On the $L_\infty$ formulation of Chern-Simons theories

TL;DR

This paper explores the use of $L_ abla$ algebras and FDAs in formulating Chern--Simons theories, highlighting their duality and extending the framework to higher-degree forms in gauge theories.

Contribution

It reviews the duality between FDAs and $L_ abla$ algebras and extends the $L_ abla$ formulation of Chern--Simons theories to FDA-based gauge theories with higher-degree forms.

Findings

Established the dual relation between FDAs and $L_ abla$ algebras.

Formulated standard Chern--Simons theories using $L_ abla$ algebras.

Extended the formulation to FDA-based gauge theories with higher-degree forms.

Abstract

$L_{\infty}$ algebras have been recently studied as algebraic frameworks in the formulation of gauge theories in which the gauge symmetries and the dynamics of the interacting theories are contained in a set of products acting on a graded vector space. On the other hand, FDAs are differential algebras that generalize Lie algebras by including higher-degree differential forms on their differential equations. In this article, we review the dual relation between FDAs and $L_{\infty}$ algebras. We study the formulation of standard Chern--Simons theories in terms of $L_{\infty}$ algebras and extend the results to FDA-based gauge theories. We focus on two cases, namely a flat (or zero-curvature) theory and a generalized Chern--Simons theory, both including high-degree differential forms as fundamental fields.