The Quiver Matrix Model and 2d-4d Conformal Connection

Abstract

We review the quiver matrix model (the ITEP model) in the light of the recent progress on 2d-4d connection of conformal field theories, in particular, on the relation between Toda field theories and a class of quiver superconformal gauge theories. On the basis of the CFT representation of the beta deformation of the model, a quantum spectral curve is introduced as << det (x- i g_s \partial ϕ(z)) >>=0 at finite N and for beta \neq 1. The planar loop equation in the large N limit follows with the aid of W_n constraints. Residue analysis is provided both for the curve of the matrix model with the "multi-log" potential and for the Seiberg-Witten curve in the case of SU(n) with 2n flavors, leading to the matching of the mass parameters. The isomorphism of the two curves is made manifest.

37 pages; v2: version to appear in Prog. Theor. Phys. Title changed. Isomorphism of the SU(n) spectral curve and the SW curve of Witten-Gaiotto form as well as the matching of the mass parameters more fully given

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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
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• On KP-integrable Hurwitz functions
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• A direct proof of AGT conjecture at beta = 1
• Matrix model version of AGT conjecture and generalized Selberg integrals
• Explicit examples of DIM constraints for network matrix models
• 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
• The ABCDEFG of Instantons and W-algebras
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• The matrix model version of AGT conjecture and CIV-DV prepotential
• On AGT Conjecture for Pure Super Yang-Mills and W-algebra
• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• N=2 gauge theories and degenerate fields of Toda theory
• Towards a proof of AGT conjecture by methods of matrix models
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• On "Dotsenko-Fateev" representation of the toric conformal blocks
• Check of AGT Relation for Conformal Blocks on Sphere
• 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
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• Decomposing Nekrasov Decomposition
• Holomorphic blocks and the 5d AGT correspondence
• 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
• S-duality as a beta-deformed Fourier transform
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• Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions
• Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings
• N=2 Generalized Superconformal Quiver Gauge Theory
• Deformed Seiberg-Witten Curves for ADE Quivers
• On elementary proof of AGT relations from six dimensions
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• Torus HOMFLY as the Hall-Littlewood Polynomials
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• Matrix model from N = 2 orbifold partition function
• Generalized matrix models and AGT correspondence at all genera
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• On determinant representation and integrability of Nekrasov functions
• S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
• S-duality as Fourier transform for arbitrary ${\epsilon }_1,{\epsilon }_2$
• Seiberg-Witten curve via generalized matrix model
• β-deformed matrix model and Nekrasov partition function
• Complete solution to Gaussian tensor model and its integrable properties
• On $W$-representations of $\beta$- and $q,t$-deformed matrix models
• Seiberg-Witten curves and double-elliptic integrable systems
• Large N limit of beta-ensembles and deformed Seiberg-Witten relations
• Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants
• Large N techniques for Nekrasov partition functions and AGT conjecture
• Discrete Painleve system and the double scaling limit of the matrix model for irregular conformal block and gauge theory
• Superintegrability as the hidden origin of Nekrasov calculus
• Triality in SU(2) Seiberg-Witten theory and Gauss hypergeometric function
• ε-Corrected Seiberg-Witten Prepotential Obtained From Half Genus Expansion in beta-Deformed Matrix Model
• Discrete Painleve system for the partition function of $N_f=2$ $SU(2)$ supersymmetric gauge theory and its double scaling limit
• Argyres-Douglas theories and Liouville Irregular States
• Comment on integrability in Dijkgraaf-Vafa beta-ensembles
• Cauchy formula and the character ring
• Orthogonal Polynomials in Mathematical Physics
• Irregular conformal block, spectral curve and flow equations
• Matching branches of non-perturbative conformal block at its singularity divisor
• Challenges of beta-deformation
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• q-Painleve equation from Virasoro constraints
• q-Vertex Operator from 5D Nekrasov Function
• On $6j$-symbols for symmetric representations of $U_q({\mathfrak{su}}_N)$
• The ABCDEFG of Little Strings
• The integral representation of solutions of KZ equation and a modification by $\mathcal{K}$ operator insertion
• Check-Operators and Quantum Spectral Curves
• Elliptic $q,t$ matrix models
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• On generalized Macdonald polynomials
• Faces of matrix models
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• Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models
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• $\beta$-WLZZ models from $\beta$-ensemble integrals directly
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