1 Introduction

# Gauge/Vortex duality and AGT

Gauge/Vortex Duality and AGT

Mina Aganagic and Shamil Shakirov

Center for Theoretical Physics, University of California, Berkeley, USA

Department of Mathematics, University of California, Berkeley, USA

Abstract

AGT correspondence relates a class of 4d gauge theories in four dimensions to conformal blocks of Liouville CFT. There is a simple proof of the correspondence when the conformal blocks admit a free field representation. In those cases, vortex defects of the gauge theory play a crucial role, extending the correspondence to a triality. This makes use of a duality between 4d gauge theories in a certain background, and the theories on their vortices. The gauge/vortex duality is a physical realization of large duality of topological string which was conjectured in [1] to provide an explanation for AGT correspondence. This paper is a review of [2], written for the special volume edited by J. Teschner.

## 1. Introduction

Large duality plays the central role in understanding dynamics of physical string theory. This duality is inherited by the simpler, topological string with target space a Calabi-Yau three-fold [3, 4, 5]. The topological large duality, like the large duality of the physical string theory, relates the gauge theory on D-branes to closed topological string on a different background. In the topological string case, the duality is in principle tractable, since topological string is tractable.

In some cases, study of topological string theory is related to studying supersymmetric gauge theory in 4d with supersymmetry, see e.g. [6, 7] and [V:13]. It is natural to ask what the large duality of topological string theory means in gauge theory terms. We will see that the large duality of topological string becomes a gauge/vortex duality [8, 9, 10] which relates a 4d gauge theory in a variant of 2d background with flux, and the theory living on its vortices.1 The vortices in the gauge theory play the role of D-branes of the topological string. In fact, the gauge theory duality implies the topological string duality, but not the other way around.

What does this have to do with AGT correspondence [16]? As we will review, [1] conjectured that large duality of topological string provides a physical explanation for AGT correspondence, under certain conditions: Conformal block should admit free field representation, and Liouville theory should have central charge to correspond to topological string.

We interpret this purely in the gauge theory language, in the context of the gauge/vortex duality, and show that this leads to a proof of correspondence in a fairly general setting. The partition function of the 4d gauge theory associated in [17, 18] to a genus zero Riemann surface with arbitrary number of punctures equals the conformal block of Liouville theory with arbitrary central charge , on the same surface. The free field representation of conformal blocks implies Coulomb moduli are quantized, but all other parameters remain arbitrary. The crucial role vortices play, extends AGT correspondence to a triality – between the gauge theory, its vortices, and Liouville theory. The striking aspect of this result, which appeared first in [2], is the simplicity of the proof. While in this review we focus on the simplest variant of AGT correspondence, relevant for Liouville theory, same ideas apply for more general Toda CFTs (Liouville theory corresponds to Toda). The generalization to Toda case can be found in [19].2

## 2. Background

Alday, Gaiotto and Tachikawa [16] conjectured a correspondence between conformal blocks of Liouville CFT and partition functions of a class of four-dimensional theories, in 4d -background [6]. The 4d theories are conformal field theories with supersymmetry defined in [17, 18] (see also [V:1]) in terms of a pair of M5 branes wrapping a Riemann surface , which we will call the Gaiotto curve. Specifying both the conformal block and the 4d theory in this class, involves a choice of the curve with punctures, data at the punctures and pants decomposition. The conjecture is often referred to as 4d/2d correspondence.

### 2.1. 4d Gauge Theory

Let be the Seiberg-Witten curve of ,

 (2.1) Σ:p2+ϕ(2)(z)=0.

with meromorphic one form . is a double cover of , is a local coordinate on , and is a degree 2 differential on , whose choice specifies the IR data of the theory (the point on the Coulomb branch). Specifying the UV data of the theory requires fixing the behavior of the Seiberg-Witten differential near the punctures.

At a puncture at , the has a pole of order , with residues

 p∼±αiz−zi

on the two sheets. These lead to second order poles of . In the gauge theory, ’s and ’s are the UV data; the mass parameters and the gauge couplings. also depends on the IR data of the gauge theory, the choice of Coulomb branch moduli. These are associated to the sub-leading behavior of the near the punctures.

Let

 ZT4d(Σ)

be the partition function of the theory, in 4d -background. Given a gauge theory description of , can be computed using results of Nekrasov in [6] (see also [V:3]). In addition to the geometric parameters entering , depends on

 ϵ1,ϵ2,

the two parameters of the background [6]. can in principle depend on data beyond the geometry of ; different choices of the pants decomposition can lead to different descriptions of the theory with different but related ’s.

### 2.2. 2d Liouville CFT

The Liouville CFT has a representation in terms of a boson :

 SLiouv.=∫dzd¯z√g[gz¯z∂zϕ∂¯zϕ+QϕR+e2bϕ].

