Refined $\mathrm{SU}(3)$ Vafa-Witten invariants and modularity
Lothar Göttsche, Martijn Kool
TL;DR
This work formulates a refined $ ext{SU}(3)$ Vafa-Witten invariant for smooth surfaces with $b_1=0$ and $p_g>0$, articulating a detailed conjecture that interlaces instanton and monopole branches within Tanaka–Thomas’s algebro-geometric framework. It proves refined $S$-duality modularity by carefully tracking the rank-3 instanton and monopole contributions through Mochizuki’s formula, eleven universal building blocks, and lattice-theoretic theta functions, while providing extensive computational evidence via fixed-point localization on Hilbert schemes. The paper also extends the rank-2 program to rank-3, presents a robust modularity check across ranks using Seiberg-Witten invariants, Gauss/Dedekind sums, and Dedekind-type identities, and explores consequences for minimal surfaces, canonical-divisor decompositions, and blow-ups. Overall, it offers a cohesive, testable framework for refined VW invariants with concrete predictions and verifications, advancing the understanding of modularity in higher-rank Vafa-Witten theory.
Abstract
We conjecture a formula for the refined $\mathrm{SU}(3)$ Vafa-Witten invariants of any smooth surface $S$ satisfying $H_1(S,\mathbb{Z}) = 0$ and $p_g(S)>0$. The unrefined formula corrects a proposal by Labastida-Lozano and involves unexpected algebraic expressions in modular functions. We prove that our formula satisfies a refined $S$-duality modularity transformation. We provide evidence for our formula by calculating virtual $χ_y$-genera of moduli spaces of rank 3 stable sheaves on $S$ in examples using Mochizuki's formula. Further evidence is based on the recent definition of refined $\mathrm{SU}(r)$ Vafa-Witten invariants by Maulik-Thomas and subsequent calculations on nested Hilbert schemes by Thomas (rank 2) and Laarakker (rank 3).