Contents

March 2011

{centering} Quantization of Integrable Systems and a 2d/4d Duality

Nick Dorey and Sungjay Lee
DAMTP, Centre for Mathematical Sciences
Cambridge CB3 0WA, UK
and Timothy J. Hollowood
Department of Physics
Swansea University
Swansea SA2 8PP, UK

Abstract
We present a new duality between the F-terms of supersymmetric field theories defined in two- and four-dimensions respectively. The duality relates supersymmetric gauge theories in four dimensions, deformed by an -background in one plane, to gauged linear -models in two dimensions. On the four dimensional side, our main example is SQCD with gauge group and fundamental flavours. Using ideas of Nekrasov and Shatashvili, we argue that the Coulomb branch of this theory provides a quantization of the classical Heisenberg spin chain. Agreement with the standard quantization via the Algebraic Bethe Ansatz implies the existence of an isomorphism between the chiral ring of the 4d theory and that of a certain two-dimensional theory. The latter can be understood as the worldvolume theory on a surface operator/vortex string probing the Higgs branch of the same 4d theory. We check the proposed duality by explicit calculation at low orders in the instanton expansion. One striking consequence is that the Seiberg-Witten solution of the 4d theory is captured by a one-loop computation in two dimensions. The duality also has interesting connections with the AGT conjecture, matrix models and topological string theory where it corresponds to a refined version of the geometric transition.

## 1 Introduction

Supersymmetric gauge theories in two dimensions exhibit intriguing similarities to their four dimensional counterparts which have been noted many times in the past. Their common features include the existence of protected quantities with holomorphic dependence on F-term couplings and a related spectrum of BPS states which undergo non-trivial monodromies and wall-crossing transitions in the space of couplings/VEVs. In this paper we will propose a precise duality between specific theories in four- and two-dimensions (henceforth denoted Theory I and Theory II respectively). The duality applies to the large class of four dimensional theories with supersymmetry which can be realised by the standard quiver construction as in [1]. As our main example we have,

Theory I: Four-dimensional SQCD with gauge group , hypermultiplets in the fundamental representation with masses and hypermultiplets in the anti-fundamental with masses . The theory is conformally invariant in the UV with marginal coupling .

For some purposes it will also be useful to consider the corresponding gauge theory. We consider Theory I in the presence of a particular 1 Nekrasov deformation with parameter which preserves supersymmetry in an subspace of four-dimensional spacetime. The resulting effective theory in two dimensions is characterised by a (twisted) superpotential, with holomorphic dependence on (twisted) chiral superfields. The superpotential receives an infinite series of corrections from perturbation theory and instantons which encode the four-dimensional origin of the theory. It has an -dimensional lattice of stationary points corresponding to supersymmetric vacua of the deformed theory. These are determined by the F-term equation,

 →a=→mF−→nϵ →n=(n1,…,nL)∈ZL

where are the usual special Kähler coordinates on the Coulomb branch of the four-dimensional theory. A generic point on the Coulomb branch of the undeformed theory can be recovered in an appropriate , limit.

We will propose an exact duality of Theory I to a surprisingly simple model defined in two-dimensions which holds for all positive values of the integers introduced above;

Theory II: Two-dimensional supersymmetric Yang-Mills theory with gauge group with chiral multiplets in the fundamental representation with twisted masses and chiral multiplets in the anti-fundamental with twisted masses . In addition the theory has a single chiral multiplet in the adjoint representation with mass . The FI parameter and 2d vacuum angle combine to form a complex marginal coupling .

Theory II has a twisted effective superpotential which is one-loop exact [2]. In both Theory I and Theory II, the superpotential determines the chiral ring of supersymmetric vacuum states.

Claim: The chiral rings of Theory I and Theory II are isomorphic. In particular, there is a - correspondence between the supersymmetric vacua of the two theories and, with an appropriate identification of complex parameters, the values of the twisted superpotentials coincide in corresponding vacua (up to a vacuum-independent additive constant),

 W(I) on−shell≡ W(II)

The rank of the 2d gauge group is identified in terms of the 4d parameters according to . Thus, when is small, low values of correspond to points near the Higgs branch root of the 4d theory. The deformation parameter of Theory I is identified with adjoint mass of Theory II. The explicit map between the remaining parameters takes the form,

 ^τ=τ+12(N+1) ,→MF=→mF−32→ϵ ,→MAF=→mAF+12→ϵ . (1.1)

where . Further details of the map between the chiral rings of the two theories is given in Subection 2.5 below.

