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Gerbes of chiral differential operators

Wednesday 11th December 2002 - 16:30 to 17:30
INI Seminar Room 1
Session Title: 
Elliptic cohomology and chromatic phenomena
In this talk we give a complete classification of a certain important class of vertex algebras, the so called algebras of chiral differential operators (cdo for short). These were introduced by Beilinson and Drinfeld, motivated by the ideas from the conformal field theory. It turned out, that "almost classical" structure of what we call the vertex algebroid controls the world of cdo. This structure consists of two part. The first is a structure of an algebroid Lie, and the other is its extension, both derived from the major identities held in chiral algebras. These extra structures can be fit into a complex of vector spaces which is a direct generalization of the De Rham-Chevalley complex for Lie algebras. The classes of equivalences of cdo are in one to one correspondence with the third cohomology group of this complex. We also provide the description of each isomorphism class.

This allows us to study the sheafication of a an important class of cdo over a given manifold X. This is a cdo defined by the Heisenberg algebra and the Clifford algebra, or the free bosons and the free fermions. It turns out that there is a characteristic class which is an obstraction to glueing these local pieces into a sheaf. It has two components, first, is the Atiyah class, and the other is the Simons term. There is a projection of this class into cohomology of X, and the image of this projection is the second component of the Chern character of X.

We also describe these algebras for specific classes of manifolds, namely algebraic groups and homogeneous spaces and a conjectural connection of these algebras defined for hypersurfaces to the Vafa orbifold models.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons