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.

arXiv comment30 pages, no figures, added references

References in corpus (21)

• Introduction to sh Lie algebras for physicists
• An Invitation to Higher Gauge Theory
• $L_{\infty }$ Algebras and Field Theory
• Higher dimensional Chern-Simons supergravity
• $L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
• The sh Lie structure of Poisson brackets in field theory
• Transgression forms and extensions of Chern-Simons gauge theories
• Homotopy algebras inspired by classical open-closed string field theory
• 3-Cocycles, Non-Associative Star-Products and the Magnetic Paradigm of R-Flux String Vacua
• Gauge Supergravities for all Odd Dimensions
• Leibniz Gauge Theories and Infinity Structures
• Free Differential Algebras: Their Use in Field Theory and Dual Formulation
• Differential geometry construction of anomalies and topological invariants in various dimensions
• $L_{\infty }$ algebras for extended geometry from Borcherds superalgebras
• The $L_{\infty }$ structure of gauge theories with matter
• $L_{\infty }$ algebras and Tensor Hierarchies in Exceptional Field Theory and Gauged Supergravity
• Extension of Chern-Simons forms and new gauge anomalies
• $L_{\infty }$-Algebras of Einstein-Cartan-Palatini Gravity
• Extended gauge theory and gauged Free Differential Algebras
• Higher form gauge fields and their nonassociative symmetry algebras
• Gauge-invariant theories and higher-degree forms

Cited by in corpus (1)

• Extended Bargmann FDA and non-relativistic gravity