The initial motivation for this duality comes from the mysterious connection between supersymmetric gauge theories and quantum integrable systems developed in a remarkable series of papers by Nekrasov and Shatashvili (NS) [3, 4]. These authors propose a general correspondence in which the space of supersymmetric vacua of a theory with supersymmetry is identified with the Hilbert space of a quantum integrable system. The generators of the chiral ring are mapped to the commuting conserved charges of the integrable system. The twisted superpotential itself corresponds to the so-called Yang-Yang potential which is naturally thought of as a generating function for the conserved charges. The ideas of [4] also extend the well known connection between supersymmetric gauge theory in four-dimensions and classical integrable systems [6, 7, 8] which is reviewed in Section 2 below. In particular they propose that the introduction of a Nekrasov deformation in one plane, breaking four-dimensional supersymmetry to an subalgebra, corresponds to a quantization of the corresponding classical integrable system with the deformation parameter playing the role of Planck’s constant .

Our main observation is that the same quantum integrable system arises in two different contexts. In the case , it has been known for some time [9] that Theory I corresponds to a classical Heisenberg spin chain with spins whose Poisson brackets provide a representation of at each site. After imposing appropriate reality conditions, we will adapt the ideas of [4] and argue that introducing non-zero corresponds to a specific quantization of this system2 in which the classical spins at each site are replaced by quantum operators acting in a highest-weight representation of . The resulting quantum chain is integrable and can be diagonalised exactly using the Quantum Inverse Scattering method which leads to a simple set of rational Bethe Ansatz equations. However, precisely these equations also arise as the F-term equations of Theory II and the corresponding twisted superpotential, , coincides with the Yang-Yang potential of the spin chain [3]. This is not a coincidence: with the identification of parameters proposed above, Theory II can be identified as the worldvolume theory of a vortex string or surface operator probing the Higgs branch of Theory I3. As we discuss below, this connection suggests a physical explanation of the correspondence along the lines of [11, 12] as well as relations to several other recent developments.

Equivalence between the NS quantization and the standard quantization of the spin chain implies the duality between Theories I and II proposed above. In Section 3 below we test the duality by an explicit calculation of in each vacuum including classical, perturbative contributions as well as non-perturbative contributions up to second order in the four-dimensional instanton expansion. The corresponding computation of involves an iterative solution of the one-loop exact F-term equations in powers of the parameter . The calculation yields precise agreement in all vacua when the parameters are identified according to (1.1).

One interesting consequence of the proposed duality is that the full Seiberg-Witten solution of Theory I can be recovered by taking an , limit of the one-loop exact F-term equations of Theory II. In fact this is just the standard semiclassical limit of the non-compact spin chain (see e.g. [36, 38]) where the Bethe roots condense to form branch cuts in the spectral plane. In our context the resulting double cover of the complex plane is the Seiberg-Witten curve. This is very reminiscent of the Dijkgraaf-Vafa matrix model [10] approach where the eigenvalues of an matrix condense to form the branch cuts of the Seiberg-Witten curve. An important difference is that, in the NS limit, the Bethe Ansatz equations provide an exact solution of the system even at finite (see point 4 below).

In a forthcoming paper [52] we will also sketch a similar correspondence for a larger class of quiver theories in four (as well as five and six) dimensions. A common feature is that quantization leads to spin chains based on highest weight representations of Lie algebras where the ferromagnetic ground-state of the spin chain corresponds to the root of the gauge theory Higgs branch. In all cases, the Algebraic Bethe Ansatz leads to simple rational, (trigonometric, elliptic) equations which coincide with the F-term equations of a dual two-dimensional theory. In forthcoming work [52] we will show how the correspondence can be proved by directly relating the Bethe Ansatz equation to the saddle-point equation describing the instanton density in the Nekrasov-Shatashvili limit [4] (see also the recent papers [50, 51]).

There are also several interesting connections to other developments in supersymmetric gauge theory both new and old:

1: As already mentioned, there is a simple physical relation between the two theories which holds with identifications between the parameters described above: Theory II can be understood as the theory on the world-volume of a vortex string or surface operator [27, 28] probing the Higgs branch of the same four-dimensional theory. More precisely, Theory II is a gauged linear -model for the moduli space4 of non-abelian vortices of Theory I [13] (See also [29, 30]). The proposed duality relates the world-volume theory of vortices to the bulk theory (i.e. Theory I) on its Coulomb branch. As usual, Higgs phase vortices carry quantized magnetic flux. Interestingly, the dual Coulomb branch vacua of Theory I also exhibit quantized magnetic fluxes in the presence of the -deformation.

2: The duality provides an analog of the geometric transition of Gopakumar and Vafa [31] for the “refined” topological string [45] with refinement parameters in the Nekrasov-Shatashvili (NS) limit . Theories I and II correspond to the closed and open string sides of the transition respectively.

3: The duality can also be understood in terms of the conjecture [26] relating four dimensional supersymmetric gauge theory with Liouville theory. We will argue that in certain cases the proposed duality is equivalent to the conjecture of Alday et al [27] that a surface operator in gauge theory corresponds to a particular degenerate operator in Liouville theory.

4: Recent work [32] has advocated a further duality between supersymmetric gauge theories in a four dimensional -background and matrix models. In particular, the Nekrasov partition function of Theory I should have a matrix integral representation. Here we identify the dimension of matrix as the rank of the gauge group of Theory II. We conjecture that the resulting integral over the matrix eigenvalues can be evaluated explicitly by a saddle point in the NS limit even at finite . The saddle-point equations are precisely the Bethe Ansatz equations of the spin chain and the free energy is equal to the prepotential of the gauge theory. This should also have interesting consequences for refined open topological string amplitudes in the limit .

5: Some time ago a duality was proposed [33, 34] relating the BPS spectrum of a two-dimensional theory and that of a corresponding four-dimensional gauge theory at the root of its Higgs branch. The correspondence is further studied in [35] with care given to precise supermultiplet counting together with their wall-crossing behaviour. We show that the present proposal reduces to the earlier one in the limit .

The rest of the paper is organised as follows. In Section 2, we review the basic features of Theory I, its relation to the classical Heisenberg spin chain in the case and its quantization for . We also introduce Theory II and review its realisation on the worldvolume of a vortex string/surface operator and provide the precise statement of our duality conjecture. Section 3 is devoted to a detailed check of this proposal. Discussion of our results, generalisations and connections to topological strings, matrix models and Liouville theory are presented in Section 4.

## 2 Supersymmetric Gauge Theory and Integrable Systems

In this Section we will begin reviewing the relevant feature of the four-dimensional theory introduced in Section 1. As above, we focus on four-dimensional Super QCD with gauge group and hypermultiplets. For an gauge theory, hypermultiplets in the fundamental and anti-fundamental representations of the gauge group are essentially equivalent. It is nevertheless convenient to focus on the case where half of the hypermultiplets are in the fundamental representation and the rest in the anti-fundamental representation. The duality discussed below will apply equally to the corresponding gauge theory which differs from the theory by an additional factor which is IR free. The fundamental and anti-fundamental hypermultiplet masses are denoted by and with respectively. The theory is conformally invariant in the UV with marginal coupling .

The exact low-energy solution of the undeformed theory is governed by the corresponding Seiberg-Witten curve,

 L∏l=1(v−~ml)t2−2L∏l=1(v−ϕl)t−h(h+2)L∏l=1(v−ml)=0 .h(τ)=−2qq+1 , (2.1)

where is the factor associated with a four-dimensional Yang-Mills instanton. Here denote the classical eigenvalues of the adjoint scalar field in the vector multiplet. As the gauge group is we impose a traceless condition .

The masses of BPS states in the four-dimensional theory are determined by a meromorphic differential, on the curve. A standard basis of A- and B-cycles on the Seiberg-Witten curve, with can be defined in the weak-coupling limit . Electric and magnetic central charges are determined by the periods of in this basis,

where with similar notation for other -component vectors. For theories with matter the Seiberg-Witten differential also has simple poles at the points , with residues and respectively. It is convenient to introduce additional cycles and encircling these poles so that the corresponding periods of are,

 →mF=12πi∮→CFλSW, →mAF=12πi∮→CAFλSW .

The standard IIA brane construction of Theory I at a generic point on its Coulomb branch is shown in Figure (2.1). Here we follow the conventions of [1]. Each horizontal line corresponds to a D4 brane. In the figure each D4 is labelled by the corresponding value of the complex coordinate . In the following, the point on the Coulomb branch where it touches the Higgs branch will have a particular significance. The configuration corresponding to this Higgs branch root is shown in Figure (2.2). At the root of baryonic Higgs branch, the cycles degenerate leading to additional massless hypermultiplets. This corresponds to a factorization of the Seiberg-Witten curve. More precisely, when ’s are tuned to satisfy a relation

 −hL∏l=1(v−~ml)+(h+2)L∏l=1(v−ml)=2L∏l=1(v−ϕl) , (2.3)

the Seiberg-Witten curve becomes degenerate

 [L∏l=1(v−~ml)t−(h+2)L∏l=1(v−ml)]×[t+h]=0 , (2.4)

and . We will soon explain a correspondence between the root of baryonic Higgs branch and ferromagnetic vacuum of the integrable model.

### 2.1 The classical integrable system

We now review the connection between supersymmetric gauge theories in four dimensions and complex classical integrable systems. We begin by introducing the Heisenberg spin chain.

We will consider a chain of complex “spins” [36, 37, 38] corresponding to classical variables, , , for with Poisson brackets:

 {L+l,L−m}=2iδlmL0m {L0l,L±m}=±iδlmL±m . (2.5)

Here , and are indices in the Lie algebra . The spins at each site have a fixed value of the quadratic Casimir,

 L+lL−l+(L0l)2 = J2l . (2.6)

Integrability of the classical spin chain starts from an auxiliary linear problem based on the Lax matrix,

 Ll(x) = (x+iL0liL+liL−lx−iL0l) , (2.7)

where is a spectral parameter. A tower of commuting conserved quantities are obtained by constructing the corresponding monodromy matrix,

 T(x) = VL∏l=1Ll(x−θl) ,

where we have included inhomogeneities , , at each site and a diagonal “twist” matrix,

 V = (−h00h+2) .

