Discrete Painlevé 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 τ -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 τ -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 τ -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.

Virasoro Constraints Revisited

“…However, given a generic solution to the Virasoro constraints it is not obvious (and in general not true) that this also solves Hirota bilinear equations. One can conclude that the moduli space of solutions of Virasoro constraints and that of Hirota relations intersect along a subspace containing the matrix model solutions but the precise relation between the two is still not completely understood (see [3] for a review).…”

“…In Sect. 3 we move on to discuss non-homogeneous Virasoro constraints, where we provide the solution when the eigenvalue model is allowed to have a domain of integration with nonempty boundary. We discuss the specific examples in which the domain of integration is taken to be an hypercube or an orthant in R N .…”

A slow review of the AGT correspondence

“…An unrelated application of Z inst and qW N conformal blocks is to construct solutions of q-Painlevé equations [453][454][455][456], as in the 4d case.…”

Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

“…The topological strings in turn engineer theories of class S when formulated on certain toric Calabi-Yau manifolds [35,36]. In fact, it turns out that these isomonodromic deformations underlie topological strings only in the geometric engineering limit, where we have a theory of class S. The full topological string partition function itself, as computed with the (unrefined) topological vertex [37], are instead related to tau function of q-Painlevé equations [38][39][40][41], and q-Virasoro conformal blocks [42][43][44][45]. In fact, the connection with isomonodromy problems goes beyond the perturbative setting of the topological vertex, making contact with the nonperturbative proposal of [46] for the Topological String partition function (see also [47] for recent developments).…”