TQFTs in Spectra

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DMS-0116288Jack MoravaThe central idea of this proposal is a mathematical definition for the physicists' notion of topological gravity (analogous to Segal's mathematical definition of conformal field theory) as a representation of a monoidal category with manifolds as objects, using the geometric realization of a category of cobordisms between those objects as its morphism spaces; these spaces are unions of classifying spaces for the diffeomorphism groups of the cobordisms. In two dimensions, the resulting category is quite similar to that considered by Segal, but it generalizes very naturally, e.g. to four dimensions, where it has close connections with classical general relativity; but there is also a version for topological (i.e, non-smooth) four-manifolds, lacking any clear classical analog. Donaldson and Seiberg-Witten theory fit naturally into this framework, which predicts that such invariants should have higer-order `gravitational descendants', e.g. higher-codimension versions of the wall-crossing obstructions. In dimension two, this formalism fits in well on the one hand with work of Madsen and Tillmann on Mumford's conjecture, and on the other with the theory of a `large' quantum cohomology studied by Kontsevich, Manin, Witten, and others. One concrete goal of the projectis to construct a cohomological theory related to the Atiyah-Patodi-Singer eta-invariant of a three-manifold, as Casson's invariant is related to Floer homology. Classical mechanics studies the trajectories of point particles in a smooth geometric background, and much recent work in string theory can be formulated in similar terms, with the background replaced by the (infinite-dimensional) space of smooth loops in some ambient manifold. However, the mathematics of these free loopspaces is quite challenging, and their topology (not to mention their geometry) is not yet well-understood.An added complication is that the models studied in quantum field theory involve topology change in a conceptually intrinsic way, and thus seem often to call, not for the free loopspace itself, but for a suitablecompletion with nice properties -- whose nature is still being worked out. This proposal suggests that the desired completion is an analog, for free loopspaces, of the dual of a finite-dimensional smooth manifold (as studied in the 1960's by Whitehead, Spanier, Atiyah, and others). This point of view seems compatible with work of Chas and Sullivan on string topology, with work of Cohen, Jones, and Segal on Floer homotopy type, and with work of Ando and myself on the Witten genus.