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

## TL;DR

This paper explores the integrability properties of 5-dimensional matrix models, revealing connections between discrete Painlevé equations, Miwa variables, and string equations, and their implications for conformal blocks and hypergeometric functions.

## Contribution

It introduces a novel framework linking discrete Painlevé equations with Miwa variables and string equations in 5d matrix models, extending understanding of their integrability and differential equations.

## Key findings

- Discrete Painlevé equations relate to Miwa variables in 5d models.
- String equations become finite-difference equations in Miwa variables.
- Continuous limit yields differential Painlevé VI equations.

## 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 Painlev\'e 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 $q\longrightarrow 1$ to $4d$ one should consider the Miwa variables with non-unit multiplicities, what finally converts the simple discrete Painlev\'e $q$-PVI into sophisticated differential Painlev\'e VI equations, which will be considered elsewhere.