String topology and configuration spaces of two points

Abstract

Given a closed manifold $M$. We give an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct. In the simply-connected case, this admits a particularly nice description in terms of a Poincar\'e duality model of the manifold, and involves the configuration space of two points on $M$. We moreover, construct an $IBL_\infty$-structure on (a model of) cyclic chains on the cochain algebra of $M$, such that the natural comparison map to the $S^1$-equivariant loop space homology intertwines the Lie bialgebra structure on homology. The construction of the coproduct/cobracket depends on the perturbative partition function of a Chern-Simons type topological field theory. Furthermore, we give a construction for these string topology operations on the absolute loop space (not relative to constant loops) in case that $M$ carries a non-vanishing vector field and obtain a similar description. Finally, we show that the cobracket is sensitive to the manifold structure of $M$ beyond its homotopy type. More precisely, the action of ${\rm Diff}(M)$ does not (in general) factor through ${\rm aut}(M)$.