A perfect obstruction theory for SU(2)-Higgs pairs
Simon Schirren
TL;DR
This work advances the theory of Vafa–Witten invariants for rank-2 Higgs sheaves on a smooth projective surface by replacing the classical C^×-localisation with an intrinsic Z/2-involution. Central to the approach is a spectral reformulation on X=Tot(K_S) and a lifting of the involution to the full tangent–obstruction complex, allowing a fixed-part obstruction theory on the SU(2) locus N^⊥. The authors develop a determinant/tracesplitting and trace-shift mechanism to decompose deformations into fixed and moving parts, showing that the fixed part yields a perfect obstruction theory on N^⊥ and a well-defined virtual cycle. The resulting framework enables a robust localisation computation of VW-type invariants and offers pathways to extend Vafa–Witten theory to other Lie algebras through similar fixed-point analyses. Overall, the paper provides a rigorous,-local-to-global strategy to obtain meaningful invariants from non-compact moduli by using an intrinsic involution-based localisation, with potential broad impact in gauge-theoretic enumerative geometry.
Abstract
We present a new method for constructing virtual cycles for rank-2 Higgs sheaves $(E,φ)$ on a smooth projective surface $S$. Using this, we redefine the $\mathbf{SU}(2)$-perfect obstruction theory previously constructed by Tanaka-Thomas. The key step in our construction involves modifying the $\mathbf{C}^\times$-localisation formula of Graber-Pandharipande by replacing the torus action with an involution $(E,φ) \mapsto (E^*,-φ^*)$.