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

Abstract

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.

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“…Definition 4.1. [Segal 2008] Let L = d i=0 L i be a finite dimensional L ∞ algebra over k, with its L ∞ products denoted by µ k . Define L to be the graded vector space…”

“…Step two is based on the cyclic completion (see Theorem 4.2) and boundedness of L ∞ products (see Theorem 4.4). Theorem 4.2 was first proved by Aspinwall and Fidkowski [2006] and later reproved in a much more general setting by Segal [2008]. The terminology cyclic completion is due to Segal.…”

“…Theorem 4.2 [Aspinwall and Fidkowski 2006;Segal 2008]. The Yoneda algebra L ω is the cyclic completion of the Yoneda algebra L.…”