Quasi-Hamiltonian Geometry of Meromorphic Connections

TL;DR

This paper introduces new complex quasi-Hamiltonian spaces associated with meromorphic connections, providing a finite-dimensional symplectic framework for monodromy and Stokes data, and offers a novel proof of their symplectic properties.

Contribution

It constructs new quasi-Hamiltonian G-spaces from meromorphic connections, extending previous models and enabling a finite-dimensional symplectic description of monodromy data.

Findings

New examples of quasi-Hamiltonian G-spaces are constructed.

The symplectic structure on monodromy and Stokes data is established.

A new proof of the symplectic nature of isomonodromic deformations is provided.

Abstract

For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal G-bundles over a disc, and they generalise the conjugacy class example of Alekseev, Malkin and Meinrenken (which appears in the simple pole case). Using the `fusion product' in the theory this gives a finite dimensional construction of the natural symplectic structures on the spaces of monodromy/Stokes data of meromorphic connections over arbitrary genus Riemann surfaces, together with a new proof of the symplectic nature of isomonodromic deformations of such connections.