In a joint work with J. Fröhlich and B. Pedrini, we study the relation between Chas and Sullivan's ``string topology'' and topological field theories. Namely, we consider symplectic reduction from (a graded version of) the space of Lie algebra valued forms on a manifold M and show that the Poisson algebra of functions on the reduced phase space contains a subalgebra (``generalized Wilson loops'') whose elements are labeled by homology classes in the loop space of M. More precisely, we construct a map from the reduced phase space to the S1-equivariant cohomology of the loop space of M and show that it is a Lie algebra homomorphism (where on the first space the Lie algebra structure is induced from the Poisson structure and in the second space it is Chas and Sullivan's). This generalizes in higher dimensions works of Andersen, Reshetikhin and Mattes relating Wilson loops in Chern-Simons theory to the Goldman bracket.
Math-Physics Joint Seminar
Monday, February 24, 2003 - 2:00pm
Alberto Cattaneo
Institut fuer Mathematik, Universitaet Zuerich