119642357
On the integrability of four dimensional $ \mathcal{N}=2 $ gauge theories in the omega background
A bstractWe continue to investigate the relationship between the infrared physics of $ \mathcal{N}=2 $ supersymmetric gauge theories in four dimensions and various integrable models such as Gaudin, Calogero-Moser and quantum spin chains. We prove interesting dualities among some of these integrable systems by performing different, albeit equivalent, quantizations of the Seiberg-Witten curve of the four dimensional theory. We also discuss conformal field theories related to $ \mathcal{N}=2 $ 4d gauge theories by the Alday-Gaiotto-Tachikawa (AGT) duality and the role of conformal blocks of those CFTs in the integrable systems. As a consequence, the equivalence of conformal blocks of rank two Toda and Novikov-Wess-Zumino-Witten (WZNW) theories on the torus with punctures is found.
118182086
THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE
In this paper we shall study a special class of solutions of the self-dual Yang-Mills equations. The original self-duality equations which arose in mathematical physics were defined on Euclidean 4-space. The physically relevant solutions were the ones with finite action-the so-called 'instantons'. The same equations may be dimensionally reduced to Euclidean 3-space by imposing invariance under translation in one direction. These equations also have physical relevance-the solutions which have finite action in three dimensions are the 'magnetic monopoles'. If we take the reduction process one step further and consider solutions which are invariant under two translations, we obtain a set of equations in the plane. Here, however, there is no clear physical meaning and, indeed, attempts to find finite action solutions have failed. Nevertheless, these are the equations we shall consider. Despite the lack of interesting solutions in R2, the equations have the important property-conformal invariance-which allows them to be defined on manifolds modelled on R2 by conformal maps, namely Riemann surfaces. We shall consider here solutions of the self-duality equations defined on a compact Riemann surface. There are in fact solutions, as we shall show, and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study. It brings together in a harmonious way the subjects of Riemannian geometry, topology, algebraic geometry, and symplectic geometry. Illuminating all these facets of the same object accounts for the length of this paper. The self-duality equations are equations from gauge theory; geometrically they are defined in terms of connections on principal bundles. While the group of the principal bundle may be chosen arbitrarily for the equations to make sense, we restrict attention here to the simplest case of SU(2) or SO(3). There are two reasons for this. The first, and most obvious, is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general Lie group of higher rank. The second reason is that solutions for SU(2) have an intimate relationship with the internal structure of the Riemann surface. As a consequence of results we shall prove about solutions to the self-duality equations, we learn something about the moduli space of complex structures on the surface itself, namely Teichmuller space.
119324749
Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories
Seiberg-Witten geometry of mass deformed N=2 superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space M of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space of holomorphic G-bundles on a (possibly degenerate) elliptic curve defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group G. The integrable systems underlying, or, rather, overlooking the special geometry of M are identified. The moduli spaces of framed G-instantons on R^2xT^2, of G-monopoles with singularities on R^2xS^1, the Hitchin systems on curves with punctures, as well as various spin chains play an important role in our story. We also comment on the higher dimensional theories. In the companion paper the quantum integrable systems and their connections to the representation theory of quantum affine algebras will be discussed
14877553
Quantization of Integrable Systems and Four Dimensional Gauge Theories
We study four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. The epsilon-parameter of the Omega-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L^2-spectrum. We very briefly discuss the quantization of Hitchin system.
121238466
Dual moment maps into loop algebras
AbstractMoment maps are defined from the space of rank-r deformations of a fixedn xn matrixA to the duals
$$(\widetilde{g1}(r)^ + )^* , (\widetilde{g1}(n)^ + )^* $$
 of the positive half of the loop algebras
$$\widetilde{g1}(r),\widetilde{g1}(n)$$
. These maps are shown to give rise to the same invariant manifolds under Hamiltonian flow obtained through the Adler-Kostant-Symes theorem from the rings
$$I(\widetilde{g1}(r)^* ),I(\widetilde{g1}(n)^* )$$
 of invariant functions. This gives a dual characterization of integrable Hamiltonian systems as isospectral flow in the two loop algebras.
17930740
Bispectral and (glN, glM) dualities, discrete versus differential
We define integral transforms which establish a correspondence between spaces of quasi-exponentials and spaces of quasi-polynomials under certain conditions. As a corollary of the properties of our transforms we obtain a correspondence between solutions to the Bethe ansatz equations of two (glN,glM) dual quantum integrable models: one is the special trigonometric Gaudin model and the other is the special XXX model.
