Open-closed homotopy algebra in mathematical physics

TL;DR

This paper explores the structure of open-closed homotopy algebras (OCHAs) derived from string field theory, clarifying their mathematical relations and providing a minimal model for deformation of open string structures by closed strings.

Contribution

It introduces an explicit construction of OCHAs from string theory, connects them with deformation quantization and mirror symmetry, and presents a minimal model for their algebraic structure.

Findings

OCHAs derived from string field theory relate to deformation quantization.

Explicit minimal model for OCHAs is provided.

OCHAs offer a framework for deforming open string structures with closed strings.

Abstract

In this paper we discuss various aspects of open-closed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach's open-closed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach's quantum open-closed string field theory. We clarify the explicit relation of an OCHA with Kontsevich's deformation quantization and with the B-models of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of open-closed string field theory. We show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures ($A_\infty$-algebras) by closed strings ($L_\infty$-algebras).