On parity-odd sector in metric-affine theories

TL;DR

This paper classifies all quadratic parity-violating curvature terms in four-dimensional metric-affine gravity, providing bases and analyzing special geometries, advancing understanding of parity violation in gravitational theories.

Contribution

It identifies 12 independent parity-odd curvature scalars in metric-affine gravity and offers explicit bases, including the Pontryagin invariant removal, for the first time.

Findings

12 independent parity-odd scalars identified

Pontryagin invariant can eliminate one scalar

Bases provided in component and differential form notation

Abstract

We undertake the construction of quadratic parity-violating terms involving the curvature in the four-dimensional metric-affine gravity. We demonstrate that there are only 12 linearly independent scalars, plus an additional one that can be removed by using the Pontryagin invariant. Several convenient bases for this sector are provided in both components and differential form notation. We also particularize our general findings to some constrained geometries like Weyl-Cartan and metric-compatible connections.