Monopole contributions to refined Vafa-Witten invariants

TL;DR

This paper analyzes the monopole contribution to refined Vafa-Witten invariants, proving universality results, and confirming conjectures for ranks 2 and 3 through theoretical and computational methods.

Contribution

It establishes a universality result for the generating series of Higgs pairs and confirms conjectures for specific ranks using toric computations.

Findings

Universality of the generating series for Higgs pairs with 1-dimensional weight spaces.

Complete monopole contribution for prime rank cases.

Agreement of computational results with existing conjectures for rank 2 and 3.

Abstract

We study the monopole contribution to the refined Vafa-Witten invariant, recently defined by Maulik and Thomas [13]. We apply results of Gholampour and Thomas [7] 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 G\"ottsche and Kool [10] for rank 2 and 3.