Homotopy algebra of open-closed strings

TL;DR

This paper surveys open-closed homotopy algebras, their geometric background involving Riemann surface compactifications, and their connections to deformation theory, highlighting Merkulov's geometric A_infty-structure as a key example.

Contribution

It presents a comprehensive overview of open-closed homotopy algebras, including new insights into Merkulov's geometric A_infty-structure and their relation to deformation theory.

Findings

Merkulov's geometric A_infty-structure as an example of OCHA

Connections between open-closed homotopy algebras and deformation theory

Geometric interpretation via Riemann surface compactifications

Abstract

This paper is a survey of our previous works on open-closed homotopy algebras, together with geometrical background, especially in terms of compactifications of configuration spaces (one of Fred's specialities) of Riemann surfaces, structures on loop spaces, etc. We newly present Merkulov's geometric A_infty-structure [Internat. Math. Res. Notices (1999) 153--164, arxiv:math/0001007] as a special example of an OCHA. We also recall the relation of open-closed homotopy algebras to various aspects of deformation theory.