Isomonodromic Deformations: Confluence, Reduction and Quantisation

Ilia Gaiur, Marta Mazzocco*, Vladimir Rubtsov

*Corresponding author for this work

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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 truncated current algebra, also called generalised Takiff 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 τ-function. In order to achieve this, we study the confluence cascade of r+1 simple poles to give rise to a singularity of arbitrary Poincaré rank r as a Poisson morphism and explicitly compute the isomonodromic Hamiltonians. In loving memory of Igor Krichever. A great man and outstanding mathematician.
Original languageEnglish
Pages (from-to)1385-1461
Number of pages77
JournalCommunications in Mathematical Physics
Volume400
Issue number2
Early online date1 Mar 2023
DOIs
Publication statusPublished - Jun 2023

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