Vertex algebras and geometry

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Algebras of chiral differential operators, a concept around which this project is centered, are in this class and were introduced by Schechtman, Vaintrob, and the principal investigator some 10 years ago. Here is the list of the main topics addressed in this project: 1. Gerbes of asymptotic chiral differential operators on smooth algebraic varieties. 2. Asymptotic chiral differential operators and localization of affine W-algebras. W-algebras and singularity theory. 3. Asymptotic chiral differential operators on singular affine algebraic varieties. Semi-infinite induction functor and string theory.The present project belongs in the interface of mathematics, especially its algebro-geometric part, and modern quantum field theory. Classical mechanics is adequately described by such algebraic structures as Poisson algebras, quantization is usually satisfactorily reflected in geometric representation theory. The passage to fields drastically changes the landscape and, in particular, leads to such modern and still poorly understood concepts as a vertex Poisson algebra and a vertex algebra.