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On $GL(1|1)$ Higgs bundles

Anton M. Zeitlin

TL;DR

The paper develops a complete framework for GL(1|1) Higgs bundles on a compact Riemann surface, providing explicit moduli descriptions for SL(1|1) and GL(1|1) cases via Picard-type and cohomological data, and detailing the role of the Berezinian. It proves a super Narasimhan--Seshadri-type correspondence by relating stable Higgs bundles to flat $SL(1|1)$-connections and establishes a super nonabelian Hodge-type equivalence, with Hitchin equations solved in the super context. On $\mathbb{P}^1$ it derives the GL(1|1) Hitchin system and shows its integrable structure connects to the Garnier system, including a quantum Gaudin (super) model after parabolic augmentation and quantization. The work lays explicit groundwork for supergeometric analogues of classical Higgs-bundle theory and points toward extensions to higher rank supergroups and super Teichmüller-type structures.

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.

On $GL(1|1)$ Higgs bundles

TL;DR

The paper develops a complete framework for GL(1|1) Higgs bundles on a compact Riemann surface, providing explicit moduli descriptions for SL(1|1) and GL(1|1) cases via Picard-type and cohomological data, and detailing the role of the Berezinian. It proves a super Narasimhan--Seshadri-type correspondence by relating stable Higgs bundles to flat -connections and establishes a super nonabelian Hodge-type equivalence, with Hitchin equations solved in the super context. On it derives the GL(1|1) Hitchin system and shows its integrable structure connects to the Garnier system, including a quantum Gaudin (super) model after parabolic augmentation and quantization. The work lays explicit groundwork for supergeometric analogues of classical Higgs-bundle theory and points toward extensions to higher rank supergroups and super Teichmüller-type structures.

Abstract

We investigate the moduli space of holomorphic Higgs bundles over a compact Riemann surface. The supergroup , 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 and its integrability.
Paper Structure (17 sections, 19 theorems, 149 equations)

This paper contains 17 sections, 19 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. The decomposition of $GL(1|1)$ supergroup
  4. Transition Functions for $GL(1|1)$ bundles and cocycle conditions
  5. $\mathop{\mathrm{\mathrm{SL}}}\nolimits(1|1)$ bundles
  6. Generalization to $GL(1|1)$
  7. Higgs field
  8. Behavior of the Supertrace
  9. Diagonalization of Higgs field
  10. Transformations of Higgs field
  11. $\mathfrak{sl}(1|1)$ Higgs field
  12. General $\mathop{\mathrm{\mathrm{GL}}}\nolimits(1|1)$ Higgs bundles
  13. Hitchin equations for $SL(1|1)$ bundles
  14. Flat $\mathop{\mathrm{\mathrm{SL}}}\nolimits(1|1)$-connections and their parametrization
  15. $\mathop{\mathrm{\mathrm{GL}}}\nolimits(1|1)$ Hitchin system on $\mathbb{P}^1$, Garnier and Gaudin models
  16. ...and 2 more sections

Key Result

Proposition 3.1

The product $g(h_1, \alpha_1, \beta_1) \cdot g(h_2, \alpha_2, \beta_2) = g(h, \alpha, \beta)$ satisfies: Moreover, the parametrization of the inverse element is:

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.2
  • ...and 27 more