Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 1

The Tanaka–Thomas’s Vafa–Witten invariants via surface Deligne–Mumford stacks

Pages: 503 – 573

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n1.a13


Yunfeng Jiang (Department of Mathematics, University of Kansas, Lawrence, Ks., U.S.A.)

Promit Kundu (Department of Mathematics, University of Kansas, Lawrence, Ks., U.S.A.)


We provide a definition of Vafa–Witten invariants for projective surface Deligne-Mumford stacks, generalizing the construction of Tanaka–Thomas on the Vafa–Witten invariants for projective surfaces inspired by the $S$-duality conjecture. We give calculations for a root stack over a general type quintic surface, and quintic surfaces with ADE singularities. The relationship between the Vafa–Witten invariants of quintic surfaces with ADE singularities and the Vafa–Witten invariants of their crepant resolutions is also discussed.


The first author would like to thank Hong Kong University of Science and Technology for hospitality where part of the work is done. This work is partially supported by NSF DMS-1600997.

Received 25 February 2020

Accepted 29 December 2020

Published 11 April 2021