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.
This research project belongs in the interface of modern physics, especiallywhat is known as string theory and mathematics, especially algebra and algebraic geometry. For a physicist the esults obtained can be interpreted as statements about nalf-twisted Witten's model, for a mathematician, Malikov obtains localization of modules over certain infinite dimensional algebras. The localization idea belongs to Beilinson and Bernstein (and independentlyto Brylinski and Kashiwara.) They suugested that a rather unsophisticated algebra of differential operators contains various important and much more convoluted Lie algebras. In particular, once an algebra of differential operators acts on a vector space, all the algebras it contains start operating there too. Algebras of differential operators being intrisically geometric objects, this creates an avenue for applications of geometrical methods to algebra. This idea has proven remarkably powerful. In the present project, this is done for a class of more complicated algebras that have originated in modern physics and have to with replacing a point witha segment of a line, i.e., a string. During the period 2008--2010 Malikov has supervised one graduate student, D.Chebotarov, and one undergraduate, and taught two graduate graduate courses on representation theory and D-modules. Over this period of time Chebotarov has authored three research papers and is now on the verge of obtaining hs Ph.D. All these activities have to a great extent been influenced by the Malikov research activity. The students attending those classes and otherwise interacting with Malikov had therefore an appealing opportunity to ``breath the air of live science''.
