Vertex Algebras and the Equivariant Lie Algebroid Cohomology

Abstract: A vertex-algebraic analogue of the Lie algebroid complex is constructed, which generalizes the "small" chiral de Rham complex on smooth manifolds. The notion of VSA-inductive sheaves is also introduced. This notion generalizes that of sheaves of vertex superalgebras. The complex mentioned above is constructed as a VSA-inductive sheaf. With this complex, the equivariant Lie algebroid cohomology is generalized to a vertex-algebraic analogue, which we call the chiral equivariant Lie algebroid cohomology. In fact, the notion of the equivariant Lie algebroid cohomology contains that of the equivariant Poisson cohomology. Thus the chiral equivariant Lie algebroid cohomology is also a vertex-algebraic generalization of the equivariant Poisson cohomology. A special kind of complex is introduced and its properties are studied in detail. With these properties, some isomorphisms of cohomologies are developed, which enables us to compute the chiral equivariant Lie algebroid cohomology in some cases. Poisson-Lie groups are considered as such a special case.