On solutions of the Fuji-Suzuki-Tsuda system

TL;DR

This paper generalizes known results for Painlevé VI to the Fuji-Suzuki-Tsuda system, deriving new determinant and series representations of its tau function, and explores algebraic braid group dynamics and the AGT-W relation.

Contribution

It introduces higher rank generalizations of tau functions, hypergeometric kernels, and studies algebraic braid group dynamics in the context of the Fuji-Suzuki-Tsuda system.

Findings

Derived Fredholm determinant and series representations of tau functions.

Extended results to multivariate and higher rank systems.

Connected isomonodromic deformations to the AGT-W relation.

Abstract

We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlev\'e VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for $c=N-1$.