String topology and homotopy Frobenius algebras

The ultimate goal of this action is to establish that chain-level string topology is not a homotopy invariant.

This is achieved by showing that chain-level string topological structures are induced by a homotopy Frobenius structure on the cochain algebra and by connecting the homotopy Frobenius structure with known invariants from quantum field theory. This is broken down into four independent work packages.

The first goal is to show that from a Chern-Simons type partition function one can construct a homotopy Frobenius algebra and show that this is essentially an equivalence between the relevant deformation spaces.

The second goal is to algebraically construct string topology operations on the Hochschild homology of a homotopy Frobenius algebra.

The third goal compares the induced structure on the cyclic homology with the known homotopy involutive Lie bialgebra structure.

And ultimately, the fourth goal is to compare the algebraically constructed operations with geometric ones on the loop space under the comparison map given by Chen's iterated integrals.