On counting associative submanifolds and Seiberg–Witten monopoles

Abstract

Building on ideas from [DT ; DS ; Wal ; Hay ], 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. of solutions to the Seiberg-Witten equation on them. It is known that counting associatives by themselves does not lead to an invariant, because the following situations may arise along a 1parameter family of G 2 -metrics:. An embedded associative submanifold develops a self-intersection. Out of this self-intersection a new associative submanifold is created, as shown by Nordström [Nor ]. Topologically, this submanifold is a connected sum.. By analogy with special Lagrangians in Calabi-Yau 3-folds [Joy , Section ], it has been conjectured that it is possible for three distinct associative submanifolds to degenerate into a singular associative submanifold with an isolated singularity modeled on the cone over T 2 [Wal , p. ; Joy , Conjecture . ]. Topologically, these three submanifolds form a surgery triad.We will argue that known vanishing results and surgery formulae for the Seiberg-Witten invariants of 3-manifolds [MT , Proposition . and Theorem . ], show that the count of associatives weighted by solutions to the Seiberg-Witten equation is invariant under transitions ( ) and ( ), assuming that all connected components of the associative submanifolds in question have b 1 > 1. This restriction is needed in order to be able to avoid reducible solutions and obtain a well-defined Seiberg-Witten invariant as an integer. 1 We know of no natural assumption that would ensure that this restriction holds for all relevant associative submanifolds. Hence, the Haydys-Walpuski proposal cannot yield an invariant which is just an integer. One can define a topological invariant using the Seiberg-Witten equation for any compact, oriented 3-manifold. This invariant, however, is not a number but rather a homology group, called monopole Floer homology [MW ; Man ; KM ; Frø ]. The behavior of monopole Floer homology under connected sum and in surgery triads is well-understood [KMOS , Theorem . ; BMO; Lin , Theorem ]. We will explain how to construct a chain complex associated with a G 2 -manifold using the monopole chain complexes of associative submanifolds. The homology of this chain complex might be invariant under transitions ( ) and ( ).The discussion so far only involved the classical Seiberg-Witten equation. There is a further transition that might spoil the invariance of the proposed homology group:. Along generic 1-parameter families of G 2 -metrics, somewhere injective immersed associative submanifolds can degenerate by converging to a multiple cover.We will explain why this phenomenon occurs and that it can change the number of associatives, eve...

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