A virtual structure for symplectic Higgs bundles

TL;DR

This paper develops a perfect obstruction theory for moduli of symplectic Higgs sheaves on projective surfaces, potentially enabling virtual counts and invariants related to symplectic gauge theories.

Contribution

It introduces a minimality assumption on the Chern character that ensures all sheaves are locally free, facilitating the definition of a virtual fundamental class.

Findings

Established a perfect obstruction theory for symplectic Higgs sheaves

Ensured all sheaves are locally free under the minimality assumption

Potentially enables computation of symplectic Vafa-Witten invariants

Abstract

We define a perfect obstruction theory for a moduli of symplectic Higgs sheaves $(E,\phi)$ on projective surfaces $S$. Key to this is a minimality assumption on $\textrm{ch}(E)$ that forces all $E$ to be locally free. This might have implications to define a virtual count and $Sp(r)$-Vafa-Witten invariants.