Gerbes on $G_2$ Manifolds

TL;DR

This paper explores the analogues of divisors and line bundles on $G_2$-manifolds using coassociative submanifolds and gerbe connections, and constructs a related map, including examples in twisted connected sum $G_2$-manifolds.

Contribution

It introduces a novel framework linking coassociative submanifolds and gerbe connections on $G_2$-manifolds, extending classical complex geometry concepts.

Findings

Defined an analogue of the divisor group using coassociative submanifolds

Formulated a gauge theoretical equation for gerbe connections on $G_2$-manifolds

Constructed coassociative submanifolds in twisted connected sum $G_2$-manifolds

Abstract

On a projective complex manifold, the Abelian group of Divisors maps surjectively onto that of holomorphic line bundles (the Picard group). On a $G_2$-manifold we use coassociative submanifolds to define an analogue of the first, and a gauge theoretical equation for a connection on a gerbe to define an analogue of the last. Finally, we construct a map from the former to the later. Finally, we construct some coassociative submanifolds in twisted connected sum $G_2$-manifolds.