Discrete Painleve equation, Miwa variables, and string equation in 5d matrix models

Abstract

The modern version of conformal matrix model (CMM) describes conformal blocks in the Dijkgraaf-Vafa phase. Therefore it possesses a determinant representation and becomes a Toda chain $\tau$-function only after a peculiar Fourier transform in internal dimensions. Moreover, in CMM Hirota equations arise in a peculiar discrete form (when the couplings of CMM are actually Miwa time-variables). Instead, this integrability property is actually independent on the measure in the original hypergeometric integral. To get hypergeometric functions, one needs to pick up a very special $\tau$-function satisfying an additional "string equation". Usually, its role is played by the lowest $L_{-1}$ Virasoro constraint, but, in the Miwa variables, it turns into a finite-difference equation with respect to the Miwa variables. One can get rid of these differences by rewriting the string equation in terms of some double ratios of the shifted $\tau$-functions, and then these ratios satisfy more sophisticated equations equivalent to the discrete Painlevé equations by M. Jimbo and H. Sakai ($q$-PVI equation). They look much simpler in the $q$-deformed ($"5d"$) matrix model, while in the "continuous" limit $q⟶1$ to $4d$ one should consider the Miwa variables with non-unit multiplicities, what finally converts the simple discrete Painlevé $q$-PVI into sophisticated differential Painlevé VI equations, which will be considered elsewhere.

arXiv comment13 pages

References in corpus (39)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• 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
• On Geometry and Matrix Models
• Integrability and Matrix Models
• On AGT relation in the case of U(3)
• An Algorithm for the Microscopic Evaluation of the Coefficients of the Seiberg-Witten Prepotential
• Towards unified theory of $2d$ gravity
• Collective fields, Calogero-Sutherland model and generalized matrix models
• Conformal Matrix Models as an Alternative to Conventional Multi-Matrix Models
• A $q$-anaolg of the sixth Painlevé equation
• Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions
• Conformal blocks as Dotsenko-Fateev Integral Discriminants
• The Quiver Matrix Model and 2d-4d Conformal Connection
• Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models
• Excited States of Calogero-Sutherland Model and Singular Vectors of the $W_N$ Algebra
• $2d$ gravity and matrix models. I. $2d$ gravity
• 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
• Unification of All String Models with $c<1$
• Penner Type Matrix Model and Seiberg-Witten Theory
• Matrix model version of AGT conjecture and generalized Selberg integrals
• Geometric Aspects of Painlevé Equations
• Free fermions and tau-functions
• Seiberg-Witten theory, matrix model and AGT relation
• Solving Virasoro Constraints in Matrix Models
• Unified description of correlators in non-Gaussian phases of Hermitean matrix model
• An A_r threesome: Matrix models, 2d CFTs and 4d N=2 gauge theories
• Partition Functions of Matrix Models as the First Special Functions of String Theory. II. Kontsevich Model
• Painlevé VI connection problem and monodromy of c=1 conformal blocks
• Cluster integrable systems, q-Painleve equations and their quantization
• Matrix Models vs. Matrix Integrals
• On determinant representation and integrability of Nekrasov functions
• Argyres-Douglas theories, Painlevé II and quantum mechanics
• On a c-number quantum $\tau$-function
• Comment on integrability in Dijkgraaf-Vafa beta-ensembles
• Challenges of Matrix Models
• q-Painleve equation from Virasoro constraints
• q-Vertex Operator from 5D Nekrasov Function

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