Refined $\mathrm{SU}(3)$ Vafa–Witten invariants and modularity

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 $p_g (S) \gt 0$. 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 $\chi_y$-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 $3$).

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This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while M.K. was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 Semester.

Received 9 August 2018

Published 5 November 2019