Associative submanifolds in Joyce's generalised Kummer constructions

TL;DR

This paper constructs associative submanifolds within $G_2$-manifolds created via Joyce's generalized Kummer method, showing their volume diminishes as the manifolds approach orbifold limits, supporting a theoretical prediction.

Contribution

It provides explicit examples of associative submanifolds in Joyce's $G_2$-manifolds and analyzes their volume behavior near orbifold limits.

Findings

Associative submanifolds' volume tends to zero near orbifold limits

Supports Halverson and Morrison's prediction about volume behavior

Constructs explicit examples within Joyce's framework

Abstract

This article constructs examples of associative submanifolds in $G_2$-manifolds obtained by resolving $G_2$-orbifolds using Joyce's generalised Kummer construction. As the $G_2$-manifolds approach the $G_2$-orbifolds, the volume of the associative submanifolds tends to zero. This partially verifies a prediction due to Halverson and Morrison.