Open/Closed String Topology and Moduli Space Actions via Open/Closed Hochschild Actions

TL;DR

This paper extends string topology to open/closed sectors, revealing new algebraic structures and moduli space actions that incorporate punctures and dualities, advancing the understanding of open/closed string interactions.

Contribution

It introduces an extension of correlation functions to open/closed cases, establishing actions of an open/closed Sullivan PROP and moduli space with novel structures and conditions.

Findings

Open/closed Sullivan PROP actions established

New structures arise from open sector considerations

Moduli space restricted and covered to include punctures

Abstract

In this paper we extend our correlation functions to the open/closed case. This gives rise to actions of an open/closed version of the Sullivan PROP as well as an action of the relevant moduli space. There are several unexpected structures and conditions that arise in this extension which are forced upon us by considering the open sector. For string topology type operations, one cannot just consider graphs, but has to take punctures into account and one has to restrict the underlying Frobenius algebras. In the moduli space, one first has to pass to a smaller moduli space which is closed under open/closed duality and then consider covers in order to account for the punctures.