Topological Strings and Quantum Black Holes · 2018-07-26 · LAHORE UNIVERSITY OF MANAGEMENT...

LAHORE UNIVERSITY OF MANAGEMENT SCIENCES

Topological Strings and Quantum Black

Holes

by

Asad Hussain

A thesis submitted in partial fulfillment for the

degree of Bachelor of Science

in the

Department of Physics

School Of Science & Engineering

July 2018

University Web Site URL Here (include http://)

Faculty Web Site URL Here (include http://)

Department or School Web Site URL Here (include http://)

LAHORE UNIVERSITY OF MANAGEMENT SCIENCES

Abstract

Department of Physics

School Of Science & Engineering

by Asad Hussain

Quantizing a Black hole, or atleast getting microstate counts within the domain of some theory

of quantum gravity is an important way to learn about the new theories of QG and test them.

One such ”quantization” is put forward by the OSV Conjecture, which conjectures a relation

of the type: ZBH = |Ztop|2. Here, ZBH stands for the partition function of 4-d BPS Black

Hole in Type IIB String theory compactified on a Calibi Yau, and Ztop is the Partition function

for the Topological String on a Calibi Yau target space with the CY moduli on the attractor

point of the N=2 SUGRA moduli fields. In fact, the OSV Conjecture goes on to assert that

Ztop is to be interpreted as the black hole wavefunction, Ψ(p + iφπ ,W2). Thus, if correct, we

have the interesting instance of the wavefunction of a BPS black hole (over the space of it’s

possible charges) in string theory. We will cover the necessary background to understand this

relation. This will include studying String Theory, Witten-type Topological QFTs, A/B Model

topological strings, Couplings of Higher derivative N=2 supergravity, and Special Geometry

University Web Site URL Here (include http://)

Faculty Web Site URL Here (include http://)

Department or School Web Site URL Here (include http://)

Contents

Abstract i

Preface 1

1 Introduction: The Problem with Black Hole Microstates 2

1.1 No Hair Theorem & Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Sigma Models: String Theory 4

2.1 Generalizing a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Relativistic Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 A String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Topological String Theory 9

3.1 Supersymmetric Quantum Mechanics: d = 0 + 1 Sigma Model on M . . . . . . . 9

3.1.1 SQM where M is a Riemannian Manifold . . . . . . . . . . . . . . . . . . 10

3.2 Cohomological Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 General Cohomological Field Theories . . . . . . . . . . . . . . . . . . . . 14

3.2.2 Descent Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.3 2d Cohomological Field Theories . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 N = (2, 2) SCFTs with Topological Twist . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1 An example: A-Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Ooguri, Strominger, Vafa (OSV) conjecture: ZBH = |Ztop|2 23

4.1 Macroscopic Entropy from Stabilization Equations . . . . . . . . . . . . . . . . . 23

4.1.1 Form of N = 2, d = 4 Supergravity . . . . . . . . . . . . . . . . . . . . . . 24

4.1.2 Attractor Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.3 Higher Order Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.4 Reinterpreting the Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.5 What’s the phase space? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.6 Comparing with Topological String Amplitudes . . . . . . . . . . . . . . . 28

A Appendix: Differential Forms & Action Principles 31

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A.2 Choosing from a space of trajectories . . . . . . . . . . . . . . . . . . . . . . . . . 31

ii

Contents iii

A.3 Mathematical Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

A.3.1 Some Differential Form preliminaries . . . . . . . . . . . . . . . . . . . . . 32

A.3.2 Inner Product on Differential Forms . . . . . . . . . . . . . . . . . . . . . 33

A.3.3 Variation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A.3.4 Deriving Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 36

A.3.5 Conjugate Momentum One form and the E-L equation in the form language 37

A.4 Application to known problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A.4.1 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Bibliography 41

Preface

Ever since Bekenstein-Hawking Entropy was discovered, we had a thermodynamic formula for

black hole entropy. But since entropy is proportional to the log of the micro-states, we should be

able to give a count of these micro-states of a black hole. Answers about Black Hole Statistical

Mechanics as opposed to thermodynamics are in the domain of some quantum theory of gravity.

We will be answering such question in the domain of string theory.

We will first foray through some Witten Type Topological Quantum Field Theories, and prove

that ”twisted” versions of N = (2,2) Supersymmetric sigma models (on Kahler Manifolds) follow

A & B Models of Witten Type TQFTs. We will prove this for the A model. One can then

promote the metric to a field that we will sum over in the path integral, and thus one will

have have created Topological String Theory, by essentially integrating over the moduli space

of Riemann Surfaces. This will define the Partition function for this theory.

We then use a result by I. Antoniadis, E. Gava, K. Narain and T. Taylor which relates the

scattering amplitudes of a Type II-B String compactified on a Calibi Yau manifold with the genus

g free energy of the topological string. These scattering amplitudes appear in the effective action

of this theory as the coupling constants which couple to the higher derivative terms containing

the Riemann Tensor. The full effective action then contains topological string amplitudes. Using

Wald’s method one can compute the entropy of a black hole solution in Type IIB Supergravity

subject to higher derivative corrections.

We will then discover how the formula for the entropy of the full higher derivative Type IIB

supergravity is reinterpreted by Ooguri, Strominger and Vafa and a conjectured relation of the

type: ZBH = |Ztop|2 is introduced. The Ztop can be reinterpreted as the Wave function of the

black hole over the phase space of black hole charge configurations.

1

Chapter 1

Introduction: The Problem with

Black Hole Microstates

Black Holes are one of the most curious objects in theoretical physics. They are the purest

objects one can find, and the closest thing to a point particle that exists. According to pure

classical General Relativity, Black holes have this curious property that all black holes with

the same charge, mass and angular momentum are alike, and there are no internal degrees of

freedom in them. This is known as the no-hair theorem. This fact about black holes is fatal to

one of the key tenants of physics: The 2nd Law of Thermodynamics.

1.1 No Hair Theorem & Entropy

Consider a universe, just empty space-time and introduce 2 objects: A black hole, and some

gas with some amount of entropy. The phase space of the whole system can be written as

HGas ⊗HBH (1.1)

as the system evolves, the book gravitates towards the black hole, and eventually falls into the

black hole. The only difference to the black hole that occurs is that only it’s mass has increased,

and maybe it’s angular momentum.

For example, consider a charged black hole with the following solution:

ds2 = −f(ρ) dt2 + f−1(ρ) dρ2 + ρ2 dΩ2 (1.2)

2

Chapter 1 Introduction: The Problem with Black Hole Microstates 3

F = p sin θdθ ∧ dφ+ qdt ∧ dρρ2

(1.3)

Here p, and q give the total magnetic and electric charge of the system at infinity and one can

calculate the mass of the black hole using the ADM mass. The ADM mass can be computed

through an integral over S2∞. And one can integrate the F and ∗F over S2

∞ to get the electric

and magnetic charges of the Black Hole.

It’s macro states can be inferred by integrals over S∞. But the problem is that, there is only

one solution for each macrostate. Since the solution space is isomorphics to the phase space, we

have that we get one macrostate per phase space point, which is a definition of a microstate.

Hence, for classical GR+EM, we have:

SBH = ln(1) = 0 (1.4)

So since the entropy of every classical black hole is 0. Any system that falls into a black hole

loses all it’s entropy, and the black hole gains none of it - a strict violation of the second law of

thermodynamics.

