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.

47 pages; v2: references added, typos corrected; v3: a version to appear in Nuclear Physics B

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• Asymptotically free N=2 theories and irregular conformal blocks
• Loop operators and S-duality from curves on Riemann surfaces
• Seiberg-Witten theory, matrix model and AGT relation
• Gauge Theory Wilson Loops and Conformal Toda Field Theory
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Cited by in corpus (96)

• Quantum Geometry of Refined Topological Strings
• Wild Quiver Gauge Theories
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• Proving AGT conjecture as HS duality: extension to five dimensions
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• A direct proof of AGT conjecture at beta = 1
• Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
• Gauge Theories on ALE Space and Super Liouville Correlation Functions
• Seiberg-Witten theory as a Fermi gas
• Vertices, Vortices & Interacting Surface Operators
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• Instantons on ALE spaces and Super Liouville Conformal Field Theories
• AGT on the S-duality Wall
• Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• Spectral Duality in Integrable Systems from AGT Conjecture
• Quantization of Integrable Systems and a 2d/4d Duality
• Holographic thermal correlators from supersymmetric instantons
• A New 2d/4d Duality via Integrability
• 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
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• The Liouville side of the Vortex
• Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble
• Instanton partition functions in N=2 SU(N) gauge theories with a general surface operator, and their W-algebra duals
• A slow review of the AGT correspondence
• S-duality as a beta-deformed Fourier transform
• Exact WKB methods in $SU(2)$ $N_f=1$
• Deformed Seiberg-Witten Curves for ADE Quivers
• Exact WKB Analysis of N = 2 Gauge Theories
• Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String
• N=1 Geometries via M-theory
• Classical irregular block, N=2 pure gauge theory and Mathieu equation
• Mathieu equation and Elliptic curve
• Quantum Hitchin Systems via beta-deformed Matrix Models
• Liouville theory, N=2 gauge theories and accessory parameters
• Generalized matrix models and AGT correspondence at all genera
• The Stringy Instanton Partition Function
• S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
• Exponential Networks, WKB and Topological String
• Exact thermal correlators of holographic $CFT$s
• W(1+infinity) algebra as a symmetry behind AGT relation
• BPS relations from spectral problems and blowup equations
• Seiberg-Witten curve via generalized matrix model
• β-deformed matrix model and Nekrasov partition function
• Quantum geometry from the toroidal block
• Quasinormal Modes of C-metric from SCFTs
• Quantum toroidal algebras and solvable structures in gauge/string theory
• Seiberg-Witten curves and double-elliptic integrable systems
• Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization
• Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants
• Large N limit of beta-ensembles and deformed Seiberg-Witten relations
• Quantum curves and conformal field theory
• Large N techniques for Nekrasov partition functions and AGT conjecture
• Quantum periods and prepotential in $\mathcal{N}=2$ SU(2) SQCD
• Quantum Seiberg-Witten curve and Universality in Argyres-Douglas theories
• Modular and duality properties of surface operators in N=2* gauge theories
• (de)Tails of Toda CFT
• Quantum spectral problems and isomonodromic deformations
• Classical limit of irregular blocks and Mathieu functions
• Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations
• Challenges of beta-deformation
• On complete solution of the quantum Dell system
• Spectral determinants and quantum theta functions
• Baxter's T-Q equation, SU(N)/SU(2)^{N-3} correspondence and Ω-deformed Seiberg-Witten prepotential
• Deformed BPS Monopole in Omega-background
• Holographic Thermal Correlators for Hyperbolic $CFT$s
• $T$, $Q$ and periods in $SU(3)$ $\mathcal{N}=2$ SYM
• Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function
• On the Integrability of Four Dimensional N=2 Gauge Theories in the Omega Background
• Recursion relation for instanton counting for $SU(2)$ $\mathcal{N}=2$ SYM in NS limit of $\Omega$ background
• Combinatorial approach to Mathieu and Lamé equations
• VEV of Baxter's Q-operator in N=2 gauge theory and the BPZ differential equation
• The integral representation of solutions of KZ equation and a modification by $\mathcal{K}$ operator insertion
• Surface operators in 5d gauge theories and duality relations
• Quantum Seiberg-Witten periods for $\mathcal{N}=2$ $SU(N_c)$ SQCD around the superconformal point
• BPS Monopole Equation in Omega-background
• On the status of DELL systems
• On Chebyshev Wells: Periods, Deformations, and Resurgence
• Classical irregular blocks, Hill's equation and PT-symmetric periodic complex potentials
• N=2 supersymmetric QCD and elliptic potentials
• Quantum spectral curve for (q,t)-matrix model
• Classical Liouville Three-point Functions from Riemann-Hilbert Analysis
• Exact quantization conditions for the elliptic Ruijsenaars-Schneider model
• Geometric transitions, double scaling limits and gauge theories
• Matone's relation of N=2 super Yang-Mills and spectrum of Toda chain
• Spectra of elliptic potentials and supersymmetric gauge theories
• Deformed planar topological open string amplitudes on Seiberg-Witten curve
• Orbifold Vortex and Super Liouville Theory
• Chiral rings for surface operators in 4d and 5d SQCD
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket
• Deformed SW curve and the null vector decoupling equation in Toda field theory
• Solvable spectral problems from 2d CFT and N=2 gauge theories
• Entanglement Renyi entropy of two disjoint intervals for large $c$ Liouville field theory