As usual, the trace of the monodromy matrix is the generating function for a tower of conserved charges,

 2P(x) = tr2[T(x)] (2.8) = 2xL+q1xL−1+…+qL−1x+qL .

One may check starting from the Poisson brackets (2.5) that the conserved charges, , are in involution: , , which establishes the Liouville integrability of the chain. The lowest charge can be set to zero by a linear shift of the spectral parameter and we will do so in the following.

As for any integrable Hamiltonian system, the exact classical trajectories of the spins can be found by a canonical transformation to action-angle variables. Using standard methods, the action variables are identified as the moduli of a spectral curve defined by the equation,

 F(x,y) = det(y12−T(x))=0 ,

while the angle variables naturally parameterise the Jacobian variety More explicitly the curve takes the form,

 ΓL:y2−2P(x)y−h(h+2)K(x)~K(x) = 0 , (2.9)

where,

 K(x)=L∏l=1(x−θl−iJl) ~K(x)=L∏l=1(x−θl+iJl) .

This curve is an equivalent form of the Seiberg-Witten curve (2.1) for Theory I provided we make the identifications,

 →mF=→θ+i→J →mAF=→θ−i→J ,

and

 2P(x) = 2xL+q2xL−2+…+qL−1x+qL = 2L∏l=1(x−ϕl).

Also is identified with the twist parameter of the spin chain denoted by the same letter. The relation between (2.1) and (2.9) corresponds to the holomorphic change of variables, , .

With this identification different values of the moduli of Theory I are associated with different values of the integrals of motion of the complex spin chain. One point of particular interest is the ferromagnetic vacuum of the chain where each spin is in its classical groundstate: , . It is easy to check that this corresponds to the root of the Higgs branch in the gauge theory where the VEVs take the values .

### 2.2 Nekrasov-Shatashvili Quantization

We now turn our attention to the gauge theory in the presence of the so-called -background. This deformation, which breaks four-dimensional Lorentz invariance, is specified by parameters and . The F-terms of the deformed theory are determined by the Nekrasov partition function [21, 22],

 Z(→a,ϵ1,ϵ2) .

In this paper we will be mainly concerned with the Nekrasov-Shatashvili limit with held fixed, where the deformation is restricted to one plane in . We define a quantum prepotential in this limit as,

 F(→a,ϵ) = limϵ2→0[ϵ1ϵ2logZ(→a,ϵ1,ϵ2)|ϵ1=ϵ] .

In the further limit , the quantum prepotential reduces to the familiar prepotential of the undeformed theory: . Following [39, 41], the quantum prepotential can be obtained by a suitable deformation of the Seiberg-Witten differential appearing in (2.2),

 λ(ϵ) = λSW+O(ϵ) ,

with periods,

such that,

For convenience we will suppress the dependence of the deformed central charges from now on and denote them simply as and .

For , the four-dimensional supersymmetry is broken down to supersymmetry two dimensions. The zero modes of the vector multiplet in the four-dimensional low energy theory give rise to a field strength multiplet in two dimensions. This multiplet includes the gauge field strength in the undeformed directions and the scalar fields which parametrize the 4d Coulomb branch. Thus is the lowest component of a twisted chiral superfield in superspace. This superfield inherits a twisted superpotential from the partition function of the four-dimensional theory. The resulting twisted superpotential is a multi-valued function on the Coulomb branch,

 W(I)(→a,ϵ) = 1ϵF(→a,ϵ)−2πi→k⋅→a (2.12)

where the integer-valued vector5 corresponds to the choice of branch. This choice corresponds to the freedom to shift the 2d vacuum angle associated with each factor in the low energy gauge group by an integer multiple of . This is equivalent to introducing a constant electric field in spacetime which is then screened by pair creation [19, 2]. The vector thus specifies the choice of a quantized two-dimensional electric flux in the Cartan subalgebra of .

The main claim of [4] is that is the Yang-Yang potential for a quantization of the classical integrable system described above, in which the effective Planck constant is proportional to the deformation parameter . This requires further explanation: corresponding to a complex classical integrable system there can be several choices of reality condition which yield inequivalent real integrable systems. Each of these real systems gives rise upon quantization to different quantum integrable systems. As discussed in [4], the twisted superpotential given above gives rise to a particular quantization which they refer to as Type A which we now review.