In 1974, Hawking combined Quantum Field Theory and General Relativity and predicted that

black holes would be black bodies with a non zero temperature, from which one could compute

the entropy of a black hole, and he derived the famous relation:

SBH =A

4(1.5)

So in the semiclassical regime, Black Holes do have entropy and it is propotional to the area of

the horizon atleast for the Schwarzschild Black Hole.

This relation is a great testing ground for quantum gravity theories. No matter what Quantum

Gravity one makes, it should be able to give the Area law for Black Hole Entropy, atleast to

leading order.

In this thesis, we explore how string theory has expanded in this area and gotten close agreement.

Infact, we use results from topological strings to find the full partition function of a specific black

hole: N = 2,d = 4 BPS Black Hole.

Chapter 2

Sigma Models: String Theory

One direction to pursue in creating a new theory of Quantum Gravity is to consider string

theory. String theory is a generalization of the concept of a particle. Instead of the fundamental

constituents of the universe being particles, 0-dimensional objects that trace out word-lines in

spacetime, we consider them being strings: 1-dimensional objects that stretch out 2-dimensional

sheets over space time. Most of the content of this chapter is my own interpretation using

differential forms and sigma models but some aspects are borrowed from ref. [3].

2.1 Generalizing a Particle

A nice way to model particles is using a sigma model. A sigma model σ(T,M) from T ontoMis a way of constructing theories or atleast configuration spaces to define your theory.

The sigma model configuration space C is simply the space of all maps:

C = σ : T →M (2.1)

Now we need a way to match each point in the configuration space to a number i.e. an action.

The most natural way to go about this is to consider the geometric objects available to us in

the target space, perform some sort of pull back onto T and use that to create a number.

Let’s consider a particle and see how it’s a sigma model. We can think of a particle as just

world lines through spacetime. Where the action is the length of the

4

Chapter 2 Sigma Models: String Theory 5

2.2 Relativistic Particle

The path for a relativistic particle is a curve through space time. So each point in configuration

space is actually given by some path P through spacetime, where P is one dimensional. The

sigma model we can construct is:

X : P →M (2.2)

We can construct the Lagrangian, by pulling back the metric g from the ambient spaceM onto

P. We can then construct from it a volume form, which we can integrate over P.

S =

∫Pω[X∗g] (2.3)

We introduce the coordinates and parametrization of the curve P in M as:

P : xµ(s) for s ε IR (2.4)

and we perform the pull back.

ωM =√−det[dxµ ⊗ dxµ] dx0 ∧ dx1 ∧ dx2 ∧ dx3

ω[X∗gM] =√−det[X∗dxµ ⊗X∗dxµ] ds

ω[X∗gM] =√−det[xµxµds⊗ ds] ds

ω[X∗gM] =√−xµxµ ds

(2.5)

Hence our action is:

S =

∫P

√−xµxµ ds (2.6)

This just says that the action of a configuration is the length of the string.

2.3 A String

A string can be written as a sigma model with the target space being, M. We will call the

worldsheet Σ, which is going to be the place on which our theory will exist. Hence the Lagrangian

will become:


Σ→M (2.7)

We can pull back the theory and get the Nambu Gotu action:

L =√

det[Σ∗gµνdxµ ⊗ dxν ]dτ ∧ dσ (2.8)

This evaluates to:

L =√

(X ′µX ′µ)(XµXµ)− (XµX ′µ)2dτ ∧ dσ (2.9)

We can again use the same trick and write this as a non-linear sigma model but with the metric

of the worldsheet kept free, and not constrained to equal that from the pull back. No kinetic

term for the worldsheet metric. Which means it’s simply a gauge fixing term to allow us to

simplify our Lagrangian.

The new lagrangian gives the Polyakov Action:

S =

∫ΣdXµ ∧ ∗dXµ =

∫Σgµν

∂Xµ

∂σa∂Xν

∂σbhab√hdσ0 ∧ dσ1 (2.10)

We can add a super-symmetric part to the action as well, with no interaction term. We can

solve them as a Quantum system separately, since they are different fields with no interaction,

and in the end just tensor the two hilbert spaces.

We can now find the variation of the action, which we derived a ’form’ formula for in the

Appendix:

δS =

∫ΣδdXµ ∧ ∗dXµ + dXµ ∧ ∗δdXµ −

1

2(T · h) ∧ ω (2.11)

=

∫Σ

2dδXµ ∧ ∗dXµ −1

2(Tabδh

ab) ∧ ω (2.12)

=

∫Σ

2δXµ ∧ ∗d†dXµ −1

2(Tabδh

ab) ∧ ω +

∫∂Σ

2δXµ ∧ ∗dXµ = 0 (2.13)

There is a lot of information here. This tells us that there are three conditions that must be

full filled separately to make all terms 0:

1. Equations of Motion:

d†dXµ = 0 on Σ (2.14)


2. Either:

δXµ = 0 on ∂Σ (2.15)

. This is the Dirichlet Boundary Condition.

Or:

∗ dXµ = 0 on ∂Σ (2.16)

. This is the Nuemman Boundary Condition

3. The Virasoro Constraint:

Tab = ∂aXµ∂bXµ −

1

2habh

cd∂cXµ∂dXµ = 0 (2.17)

or equivalently:

T = Tabdσa ∧ ∗dσb = dXµ ∧ ∗dXµ −

1

2ω ∧ ∗[dXµ ∧ ∗dXµ] = 0 (2.18)

We will leave choosing from the options in 2 for a later chapter on D-Branes. For Now, we will

only consider those world-sheets, where ∂Σ = Aτ→∞∪Bτ→−∞. Essentially, this means that the

boundary of the worldsheet exists only at the two infinities, and for a time slice at τ ∈ (−∞,∞)

the space-like section of the worldsheet must have no boundary. String like these are called

closed.

The equations of motion are simply the solutions to the wave equation since d†d is simply the

D’Lambertian on M.

Before we move on, some redundancies have to be removed.

We can see that the worldsheet is invariant under the following transformations:

1. Weyl Invariance: hab → eωhab. (1 dof per point)

2. World Sheet reparameterization: (τ, σ)→ (τ′(τ, σ), σ

′(τ, σ)). (2 dof per point)

3. Poincare Transformations of the target manifold M if M is minkowski. Which we will

assume from now on.

Since hab has 3 degrees of freedom per point on Σ. We can fix it to the flat metric. i.e. hab = ηab.

Once we know that the metric on Σ is minkowski, we know the D’Lambertian on M takes a

simple form of the flat space wave equations.

We shift to coordinates σ+ = and σ− and then expand the fields Xµ in terms of their modes:


Xµ = Xµ0 + τPµ0 +

i

2

∑n 6=0

1

n(αµne

−inσ+ + αµne−inσ−) (2.19)

Now the constraints also need to be imposed which we derived earlier, specifically:

Tab = 0 (2.20)

imposing these constraints gives us:

T++(σ+) =1

2(∂XL

∂σ+)2 =

∑n

Lne2inσ+

= 0 (2.21)

and

T−−(σ−) =1

2(∂XL

∂σ−)2 =

∑n

Lne2inσ− = 0 (2.22)

where we have defined our Ln and Ln as:

Ln =1

2

∑m

˜αn−mµαmµ (2.23)

Ln =1

2

∑m

αµn−mαmµ (2.24)

Hence we have an appropriate expansion for a classical string.

