L∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism. Issue 7 (8th July 2019) - Jurčo, Branislav; Raspollini, Lorenzo; Sämann, Christian; Wolf, Martin
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 ∞ ‐algebra and how quasi‐isomorphisms between L ∞ ‐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. Abstract : The authors 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, it is shown how a field theory gives rise to an L ∞ ‐algebra and how quasi‐isomorphisms between L ∞ ‐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. Most importantly, higher (categorified) Chern–Simons theories are discussed in great detail and some useful shortcuts in usually rather involved computations are given.