Consider a conformal block on with insertions of primaries with momenta at points :

 B(α,z)=⟨Vα0(z0)⋯Vαℓ(zℓ)Vα∞(∞)⟩,

where

 Vα(z)=exp(−αbϕ(z))

is the vertex operator of a primary with momentum . Above, is the background charge, ; Liouville theory with this background charge has central charge . In addition to momenta and positions of the vertex operators inserted, the conformal block depends on the momenta in the intermediate channels; in denoting the conformal block by we have suppressed the dependence on the latter.

### 2.3. The correspondence

The conjecture of [16] is that the partition function computes a conformal block of Liouville CFT on :

 ZT4d(Σ)=B(α,z),

where is related to two parameters by

 b=√ϵ1ϵ2,

while the parameters , of map to the corresponding parameters in the conformal block and the Coulomb branch parameters map to the momenta in intermediate channels.

## 3. AGT and Large N Duality

In [1] Dijkgraaf and Vafa explained the correspondence, in a particular case of the self-dual -background,

 (3.1) ϵ1=gs=−ϵ2,

in terms of a large duality in topological string theory. The argument of [1] has three parts, which we will now describe. As everywhere else in this review, we will focus on the case when the Gaiotto curve is genus zero. One can extend the argument more generally [1], as all the ingredients generalize to a double cover of an arbitrary genus Riemann surface .

### 3.1. The Physical and the Topological String

The gauge theory partition function in the self-dual -background is conjectured in [1] to be the same as the partition function

 Z(YΣ)

of the topological B-model on a Calabi-Yau manifold , with topological string coupling . The Calabi-Yau is a hyper surface

 (3.2) YΣ:p2+ϕ(2)(z)=uv,

with holomorphic three-zero form . The geometry of and the Seiberg-Witten curve (2.1) are closely related: the latter is recovered from the former by setting or to zero.

This is a consequence of two facts. First, one observes that IIB string theory on is dual to M-theory with an M5 brane wrapping .3 This gives us another way to obtain the same 4d, theory . Second, the partition function of IIB string theory on times the self-dual background is the same as the topological B-model string partition function on [6, 25, 26]. Thus, one can simply identify the physical and the topological string partition functions

 (3.3) ZT4d(Σ)=Z(YΣ).

The power of this observation is that the topological B-model partition function is well defined even when the Nekrasov partition function is not – because for example, the gauge theory lacks a Lagrangian description. It is also important that sometimes one and the same topological string background gives rise to several different Lagrangian descriptions for one and the same theory – for example, with four fundamentals vs. with fundamentals. The former is the theory which is usually associated in the AGT literature to Liouville theory on the sphere with punctures; the latter is the one that naturally comes out from our approach.

### 3.2. Large N Duality in Topological String

Next, [1] show that the B-model on has a dual, holographic description, in terms of topological B-model branes on a different Calabi-Yau, related to , by a geometric transition. Let us first describe the Calabi-Yau that results. Then, we will explain the duality.

#### A Geometric Transition

By varying Coulomb branch moduli of we can get the Seiberg-Witten curve to degenerate. Let us call the degenerate curve that results the -curve:

 (3.4) S:p2−(W′(z))2=0.

Here

 W′(z)=ℓ∑i=0αiz−zi,

is determined by keeping the behavior of the Seiberg-Witten differential fixed at the punctures. The -curve describes the degeneration of the Seiberg-Witten curve to two components, . Correspondingly, a single M5 brane wrapping breaks into two branes, wrapping the two components.

The -curve corresponds to a singular Calabi-Yau :

 (3.5) YS:p2−(W′(z))2=uv,

with singularities at equal to zero and points in the -plane where

 W′(z)=0.

The Calabi-Yau we need is obtained by blowing up the singularities. One can picture this by viewing as a family of surfaces, one for each point in the -plane. At every there is an in the surface whose area is proportional to , The singularity occurs where the shrinks. After blowing up, we get a family of ’s of non-zero area, one at each point in the plane, and all homologous to each other. The minimal area ’s are where the singularities were – at points in the plane with .

The geometric transition trades for the blowup of . For economy of notations, we will denote and its blowup in the same way, since their complex structure is the same, given by (3.5).

#### Large N Duality

The B-model on has a holographic description in terms of B-model on (the blowup of ) with topological B-model D-branes wrapping the class. The branes get distributed between the minimal ’s at points in the -plane where vanishes. This breaks the gauge group from to , with . The Coulomb-branch moduli of get related to t’Hooft couplings in the theory on B-branes. The remaining parameters, and the topological string coupling are the same on both sides. This is the topological B-string version of gauge/gravity duality [4].

The large duality relates the closed topological string partition function of the B-model on , and thus the partition function , to partition function of the topological B-branes on (the blowup of) ,

 Z(YΣ)=Z(YS;N).

The right hand side depends not only on the net number of branes, but also how they are split between the different ’s.

The partition function of B-type branes wrapping the in a Calabi-Yau of the form of (3.5) was found in [4]. It equals

 (3.6) 1vol(U(N))∫dΦexp(TrW(Φ)/gs),

where is the volume of . The integral is a holomorphic integral, over complex matrices . In evaluating it, one has to pick a contour, ending at a critical point of the potential. In the present case,

 W(x)=∑iαilog(x−zi).

