A Spinorial Hopf Differential for Associative Submanifolds

Given a CMC surface in $R^3$, its traceless second fundamental form can be viewed as a holomorphic section called the Hopf differential. By analogy, we show that for an associative submanifold of a 7-manifold $M^7$ with $G_2$-structure, its traceless second fundamental form can be viewed as a twisted spinor. Moreover, if $M$ is $R^7$, $T^7$, or $S^7$ with the standard $G_2$-structure, then this twisted spinor is harmonic. Consequently, every non-totally-geodesic associative 3-fold in $R^7$, $T^7$, and $S^7$ admits non-vanishing harmonic twisted spinors. Analogous results hold for special Lagrangians in $R^6$ and $T^6$, coassociative 4-folds in $R^7$ and $T^7$, and Cayley 4-folds in $R^8$ and $T^8$.