Semistable refined Vafa-Witten invariants
Henry Liu
TL;DR
The paper constructs semistable refined Vafa–Witten invariants on a smooth projective surface by extending Joyce’s wall-crossing formalism to equivariant K-theory and to moduli with symmetric obstruction theories. It introduces symmetrized pullbacks to transfer symmetry along smooth morphisms and uses master-space localization to derive a wall-crossing formula that shows independence of the auxiliary Joyce–Song parameter $k$ for refined invariants defined via Joyce–Song pairs. A quiver-framed-stack framework is developed to relate framed and unframed invariants, and a Lie-algebraic recursion is established to define semistable refined invariants in a k-independent way, yielding a quantized Joyce–Song-type expression for semistable VW invariants that reduces to the known stable case when there are no strictly semistable objects. The construction relies on a careful analysis of fixed loci, obstruction theories, and cosection localization, with vanishings governed by $H^1(O_S)$ and $H^2(O_S)$. Overall, the work provides a robust, wall-crossing–consistent scheme for refined semistable Vafa–Witten invariants in the equivariant K-theoretic setting and lays groundwork for further wall-crossing explorations.
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.