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Mathematical Physics
Title: Hamiltonian representation of isomonodromic deformations of general rational connections on $\mathfrak{gl}_2(\mathbb{C})$
(Submitted on 9 Dec 2022 (v1), last revised 12 Apr 2024 (this version, v6))
Abstract: In this paper, we study and build the Hamiltonian system attached to any $\mathfrak{gl}_2(\mathbb{C})$ meromorphic connection with 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.
Submission history
From: Olivier Marchal [view email][v1] Fri, 9 Dec 2022 13:04:37 GMT (85kb,D)
[v2] Tue, 7 Feb 2023 20:04:45 GMT (89kb,D)
[v3] Fri, 10 Mar 2023 10:09:46 GMT (67kb,D)
[v4] Thu, 20 Apr 2023 07:05:18 GMT (69kb,D)
[v5] Wed, 22 Nov 2023 08:31:18 GMT (105kb,D)
[v6] Fri, 12 Apr 2024 14:30:28 GMT (83kb,D)
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