Virtual refinements of the Vafa-Witten formula
Lothar Göttsche, Martijn Kool
TL;DR
The paper formulates a conjectural, universal description for the generating function of virtual $\chi_y$-genera of rank $2$ instanton moduli spaces on surfaces with holomorphic $2$-forms, via Mochizuki's wall-crossing formula, Seiberg–Witten invariants, and seven universal series $A_i$. It translates descendent Donaldson invariants into a structure controlled by these universal series, computable on toric surfaces, and checks the conjecture through extensive toric localization and multiple geometric families including K3, elliptic, blow-ups, and branched covers. The work connects to Vafa–Witten and Dijkgraaf–Park–Schroers physics, and extends to $ ext{μ}$-class refinements, offering blow-up and disconnected-canonical-divisor consequences. A broad numerical program supports the universality and conjectural framework, with detailed verifications across many geometries and orders. The results provide a cohesive bridge between enumerative geometry on moduli spaces and physical invariants, anchored by universal data and precise modular-structure predictions.
Abstract
We conjecture a formula for the generating function of virtual $χ_y$-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 $χ_y$-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 $μ$-classes, thereby interpolating between Vafa-Witten's formula and Witten's conjecture for Donaldson invariants.