Mathematical Physics Studies

Jörg Teschner Editor

New Dualities of Supersymmetric Gauge Theories

Mathematical Physics Studies

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Jörg TeschnerEditor

New Dualitiesof Supersymmetric GaugeTheories

123

EditorJörg TeschnerString Theory GroupDESYHamburgGermany

ISSN 0921-3767 ISSN 2352-3905 (electronic)Mathematical Physics StudiesISBN 978-3-319-18768-6 ISBN 978-3-319-18769-3 (eBook)DOI 10.1007/978-3-319-18769-3

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Contents

Exact Results on N ¼ 2 Supersymmetric Gauge Theories . . . . . . . . . . 1Jörg Teschner

Families of N ¼ 2 Field Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Davide Gaiotto

Hitchin Systems in N ¼ 2 Field Theory . . . . . . . . . . . . . . . . . . . . . . . 53Andrew Neitzke

A Review on Instanton Counting and W-Algebras . . . . . . . . . . . . . . . 79Yuji Tachikawa

β-Deformed Matrix Models and 2d/4d Correspondence . . . . . . . . . . . . 121Kazunobu Maruyoshi

Localization for N ¼ 2 Supersymmetric Gauge Theoriesin Four Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Vasily Pestun

Line Operators in Supersymmetric Gauge Theoriesand the 2d-4d Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Takuya Okuda

Surface Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Sergei Gukov

The Superconformal Index of Theories of Class S . . . . . . . . . . . . . . . 261Leonardo Rastelli and Shlomo S. Razamat

A Review on SUSY Gauge Theories on S3 . . . . . . . . . . . . . . . . . . . . . 307Kazuo Hosomichi

v

3d Superconformal Theories from Three-Manifolds . . . . . . . . . . . . . . 339Tudor Dimofte

Supersymmetric Gauge Theories, Quantization of Mflat,and Conformal Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375Jörg Teschner

Gauge/Vortex Duality and AGT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419Mina Aganagic and Shamil Shakirov

B-Model Approach to Instanton Counting . . . . . . . . . . . . . . . . . . . . . 449Daniel Krefl and Johannes Walcher

vi Contents

Exact Results on N = 2 SupersymmetricGauge Theories

Jörg Teschner

The following is meant to give an overview over our special volume. The first threeSects. 1–3 are intended to give a general overview over the physical motivationsbehind this direction of research, and some of the developments that initiated thisproject. These sections are written for a broad audience of readers with interest inquantum field theory, assuming only very basic knowledge of supersymmetric gaugetheories and string theory. This will be followed in Sect. 4 by a brief overview overthe different chapters collected in this volume, while Sect. 5 indicates some relateddevelopments that we were unfortunately not able to cover here.

Due to the large number of relevant papers the author felt forced to adopt a veryrestrictive citation policy. With the exception of very few original papers only reviewpapers will be cited in Sects. 1 and 2. More references are given in later sections, butit still seems impossible to list all papers on the subjects mentioned there. The authorapologises for any omission that results from this policy.

1 Background, History and Context

1.1 Strong Coupling Behavior of Gauge Theories

Gauge theories play a fundamental role in theoretical particle physics. They describein particular the interactions that bind the quarks into hadrons. It is well under-stood how these interactions behave at high energies. This becomes possible due tothe phenomenon of asymptotic freedom: The effective strength of the interactions

A citation of the form [V:x] refers to article number x in this volume.

J. Teschner (B)

DESY Theory, Notkestr. 85, 22603 Hamburg, Germanye-mail: teschner@mail.desy.de

© Springer International Publishing Switzerland 2016J. Teschner (ed.), New Dualities of Supersymmetric Gauge Theories,Mathematical Physics Studies, DOI 10.1007/978-3-319-18769-3_1

1

2 J. Teschner

depends on the energy scale, and goes to zero for large energies. It is much less wellunderstood how the interactions between quarks behave at low energies: The exper-imental evidence indicates that the interactions become strong enough to preventcomplete separation of the quarks bound in a hadron (confinement). The theoreticalunderstanding of this phenomenon has remained elusive.

