Integral formulas for quantum isomonodromic systems

TL;DR

This paper introduces integral formulas as particular solutions to quantum isomonodromic systems, which are quantizations of Hamiltonian systems related to isomonodromic deformations, and suggests their connection to instanton partition functions in supersymmetric gauge theories.

Contribution

It provides explicit integral formulas generalizing hypergeometric functions for quantum isomonodromic systems, linking them to gauge theory partition functions.

Findings

Integral formulas as solutions to quantum isomonodromic systems

Generalization of hypergeometric functions

Potential connection to instanton partition functions

Abstract

We conisder time-dependent Schr\"odinger systems, which are quantizations of the Hamiltonian systems obtained from a similarity reduction of the Drinfeld-Sokolov hierarchy by K. Fuji and T. Suzuki, and a similarity reduction of the UC hierarchy by T.Tsuda, independently. These Hamiltonian systems describe isomonodromic deformations for certain Fuchsian systems. Thus, our Schr\"odinger systems can be regarded as quantum isomonodromic systems. Y. Yamada conjectured that our quantum isomonodromic systems determine instanton partition functions in N=2 SU(L) gauge theory. The main purpose of this paper is to present integral formulas as particular solutions to our quantum isomonodromic systems. These integral formulas are generalizations of the generalized hypergeometric function.