Conserved currents in the Cartan formulation of general relativity

TL;DR

The paper develops a BRST-cohomology-based framework to construct local, on-shell closed co-dimension 2 forms (surface charges) in the Cartan formulation of general relativity and demonstrates their exact equivalence to the corresponding charges in the metric formulation. By treating the Lorentz connection as an auxiliary field, the authors show how to build these forms from reducibility parameters and how the Cartan variables reduce to the metric variables on-shell. They compute the explicit co-dimension 2 forms for the Cartan theory and verify that, after eliminating auxiliary fields, the resulting charges coincide with the standard metric formulation expressions. This work clarifies the interplay between first-order (Cartan) and second-order (metric) formulations for conserved charges and provides a robust, cohomological foundation for surface charges in general relativity.

Abstract

We derive the expressions for the local, on-shell closed co-dimension 2 forms in the Cartan formulation of general relativity and explicitly show their equivalence to those of the metric formulation.