We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck's notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DDer(A), the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A), a certain Fock space' associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)), of differential operators, is filtered and gr D(F(A)), the associated graded algebra, is commutative in some twisted' sense. The resulting double Poisson structure on gr D(F(A)) is closely related to the one introduced by Van den Bergh. Specifically, we prove that gr D(F(A))=F(T_A(DDer(A)), provided A is smooth. It is crucial for our construction that the Fock space F(A) carries an extra-structure of a wheelgebra, a new notion closely related to the notion of a wheeled PROP. There are also notions of Lie wheelgebras, and so on. In that language, D(F(A)) becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on DDer(A) gives rise to a second order (wheeled) differential operator, a noncommutative analogue of the BV-operator.
One may also form a directed analogue of the construction of @cite_4 , replacing modular operads with wheeled PROPs (which include commutative wheelgebras), by replacing undirected graphs with directed graphs. Here, one can additionally keep track of a genus grading at vertices.
@cite_4
2043706520
I describe the noncommutative Batalin-Vilkovisky geometry as- sociated naturally with arbitrary modular operad. The classical limit of this geometry is the noncommutative symplectic geometry of the corresponding tree-level cyclic operad. I show, in particular, that the algebras over the Feyn- man transform of a twisted modular operad P are in one-to-one correspondence with solutions to quantum master equation of Batalin-Vilkovisky geometry on the affineP manifolds. As an application I give a construction of character- istic classes with values in the homology of the quotient of Deligne-Mumford moduli spaces. These classes are associated naturally with solutions to the quantum master equation on affineS(t) manifolds, where S(t) is the twisted modular Det operad constructed from symmetric groups, which generalizes the cyclic operad of associative algebras.