On the L∞ formulation of Chern-Simons theories

Abstract

L∞ algebras have been largely 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 in their differential equations. In this article, we review the dual relation between FDAs and L∞ algebras. We study the formulation of standard Chern-Simons theories in terms of L∞ 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.