An energy functional on the universal spinor bundle

We study an energy functional on the universal spinor bundle over a closed $n$-dimensional spin manifold $M$. The critical points of this functional, which is modelled on the total torsion functional of $G_2$-structures in seven dimensions, are pairs of Ricci-flat metrics and real parallel spinor fields provided that $n$ equals $3$ or $7$. We then modify the functional to obtain the analogue in arbitrary dimensions. Finally we apply the universal spinor bundle approach to solve some ODEs problems concerning $G_2$-structures.