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

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This proposal consists of several projects using algebraic topological techniques to study questions arising from geometric topology and geometry. Among the most exciting and active areas of this type of study are ``String Topology", the study of the homotopy type of moduli spaces, and the study of topological field theories. The projects in this proposal, some long term, involve investigations into these new and very interesting areas of topology. In this proposal, there are projects that will apply string topology to geometry, including the development of ``Quantum String topology" of a symplectic manifold with M. Schwarz, the study of the cobordism type of moduli spaces of holomorphic curves with I. Madsen, and the relation of the string topology of a manifold to the ``Fukaya-Seidel" category of the cotangent bundle with its canonical symplectic structure,with C. Teleman and A. Blumberg. There are also projects that will to continue to study and develop the homotopy theoretic structure of this theory, including the construction and study of ``higher order" string topology operations with J.D.S. Jones, and the construction and study of the ``string topology $A infinity category" of a manifold, and its Hochschild homology, with A. Blumberg and C. Teleman. Other projects include the study of characteristic classes of conformal field theories, with Madsen, and a project with N. Kitchloo, to address questions asked 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 new 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. In this proposal this theory is studied in a variety of contexts, including the mathematical formalisms of Quantum Field Theory. Moreover the several projects understanding the relationship between string topology and more geometric theories are pursued.