A virtual structure for symplectic Higgs bundles
Simon Schirren
TL;DR
The article develops a framework to define a perfect obstruction theory for moduli of symplectic Higgs bundles on a projective surface, leveraging a minimality condition on the Chern character to ensure local freeness of the underlying bundle. It introduces an involution iota: (E,φ) ↦ (E^*, -φ^*) whose fixed locus splits into orthogonal and symplectic components, and translates the problem to the spectral side via 𝔈_φ on X = Tot(K_S). By constructing an iota-equivariant symmetric obstruction theory and employing equivariant localization, the authors define a virtual cycle on the Sp(r) fixed locus, providing a pathway to Sp(r) Vafa-Witten-type invariants. The framework hinges on lifting iota to spectral data and a careful treatment of duals, universal families, and the virtual differential, with an eye toward generalizations beyond the minimal c2 constraint and connections to symmetric virtual structures. Overall, the work lays a rigorous foundation for Sp(r) VW-type enumerative invariants via fixed-point localization in the moduli of Higgs sheaves on surfaces.
Abstract
We define a perfect obstruction theory for a moduli of symplectic Higgs sheaves $(E,φ)$ 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.