Topology of Orbifolds, Vertex Algebras and Quantum Cohomology

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The first part of the project is concerned with topological and string-theoreticinvariants of orbifolds. Based on physical predictions, Chen and Ruan introduceda new cohomology theory for orbifolds. The object of study here will be algebraicstructures on the orbifold cohomology and relations with other cohomology theories. Applications and generalizations of the chiral de Rham complex, acanonical sheaf of vertex algebras associated to a manifold, are in the focus ofthe second part of the project. In particular, vertex algebras and modulesrelated to general orbifolds and their relation with orbifold cohomology,elliptic genus and quantum cohomology will be investigated. The aim of the thirdpart of the project is to study geometric and algebraic structures related tocohomological field theories of spin type and the corresponding spin analogs ofquantum cohomology and Gromov-Witten invariants. This part is related to aconjecture by E.Witten stating that the generating function of intersectionnumbers on the moduli spaces of Riemann surfaces with r-spin structures is atau-function of the r-KdV hierarchy.Ideas coming from physical models of quantum field theory are responsible formany exciting recent developments in mathematics, in particular in topology andgeometry. This project studies several geometric problems inspired by stringtheory, the current candidate for the theory unifying all kinds of physicalforces. Because of lack of experimental tools, physicists working in stringtheory are testing their methods on sophisticated mathematical models, and theirresults often indicate the existence of conjectural ties between variousseemingly remote mathematical objects. The goal of the current project is amathematical study of some of these conjectures and interactions. Besidesproducing new mathematical results, it may lead to a mathematical foundation forsome physical models and a justification of physical ways of analyzing them.