Hamiltonian representation of isomonodromic deformations of general rational connections on $\mathfrak{gl}_2(\mathbb{C})$

TL;DR

This paper develops a Hamiltonian framework for isomonodromic deformations of $rak{gl}_2(C)$ meromorphic connections, including Lax pairs, Darboux coordinates, and reductions, connecting to Painlevé equations and topological recursion.

Contribution

It introduces a Hamiltonian system for general $rak{gl}_2(C)$ connections with arbitrary poles, including explicit Lax pairs, Darboux coordinates, and a reduction to essential isomonodromic deformations.

Findings

Constructed Hamiltonian systems for $rak{gl}_2(C)$ connections.

Connected the framework to Painlevé equations for genus 1 spectral curves.

Linked the results to topological recursion and quantization of spectral curves.

Abstract

In this paper, we study and build the Hamiltonian system attached to any $\mathfrak{gl}_2(\mathbb{C})$ meromorphic connection with an arbitrary number of non-ramified poles of arbitrary degrees. In particular, we propose the Lax pairs and Hamiltonian evolutions expressed in terms of irregular times and monodromies associated to the poles as well as $g$ Darboux coordinates defined as the apparent singularities arising in the oper gauge. Moreover, we also provide a reduction of the isomonodromic deformations to a subset of $g$ non-trivial isomonodromic deformations. This reduction is equivalent to a map reducing the set of irregular times to only $g$ non-trivial isomonodromic times. We apply our construction to all cases where the associated spectral curve has genus 1 and recover the standard Painlev\'{e} equations. We finally make the connection with the topological recursion and the quantization of classical spectral curve from this perspective.