Gauge theory in dimension $7$

TL;DR

This paper explores gauge theory on 7-dimensional manifolds with $G_2$ structures, examining their relation to supergravity, and studies the deformation theory of associative submanifolds with boundary conditions, revealing elliptic deformation equations.

Contribution

It reviews $G_2$-manifolds and their gauge bundles, and develops the deformation theory of associative submanifolds with boundary conditions, including elliptic equations and topological index formulas.

Findings

Deformation space characterized by elliptic equations

Index given by a topological formula

Relation to supergravity discussed

Abstract

We first review the notion of a $G_2$-manifold, defined in terms of a principal $G_2$ ("gauge") bundle over a $7$-dimensional manifold, before discussing their relation to supergravity. In a second thread, we focus on associative submanifolds and present their deformation theory. In particular, we elaborate on a deformation problem with coassociative boundary condition. Its space of infinitesimal deformations can be identified with the solution space of an elliptic equation whose index is given by a topological formula.