Deformation quantization of moduli spaces of Higgs bundles on a Riemann surface with translation structure

TL;DR

This paper constructs a natural deformation quantization of a subset of the moduli space of semistable Higgs bundles on a Riemann surface with a translation structure, leveraging the surface's geometric properties.

Contribution

It introduces a novel deformation quantization method for Higgs bundle moduli spaces using the translation structure of the underlying Riemann surface.

Findings

Deformation quantization is achieved on a Zariski open subset of the moduli space.

The construction utilizes the translation structure on the Riemann surface.

The moduli space considered is a complex symplectic manifold.

Abstract

Let X be a compact connected Riemann surface of genus g > 0 equipped with a nonzero holomorphic 1-form. Let M denote the moduli space of semistable Higgs bundles on X of rank r and degree r(g-1)+1; it is a complex symplectic manifold. Using the translation structure on the open subset of X where the 1-form does not vanish, we construct a natural deformation quantization of a certain nonempty Zariski open subset of M.