On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles

Abstract

We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block through the beta-ensemble resolvents and to its asymptotics in the limit of large dimensions (both external and intermediate) taken asymmetrically in terms of the deformation epsilon-parameters. The next-to-leading term in the asymptotics defines the generating differential in the Bohr-Sommerfeld representation of the one-parameter deformed Seiberg-Witten prepotentials (whose full two-parameter deformation leads to Nekrasov functions). This generating differential is also shown to be the one-parameter version of the single-point resolvent for the corresponding beta-ensemble, and its periods in the perturbative limit of the gauge theory are expressed through the ratios of the Harish-Chandra function. The Shrödinger/Baxter equations, considered earlier in this context, directly follow from the differential equations for the degenerate conformal block. This provides a powerful method for evaluation of the single-deformed prepotentials, and even for the Seiberg-Witten prepotentials themselves. We mostly concentrate on the representative case of the insertion into the four-point block on sphere and one-point block on torus.

28 pages

References in corpus (125)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• N=2 dualities
• Solutions Of Four-Dimensional Field Theories Via M Theory
• Supersymmetric Yang-Mills Systems And Integrable Systems
• Matrix Models, Topological Strings, and Supersymmetric Gauge Theories
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• 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
• Supersymmetric vacua and Bethe ansatz
• Instantons and recursion relations in N=2 Susy gauge theory
• Integrating Over Higgs Branches
• Wall-crossing, Hitchin Systems, and the WKB Approximation
• Integrable systems and supersymmetric gauge theory
• D-particle bound states and generalized instantons
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• Quantum integrability and supersymmetric vacua
• The Omega Deformation, Branes, Integrability, and Liouville Theory
• S-duality and 2d Topological QFT
• On AGT relation in the case of U(3)
• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• 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
• Five Dimensional Gauge Theories and Relativistic Integrable Systems
• The Virtue of Defects in 4D Gauge Theories and 2D CFTs
• Higher genus correlators for the hermitian matrix model with multiple cuts
• Gravitational corrections in supersymmetric gauge theory and matrix models
• Gauge Theory Loop Operators and Liouville Theory
• Integrability and Seiberg-Witten Theory: Curves and Periods
• On non-conformal limit of the AGT relations
• Asymptotically free N=2 theories and irregular conformal blocks
• Breakdown of universality in multi-cut matrix models
• Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model
• Conformal Matrix Models as an Alternative to Conventional Multi-Matrix Models
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Para-Liouville/Toda central charges from M5-branes
• 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
• Laughlin's wave functions, Coulomb gases and expansions of the discriminant
• Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory
• Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• Integrable Structures in Supersymmetric Gauge and String Theory
• Hitchin systems, N=2 gauge theories and W-gravity
• RG Equations from Whitham Hierarchy
• Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model
• Prepotential and the Seiberg-Witten Theory
• The Effective Prepotential of N=2 Supersymmetric SU(N_c) Gauge Theories
• N=2 Supersymmetric QCD and Integrable Spin Chains: Rational Case N_f < 2N_c
• Matrix models vs. Seiberg-Witten/Whitham theories
• The Power of Nekrasov Functions
• Matrix model version of AGT conjecture and generalized Selberg integrals
• BGWM as Second Constituent of Complex Matrix Model
• Multiscale N=2 SUSY field theories, integrable systems and their stringy/brane origin -- I
• Extended Seiberg-Witten Theory and Integrable Hierarchy
