A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula
Lothar Göttsche, Martijn Kool
TL;DR
The paper develops a rank-2 analogue of the DMVV framework, proposing a modular-form–driven formula for the virtual elliptic genus and a universal cobordism structure for moduli of rank-2 sheaves on surfaces of general type. It builds a bridge from geometric invariants to modular objects via Shen’s virtual cobordism description, Mochizuki’s universal formulas, and toric localization, reducing the problem to seven universal functions determined by toric models. The authors prove a sequence of interlocking conjectures—elliptic genus, cobordism, and their generalizations—by expressing invariants through descendent Donaldson invariants, Hilbert-scheme data, and SW invariants, and verify them in a broad array of examples including K3, elliptic surfaces, double covers, and quintic hypersurfaces. The work suggests blow-up formulas and deep universal structures for rank-2 moduli, linking virtual invariants to quasi-Jacobi lifts and broadening the scope of modular phenomena in algebraic geometry with potential insights for physics-inspired counts.
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 $χ_{10}$ 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 $χ(\mathcal{O}_S)$ 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 $χ_y$-genera. Similar techniques allow us to verify our new conjectures in many cases.