U))⊗ Ec(U)∨

it has a natural structure of a differentiable pro-vector space.

A.7.5. Functions with smooth first derivative.

A.7.5.1 Definition. A function Φ ∈ O(Ec(U)) has smooth first derivative if dΦ, which isa priori an element of

Ω1(Ec(U)) = O(Ec(U))⊗ E!(U)


is an element of the subspace

O(Ec(U))⊗ E !(U).

Note that we can identify, concretely, O(Ec(U))⊗ E !(U) with the space

∏n

Symn E!(U)⊗ E !(U)

and

Symn E!(U)⊗ E !(U) ⊂ E

!(U)⊗n ⊗ E !(U).

(Spaces of the form E (U)⊗ E (U) were described concretely above.)

Thus O(Ec(U)) ⊗ E !(U) is a differentiable pro-vector space. It follows that thespace of functionals with smooth first derivative is a differentiable pro-vector space,since it is defined by a fiber diagram of such objects.

An even more concrete description of the space O sm(Ec(U)) of functionals withsmooth first derivative is as follows.

A.7.5.2 Lemma. A functional Φ ∈ O(Ec(U)) has smooth first derivative if each of its Taylorcomponents

DnΦ ∈ Symn E!(U) ⊂ E

!(U)⊗n

lies in the intersection of all the subspaces

E!(U)⊗k ⊗ E !(U)⊗ E

!(U)⊗n−k−1

for 0 ≤ k ≤ n− 1.

PROOF. The proof is a simple calculation.

Note that the space of functions with smooth first derivative is a subalgebra ofO(Ec(U)). We will denote this subalgebra by O sm(Ec(U)). Again, the space of func-tions with smooth first derivative is a differentiable pro-vector space, as it is definedas a fiber product.

We can also define the space of functions on E (U) with smooth first derivative, byrequiring that the exterior derivative lies in

O(E (U))⊗ E !c (U) ⊂ Ω1(E (U)).

A.8. DERIVATIONS 237

A.7.6. Functions with smooth first derivative and proper support. We are par-ticularly interested in those functions which have both smooth first derivative andproper support. We will refer to this subspace as OP,sm(Ec(U)). The differentiablestructure on OP,sm(Ec(U)) is, again, given by viewing it as defined by the fiber dia-gram

OP,sm(Ec(U)) → O(E (U))⊗ E !(U)

↓ ↓O(Ec(U)) → O(Ec(U))⊗ E

!(U)

We have inclusions

O sm(E (U)) ⊂ OP,sm(Ec(U)) ⊂ O sm(Ec(U)),

where each inclusion has dense image.

A.8. Derivations

As before, let M be a manifold, E a graded vector bundle on M, and U an opensubset of M. In this section we will define derivations of algebras of functions onE (U).

To start with, recall that, for V a finite dimensional vector space (which we treatas a formal scheme) and O(V) = ∏ Symn V∨ the algebra of formal power series onV, we identify the space of continuous derivations of O(V) with O(V)⊗V. We viewthese derivations as the space of vector fields on V and use the notation Vect(V).

In a similar way, we can define the space of vector fields Vect(E (U)) of vectorfields on E (U) as

Vect(E (U)) = O(E (U))⊗ E (U) = ∏n

(Symn(E

!c(U))⊗ E (U)

),

using the completed projective tensor product. We have already seen (section A.7)how to define the structure of diffeological pro-vector space on spaces of this nature.

In concrete terms, the Taylor expansion of an element of X ∈ Vect(E (U)) is givenby a sequence of continuous symmetric multilinear maps

DnX : E (U)× · · · × E (U)→ E (U).


More generally, if M is a smooth manifold and if X : M → Vect(E (U)) is a smoothmap, then the Taylor expansion of X is a sequence of continuous symmetric multilin-ear maps

E (U)× · · · × E (U)→ E (U)⊗ C∞(M) = Γ(U ×M, E∣∣U).

In this section we will show the following.

A.8.0.1 Proposition. Vect(E (U)) has a natural structure of Lie algebra in the multicate-gory of diffeological pro-vector spaces. Further, O(E (U)) has an action of the Lie algebraVect(E (U)) by derivations, where the structure map Vect(E (U))×O(E (U))→ O(E (U))

is smooth.

PROOF. To start with, let’s look at the case of a finite-dimensional vector space V,to get an explicit formula for the Lie bracket on Vect(V), and the action of Vect(V) onO(V). Then, we will see that these formulae make sense when V = E (U).

Let X ∈ Vect(X), and let us consider the Taylor components DnX, which are mul-tilinear maps

V × · · · ×V → V.

Our conventions are such that

Dn(X)(v1, . . . , vn) =

(∂

∂v1. . .

∂

∂vnX)(0) ∈ V

Here, we are differentiating vector fields on V using the trivialization of the tangentbundle to this formal scheme arising from the linear structure.

