Isomonodromic deformations: Confluence, Reduction $\&$ Quantisation

TL;DR

This paper explores the Hamiltonian structure of isomonodromic deformations of differential systems with poles, aiming to develop confluent versions of Knizhnik--Zamolodchikov equations and connect their solutions to isomonodromic tau-functions.

Contribution

It introduces a Hamiltonian framework for isomonodromic deformations with arbitrary pole orders and constructs confluent versions of key equations, linking them to tau-functions.

Findings

Explicit computation of isomonodromic Hamiltonians for arbitrary Poincaré rank

Development of a Poisson morphism for pole confluence

Connection of solutions to isomonodromic tau-functions

Abstract

In this paper we study the isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the Takiff algebra (i.e. truncated current algebra). Our motivation is to produce confluent versions of the celebrated Knizhnik--Zamolodchikov equations and explain how their quasiclassical solution can be expressed via the isomonodromic $\tau$-function. In order to achieve this, we study the confluence cascade of $r+ 1$ simple poles to give rise to a singularity of arbitrary Poincar\'e rank $r$ as a Poisson morphism and explicitly compute the isomonodromic Hamiltonians.