Deformations of Topological Open Strings

TL;DR

This work presents a comprehensive framework linking deformations of topological open strings to the Hochschild cohomology of the open-string algebra, thereby realizing a physical instance of the Deligne theorem as a two-dimensional closed-string structure. By analyzing mixed bulk-boundary correlators and their Ward identities, it uncovers a Gerstenhaber (and more generally $G_\infty$) algebraic structure governing deformations, and recasts these deformations in terms of the operad of little discs and $d$-algebras. The paper grounds the theory in concrete gauge-theory realizations, notably Chern–Simons and holomorphic Chern–Simons theories, and demonstrates how the A- and B-models on the disc realize the deformation complex as Hochschild cohomology of the open-string algebra. It outlines broad generalisations to higher dimensions, AdS/CFT, and open/closed dualities, and points to fruitful future directions in formality, integrability, and the interplay between boundary conditions and bulk moduli. Overall, it provides a rigorous algebraic backbone for open-closed string deformations with explicit links to gauge theory and topological field theories.

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.

Figures (8)

• Figure 1: Closed string interactions.
• Figure 2: Open string interactions.
• Figure 3: Mixed interactions.
• Figure 4: Deformation of the product.
• Figure 5: Factorisation of the BRST operator.