Pure and Applied Mathematics Quarterly
Volume 15 (2019)
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
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