Deformations of associative submanifolds with boundary

TL;DR

This paper studies the infinitesimal deformations of associative submanifolds with boundary in $G_2$-manifolds, characterizing the deformation space via elliptic problems and computing the index explicitly.

Contribution

It establishes the elliptic nature of the deformation problem and provides a formula for the index, including explicit examples of non-trivial index values.

Findings

Deformation space characterized by an elliptic problem.

Index formula involving boundary topology and Chern class.

Explicit examples of non-trivial index provided.

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 $TX_{|\partial Y}$ 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.