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

Indranil Biswas

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.

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

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.

Paper Structure

This paper contains 8 sections, 4 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Constant symplectic form and its quantization
  3. Deformation quantization
  4. Moyal--Weyl deformation quantization
  5. One-form on Riemann surfaces
  6. Moduli space of Higgs bundles and deformation quantization
  7. A description of moduli spaces of Higgs bundles
  8. Deformation quantization of the moduli space

Key Result

Lemma \oldthetheorem

For any $\textbf{f},\, \textbf{g}\, \in\, {\mathcal{A}}(V)$,

Theorems & Definitions (6)

  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem