Chern-Simons theory and string topology

Abstract

We construct chain-level $S^1$-equivariant string topology for each simply connected closed manifold. This amounts to constructing a Maurer-Cartan element for the canonical involutive Lie bialgebra (IBL) structure on the dual cyclic bar complex of its de Rham cohomology which is unique up to ${\rm IBL}_\infty$ gauge equivalence. The construction involves integrals over configuration spaces associated to trivalent ribbon graphs, which can be seen as a version of perturbative Chern-Simons theory in arbitrary dimension.