Symmetry algebra in gauge theories of gravity

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 group. 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. (paper)