Chapter 3

Topological String Theory

Topological String theory is a great tool in our toolbox that helps us compute multiple things in

string theory. In this chapter, we will first explore the interesting world of topological quantum

field theories, or TQFTs. Here, we will see that the observables of TQFTs are all topological in

nature. The observables calculate topological invariants of the manifold M on which they are

defined (or the Target space if the TQFT is some sort of sigma model).

From there one can move to constructing topological string theory, where now, the metric g

of the manifold M is integrated over as well in the path integral of the TQFT. In addition to

which, once one sums over all topologies (appropriately defined), the final theory will become a

topological string theory.

We will begin by exploring Supersymmetric Quantum Mechanics on Riemannian Manifolds.

This chapter borrows from ref. [4] and chapter 10,12 and 16 of ref. [5].

3.1 Supersymmetric Quantum Mechanics: d = 0+1 Sigma Model

on M

The configuration space for our model consists of all maps φ defined as:

φ : R→M (3.1)

We assume thatM is equipped with at least a riemannian metric g, and hence has all the usual

things like the Riemann tensor RIJKL.

In the case that M is Kahler it comes equipped with a complex structure J . Using which we

can define the Kahler form ω such that ω(X,Y ) = g(JX, Y ) = igijdzi ∧ dzj

9

Chapter 3 Topological String Theory 10

3.1.1 SQM where M is a Riemannian Manifold

We can write down our map in a particular coordinate chart of M by using the coordinate

function xI , by composing them: φI = xI φ.

Since we want some degree of super-symmetry, we need to construct our fermions. It was easy

to construct the bosons as simply embedding coordinates. We can construct our fermions as

the complexified tangent bundle of M pulled back onto φ. Hence:

ψ, ψ ∈ Γ(R, φ∗TM⊗ C) (3.2)

so we can now construct our co-variant derivatives on the bundle that ψ exists in. This will

simply be done by noting the fact that the connection on the bundle on R is the one inherited

from M by pullback. Hence:

d~ψ = dψI ~eI + ψJφ∗[ωIJ ]~eI = dψI ~eI + ψJωIJK∂φK

∂tdt~eI (3.3)

We can construct the Lagrangian for the theory by first connecting the dot product on the

inherited bundle through

~ψ1 · ~ψ2 = ψI1ψJ2 gIJ (3.4)

and define the operation ∧ on vector valued one forms:

~α∧~β = (αI ∧ βJ)(~eI · ~eJ) (3.5)


Now we can write the Lagrangian:

L = dφI ∧ dφI +i

2(~ψ∧d~ψ − d~ψ∧~ψ)− 1

2RIJKLψ

I ψJψKψLdt (3.6)

opened up, it is:

L = gIJ φI φJdt+i

2gIJ(ψI∇tψJ −∇tψIψJ)dt− 1

2RIJKLψ

I ψJψKψLdt (3.7)

Now that we’ve defined our fermion fields and the bundle it comes from, we can start considering

the super symmetry transformations.

The following transformations do not change the Lagrangian of the system, and constitute the

set of supersymmetry transformations that this Lagrangian has:

δφI = εψI − εψI (3.8)

δψI = ε(iφ− ωIJKψJψK) (3.9)

δψI = ε(−iφ− ωIJKψJ ψK) (3.10)

From the Noether’s theorum we can derive the conserved charge for these transformations as:

Q = igIJ ψI φJ (3.11)

Q = −igIJψI φJ (3.12)

Another symmetry available is the phase rotation for the fermions:

ψI → e−iγψI (3.13)

ψI → eiγψI (3.14)

and the Noether charge for which is then:

F = gIJ ψIψJ (3.15)

And the conjugate momenta for out fields are pI = ∂L∂φ

= gIJ φJ = φI , and πψI = igIJ ψJ .


Once, we have those we can impose commutation relations and promote our variables to local

operator valued fields. We will fix the operator ambiguity later. The commutation relations

are:

[φI , pJ ] = iδIJ (3.16)

ψI , ψJ = gIJ (3.17)

and we set all other commutators to vanish.

We then define the Hamiltonian operator ordering in such a way that

Q, Q = 2H (3.18)

We can find how the Q and Q are charged under F :

[F,Q] = Q, [F, Q] = −Q (3.19)

Hence:

[H,F ] = [Q, Q, F ]

= Q[Q, F ] + [Q,F ]Q+ Q[Q,F ] + [Q, F ]Q

= QQ−QQ− QQ+ QQ

= 0

(3.20)

[H,F ] = 0 (3.21)

Of course these our operator valued fields. And we would like some sort representation for

these? Crazy! One would think that we shouldn’t go this route, and give up already. Finding a

representation for operator valued fields without any mechanism to decouple the commutation

relations to a manageable one (a.k.a a bunch of harmonic oscillators) would be impossible.

Especially when the only way we really know how to quantize in any standard way is to just

convert every problem we know to a bunch of decoupled harmonic oscillators.

But there is in fact a representation for all these operators.


The representation of states of this theory is actually the complexified space of differential

forms on M !:

H = Ω(M)⊗ C (3.22)

and this space of ”wavefunctions” or states, has an inner product defined as:

〈ψ|ψ′〉 =

∫Mψ ∧ ∗ψ′ (3.23)

Here, ψ are differential forms, mind you. And finally the observables are operators defined as:

φI = xI× (3.24)

pI = −i∇I (3.25)

ψI = dxI∧ (3.26)

ψI = gIJ i ∂

∂xJ(3.27)

where i ∂

∂xJis the operator that acts on differential forms and contracts them with the vector

∂∂xJ

. It gives 0 for all 0-forms (functions) on M

Now let’s define our lowest weight state in our representation. We simply define this state as

|0〉 such that ψI |0〉 = 0, and then define the rest from there. We can then make the following

connection between states and differential forms:

|0〉 ⇐⇒ 1 (3.28)

ψI |0〉 ⇐⇒ dxI (3.29)

ψI ψJ |0〉 ⇐⇒ dxI ∧ dxJ (3.30)

...⇐⇒ ... (3.31)

ψ1ψ2...ψn|0〉 ⇐⇒ dx1 ∧ dx2 ∧ ...dxn (3.32)


3.2 Cohomological Field Theories

3.2.1 General Cohomological Field Theories

A cohomological field theory, is a QFT that follows the following rules:

1. It has a Fermionic symmetry generator that squares to 0.

Q2 = 0 (3.33)

2. All ”physical” operators of the theory have the property that:

Q,Oi = 0 (3.34)

3. The vaccuum is invariant under the fermionic symmetry

Q|0〉 = 0 (3.35)

4. The energy momentum tensor is ’Q-exact’:

Tαβ =δS

δgαβ= Q,Gαβ (3.36)

One thing we can immediately prove is that any correlator with the Q term in it will equal to

0. We just have to anticommute the Q till it hits the vacuum.

〈0|Oi1 · · · OijQΛOij+1 · · · Oin |0〉 = ±〈0|Oi1 · · ·QOijΛOij+1 · · · Oin |0〉

= ±〈0|QOi1 · · · OijΛOij+1 · · · Oin |0〉

= 0 (3.37)

Hence:

〈Oi1 · · · OijQ,ΛOij+1 · · · Oin〉 = 0 (3.38)

Using the 4th requirement of being a cohomological field theory we can prove an astonishing

fact:


δ

δhαβ〈Oi1 · · · Oin〉 =

δ

δhαβ

(∫DφOi1 · · · OineiS[φ]

)= i

∫DφOi1 · · · Oin

δS

δhαβeiS[φ]

= i〈Oi1 · · · OinQ,Gαβ〉

= 0 (3.39)

This is quite interesting. Varying the metric does absolutely nothing to any of the correlators of

the theory. Remember, the time-ordering of different points on a manifold are defined precisely

by hαβ on the manifold and if varying with respect to the metric changes nothing of the correla-

tors, then the time ordering of the points within a correlator shouldn’t really mater! Infact, we

shouldn’t even worry about relative positions of local operator insertions in the path integrals,

they can always be moved around by changing the metric.

