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Equivariant K-theory and refined Vafa-Witten invariants

Richard P. Thomas

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 $χ_{-t}^{}$-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.

Equivariant K-theory and refined Vafa-Witten invariants

Abstract

In [MT2] the Vafa-Witten theory of complex projective surfaces is lifted to oriented -equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in invariant under which specialise to numerical Vafa-Witten invariants at . On the "instanton branch" the invariants give the virtual -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.

Paper Structure

This paper contains 12 sections, 13 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. $K$-theoretic virtual cycles
  3. Virtual cycle and virtual structure sheaf
  4. Twisted virtual structure sheaf
  5. Localisation
  6. Refinement
  7. Shifted cotangent bundles
  8. $K$-theoretic invariants from degeneration loci
  9. $K$-theoretic degeneracy loci
  10. Application to Vafa-Witten theory
  11. Refined Vafa-Witten invariants: stable case
  12. tocSome calculations for $K_S>0$tocSome calculations for $\mathbf{K_S>0}$

Key Result

Theorem 1

Let $M$ be a quasi-projective $T$-scheme with compact $T$-fixed locus, and a $T$-equivariant symmetric perfect obstruction theory. Then the refined invariant of Definition defdef is a rational function of $t^{\frac{1}{2}}$, invariant under $t^{\frac{1}{2}}\leftrightarrow t^{-\frac{1}{2}}$. It is deformation invariant and has poles only at roots of unity and the origin, but not at $t=1$. Specialis

Theorems & Definitions (21)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Proposition 2.6
  • proof
  • Proposition 2.13
  • proof
  • Definition 2.19
  • ...and 11 more