On counting associative submanifolds and Seiberg-Witten monopoles
Aleksander Doan, Thomas Walpuski
TL;DR
This paper proposes a new Floer homology theory for G2-manifolds, generated by associative submanifolds and Seiberg-Witten solutions, linking geometric transitions to algebraic invariants.
Contribution
It introduces a novel construction of Floer homology groups for G2-manifolds based on associative submanifolds and Seiberg-Witten monopoles, extending previous ideas.
Findings
Outline of a Floer homology construction for G2-manifolds.
Connection between associative submanifolds and Seiberg-Witten equations.
Relation to stable pair invariants in Calabi-Yau geometry.
Abstract
Building on ideas from [DT98; DS11; Wal17; Hay17], we outline a proposal for constructing Floer homology groups associated with a G2-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.
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