Diagonalizing and integrating over the angles, the integral reduces to

 (3.7) Z(YS;N)=1N!∫dNx∏I

Here is the order of the Weyl group that remains as a group of gauge symmetries.

The claim is that large expansion of the integral equals topological B-model partition function on (3.2). At the level of planar diagrams this can be seen as follows. In the matrix integral, define an operator

 (3.8) ∂ϕ(z)=W′(z)+gs∑I1z−xI,

where are the eigenvalues of . The expectation value of

 T(z)=(∂ϕ)2

computed in the matrix theory captures the geometry of the underlying Riemann surface by identifying in (2.1) with

 ϕ(2)(z)=⟨T(z)⟩.

There are two limits in which a classical geometry emerges from this. First, by simply sending to zero we recover the -curve, since then . But, there is also a new classical geometry that emerges at large . Letting ’s go to infinity, keeping fixed we get

 ⟨T(z)⟩∼(W′(z))2+f(z),

with

 f(z)=⟨gs∑IW′(z)−W′(xI)z−xI⟩.

From the form of the potential , it follows that has the form

 f(x)=∑iμix−zi

with at most single poles. Thus, the branes deform the geometry of the Calabi-Yau we started with. The resulting Calabi-Yau is exactly of the form (3.2), corresponding to the Seiberg-Witten curve in (2.1) at a generic point of its moduli space.

The large duality is expected to hold order by order in the expansion; we just gave evidence it holds in the planar limit (the full proof of the correspondence in the planar limit is easy to give along these lines, see [4]). The good variable in the large limit turns out to be the chiral operator we defined in (3.8). The field , is in fact the string field of the B-model.

The B-model string field theory, called Kodaira-Spencer theory of gravity, was constructed in [27], capturing variations of complex structure. For Calabi-Yau manifolds of the form (3.2) the Kodaira-Spencer theory becomes a two dimensional theory on the curve . The theory describes variations of complex structures of , so the Kodaira-Spencer field can be identified with fluctuations of the holomorphic form of the Calabi-Yau. For fluctuations of the form are equivalent to fluctuations of the meromorphic form on :

 δλ=δpdz=∂ϕdz.

The Kodaira-Spencer field is a chiral boson which lives on . When is a double cover of a curve , a single boson on is really a pair of bosons , on , one corresponding to each sheet. The field that arises in the matrix model in (3.8) can be thought of as off diagonal combination of the two. The diagonal combination is a center of mass degree of freedom and decouples from the dynamics of the branes.4

### 3.3. Topological D-branes and Liouville Correlators

To complete the argument, [1] observe that the B-brane partition function equals the Liouville correlator at , when written in the free-field or Dotsenko-Fateev representation [28, 29],

 (3.9) Z(YS;N)=B(α/gs,z;N)|c=1.

One treats the Liouville potential as a perturbation and computes the correlator in the free boson CFT

 (3.10) B(α,z;N)=⟨Vα1(z1)…Vαℓ(zℓ)Vα∞(∞)∮dx1S(x1)⋯∮dxNS(xN)⟩0,

where we took the chiral half. Here, is the screening charge

 S(z)=e2bϕ(z),

whose insertions come from bringing down powers of the Liouville potential. It follows that (3.10) vanishes unless

 α∞b+ℓ∑i=0αib=2bN+RQ,

constraining the net charge of the vertex operator insertions to be the number of screening charge integrals. This constraint can be found directly from the path integral, by integrating over the zero modes of the bosons [28, 29, 30]. We will place a vertex operator at infinity of the plane, and then the equation determines the momentum of the operator at infinity in terms of the momenta of the remaining vertex operators at finite points and numbers of screening charge integrals.

An integral expression for the expectation value of the correlator in (3.10) is easy to obtain, for example, by using the free boson mode expansion

 bϕ(z)=ϕ0+h0logz+∑k≠0hkz−kk,

where is a constant, and satisfy the standard algebra

 (3.11) [hk,hm]=−b22 kδk+m,0

where . From this one obtains the two point functions:

 ⟨Vα(z)Vα′(z′)⟩=(z−z′)−αα′2b2, ⟨Vα(z)S(z′)⟩=(z−z′)α, ⟨S(z)S(z′)⟩=(z−z′)−2b2.

The final result is that (3.10) equals

 B(α,z;N)=rN! ∫∏dNx∏i,I(xI−zi)αi∏J≤I(xI−xJ)−2b2,

where the integrals are over the position of screening charge insertions and

 r=∏i,j(zi−zj)−αiαj2b2

is a constant, independent on the integration variables. This is the free-field -ensemble (with ) reviewed in [V:7].

Setting (taking in Liouville CFT) and rescaling by , it follows immediately that the free field expression for the conformal block agrees with the partition function of B-branes in topological string on as we claimed in (3.9). Moreover, in the large limit, the holomorphic part of the Liouville field can be identified with the matrix model operator (3.8). This completes the argument of [1].