When the interactions are weak one may approximate the resulting effects rea-sonably well using perturbation theory, as can be developed systematically using theexisting Lagrangian formulations. However, the calculation of higher order effects inperturbation theory gets cumbersome very quickly. It is furthermore well-known thatadditional effects exist that can not be seen using perturbation theory. Exponentiallysuppressed contributions to the effective interactions are caused, for example, by theexistence of nontrivial solutions to the Euclidean equations of motion called instan-tons. The task to understand the strong coupling behavior of gauge theories looksrather hopeless from this point of view: It would require having a complete resum-mation of all perturbative and non perturbative effects. Understanding the strongcoupling behaviour of general gauge theories remains an important challenge forquantum field theory. However, there exist examples in which substantial progresshas recently been made on this problem: Certain important physical quantities likeexpectation values of Wilson loop observables can even be calculated exactly. Whatmakes these examples more tractable is the existence of supersymmetry. It describesrelations between bosons and fermions which may imply that most quantum correc-tions from bosonic degrees of freedom cancel against similar contributions comingfrom the fermions. Whatever remains may be exactly calculable.

Even if supersymmetry has been crucial for getting exact results up to now, it seemslikely that some of the lessons that can be learned by analysing supersymmetric fieldtheories will hold inmuch larger generality. Onemay in particular hope to deepen ourinsights into the origin of quantum field theoretical duality phenomena by analysingsupersymmetric field theories, as will be discussed in more detail below. As anotherexample let us mention that it was expected for a long time that instantons play a keyrole for the behaviour of gauge theories at strong coupling. This can nowbe illustratedbeautifully with the help of the new exact results to be discussed in this volume. Webelieve that the study of supersymmetric field theories offers a promising path toenter into the mostly unexplored world of non-perturbative phenomena in quantumfield theory.

1.2 Electric-Magnetic Duality Conjectures

It is a hope going back to the early studies of gauge theories that there may existasymptotic strong coupling regions in the gauge theory parameter space in which aconventional (perturbative) description is recovered using a suitable new set of fieldvariables. This phenomenon is called a duality. Whenever this occurs, one may getaccess to highly nontrivial information about the gauge theory at strong coupling.

Exact Results on N = 2 Supersymmetric Gauge Theories 3

For future reference let us formulate a bit more precisely what it means to have aduality. Let us consider a family {Fz; z ∈ M} of quantum theories having a modulispaceMof parameters z. Thequantum theoryFz is for eachfixedvalue of z abstractlycharacterised by an algebra of observables Az and a linear functional on Az whichassigns to each observable O ∈ Az its vacuum expectation value 〈O〉z . We say that{Fz; z ∈ M} is a quantum field theory with fields � and action Sτ [�] depending oncertain parameters τ (like masses and coupling constants) if there exists a point z0in the boundary of the moduli spaceM, a coordinate τ = τ (z) in the vicinity of z0,and a map O assigning to each O ∈ Az a functional OO,τ [�] such that

〈 O 〉z �∫

[D�] e−Sτ [�] OO,τ [�], (1.1)

where � means equality of asymptotic expansions around z0 and the right hand sideis defined in terms of the action S[�] using path integral methods.

We say that a theory with fields �, action Sτ [�] and parameters τ is dual toa theory characterised by similar data S′

τ ′ [�′] if there exists a family of quantumtheories {Fz; z ∈ M} with moduli space M having boundary points z0 and z′

0 suchthat the vacuum expectation values of Fz have an asymptotic expansion of the form(1.1) near z0, and also an asymptotic expansion

〈 O 〉z �∫

[D�′]′ e−S′τ ′ [�′] O′

O,τ ′ [�′], (1.2)

near z′0, with O′ being a map assigning to each O ∈ Az a functional O′

O,τ ′ [�′].A class of long-standing conjectures concerning the strong coupling behavior of

gauge theories are referred to as the electric-magnetic duality conjectures. Some ofthese conjectures concern the infrared (IR) physics as described in terms of low-energy effective actions, others are about the full ultraviolet (UV) descriptions ofcertain gauge theories. The main content of the first class of such conjectures is mosteasily described for theories having an effective description at low energies involvingin particular an abelian gauge field A and some chargedmatter q. The effective actionS(A, q; τIR) will depend on an effective IR coupling constant τIR. The phenomenonof an electric magnetic duality would imply in particular that the strong couplingbehavior of such a gauge theory can be represented using a dual action S′(A′, q ′; τ ′

IR)

that depends on the dual abelian gauge field A′ related to A simply as

F ′μν = 1

2εμνρσ Fρσ. (1.3)

The relation between the dual coupling constant τ ′IR and τIR is also conjectured to be

very simple,

τ ′IR = − 1

τIR

. (1.4)

4 J. Teschner

The relation expressing q ′ in terms of q and Amay be very complicated, in general. Inmany cases one expects thatq ′ is thefield associated to solitons, localized particle-likeexcitations associated to classical solutions of the equations ofmotion of S(A, q; τIR).Such solitons are usually very heavy at weak coupling but may become light at strongcoupling where they may be identified with fundamental particle excitations of thetheory with action S′(A′, q ′; τ ′

IR).For certain theories there exist even deeper conjectures predicting dualities

between different perturbative descriptions of the full ultraviolet quantum field theo-ries. Such conjectures, often referred to as S-duality conjectures originated from theobservations of Montonen and Olive [MoOI, GNO], and were subsequently refinedin [WO, Os], leading to the conjecture of a duality between the N = 4 supersym-metric Yang-Mills theory with gauge group G and coupling τ one the one hand, andthe N = 4 supersymmetric Yang-Mills theory with gauge group LG and coupling−1/nGτ on the other hand. LG is the Langlands dual of a group G having as Cartanmatrix the transpose of the Cartan matrix of G, and nG is the lacing number1 of theLie algebra of G.

A given UV action S can be used to define such expectation values perturbatively,as well as certain non-perturbative corrections like the instantons. The question iswhether all perturbative and non perturbative corrections can be resummed to get thecross-over to the perturbation theory defined using a different UV action S′.

A non-trivial strong-coupling check for the S-duality conjecture in the N = 4supersymmetric Yang-Mills theory was performed in [VW].2 Generalised S-dualityconjectures have been formulated in [Ga09] (see [V:2] for a review) for a large class ofN = 2 supersymmetric gauge theories which are ultraviolet finite and therefore havewell-defined bare UV coupling constants τ . It is of course a challenge to establishthe validity of such conjectures in any nontrivial example.

1.3 Seiberg-Witten Theory

A breakthrough was initiated by the discovery of exact results for the low energyeffective action of certain N = 2 supersymmetric gauge theories by Seiberg andWitten [SW1, SW2]. There are several good reviews on the subject, see e.g. [Bi, Le,Pe97, DPh, Tac] containing further references.3

The constraints of N = 2 supersymmetry restrict the low-energy physics con-siderably. As a typical example let us consider a gauge theory with SU (M) gaugesymmetry. The gauge field sits in a multiplet ofN = 2 supersymmetry containing ascalar field φ in the adjoint representation of SU (M).N = 2 supersymmetry allows

1The lacing number nG is equal to 1 is the Lie-algebra of G is simply-laced, 2 if it is of type Bn ,Cn and F4, and 3 if it is of type G2.2The result of [Sen] furnishes a nontrivial check of a prediction following from theMontonen-Oliveconjecture.3A fairly extensive list of references to the early literature can be found e.g. in [Le].

Exact Results on N = 2 Supersymmetric Gauge Theories 5

parametric families of vacuum states. The vacuum states in the Coulomb branch canbe parameterised by the vacuum expectation values of gauge-invariant functions ofthe scalars like u(k) := 〈Tr(φk)〉, k = 2, . . . M . For generic values of these quantitiesone may describe the low-energy physics in terms of a Wilsonian effective actionSeff [A] which is a functional of d = M − 1 vector multiplets Ak , k = 1, . . . , d, hav-ing scalar components ak and gauge group U (1)k , respectively. The effective actionSeff [A] turns out to be completely determined by a single holomorphic functionF(a)

of d variables a = (a1, . . . , ad) called the prepotential. It completely determines the(Wilsonian) low energy effective action as Seff = Seff

bos + Sefffer , where

Seffbos = 1

4π

∫d4x

(Im(τ kl)∂μak∂

μal + 1

2Im(τ kl)Fk,μν Fμν

l + 1

2Re(τ kl)Fk,μν Fμν

l

),

(1.5)

while Sefffer is the sum of all terms containing fermionic fields, uniquely determined

by N = 2 supersymmetry. The a-dependent matrix τ kl(a) in (1.5) is the matrix ofsecond derivatives of the prepotential,

τ kl(a) := ∂ak ∂alF(a). (1.6)

Based on physically motivated assumptions about the strong coupling behaviorof the gauge theories under consideration, Seiberg and Witten proposed a precisemathematical definition of the relevant functions F(a) for M = 2. This type ofdescription was subsequently generalised to large classes ofN = 2 supersymmetricgauge theories including the cases with M > 2.