119169817
Spectral Duality Between Heisenberg Chain and Gaudin Model
In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N-site GLk Heisenberg chain is dual to the special reduced k + 2-points glN Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.
55929989
Spectral duality in integrable systems from AGT conjecture
We describe relationships between integrable systems with N degrees of freedom arising from the Alday-Gaiotto-Tachikawa conjecture. Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced glN Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the Alday-Gaiotto-Tachikawa relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrödinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2 × 2 and N × N representations of the Toda chain and the famous Adams-Harnad-Hurtubise duality.
119573641
BPS states in omega background and integrability
A bstractWe reconsider string and domain wall central charges in $ \mathcal{N} $ = 2 supersymmetric gauge theories in four dimensions in presence of the Omega background in the Nekrasov-Shatashvili (NS) limit. Existence of these charges entails presence of the corresponding topological defects in the theory — vortices and domain walls. In spirit of the 4d/2d duality we discuss the worldsheet low energy effective theory living on the BPS vortex in $ \mathcal{N} $ =2 Supersymmetric Quantum Chromodynamics (SQCD). We discuss some aspects of the brane realization of the dualities between various quantum integrable models. A chain of such dualities enables us to check the AGT correspondence in the NS limit.
119318002
Quantization of integrable systems and a 2d/4d duality
We present a new duality between the F-terms of supersymmetric field theories defined in two-and four-dimensions respectively. The duality relates $ \mathcal{N} = 2 $ super-symmetric gauge theories in four dimensions, deformed by an Ω-background in one plane, to $ \mathcal{N} = \left( {2,2} \right) $ gauged linear σ-models in two dimensions. On the four dimensional side, our main example is $ \mathcal{N} = 2 $ SQCD with gauge group G = SU(L) and NF = 2 L fundamental flavours. Using ideas of Nekrasov and Shatashvili, we argue that the Coulomb branch of this theory provides a quantization of the classical Heisenberg SL(2) spin chain. Agreement with the standard quantization via the Algebraic Bethe Ansatz implies the existence of an isomorphism between the chiral ring of the 4 d 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 4 d 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 4 d 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.
119588455
A new 2d/4d duality via integrability
We prove a duality, recently conjectured in arXiv:1103.5726, which relates the F-terms of supersymmetric gauge theories defined in two and four dimensions respectively. The proof proceeds by a saddle point analysis of the four-dimensional partition function in the Nekrasov-Shatashvili limit. At special quantized values of the Coulomb branch moduli, the saddle point condition becomes the Bethe Ansatz Equation of the SL(2) Heisenberg spin chain which coincides with the F-term equation of the dual two-dimensional theory. The on-shell values of the superpotential in the two theories are shown to coincide in corresponding vacua. We also identify two-dimensional duals for a large set of quiver gauge theories in four dimensions and generalize our proof to these cases.
204933923
Multiscale N=2 SUSY field theories, integrable systems and their stringy/brane origin -- I
We discuss supersymmetric Yang-Mills theories with the multiple scales in the brane language. The issue concerns N=2 SUSY gauge theories with massive fundamental matter including the UV finite case of $n_{f}=2n_c$, theories involving products of SU(n) gauge groups with bifundamental matter, and the systems with several parameters similar to $\Lambda_{QCD}$. We argue that the proper integrable systems are, accordingly, twisted XXX SL(2) spin chain, $SL(p)$ magnets and degenerations of the spin Calogero system. The issue of symmetries underlying integrable systems is addressed. Relations with the monopole systems are specially discussed. Brane pictures behind all these integrable structures in the IIB and M theory are suggested. We argue that degrees of freedom in integrable systems are related to KK excitations in M theory or D-particles in the IIA string theory, which substitute the infinite number of instantons in the field theory. This implies the presence of more BPS states in the low-energy sector.
