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

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.

Closed/open string diagrammatics

“…In particular for closed strings, it can be shown [73] that there is a natural chain complex of open or relative cells, which calculates the homology of the moduli spaces, i.e., the homology of Arc # (F, β ∅ ) which can be given the structure of an operad. In effect, these spaces are graded by the number of arcs in an arc family, and the corresponding filtration is preserved by the gluing operations when viewed as operations on filtered families.…”

Topological conformal field theories and gauge theories

“…A different construction of closed TCFTs, staring with a cyclic A 1 algebra, was previously given by Kontsevich and Soibelman [11; 12; 14]. Recent work of Tradler and Zeinalian [24] and Kaufmann [9] give new approaches to this problem.…”

Modular envelopes, OSFT and nonsymmetric (non-$\Sigma$) modular operads

“…The computations in this section, namely in Example 40, can be used to lift the commutativity assumption in [22, Section 4.1], cf. namely Remark 4.1 of [22], proving that TFT's as functors from the category of cobordisms to vector spaces are equivalent to Frobenius algebras.…”