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

Abstract

In this paper, we study and build the Hamiltonian system attached to any $\mathfrak{gl}_2(\mathbb{C})$ rational connection with arbitrary number of non-ramified poles of arbitrary degrees. In particular, we propose the Lax pairs expressed in terms of irregular times associated to the poles and a map reducing the space of irregular times to isomonodromic times. We apply the general theory 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.Comment: 67 pages + appendice