The F-term equations coming from (2.12) can written in terms of the deformed magnetic central charge using (2.11) as follows,

This corresponds to a quantization condition for the conserved charges of the integrable system, each point of the lattice corresponds to a different quantum state. The values of the commuting conserved charges in each state are encoded in the on-shell value of the superpotential . Each branch of this multi-valued function corresponds to a supersymmetric vacuum and to a state of the quantum integrable system Differentiating with respect to the parameters yields the vacuum expectation values of chiral operators in the corresponding vacuum. For example,

To extract the VEVs of higher dimension chiral operators for we should deform the prepotential of the UV theory with appropriate source terms [18].

Theories with supersymmetry in four dimensions exhibit two distinct manifestations of electromagnetic duality. First there is the low-energy electromagnetic duality which provides an alternative description of the low-energy effective theory at any point on the Coulomb branch in terms of a dual field with dual prepotential , which is related to the original prepotential by Legendre transform6 .

In an gauge theory, the full group of low-energy duality transformation includes a copy of which acts linearly on the central charges . For Theory I, there are also additional additive transformations involving shifts of the central charges by integer multiples of the mass parameters , [20].

In [4], the authors proposed that these different formulations of the low-energy theory give rise upon deformation to non-zero , to different quantizations of the same complex classical integrable system. In particular, performing the basic electric-magnetic duality transformation we obtain a dual superpotential,

whose F-term equations give rise to dual quantization conditions, denoted Type B in [4],

The discrete choice of vacuum now corresponds to a choice of magnetic flux in the Cartan subalgebra of the gauge group. As a consequence of (2.15), we note that the two superpotentials and are actually equal as multi-valued functions when evaluated on-shell. This is true for both quantization conditions (2.13) and (2.16).

Summarising the above, the Type A and B quantization conditions can be written as,

 12πi∮→Aλ(ϵ)∈ϵZL and 12πi∮→Bλ(ϵ)∈ϵZL ,

respectively. Other quantizations corresponding to other transformations in the duality group naturally correspond to period conditions for different choices of basis cycles.

The theory with gauge group and hypermultiplets exhibits another form of electric-magnetic duality. The exact S-duality of the theory relates electric and magnetic observables of the theory at different values of the marginal coupling . In the present context it implies a non-trivial duality between Type A and Type B quantization at different values of .

### 2.3 Quantization via the Bethe Ansatz

In this section we will review the standard approach to quantizing the Heisenberg spin chain (see e.g. [23, 24, 36, 38]). Starting from the Poisson brackets (2.5) we make the usual replacement of the classical variables , at each site by operators , obeying commutation relations,

 [^L+l,^L−m]=−2ℏδlm^L0m [^L0l,^L±m]=∓ℏδlm^L±m , (2.17)

for . The spins at each site each commute with the Casimir operator,

 ^L2l = 12(^L+l^L−l+^L−l^L+l)+(^L0l)2=sl(sl−1)ℏ2.

Depending on the value of , these operators can act on representations of either or . We will focus on the latter case. If we choose the spins can be chosen to act in the principal discrete series representation7 of ,

 D+s = {|s,μ⟩,μ=s,s+1,s+2,…}

These are highest weight representations of and the resulting spin chain admits a tower of commuting charges which can be simultaneously diagonalised. By restricting to a representation with at the ’th site we are defining a real quantum integrable system. An important feature of this non-compact spin chain is that it has a semiclassical limit where each spin is highly excited. As we discuss below, the relation to the complex integrable system we discussed in subsection 2.1 becomes clear in this limit.

As we are dealing with a spin chain based on highest weight representations the Algebraic Bethe Ansatz [23] is applicable and can be used to find the exact spectrum of the model [24]. We now review the solution of the spin chain using an alternative approach based on the Baxter equation. We start by defining a quantum version of the classical Lax matrix (2.7) which takes the form,

 ^Ll(x) = ⎛⎝x+i^L0li^L+li^L−lx−i^L0l⎞⎠ (2.18)

and we define the quantum transfer matrix for an inhomogeneous chain with twisted boundary conditions by,

 ^T(x) = tr2[V→∏l=1^Ll(x−θl)] = 2xL+^q2xL−2+…+^qL−1x+^qL ,

where , , are a set of mutually commuting conserved charges which are the quantized versions of the corresponding classical charges .

