Virtual Refinements of the Vafa–Witten Formula

Abstract

We conjecture a formula for the generating function of virtual χ y -genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten.These virtual χ y -genera can be written in terms of descendent Donaldson invariants. Using T. Mochizuki's formula, the latter can be expressed in terms of Seiberg-Witten invariants and certain explicit integrals over Hilbert schemes of points. These integrals are governed by seven universal functions, which are determined by their values on P 2 and P 1 × P 1 . Using localization we calculate these functions up to some order, which allows us to check our conjecture in many cases.In an appendix by H. Nakajima and the first named author, the virtual Euler characteristic specialization of our conjecture is extended to include µ-classes, thereby interpolating between Vafa-Witten's formula and Witten's conjecture for Donaldson invariants.

Atiyah class and sheaf counting on local Calabi Yau fourfolds

“…The "type II" fixed loci consist of sheaves F with scheme theoretic support on the divisor 2S ⊂ Y, whose connected components are naturally isomorphic to twisted nested Hilbert schemes, as reviewed in detail in Section 5.1. By analogy with similar previous results [15,19], universality results for residual virtual integrals of type I and II are proven in Propositions 5.1 and 5.3. These are valid for any pair (S, D) as above.…”

Refined $\mathrm{SU}(3)$ Vafa–Witten invariants and modularity

“…7.5.2], the latter can be expressed in terms of Seiberg-Witten invariants and integrals over products of Hilbert schemes of points on S. We show that these integrals are determined by their values on S = P 2 and P 1 × P 1 , which can be calculated by localization. A similar strategy was employed in the rank 2 case in [GK1,GK2] (which in turn was inspired by [GNY1,GNY3]).…”

“…When deg K S < 0 or K S ∼ = O S , Vafa-Witten invariants are Euler characteristics of smooth moduli spaces (assuming "stable equals semistable"). Modularity of generating functions of Euler characteristics of smooth moduli spaces of sheaves has been verified by direct calculation in many examples; mostly for rank 2 (see references in [GK1]). For some higher rank calculations, see [BN, Koo, Man, Moz, Wei].…”

“…• χ vir y (M H S (2, c 1 , c 2 )) for roughly c 2 ≤ 7, and c 2 ≤ 30 when y = −1 [GK1], 1 • monopole contribution to VW H S (2, L, c 2 ) for c 2 ≤ 3 in [TT1]. In fact, conjectural formulae exist in both cases following from (generalizations of) a formula from Vafa-Witten [VW,(5.38)].…”

“…Our first conjecture concerns the virtual χ y -genera of M H S (3, c 1 , c 2 ) when there are no rank 3 strictly Gieseker H-semistable sheaves on S with Chern classes c 1 , c 2 . The rank 1 case was covered by [GS] and, as just mentioned, a conjectural formula for the rank 2 case was proposed in [GK1].…”