Vertex algebras and geometry of manifolds

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The purpose of this project is to further the understanding of therelation between vertex algebras and the geometry of manifolds.This relation, discovered relatively recently, is an infinitedimensional version of the classical relation between theoreticalphysics and mathematics. Indeed, quantum mechanics of a particleon a manifold is a part of analysis on manifolds. Similarly,super-symmetric quantum mechanics is essentially equivalent tothe study of the de Rham complex on manifolds. Attempts to carrythis over to the case of the space of loops on a manifold havecreated an entirely new Universe of mathematical and physicalconcepts. One such concept, vertex algebra, a child of stringtheory and infinite dimensional representation theory, has alreadyproved indispensable in areas as remote form mathematical physicsas abstract finite group theory. Relatively recently, thanks towork of Gorbounov and Malikov among others, vertex algebras havebeen fruitfully applied to several chapters of geometry includingelliptic genus and mirror symmetry.String theory, a part of quantum field theory and the onlyexisting candidate for the "theory of everything", while havingnot yet achieved its ultimate goal, has enriched modernmathematics and physics with a dazzling array of new concepts andnew relations among old concepts. Vertex algebra, a concept thatsimultaneously generalizes the notions of Lie and commutativealgebra, is one conspicuous example. It has long since foundapplications in areas as old and diverse as representation theory,group theory, combinatorics, and number theory. The presentproject is devoted to the exploration of a recently discoveredrelation to geometry of higher dimensional spaces.