Linear Transformations on Affine-Connections

TL;DR

This paper presents a theorem in Metric-Affine Geometry that characterizes invariance of functionals under affine connection transformations, with applications to invariant actions in Metric-Affine Gravity and constraints on hypermomentum.

Contribution

It introduces a simple theorem linking invariance of functionals to their $ abla$-variation tensor symmetries, applicable to Metric-Affine Gravity.

Findings

Derived invariant actions under specific affine connection transformations.

Established conditions on hypermomentum for invariance.

Provided a method to generate invariant quantities in Metric-Affine Geometry.

Abstract

We state and prove a simple Theorem that allows one to generate invariant quantities in Metric-Affine Geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We show that the initial functional is invariant under the aforementioned group of transformations iff its $\Gamma$-variation produces tensor of a given symmetry. Conversely if the tensor produced by the $\Gamma$-variation of the functional respects a certain symmetry then the functional is invariant under the associated transformation of the affine connection. We then apply our results in Metric-Affine Gravity and produce invariant actions under certain transformations of the affine connection. Finally, we derive the constraints put on the hypermomentum for such invariant Theories.