Vassily Gorbounov

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.

Quantum integrability and generalised quantum Schubert calculus

Abstract We introduce and study a new mathematical structure in the generalised (quantum) cohomology theory for Grassmannians. Namely, we relate the Schubert calculus to a quantum integrable system known in the physics literature as the asymmetric six-vertex model. Our approach offers a new perspective on already established and well-studied special cases, for example equivariant K-theory, and in addition allows us to formulate a conjecture on the so-far unknown case of quantum equivariant K-theory.

Homotopy Gerstenhaber Structures and Vertex Algebras

We provide a simple construction of a G ∞ -algebra structure on an important class of vertex algebras V, which lifts the Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications to algebraic topology: the construction of a sheaf of G ∞  algebras on a Calabi–Yau manifold M, extending the operations of multiplication and bracket of functions and vector fields on M, and of a Lie ∞  structure related to the bracket of Courant (Trans Amer Math Soc 319:631–661, 1990).

Some Remarks on landau-Ginzburg Potentials for Odd-Dimensional Quadrics

We study the possibility of constructing a Frobenius manifold for the standard Landau-Ginzburg model of odd-dimensional quadrics

Homological Algebra and Divergent Series

Q_{2n+1}

Schubert calculus and singularity theory

and matching it with the Frobenius manifold attached to the quantum cohomology of these quadrics. Namely, we show that the initial conditions of the quantum cohomology Frobenius manifold of the quadric can be obtained from the suitably modified standard Landau-Ginzburg model.

Gerbes of chiral differential operators. II. Vertex algebroids

We study some features of infinite resolutions of Koszul algebras motivated by the developments in the string theory initiated by Berkovits.

Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra

Abstract Schubert calculus has been in the intersection of several fast developing areas of mathematics for a long time. Originally invented as the description of the cohomology of homogeneous spaces, it has to be redesigned when applied to other generalized cohomology theories such as the equivariant, the quantum cohomology, K -theory, and cobordism. All this cohomology theories are different deformations of the ordinary cohomology. In this note, we show that there is, in some sense, the universal deformation of Schubert calculus which produces the above mentioned by specialization of the appropriate parameters. We build on the work of Lerche Vafa and Warner. The main conjecture these authors made was that the classical cohomology of a Hermitian symmetric homogeneous manifold is a Jacobi ring of an appropriate potential. We extend this conjecture and provide a simple proof. Namely, we show that the cohomology of the Hermitian symmetric space is a Jacobi ring of a certain potential and the equivariant and the quantum cohomology and the K -theory is a Jacobi ring of a particular deformation of this potential. This suggests to study the most general deformations of the Frobenius algebra of cohomology of these manifolds by considering the versal deformation of the appropriate potential. The structure of the Jacobi ring of such potential is a subject of well developed singularity theory. This gives a potentially new way to look at the classical, the equivariant, the quantum and other flavors of Schubert calculus.