119672121
On three dimensional quiver gauge theories and integrability
A bstractIn this work we compare different descriptions of the space of vacua of certain three dimensional $ \mathcal{N}=4 $ superconformal field theories, compactified on a circle and mass-deformed to $ \mathcal{N}=2 $ in a canonical way. The original $ \mathcal{N}=4 $ theories are known to admit two distinct mirror descriptions as linear quiver gauge theories, and many more descriptions which involve the compactification on a segment of four-dimensional $ \mathcal{N}=4 $ super Yang-Mills theory. Each description gives a distinct presentation of the moduli space of vacua. Our main result is to establish the precise dictionary between these presentations. We also study the relationship between this gauge theory problem and integrable systems. The space of vacua in the linear quiver gauge theory description is related by Nekrasov-Shatashvili duality to the eigenvalues of quantum integrable spin chain Hamiltonians. The space of vacua in the four-dimensional gauge theory description is related to the solution of certain integrable classical many-body problems. Thus we obtain numerous dualities between these integrable models.
122644424
TYPE IIB SUPERSTRINGS, BPS MONOPOLES, AND THREE-DIMENSIONAL GAUGE DYNAMICS
We propose an explanation via string theory of the correspondence between the Coulomb branch of certain three-dimensional supersymmetric gauge theories and certain moduli spaces of magnetic monopoles. The same construction also gives an explanation, via SL(2, Z) duality of Type IIB superstrings, of the recently discovered “mirror symmetry” in three dimensions. New phase transitions in three dimensions as well as new infrared fixed points and even new coupling constants not present in the known Lagrangians are predicted from the string theory construction. An important role in the construction is played by a novel aspect of brane dynamics in which a third brane is created when two branes cross.
119330850
Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories
In the last decade it became clear that methods and techniques based on supersymmetry provide deep insights in quantum chromodynamics and other supersymmetric and non-supersymmetric gauge theories at strong coupling. In this review we summarize major advances in the critical (Bogomol'nyi-Prasad-Sommerfeld-saturated, BPS for short) solitons in supersymmetric theories and their implications for understanding basic dynamical regularities of non-supersymmetric theories. After a brief introduction in the theory of critical solitons (including a historical introduction) we focus on three topics: (i) non-Abelian strings in N=2 and confined monopoles; (ii) reducing the level of supersymmetry; and (iii) domain walls as D brane prototypes.
15459761
Liouville Correlation Functions from Four-Dimensional Gauge Theories
We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of $${\mathcal{N}=2}$$ SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0, 1.
119441269
H3+-WZNW correlators from Liouville theory
We prove that arbitrary correlation functions of the H3+-WZNW model on a sphere have a simple expression in terms of Liouville theory correlation functions. This is based on the correspondence between the KZ and BPZ equations, and on relations between the structure constants of Liouville theory and the H3+-WZNW model. In the critical level limit, these results imply a direct link between eigenvectors of the Gaudin hamiltonians and the problem of uniformization of Riemann surfaces. We also present an expression for correlation functions of the SL(2)/U(1) gauged WZNW model in terms of correlation functions in Liouville theory.
1807250
H+3 WZNW model from Liouville field theory
There exists an intriguing relation between genus zero correlation functions in the H+3 WZNW model and in Liouville field theory. We provide a path integral derivation of the correspondence and then use our new approach to generalize the relation to surfaces of arbitrary genus g. In particular we determine the correlation functions of N primary fields in the WZNW model explicitly through Liouville correlators with N+2g−2 additional insertions of certain degenerate fields. The paper concludes with a list of interesting further extensions and a few comments on the relation to the geometric Langlands program.
115176676
Wall-crossing, Hitchin Systems, and the WKB Approximation
We consider BPS states in a large class of d = 4,N = 2 eld theories, obtained by reducing six-dimensional (2; 0) superconformal eld theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S 1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramication. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in (1). It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.
16087486
Loop and surface operators in $ \mathcal{N} = 2 $ gauge theory and Liouville modular geometry
Recently, a duality between Liouville theory and four dimensional $ \mathcal{N} = 2 $ gauge theory has been uncovered by some of the authors. We consider the role of extended objects in gauge theory, surface operators and line operators, under this correspondence. We map such objects to specific operators in Liouville theory. We employ this connection to compute the expectation value of general supersymmetric ’t Hooft-Wilson line operators in a variety of $ \mathcal{N} = 2 $ gauge theories.
16122549
Phases of N = 2 theories in two dimensions
By looking at phase transitions which occur as parameters are varied in supersymmetric gauge theories, a natural relation is found between sigma models based on Calabi-Yau hypersurfaces in weighted projective spaces and Landau-Ginzburg models. The construction permits one to recover the known correspondence between these types of models and to greatly extend it to include new classes of manifolds and also to include models with (0,2) world-sheet supersymmetry. The construction also predicts the possibility of certain physical processes involving a change in the topology of space-time.
