Symmetry algebra in gauge theories of gravity. (English) Zbl 1476.83022
Summary: 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.
MSC:
| 83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 58A15 | Exterior differential systems (Cartan theory) |
| 22E43 | Structure and representation of the Lorentz group |
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