Equivariant K-theory and refined Vafa-Witten invariants

Abstract

In [MT2] the Vafa-Witten theory of complex projective surfaces is lifted to oriented C * -equivariant cohomology theories. Here we study the K-theoretic refinement. It gives rational functions in t 1/2 invariant under t 1/2 ↔ 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, 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 [La1] proves universality results for the invariants.

Verlinde formulae on complex surfaces: K-theoretic invariants

“…By adapting an argument from [GNY2] combined with a new formula for twisted elliptic genera of Hilbert schemes of points on surfaces, the first named author proves Conjecture 1.2 for K3 surfaces in [Got2]. By adapting an argument of [Laa2] combined with the above-mentioned formula for twisted elliptic genera of Hilbert schemes of points on surfaces, we prove the following (where the formula for C 1 was previously determined in [Tho,Laa2]):…”

“…In other words, only universally thickened nestings Z 0 = Z 1 contribute to the invariants. 11 This fact is explained geometrically using cosection localization in [Tho,Sect. 5.3].…”

“…where (•) ≥0 denotes the part with non-negative C * -weight and r ≥0 is its rank. Moreover, for any complex E we have [Tho,(2.28)]…”

“…where K vir N is a choice of square root of K vir N = det(Ω vir N ). Over the fixed locus N C * , this choice of square root is canonical [Tho,Prop. 2.6].…”

“…Here Λ −1 (•) is introduced in Section 2 and y is related to t := c C * 1 (t) by y = e t . The Nekrasov-Okounkov twist ensures that these invariants are unchanged under y ↔ y −1 [Tho,Prop. 2.27].…”