Another way to see this is to remember that the metric is what gives a manifold it’s structure.

What differentiates geometry from topology is the existence of a metric. Now if the correlators

donot care about the metric at all, then the correlators must be topological quantities. Hence

this is one example of a topological field theory.

Now, this is all great, but how do we construct lagrangians that have exact energy momentum

tensors? This is done by making sure that the Lagrangian itself can be written as a Q-exact

operator:

L = Q,V =⇒ S = Q,∫MV (3.40)

For a theory like this we can also consider:

d

d~〈Oi1 · · · Oin〉 =

d

d~

(∫DφOi1 · · · Oine

i~S[φ]

)= 〈Oi1 ..Something Q-exact..Oin〉 = 0.

Hence, weirdly, the correlators do not even depend on ~! They are completely equivalent t in

the classical limit.

3.2.2 Descent Equations

Consider the Operator:

Pα = Q,Gα (3.41)


Here, Gα = Gα0. And hence, Pα = Tα0

consider some scalar operator O(0)

O(1)α = iGα,O(0), (3.42)

where O(0)(x) is a scalar physical operator: Q,O(0)(x) = 0. Now, let’s see how this field

transforms under infinitesimal translates generated by the momentum operator:

d

dxαO(0) = i[Pα,O(0)]

= i[Q,Gα,O(0)]

= ±iGα,O(0), Q − iO(0), Q, Gα

= Q,O(1)α . (3.43)

We can write it as an operator valued differential form.

O(1) = O(1)α dxα = dO(0) = Q,O(1) (3.44)

We can keep on constructing higher order forms in this way, up until the point that it becomes

an n-dimensional form. Applying the d operator after that would trivially lead to 0.

Q,O(0) = 0

Q,O(1) = dO(0)

Q,O(2) = dO(1)

· · · =

Q,O(n) = dO(n−1)

0 = dO(n). (3.45)

We can ’deform’ the Lagrangian with terms like taO(n)a since the operator is physical, which

means that:

Q,

∫MO(n)

= 0. (3.46)


3.2.3 2d Cohomological Field Theories

The worldsheet of the string is 2 dimensional, hence it is prudent to study 2 dimensional

versions of the topological QFTs we’re trying to create, since after all, Topological Strings

have topological QFTs on the worldsheet.

QFT amplitudes can be formulated in the following way. Filterate your manifold Minto space-

like slices by M = ∪tΣt. Each space-like slice, gets a Hilbert Space. Consider the basis vectors

|φ(x)〉 for Ht being the configurations of all the fields over Σt.

An assumption in QFT is that the hilbert space at time t1, Ht1 is isomorphic to the Hilbert

space at time t2, Ht2 . So the incoming and outgoing states can be identified as part of the same

Hilbert space, and we can evaluate all the operator products we want.

On the other side of this map is the fact that if we want to evaluate the amplitude, we have

to compute the path integral, with the boundary conditions we had identified with the basis

vectors:

∫ BC2

BC1Dφ · · · eiS[φ] = 〈BC1|T (· · · ) |BC2〉. (3.47)

φ(t1)|BC1〉 = f(t1)|BC1〉. (3.48)

Now let’s take the case that we have a CFT and we want to compute the amplitude:

〈Oa|Ob(x2)Oc(x3)|Od〉cylinder, (3.49)

Consider doing this calculation on a cylinder, S1×R. We can take the state to correspond to a

hemisphere with the operator corresponding to that state inserted there. Then once we do the

path integral over the hemisphere, we need to specify the boundary conditions on the boundary

of the hemisphere (the circle). Since this is an object that acts on boundary conditions and

gives a number, this corresponds to a bra vector. Similarly one can define a ket vector with

the orientation of the hemisphere reversed. ”Glueing” two of these manifolds (with undefined

boundary conditions) together implies that the boundaries are defined to be point wise equal

and then all states on the Ht restricted to the common boundary are integrated over. This

defines a product structure.

Now since our theory is topological, we can stretch our manifolds as much as we like. So let’s take

the space between two operator insertions, and stretch them out infinitely. This will correspond


to a vaccum amplitude between these two points. Any distance can also be contacted down to

a close distance. So we can linearly break up our operator insertions by:

〈O1 · · · On〉Σ =∑a,b

〈O1 · · · OiOa〉Σ1ηab〈ObOi+1 · · · On〉Σ2 , (3.50)

〈O1 · · · On〉Σ =∑A,B

〈O1 · · · Oi|OA〉Σ1ηAB〈OB|Oi+1 · · · On〉Σ2 . (3.51)

Q (O1 . . .Oi|0〉) = 0, (3.52)

We can start with trying to figure out what ηµν is. We can consider the two operator genus 0

vacuum:

Cab = 〈OaOb〉 (3.53)

This results in

Cab = CacηcdCdb. (3.54)

〈O1 · · · On〉Σ = (−1)Faηab∑a,b

〈OaObO1 · · · On〉Σ′ , (3.55)

One can see by playing this game of opening up products, that all products can be deduced by

the genus 0 amplitude of a triple product:

cabc ≡ 〈OaObOc〉0, (3.56)

We can now separate a general genus Σ product:

〈· · · OaOb · · · 〉Σ =∑c,d

〈· · · Oc · · · 〉Σηcdcabd

=∑c

〈· · · Oc · · · 〉Σcabc, (3.57)

We can use this and define an operator algebra:


OaOb =∑c

cabcOc. (3.58)

And we finally have our operator algebra for our TQFTs using which we can calculate any

amplitude for our theory.

As a final side note: if we deform our cohomological field theory Lagrangian by terms of the

form taO(2)a. Then calculation of the td derivative gives us:

dcabcdtd

= 〈OaObOc∫O(2)d 〉, (3.59)

Which can give us the effect of all deformations of the Lagrangian on the cabc.

3.3 N = (2, 2) SCFTs with Topological Twist

Supersymmetry generators are usually spin 12 representations of the Lorentz group. In two

dimensions the Lorentz group is simply SO(2) = U(1), and the spin 12 representation is reducible

since each spinor can be split up into two components which transform with opposite charge

under this U(1). That is why, we use the notation N = (p, q) when there are p irreducible

spinor supercharges with positive U(1)-charge and q with negative U(1)-charge. In this chapter

we will be considering N = (2, 2) supersymmetry - but twisted.

