An energy functional on the universal spinor bundle

TL;DR

This paper introduces an energy functional on the universal spinor bundle, identifying critical points as Ricci-flat metrics with parallel spinors in dimensions 3 and 7, and extends the approach to arbitrary dimensions.

Contribution

It defines a new energy functional on the universal spinor bundle, characterizes its critical points, and generalizes the functional to all dimensions, applying it to G2-structure ODE problems.

Findings

Critical points correspond to Ricci-flat metrics with parallel spinors in dimensions 3 and 7.

Modified functional extends the framework to arbitrary dimensions.

Application to G2-structure ODEs demonstrates practical utility.

Abstract

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.