Gerbes of chiral differential operators. II: Vertex algebroids
In this paper the authors study a certain class of vertex algebras, which they call vertex algebras of differential operators (since for the class of vertex algebras these algebras are analogs of the classical algebras of algebraic differential operators). The following result is one of the main results of the paper: Let \(A\) be a smooth algebra over a field \(\mathbf k\) of characteristic \(0\) and \(T(A)\) be the algebra of \(\mathbf k\)-derivations of \(A\). Assume that \(T(A)\) admits an \(A\) base consisting of commutative vector fields. Then there is an action of some (very complicated) abelian group, constructed in terms of \(A\), on the groupoid of all vertex \(A\)-algebroids, which restricts to an equivalence of categories for every fixed \(A\)-algebroid. Sheafifying the construction of such action one gets a sheaf of groupoids, which gives rise to a certain characteristic class, whose description in terms of Pontryagin-Chern-Simons class constitutes the second main result of the paper. Several applications of the above theory in the case of curves are given. The paper finishes with the study of the conformal structures on the vertex algebras of differential operators. Part I, cf. Math. Res. Lett. 7, No. 1, 55--66 (2000; Zbl 0982.17013).
scientific article; zbMATH DE number 2101936