So we start with the commutation relations of the susy generators:

Q+ +Q−,Q+ −Q− = 2H

Q+ +Q−,Q+ +Q− = 2P,

Q+ +Q−,Q+ −Q− = 2H

Q+ +Q−,Q+ +Q− = 2P, (3.60)

Now we define new operators:

QA = Q+ +Q−

QB = Q+ +Q− (3.61)


By observation one can conclude that these square to 0, and that the Hamiltonian and mo-

mentum are still exact with respect to the newly defined operators. We can now check the

commutation relations of the newly defined operators with the Lorentz Generator M .

Remember the Lorentz Group on the worldsheet is U(1), and hence can be represented with

only one generator. M

Now on flat space, susy transformations work just fine. What you need to do the transformation

is some sort of spinor:

δΦi = ε+Q+Φi (3.62)

For the symmetry to be global, ε+ has to be a covariantly constant spinor. So if the image of

the worldsheet has an inherited metric such that there exists a covariantly constant spinor, all

is good. But let’s say we wanted to get straight to string theory, and integrate over all metrics

on the worldsheet, we’d hit a problem. When we try to integrate over all possible metrics, we

will get some metrics that simply donot admit any covariantly constant spinors. So symmetries

generated by spin half representations of the Lorentz group cannot lend themselves easily to a

”integrate over all metrics approach”.

It’s interesting to note that this problem does not plague bosonic symmetries. Global Bosonic

symmetries usually have an infintiesimal parameter which is a number. So if you have some

sort of a symmetry generated by a scalar, and you would like for it to be a global symmetry,

then you can construct it easily.

So if we can construct some method by which we can maybe change our Lorentz group so that

Q could become a scalar, that would be great. We could then make the supersymmetry global

no matter what the worldsheet metric.

So let’s do just that. Let’s redefine the Lorentz group generators and change them just a bit

MA = M − FV or MB = M − FA, (3.63)

and then define these new MA and MB to be THE Lorentz generators of the space. Here, FV

and FA are the vector and axial charges which come from the R symmetry present in N = (2, 2)

supersymmetry. R symmetry is recalled here:

RV (α) : (θ+, θ+

)→ (e−iαθ+, eiαθ+

), (θ−, θ−

)→ (e−iαθ−, eiαθ−

)

RA(α) : (θ+, θ+

)→ (e−iαθ+, eiαθ+

), (θ−, θ−

)→ (eiαθ−, e−iαθ−

). (3.64)


Now we can use the commutators of the F charges with the Q charges, and then see how Q

transforms under a twisted M .

The commutation relations are:

[FV ,Q±] = Q±

[FV ,Q±] = −Q±

[FA,Q±] = ±Q±

[FA,Q±] = ∓Q±. (3.65)

And once we use these to compute the charges of Qs under changed Ms we get:

[MA,Q+] = −2Q+ [MB,Q+] = −2Q+

[MA,Q−] = 0 [MB,Q−] = 2Q−

[MA,Q+] = 0 [MB,Q+] = 0

[MA,Q−] = 2Q− [MB,Q−] = 0. (3.66)

So we can see, that under MA, QA transforms as a scalar. For MB, QB transforms as a scalar.

So we have finally succeeding in ”twisting” and making sure that arbitrary metrics don’t kill

global supersymmetry.

3.3.1 An example: A-Twist

If we decide to twist the system and make MA the lorentz generator, we get our fields as sections

of different bundles than they were before. So the new bundles for the supersymmetric fields

are:

ψi+ ≡ ψiz ∈ Ω1,0 ⊗ φ∗(T (1,0)M)

ψi− ≡ χi ∈ φ∗(T (1,0)M)

ψi+ ≡ χı ∈ φ∗(T (0,1)M)

ψi− ≡ ψız ∈ Ω0,1 ⊗ φ∗(T (0,1)M), (3.67)


And, since we know the bundles, and the fact that it’s a sigma model, we can easily construct

the lagrangian:

L = −2t(gi.zφ

i.zφ

j+ gi.zφ

i.zφ

j+ igiψ

iz∆zχ

+ igiψz∆zχ

i− +

1

2Riklψ

izψ

zχ

kχl)

It turns out, it’s almost possible to write this Lagrangian in the form:

L′ = −itQA, V (3.68)

Just use:

V = gi

(ψiz.zφ

j+.zφ

iψz

). (3.69)

However the matching isn’t exact. They differ by some terms, which, after adding and sub-

tracting the fermionic equations of motion, we get:

L′ = L− 2tgi(.zφ

i.zφ

j − .zφi.zφ

j). (3.70)

But the left hand side above is just the pull back of the target space kahler form.

S − S′ = 2t

∫Σd2zgi

(.zφ

i.zφ

j − .zφi.zφ

j)= t

∫Σφ∗(ω)

= t

∫φ(Σ)

ω (3.71)

So S−S′ = tω ·β. Here β is the 2nd homology class of the worldsheet image in the Target Space.

Since this term that we have calculated is completely topological in nature we can clearly see

that we have found an example of a cohomological field theory.

Chapter 4

Ooguri, Strominger, Vafa (OSV)

conjecture: ZBH = |Ztop|2

In this chapter we will focus on the OSV conjecture as it was first introduced by Ooguri,

Strominger and Vafa in ref. [10]. For the background on n = 2, d = 4 supergravity and its

relation to special geometry, we have borrowed from ref. [8] and ref. [9].

4.1 Macroscopic Entropy from Stabilization Equations

So we first consider N = 2, d = 4 supergravity, and what we really want to do is study the

couplings of the Riemann Tensor Rµνρσ. The reason is, that eventually we want to be able to

extract the entropy for this theory, from Wald’s entropy formula, even when Higher derivative

R corrections (coming from the conformality condition of the Type IIB String)to N = 2, d = 4

supergravity are taken into account.

If we want to study the couplings of the reimann tensor there is a simpler way of formulating

this system, in terms of what’s called a special geometry.

Before we begin it was shown in ref. [1, 2] that, all higher derivative F-type term corrections to

this SUGRA come in the form:

∫d4xd4θ(WabW

ab)gFg(XΛ) (4.1)

We can formulate N = 2, d = 4 SUGRA as a guage theory and it’s parameters take values in

bundles over that space:

23

Chapter 4 Ooguri, Strominger, Vafa (OSV) conjecture: ZBH = |Ztop|2 24

4.1.1 Form of N = 2, d = 4 Supergravity

A general ungauged N = 2 supergravity theory in 4 dimensions may be obtained by combining

massless supersymmetric multiplets with spin less or equal to 2:

i) The gravity multiplet, which has the graviton gµν , two gravitini ψαµ and one Abelian gauge

field Aµ known as the graviphoton;

ii) nV vector multiplets: each containing an Abelian gauge field Aµ, two gaugini λα and one

complex scalar. The complex scalars zi come as sections of a bundle with the fibre being

a projective special Kahler manifoldMV of real dimension 2nV .(nV = h2,1(CY ) for Calibi

