Vanishing theorems for associative submanifolds

TL;DR

This paper establishes a vanishing theorem for associative submanifolds in G_2 manifolds, ensuring the smoothness of their deformation moduli spaces under certain boundary conditions.

Contribution

The authors apply Bochner's technique to prove a vanishing theorem that guarantees local smoothness of the associative deformation moduli space with boundary conditions.

Findings

Vanishing theorem for associative submanifolds

Ensures local smoothness of the moduli space

Applicable to manifolds with G_2 holonomy

Abstract

Let M^7 a manifold with holonomy in G_2, and Y^3 an associative submanifold with boundary in a coassociative submanifold. In [5], the authors proved that M_{X,Y}, the moduli space of its associative deformations with boundary in the fixed X, has finite virtual dimension. Using Bochner's technique, we give a vanishing theorem that forces M_{X,Y} to be locally smooth.