Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 4

Special Issue in Honor of Simon Donaldson: Part 2 of 2

Guest Editor: Richard Thomas (Imperial College London)

On counting associative submanifolds and Seiberg–Witten monopoles

Pages: 1047 – 1133

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n4.a4


Aleksander Doan (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.; and Trinity College, Cambridge, United Kingdom)

Thomas Walpuski (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)


Building on ideas from [DT98, DS11, Wal17, Hay17], we outline a proposal for constructing Floer homology groups associated with a $G_2$-manifold. These groups are generated by associative submanifolds and solutions of the ADHM Seiberg–Witten equations. The construction is motivated by the analysis of various transitions which can change the number of associative submanifolds. We discuss the relation of our proposal to Pandharipande and Thomas’ stable pair invariant of Calabi–Yau $3$-folds.


associative submanifolds, Donaldson–Thomas theory, Floer homology, $G^2$-manifolds, monopoles, Seiberg–Witten equations, Stable pair invariants

2010 Mathematics Subject Classification

Primary 53C38, 57R57, 57R58. Secondary 14N35, 53C07, 53C25, 53C26, 53C29, 53C40.

This material is based upon work supported by the National Science Foundation under Grant No. 1754967, and by the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics.

Received 22 December 2017

Published 20 March 2020