Symplectic geometry of a moduli space of framed Higgs bundles

TL;DR

This paper constructs a natural holomorphic symplectic structure on a moduli space of framed Higgs bundles over a Riemann surface, extending the known Poisson structure on the space of twisted Higgs bundles.

Contribution

It introduces a new holomorphic symplectic manifold structure on the moduli space of framed Higgs bundles, enhancing the existing Poisson structure on twisted Higgs bundle moduli.

Findings

The moduli space of framed Higgs bundles admits a natural holomorphic symplectic structure.

The symplectic structure on the framed Higgs moduli space is explicitly investigated.

The associated Hitchin system for the framed Higgs bundles is analyzed.

Abstract

Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${\mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known that ${\mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure which is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${\mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold which is called here ${\mathcal M}_H(r,d)$. This ${\mathcal M}_H(r,d)$ is constructed by trivializing, over $D$, the restriction of the vector bundles underlying the $D$-twisted Higgs bundles; such objects are called here as framed Higgs bundles. We also investigate the symplectic structure on the moduli space ${\mathcal M}_H(r,d)$ of framed Higgs bundles as well as the Hitchin system associated to it.