Portal de Periódicos da CAPES

A rank $2$ Dijkgraaf–Moore–Verlinde–Verlinde formula

2019; Volume: 13; Issue: 1 Linguagem: Inglês

10.4310/cntp.2019.v13.n1.a6

1931-4531

Lothar Göttsche, Martijn Kool,

Homotopy and Cohomology in Algebraic Topology

We conjecture a formula for the virtual elliptic genera of moduli spaces of rank 2 sheaves on minimal surfaces S of general type.We express our conjecture in terms of the Igusa cusp form χ 10 and Borcherds type lifts of three quasi-Jacobi forms which are all related to the Weierstrass elliptic function.We also conjecture that the generating function of virtual cobordism classes of these moduli spaces depends only on χ(O S ) and K 2 S via two universal functions, one of which is determined by the cobordism classes of Hilbert schemes of points on K3.We present generalizations of these conjectures, e.g. to arbitrary surfaces with p g > 0 and b 1 = 0.We use a result of J. Shen to express the virtual cobordism class in terms of descendent Donaldson invariants.In a prequel, we used T. Mochizuki's formula, universality, and toric calculations to compute such Donaldson invariants in the setting of virtual χ ygenera.Similar techniques allow us to verify our new conjectures in many cases.We use Mochizuki's notation [Moc]: SW(a) stands for SW(2a-K S ) with SW(b) being the usual Seiberg-Witten invariant in class b ∈ H 2 (S, Z).3 In Remark 7.1 of Section 7, we motivate how we initially found the formula of Conjecture 1.1.