An open-closed string analogue of Hochschild cohomology

TL;DR

This paper establishes a new algebraic framework linking open-closed homotopy algebras to Hochschild cohomology, revealing a Gerstenhaber algebra structure and extending the Hochschild differential in an open-closed context.

Contribution

It introduces an open-closed Hochschild cochain complex and demonstrates its Gerstenhaber algebra structure, extending Hochschild theory to open-closed homotopy algebras.

Findings

Hochschild cohomology admits a Gerstenhaber algebra structure in the open-closed setting

Development of open-closed brace relations and concise OCHA descriptions

Extension of Hochschild differential to an A-infinity structure in open-closed context

Abstract

We prove that every open-closed homotopy algebra, introduced by Kajiura and Stasheff (arXiv: archive/0410291), naturally gives rise to an open-closed version of Hochschild cochain complex whose cohomology admits a canonical Gerstenhaber algebra structure. We also develop the open-closed brace relations, provide a concise description of OCHAs, and establish an A-infinity structure that extends the open-closed Hochschild differential.