The gauge symmetries of first-order general relativity with matter fields

TL;DR

This paper explores the extension of internal gauge symmetries in first-order general relativity with matter fields, revealing how these symmetries depend on matter content, energy-momentum, and fermion currents across different dimensions.

Contribution

It extends the known gauge symmetries of the Palatini and Holst actions to include matter couplings, detailing their dependence on energy-momentum tensors and fermion currents in various dimensions.

Findings

Extended gauge symmetry depends on matter energy-momentum tensor.

Fermions introduce additional terms proportional to axial currents.

Symmetry extensions incorporate the Immirzi parameter and matter couplings.

Abstract

In $n$-dimensional spacetimes ($n>3$), there exists an internal gauge symmetry of the Palatini action with a cosmological constant that is the natural generalization of the so-called "local translations" of three-dimensional general relativity. We report the extension of this symmetry to include the minimal coupling of Yang-Mills and fermion fields to the Palatini action with a cosmological constant. We show that, as in the case of three-dimensional local translations, the extended symmetry depends on the energy-momentum tensor of the corresponding matter field and, for fermions, it contains an additional term that in four dimensions is proportional to the axial fermion current. We also report the extension of the analog of this internal gauge symmetry for the Holst action with a cosmological constant by incorporating minimally coupled scalar and Yang-Mills fields, as well as a non-minimally coupled fermion field. In the last case, the extended symmetry is affected by both the Immirzi parameter and the energy-momentum tensor and, for fermions, it also depends on the axial fermion current.