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

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.