118644168
Conformal Algebra and Multipoint Correlation Functions in 2d Statistical Models - Nucl. Phys. B240, 312 (1984)
Based on the conformal algebra approach, a general technique is given for the calculation of multipoint correlation functions in 2D statistical models at the critical point. Particular conformal operator algebras are found for operators of the 2D q-component Potts model (1 < q < 4), and the O(N) model (0 < N < 2) at the critical point. A number of four-point correlation functions are calculated for these models.
119612030
Knot Invariants from Four-Dimensional Gauge Theory
It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solutions, without reference to any quantum dualities. After suitably perturbing the equations to make their behavior more generic, we are able to get a fairly clear understanding of how the Jones polynomial emerges. The main ingredient in the argument is a link between the four-dimensional gauge theory equations in question and conformal blocks for degenerate representations of the Virasoro algebra in two dimensions. Along the way we get a better understanding of how our subject is related to a variety of new and old topics in mathematical physics, ranging from the Bethe ansatz for the Gaudin spin chain to the M-theory description of BPS monopoles and the relation between Chern-Simons gauge theory and Virasoro conformal blocks.
111384882
Generalized matrix models and AGT correspondence at all genera
We study generalized matrix models corresponding to n-point Virasoro conformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT correspondence, these describe four dimensional $ \mathcal{N} = 2 $ SU(2)n+3g−3 gauge theories with generalized quiver diagrams. We obtain the generalized matrix models from the perturbative evaluation of the Liouville correlation functions and verify the consistency of the description with respect to degenerations of the Riemann surface. Moreover, we derive the Seiberg-Witten curve for the $ \mathcal{N} = 2 $ gauge theory as the spectral curve of the generalized matrix model, thus providing a check of AGT correspondence at all genera.
119321314
Wild quiver gauge theories
A bstractWe study $ \mathcal{N} = {2} $ supersymmetric SU(2) gauge theories coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional A1 (2,0) theory on Riemann surfaces with irregular punctures. These are naturally associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We propose that the prepotential of these gauge theories on the Ω-background can be obtained from the corresponding irregular conformal blocks on the Riemann surfaces via a generalization of the coherent state construction to the case of higher order singularities.
119685337
Affine SL(2) Conformal Blocks from 4d Gauge Theories
We study Nekrasov’s instanton partition function of four-dimensional $${\mathcal{N}=2}$$ gauge theories in the presence of surface operators. This can be computed order by order in the instanton expansion by using results available in the mathematical literature. Focusing in the case of SU(2) quiver gauge theories, we find that the results agree with a modified version of the conformal blocks of affine SL(2) algebra. These conformal blocks provide, in the critical limit, the eigenfunctions of the corresponding quantized Hitchin Hamiltonians.
12434330
A& B model approaches to surface operators and Toda thoeries
It has recently been argued [1] that the inclusion of surface operators in 4d$ \mathcal{N} = 2 $ SU(2) quiver gauge theories should correspond to insertions of certain degenerate operators in the dual Liouville theory. So far only the insertion of a single surface operator has been treated (in a semi-classical limit). In this paper we study and generalise this proposal. Our approach relies on the use of topological string theory techniques. On the B-model side we show that the effects of multiple surface operator insertions in 4d$ \mathcal{N} = 2 $ gauge theories can be calculated using the B-model topological recursion method, valid beyond the semi-classical limit. On the mirror A-model side we find by explicit computations that the 5d lift of the SU(N) gauge theory partition function in the presence of (one or many) surface operators is equal to an A-model topological string partition function with the insertion of (one or many) toric branes. This is in agreement with an earlier proposal by Gukov [2, 2, 3]. Our A-model results were motivated by and agree with what one obtains by combining the AGT conjecture with the dual interpretation in terms of degenerate operators. The topological string theory approach also opens up new possibilities in the study of 2d Toda field theories.
1807250
H+3 WZNW model from Liouville field theory
There exists an intriguing relation between genus zero correlation functions in the H+3 WZNW model and in Liouville field theory. We provide a path integral derivation of the correspondence and then use our new approach to generalize the relation to surfaces of arbitrary genus g. In particular we determine the correlation functions of N primary fields in the WZNW model explicitly through Liouville correlators with N+2g−2 additional insertions of certain degenerate fields. The paper concludes with a list of interesting further extensions and a few comments on the relation to the geometric Langlands program.
