Higher string topology operations

TL;DR

This paper extends string topology operations on the free loop space of a manifold to a topological conformal field theory framework, linking homology operations to moduli spaces of Riemann surfaces.

Contribution

It introduces a novel extension of string topology operations to a topological conformal field theory setting, connecting homology of loop spaces with moduli space structures.

Findings

Operations parameterized by moduli space homology

Extension of Chas and Sullivan's product to a conformal field theory

Establishment of a new algebraic structure on loop space homology

Abstract

Chas and Sullivan have defined an intersection-type product on the homology of the free loop space LM of an oriented manifold M. In this paper we show how to extend this construction to a topological conformal field theory of degree d. In particular, we get operations on the homology of LM which are parameterized by the homology of the moduli space of open-closed Riemann surfaces.