$L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism

Abstract

We review in detail the Batalin-Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an $L_{\infty }$-algebra and how quasi-isomorphisms between $L_{\infty }$-algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern-Simons theories and give some useful shortcuts in usually rather involved computations.

arXiv commentv4: 131 pages, minor improvements, typos fixed, references added

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• Kinematic Lie Algebras From Twistor Spaces
• $L_{\infty }$-algebras and the perturbiner expansion
• Loop Amplitudes and Quantum Homotopy Algebras
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• Adjusted Parallel Transport for Higher Gauge Theories
• Perspectives on the pure spinor superfield formalism
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• Pictures from Super Chern-Simons Theory
• $L_{\infty }$-Algebras of Einstein-Cartan-Palatini Gravity
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