The A∞ deformation theory of a point and the derived categories of local Calabi–Yaus

Let A be an augmented algebra over a semi-simple algebra S. We show that the Ext algebra of S as an A-module, enriched with its natural A-infinity structure, can be used to reconstruct the completion of A at the augmentation ideal. We use this technical result to justify a calculation in the physics literature describing algebras that are derived equivalent to certain non-compact Calabi–Yau three-folds. Since the calculation produces superpotentials for these algebras we also include some discussion of superpotential algebras and their invariants.