Chern-Simons theory and string topology

TL;DR

This paper develops a chain-level $S^1$-equivariant string topology framework for simply connected closed manifolds, linking Chern-Simons theory, graph integrals, and algebraic structures on cohomology.

Contribution

It constructs a Maurer-Cartan element for the IBL structure on the dual cyclic bar complex, providing a novel algebraic approach to string topology via perturbative Chern-Simons theory.

Findings

Construction of $S^1$-equivariant string topology for manifolds

Identification of a unique Maurer-Cartan element up to gauge equivalence

Connection between configuration space integrals and algebraic structures

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.