Yau Compactification)

iii) nH hypermultiplets: each consisting of two complex scalars and two hyperinis ψ, ψ. The

scalars are sections of a bundle with the fibre being a quaternionic-Kahler space MH of

real dimension 4nH . (nH = h1,1(CY ) + 1 for Calibi Yau Compactification)

If this Supergravity theory was the one that came from compactification of Type II String theory

over a Calibi Yau threefold:

M× CY3 (4.2)

Then the number of vector multiplets formed depend on the three fold that we compactified on

by:

nV = h2,1(CY3) (4.3)

nH = h1,1(CY3) (4.4)

We can write the Bosonic part of the SUGRA Lagrangian:

The Bosonic Part of N = 2 Supergravity is:

L =√−g[

R

2+ gij∂µz

i∂µzj ] +1

4(Im[NIJ ]F I ∧ ∗F J)− 1

8(Re[NIJ ]F I ∧ F J) (4.5)

where F I are the field stregnths of the nV one-form fields in the vector multiplet, and one one-

form field being the Graviphoton field. gij is the metric for the projective Kahler manifold for

the vector multiplets, which can be easily derived from the Kahler potential by the definition

of the potential

gij =∂K(z, z)

∂zi∂zj(4.6)


In this projective Kahler space, the coordinate charts are given in a redundant fashion with

2nv + 2 coordinates (XI , FI) that transforms under local symplectic transformations Sp(2nv +

2,R).

The Kahler potential in this chart is defined as:

K = −log[iXI∂F

∂XI− iXI ∂F

∂XI]] (4.7)

and the NIJ in the Lagrangian is called the period matrix of the projective Kahler manifold,

and has the form in this chart as:

NIJ = FIJ(X) + 2iImFIKImFJLX

KXL

ImFKLXKXL(4.8)

4.1.2 Attractor Mechanism

Let’s look at the particular case of a spherically symmetric extremal black hole in this SUGRA

theory. It was derived in [6,7] that for spherically symmetric extremal black hole solutions to

this theory, the values of the moduli fields (XΛ, FΛ) at infinity are inconsequential, and the

moduli fields eventually settle to fixed values at the Horizon of the black hole. The values are

called attractor points, and the exact value depends only the black hole charges (pΛ, qΛ).

The attractor equations derived in [6,7] relate the two:

pΛ = Re[CXΛ] (4.9)

qΛ = Re[CF0Λ] (4.10)

Since this is an extremal BPS black hole, it satisfies the BPS condition and hence, the entropy

has quite a simple form:

SBH =1

4Area = π|Q2| (4.11)

Here Q = Qm + iQe is some complex combination of magnetic and electric charges normalized

so that |Q| = rH , where rH is the radius of the horizon sphere. From this we can find that:

|Q2| = i

2(qΛCX

Λ − pΛCF0Λ) (4.12)


The black hole charges are actually indicative of the geometry of the calibi yau at the horizon.

The choice of electric and magnetic charges is equivilant to some choice of symplectic basis of

3 cycles from the homology group of the calibi yau. Symplectic basis implies that these basis

have the following relations:

AΛ ∩AΣ = 0

AΛ ∩BΣ = δΛΣ

BΛ ∩BΣ = 0. (4.13)

From these we can get the charges:

pΛ =

∫AΛ

ReΩ = Re[CXΛ] (4.14)

qΛ =

∫BΛ

ReΩ = Re[CF0Λ] (4.15)

where ReΩ = 12(Ω + Ω). In terms of this choice the black hole entropy can be written as

SBH =π

4

∫CY

Ω ∧ Ω (4.16)

4.1.3 Higher Order Corrections

String loop higher order corrections (looking at just the F-Terms) are written as:

F (XΛ,W 2) =∑h

Fh(XΛ)W 2h (4.17)

Surprisingly, the attractor value of W 2 is given by simply:

C2W 2 = 256 (4.18)

And these general F-terms follow the attractor equations, which are exact.

pΛ = Re[CXΛ] (4.19)

qΛ = Re[CFΛ(XΛ,256

C2)] (4.20)


Then it was derived using Wald’s formula in [8,9,10,11,21] that the entropy for given black hole

charges followed the equation:

SBH =πi

2(qΛCX

Λ − pΛCFΛ) +π

2Im[C3∂CF ] (4.21)

4.1.4 Reinterpreting the Formula

We can now cast the formula into a much simpler form by introducing a field φ. We will later

see what this field stands for.

We will define the imaginary part of CXΛ to be φΛ:

CXΛ = pΛ +i

πφΛ (4.22)

And now we will define a new F where F is a function of φΛ and pΛ instead of pΛ and qΛ

F(φ, p) = −π Im

[C2F

(XΛ,

256

C2

)]= −πIm

[F(CXΛ, 256

)]= −πIm

[F

(pΛ +

i

πφΛ, 256

)](4.23)

One can get another relation with qΛ, by remembering that FΛ = ∂ΛF which comes from the

definition of a projective kahler manifold. We can write qΛ as:

qΛ =1

2(CFΛ + CFΛ) = − ∂

∂φΛF(φ, p) (4.24)

Using the homogeneity relation XΛ∂ΛF (XΛ,W 2) +W∂WF (XΛ,W 2) = 2F (XΛ,W 2), and the

attractor equations, we get:

SBH(q, p) = F(φ, p)− φΛ ∂

∂φΛF(φ, p) (4.25)

4.1.5 What’s the phase space?

The equation

SBH(q, p) = F(φ, p)− φΛ ∂

∂φΛF(φ, p) (4.26)


Can be seen as the legendre transform of the free energy of the system. Hence, now we have

moved from a microcanonical ensemble, to a mixed ensemble where we have a microcanonical

ensemble of magnetic charges pΛ and a canonical ensemble of electric charges, which is given by

the thermodynamic conjugate variable. That is the electric potential φΛ.

To do a legendre transformation on the partition function, you institute a laplace transformation

on it:

ZBH(φΛ, pΛ) =∑qΛ

Ω(pΛ, qΛ)e−φΛqΛ (4.27)

These give us the integer black hole degenerecies Ω(pΛ, qΛ).

We can write the charger

Q2 = ipΛ

4(CFΛ − CFΛ) +

φΛ

4π(CFΛ + CFΛ). (4.28)

We can also clearly see that:

ZBH(φΛ, pΛ) = exp[F(φΛ, pΛ)

](4.29)

We will now fix our guages such that K = 0, C = 2Q and W 2 = 64Q2 at the attractor point. By

our earlier definition of |Q2| we get:

C = 2Q, XΛ =pΛ + iφΛ/π

2Q(4.30)

We can finally plug it in and get the Partition function of our black hole:

lnZBH = −4πQ2Im

[∑h

Fh

(pΛ + iφΛ/π

2Q

)(8

Q

)2h]. (4.31)

4.1.6 Comparing with Topological String Amplitudes

The topological String Partition function, can be written in terms of the topological string free

energy as:

Ztop(tA, gtop) = exp[Ftop(tA, gtop)

], (4.32)

and the topological free energy has an genus h expansion, weighted by g2h−2top .


