Wild genus-zero quantum de Rham spaces

TL;DR

This paper develops a deformation-quantization framework for wild de Rham spaces on the Riemann sphere, linking meromorphic connections with quantum Hamiltonian reduction of coadjoint orbits.

Contribution

It introduces a novel quantization method for de Rham spaces using quantum Hamiltonian reduction and analyzes stability conditions for meromorphic connections.

Findings

Constructed quantum Hamiltonian reduction of coadjoint orbits.

Established conditions for generic stability of connections.

Proved the moment map is faithfully flat under certain conditions.

Abstract

The wild de Rham spaces parameterize isomorphism classes of (stable) meromorphic connections, defined on principal bundles over wild Riemann surfaces. Working on the Riemann sphere, we will deformation-quantize the standard open part of de Rham spaces, which corresponds to the moduli of linear ordinary differential equations with meromorphic coefficients. We treat the general untwisted/unramified case with nonresonant semisimple formal residue, for any polar divisor and reductive structure group. The main ingredients are: (i) constructing the quantum Hamiltonian reduction of a (tensor) product of quantized coadjoint orbits in dual truncated-current Lie algebras, involving the corresponding category-O Verma modules; and (ii) establishing sufficient conditions on the coadjoint orbits, so that generically all meromorphic connections are stable, and the (semiclassical) moment map for the gauge-group action is faithfully flat.