Equivariant K-theory and refined Vafa-Witten invariants

TL;DR

This paper advances the understanding of Vafa-Witten invariants by developing their K-theoretic refinement, revealing new modular and Jacobi form structures, and confirming conjectures through explicit calculations on surfaces.

Contribution

It introduces the K-theoretic refinement of Vafa-Witten invariants, establishes their modular properties, and proves conjectures related to their structure on complex surfaces.

Findings

Refined invariants are rational functions invariant under t^{1/2}↔t^{-1/2}.

Calculations on K3 surfaces produce Jacobi forms refining modular forms.

Degeneracy loci classes are computed via Eagon-Northcott complexes, matching refined invariants.

Abstract

In [MT2] the Vafa-Witten theory of complex projective surfaces is lifted to oriented $\mathbb C^*$-equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in $t^{1/2}$ invariant under $t^{1/2}\leftrightarrow t^{-1/2}$ which specialise to numerical Vafa-Witten invariants at $t=1$. On the "instanton branch" the invariants give the virtual $\chi_{-t}^{}$-genus refinement of G\"ottsche-Kool. Applying modularity to their calculations gives predictions for the contribution of the "monopole branch". We calculate some cases and find perfect agreement. We also do calculations on K3 surfaces, finding Jacobi forms refining the usual modular forms, proving a conjecture of G\"ottsche-Kool. We determine the K-theoretic virtual classes of degeneracy loci using Eagon-Northcott complexes, and show they calculate refined Vafa-Witten invariants. Using this Laarakker [Laa] proves universality results for the invariants.