Monopole contributions to refined Vafa–Witten invariants

Abstract

We study the monopole contribution to the refined Vafa–Witten invariant recently defined by Maulik and Thomas (work in progress). We apply the results of Gholampour and Thomas (to appear in Compos. Math.) to prove a universality result for the generating series of contributions of Higgs pairs with $1$–dimensional weight spaces. For prime rank, these account for the entire monopole contribution by a theorem of Thomas. We use toric computations to determine part of the generating series and find agreement with the conjectures of Göttsche and Kool (Pure Appl. Math. Q. 14 (2018) 467–513) for ranks $2$ and $3$.