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On counting associative submanifolds and Seiberg-Witten monopoles. (English) Zbl 1436.53035

Summary: Building on ideas from [Zbl 0926.58003; Zbl 1256.53038; Zbl 1391.53029; A. Heydys, “\(G_2\) instantons and the Seiberg-Witten monopoles”, Preprint, arXiv:1703.06329], 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.

MSC:

53C38 Calibrations and calibrated geometries
57R57 Applications of global analysis to structures on manifolds
57R58 Floer homology
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C29 Issues of holonomy in differential geometry
53C40 Global submanifolds
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