# The Weitzenb\"ock formula for the Fueter-Dirac operator

## TL;DR

This paper derives a Weitzenb"ock formula for the Fueter-Dirac operator on associative submanifolds in G_2-manifolds, leading to rigidity results and new examples under curvature conditions.

## Contribution

It introduces a new Weitzenb"ock formula for the Fueter-Dirac operator and applies it to prove rigidity and construct examples of associative submanifolds.

## Key findings

- Established a vanishing theorem for associative submanifolds.
- Provided alternative proofs for known rigidity results.
- Constructed a new example of a rigid associative in a compact G_2-manifold.

## Abstract

We find Weitzenb\"ock formula for the Fueter-Dirac operator which controls the infinitesimal deformations of an associative submanifold in a $7$--manifold with a $G_2$--structure. We establish a vanishing theorem to conclude rigidity under some positivity assumptions on curvature, which are particularly mild in the nearly parallel case. As applications, we give a different proof of rigidity for one of Lotay's associatives in the round $7$-sphere from those given by Kawai. We also provide simpler proofs of previous results by Gayet for the Bryant-Salamon metric. Finally, we obtain an original example of a rigid associative in a compact manifold with locally conformal calibrated $G_2$-structure obtained by Fernandez-Fino-Raffero.