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

TL;DR

This paper explores the mathematical structure of classical field theories using $L_ty$-algebras within the Batalin-Vilkovisky formalism, highlighting their role in understanding gauge theories and classical equivalences.

Contribution

It provides a detailed review of the BV formalism, emphasizing higher algebraic structures, and introduces new insights into higher Chern-Simons theories and computational techniques.

Findings

Field theories induce $L_ty$-algebras

Quasi-isomorphisms correspond to classical equivalences

New results on higher Chern-Simons theories

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.