On some generalizations of Batalin-Vilkovisky algebras

AbstractWe 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 Δ is a differential operator of order ≤ r if an inductively defined (r+1)-linear form Φr+1Δ 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 Δ is an odd, square zero, second order differential operator on A, Φ2Δ 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 Δ 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 1.(i) classical BV algebras,2.(ii) vertex operator BV algebras, and3.(iii) generalized BV algebras, relating it to deformations of differential graded algebras