On some generalizations of Batalin-Vilkovisky algebras

Abstract

We define the concept of higher order differential operators on a general noncommutative, nonassociative superalgebra A, and show that a vertex operator superalgebra (VOSA) has plenty of them, namely modes of vertex operators. A linear operator ∆ is a differential operator of order ≤ r if an inductively defined (r + 1)-linear form Φ r+1 ∆ with values in A is identically zero. These forms resemble the multilinear string products of Zwiebach. When A is a "classical" (i.e. supercommutative, associative) algebra, and ∆ is an odd, square zero, second order differential operator on A, Φ 2 ∆ defines a "Batalin-Vilkovisky algebra" structure on A. Now that a second order differential operator makes sense, we generalize this notion to any superalgebra with such an operator, and show that all properties of the classical BV bracket but one continue to hold "on the nose". As special cases, we provide several examples of classical BV algebras, vertex operator BV algebras, and differential BV algebras. We also point out connections to Leibniz algebras and the noncommutative homology theory of Loday. Taking the generalization process one step further, we remove all conditions on the odd operator ∆ to examine the changes in the basic properties of the bracket. We see that a topological chiral algebra with a mild restriction yields a classical BV algebra in the cohomology. Finally, we investigate the quantum BV master equation for (i) classical BV algebras, (ii) vertex operator BV algebras, and (iii) generalized BV algebras, relating it to deformations of differential graded algebras.

Chicken or egg? A hierarchy of homotopy algebras

“…It has been pointed out to us by Stasheff that physicists use the term "weak A ∞ " to refer to similar algebras with nonzero element m (0) so that m (1) is no longer square-free; our use of the traditional phrase "weakly homotopy" is not intended to revoke this condition.…”

“…For example, the partition (1|1) occurs as a term only in the partition products (1) * (1|1), (1|1) * (1), (2) * (1|0), and (2) * (0|1), but nowhere else. Therefore, the sub-identity corresponding to the partition (1|1) is written as {m (1) }{m (1|1) } ± {m (1|1) }{m (1) } ± {m (2) }{m (1|0) } ± {m (2) }{m (0|1) } {a|b} = 0.…”

“…The notion of a higher order differential operator B ∈ Hom(A, A) for a noncommutative, nonassociative algebra (A, m (2) ) was introduced by the author in [1]. The definition was modeled after Koszul's [15] for commutative, associative algebras.…”

“…A weakly homotopy Batalin-Vilkovisky algebra (or differential BV-algebra, as it has been called in literature) should be a super graded vector space with a number of higher multilinear maps, in particular an odd, square-zero operator m (1) that is a derivation of some even, bilinear product m (2) , and an odd, linear, square-zero operator B that is a second-order differential operator with respect to m (2) . We require m (2) be super commutative and associative up to homotopy (although homotopy associativity is needed only to fill out the definition of the classical algebra in the cohomology), and the commutator [B, m (1) ] be a diagonalizable operator L (possibly zero) that commutes with the two degree operators.…”

“…A familiar example of coexistence of these phenomena is that we regard an A ∞ algebra to be a homotopy associative algebra because the map m (2) descends to an ordinary associative algebra map in the m (1) -cohomology, but also because its mega map m satisfies the same identity as m (2) , namely that m (2) composed with itself is zero. Thus certain homotopy Gerstenhaber algebras are given that name through the anti-symmetrization of their special bilinear maps m (1|1) descending to a G-bracket; yet others could be called G-algebras because their mega maps would be composed of higher brackets satisfying certain familiar identities, one of which directly descends to a G-bracket.…”