Cohomology of Lie conformal algebroids

TL;DR

The paper develops a systematic LCAd cohomology theory and links it to Poisson vertex algebra (PVA) cohomology via the Kähler differential LCAd $\Omega(\mathcal{V})$. It defines LCAd structures, their representations, and three cohomology flavors (basic, reduced, variational), and then proves an explicit isomorphism $H_{\mathrm{var}}(\mathcal{V},M) \cong H_{\mathrm{LCAd}}(\Omega(\mathcal{V}),M)$ (with corresponding basic and reduced variants). This establishes a bridge between deformations and derivations of PVAs and their LCAd analogues, enabling transfer of results between PVA and LCAd cohomology theories. The framework sets the stage for deformation theory and quantization of PVAs and their algebroid counterparts, hinting at broader connections to chiral algebroids in vertex algebra theory.

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.

Theorems & Definitions (76)

• Definition 2.1
• Lemma 2.2
• proof
• Example 2.3
• Example 2.4
• Example 2.5
• Remark 2.6
• Example 2.7
• Proposition 2.8
• proof : Proof of Prop.\ref{['prop:gauge']}