Moduli space actions on the Hochschild co-chains of a Frobenius algebra II: Correlators

TL;DR

This paper demonstrates how moduli space actions and Sullivan Chord diagram-based dg-PROPs act on Hochschild co-chains of Frobenius algebras, connecting geometric structures to algebraic operations and loop space homology.

Contribution

It introduces discretized operadic and PROPic structures and operadic correlation functions to realize these actions explicitly.

Findings

Cell model of moduli space acts on Hochschild co-chains

dg-PROP action of Sullivan Chord diagrams on Hochschild co-chains

Actions induce operations on loop space homology

Abstract

This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co--cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply--connected manifold. In this second part, we discretize the operadic and PROPic structures of the first part. We also introduce the notion of operadic correlation functions and use them in conjunction with operadic maps from the cell level to the discretized objects to define the relevant actions.