Differential operators and BV structures in noncommutative geometry

Abstract

We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operators on a commutative algebra in such a way that derivations of the commutative algebra are replaced by ${\mathbb{D}{\rm er}(A)}$, the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on ${\mathcal{F}(A)}$, a certain ‘Fock space’ associated to any noncommutative algebra A in a functorial way. The corresponding algebra ${\mathcal{D}(\mathcal{F}(A))}$ of differential operators is filtered and gr ${\mathcal{D}(\mathcal{F}(A))}$, the associated graded algebra, is commutative in some ‘wheeled’ sense. The resulting ‘wheeled’ Poisson structure on gr ${\mathcal{D}(\mathcal{F}(A))}$ is closely related to the double Poisson structure on ${T_{A} \mathbb{D}{\rm er}(A)}$ introduced by Van den Bergh. Specifically, we prove that gr ${\mathcal{D}(\mathcal{F}(A))\cong\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)),}$ provided the algebra A is smooth. Our construction is based on replacing vector spaces by the new symmetric monoidal category of wheelspaces. The Fock space ${\mathcal{F}(A)}$ is a commutative algebra in this category (a “commutative wheelgebra”) which is a structure closely related to the notion of wheeled PROP. Similarly, we have Lie, Poisson, etc., wheelgebras. In this language, ${\mathcal{D}(\mathcal{F}(A))}$ 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 ${\mathbb{D}{\rm er}(A)}$ gives rise to a second-order (wheeled) differential operator, a noncommutative analogue of the Batalin-Vilkovisky (BV) operator, that makes ${\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)))}$ a BV wheelgebra. In the final section, we explain how the wheeled differential operators ${\mathcal{D}(\mathcal{F}(A))}$ produce ordinary differential operators on the varieties of n-dimensional representations of A for all n ≥ 1.