Covariant Lie Derivatives and (Super-)Gravity

TL;DR

This paper provides a comprehensive mathematical framework for covariant Lie derivatives on (super-)manifolds, clarifying their role in gauge theories and supergravity, and justifying traditional methods in complex topological settings.

Contribution

It offers a complete account of covariant Lie derivatives, introduces two key examples in supergravity, and justifies traditional Lie derivative usage in non-trivial topologies.

Findings

Unified treatment of covariant Lie derivatives in gauge and supergravity theories

Explicit examples of covariant Lie derivatives in supergravity contexts

Rigorous justification for traditional Lie derivative use in topologically complex spacetimes

Abstract

The slightly subtle notion of covariant Lie derivatives of \textit{bundle-valued} differential forms is crucial in many applications in physics, notably in the computation of conserved currents in gauge theories, and yet the literature on the topic has remained fragmentary. This note provides a complete and concise mathematical account of covariant Lie derivatives on a spacetime (super-)manifold $M,$ defined via choices of lifts of spacetime vector fields to principal $G$-bundles over it, or equivalently, choices of covariantization correction terms on spacetime. As an application in the context of (super-)gravity, two important examples of covariant Lie derivatives are presented in detail, which have not appeared in unison and direct comparison: $\bf{(i)}$ The natural covariant Lie derivative relating (super-)diffeomorphism invariance to local translational (super-)symmetry, and $\bf{(ii)}$ the Kosmann Lie derivative relevant to the description of isometries of (super-)gravity backgrounds. Finally, we use the latter to rigorously justify the usage of the traditional (non-covariant) Lie derivative on coframes and associated fields in dimensional reduction scenarios along abelian $G$-fibers, an issue which has thus far remained open for topologically non-trivial spacetimes.