Deformations of associative submanifolds with boundary

Abstract

Let $M$ be a topological $G_2$-manifold. We prove that the space of infinitesimal associative deformations of a compact associative submanifold $Y$ with boundary in a coassociative submanifold $X$ is the solution space of an elliptic problem. For a connected boundary $\partial Y$ of genus $g$, the index is given by ${\int }_{\partial Y}{c}_1({\nu }_X)+1-g$, where ${\nu }_X$ denotes the orthogonal complement of $T\partial Y$ in $T{X}_{|\partial Y}$ and $c_1({\nu }_X)$ the first Chern class of ${\nu }_X$ with respect to its natural complex structure. Further, we exhibit explicit examples of non-trivial index.

19 pages

References in corpus (8)

• Mirror Symmetry, D-Branes and Counting Holomorphic Discs
• Gauge Theory in higher dimensions, II
• The exceptional holonomy groups and calibrated geometry
• Deformations in G_2 Manifolds
• Associative submanifolds of a G2 manifold
• Intersection theory of coassociative submanifolds in G_(2)-manifolds and Seiberg-Witten invariants
• Dirichlet Duality and the Nonlinear Dirichlet Problem
• Vanishing theorems for associative submanifolds

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• Vanishing theorems for associative submanifolds
• Deformations of Special Legendrian Submanifolds with Boundary (corrected version)
• Thin instantons in G_2-manifolds and Seiberg-Witten invariants
• Deformations of Compact Coassociative 4-folds with Boundary