• Loop operators and S-duality from curves on Riemann surfaces
• Seiberg-Witten theory, matrix model and AGT relation
• On Combinatorial Expansions of Conformal Blocks
• Proving the AGT relation for N_f = 0,1,2 antifundamentals
• Zamolodchikov asymptotic formula and instanton expansion in N=2 SUSY N_f=2N_c QCD
• The Dijkgraaf-Vafa prepotential in the context of general Seiberg-Witten theory
• Complex Curve of the Two Matrix Model and its Tau-function
• Proving AGT relations in the large-c limit
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• Matrix models for $\beta$-ensembles from Nekrasov partition functions
• Liouville/Toda central charges from M5-branes
• Affine sl(N) conformal blocks from N=2 SU(N) gauge theories
• DV and WDVV
• Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach
• Seiberg-Witten Curves and Integrable Systems
• On Double-Elliptic Integrable Systems. 1. A Duality Argument for the case of SU(2)
• Unified description of correlators in non-Gaussian phases of Hermitean matrix model
• Symplectic invariants, Virasoro constraints and Givental decomposition
• On AGT Conjecture for Pure Super Yang-Mills and W-algebra
• The matrix model version of AGT conjecture and CIV-DV prepotential
• Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors
• Non-conformal limit of AGT relation from the 1-point torus conformal block
• The uses of the refined matrix model recursion
• Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial
• N=2 gauge theories and degenerate fields of Toda theory
• 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
• Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
• New universal spectral correlators
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Commuting Hamiltonians from Seiberg-Witten Theta-Functions
• On "Dotsenko-Fateev" representation of the toric conformal blocks
• Coulomb integrals and conformal blocks in the AdS3-WZNW model
• CFT and topological recursion
• Matrix Models vs. Matrix Integrals
• Check of AGT Relation for Conformal Blocks on Sphere
• Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type
• Gauge Theory Wilson Loops and Conformal Toda Field Theory
• N=2 supersymmetric gauge theories with massive hypermultiplets and the Whitham hierarchy
• F-theoretic vs microscopic description of a conformal N=2 SYM theory
• Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory
• TBA for the Toda chain
• Liouville Type Models in Group Theory Framework. I. Finite-Dimensional Algebras
• On Crossing Symmmetry and Modular Invariance in Conformal Field Theory and S Duality in Gauge Theory
• Analysis of correlation functions in Toda theory and AGT-W relation for SU(3) quiver
• On relation between Nekrasov functions and BS periods in pure SU(N) case
• 5d and 6d Supersymmetric Gauge Theories: Prepotentials from Integrable Systems
• Instanton corrections in N=2 supersymmetric theories with classical gauge groups and fundamental matter hypermultiplets
• N=2 Generalized Superconformal Quiver Gauge Theory
• Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String
• On Microscopic Origin of Integrability in Seiberg-Witten Theory
• Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model
• Hitchin Equation, Irregular Singularity, and $N=2$ Asymptotical Free Theories
• Insights and Puzzles from Branes: 4d SUSY Yang-Mills from 6d Models
• On AGT description of N=2 SCFT with N_f=4
• Moore-Read Fractional Quantum Hall wavefunctions and SU(2) quiver gauge theories
• Geometric Langlands And The Equations Of Nahm And Bogomolny
• Exact solution for mean energy of 2d Dyson gas at beta = 1
• From Liouville to Chern-Simons, Alternative Realization of Wilson Loop Operators in AGT Duality
• Triality in SU(2) Seiberg-Witten theory and Gauss hypergeometric function
• Singular Phases of Seiberg-Witten Integrable Systems: Weak and Strong Coupling
• Solutions to the reflection equation and integrable systems for N=2 SQCD with classical groups
• Logarithmic potential beta-ensembles and Feynman graphs
• Matrix Models, Complex Geometry and Integrable Systems. I
• Matrix Models, Complex Geometry and Integrable Systems. II

Cited by in corpus (86)

