Mathematical Sciences: Floer Homotopy, Kontsevich-Gromov- Witten Theory, and Quantum Cohomology

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9504234 Morava The Floer homology of the free loopspace of a Kaehler manifold has been the subject of considerable attention in the last few years, in part because it defines a two-dimensional topological field theory. Recently Cohen, Jones, and Segal have defined an underlying notion of Floer homotopy type, which can be interpreted as the `universal deformation' of this topological field theory to what physicists call a theory of two-dimensional topological gravity. This Floer homotopy type is a rather mysterious MU-algebra spectrum. In this project the investigator constructs a conjectural model for it, and he probes some of the ways in which it provides an understanding of the constructions of string physicists from a homotopy-theoretical point of view. The whole subject has profound implications for the future of homotopy theory and global analysis. The theory of mechanics developed in the ninteenth century was based on 'principles of least action': a ray of light, for example, follows the path which minimizes its time of flight. The physicist Richard Feynman reinterpreted these ideas in terms of a theory of integration over the space of all possible paths; his ideas are now fundamental to our understanding of quantum mechanics. Unfortunately, the theory of such Feynman path integrals has never been made rigorous; indeed, it is now known that no naive generalization of the classical theory of integration can form an adequate basis for the integrals which arise in modern physics. In geometry, however, it has become clear recently that ideas from the theory of Feynman integrals can be used to solve classical problems of pure mathematics, and there is evidence that many geometric problems are in some sense 'tame' enough so that an analogue of the theory of Feynman integrals can be established rigorously. Although restricted in many ways, these geometrical test questions provide very clear and extremely important data for understanding the 'dyna mical' problems of direct interest to physics, and they are our best guide to a consistent theory of Feynman integrals. In this project the investigator sketches a conjectural description, in terms of algebraic topology, for a rather lapge class of topological field theories, which arise from the application of Feynman integral techniques to the geometry of complex manifolds. ***