Vertex algebras and geometry of manifolds

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The aim of this project is to further the understanding of how vertex algebras are related to geometry of manifolds. While vertex algebras, a relatively new concept in algebra, have always been understood as an essential part of conformal field theory, their relation to higher dimensional geometry, even though implicit in string theory, had been rather obscure for a while. In 1998 Malikov, Schechtman and Vaintrob proposed the notion of chiral differential operators as a direct such link. Later work has found several applications of this new theory and, rather recently, Witten and Kapustin identified algebras of chiral differential operators with the so-called Witten half-twisted model in string theory.One characteristic feature of this proposal is that it truly belongs to the interface of several disparate disciplines, such as algebra, geometry, and theoretical physics. Indeed, vertex algebras are purely algebraic objects, which only relatively recently were found to be closely related to geometry of manifolds. On the other hand, this area of research is essentially a mathematical counterpart of the physics of strings. The results of the proposal are therefore expected to have a broad impact on research community.