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\(L_{\infty}\) algebras and field theory. (English) Zbl 1371.81261

Summary: 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.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T13 Yang-Mills and other gauge theories in quantum field theory
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
18G55 Nonabelian homotopical algebra (MSC2010)
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