Thus, we can view DnX as in the endomorphism operad of the vector space V.

If A : V×n → V and B : V×m → V, let us define

A i B(v1, . . . , vn+m−1) = A(v1, . . . , vi−1, B(vi, . . . , vi+m−1), vi+m, . . . , vn+m−1).

If A, B are symmetric (under Sn and Sm, respectively), then define

A B =n

∑i=1

A i B.

Then, if X, Y are vector fields, the Taylor components of [X, Y] satisfy

Dn([X, Y]) = ∑k+l=n+1

ck,l (DkX DlY− DlY DkX)

where ck,l are combinatorial constants which are irrelevant for our purposes.

A.8. DERIVATIONS 239

Similarly, if f ∈ O(V), the Taylor components of f are multilinear maps

Dn f : V×n → C.

In a similar way, if X is a vector field, we have

Dn(X f ) = ∑k+l=n+1

c′k,l Dk(X) Dk( f ).

Thus, we see that in order to define the Lie bracket on Vect(E (U)), we need to givemaps of diffeological vector spaces

i : Hom(E (U)⊗n, E (U))×Hom(E (U)⊗m, E (U))→ Hom(E (U)⊗(n+m−1), E (U))

where here Hom indicates the space of continuous linear maps, treated as a diffeolog-ical vector space. Similarly, to define the action of Vect(E (U)) on O(E (U)), we needto define a composition map

i : Hom(E (U)⊗n, E (U))×Hom(E (U)⊗m)→ Hom(E (U)⊗n+m−1).

We will treat the first case; the second is similar.

Now, if X is an auxiliary manifold, a smooth map

X → Hom(E (U)⊗m, E (U))

is the same as a continuous multilinear map

E (U)×m → E (U)⊗ C∞(X).

Here, “continuous” means for the product topology.

This is the same thing as a continuous C∞(X)-multilinear map

Φ : (E (U)⊗ C∞(X))×m → E (U)⊗ C∞(X).

IfΨ : (E (U)⊗ C∞(X))×n → E (U)⊗ C∞(X).

is another such map, then it is easy to define Φ i Ψ by the usual formula:

Φ i Ψ(v1, . . . , vn+m−1) = Φ(v1, . . . vi−1, Ψi(vi, . . . , vm+i−1), . . . , vn+m−1)

if vi ∈ E (U)⊗ C∞(X). This map is C∞(X) linear.

Remark: It is not hard to show that Vect(E (U)), as defined above, is the space of allcontinuous derivations of the topological algebra O(E (U)); we will not need this fact.


A.9. The Atiyah-Bott lemma

In [AB67], Atiyah and Bott showed that for an elliptic complex (E , d) on a compactclosed manifold M, with E the smooth sections of a Z-graded vector bundle, there is ahomotopy equivalence (E , d) → (E , d) into the elliptic complex of distributional sec-tions. The argument follows from the existence of parametrices for elliptic operators.This result was generalized by N.N. Tarkhanov to the non-compact case [Tar87].

Let M be a smooth manifold (which, in general, will not be compact).

A.9.0.2 Definition. An elliptic complex on M is a graded vector bundle E, whose space ofsmooth sections we denote by E , together with a square-zero differential operator Q : E → E

of cohomological degree 1 possessing the following property, known as ellipticity. Let π∗Edenote the pullback bundle along the projection map for the cotangent bundle π : T∗M→ M.The symbol σ(Q) of Q is a cohomological degree 1 endomorphism of the vector bundle π∗E.We require that the complex of vector bundles (π∗E, σ(Q)) on T∗M is exact away from thezero section.

Let E denote the complex of distributional sections of E . We will endow both E

and E with their natural topologies.

A.9.0.3 Lemma (Tarkhanov, [Tar87]). There is a continuous homotopy inverse to the natu-ral inclusion

(E , Q) → (E , Q).

This is Lemma 1.7 of [Tar87]. The continuous homotopy inverse

Φ : E → E

is given by a kernel KΦ ∈ E ! ⊗ E with proper support. The homotopy S : E → E is acontinuous linear map with

[d, S] = Φ− Id .

The kernel KS for S is a distribution, that is, an element of E!⊗ E , with proper support.

PROOF. We will reproduce the proof in [Tar87]. Choose a metric on E (Hermitian,if E is a complex vector bundle) and a volume form on M. Let Q∗ be the adjoint toQ. We form the graded commutator D = [Q, Q∗]. This is an elliptic operator on eachspace E i, the cohomological degree i part of E . Thus, by standard results in the theory

A.9. THE ATIYAH-BOTT LEMMA 241

of pseudodifferential operators, there is a parametrix P for D. The kernel KP is anelement of E

! ⊗ E , and the corresponding operator P : Ec → E is an inverse for D upto smoothing operators.

