Spectral Data for SU(1,2) Higgs Bundles

TL;DR

This paper explicitly describes the Hitchin fiber of SU(1,2) Higgs bundles over a Riemann surface, identifying its structure as a fiber bundle with a GIT-quotient, and characterizes the stable locus as a coarse moduli space.

Contribution

It provides a detailed description of the Hitchin fiber for SU(1,2) Higgs bundles, including the identification of data as a Hecke transformation and the structure of the fiber as a GIT-quotient.

Findings

Hitchin fiber is a fiber bundle over Pic^d X with unirational fiber.

The fiber is a GIT-quotient of a C^×-action on (P^1)^{4g-4}.

Stable locus forms a coarse moduli space.

Abstract

In this article we give an explicit description of the Hitchin fiber of SU(1,2) Higgs bundles $(L,F,\gamma,\beta)$ over a compact Riemann surface $X$ of genus $\ge 2$ with $q=\gamma\circ\beta$ having simple zeros and Toledo invariant $\tau=2 \rm{deg} L$ satisfying $|\rm{deg} L|<g-1$. In particular we identify the data in an SU(1,2) Higgs bundle as a Hecke transformation $\iota: F\to L^{-2}K\oplus LK$ at $Z(q)$. The Hitchin fiber is identified with a fiber bundle over $\text{Pic}^d X$ with unirational fiber, which is a GIT-quotient of a $\mathbb{C}^\times$-action on $\left(\mathbb{P}^1\right)^{4g-4}$. The base parametrizes choice of the line bundle $L$ and the fiber gives parameters for the Hecke transformation. The stable locus is shown to be a coarse moduli space of the corresponding moduli functor.