119729682
M-theoretic derivations of 4d-2d dualities: from a geometric Langlands duality for surfaces, to the AGT correspondence, to integrable systems
A bstractIn part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations — which relate some cohomology of the moduli space of certain (“ramified”) G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group — can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.In part II, to the setup in part I, we introduce Omega-deformation via fluxbranes and add half-BPS boundary defects via M9-branes, and show that the celebrated AGT correspondence in [2, 3], and its generalizations — which essentially relate, among other things, some equivariant cohomology of the moduli space of certain (“ramified”) G-instantons to the integrable representations of the Langlands dual of certain affine $ \mathcal{W} $-algebras — can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.In part III, we consider various limits of our setup in part II, and connect our story to chiral fermions and integrable systems. Among other things, we derive the NekrasovOkounkov conjecture in [4] — which relates the topological string limit of the dual Nekrasov partition function for pure G to the integrable representations of the Langlands dual of an affine G-algebra — and also demonstrate that the Nekrasov-Shatashvili limit of the “fullyramified” Nekrasov instanton partition function for pure G is a simultaneous eigenfunction of the quantum Toda Hamiltonians associated with the Langlands dual of an affine G-algebra. Via the case with matter, we also make contact with Hitchin systems and the “ramified” geometric Langlands correspondence for curves.
10459077
A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories
We propose a relation between correlation functions in the 2d A(N-1) conformal Toda theories and the Nekrasov instanton partition functions in certain conformal N = 2 SU(N) 4d quiver gauge theories. Our proposal generalises the recently uncovered relation between the Liouville theory and SU(2) quivers [1]. New features appear in the analysis that have no counterparts in the Liouville case.
15050717
Affine sl(N)conformal blocks from $ \mathcal{N} = 2 $ SU(N) gauge theories
Recently Alday and Tachikawa [1] proposed a relation between conformal blocks in a two-dimensional theory with affine sl(2) symmetry and instanton partition functions in four-dimensional conformal $ \mathcal{N} = 2 $ SU(2) quiver gauge theories in the presence of a certain surface operator. In this paper we extend this proposal to a relation between conformal blocks in theories with affine sl(N) symmetry and instanton partition functions in conformal $ \mathcal{N} = 2 $ SU(N) quiver gauge theories in the presence of a surface operator. We also discuss the extension to non-conformal $ \mathcal{N} = 2 $ SU(N) theories.
122961126
On the Wess-Zumino-Witten models on the torus☆
Abstract We discuss the Ward identities of the Wess-Zumino-Witten models on Riemann surfaces and point out some ambiguities in the description of the zero modes of the currents. In the case of the torus, we show how to describe them and we write the Ward identities in such a way that they become complete. We examine in detail how the Ward identities are related to the Kubo-Martin-Schwinger condition. As an illustration of this formulation, we present a new proof of the Weyl-Kac character formula. The proof essentially relies on the mixed Virasoro × Kac-Moody Ward identities and explains the relation of the heat equation on the group manifold to the Weyl-Kac character formula.
119093787
Deformed prepotential, quantum integrable system and Liouville field theory
Abstract We study the dual descriptions recently discovered for the Seiberg–Witten theory in the presence of surface operators. The Nekrasov partition function for a four-dimensional N = 2 gauge theory with a surface operator is believed equal to the wave-function of the corresponding integrable system, or the Hitchin system, and is identified with the conformal block with a degenerate field via the AGT relation. We verify the conjecture by showing that the null state condition leads to the Schrodinger equations of the integrable systems. Furthermore, we show that the deformed prepotential emerging from the period integrals of the principal function corresponds to monodromy operation of the conformal block. We also give the instanton partition functions for the asymptotically free SU ( 2 ) gauge theories in the presence of the surface operator via the AGT relation. We find that these partition functions involve the counting of two- and four-dimensional instantons.
119213771
A 5d/3d duality from relativistic integrable system
A bstractWe propose and prove a new exact duality between the F-terms of supersymmetric gauge theories in five and three dimensions with adjoint matter fields. The theories are compactified on a circle and are subject to the Ω deformation. In the limit proposed by Nekrasov and Shatashvili, the supersymmetric vacua become isolated and are identified with the eigenstates of a quantum integrable system. The effective twisted superpotentials are the Yang-Yang functional of the relativistic elliptic Calogero-Moser model. We show that they match on-shell by deriving the Bethe ansatz equation from the saddle point of the five-dimensional partition function. We also show that the Chern-Simons terms match and extend our proposal to the elliptic quiver generalizations.