The standard problem for a quantum integrable system is to find the eigenstates of the transfer matrix,

 ^T(x)|Ψ⟩ = t(x)|Ψ⟩ ,

where the eigenvalue, , is a polynomial of degree in the spectral parameter by construction. This is accomplished by allowing the transfer matrix to act on a “wavefunction” which leads to the Baxter equation,

 −ha(x)Q(x+iℏ)+(h+2)d(x)Q(x−iℏ) = t(x)Q(x) , (2.19)

where,

 a(x)=L∏l=1(x−θl+islℏ) d(x)=L∏l=1(x−θl−islℏ)

Looking for solutions where is a polynomial of degree ,

 Q(x) = N∏j=1(x−xj)

we impose the polynomiality of to obtain equations for the zeros of ,

 L∏l=1(xj−θl−islℏxj−θl+islℏ) = qN∏k≠j(xj−xk+iℏxj−xk−iℏ) , (2.20)

for where . These are the Bethe Ansatz Equations (BAE). These equations are very well studied in the case of an untwisted homogeneous chain with , and . In particular, for this case, it is known that all the Bethe roots lie on the real axis [36]. The number of Bethe roots corresponds to the number of magnons in the corresponding state. The ferromagnetic vacuum of the spin chain is defined as an empty state , i.e., . It implies from the Baxter equation (2.19) that

 −ha(x)+(h+2)d(x)=t(x) , (2.21)

which is nothing but the condition at the root of baryonic Higgs branch (2.3). The ferromagnetic vacuum of the spin chain can therefore be identified as the Higgs branch root. The eigenvalues of the transfer matrix can easily be written in terms of the solutions of the BAE using the Baxter equation (2.19).

In the following it will also be important that the BAE corresponds to stationary points of an action function,

 Y(x) = 2πiτN∑j=1xj+N∑j=1L∑l=1[f(xj−θl+islℏ)−f(xj−θl−islℏ)] +N∑j,k=1f(xj−xk+iℏ)+iπ(N+1)N∑j=1xj ,

where and .

To understand the relation to the classical system of subsection 2.1, it will be useful to consider the semiclassical limit of the quantum chain in which the excitation numbers at each site become large. The quantum numbers which determine the spin representation at each site scale like . They are related to the classical Casimirs as . Alternatively, if are held fixed as one ends up with the classical Heisenberg magnet of spin . In either limit the quantum charges go over to Poisson commuting classical quantities .

The key feature of the semiclassical limit is that the Bethe roots condense to form cuts in the complex plane. In this limit the resolvent is naturally defined on a double cover of the -plane with two sheets joined along these cuts. The resulting Riemann surface is exactly the the spectral curve and the meromorphic differential can be identified with the Seiberg-Witten differential . For the case of the homogeneous untwisted chain of spin zero, the semiclassical limit is described in detail in Section 2.2 of [36]. In this case the Bethe roots condense to form branch cuts on the real axis. Working in the vicinity of the ferromagnetic vacuum, the curve can be represented as a double cover of the plane with real branch points8 at as shown in Figure (2.3). We also define one-cycles , surrounding each branch cut.

In the semiclassical limit, the quantum spin chain gives rise to a particular real slice of the complex classical spin chain considered above. The reality conditions select a middle-dimensional subspace of the original complex phase space. Allowing generic complex values of the moduli corresponds to working with a complexification of the spin chain in which is replaced by .

At the classical level, the moduli of the curve vary continuously. The leading semiclassical approximation the quantum spectrum arises from imposing appropriate Bohr-Sommerfeld quantisation conditions which are formulated in terms of the periods of the meromorphic differential on which coincides with the Seiberg-Witten differential ,

 12π∮αlλSW = ℏ^nl , (2.22)

for where are non-negative integers. In terms of the Bethe Ansatz, this is just the condition that each cut contains an integral number of Bethe roots. From this point of view it is obvious that the quantization conditions are unchanged by continuous variations of the parameters including the introduction of inhomogeneities and a non-trivial twist. In particular, they also apply in the weak coupling regime where the standard basis cycles of the Seiberg-Witten curve are defined. It will be useful to express the cycles appearing in the semiclassical quantization condition in this basis. The key point, mentioned above, is that the ferromagnetic vacuum of the spin chain corresponds to the Higgs branch root . In terms of the basis cycles defined above this corresponds to the point in moduli space where the cycles vanish. Thus we have .

### 2.4 Two-dimensional gauge theory and vortex strings

The next observation, following [3], is that the BAE for the Heisenberg spin chain themselves arise as the F-term equations of a certain two-dimensional gauge theory with supersymmetry which we will call Theory II. As above Theory II is a gauge theory with fundamental chiral multiplets with twisted masses and anti-fundamental chiral multiplets with twisted masses . The theory also contains an adjoint chiral multiplet with twisted mass and has a marginal complex coupling which corresponds to a background twisted chiral superfield.

We begin by focussing on the case . In this case the model can be realised on the worldvolume of D2 branes probing a configuration of intersecting NS5 and D4 branes in Type IIA string theory [48, 13] as shown9 in Figure (2.4). Since the brane-configuration is invariant under the rotations in {2,3}-, {4,5}- and {8,9}-planes, the Theory II has, at least classically, global symmetry groups as well as flavor symmetry groups . Here the FI parameter is proportional to . Classical vacua are determined by solving the D-term equations,

 L∑l=1(QlQl†−~Q†l~Ql)−[Z,Z†]=r , (2.23)

and

 L∑l=1∣∣λQl−QlMl∣∣2+L∑l=1∣∣−~Qlλ+~Ql~Ml∣∣2=0 , (2.24)

where denotes the adjoint scalar field in the vector multiplet.