Ftop(tA, gtop) =∑h

g2h−2top Ftop,h(tA) (4.33)

The tΛ can be written in terms of the XΛ.

tA =XA

X0= θA + irA, (4.34)

Now for large volume limit of the Calibi Yau, the degeneracy of Type II string states has already

been computed in [9] as:

F(CXΛ, 256

)= C2DABC

XAXBXC

X0− 1

6c2A

XA

X0+ · · ·

= (CX0)2DABCtAtBtC − 1

6c2At

A + · · · (4.35)

where

c2A =

∫Mc2 ∧ αA, (4.36)

with c2 being the second Chern class of M , and CABC = −6DABC are the four-cycle intersection

numbers.

For the topological string, expansion upto this order was computed in [17,1] and they were

found to be:

Ftop = −(2π)3i

g2top

DABCtAtBtC − πi

12c2At

A + · · · , (4.37)

Ooguri, Strominger and Vafa conjectured that this expansion is equal up to all orders. Assuming

this, we can determine the mapping between the coefficients by comparison.

The consistent definition implies a prefactor of −2iπ :

F (CXΛ, 256) = −2i

πFtop(tA, gtop) (4.38)

where gtop also had to be fixed. By comparing the first terms in , we can see that:

gtop = ± 4πi

CX0. (4.39)


Plugging (4.38) into (4.23) we get:

lnZBH = −π Im[F (CXΛ, 256)

]= Ftop + Ftop. (4.40)

where we should now write the parameters in terms of the charges, which is what we would like

to see. So by using the exact attractor equations, the moduli and gtop are:

tA =pA + iφA/π

p0 + iφ0/π, gtop = ± 4πi

p0 + iφ0/π.

and finally from 4.40 we get the relation that was part of the conjecture, in a neat form:

ZBH = |Ztop|2 (4.41)

Appendix A

Appendix: Differential Forms &

Action Principles

A.1 Introduction

The general work of a theoretical physicist can be grossly simplified into the following form:

1. Imaginesome model of a dynamic system that gives an explanation of observed phe-

nomenon.

2. Construct reasonable mathematical representations of the objects in these models.

3. Give rules by which these objects can interact and give predictions

Once 1 and 2 have been full-filled, the action principle gives a great method to get predictions,

writing the interactions of these objects as a Lagrangian on some Manifold, which basically

parameterizes the dynamical variables of the system.

A.2 Choosing from a space of trajectories

A particular trajectory of a system can be denoted by P. The space of all such trajectories

forms the space H. A particular point in H is a trajectory.

We can define functions on H. A function f on H is called a ’functional’, and is the map:

f : H → R (A.1)

31

Appendix A Appendix: Differential Forms & Action Principles 32

We assume H has a differentiable manifold structure, and has a tangent space and cotangent

space This is a highly non trivial thing to prove, and I am not sure whether it can be proved,

but we will assume it regardless, since it has yet to end up in an incorrect conclusion within the

domain of applicability in this document.

Calculus on H will essentially be the Calculus of Variations, where the variation operator δ is

analogous to the the exterior derivative operator on M, but only up to it’s action in mapping

Functional 0-forms to Functional 1-forms, higher order forms don’t have a clear meaning. These

1-forms will live in T ∗H. Vectors that live in T H are called ”variations”.

All fundamental classical laws of physics, simple mechanics or field theories, can be written as

an action principle, where S is a function on H, and the trajectory of the dynamical variable

F where δS(~η) = 0 (for all possible variations of the field ~η at F ), is the trajectory that will

actually take place, and be observed experimentally.

If we want our theories to be local on some base manifold M, then we must have that the

functional S can be written as:

S[q] =

∫ML[q] (A.2)

where L[q] is volume-form valued function on H

A.3 Mathematical Tricks

A.3.1 Some Differential Form preliminaries

Differential Forms exist on some n dimensional oriented manifoldM. P-forms can be integrated

only over p-dimensional sub manifolds ofM. For example, a 2-form can only be integrated over

a 2-dimensional surface in M.

Any n-dimensional oriented manifold M with a metric g, has one particular n-form called the

volume form ω.

If we choose some coordinates xµ on M, we have the following coordinate representations of g

and ω:

g = gµν dxµ ⊗ dxν

ω =√

(−1)s det[gµνdxµ ⊗ dxν ] dx0 ∧ ... ∧ dxn(A.3)


Here s is the number of minus signs in the signature of the metric on M.

We also define a very essential operator called the Hodge Star operator ’∗’. It is a linear operator

that maps p-forms to n− p forms and is an invertible operator, since the space of p-forms and

n− p forms have the same number of basis forms.

It is defined as an operation on p-forms:

dxµ ∧ ∗dxν = g(dxµ, dxν)ω = gµνω (A.4)

where ∗dxν is an n−p form such that (4) is true. Functions (0-forms) can pass straight through

the ∗ operator (∗α = ∗αµdxµ = αµ ∗ dxµ).

The square of the hodge star operator depends on the rank of the form it acts on. If α is a k

form on an n-dimensional manifold, with s minus signs in it’s metric:

∗ ∗ α = (−1)k(n−k)+sα (A.5)

A.3.2 Inner Product on Differential Forms

The inner product on differential forms is a very important concept. It is a bilinear operator

that maps two p-forms on M onto a number, in a symmetric, bilinear fashion.

For two p-forms α and β, we define:

(α, β)M =

∫Mα ∧ ∗β (A.6)

The inner product is symmetric. In fact, the ∧∗ operation that outputs a volume form is also

symmetric:

α ∧ ∗β = β ∧ ∗α (A.7)

We can also define integration by parts, since:

∫Md(α ∧ ∗β) =

∫Md(α) ∧ ∗β + (−1)b−1

∫Mα ∧ d(∗β) (A.8)


Where α is a p-form. (which implies that β must be a b = p + 1 form). We can define a new

operator, called the adjoint of the d operator called d∗. Such that:

(−1)b−1d∗ = ∗d∗ (A.9)

This implies that the general form of the adjoint operator is:

d∗ = (−1)n(k−1)+1+s ∗ d∗ (A.10)

Since this is true, we now have a super-important, beautiful and oft-applicable generalization

of Integration by Parts:

∫Mdα ∧ ∗β =

∫∂M

α ∧ ∗β +

∫Mα ∧ ∗d∗β (A.11)

If you want to remember only one thing from this document, remember (10). It’s the single

most important trick when varying the action.

A.3.3 Variation Operator

Let’s take an example of a scalar field on space-time. So we can make our discussion much less

abstract.

We know that a particular trajectory of the system, involves specifying the value of the field at

every point in space-time. This is equivalent to defining a function on M.

All possible functions on M form the space of all possible trajectories: H.

Hence we have:

P → φ(xµ)

H → φ(xµ)|φ ε C∞(M)(A.12)

We can clearly see that H is an infinite dimensional space.

Note that φ is a coordinate system on H. By analogy, coordinates xµ on M each give a num-

ber at every point in M. Where as the coordinate φ gives a function at every point of H


If we choose a particular field trajectory, φ0(xµ), and we would like to infinitesimally change it.

How would we do it? And what would an infinitesimal change in the space of all functions look

like?

We can take inspiration from how we performed an infinitesimal coordinate change on finite

dimensional manifolds. An infinitesimal coordinate change on a manifold is generated by some

vector field on the manifold. For some vector field ~v and a small epsilon we have:

x′µ = xµ + ε dxµ(~v) (A.13)

Completely analogously, we can construct an equation for the functions on M.

