S-duality as a beta-deformed Fourier transform

Abstract

An attempt is made to formulate Gaiotto's S-duality relations in an explicit quantitative form. Formally the problem is that of evaluation of the Racah coefficients for the Virasoro algebra, and we approach it with the help of the matrix model representation of the AGT-related conformal blocks and Nekrasov functions. In the Seiberg-Witten limit, this S-duality reduces to the Legendre transformation. In the simplest case, its lifting to the level of Nekrasov functions is just the Fourier transform, while corrections are related to the beta-deformation. We calculate them with the help of the matrix model approach and observe that they vanish for beta=1. Explicit evaluation of the same corrections from the U_q(sl(2)) infinite-dimensional representation formulas due to B.Ponsot and J.Teshner remains an open problem.

21 pages

References in corpus (82)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• N=2 dualities
• Supersymmetric Yang-Mills Systems And Integrable Systems
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Topological Strings and Integrable Hierarchies
• Integrability and Seiberg-Witten Exact Solution
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• All genus correlation functions for the hermitian 1-matrix model
• Discrete Heisenberg-Weyl Group and Modular Group
• Integrating Over Higgs Branches
• Quantum Dilogarithm
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• D-particle bound states and generalized instantons
• Vortex Counting and Lagrangian 3-manifolds
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• The Coulomb Phase of N=2 Supersymmetric QCD
• QKZ equation with |q|=1 and correlation functions of the XXZ model in the gapless regime
• The Omega Deformation, Branes, Integrability, and Liouville Theory
• Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))
• On AGT relation in the case of U(3)
• On the Quantum Moduli Space of Vacua of $N=2$ Supersymmetric $SU(N_c)$ Gauge Theories
• Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems
• Hermitean matrix model free energy: Feynman graph technique for all genera
• Issues in Topological Gauge Theory
• Darboux coordinates, Yang-Yang functional, and gauge theory
• SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls
• Liouville bootstrap via harmonic analysis on a noncompact quantum group
• Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model
• A lecture on the Liouville vertex operators
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Recursion relations in CFT and N=2 SYM theory
• On AGT conjecture
• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• A & B model approaches to surface operators and Toda theories
• The Quiver Matrix Model and 2d-4d Conformal Connection
• Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
• Unitary representations of $U_{q}(\mathfrak{sl}(2,\RR))$, the modular double, and the multiparticle q-deformed Toda chains
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links III : Compact Semi-simple Group
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• Instantons and Merons in Matrix Models
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links
• Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions
• The Power of Nekrasov Functions
• Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
• Penner Type Matrix Model and Seiberg-Witten Theory
• Matrix model version of AGT conjecture and generalized Selberg integrals
• Matrix eigenvalue model: Feynman graph technique for all genera
• BGWM as Second Constituent of Complex Matrix Model
• Quantum Wall Crossing in N=2 Gauge Theories
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Seiberg-Witten theory, matrix model and AGT relation
• On Combinatorial Expansions of Conformal Blocks
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• Zamolodchikov asymptotic formula and instanton expansion in N=2 SUSY N_f=2N_c QCD
• M-Theory of Matrix Models
• Knot invariants from rational conformal field theories
• Character expansion for HOMFLY polynomials. I. Integrability and difference equations
• Quantization of Integrable Systems and a 2d/4d Duality
• Symplectic invariants, Virasoro constraints and Givental decomposition
• An A_r threesome: Matrix models, 2d CFTs and 4d N=2 gauge theories
• Hyperelliptic curves for Supersymmetric Yang-Mills
• Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors
• Partition Functions of Matrix Models as the First Special Functions of String Theory. II. Kontsevich Model
• Towards a proof of AGT conjecture by methods of matrix models
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Nekrasov prepotential with fundamental matter from the quantum spin chain
• $N=2$ Super Yang-Mills and Subgroups of $SL(2,Z)$
• On "Dotsenko-Fateev" representation of the toric conformal blocks
• On 3d extensions of AGT relation
• Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory
• On relation between Nekrasov functions and BS periods in pure SU(N) case
• SU(N) quantum Racah coefficients & non-torus links
• Some properties of the Alday-Maldacena minimum
• Exact Electric-Magnetic Duality in N=2 Supersymmetric QCD Theories
• Seiberg-Witten curve via generalized matrix model
• New S-dualities in N=2 supersymmetric SU(2) x SU(2) gauge theory
• Duality Symmetries for N=2 Supersymmetric QCD with Vanishing beta-Functions
• Dualities in Supersymmetric Field Theories
• Quantum Dilogarithm and Tetrahedron Equation

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• Deformed N=2 theories, generalized recursion relations and S-duality
• Modular anomaly equation, heat kernel and S-duality in N=2 theories
• Painlevé VI connection problem and monodromy of c=1 conformal blocks
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux
• Knot invariants from Virasoro related representation and pretzel knots
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Matrix models for irregular conformal blocks and Argyres-Douglas theories
• Spectral dualities in XXZ spin chains and five dimensional gauge theories
• Decomposing Nekrasov Decomposition
• Wall Crossing Invariants: from quantum mechanics to knots
• S-duality as Fourier transform for arbitrary ${\epsilon }_1,{\epsilon }_2$
• S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
• Quantum geometry from the toroidal block
• On modular transformations of non-degenerate toric conformal blocks
• Modular and duality properties of surface operators in N=2* gauge theories
• Argyres-Douglas theories and Liouville Irregular States
• Matching branches of non-perturbative conformal block at its singularity divisor
• Orthogonal Polynomials in Mathematical Physics
• S-duality and the prepotential in N=2* theories (I): the ADE algebras
• Chiral observables and S-duality in N = 2* U(N) gauge theories
• Check-Operators and Quantum Spectral Curves
• Chiral trace relations in $\Omega$-deformed $\mathcal{N}=2$ theories
• Virasoro blocks and quasimodular forms
• Analytic properties of the Virasoro modular kernel
• S-duality and supersymmetry on curved manifolds
• On fusion kernel in Liouville theory
• Genus one correction to Seiberg-Witten prepotential from β-deformed matrix model
• On the Virasoro fusion and modular kernels at any irrational central charge
• On new exact conformal blocks and Nekrasov functions
• S-move matrix in the NS sector of $N=1$ super Liouville field theory