### 3.4. Discussion

The AGT conjecture, for can thus be understood as a consequence of a triality relating the closed B-model on , the holographic dual theory of -branes on the resolution of and the DF conformal blocks. The first two are conjectured to be related by large duality5 in topological string theory, the latter two by the fact that the partition function of -branes equals the DF block:

 (3.12) Z(YΣ)\lx@stackrelLargeN=Z(YS;N)=B(α/gs,z;N)|c=1.

We also used the embedding of topological string into superstring theory, which implies that the topological string partition function is the same as the physical partition function .

While this gives an explanation for the AGT correspondence in physical terms, it is by no means a proof: while the partition function of B-branes is manifestly equal to the Liouville conformal block in free field representation, the large duality is still a conjecture. The exact partition function of the B-model on is not known, so one can only attempt a proof, order by order in the genus expansion. In addition, there is a string theory argument, but no proof, that the partition function of the gauge theory and topological string partition function agree.

Thirdly, from the perspective of the 4d gauge theory, it is very natural to consider the partition function on general -background, depending on arbitrary , . Topological string on the other hand requires self-dual background, so the argument of [1] can not be extended in this case.6 In [1], it was suggested to formulate the refinement at the level of B-model string field theory. This remains to be developed better: refinement exists for any Calabi-Yau of the form ; the predictions from a naive implementation of this idea work for some, but not all choices of .

In the rest of the review, we will explain how to solve the last problem, and as it turns out the first two as problem as well, by following a different route.

The relation between topological string and superstring theory suggests one may be able to reformulate [1] in string theory language, replacing topological string branes by branes in string or M-theory. While topological string captures the case only, the full superstring or M-theory partition function makes sense for any . In fact, will will do something simpler yet: We will formulate the gauge theory analogue of [1] for any , . We will see that this approach is powerful – in fact it leads to a rigorous yet simple proof that the gauge theory partition function agrees with the free field Liouville conformal block for a sphere with arbitrary number of punctures.

The triality of relations between the 4d gauge theory, its vortices, and Liouville conformal blocks which admit free field representation implies AGT correspondence, however it stops short of the most general case. The restriction to blocks that admit free field representation means, from the 4d perspective, that the Coulomb moduli are quantized to be – arbitrary – integers, which get related to vortex charges on one hand, and numbers of screening charge integrals on the other.

## 4. Gauge/Vortex Duality

Translated to gauge theory language, the large duality of topological string theory becomes a duality between the 4d gauge theory and the 2d theory on its vortices; we will denote the later theory . Observations of relations between the two theories go back to [11, 12, 13, 14]. Recently [8, 9] proposed that the two theories are dual – indeed this is the ”other” 2d/4d relation. On the face of it, the statement is strange at best: to begin with, not even the dimensions of the 4d and the 2d theories match.

In this section we will show that, placed in a certain background, the 4d and the 2d theory describe the same physics, and thus there is good reason why their partition functions agree [10]. The large duality of [4, 1] becomes a duality between two , theories: the 4d gauge theory we started with, in a variant of 2d -background with vortex flux turned on, and the 2d theory on its vortices.

### 4.1. Higgs to Coulomb Phase Transition and Vortices

In gauge theory language, the geometric transition that relates B-model on a Calabi-Yau , first to a singular Calabi-Yau and then to a blowup of , is a Coulomb to Higgs phase transition. This follows from embedding of the B-model into IIB superstring on a Calabi-Yau, and the relation between the string theory and the gauge theory which arises in its low energy limit [31]. The same transition, in the language of M5 branes corresponds to degenerating a single M5 brane wrapping , to a pair of M5 branes wrapping two Riemann surfaces that the -curve consists of, and then separating these in the transverse directions (these are directions in the language of [32]).

The geometric transition becomes a topological string duality, as opposed to a phase transition, by adding B-branes on the in the blowup of . In terms of IIB string, the B-branes on the are D3 branes wrapping the and filling 2 of the 4 space-time directions. In terms of M5 branes, the vortices are M2 branes stretching between the M5 brane wrapping and the one wrapping . In the gauge theory on the Higgs branch, branes of string/M-theory become BPS vortices, as explained in [33, 34] and [13, 14].7

The vortices in question are non-abelian generalization of Nielsen-Olesen vortex solutions whose BPS tension is set by the value of the FI parameters. These were constructed explicitly in [13, 14]. The net BPS charge of the vortex is where is the field strength of the corresponding gauge group and the integral is taken in the 2 directions transverse to the vortex.8

### 4.2. Gauge/Vortex Duality

Consider subjecting the 4d theory to a two-dimensional -background in the two directions transverse to the vortex. We set to zero momentarily since the duality we want to claim holds for any . This is the Nekrasov-Shatashvili background studied in [39]. The 2d -background depends on the one remaining parameter, . (The equivalence of two theories is a stronger statement that the equivalence of their partition functions. The later assumes a specific background, while the former implies equivalence for any background. We will let be arbitrary once we become interested in the partition functions, as opposed to the theories themselves.)

