String Topology, Field Theories, and the Topology of Moduli Spaces

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This proposal consists of several projects using algebraic topological techniques to study questions arising in String Topology, the topology of moduli spaces, and the study of Topological Quantum Field Theories. The theory of String Topology, first introduced by Chas and Sullivan in 1999, now involves a vast array of rich structure on spaces of paths and loops of manifolds. In this proposal, there are projects that will relate string topology to geometry and topological field theory. This includes a joint project with A. Blumberg and C. Teleman that will relate string topology to the recent classification theories of topological field theories due to Costello and Hopkins-Lurie. In particular they will construct and study a string topology category of a manifold, and compute its Hochschild homology. They will investigate the ``Calabi-Yau" properties of this category. In this study interesting questions involving the role of Koszul duality in topological field theory arise, which the authors will address. Cohen and his collaborators also have a goal of comparing the symplectic field theory of the cotangent bundle of a manifold, with the string topology of that manifold. Other projects in this proposal include a collaboration with M.Schwarz on the development of the ``Quantum String topology" of a symplectic manifold, and a project with N. Kitchloo, to address conjectures by the geometers F. Lalonde and D. McDuff on the Serre spectral sequence for bundles with symplectic fibers.This proposal consists of several projects investigating the area of research known as "String Topology", as well as related questions. String topology, a theory that was first introduced by Chas and Sullivan in 1999, studies structures on spaces of paths, loops, and surfaces. This structure was motivated by formalisms in string theory in physics. The idea is to understand how loops (or paths) in a background space can evolve in time. Loops can evolve by changing in size and even breaking apart. These phenomena are measured by studying surfaces mapping to the background space, that span these loops. In this project, Cohen tends to study various aspects of the theory related to topological quantum field theories, as well as those related to symplectic geometry.