A Twisted sl_2(C) Isomonodromic-Isospectral Correspondence

TL;DR

This paper extends the isomonodromic-isospectral correspondence to twisted rank 2 meromorphic connections, providing new constructions for the Painlevé I hierarchy and linking Hamiltonians with spectral data.

Contribution

It generalizes the isomonodromic-isospectral correspondence to the twisted case for rank 2 connections, introducing new maps relating Hamiltonians and spectral coordinates.

Findings

Constructed isospectral approach for Painlevé I hierarchy

Developed two maps linking Hamiltonians and spectral data

Extended correspondence to twisted sl_2(C) connections

Abstract

The aim of this article is to generalize the isomonodromic-isospectral correspondence for meromorphic connections of rank $2$ over $\mathbb{P}^1$ to the twisted case. More specifically, the construction of the isospectral approach is provided for the Painlev\'{e} I hierarchy, then two maps are constructed, one linking the sets of isomonodromic and isospectral Hamiltonians, another linking the set of apparent singularities to a set of isospectral coordinates.