Communications in Number Theory and Physics

Volume 13 (2019)

Number 1

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

Pages: 165 – 201



Lothar Göttsche (International Centre for Theoretical Physics (ICTP), Trieste, Italy)

Martijn Kool (Mathematical Institute, University of Utrecht, The Netherlands)


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 $K^2_S$ 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 $p_g \gt 0$ 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.

Received 31 May 2018

Accepted 2 October 2018

Published 29 April 2019