Fyodor Malikov

Gerbes of chiral differential operators

This paper is a sequel to [GII]. Its aim is to “switch on an exterior vector bundle” in the framework of [GII].

ON CHIRAL DIFFERENTIAL OPERATORS OVER HOMOGENEOUS SPACES

We give a classification and construction of chiral algebras of differential op- erators over semisimple algebraic groups G and over homogeneous spaces G/N and G/P where N is a nilpotent and P a parabolic subgroup.

Lagrangian Approach to Sheaves of Vertex Algebras

We explain how sheaves of vertex algebras are related to mathematical structures encoded by a class of Lagrangians. The exposition is focused on two examples: the WZW model and the (1,1)-supersymmetric

Localization of Affine W-Algebras

\sigma

A Vertex Algebra Attached to the Flag Manifold and Lie Algebra Cohomology

-model. We conclude by showing how to construct a family of vertex algebras with base the Barannikov-Kontsevich moduli space.We explain how sheaves of vertex algebras are related to mathematical structures encoded by a class of Lagrangians. The exposition is focused on two examples: the WZW model and the (1,1)-supersymmetric σ-model. We conclude by showing how to construct a family of vertex algebras with base the Barannikov-Kontsevich moduli space thus furnishing the B-model moduli for Witten’s half-twisted model.

An Introduction to Algebras of Chiral Differential Operators

We introduce the notion of an asymptotic algebra of chiral differential operators. We then construct, via a chiral Hamiltonian reduction, one such algebra over a resolution of the intersection of the Slodowy slice with the nilpotent cone. We compute the space of global sections of this algebra, thereby proving a localization theorem for affine W-algebras at the critical level.

Vertex Algebroids over Veronese Rings

Each flag manifold carries a unique algebra of chiral differential operators. Continuing along the lines of arXiv:0903.1281 we compute the vertex algebra structure on the cohomology of this algebra. The answer is: the tensor product of the center and a subalgebra; the center is isomorphic, as a commutative associative algebra, to the cohomology of the corresponding maximal nilpotent Lie algebra; the subalgebra is the vacuum module over the corresponding affine Lie algebra of critical level and 0 central character. We next find the Friedan-Martinec-Shenker-Borisov bosonization of the cohomology algebra in case of the projective line and show that this algebra vanishes nonperturbatively, thus verifying a suggestion by Witten.

Gerbes of chiral differential operators. II. Vertex algebroids

These notes are an informal introduction to algebras of chiral differential operators. The language used is one of vertex algebras, otherwise the approach chosen is that suggested by Beilinson and Drinfeld. The prerequisites are kept to a minimum, and we even give an informal introduction to the Beilinson-Bernstein localization theory in the example of the projective line.

Chiral de Rham complex. II

We find a canonical quantization of Courant algebroids over Veronese rings. Part of our approach allows a semi-infinite cohomology interpretation, and the latter can be used to define sheaves of chiral differential operators on some homogeneous spaces including the space of pure spinors punctured at a point.