On $GL(1|1)$ Higgs bundles

TL;DR

This paper explores the moduli space of $GL(1|1)$ Higgs bundles on Riemann surfaces, extending classical Higgs bundle theory into supergeometry and analyzing the associated Hitchin system.

Contribution

It provides an explicit description of the moduli space, develops supergeometric analogues of classical theorems, and demonstrates the integrability of the Hitchin system for $GL(1|1)$.

Findings

Explicit moduli space description for $GL(1|1)$ Higgs bundles

Supergeometric analogues of Narasimhan-Seshadri and nonabelian Hodge theorems

Integrability of the Hitchin system on $ olinebreak P^1$

Abstract

We investigate the moduli space of holomorphic $GL(1|1)$ Higgs bundles over a compact Riemann surface. The supergroup $GL(1|1)$, the simplest non-trivial example beyond abelian cases, provides an ideal setting for developing supergeometric analogues of classical results in Higgs bundle theory. We derive an explicit description of the moduli space and we study the analogue of the Narasimhan-Seshadri theorem as well as the nonabelian Hodge correspondence. Furthermore, we formulate and solve the corresponding Hitchin equations, demonstrating their compatibility with fermionic contributions. As a highlight, we discuss the related Hitchin system on $\mathbb{P}^1$ and its integrability.