Isomonodromic deformations with an irregular singularity and hyperelliptic curves

Abstract

In this paper, we extend the result of Kitaev and Korotkin (1998 Int. Math. Res. Notices 17 877–905) to the case where a monodromy-preserving deformation has an irregular singularity. For the monodromy-preserving deformation, we obtain the τ-function whose deformation parameters are the positions of regular singularities and the parameter t of an irregular singularity. Furthermore, the τ-function is expressed by the hyperelliptic Θ function moving the argument and the period where t and the positions of regular singularities move z and respectively.