For , and Theory II has a classical Coulomb branch parametrized by the eigenvalues of the adjoint scalar field in the vector multiplet. In the figure this corresponds to the special case where each D2 is suspended between NS5 and NS5 and can move independently in the and directions. On the other hand, the eigenvalues of parameterise the position of D2-branes in the {2,3}-plane.

For , the theory is on a Higgs branch with , . The vector multiplet VEVs are fixed by the second D-term condition (2.24). Solutions are labelled by the number of ways of distributing the scalars between the values . Thus we specify a vacuum by choosing non-negative integers with . In the brane construction these correspond to the number of D2 branes ending on each D4 brane as shown in the figure.

The brane construction also reveals an interesting physical relation between Theory I and Theory II [12]. In the absence of the D2 branes, the worldvolume theory on the intersection of the remaining branes is precisely SQCD with gauge group and hypermultiplets with masses and and complex gauge coupling . In other words, it is Theory I in the undeformed case . To understand the relation, compare the brane configuration shown in Figure (2.1) with the configuration in Figure (2.4) (in the absence of D2 branes). The former configuration represents a generic point on the Coulomb branch of Theory I. To pass to the configuration shown in Figure (2.4), we reconnect the D4 branes on NS5 and then move NS5 away from the D4 branes in the direction. The first step corresponds to moving on the Coulomb branch to the Higgs branch root root (see Figure (2.2)), the second to moving out along the Higgs branch. The theory on the Higgs branch admits vortex strings which corresponds to the D2 branes in the Figure (2.4). Thus we identify Theory II as the worldvolume theory on vortex strings probing the Higgs branch of Theory I. For a review of such non-abelian vortex string see [14].

At this point there are several subtleties. First, to have BPS vortices with a supersymmetric worldvolume theory we must consider a four-dimensional theory with gauge group rather than . Thus, although the F-term duality described in this paper works the same for both theories, the interpretation in terms of vortex strings is only available in the case. Next, the gauge theory designated as Theory II above cannot be directly identified as the worldvolume theory of the vortex. Rather, Theory II is an gauged linear -model which flows in the IR to a non-linear -model (with twisted mass terms) whose target space is a certain Kähler manifold which is closely related to the moduli space of vortices in Theory I [13]. In the present case, where the four-dimensional theory is conformal, the target space has zero first Chern class and can therefore be expected to admit a unique Ricci-flat metric which provides a natural IR fixed-point for the worldsheet theory. In fact, the actual Kähler metric on the classical vortex moduli space differs from this target space metric. Even worse, as we are discussing semi-local vortices, some elements of the true classical moduli space metric diverge due to the non-normalisability of some of the vortex zero modes (see eg [15, 16]). Although this has not been analysed in detail, we expect that this problem is cured once the -background is reintroduced10. Indeed, as above, a Nekrasov deformation in one plane renders even the vacuum moduli of the four-dimensional gauge theory normalisable as fields in a two-dimensional effective action. Correspondingly we expect that, in the present context, the true world-volume theory of vortices flows to the same IR fixed point as Theory II. However, what we actually need here is much weaker: that the two theories agree at the level of F-terms and we will assume this is the case.

With the identification described above, the numbers labelling the classical vacua of the worldvolume theory determine the number of units of magnetic flux in each of the Cartan subalgebra of . The tension of the vortex strings is controlled by the Higgs branch VEV of the four-dimensional theory, proportional to . In the limit, , the tension diverges and the vortex strings become static surface operators of the type considered in [27].

Next we restore the Nekrasov deformation to Theory I. From a two-dimensional perspective this corresponds to a twisted mass term for fields charged under rotations in the - plane. The adjoint chiral multiplet has charge under this symmetry and this field acquires mass . Thus the adjoint chiral mass in Theory II is identified with the Nekrasov deformation parameter in Theory I. More generally, we can consider a twisted mass corresponding to - rotations mixed with the other global symmetries which act on the chiral multiplets and . In fact the relevant symmetry for the duality we consider turns out to be one under which has charge and has charge . The corresponding dictionary between the 2d and 4d masses then becomes,

 →MF=→mF−32→ϵ ,→MAF=→mAF+12→ϵ . (2.25)

where .

Having discussed the classical behaviour of Theory I, we now turn our attention to the quantum theory. Quantum effects lift the classical Coulomb and Higgs branches described above leaving only isolated supersymmetric vacua corresponding to stationary points of the twisted superpotential. Upon integrating out the matter multiplets we obtain an effective twisted superpotential on the Coulomb branch of the form,

 W(II)(λ) = 2πi^τN∑j=1λj−ϵN∑j=1L∑l=1f(λj−Mlϵ) (2.26) +ϵN∑j=1L∑l=1f(λj−~Mlϵ)+ϵN∑j,k=1f(λj−λk−ϵϵ) ,

where and denotes the two-dimensional holomorphic coupling constant. It is known that this twisted superpotential is one-loop exact. Defining

 q=exp(2πiτ)≡(−1)N+1exp(2πi^τ) ,

the resulting F-term equations can be expressed as below

 L∏l=1(λj−Mlλj−~Ml) = q∏k≠j(λj−λk−ϵλj−λk+ϵ) (2.27)

with , coincide with the BAE (2.20) with the identification of the variables with and setting,

 →MF=→θ+i→sℏ →MAF=→θ−i→sℏ

and . The marginal coupling is identified with the corresponding parameter in the Yang-Yang functional.

The space of supersymmetric ground states of Theory II is now identified with the Hilbert space of the quantum spin chain. Strictly speaking this correspondence holds when the complex parameters of Theory II satisfy the reality conditions implied by the above identifications. More generally one must consider an analytic continuation of the quantum spin chain to complex values of its parameters. The rank of the 2d gauge group corresponds to the number of magnons in the spin chain state. Pleasingly the total number of states of the spin chain containing magnons is the number of partitions of into non-negative integers which is the same as the number of classical SUSY vacua of the theory. It is important to note that these vacua, like the states of the spin chain, correspond to non-degenerate solutions of (2.27) where for all . Degenerate solutions are points where the Coulomb branch effective description breaks down and any corresponding vacuum states would have unbroken non-abelian gauge symmetry. The absence of such vacua is consistent with the results of [17].

To summarise the above discussion, Theory II corresponds to the worldvolume theory of vortex strings probing the Higgs branch of Theory I. The tension of these strings is controlled by the Higgs branch VEV of the four-dimensional theory which corresponds to a D-term in superspace [12]. The F-term equations of motion of Theory II coincide with the BAE for the Heisenberg spin chain. This conclusion is independent of D-term couplings and therefore hold equally well in the limit of infinite tension where the vortex string becomes a surface operator.

### 2.5 The duality proposal

As mentioned above starting from a complex classical integrable system there can be many inequivalent reality conditions and many different quantizations. In subsections 2.1 and 2.3 we have discussed the particular reality conditions which lead to the Heisenberg spin chain and the standard quantization which leads to an integrable quantum spin chain. On the other hand we have reviewed the Nekrasov-Shatashvili quantization which produces a family of reality conditions and quantizations related by electric-magnetic duality transformations. It is natural to ask whether one of this family corresponds to the standard quantization of the spin chain. To start with we can address this question in the semiclassical regime of small . In this case the quantization conditions of the NS proposal take the generic form,

 ∮→AλSW ∈ ϵZ ,

for some choice of one-cycles on which are the image under the low-energy duality group of the basis cycles appearing in the Type A quantization condition. We obtain agreement with the semiclassical limit of the chain if and the basis cycles are chosen as .

With the choices outlined above, the NS quantization condition coincides with the standard one at leading semiclassical order. In a generic quantum system it is easy to imagine different quantizations which agree at leading order in . However, for a real classical integrable system with a given symplectic form, quantizations which preserve integrability are very special and we do not know of an example where two distinct quantizations coincide at leading semiclassical order. For this reason, we conjecture that the agreement between the NS quantization and the standard one persists to all orders.

The Hilbert space resulting from the NS quantization corresponds to the space of SUSY vacua determined by the superpotential,

which is obtained by applying a duality transformation to the superptotential given in (2.12). In addition to the standard electric-magnetic duality, the transformation includes the shift , which is also part of the low-energy duality group [20]. The dual superpotential leads to the F-term equations,

 →a−→mF ∈ ϵZL ,

while the Hilbert space of the standard quantization coincides with the space of SUSY vacua with superpotential (2.26) whose F-term equations coincide with the BAE. Equivalence between the two quantization schemes leads to the following duality conjecture,

Proposal The twisted chiral rings of Theory I and Theory II are isomorphic. This means in particular that the stationary points of the twisted superpotentials (2.28) and (2.26) are in one-to-one correspondence and, with appropriate identifications between the complex parameters, these superpotentials take the same value in corresponding vacua;

 W(I) on−shell≡ W(II)

where equality holds up to an additive vacuum-independent constant. Here, we the fact that and are equal on-shell as multi-valued functions. In the next section we will test this proposal by explicit calculation on both sides.

To complete the map between the chiral rings of the two theories we should also give formulae for the VEVs of the tower of chiral operators11 in each vacuum state in terms of the corresponding set of Bethe roots . Given the correspondence between these operators and the conserved charges of the classical spin chain which holds for , it is natural to conjecture that, in the deformed theory, they are related to the corresponding conserved charges of the quantum spin chain;

 ⟨{λj}∣∣ ∣∣L∏