The Weitzenbock formula for the Fueter–Dirac operator

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