On determinant representation and integrability of Nekrasov functions

Abstract

Conformal blocks and their AGT relations to LMNS integrals and Nekrasov functions are best described by "conformal" (or Dotsenko-Fateev) matrix models, but in non-Gaussian Dijkgraaf-Vafa phases, where different eigenvalues are integrated along different contours. In such matrix models, the determinant representations and integrability are restored only after a peculiar Fourier transform in the numbers of integrations. From the point of view of conformal blocks, this is Fourier transform w.r.t. the intermediate dimensions and this explains why such quantities are expressed through tau-functions in Miwa parametrization, with external dimensions playing the role of multiplicities. In particular, these determinant representations provide solutions to the Painlevé VI equation. We also explain how this pattern looks in the pure gauge limit, which is described by the Brezin-Gross-Witten matrix model.

arXiv comment15 pages

References in corpus (67)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• 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
• On non-conformal limit of the AGT relations
• Collective fields, Calogero-Sutherland model and generalized matrix models
• Towards unified theory of $2d$ gravity
• Asymptotically free N=2 theories and irregular conformal blocks
• 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
• Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
• The Quiver Matrix Model and 2d-4d Conformal Connection
• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models
• Excited States of Calogero-Sutherland Model and Singular Vectors of the $W_N$ Algebra
• Proving AGT conjecture as HS duality: extension to five dimensions
• $2d$ gravity and matrix models. I. $2d$ gravity
• Generation of Matrix Models by W-operators
• How instanton combinatorics solves Painlevé VI, V and III's
• On KP-integrable Hurwitz functions
• Ding-Iohara-Miki symmetry of network matrix models
• 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
• BGWM as Second Constituent of Complex Matrix Model
• Isomonodromic tau-functions from Liouville conformal blocks
• Geometric Aspects of Painlevé Equations
• Seiberg-Witten theory, matrix model and AGT relation
• On Painlevé/gauge theory correspondence
• On the complete perturbative solution of one-matrix models
• 5D partition functions, q-Virasoro systems and integrable spin-chains
• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• Correlators in tensor models from character calculus
• Ward identities and combinatorics of rainbow tensor models
• Matrix Models for Random Partitions
• Gauge Invariants, Correlators and Holography in Bosonic and Fermionic Tensor Models
• From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group
• Unitary Integrals and Related Matrix Models
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Bilinear equations on Painleve tau functions from CFT
• The Multicritical Kontsevich-Penner Model
• Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks
• On "Dotsenko-Fateev" representation of the toric conformal blocks
• Matrix Models vs. Matrix Integrals
• Decomposing Nekrasov Decomposition
• An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativistic and Elliptized Integrable Systems
• AGT, Burge pairs and minimal models
• Quiver Matrix Model and Topological Partition Function in Six Dimensions
• On elementary proof of AGT relations from six dimensions
• Five-dimensional SU(2) AGT conjecture and recursive formula of deformed Gaiotto state
• On Finite-Volume Gauge Theory Partition Functions
• AGT, N-Burge partitions and W_N minimal models
• Random Matrix Theory and the Sixth Painlevé Equation
• Effective QCD Partition Function in Sectors with Non-Zero Topological Charge and Itzykson-Zuber Type Integral
• Pure $SU(2)$ gauge theory partition function and generalized Bessel kernel
• CFT approach to the $q$-Painlevé VI equation
• Comment on integrability in Dijkgraaf-Vafa beta-ensembles
• Matching branches of non-perturbative conformal block at its singularity divisor
• QCD, Wick's Theorem for KdV $\tau$-functions and the String Equation
• Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory
• q-Vertex Operator from 5D Nekrasov Function
• Character Expansions in Physics
• An Elliptic Garnier System from Interpolation
• Conformal blocks and Painlevé functions
• New results in $\mathcal{N}=2$ theories from non-perturbative string

Cited by in corpus (17)

• A slow review of the AGT correspondence
• Argyres-Douglas theories, Painlevé II and quantum mechanics
• Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations
• Painleve equations from Nakajima-Yoshioka blowup relations
• Discrete Painleve system and the double scaling limit of the matrix model for irregular conformal block and gauge theory
• Discrete Painleve system for the partition function of $N_f=2$ $SU(2)$ supersymmetric gauge theory and its double scaling limit
• Instantons to the people: the power of one-form symmetries
• Orthogonal Polynomials in Mathematical Physics
• q-Painleve equation from Virasoro constraints
• Resurgence, Painleve Equations and Conformal Blocks
• On $6j$-symbols for symmetric representations of $U_q({\mathfrak{su}}_N)$
• $t{t}^{\ast }$ Toda equations for surface defects in $\mathcal{N}=2$ SYM and instanton counting for classical Lie groups
• Elliptic $q,t$ matrix models
• Cut-and-join operators and Macdonald polynomials from the 3-Schur functions
• Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models
• AGT correspondence, (q-)Painlevè equations and matrix models
• Theory space of one unitary matrix model and its critical behavior associated with Argyres-Douglas theory