As in [39], we view this partial -background as a kind of compactification: it results in a 2d theory with infinitely many massive modes, with masses spaced in multiples of . The background also preserves only out of the supercharges. Under conditions which we will spell out momentarily, the effective 2d theory that we get is equivalent to the theory on its vortices. The condition that is clearly necessary is that we turn on vortex flux. We assume it is also sufficient.

The vortex charge is where labels a gauge field in the IR, and is the corresponding field strength. Here, is the cigar, the part of the 4d space time with 2d deformation on it. It is parameterized by one complex coordinate, which we will call . Without the deformation, turning on would be introducing singularities in space-time which one would interpret in terms of surface operator insertions [35]. In background, one can turn on the vortex flux without inserting additional operators – in fact, the only effect of the flux is to shift the effective values of the Coulomb branch moduli. Let us explain this in some detail.

In the background, gets rotated with rotation parameter , in such a way that the origin is fixed. The best way to think of the theory that results [40, 39] is in terms of deleting the fixed point of the rotation, and implementing a suitable boundary condition. Because the disk is non-compact, we really need two boundary conditions: one at the origin of the plane and one at infinity. Turning on flux simply changes the boundary condition we impose at the origin. Without vortices, one imposes the boundary condition [40] that involves setting , where is the connection of -th gauge field along . With units of vortex flux on , we need instead .

In the -background, the 4d theory in the presence of units of vortex flux and with Coulomb branch scalar turned on is equivalent to studying the theory without vortices, at , but with shifted by

 ai→ai+Niϵ.

This comes about because in the background, always appears in the combination [40]

 ai+ϵwDi,w,

where is the covariant derivative along the -plane traverse to the vortex. Thus, in the background, at the level of F-terms, turning on vortex flux is indistinguishable from the shift the effective values of the Coulomb branch moduli.9

The 4d theory placed in 2d -background, with vortex flux turned on has an effective description studied in [39, 40] in terms of the theory with supersymmetry with massive modes integrated out. The theory has a non-zero superpotential ,where is the effective superpotential derived in [39], and the shift by is due to the flux we turned on. The critical points of the superpotential correspond to supersymmetric vacua of the theory. In the A-type quantization, considered in [39], the vacua are at or, equivalently, at . In the B-type quantization, they are at [41, 8, 9]. Choosing , for all is the vacuum at the intersection of the Higgs and the Coulomb branch. Choosing corresponds to putting the theory at the root of the Higgs branch – but in the background of units of flux.10

There is a second description of the same system. If we place the theory at the root of the Higgs branch, the 4d theory has vortex solutions of charge even without the -deformation. These are the non-abelian Nielsen-Olsen vortices of [13, 14]. We get a second theory with supersymmetry – this is the theory on vortices themselves. In the theory on the vortex, the only effect of the -deformation is to give the scalar, parameterizing the position of the vortex in the -plane, twisted mass . From this perspective, turning on is necessary since it removes a flat direction (position of vortices in the trasverse space).

Similarity of the two theories at the level of the BPS spectrum was observed in [11, 12, 13, 14, 15]. For a class of theories, this duality was first proposed in [8, 9], motivated by study of integrability. The physical explanation for gauge/vortex duality we provided implies the duality should be general, and carry over to many other systems.11

### 4.3. Going up a Dimension

The duality between , in the variant of the 2d -background we described above, and lifts to a duality in one higher dimension, between a pair of theories, and , compactified on a circle. We will prove the stronger, higher dimensional version, of the duality. lifts to a five-dimensional theory with supersymmetry. From 4d perspective, one gets a theory with infinitely many Kaluza-Klein modes. One can view this theory as a deformation of , depending on one parameter, the radius of the circle. Note that is not simply placed in a product of 2d -background times a circle – rather the background is a circle fibration

 (D×S1)t,

where as one goes around the D rotates by , sending .12 Similarly, the 2d theory on the vortex, lifts to a 3d theory , on a circle of the same radius. The claim is that the two , theories we get in this way are dual, where the duality holds at least at the level of -type terms. In the limit when goes to zero, the KK tower is removed, and we recover the theories we started with.

In the next section we will prove the duality by showing that partition functions of the two theories agree. When we compute the partition function of the 5d theory, we submit it to the full Nekrasov background depending on both and . This is the background

 (4.1) (D×C×S1)q,t,

where as one goes around the , we simultaneously rotate by , and by . In the 3d theory on vortices, is a twisted mass, but is a parameter of the background along the vortex world volume. The background for is fixed once we choose the background for , simply by the 5d origin of the vortices. is compactified on

 (4.2) (C×S1)q.

As we go around the , rotates by , and we turn on a Wilson line for a global symmetry rotating the adjoint scalar (and thus giving it mass ).