• Quantum Geometry of Refined Topological Strings
• Wild Quiver Gauge Theories
• Proving AGT conjecture as HS duality: extension to five dimensions
• Instanton counting with a surface operator and the chain-saw quiver
• A direct proof of AGT conjecture at beta = 1
• Ding-Iohara-Miki symmetry of network matrix models
• M5-branes, toric diagrams and gauge theory duality
• Gauge Theories on ALE Space and Super Liouville Correlation Functions
• Vertices, Vortices & Interacting Surface Operators
• Instantons on ALE spaces and Super Liouville Conformal Field Theories
• Spectral Duality in Integrable Systems from AGT Conjecture
• Spectral Duality Between Heisenberg Chain and Gaudin Model
• Quantization of Integrable Systems and a 2d/4d Duality
• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• 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
• Rainbow tensor model with enhanced symmetry and extreme melonic dominance
• Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux
• Quantum Painleve-Calogero Correspondence
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Holographic interpretation of 1-point toroidal block in the semiclassical limit
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• Anomaly in RTT relation for DIM algebra and network matrix models
• Knot invariants from Virasoro related representation and pretzel knots
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• The Liouville side of the Vortex
• Spectral Networks with Spin
• Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble
• The toroidal block and the genus expansion
• Transformations of Spherical Blocks
• A slow review of the AGT correspondence
• Non-Perturbative Quantum Geometry III
• S-duality as a beta-deformed Fourier transform
• Exact WKB Analysis of N = 2 Gauge Theories
• Holographic duals of large-c torus conformal blocks
• Quantum Hitchin Systems via beta-deformed Matrix Models
• Non-Perturbative Quantum Geometry II
• Generalized matrix models and AGT correspondence at all genera
• Non-Perturbative Quantum Geometry
• Wall Crossing Invariants: from quantum mechanics to knots
• S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
• Quantum geometry from the toroidal block
• Seiberg-Witten curves and double-elliptic integrable systems
• Classical Virasoro irregular conformal block
• Superintegrability as the hidden origin of Nekrasov calculus
• Quantum curves and conformal field theory
• Modular and duality properties of surface operators in N=2* gauge theories
• Classical Virasoro irregular conformal block II
• Non-perturbative studies of N=2 conformal quiver gauge theories
• Challenges of beta-deformation
• On complete solution of the quantum Dell system
• Matching branches of non-perturbative conformal block at its singularity divisor
• Irregular conformal block, spectral curve and flow equations
• Quantization of Harer-Zagier formulas
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function
• $T$, $Q$ and periods in $SU(3)$ $\mathcal{N}=2$ SYM
• On the large $\Omega$-deformations in the Nekrasov-Shatashvili limit of $\mathcal{N}=2^{\ast }$ SYM
• Exact partition functions for the $\Omega$-deformed $\mathcal{N}=2^{\ast }$ $SU(2)$ gauge theory
• Surface operators in 5d gauge theories and duality relations
• VEV of Baxter's Q-operator in N=2 gauge theory and the BPZ differential equation
• On the status of DELL systems
• Check-Operators and Quantum Spectral Curves
• Semiclassical torus blocks in the t-channel
• On Chebyshev Wells: Periods, Deformations, and Resurgence
• Chiral trace relations in $\Omega$-deformed $\mathcal{N}=2$ theories
• Refined toric branes, surface operators and factorization of generalized Macdonald polynomials
• N=2 supersymmetric QCD and elliptic potentials
• Exact partition functions for deformed $\mathcal{N}=2$ theories with $N_f=4$ flavours
• Quantum spectral curve for (q,t)-matrix model
• Classical Liouville Three-point Functions from Riemann-Hilbert Analysis
• Non-stationary difference equation for q-Virasoro conformal blocks
• Deformed planar topological open string amplitudes on Seiberg-Witten curve
• Orbifold Vortex and Super Liouville Theory
• S-move matrix in the NS sector of $N=1$ super Liouville field theory
• Super-spectral curve of irregular conformal blocks
• On a path integral representation of the Nekrasov instanton partition function and its Nekrasov--Shatashvili limit
• Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlevé Equation
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket
• Chiral rings for surface operators in 4d and 5d SQCD
• Deformed SW curve and the null vector decoupling equation in Toda field theory
• Irregular Conformal States and Spectral Curve: Irregular Matrix Model Approach
• 5D AGT conjecture for circular quivers
• The light asymptotic limit of conformal blocks in Toda field theory
• AGT correspondence, (q-)Painlevè equations and matrix models