Discrete matrix models for partial sums of conformal blocks associated to Painlevé transcendents

Abstract

A recently formulated conjecture of Gamayun, Iorgov and Lisovyy gives an asymptotic expansion of the Jimbo--Miwa--Ueno isomonodromic $\tau$-function for certain Painlevé transcendents. The coefficients in this expansion are given in terms of conformal blocks of a two-dimensional conformal field theory, which can be written as infinite sums over pairs of partitions. In this note a discrete matrix model is proposed on a lattice whose partition function can be used to obtain a multiple integral representation for the length restricted partial sums of the Painlevé conformal blocks. This leads to expressions of the partial sums involving Hänkel determinants associated to the discrete measure of the matrix model, or equivalently, Wronskians of the corresponding moment generating function which is shown to be of the generalized hypergeometric type.

arXiv comment16 pages. v2: improved presentation, typos corrected

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Cited by in corpus (3)

• Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions
• A slow review of the AGT correspondence
• Exact partition functions for deformed $\mathcal{N}=2$ theories with $N_f=4$ flavours