The pre-symplectic geometry of opers and the holonomy map

TL;DR

This paper explores the geometric structure of opers on surfaces, showing how their moduli space relates holomorphically to flat bundles and extends symplectic structures from complex projective structures.

Contribution

It constructs the moduli space of marked opers as a holomorphic fiber bundle over Teichmüller space and proves the holonomy map is a holomorphic immersion, extending symplectic structures.

Findings

Holomorphic fiber bundle structure over Teichmüller space

Holomorphic immersion of the holonomy map

Extension of symplectic structure to pre-symplectic structure

Abstract

In this paper, we construct the moduli space of marked oper structures on a closed, oriented smooth surface of negative Euler characteristic as a holomorphic fiber bundle over Teichm\"{u}ller space. We prove that the holonomy map from the space of marked oper structures to the moduli space of reductive flat bundles is a holomorphic immersion, generalizing the known results for the moduli space of marked complex projective structures. Finally, we prove that the symplectic structure on the moduli space of marked complex projective structures extends to a pre-symplectic structure on the moduli space of marked opers whose reduced phase space is the space of marked complex projective structures.