S-duality

The Gaiotto construction ofN “ 2 superconformal field theories gives a nice geometric interpretation of the S-duality ofN “2 gauge theories. One of the prime examples of the theories exhibit S-duality isSUp2q gauge theory with 4 fundamental hypermultiplets. It has vanishing beta function, and believed to be exactly conformal. It can be obtained from 2 M5-branes wrapped on the 4-punctured sphere.

---> x x x

x x

Figure 2.8: A weakly coupled description of the corresponding theory can be read off from going to a degenerate limit. Once this is done, a hhin tube corresponds to a gauge group, and punctures correspond to hypermultiplets. This givesSUp2qgauge theory withN_f “4 hypermultiplets

Figure 2.9: The quiver diagram corresponding to the 4-punctured sphere UV-curve. It can be easily read from considering the degenerate limit of the curve.

Note that any punctured Riemann surface can be constructed by gluing pair of pants.

The basic building block is the three-punctured sphere. It has a trifundamental hypermul-

x x

x x

x x

Figure 2.10: Obtaining a new curve from gluing pair of pants

tiplet which transform asp2, 2, 2q under the flavor symmetry SUp2q₁ˆSUp2q₂ˆSUp2q₃. Connecting two punctures corresponds to gauging the flavor symmetry. S-dualities can be

2

2

2 2

2 2

Figure 2.11: Gauging the flavor symmetry group can be thought of as gluing the pair of pants

understood as different choices of the degeneration limit coming from the single Riemann surfaceC_.

--->

x x x

x x

x

xx x

x

a b

c d

a c

b d

Figure 2.12: S-duality on theSUp2qtheory with 4 hypermultiplets. It can be understood as taking different degenerate limit. Since the both curves on the left and right are the same but rotated, the effective 4-dimensional theory has to be the same.

Since the S-duality can be concisely understood as an operation on the Riemann sur- face, it is natural to expect that S-duality is reflected in the 4d/2d correspondence as well.

In the 2d side, the partition function of 4d gauge theory is given by a correlation function on 2d theory. Indeed, the S-duality of the theory on the 4-punctured sphere is reflected as the channel duality of the correlation function. The theory corresponding to the one- punctured torus isN “2^˚theory. When the mass of adjoint hypermultiplet is set to zero, it is nothing butN “4 theory. The S-duality can be thought as a modular invariance of the 2d CFT. In practice, it is very hard to evaluate the correlation function of 2d theory exactly.

Therefore the direct check of S-duality from 4d/2d correspondence is still difficult, but we

can argue the duality on a more general ground.

Chapter 3

ABCDEFG of Instanton counting

In this chapter, we study methods of instanton counting to solve 4-dimensional gauge the- ories withN “2 supersymmetry for various gauge groups and matter fields. Especially in this chapter

• we find a uniform expression of 1-instanton partition function for arbitrary gauge groups in terms of root lattice [3],

• we find renormalization scheme dependence ofN “ 2 SCFT and interpret it geo- metrically in terms of a map between two Gaiotto curves [1],

• we derive the contour integral formula for the half-hypermultiplets [1, 2].

3.1 Nekrasov’s solution to N _{“ 2} gauge theory

Nekrasov partition function was introduced in [28], as a culmination of a long series of works, e.g.,[50, 51, 52] on the instanton calculation of the non-perturbative effects inN “2 gauge theory.

At low energies the four-dimensional N “ 2 gauge theory is governed by the pre- potentialF₀, which determines the metric on the Coulomb branch of the gauge theory.

Classically, the metric on the Coulomb branch is flat and the prepotential

F₀^clas“2πiτ_UV~a¨~a, (3.1)

is proportional to the microscopic coupling constantτ_UV. At the quantum level the pre- potential receives both one-loop and non-perturbative instanton corrections, which give

corrections to the metric on the Coulomb moduli space. The instanton corrections to the prepotential can be computed as equivariant integrals over the instanton moduli space [28]. Let us briefly sketch how this comes about. We will discuss in much more detail in section 3.2.

Instantons onR⁴are solutions of the self-dual instanton equation

F_A^` “0. (3.2)

The instanton moduli spaceM^Gparametrizes these solutions up to gauge transformations that leave the fiber at infinity fixed. The componentsM^G_k of the instanton moduli space are labeled by the topological instanton numberk“1{8π²ş

F_A^F_A. The instanton corrections to the prepotential for the pureN “2 gauge theory are captured by the instanton partition function

Z^inst“ ÿ

k

q^k

¿

M^G_k

1, (3.3)

where ű

1 formally computes the volume of the moduli space. The parameter q can be considered as a formal parameter which counts the number of instantons. Physically, it is identified with a powerq “ Λ^b0 of the dynamically generated scale Λ, when the gauge theory is asymptotically free. The powerb₀is determined by the one-loop β-function. It is identified with an exponentq“expp2πiτ_UVqof the microscopic couplingτ_UVwhen the beta-function of the gauge theory vanishes.

If we introduce hypermultiplets to the pureN _“2 gauge theory, the instanton correc- tion to the prepotential are instead determined by solutions of the monopole equations

F_A,µν<sup>`</sup> ` i

2 q_αΓ_µν^α_βq^β “0, (3.4)

ÿ

µ

Γ^µ_αα9 D_A,µq^α“0.

In these equations Γ^µ are the Clifford matrices and ř

µΓ^µD_A,µ is the Dirac operator in the instanton background for the gauge field A. Although there are no positive chiral- ity solutions to the Dirac equation, the vector space of negative chirality solutions is k- dimensional. Because this vector space depends on the gauge backgroundA, it is useful

to view it as ak-dimensional vector bundle over the instanton moduli spaceM_k^G. We will call this vector bundleV. More precisely, since the solutions to the Dirac equations are naturally twisted by the half-canonical line bundleLoverR⁴we will denote it byV _bL.

