Enumerative Geometry and Topology of Moduli Spaces

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DMS-0401128Michael Thaddeus This project encompasses several variations on the theme of quantum cohomology, aiming to construct interesting moduli spaces and study their enumerative geometry. The bulk of the project concerns the space of holomorphic maps from a curve to a loop group. This is finite-dimensional and can be compactified like a space of stable maps, allowing Gromov-Witten invariants of loop groups to be defined and calculated. This should allow a fruitful generalization of the very active field of quantum cohomology of homogeneous spaces to the infinite-dimensional setting of affine Lie groups. Another variation of quantum cohomology is the stringy or orbifold cohomology of Chen-Ruan: a version of the Weil conjectures for this theory will be sought, and the role of a B-field or grebe investigated. Yet another is the quantum Riemann-Roch theory recently developed by Coates, which will be applied to symmetric products of curves.This is research in algebraic geometry, one of the most classical parts of mathematics, concerned with finding solutions of polynomial equations. But it is informed by some of the latest ideas in the theory of holomorphic curves, and especially the Gromov-Witten theory, which seeks to count the numbers of curves on a given surface (or a space of higher dimension) of a given shape or position. This in turn is largely motivated by quantum field theory, and it is hoped that the project will ultimately make contact with ideas from physics.