Moduli of Spin Curves, Geometric Vertex Algebras and Quantum Cohomology

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Abstract for DMS - 0104397.The investigator will study two groups of problems related tophysical models of topological string theory and two-dimensionalgravity and their interaction with geometry and other branches ofmathematics. Based on physical ideas and methods, E.Wittenconjectured that certain topological invariants of the moduli spacesof Riemann surfaces with higher spin structures assemble in agenerating function that can be determined by an infinite system ofdifferential equations (the so-called Gelfand-Dickey hierarchy).The goal of the first part of the proposal is to study geometric andalgebraic structures related to this conjecture by using techniquesfrom the theory of Gromov-Witten invariants and quantumcohomology. Besides exposing deep connections between seeminglyunrelated areas of geometry and analysis, a goal of this part of theproject is to develop tools for analyzing more general cohomologicalfield theories of spin type and the corresponding spin analogs ofquantum cohomology and Gromov-Witten invariants. The second part ofthe proposal is related to applications and generalizations of theso-called chiral de Rham complex, a canonical sheaf of vertexalgebras introduced in joint work with Malikov and Schechtman. Themain goal here is to find a rigorous geometric foundation for thephysical models of two-dimensional superconformal field theory onsmooth varieties and orbifolds.String theory is a proposed physical theory which attempts to unifyall kinds of forces and, in particular, combines Einstein's theoryof gravity with the quantum theory. Because this theory cannot yetbe verified on experiments, physicists working in string theory aretesting it on sophisticated mathematical models. Often the samephysical quantity can be computed by more than one method whichgives answers expressed in terms of very different mathematicalstructures. This suggests the existence of deep ties between variousseemingly remote parts of mathematics and leads to formulation ofnon-trivial conjectures connecting different mathematical objects,usually geometric and analytical in nature. The goal of the currentproject is a mathematical study of some of these conjectures andinteractions. Besides producing new mathematical results, it shouldgive mathematical verification of some physical models and providemore direct links between mathematical and physical methods ofanalysis of these models.