Isomonodromic deformatiion with an irregular singularity and hyperelliptic curve

TL;DR

This paper extends isomonodromic deformation theory to include irregular singularities, deriving a tau function expressed via hyperelliptic theta functions that depend on singularity positions and deformation parameters.

Contribution

It generalizes previous results to irregular singularities and explicitly constructs the tau function using hyperelliptic theta functions with moving arguments and periods.

Findings

Derived tau function for irregular singularities

Expressed tau function via hyperelliptic theta functions

Connected deformation parameters with hyperelliptic curve data

Abstract

In this paper, we extend the result of Kitaev and Korotkin to the case where a monodromy-preserving deformation has an irregular singularity. For the monodromy-preserving deformation, we obtain the $\tau$-function whose deformation parameters are the positions of regular singularities and the parameter $t$ of an irregular singularity. Furthermore, the $\tau$-function is expressed by the hyperelliptic $\Theta$ function moving the argument $\z$ and the period $\B,$ where $t$ and the positions of regular singularities move $z$ and $\B,$ respectively.