## 5. Building up Triality

When is a lift of the M5 brane theory of section 2 to a one higher dimensional theory on a circle of radius , the gauge/vortex duality extends to a triality. The triality is a correspondence between the 5d gauge theory , the 3d theory on its vortices , both on a circle of radius and a -deformation of Liouville conformal block. As goes to zero, the deformation goes away and we recover the conformal blocks of Liouville. The -deformation of the Virasoro algebra was defined in [45, 46], and studied further and as well as extended to W-algebras in [47].

The triality comes about because the partition function of the vortex theory will turn out to equal the -deformed Liouville conformal block,

 (5.1) ZV3d=Bq,

analogously to the way the partition function of topological D-branes was the same as the conformal block of Liouville at . The relation between and is the gauge/vortex duality. The duality implies that their partition functions are equal,

 (5.2) ZT5d=ZV3d.

The left hand side is computed on (4.1) and the right hand side, by restriction, on (4.2). Thus, combining the two relations, we get a relation between -deformation of the partition function of and the -deformation of the Liouville conformal block,

 (5.3) ZT5d=ZV3d=Bq.

In a limit, both deformations go away and we recover the relation between a partition function of the 4d, theory and the ordinary Liouville conformal block . We will prove this for the case when is a sphere with any number of punctures. The equality in (5.2), as we anticipated on physical grounds, holds for special values of Coulomb branch moduli – those corresponding to placing the 5d theory at a point where the Higgs branch and Coulomb branches meet, and turning on fluxes. By taking the large flux limit, where goes to infinity, goes to zero keeping their product fixed, all points of the Coulomb branch and arbitrary conformal blocks get probed in this way.

In the rest of the section we will spell out the details of the theories involved, and their partition functions. Then, in the next section, we will prove their equivalence.

### 5.1. The 5d Gauge Theory T5d

The 5d theory per definition reduces to, as we send to zero, the 4d theory arising from a pair of M5 branes wrapping a genus zero curve with punctures.

The theory turns out to be very simple: at low energies it is described by a gauge theory with hyper-multiplets: hypermultiplets in fundamental representation, in anti-fundamental, and 5d Chern-Simons level zero.13 Except for , the gauge theory theory is different from the generalized quiver of [17]. This is nothing exotic: there are different ways to take to zero limit, and different limits can indeed result in inequivalent theories. At finite , the theory we get is unique, but with possibly more than one description.

The Coulomb branch of the 4d theory is described by a single M5 brane wrapping the 4d Seiberg-Witten curve (2.1). The Seiberg-Witten curve of compactified on a circle can be written as

 (5.4) Σ:Q+(ex)ep+P(ex)+Q−(ex)e−p=0,

with the meromorphic one form equal to (see, e.g. [50]). We will denote both the 4d and the 5d Seiberg Witten curves by the same letter, even though the curves are inequivalent; it should be clear from the context which one is meant. Here, are polynomials of degree in ,

 Q±(ex)=e±ζ/2ℓ∏i=1(1−ex/f±,i),

and is a polynomial of degree in . At points where the Higgs and the Coulomb branch meet, degenerates to:

 (5.5) S:(Q+(ex)ep−Q−(ex))(e−p−1)=0.

The 5d Seiberg-Witten curve in (5.4) and the S-curve in (5.5) reduce to the 4d ones in (2.1), and (3.4), by taking the to zero limit. The limit one needs corresponds to keeping and fixed and taking

 (5.6) f+,i=zi,   f−,i=zi qαi.

Finally, one defines , and replaces by to get (3.4), the curve with its canonical one form . Note that one of the punctures we get is automatically placed at .14

#### Partition function in Ω-background

The 5d -background is defined as a twisted product

 (5.7) (C×C×S1)q,t,

where as, one goes around the , one rotates the two complex planes by and (the first copy of is what we called before). These are paired together with the 5d symmetry twist by , to preserve supersymmetry. The 5d gauge theory partition function in this background is the trace

 (5.8) ZT5d(Σ)=Tr(−1)Fg5d,

corresponding to looping around the circle in (5.7). Insertion of turns the partition function of the theory to a supersymmetric partition function. One imposes periodic identifications with a twist by where is a product of simultaneous rotations: the space-time rotations by and , the -symmetry twist, flavor symmetry rotations , and gauge rotation by for the ’th factor. The latter has the same effect as turning on a Coulomb-branch modulus (see [44] for a review). The partition function of in this background is computed in [6], using localization. The partition function is a sum

 (5.9) ZT5d(Σ)=r5d∑→RI5d→R,

over -touples of 2d partitions

 →R=(R1,…,Rℓ),

labeling fixed points in the instanton moduli space. The instanton charge is the net number of boxes in the ’s. The coefficient contains the perturbative and the one loop contribution to the partition function.