By multiplying KP by a smooth function on M×M which is 1 in a neighborhoodof the diagonal, we can assume that KP has proper support. This means that P willextend to a map E → E and will still be a parametrix: thus P D and D P both differfrom the identity by smoothing operators.

The homotopy S is now defined by

S = Q∗P.

Note that the homotopy inverse Φ : E → E and the homotopy S : E → E we haveconstructed are maps of differentiable vector spaces (as well as being continuous).

APPENDIX B

Extending a factorization algebra from a basis

In this appendix we will prove the following technical proposition, stated in sec-tion 3.6. The reader should refer to that section for the notation.

B.0.0.4 Proposition. Let U be a factorizing basis of a space X, closed under finite intersection.Let F be a U-factorization algebra, as defined in section 3.6. Let iU∗ (F ) be the prefactorizationalgebra which sends an open subset V ⊂ V to

iU∗ (F )(V) = C(UV ,F

where UV is the factorizing cover of V consisting of sets in U which are subsets of V.

Then, iU∗ (F ) is a factorization algebra whose restriction to open sets in the cover U isquasi-isomorphic to F .

PROOF. We need to check that if W is a factorizing cover of V ⊂ X, then

iU∗ (F )(V) ' C(W, iU∗ (F )).

Before we prove this, we need a lemma. Let UW be the cover of V consisting ofopen sets in U which are subordinate to W.

B.0.0.5 Lemma. For any U-prefactorization algebra F , the natural map

C(W, iU∗ (F ))→ C(UW,F )


PROOF. Before we check this, let us recall the notation we used when discussingCech complexes. Let PU denote the set of subsets α ⊂ U, where for each distincti, j ∈ α, Ui and Uj are disjoint. If α ∈ PU we will let

Uα = qi∈αUi.

243

244 B. EXTENDING A FACTORIZATION ALGEBRA FROM A BASIS

If α1, . . . , αk ∈ PU, we will let

F (α1, . . . , αk) = ⊕i1∈α1,...,ik∈αkF (Ui1 ∩ · · · ∩Uik).

With this notation, if W ⊂ M, then

iU∗ (F )(W) = ⊕α1,...,αr∈UWF (α1, . . . , αr)[r− 1]

where UW refers to the cover of W consisting of open sets in U which lie in W.

Let us define a filtration on iU∗ (F ) by saying that

FiiU∗ (F ) = ⊕r≤i ⊕α1,...,αr∈UW F (α1, . . . , αr)[r− 1].

This filters iU∗ (F ) as a prefactorization algebra.

There is a natural map

C(W, iU∗ (F ))→ C(UW,F ).

Let us filter C(W, iU∗ (F )) by the filtration coming from iU∗ (F ). Let us filter C(UW,F )in the same way that we filtered iU∗ (F ). The map preserves the filtration.

Thus, to prove that this map is a quasi-isomorphism, it suffices to show that it ison the associated graded.

The complex Grn C(W, iU∗ (F )) breaks up as a direct sum of pieces correspondingto tuples α1, . . . , αn ∈ PUW, as follows. If β ∈ PW and α ∈ PUW, say α ⊂ β if Uα ⊂ U′β.Then,

Grn C(W, iU∗ (F ))

=⊕

α1,...,αn∈PU

F (α1, . . . , αn)[n− 1]⊗

⊕β1,...,βm∈PWαi⊂β j all i,j

C · (β1, . . . , βk)

.

Here (β1, . . . , βk) denotes a vector in degree −k. This is a direct sum decomposition ofcochain complexes.

On the other hand,

Grn C(UW,F ) = ⊕α1,...,αn∈PUWF (α1, . . . , αn)

B. EXTENDING A FACTORIZATION ALGEBRA FROM A BASIS 245

Thus, to prove the lemma, we need to verify that the complex

⊕β1,...,βm∈PWαi⊂β j all i,j

C · (β1, . . . , βk)

has homology C if all αi ∈ PUW, and zero otherwise.

It is clear that the complex is zero if all αi are not in PUW. So let us assume that allαi are in PUW. Then, the complex is simply the simplicial chain complex on the infinitesimplex with vertices β ∈ PU such that ∪Uαi ⊂ Uβ. This is of course contractible.

It remains to shows that the natural map

C(UW,F )→ C(UV ,F )

is a quasi-isomorphism. (Here, as before, UV refers to the cover of V consisting of setsin U which lie in V).

To see that this map is a quasi-isomorphism, observe that by another applicationof the sub-lemma there is a quasi-isomorphism

C(U, iUW∗ (F )) ' C(UW,F ).

Here iUW∗ refers to the prefactorization algebra on V obtained by extending F , as be-fore, but now considered as a UW-factorization algebra.

Now the fact that F is a U-factorization algebra implies that, for all U ∈ U, thenatural map

C(UW ∩ UU ,F )→ F (U)


It follows that the natural map

C(U, iUW∗ (F ))→ C(U,F )

is a quasi-isomorphism, as desired.

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