Abstract
We conjecture a formula for the generating function of virtual chi _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 chi _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 {mathbb {P}}^2 and {mathbb {P}}^1 times {mathbb {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 mu -classes, thereby interpolating between Vafa–Witten’s formula and Witten’s conjecture for Donaldson invariants.
Highlights
Let S be a smooth projective complex surface with b1(S) = 0 and polarization H
There is a C∗-scaling action on the Higgs field and Tanaka-Thomas define “SU(r ) Vafa–Witten invariants” by virtual localization with respect to this action. They show that the contribution to the invariant of the components corresponding to φ = 0 are precisely the virtual Euler characteristics ZS,c1 (x) that we study
The strong form of Mochizuki’s formula and Conjecture 5.1 imply a version of Conjecture 1.1 for arbitrary blow-ups of surfaces S with b1(S) = 0, pg(S) > 0, and Seiberg–Witten basic classes 0 and KS = 0 (Proposition 5.6)
Summary
There is a C∗-scaling action on the Higgs field and Tanaka-Thomas define “SU(r ) Vafa–Witten invariants” by virtual localization with respect to this action They show that the contribution to the invariant of the components corresponding to φ = 0 are precisely the virtual Euler characteristics ZS,c1 (x) that we study (though Tanaka-Thomas’s invariants are defined for any rank). We use the virtual Hirzebruch-Riemann-Roch formula to express χ −y(M) in terms of certain descendent Donaldson invariants (Proposition 2.1). The strong form of Mochizuki’s formula and Conjecture 5.1 imply a version of Conjecture 1.1 for arbitrary blow-ups of surfaces S with b1(S) = 0, pg(S) > 0, and Seiberg–Witten basic classes 0 and KS = 0 (Proposition 5.6). Nakajima conjecture a formula unifying the virtual Euler characteristic specialization of Conjecture 5.7 and Witten’s conjecture for Donaldson invariants.
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