An algebra of differential operators is the enveloping algebra of a Lie algebroid T of vector fields. Similarly, a vertex algebra of differential operators is the enveloping algebra of a vertex algebroid, which is a Lie algebroid equipped with certain complementary differential operators. These operators should satisfy some complicated identities, these identities being a corollary of the Borcherd’s axioms of a vertex algebra.
In this note we attempt to shed some light at the definition of a vertex algebroid, by proposing a new, equivalent definition which has nothing to do with the axioms of a vertex algebra and uses only classical objects such as complexes of De Rham, Hochschild and Koszul. This point of view works nicely for Calabi-Yau structures as well and opens the way to higher dimensional generalisations.
