On determinant representation and integrability of Nekrasov functions

TL;DR

This paper explores the determinant representation and integrability of Nekrasov functions through matrix models, Fourier transforms, and tau-functions, revealing their connection to Painlevé VI and the pure gauge limit.

Contribution

It introduces a novel Fourier transform approach in non-Gaussian matrix models to restore determinant representations and integrability of Nekrasov functions, linking them to tau-functions and Painlevé equations.

Findings

Determinant representations are restored via a Fourier transform in eigenvalue integration contours.

Nekrasov functions are expressed through tau-functions in Miwa parametrization.

Solutions to Painlevé VI are obtained from these determinant representations.

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 Painlev\'e VI equation. We also explain how this pattern looks in the pure gauge limit, which is described by the Brezin-Gross-Witten matrix model.