  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 $T^8$.
