The odd origin of Gerstenhaber brackets, Batalin-Vilkovisky operators, and master equations

Abstract

Abstract. Using five basic principles we treat Gerstenhaber/Lie brackets, BV operators and Master equations appearing in mathematical and physical contexts in a unified way. The different contexts for this are given by the different types of (Feynman) graphs that underlie the particular situation.Two of the maxims we bring forth are (1) that extending to the non-connected graphs gives a commutative multiplication forming a part of the BV structure and (2) that there is a universal odd twist that unifies and explains seemingly ad hoc choices of signs, and is responsible for the BV operator being a differential.Our treatment results in uniform, general theorems. These allow us to prove new results and recover and connect many constructions that have appeared independently throughout the literature. The more general point of view also allows us to disentangle the necessary from the circumstantial.

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

“…(1) This type of non-connected version of (co)-operads is one of the variations for non-connected operads studied in detail in [KWZ12].…”

Comprehensive factorisation systems

“…[25], where this Feynman category has been denoted G ctd . Since trees are connected we have a Feynman functor j : F cyc → F ctd .…”

Quantizing Deformation Theory II

“…Again, Barannikov in [Bar07], cf. [KWZn15], constructs, a bi-dg-Lie-algebra structure (g, [−, −], d, ∆), see Example 2.5, on a shifted direct sum . As in the previous example, this leads to a BV ∞ -algebra of a particular type, with all the BV operators ∆ n , except ∆ 2 , being derivations.…”