Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions

Abstract

We give a concise summary of the impressive recent development unifying a number of different fundamental subjects. The quiver Nekrasov functions (generalized hypergeometric series) form a full basis for all conformal blocks of the Virasoro algebra and are sufficient to provide the same for some (special) conformal blocks of W-algebras. They can be described in terms of Seiberg-Witten theory, with the SW differential given by the 1-point resolvent in the DV phase of the quiver (discrete or conformal) matrix model (β-ensemble), dS = ydz + O(ε^2) = \sum_p ε^{2p} ρ_β^{(p|1)}(z), where εand βare related to the LNS parameters ε_1 and ε_2. This provides explicit formulas for conformal blocks in terms of analytically continued contour integrals and resolves the old puzzle of the free-field description of generic conformal blocks through the Dotsenko-Fateev integrals. Most important, this completes the GKMMM description of SW theory in terms of integrability theory with the help of exact BS integrals, and provides an extended manifestation of the basic principle which states that the effective actions are the tau-functions of integrable hierarchies.

14 pages

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• N=2 dualities
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• A Perturbative Window into Non-Perturbative Physics
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• The Quiver Matrix Model and 2d-4d Conformal Connection
• Recursive representation of the torus 1-point conformal block
• N=1 and N=2 Geometry from Fluxes
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• A matrix model for simple Hurwitz numbers, and topological recursion
• Hitchin systems, N=2 gauge theories and W-gravity
• Six-dimensional D_N theory and four-dimensional SO-USp quivers
• Instantons and Merons in Matrix Models
• Generation of Matrix Models by W-operators
• The Power of Nekrasov Functions
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Cited by in corpus (116)

• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• Proving AGT conjecture as HS duality: extension to five dimensions
• Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• On KP-integrable Hurwitz functions
• A direct proof of AGT conjecture at beta = 1
• Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
• Ding-Iohara-Miki symmetry of network matrix models
• Explicit examples of DIM constraints for network matrix models
• Matrix model version of AGT conjecture and generalized Selberg integrals
• M5-branes, toric diagrams and gauge theory duality
• Seiberg-Witten theory, matrix model and AGT relation
• On the complete perturbative solution of one-matrix models
• Proving the AGT relation for N_f = 0,1,2 antifundamentals
• Matrix models for $\beta$-ensembles from Nekrasov partition functions
• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• Quantization of Integrable Systems and a 2d/4d Duality
• The matrix model version of AGT conjecture and CIV-DV prepotential
• Deformed N=2 theories, generalized recursion relations and S-duality
• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• The uses of the refined matrix model recursion
• 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
• Gauge/Liouville Triality
• On "Dotsenko-Fateev" representation of the toric conformal blocks
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• Coherent states in quantum ${\mathcal{W}}_{1+\infty }$ algebra and qq-character for 5d Super Yang-Mills
• Check of AGT Relation for Conformal Blocks on Sphere
• Gauge Theory Wilson Loops and Conformal Toda Field Theory
• Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Matrix models for irregular conformal blocks and Argyres-Douglas theories
• Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble
• Decomposing Nekrasov Decomposition
• Holomorphic blocks and the 5d AGT correspondence
• RG-Whitham dynamics and complex Hamiltonian systems
• A slow review of the AGT correspondence
• From Symmetric Product CFTs to ${\mathrm{AdS}}_3$
• Analysis of correlation functions in Toda theory and AGT-W relation for SU(3) quiver
• 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space
• Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings
• S-duality as a beta-deformed Fourier transform
• Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions
• On relation between Nekrasov functions and BS periods in pure SU(N) case
• Non-Perturbative Topological Strings And Conformal Blocks
• N=2 Generalized Superconformal Quiver Gauge Theory
• Deformed Seiberg-Witten Curves for ADE Quivers
• From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators
• On elementary proof of AGT relations from six dimensions
• Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model
• Hitchin Equation, Irregular Singularity, and $N=2$ Asymptotical Free Theories
• Torus HOMFLY as the Hall-Littlewood Polynomials
• Quantum Hitchin Systems via beta-deformed Matrix Models
• On AGT description of N=2 SCFT with N_f=4
• (q,t)-KZ equation for Ding-Iohara-Miki algebra
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• Generalized matrix models and AGT correspondence at all genera
• On determinant representation and integrability of Nekrasov functions
• Wall Crossing Invariants: from quantum mechanics to knots
• Exact solution for mean energy of 2d Dyson gas at beta = 1
• S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
• S-duality as Fourier transform for arbitrary ${\epsilon }_1,{\epsilon }_2$
• Holomorphic field realization of SH${}^c$ and quantum geometry of quiver gauge theories
• β-deformed matrix model and Nekrasov partition function
• Seiberg-Witten curve via generalized matrix model
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• Complete solution to Gaussian tensor model and its integrable properties
• On modular transformations of non-degenerate toric conformal blocks
• Seiberg-Witten curves and double-elliptic integrable systems
• Large N limit of beta-ensembles and deformed Seiberg-Witten relations
• Large N techniques for Nekrasov partition functions and AGT conjecture
• Superintegrability as the hidden origin of Nekrasov calculus
• Triality in SU(2) Seiberg-Witten theory and Gauss hypergeometric function
• Argyres-Douglas theories and Liouville Irregular States
• Cauchy formula and the character ring
• Comment on integrability in Dijkgraaf-Vafa beta-ensembles
• On Factorization of Generalized Macdonald Polynomials
• Quantization of Harer-Zagier formulas
• Matching branches of non-perturbative conformal block at its singularity divisor
• Parametric dependence of irregular conformal block
• Orthogonal Polynomials in Mathematical Physics
• Challenges of beta-deformation
• On the large $\Omega$-deformations in the Nekrasov-Shatashvili limit of $\mathcal{N}=2^{\ast }$ SYM
• Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function
• q-Vertex Operator from 5D Nekrasov Function
• Deformed BPS Monopole in Omega-background
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• q-Painleve equation from Virasoro constraints
• Superintegrability of matrix Student's distribution
• On $6j$-symbols for symmetric representations of $U_q({\mathfrak{su}}_N)$
• BPS Monopole Equation in Omega-background
• Exact partition functions for the $\Omega$-deformed $\mathcal{N}=2^{\ast }$ $SU(2)$ gauge theory
• The integral representation of solutions of KZ equation and a modification by $\mathcal{K}$ operator insertion
• Check-Operators and Quantum Spectral Curves
• On generalized Macdonald polynomials
• Beta Deformation and Superpolynomials of (n,m) Torus Knots
• Elliptic $q,t$ matrix models
• Faces of matrix models
• Perturbation of multi-critical unitary matrix models, double scaling limits, and Argyres-Douglas theories
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• An(1) Affine Quiver Matrix Model
• Irreducible representations of simple Lie algebras by differential operators
• Quantum spectral curve for (q,t)-matrix model
• Classical Liouville Three-point Functions from Riemann-Hilbert Analysis
• Penner Type Ensemble for Gauge Theories Revisited
• Genus one correction to Seiberg-Witten prepotential from β-deformed matrix model
• Geometric transitions, double scaling limits and gauge theories
• Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models
• Deformed planar topological open string amplitudes on Seiberg-Witten curve
• Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models
• On a path integral representation of the Nekrasov instanton partition function and its Nekrasov--Shatashvili limit
• Construction of irregular conformal/W block and flavor mass relations of $\mathcal{N}=2$ SUSY gauge theory from the $A_{n-1}$ quiver matrix model
• $\beta$-WLZZ models from $\beta$-ensemble integrals directly