Semistable refined Vafa-Witten invariants

TL;DR

This paper constructs semistable refined Vafa-Witten invariants for smooth complex surfaces, confirming a key conjecture by extending Joyce's wall-crossing formalism to equivariant K-theory and moduli stacks.

Contribution

It extends Joyce's wall-crossing formalism to equivariant K-theory and symmetric obstruction theories, enabling the construction of refined invariants for complex surfaces.

Findings

Construction of semistable refined Vafa-Witten invariants

Proof of the main conjecture from arXiv:1810.00078

Introduction of the symmetrized pullback for symmetric obstruction theories

Abstract

For any smooth complex projective surface $S$, we construct semistable refined Vafa-Witten invariants of $S$ which prove the main conjecture of arXiv:1810.00078. This is done by extending part of Joyce's universal wall-crossing formalism to equivariant K-theory, and to moduli stacks with symmetric obstruction theories, particularly moduli stacks of sheaves on Calabi-Yau threefolds. An important technical tool which we introduce is the symmetrized pullback, along smooth morphisms, of symmetric obstruction theories.