Cohomology of Lie conformal algebroids

TL;DR

This paper explores the structure and cohomology of Lie conformal algebroids, establishing connections with Poisson vertex algebras and introducing new constructions and isomorphisms in their cohomology theories.

Contribution

It introduces several constructions of Lie conformal algebroids and establishes an isomorphism between their cohomology and that of Poisson vertex algebras.

Findings

Constructed various examples of LCAd, including conformal derivations and gauge LCAd.

Established a cohomology isomorphism between LCAd and Poisson vertex algebras.

Linked cohomology theories of LCAd with those of PVAs.

Abstract

We study Lie conformal algebroids (LCAd) and their representations using the language of lambda-brackets and Lie conformal algebras. We describe several general constructions, such as the LCAd of conformal derivations CDer(A) of a differential algebra A, the gauge LCAd G(A,M) associated to a differential algebra A and its module M, the current LCAd F^ of a Lie algebroid F, and the LCAd structure of the space Omega(V) of Kahler differentials over a Poisson vertex algebra (PVA) V. We develop the cohomology theories of LCAd and we relate them to the corresponding cohomology theories of PVA. In particular, we find an isomorphism between the cohomology of a PVA V with coefficients in a module M and the corresponding cohomology of the LCAd Omega(V) with coefficients in the same module.