Intersection theory of coassociative submanifolds in G2-manifolds and Seiberg-Witten invariants
نویسنده
چکیده
We study the problem of counting instantons with coassociative boundary condition in (almost) G2-manifolds. This is analog to the open GromovWitten theory for counting holomorphic curves with Lagrangian boundary condition in Calabi-Yau manifolds. We explain its relationship with the Seiberg-Witten invariants for coassociative submanifolds. Intersection theory of Lagrangian submanifolds is an essential part of the symplectic geometry. By counting the number of holomorphic disks bounding intersecting Lagrangian submanifolds, Floer and others defined the celebrated Floer homology theory. It plays an important role in mirror symmetry for Calabi-Yau manifolds and string theory in physics. In M-theory, Calabi-Yau threefolds are replaced by seven dimensional G2-manifolds M (i.e. oriented Octonion manifolds [18]). The analog of holomorphic disks (resp. Lagrangian submanifolds) are instantons or associative submanifolds (resp. coassociative submanifolds or branes) inM [17]. An important project is to count the number of instantons with coassociative boundary conditions. In particular we want to study the following problem. Problem: Given two nearby coassociative submanifolds C and C in a (almost) G2-manifold M . Relate the number of instantons in M bounding C ∪ C to the Seiberg-Witten invariants of C. The basic reason is a coassociative submanifold C which is infinitesimally close to C corresponds to a symplectic form on C which degenerates along C ∩ C. Instantons bounding C ∪ C would become holomorphic curves on C modulo bubbling. By the work of Taubes, we expect that the number of such instantons is given by the Seiberg-Witten invariant of C. In this paper we treat the special case when C and C are disjoint, i.e. C is a symplectic four manifold. Recall that Taubes showed that the Seiberg-Witten invariants of such a C is given by the Gromov-Witten invariants [23] of C. Our main result is following theorem.
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