Deformations of associative submanifolds with boundary

Let M be a topological G2-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 ∂Y of genus g, the index is given by ∫∂Yc1(νX)+1−g, where νX denotes the orthogonal complement of T∂Y in TX|∂Y and c1(νX) the first Chern class of νX with respect to its natural complex structure. Further, we exhibit explicit examples of non-trivial index.