φ′

= φ + ε δφ(~η) (A.14)

Where we see that we had to construct two things:

Firstly, a vector field ~η on H, which is what we call a ’variation’. It has the information about

how to explicitly change the function point by point, for each given function, which is a lot of

information to provide. Thankfully, we rarely have to provide it for our purposes, because laws

of physics ask a particular functional to be 0 for all variations at a particular trajectory.

Secondly, a one-form equivalent of the dxµ onM, called a ”functional one-form” δφ on H ( δφi

if there’s more than one field in the picture). This functional one-form, ’eats’ a variation, and

outputs a function on M, which is multiplied by a small ε and then added to the old function

we wish to change infinitesimally.

It is clear that ~η lives in T H, whereas δφ lives in T ∗H

Now can we a construct a δ operator analogous to the d operator.

We can look at how the d operator acts on the functions on M:

df =∂f

∂xµdxµ (A.15)

Now if we consider a function S[φi] on H, called a ”functional”, which maps trajectories onto

numbers, and notice that (15) has some derivative with respect to the coordinates, times the

one-forms induced from the coordinates, we can guess the following relation:

δS =δS

δφiδφi (A.16)


Where, δδφi

are functional derivatives with respect to the fields, and δφi are the functional one-

forms

However the variational operators have a bit more structure, when they interact with functionals

that are the result of integrals of volume forms.

For example, imagine a functional that’s defined by some function on M by the following way:

S[φ] =

∫ML[φ]ω (A.17)

Then we have the following rule for acting on the expression with δ:

δS[φ] =

∫Mδ(L[φ]ω)

δS[φ] =

∫Mδ(L[φ])ω Assume ω independent of φ

δS[φ] =

∫M

(∂L[φ]

∂φδφ+

∂L[φ]

∂(∂µφ)δ(∂µφ) + ...)ω

(A.18)

In physics we always use those dynamical quantities such that the laws don’t depend on the

variation of the second derivatives of the quantities. I.e. The series truncates at the second

term in (18)

The most important take away from this is that the variational operator commutes with the

exterior derivative operator and the hodge star operator:

δ(∗dα) = ∗δ(dα) = ∗dδα (A.19)

A.3.4 Deriving Equations of Motion

By applying the rule in (19) and Integration by parts repeatedly, we can vary any Lagrangian

and collect all the δφ terms on one side. Then we know, that since δφ(~η) 6= 0 for all variations

~η, the terms that are multiplying the δφ term will go to 0. Which gives our equations of motion.

An example for a scalar field theory action proceeds like this:

Let’s say we have the typical Scalar field Lagrangian with a kinetic term.


S[φ] =

∫M∂µφ∂

µφ√−gdnx =

∫Mdφ ∧ ∗dφ

δS[φ] =

∫Mδ(dφ ∧ ∗dφ) = 0 Apply Product rule and δ ∗ α = ∗δα

=

∫Mδ(dφ) ∧ ∗dφ+ dφ ∧ ∗δ(dφ) Now use symmetry of ∧∗ operator

= 2

∫Mdδφ ∧ ∗dφ Now use Integration by parts

= 2

∫Mδφ ∧ ∗d∗dφ+ 2

∫∂M

δφ ∧ ∗dφ = 0

(A.20)

Hence: d∗dφ = 0 on M (our equation of motion for φ) and either φ is fixed on ∂M (which

means δφ was 0 since there was no variation on the boundary) or dφ = 0 on ∂M. This produces

the system with the boundary conditions.

d∗dφ = 0 is simply the Klein Gordon equation for a general space-time, as are the well known

equations of motion for the given Lagrangian.

A.3.5 Conjugate Momentum One form and the E-L equation in the form

language

We can define the generalized momentum one form and the generalized force function, in the

following way.

Given a Lagrangian function L[φ] we can write the action as:

S =

∫ML[φ]ω (A.21)

We find the functional one-form δS:

δS[φ] =

∫MδL[φ]ω + L[φ]δω

=

∫M

δL[φ]

δφδφω +

δL[φ]

δgµνδgµνω + L[φ]δω

=

∫M

(∂L[φ]

∂φδφ+

∂L[φ]

∂(∂µφ)δ(∂µφ)

)ω +

δL[φ]

δgµνδgµνω + L[φ](−1

2ωgµνδg

µν)

=

∫M

(∂L[φ]

∂φδφ+

∂L[φ]

∂(∂µφ)dxµ ∧ ∗δ(dφ)

)ω − 1

2

(gµνL[φ]− 2

δL[φ]

δgµν

)δgµνω

(A.22)


We now define the generalized momentum one-form:

P =∂L[φ]

∂(∂µφ)dxµ (A.23)

We also define the generalized force function:

F =∂L[φ]

∂φ(A.24)

And finally we define the Energy Momentum Tensor (Or the stress-energy tensor) as:

Tµν = gµνL[φ]− 2δL[φ]

δgµν(A.25)

The general equation for the functional one-form δS is:

δS =

∫M

Fωδφ+ P ∧ ∗d(δφ)− 1

2(T · δg)ω (A.26)

where T · δg = Tµνδgµν

A.4 Application to known problems

A.4.1 Electromagnetism

In electromagnetism, again, our degree of freedom is the 4-potential one-form A = Aµdxµ, and

it’s corresponding lagrangian is:

L[q] = FµνFµν = F ∧ ∗F = dA ∧ ∗dA (A.27)

We can vary this action easily, by remembering that the variation operator δ, commutes with

the d operator.


δS[A] =

∫MδL[A]

=

∫Mδ(dA ∧ ∗dA)

=

∫MdδA ∧ ∗dA+

∫MdA ∧ ∗dδA

= 2

∫M

(dδA ∧ ∗dA)

= 2

∫M

(δA ∧ ∗d∗dA) +

∫∂M

(δA ∧ ∗dA) = 0

(A.28)

The adjoint of the exterior derivative d∗ = (−1)k(n−k)+s ∗ d∗, where d∗ acts on a k form, Mhas n dimensions and s of them have a -1 signature. Since (3) is 0 for all variations that the

functional one-form δA could act on, we have that: d∗dA = 0 on M and that dA = 0 on ∂M ,

which is the boundary.

These are exactly equivalent to maxwells equations without a source.

To find equations for Electromagnetism with a source, add a term to the Lagrangian:

S[A] =

∫ML[A] =

∫M

(dA ∧ ∗dA− 8πA ∧ ∗J) (A.29)

Then we vary the action like we did last time:

δS[A] =

∫MδL[A]

=

∫Mδ(dA ∧ ∗dA)−

∫M

8πδA ∧ ∗J

= 2

∫M

(dδA ∧ ∗dA)−∫M

8πδA ∧ ∗J

=

∫M

(δA ∧ ∗(2d∗dA− 8πJ)) +

∫∂M

(δA ∧ ∗dA) = 0

(A.30)

Now the equation of motion is:

d∗dA = 4πJ

∗d ∗ dA = 4πJ

∗ ∗ d ∗ dA = 4π ∗ J

d ∗ dA = 4π ∗ J

(A.31)


Hence we have the maxwell Equations with source:

F = dA

d ∗ F = 4π ∗ J(A.32)

Bibliography

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theory Nucl. Phys. B 413, 162 (1994),

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