The contribution

 I5d→R=qζ|→R| zV,→R×zH,→R×zH†,→R

of each fixed point is a product over the contributions of the vector multiplets, the fundamental and anti-fundamental hypermultiplets , in . The instanton counting parameter, related to the gauge coupling of the theory, is . depends on Coulomb branch moduli encoded in , and the parameters related to the masses of the hypermultiplets. The vector multiplet contributes

 zV,→R=∏1≤a,b≤ℓ[NRaRb(ea/eb)]−1.

The fundamental hypermultiplets contribute

 zH,→R=∏1≤a≤ℓ∏1≤b≤ℓN∅Rb(vfa/eb),

and the anti-fundamentals give

 zH†,→R=∏1≤a≤ℓ∏1≤b≤ℓNRa∅(vea/fb+ℓ).

The basic building block is the Nekrasov function

 NRP(Q)=∞∏i=1∞∏j=1φ(QqRi−Pjtj−i+1)φ(QqRi−Pjtj−i) φ(Qtj−i)φ(Qtj−i+1),

with being the quantum dilogarithm [51, 2]. Furthermore, , and as before (we use the conventions of [52]). In what follows, it is good to keep in mind that there is no essential distinction between the fundamental and anti-fundamental hypermultiplets.15 In keeping with this, it is natural to think of all the matter multiplets at the same footing, and write the partition function, say, in terms of the fundamentals alone, whose masses run over values, , with .

### 5.2. The Vortex Theory V3d

The non-abelian generalization of Nielsen-Olesen vortices was found in [13, 14]. In particular, starting with a bulk non-abelian gauge theory like , with supercharges, gauge symmetry and hypermultiplets in fundamental representation, they constructed the theories living on its half BPS vortex solutions. The theory on charge vortices is very simple: it is a gauge theory with supercharges, with chiral multiplets in fundamental, and in anti-fundamental representation, as well as a chiral multiplet in the adjoint representation. The theory has a flavor symmetry rotating the chiral and anti-chiral multiplets separately. This symmetry prevents their superpotential couplings. Since is five dimensional, the theory on its vortices is three dimensional theory, which we will denote . Presence of the 2d background transverse to the vortex gives the adjoint chiral field twisted mass . In addition, the theory is compactified on a circle of radius . The masses of hypermultiplets of get related to the twisted masses of the chiral multiplets in . We will see the precise relation momentarily.

#### Partition function in Ω-background

We compactify on the 3d background:

 (C×S1)q.

As we go around the we simultaneously rotate the complex plane by and twist by the -symmetry, to preserve supersymmetry. The partition function of the theory in this background in computes the index

 (5.10) ZV3d(S;N)=Tr(−1)Fg3d,

where is a product of space-time rotation by , an symmetry transformation by , as well as the global symmetry rotation by . The partition function of the theory can be computed by first viewing the symmetry as a global symmetry: in this case, since the theory is not gauged, and due to the 3d background, the index in (5.10) is simply a product of contributions from matter fields and the -bosons, all depending on the Coulomb branch parameters .

The contribution of the flavor in the fundamental representation is

 (5.11) ΦF(x)=∏1≤I≤Nφ(eRxI−Rm−)φ(eRxI−Rm+),

where are the twisted masses. The right hand side is written in terms of Faddeev-Kashaev quantum dilogarithms [51, 2],

 φ(z)=∞∏n=0(1−qnz).

There are different ways to show this, for example, one can reduce the 3d theory down to quantum mechanics on the circle and integrate out a tower of massive states. Alternatively, the index can be obtained by counting holomorphic functions on the target space of the quantum mechanics, see [44]. We can think of the flavor in the fundamental representation in one of two equivalent ways: it is a pair of chiral multiplets, one in the fundamental and the other in the anti-fundamental representation. Alternatively, it contains a chiral multiplet and an anti-chiral multiplet, but both transform in the fundamental representation. The above way of writing is adapted to the second viewpoint.

The vector multiplet, the adjoint chiral field and the -bosons, give a universal contribution for any gauge group:

 (5.12) ΦV(x)=∏1≤I

The numerator is due to the W-bosons, and the denominator to the adjoint of mass scalar of mass . Finally, since the gauge group is gauged, we integrate over ’s. This simply projects to gauge invariant functions of the moduli space,

 (5.13) ZV3d(S;N)=1N!∫dNxΦV(x)ℓ∏a=1ΦFa(x)eζTrx/ℏ.

The integrand is a product including all contributions of the massive BPS particles in the theory, the bosons, flavors ’s, and the adjoint. The exponent contains the classical terms, the FI parameter , and the Chern-Simons level which is zero in our case. If the gauge symmetry were just a global symmetry, ’s would have been parameters of the theory and the partition function of the theory would have been the integrand. Gauging the symmetry corresponds to simply integrating16 over .

We need to determine the contour of integration to fully specify the path integral. The choice of a contour in the matrix model corresponds to the choice of boundary conditions at infinity in the space where the gauge theory lives [65]. At infinity, fields have to approach a vacuum of the theory. For small and , the vacua are the critical points of

 W(x)=ℓ∑a=1 logφ(eRx−Rm−,a)φ(eRx−Rm+,a).

