Moduli Spaces: Their Topology and Representations

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9803427 Kimura This project's goal is to study the "representation ring," in the appropriate sense, of the (homology groups of the) moduli space of curves and its close cousins. Such objects are rigorous constructions of nontrivial topological quantum field theories. In the case of the moduli space of curves, the corresponding representations are cohomological field theories (in the sense of Kontsevich and Manin), the most dramatic examples of which arise from Gromov-Witten invariants of a smooth projective variety. The investigator will study properties of canonical families of such structures arising from the geometry of these moduli spaces, using techniques borrowed from homotopy theory (operads), algebraic geometry (moduli spaces), and quantum field theory (Feynman diagrams). Recently, much of the interaction between mathematics and theoretical physics has revolved about topological quantum field theories, quantum field theories whose observables correspond to topological invariants. A rigorous construction of a nontrivial class of such theories is now possible through the study of Gromov-Witten invariants, topological invariants associated to a certain space of maps from Riemann surfaces into a Kahler manifold. The purpose of this project is to study natural families of such theories (and their relatives) that arise from the geometry of these spaces and to understand their properties under natural operations suggested by the geometry. Success should shed light on both the mathematics and the physics involved. ***