Gerbes of chiral differential operators. II: Vertex algebroids
type: Item
instance of: scholarly article
title: Gerbes of chiral differential operators. II: Vertex algebroids
zbMATH Open document ID: 1056.17022
author: V. Gorbounov
author: Fyodor Malikov
author: Vadim Schechtman
published in: Inventiones Mathematicae
publication date: 2004-09-22T00:00:00Z
review text: 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).
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 14F40
zbMATH DE Number: 2101936
zbMATH Keywords: vertex algebra
zbMATH Keywords: algebroid
zbMATH Keywords: vector bundle
zbMATH Keywords: torseur
zbMATH Keywords: gerbe
zbMATH Keywords: Calabi-Yau structure
zbMATH Keywords: enveloping algebra
MaRDI profile type: Publication
full work available at URL: s00222-003-0333-4
OpenAlex ID: W1992886027
DOI: 10.1007/S00222-003-0333-4
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description: scientific article; zbMATH DE number 2101936