There are vacua of both before and after the -deformation. Splitting the eigenvalues so that of them approach the -th critical point, we break the gauge group,

 U(N)→U(N1)×…×U(Nℓ).

We can think of all the quantities appearing in the potential as real; then the integration is along the real axis. To fully specify the contour of integration, we need to prescribe how we go around the poles in the integrand. The integral can be computed by residues, with slightly different prescriptions for how we go around the poles for the different gauge groups. In this way, we get distinct contours , and with them the partition function,

 ZV3d(S;N)=1∏ℓa=1Na!∮CN1,…,NℓdNxΦV(x);ℓ∏a=1ΦFa(x)e−ζTrx/ℏ.

Dividing by corresponds to dividing by the residual gauge symmetry, permuting the eigenvalues in each of the vacua. For this is a topological string partition function of the B-model on studied in [66], and related to Chern-Simons theory. The partition function is the partition function of refined Chern-Simons theory [54], with observables inserted.

We will show that the partition function of is nothing but the -deformation of the free-fieldfree field conformal block of the Liouville CFT on a sphere with punctures. Since the deformation of Liouville CFT might be not familiar, let us review it.

### 5.3. q-Liouville

In this section, we will show that the free field integrals of a -deformed Liouville conformal field theory [45, 46, 67] have a physical interpretation. They are partition functions of the 3d gauge theory, which we will called , in the 3d -background . The equivalence of the -Liouville conformal block and the gauge theory partition function is manifest. The screening charge integrals of DF are the integrals over the Coulomb branch of the gauge theory. Inserting the Liouville vertex operators corresponds to coupling the 3d gauge theory to a flavor. The momentum and position of the puncture are given by the real masses of the two chirals within the flavor.

The -deformed Virasoro algebra is written in terms of the deformed screening charges

 S(z)= :exp⎛⎝2ϕ0+2h0logz+∑k≠01+(t/q)kkhkz−k⎞⎠:,

where

 [hk,hm]=11+(t/q)k1−tk1−qk mδk+m,0.

The defining property of the generators of the deformed Virasoro-algebra, is that they commute with the integrals of the screening charges . The primary vertex operators get deformed as well. The vertex operator carrying momentum becomes:

 Vα(z)= :exp⎛⎝−αb2ϕ0−αb2h0logz+∑k≠01−q−αkk(1−t−k)hkz−k⎞⎠:.

Note, that these operators manifestly become the usual Liouville operators in the limit where go to , by sending to zero.

Just as before, using these commutation relations, one computes the correlator and obtains the following free field integral:

 (5.14) Bq(α,z;N)=r∏ℓa=1Na!∮C1,…,CℓdNyΔ2q,t(y)ℓ∏a=0Va(y;za),

where the measure is the -deformed Vandermonde

 Δ2q,t(y)=∏1≤I≠J≤Nφ(yI/yJ)φ(tyI/yJ),

and the potential equals

 Va(y;za)=N∏I=1φ(qαaza/yI)φ(za/yI).

In particular, using the properties of the quantum dilogarithm, it is easy to find that As in the undeformed case, the relation holds up to a constant of proportionality . In this paper, we avoid detailed consideration of this normalization constant. The meaning of the constant , on the Liouville side, is to account for all possible two-point functions between the vertex operators . Like in the undeformed case, the eigenvalues are grouped into sets of size , , by the choice of contours they get integrated over.17

## 6. Gauge/Liouville Triality

In what follows, we will prove that there is a triality that relates the 5d and 3d gauge theories and , compactified on a circle, and -deformation of Liouville conformal blocks. We will show this in two steps.

### 6.1. q-Liouville and V3d

The first step is to show that -deformation of the Liouville conformal block (5.14), corresponding to a sphere with punctures equals the partition function of :

 ZV3d(S;N)=Bq(α,z;N).

This follows immediately by a simple change of variables that sets

 (6.1) za=e−Rm+,a,qαa=eRm+,a−Rm−,a,y=e−Rx.

The insertion of a primary vertex operator in Liouville gets related to coupling the 3d gauge theory on the vortex to a flavor: the mass splitting is related to Liouville momentum, the mass itself to the position of the vertex operator. The puncture at arises from the Fayet-Iliopolous potential, if we set .

### 6.2. V3d and T5d: Gauge/Vortex Duality

The second step is to show that the partition function of the 5d gauge theory and partition function of its vortices, described by the 3d gauge theory agree

 ZV3d(S,N)=ZT5d(Σ).

For this we place at the point where the Coulomb and Higgs branches of meet, with as before, and degenerates to . In addition we turn on units of vortex flux.18 In the -background this is equivalent to not turning on flux and shifting the Coulomb-branch parameters of so that

 ZT5d(Σ)=r5d∑→RI5d→R

is evaluated at

 (6.2) ea=tNafa/v,

where runs form to . Here, are the masses of of the hypermultiplets, and the integer shifts correspond to