On the relation between open and closed topological strings

TL;DR

The authors propose that closed-string data in topological string theories can be reconstructed from open-string data by identifying the closed sector with the Hochschild cohomology $HH^*(\mathcal{A})$ of the D-brane category. They explicitly compute $HH^*(\mathcal{A})$ for Landau-Ginzburg models on affine space and certain orbifolds, showing it reproduces the Jacobi ring $J_W$, bulk-boundary maps, and key disk correlators via the diagonal brane construction. They demonstrate consistency with known Yukawa couplings and bulk-boundary relations, and illustrate the approach across minimal models and Gepner-type CY examples, including CY 0-folds and higher-dimensional Fermat hypersurfaces. The results support a uniform, algebraic route to open-closed correlators and suggest avenues for generalizing to higher-genus theories and potentially bosonic strings, with practical implications for computing superpotentials in Gepner models.

Abstract

We discuss the relation between open and closed string correlators using topological string theories as a toy model. We propose that one can reconstruct closed string correlators from the open ones by considering the Hochschild cohomology of the category of D-branes. We compute the Hochschild cohomology of the category of D-branes in topological Landau-Ginzburg models and partially verify the conjecture in this case.