Instanton corrections to theN “2 gauge theory, withN_f hypermultiplets in the funda- mental representation of the gauge group, are computed by the instanton partition func- tion

Z^inst “ÿ

k

q^k

¿

M^G_k

epVbLbMq, (3.5)

which is the integral of the Euler class of the vector bundleV_bLof solutions to the Dirac equation over the moduli spaceM^G_k. The flavor vector space M“C^Nf encodes the num- ber of hypermultiplets in the gauge theory.

A difficulty in the evaluation of the instanton partition functions (3.3) and (3.5) is that the instanton moduli space M^G_k both suffers from an UV and an IR non-compactness.

Instantons can become arbitrary small, as well as move away to infinity inR⁴. The IR non- compactness can be solved by introducing theΩ-background, which refers to the action of the torus

T²_e1,e2 “Up1q_e1ˆUp1q_e2 (3.6) onR⁴ “ C‘Cby a rotationpz₁,z₂q ÞÑ pe^ie1z₁,e^ie2z₂qaround the origin with parameters e₁,e2 PC. If we localize the instanton partition function equivariantly with respect to the T²_e1,e2-action, only instantons at the fixed origin will contribute, so that we can ignore the instantons that run off to infinity. The UV non-compactness can be cured for gauge group UpNq by turning on an FI parameter. For Sp and SO gauge groups it is shown in [53]

how to evaluate the instanton integrals, while implicitly curing the UV non-compactness of the instanton moduli space. Note that this effectively means that we have introduced a renormalization scheme.

Apart from the torusT²_e1,e2 there are a few other groups that act on the instanton moduli spaceM^G_k. Their actions can be understood best from the famous ADHM construction of the instanton moduli space [54]. This construction gives the moduli space as the quotient of the solutions of the ADHM equations by the so-called dual groupG^D_k with Cartan torusT^k_φi

whose weights we will callφ_i. There is a also natural action of the Cartan torusT_~^N_a of the framing groupGon the ADHM solution space, whose weights are given by the Coulomb branch parameters~a. Last, if the theory contains hypermultiplets, there is furthermore an action of the Cartan T_m^N_~^f of the flavor symmetry group acting on M, whose weights correspond to the massesm~ of the hypers.

In total, we want to compute the partition function equivariantly with respect to the torus

T“T²_e1,e2ˆT_~^N_a ˆT^k_φiˆT^N_~m^f, (3.7) which comes down to computing the equivariant character of the action of those four tori.

This results in a rational functionz^kpφ_i,~a,~m,e₁,e₂qof the weights. From the construction of the Dirac bundle it is clear thatz^kfactorizes if there are multiple hypers. Finally, we need to take into account the ADHM quotient. This we do by integrating over the dual group G_k^D. In total the instanton partition function is given by the integral

Z^inst_k “ ż ź

i

dφ_iz^k_gaugepφ_i,~a,e₁,e2qz^k_matterpφ_i,~a,~m,e₁,e2q, (3.8)

where all the N “ 2 multiplets in the gauge theory give a separate contribution. The instanton partition function of in principle anyN _“ 2 gauge theory with a Lagrangian prescription can be computed in this way. We will derive the explicit expressions in section 3.2

The integrand of (3.8) will have poles on the real axis. To cure this we will introduce small positive imaginary parts for the equivariance parameters. At least for asymptotically free theories we can then convert (3.8) into a contour integral, so that the problem reduces to enumerating poles and evaluating their residues. ForUpNqtheory the poles are labeled byNYoung diagrams with in totalkboxes [28, 55, 56]: one way of phrasing this is that the UpNqinstanton splits intoNnon-commutativeUp1qinstantons.

For theSppNqorSOpNqtheory it is not that simple to enumerate the poles of the con- tour integrals. Furthermore, not only the fixed points of the gauge multiplet are more com- plicated, but (in contrast to theUpNqtheory) also matter multiplets contribute additional poles. As an example, in appendix B we devise a technique to enumerate all the poles for

anSppNqgauge multiplet. Each pole can still be expressed as a generalized diagram with signs, but the prescription is much more involved than in theUpNqcase.

The instanton partition functionZ^inst in theΩ-background obviously depends on the equivariant parameterse₁ande₂. In fact, it is rather easy to see that the series expansion of logpZ^instqstarts out with a term proportional to _e¹

1e2, which is the regularized volume of theΩ-background. Even better,Z^inst has a series expansion¹

Z^inst“expF^inst“exp

¨

˝

8

ÿ

g“0

¯

h^2g´2F_g^instpβq

˛

‚, (3.9)

in terms of the parameter ¯h² “ ´e₁e₂ andβ “ ´^e_e¹

2. We call the exponent of the instanton partition function the instanton free energyF^inst. As our notation suggests, we recover the non-perturbative instanton contribution to the prepotentialF₀from the leading contribu- tion of the exponent when ¯hÑ 0. This has been showed in [23, 55, 60]. Let us emphasize that the prepotentialF₀ does not depend on the parameterβ. The higher genus free en- ergiesF_gě1pβq compute gravitational couplings to theN “ 2 gauge theory, and play an important role in, for example, (refined) topological string theory.

To recover the full prepotential, we need to add classical and 1-loop contributions to the instanton partition function. We call the complete partition function

Z^Nek “Z^clasZ^1´loopZ^inst (3.10)

theNekrasov partition function.