120475884
The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem
It is shown that the Knizhnik-Zamolodchikov (KZ) equation (and corresponding vector bundle) can be viewed as a quantization of the isomonodromy problem for differential equations with several singular points.
121306589
Conformal and current algebras on a general Riemann surface
Abstract Starting from a path integral formulation, Ward identities are derived for conformal and current algebras on a general Riemann surface. An n+1-point amplitude with energy-momentum tensor insertion is related by the Ward identity to an n-point amplitude and its derivative with respect to the modular parameters. Similarly, an n+1-point function with current insertion is related to an n-point function and its derivative with respect to an harmonic background gauge field. We discuss how conformal and current algebras, defined in each coordinate patch, are glued together to form a global algebraic structure on the Riemann surface.
33700
Conformal blocks on elliptic curves and the Knizhnik-Zamolodchikov-Bernard equations
We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as a subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes. In some cases, explicit calculations are possible and confirm the conjecture. The Friedan-Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard's generalization of the Knizhnik-Zamolodchikov equation.
40080265
Walls, lines, and spectral dualities in 3d gauge theories
A bstractIn this paper we analyze various half-BPS defects in a general three dimensional $ \mathcal{N} $ = 2 supersymmetric gauge theory $ \mathcal{T} $. They correspond to closed paths in SUSY parameter space and their tension is computed by evaluating period integrals along these paths. In addition to such defects, we also study wall defects that interpolate between $ \mathcal{T} $ and its SL(2, $ \mathbb{Z} $) transform by coupling the 3d theory to a 4d theory with S-duality wall. We propose a novel spectral duality between 3d gauge theories and integrable systems. This duality complements a similar duality discovered by Nekrasov and Shatashvili. As another application, for 3d $ \mathcal{N} $ = 2 theories associated with knots and 3-manifolds we compute periods of (super) A-polynomial curves and relate the results with the spectrum of domain walls and line operators.
8189544
Correlation functions in conformal Toda field theory I
Two-dimensional sl(n) quantum Toda field theory on a sphere is considered. This theory provides an important example of conformal field theory with higher spin symmetry. We derive the three-point correlation functions of the exponential fields if one of the three fields has a special form. In this case it is possible to write down and solve explicitly the differential equation for the four-point correlation function if the fourth field is completely degenerate. We give also expressions for the three-point correlation functions in the cases, when they can be expressed in terms of known functions. The semiclassical and minisuperspace approaches in the conformal Toda field theory are studied and the results coming from these approaches are compared with the proposed analytical expression for the three-point correlation function. We show, that in the framework of semiclassical and minisuperspace approaches general three-point correlation function can be reduced to the finite-dimensional integral.
118959569
W symmetry in conformal field theory
Abstract We review various aspects of W algebra symmetry in two-dimensional conformal field theory and string theory. We pay particular attention to the construction of W algebras through the quantum Drinfeld-Sokolov reduction and through the coset construction.
15392620
Spectral curves for super-Yang-Mills with adjoint hypermultiplet for general simple Lie algebras☆
The Seiberg-Witten curves and differentials for N = 2 supersymmetric Yang-Mills theories with one hypermultiplet of mass m in the adjoint representation of the gauge algebra G, are constructed for arbitrary classical or exceptional simple G (except G2. The curves are obtained from the recently established Lax pairs with spectral parameter for the (twisted) elliptic Calogero-Moser integrable systems associated with the algebra G. Curves and differentials are shown to have the proper group theoretic and complex analytic structure, and to behave as expected when m tends either to 0 or to oo. By way of example, the prepotential for G = SU (N) evaluated with these techniques, is shown to agree with standard perturbative results. A renormalization group type equation relating the prepotential to the Calogero-Moser Hamiltonian is obtained for arbitrary G, generalizing a previous result for G = SU(N). Duality properties and decoupling to theories with other representations are briefly discussed.
118506252
Twisted masses and enhanced symmetries: the A&D series
A bstractWe study new symmetries between A and D type quiver gauge theories with different numbers of colors. We realize these gauge theories with twisted masses via a brane construction that reproduces all the parameters of the Gauge/Bethe correspondence.