The Extended Cartan Homotopy Formula and a Subspace Separation Method for Chern--Simons Theory

TL;DR

This paper introduces a systematic method based on the Extended Cartan Homotopy Formula to decompose Chern--Simons Lagrangians into bulk and boundary parts, facilitating analysis of gauge theories and supergravity models.

Contribution

It presents a novel subspace separation technique for Chern--Simons Lagrangians using transgression forms and the Extended Cartan Homotopy Formula, applicable to complex gauge algebra structures.

Findings

Successfully applied to five-dimensional CS Supergravity

Enables clear separation of bulk and boundary contributions

Provides a systematic approach for gauge algebra decomposition

Abstract

In the context of Chern--Simons (CS) Theory, a subspace separation method for the Lagrangian is proposed. The method is based on the iterative use of the Extended Cartan Homotopy Formula, and allows one to (1) separate the action in bulk and boundary contributions, and (2) systematically split the Lagrangian in appropriate reflection of the the subspace structure of the gauge algebra. In order to apply the method, one must regard CS forms as a particular case of more general objects known as transgression forms. Five-dimensional CS Supergravity is used as an example to illustrate the method.