Volume 24, issue 6 (2020)

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Monopole contributions to refined Vafa–Witten invariants

Ties Laarakker

Geometry & Topology 24 (2020) 2781–2828
DOI: 10.2140/gt.2020.24.2781

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.

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Vafa–Witten invariants, monopole contribution, VW invariants
Mathematical Subject Classification 2010
Primary: 14C05, 14D20, 14J80
Received: 10 February 2019
Revised: 8 July 2019
Accepted: 9 December 2019
Published: 29 December 2020
Proposed: Lothar Göttsche
Seconded: Jim Bryan, Paul Seidel
Ties Laarakker
Imperial College
United Kingdom