Equivariant K-theory and refined Vafa-Witten invariants

Abstract

In [MT2] the Vafa-Witten theory of complex projective surfaces is lifted to oriented ${\mathbb{C}}^{\ast }$-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 Göttsche-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 Göttsche-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.

Gap pointed out by Martijn Kool fixed by more general result in arXiv:2403.12741. References updated. 51 pages

References in corpus (3)

• Donaldson-Thomas invariants via microlocal geometry
• Monopole contributions to refined Vafa-Witten invariants
• Degeneracy loci, virtual cycles and nested Hilbert schemes II

Cited by in corpus (18)

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• Monopole contributions to refined Vafa-Witten invariants
• Higher rank K-theoretic Donaldson-Thomas theory of points
• K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface
• Verlinde formulae on complex surfaces: K-theoretic invariants
• Degeneracy loci, virtual cycles and nested Hilbert schemes II
• Twisted sheaves and $\mathrm{SU}(r)/{\mathbb{Z}}_r$ Vafa-Witten theory
• The homology of moduli stacks of complexes
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