Hidden gauge invariances of torsion theories: closed algebras and absence of ghosts

Abstract

We study the possible affine gauge transformations of the torsion tensor that make up Lie algebras. We find two such non-trivial structures, in which the gauge parameters are a $2$-form and a scalar. The first one gives rise to a non-abelian Lie algebra that is isomorphic to the Lorentz algebra, and which commutes with the latter. By linearizing this new gauge transformation we single out the gauge-invariant field variables on a flat background. Then, taking into account the most general power-counting renormalizable action of the torsion and imposing gauge invariance, we are able to find an invariant action that only has two free parameters. The torsion field equations of the resulting action are compatible with General Relativity in the torsionfree limit. Then, employing the spin-parity decompositions and the path integral method, we show that the theory, supplemented by the higher-derivative and Einstein-Hilbert actions, is free from kinematical ghosts, while it has a tachyonic scalar mode. Eventually, we comment on the technical difficulties that one will encounter when calculating the divergent part of the $1$-loop effective action.