Action Principles for Transgression and Chern-Simons AdS Gravities

TL;DR

This paper explores action principles for Chern-Simons AdS gravities using Transgression forms, ensuring finite conserved charges and well-defined variational principles even with non-zero torsion, relevant for Lovelock theories.

Contribution

It demonstrates that various Transgression-based action functionals for Chern-Simons AdS gravity are well-defined and produce finite physical quantities under broad conditions.

Findings

Action principles are well-defined for asymptotically AdS configurations.

Noether charges and Euclidean actions are finite with asymptotic curvature conditions.

Results are applicable to Lovelock gravity boundary regularization.

Abstract

Chern-Simons gravities are theories with a lagrangian given by a Chern-Simons form constructed from a space-time gauge group. In previous investigations we showed that, for some special field configurations that are solutions of the field equations, the extension from Chern-Simons to Transgression forms as lagrangians, motivated by gauge invariance, automatically yields the boundary terms required to regularize the theory, giving finite conserved charges and black hole thermodynamics. Further work by other researchers showed that one of the action functionals considered in the above mentioned work yields a well defined action principle in the metric (zero torsion) case and for asymptotically Anti de Sitter (AdS) space-times. In the present work we consider several action functionals for Chern-Simons AdS gravity constructed from Transgression forms, and show the action principles to be well defined and the Noether charges and Euclidean action to be finite for field configurations satisfying only that the gauge field curvature (field strength) for the AdS gauge group is asymptotically finite. For that purpose we consider an asymptotic expansion of the vielbein and spin connection that may be regarded as a perturbation of an AdS space-time, but allowing a non zero torsion. Our results are of potential interest for Lovelock gravity theories, as it has been shown that the boundary terms dictated by the transgressions for Chern-Simons gravities are also suitable to regularize Lovelock theories.