L∞ algebras and field theory

Abstract

We review and develop the general properties of L∞algebras focusing on the gauge structure of the associated field theories. Motivated by the L∞homotopy Lie algebra of closed string field theory and the work of Roytenberg and Weinstein describing the Courant bracket in this language we investigate the L∞structure of general gauge invariant perturbative field theories. We sketch such formulations for non-abelian gauge theories, Einstein gravity, and for double field theory. We find that there is an L∞algebra for the gauge structure and a larger one for the full interacting field theory. Theories where the gauge structure is a strict Lie algebra often require the full L∞algebra for the interacting theory. The analysis suggests that L∞algebras provide a classification of perturbative gauge invariant classical field theories.