On Transgression Forms and Chern--Simons (Super)gravity

TL;DR

This paper introduces a transgression form as a Lagrangian for gauge field theories, providing a systematic method to analyze their structure, symmetries, and conserved charges, with applications to Chern--Simons gravity and supergravity.

Contribution

It develops a novel approach using the Extended Cartan Homotopy Formula to split Lagrangians and analyze gauge theories, including gravity and supergravity, in a structured way.

Findings

Systematic splitting of Lagrangian into bulk and boundary parts

Application to Chern--Simons gravity and supergravity

Derivation of conserved charges and boundary conditions

Abstract

A transgression form is proposed as lagrangian for a gauge field theory. The construction is first carried out for an arbitrary Lie Algebra g and then specialized to some particular cases. We exhibit the action, discuss its symmetries, write down the equations of motion and the boundary conditions that follow from it, and finally compute conserved charges. We also present a method, based on the iterative use of the Extended Cartan Homotopy Formula, which allows one to (i) systematically split the lagrangian in order to appropriately reflect the subspaces structure of the gauge algebra, and (ii) separate the lagrangian in bulk and boundary contributions. Chern--Simons Gravity and Supergravity are then used as examples to illustrate the method. In the end we discuss some further theoretical implications that arise naturally from the mathematical structure being considered.