1. In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by A.Vaintrob and two of the authors, cf.[MSV] and [MS1]. Hopefully our result clarifies to some extent the constructions of op. cit.Recall that in [MSV] we discussed two kinds of sheaves on smooth complex algebraic (or analytic) varieties X. First, we defined the sheaf of conformal vertex superalgebras Ω ch X , called chiral de Rham algebra. These sheaves are canonically defined for an arbitrary X. Second, for some varieties X one can define a purely even counterpart of Ω ch X , a sheaf of graded vertex algebras O ch X , called a chiral structure sheaf, cf. op. cit., §5. For example, one can define O ch X for curves, and for flag spaces G/B. For an arbitrary X, there arises certain cohomological obstruction to the existence of O ch X . The infinitesimal incarnation of this obstruction is calculated in op. cit., §5, A.

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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.

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

Abstract. We interpret the equivariant cohomology algebra H * GLn×C * (T * F λ ; C) of the cotangent bundle of a partial flag variety F λ parametrizing chains of subspaces 0Under this identification the dynamical connection of [TV1] turns into the quantum connection of [BMO] and [MO]. As a result of this identification we describe the algebra of quantum multiplication on H * GLn×C * (T * F λ ; C) as the algebra of functions on fibers of a discrete Wronski map. In particular this gives generators and relations of that algebra. This identification also gives us hypergeometric solutions of the associated quantum differential equation. That fact manifests the LandauGinzburg mirror symmetry for the cotangent bundle of the flag variety.