Deformations of Topological Open Strings

TL;DR

This paper explores the algebraic structures underlying deformations of topological open string theories, highlighting their Hochschild complex and Gerstenhaber algebra properties, with applications to A- and B-models.

Contribution

It identifies the algebraic operations of the bulk sector in topological open strings using mixed correlators, providing a physical realization of the Deligne theorem.

Findings

Hochschild complex encodes deformations of open string theories

Gerstenhaber algebra structure is present in open string deformations

Application to topological A- and B-models on the disc

Abstract

Deformations of topological open string theories are described, with an emphasis on their algebraic structure. They are encoded in the mixed bulk-boundary correlators. They constitute the Hochschild complex of the open string algebra -- the complex of multilinear maps on the boundary Hilbert space. This complex is known to have the structure of a Gerstenhaber algebra (Deligne theorem), which is also found in closed string theory. Generalising the case of function algebras with a B-field, we identify the algebraic operations of the bulk sector, in terms of the mixed correlators. This gives a physical realisation of the Deligne theorem. We translate to the language of certain operads (spaces of d-discs with gluing) and d-algebras, and comment on generalisations, notably to the AdS/CFT correspondence. The formalism is applied to the topological A- and B-models on the disc.