148 research outputs found

### Vertex algebras and the Landau-Ginzburg/Calabi-Yau correspondence

We construct a spectral sequence that converges to the cohomology of the
chiral de Rham complex over a Calabi-Yau hypersurface and whose first term is a
vertex algebra closely related to the Landau-Ginburg orbifold. As an
application, we prove an explicit orbifold formula for the elliptic genus of
Calabi-Yau hypersurfaces.Comment: Latex, 50p. Some typos corrected, the page size may have been fixed.
One new result, a theorem on the vertx algebra structure of the
Landau-Ginzburg orbifold appears in sect. 5.2.18. This is the final version
to appear in the Moscow Mathematical Journa

### A vertex algebra attached to the flag manifold and Lie algebra cohomology

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.Comment: a reference adde

### Deformations of chiral algebras and quantum cohomology of toric varieties

We reproduce the quantum cohomology of toric varieties (and of some
hypersurfaces in projective spaces) as the cohomology of certain vertex
algebras with differential. The deformation technique allows us to compute the
cohomology of the chiral de Rham complex over the projective space.Comment: we use the deformation technique from the earlier version of this
note to compute the cohomology of the chiral de Rham complex over the
projective space; the two new results, Theorems 2.5A and B, are explained in
sect. 2.

### Kazhdan-Lusztig tensoring and Harish-Chandra categories

We use the Kazhdan-Lusztig tensoring to define affine translation functors,
describe annihilating ideals of highest weight modules over an affine Lie
algebra in terms of the corresponding VOA, and to sketch a functorial approach
to ``affine Harish-Chandra bimodules''.Comment: 22 pages late

### Localization of affine W-algebras

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.Comment: 36 page

### A chiral Borel-Weil-Bott theorem

We compute the cohomology of modules over the algebra of twisted chiral
differential operators over the flag manifold. This is applied to (1) finding
the character of $G$-integrable irreducible highest weight modules over the
affine Lie algebra at the critical level, and (2) computing a certain elliptic
genus of the flag manifold. The main tool is a result that interprets the
Drinfeld-Sokolov reduction as a derived functor.Comment: Some considerable reworking. A final version to appear in Adv. in
Mat

### Vertex Algebroids over Veronese Rings

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

### Rings of skew polynomials and Gel'fand-Kirillov conjecture for quantum groups

We introduce and study action of quantum groups on skew polynomial rings and
related rings of quotients. This leads to a ``q-deformation'' of the
Gel'fand-Kirillov conjecture which we partially prove. We propose a
construction of automorphisms of certain non-commutaive rings of quotients
coming from complex powers of quantum group generators; this is applied to
explicit calculation of singular vectors in Verma modules over
U_{q}(\gtsl_{n+1}).
We finally give a definition of a $q-$connection with coefficients in a ring
of skew polynomials and study the structure of quantum group modules twisted by
a $q-$connection.Comment: 25 page

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