The mathematics underlying the definition of F(a) is called special geometry. Inmany cases including the examples discussed above one may describe F(a) usingan auxilliary Riemann surface � called the Seiberg-Witten curve which in suitablelocal coordinates can be described by a polynomial equation P(x, y) = 0. Thepolynomial P(x, y) has coefficients determined by the mass parameters, the gaugecoupling constants, and the values u(k) parameterising the vacua. Associated to �

is the canonical one form λSW = ydx on �. Picking a canonical basis for the firsthomology H1(�, Z) of �, represented by curves α1, . . . ,αd and β1, . . . ,βd withintersection index αr ◦ βs = δrs one may consider the periods

ar =∫

αr

λSW, aDr =

∫βr

λSW. (1.7)

Both ar ≡ ar (u) and aDr ≡ aD

r (u), r = 1, . . . , d, represent sets of complex coor-dinates for the d-dimensional space of vacua, in our example parameterised byu = (u(2), . . . , u(M)). It must therefore be possible to express aD in terms of a. It turnsout that the relation can be expressed using a function F(a), a = (a1, . . . , ad), fromwhich the coordinates ar can be obtained via aD

r = ∂arF(a). It follows that F(a) isup to an additive constant defined by � and the choice of a basis for H1(�, Z).

6 J. Teschner

The choice of the field coordinates ak is not unique. Changing the basisα1, . . . ,αd

and β1, . . . ,βd to α′1, . . . ,α

′d and β′

1, . . . ,β′d will produce new coordinates a′

r , a′Dr ,

k = 1, . . . , d along with a new functionF ′(a′)which is the prepotential determininga dual action S′

eff [a′]. The actions Seff [a] and S′eff [a′] give us equivalent descriptions

of the low-energy physics. This gives an example for an IR duality.

1.4 Localization Calculations of SUSY Observables

Having unbroken SUSY opens the possibility to compute some important quantitiesexactly using a method called localization [W88]. This method forms the basis formuch of the recent progress in this field.

Given a supersymmetry generator Q such that Q2 = P , where P is the generatorof a bosonic symmetry. Let S = S[�] be an action such that QS = 0. Let usfurthermore introduce an auxiliary fermionic functional V = V [�] that satisfiesPV = 0. We may then consider the path integral defined by deforming the actionby the term t QV , with t being a real parameter. In many cases one can argue thatexpectation values of supersymmetric observablesO ≡ O[�], QO = 0, defined bythe deformed action, are in fact independent of t , as the following formal calculationindicates. Let us consider

d

dt

∫[D�] e−S−t QV O =

∫[D�] e−S−t QV QV O

=∫

[D�] Q(e−S−t QV V O) = 0, (1.8)

if the path-integral measure is SUSY-invariant,∫ [D�] Q(. . . ) = 0. This means that

〈O 〉 :=∫

[D�] e−S O = limt→∞

∫[D�] e−S−t QV O. (1.9)

If V is such that QV has positive semi-definite bosonic part, the only non-vanishingcontributions are field configurations satisfying QV = 0. There are cases where thespaceM of solutions of QV = 0 is finite-dimensional.4 The arguments above thenimply that the expectation values can be expressed as an ordinary integral over thespace M which may be calculable.

The reader should note that this argument bypasses the actual definition ofthe path integral in an interesting way. For the theories at hand, the definition of∫ [D�] e−S−t QV represents a rather challenging task which is not yet done. Whatthe argument underlying the localisation method shows is the following: If there isultimately any definition of the theory that ensures unbroken supersymmetry in the

4In other cases M may a union of infinitely many finite-dimensional components of increasingdimensions, as happens in the cases discussed in Sect. 1.5.

Exact Results on N = 2 Supersymmetric Gauge Theories 7

sense that∫ [D�] Q(. . . ) = 0, the argument (1.8) will be applicable, and may allow

us to calculate certain expectation values exactly even if the precise definition of thefull theory is unknown.

1.5 Instanton Calculus

The work of Seiberg and Witten was based on certain assumptions on the strongcoupling behavior of the relevant gauge theories. It was therefore a major progresswhen it was shown in [N, NO03, NY, BE] that the mathematical description forthe prepotential conjectured by Seiberg and Witten can be obtained by an honestcalculation of the quantum corrections to a certain two-parameter deformation of theprepotential to all orders in the instanton expansion.

To this aim it turned out to be very useful to define a regularisation of certainIR divergences called Omega-deformation by adding terms to the action breakingLorentz symmetry in such a way that a part of the supersymmetry is preserved [N],5

S → Sε1ε2 = S + Rε1ε2 . (1.10)

One may then consider the partition function Z defined by means of the path inte-gral defined by the action Sε1ε2 . As an example let us again consider a theory withSU (M) gauge group. This partition function Z = Z(a, m, τ ; ε1, ε2) depends on theeigenvalues a = (a1, . . . , aM−1) of the vector multiplet scalars at the infinity of R

4,the collection m of all mass parameters of the theory, and the complexified gaugecoupling τ formed out of the gauge coupling constant g and theta-angle θ as

τ = 4πi

g2+ θ

2π. (1.11)

The unbroken supersymmetry can be used to apply the localisationmethod brieflydescribed in Sect. 1.4, here leading to the conclusion that the path integral definingZ can be reduced to a sum of ordinary integrals over instanton moduli spaces. Theculmination of a long series of works6 were explicit formulae for the summandsZ (k)(a, m; ε1, ε2) that appear in the resulting infinite series7 of instanton corrections

5The regularisation introduced in [N] provides a physical interpretation of a regularisation forintegrals over instanton moduli spaces previously used in [LNS, MNS1].6The results presented in [N, NO03] were based in particular on the previous work [LNS, MNS1,MNS2]. Similar results were presented in [FPS, Ho1, Ho2, FP, BFMT]; for a review see [V:4].7The infinite series (1.12) are probably convergent. This was verified explicitly for the example ofpure SU (2) Super-Yang-Mills theory in [ILTy], and it is expected to follow for UV finite gaugetheories from the relations with conformal field theory to be discussed in the next section.

8 J. Teschner

Z(a, m, τ ; ε1, ε2) = Zpert(a, m, τ ; ε1, ε2)

(1+

∞∑k=1

qk Z (k)(a, m; ε1, ε2)

), (1.12)

with q = e2πiτ in the ultraviolet finite cases, while it is related to the runningeffective scale � otherwise. The explicitly known prefactor Zpert(a, m, τ ; ε1, ε2)is the product of the simple tree-level contribution with a one-loop determinant.The latter is independent of the coupling constants qr , and can be expressed in termsof known special functions.

In order to complete the derivation of the prepotentials proposed by Seiberg andWitten it then remained to argue that F(a) ≡ F(a, m, τ ) is related to the partitionfunction Z as

F(a, m; τ ) = − limε1,ε2→0

ε1ε2Z(a, m; τ ; ε1, ε2), (1.13)

and to derive the mathematical definition of F(a) proposed by Seiberg and Wittenfrom the exact results on Z(a, m; τ ; ε1, ε2) obtained in [N, NO03, NY, BE].

2 New Exact Results on N = 2 Supersymmetric FieldTheories

2.1 Localisation on Curved Backgrounds

Another useful way to regularise IR-divergences is to consider the quantum fieldtheory on four-dimensional Euclidean space-times M4 of finite volume. The finite-size effects encoded in the dependence of physical quantities with respect to thevolume or other parameters of M4 contain profound physical information. It hasrecently become possible to calculate some of the these quantities exactly. One may,for example, consider gauge theories on a four-sphere S4 [Pe07], or more generallyfour-dimensional ellipsoids [HH],

S4ε1,ε2

:= { (x0, . . . , x4) | x20 + ε21(x2

1 + x22 ) + ε22(x2

3 + x24 ) = 1 }. (2.1)

The spaces S4ε1,ε2

have sufficient symmetry for having an unbroken supersymmetryQ such that Q2 is the sum of a space-time symmetry plus possibly an internalsymmetry. Expectation values of supersymmetric observables on S4

ε1,ε2therefore

represent candidates for quantities that may be calculated by the localisation method.Interesting physical quantities are the partition function on S4, and the values ofWilson- and ’t Hooft loop observables. Wilson loop observables can be defined aspath-ordered exponentials of the general form Wr,i := TrP exp

[ ∮C ds (i xμ Ar

μ +|x |φr )

]. The ’tHooft loop observables Tr,i , i = 1, 2, can be defined semiclassically by

performing a path integral over field configurations with a specific singular behavior