Advances in Theoretical and Mathematical Physics

Volume 27 (2023)

Number 2

$3$-manifolds and Vafa–Witten theory

Pages: 523 – 561



Sergei Gukov (California Institute of Technology, Pasadena, Calif., U.S.A.)

Artan Sheshmani (Jefferson Laboratory, Harvard University, Cambridge, Massachusetts, U.S.A.)

Shing-Tung Yau (Yau Mathematical Science Center, Tsinghua University, Beijing, China)


We initiate explicit computations of Vafa–Witten invariants of $3$-manifolds, analogous to Floer groups in the context of Donaldson theory. In particular, we explicitly compute the Vafa–Witten invariants of $3$-manifolds in a family of concrete examples relevant to various surgery operations (the Gluck twist, knot surgeries, log-transforms). We also describe the structural properties that are expected to hold for general $3$-manifolds, including the modular group action, relation to Floer homology, infinite-dimensionality for an arbitrary $3$-manifold, and the absence of instantons.

S. G. was supported by the National Science Foundation under Grant No. NSF DMS 1664227 and by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632.

The research of A. S. was partially supported by the NSF DMS-1607871, NSF DMS-1306313, the Simons 38558, and HSE University Basic Research Program. A.S. would like to further sincerely thank the Center for Mathematical Sciences and Applications at Harvard University, and Harvard University Physics department, IMSA University of Miami, as well as the Laboratory of Mirror Symmetry in Higher School of Economics, Russian federation, for the great help and support.

The work of S.-T. Y. was partially supported by the Simons Foundation grant 38558.

Published 12 October 2023