Discrete Painlev\xe9 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 T-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 of the measure in the original hypergeometric integral. To get hypergeometric functions, one needs to pick up a very special T-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 T-functions, and then these ratios satisfy more sophisticated equations equivalent to the discrete Painleve 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 Painleve q-PVI into sophisticated differential Painleve VI equations, which will be considered elsewhere.