A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula

TL;DR

This paper conjectures a new formula for the virtual elliptic genera of rank 2 sheaves on minimal surfaces of general type, linking it to modular forms and universal functions, with verification in various cases.

Contribution

It introduces a conjectural formula involving Igusa cusp forms and Borcherds lifts for the virtual elliptic genera of rank 2 sheaves, extending to broader surface classes.

Findings

Conjectural formula for virtual elliptic genera involving modular forms.

Verification of conjectures using Donaldson invariants and toric calculations.

Extension of conjectures to surfaces with $p_g>0$ and $b_1=0$.

Abstract

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 $\chi_{10}$ 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 $\chi(\mathcal{O}_S)$ and $K_S^2$ 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 $\chi_y$-genera. Similar techniques allow us to verify our new conjectures in many cases.