Equivariant K-Theory and Refined Vafa\u2013Witten Invariants

Richard P Thomas

doi:10.1007/s00220-020-03821-1

Abstract

In Maulik and Thomas (in preparation) 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 Göttsche–Kool, extended to allow for strictly semistable sheaves. 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 (Monopole contributions to refined Vafa–Witten invariants. arXiv:1810.00385) proves universality results for the invariants.

Highlights

We would like to refine the numerical invariants [TT1, TT2] on a smooth complex polarised surface (S, OS(1)). Those numerical invariants are closely related to local DT invariants of the local Calabi-Yau threefold X = KS. ( when H 1(OS) = H 2(OS) they are precisely local DT invariants, as studied in [GSY1,GSY2] for instance.) They count certain compactly supported 2-dimensional torsion sheaves on X via localisation with respect to the obvious T = C∗ action on X
On the instanton branch M our refined Vafa–Witten invariants recover the virtual χ−t -genus refinement studied by Göttsche–Kool on surfaces with KS > 0 [GK1]
The upshot is that monopole branch contributions to refined Vafa– Witten invariants can be computed from calculations on smooth products of Hilbert schemes of S

Summary

Introduction

For threefolds X with a C∗ action, Nekrasov and Okounkov [NO] give a different refinement of DT theory via equivariant virtual K -theoretic invariants This means replacing the length of the 0-dimensional virtual cycle (the classical DT invariant) by the holomorphic Euler characteristic of the virtual structure sheaf. Nekrasov-Okounkov twist by a square root of the virtual canonical bundle of the moduli space before taking (equivariant) holomorphic Euler characteristic. This is motivated by physics, relating the ∂ operator to the Dirac operator. In this paper we specialise the general definition from [MT2] to T equivariant K -theory and explore it in more detail

Results

K -Theoretic Virtual Cycles

K -Theoretic Invariants from Degeneration Loci

Refined Vafa–Witten Invariants