$L_{\infty }$ Algebras and Field Theory

Abstract

We review and develop the general properties of $L_{\infty }$ algebras focusing on the gauge structure of the associated field theories. Motivated by the $L_{\infty }$ 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 $L_{\infty }$ 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 $L_{\infty }$ 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 $L_{\infty }$ algebra for the interacting theory. The analysis suggests that $L_{\infty }$ algebras provide a classification of perturbative gauge invariant classical field theories.

54 pages, v2: minor changes, refs. added, v3: minor corrections, version published in Fortsch.Phys

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