Symmetry algebra in gauge theories of gravity

TL;DR

This paper derives the algebra of internal symmetries and local translations in generic gauge theories of gravity, showing they form a larger gauge symmetry and applying the framework to unimodular Einstein-Cartan theory.

Contribution

It provides a general off-shell algebra of internal symmetries and local translations in gauge gravity theories, including matter coupling and nondynamical fields.

Findings

Algebra of symmetries closes off shell.

Local translations depend on internal symmetry.

Application to unimodular Einstein-Cartan theory.

Abstract

Diffeomorphisms and an internal symmetry (e.g., local Lorentz invariance) are typically regarded as the symmetries of any geometrical gravity theory, including general relativity. In the first-order formalism, diffeomorphisms can be thought of as a derived symmetry from the so-called local translations, which have improved properties. In this work, the algebra of an arbitrary internal symmetry and the local translations is obtained for a generic gauge theory of gravity, in any spacetime dimensions, and coupled to matter fields. It is shown that this algebra closes off shell suggesting that these symmetries form a larger gauge symmetry. In addition, a mechanism to find the symmetries of theories that have nondynamical fields is proposed. It turns out that the explicit form of the local translations depend on the internal symmetry and that the algebra of local translations and the internal group still closes off shell. As an example, the unimodular Einstein-Cartan theory in four spacetime dimensions, which is only invariant under volume preserving diffeomorphisms, is studied.