From β to η: a new cohomology for deformed Sasaki-Einstein manifolds

We discuss in detail the different analogues of Dolbeault cohomology groups on Sasaki-Einstein manifolds and prove a new vanishing result for the transverse Dolbeault cohomology groups H-(partial derivative) over bar((p,0))(k) graded by their charge under the Reeb vector. We then introduce a new cohomology, eta-cohomology, which is defined by a CR structure and a holomorphic function f with non-vanishing eta df. It is the natural cohomology associated to a class of supersymmetric type IIB flux backgrounds that generalise the notion of a Sasaki-Einstein manifold. These geometries are dual to finite deformations of the 4d N = 1 SCFTs described by conventional Sasaki-Einstein manifolds. As such, they are associated to Calabi-Yau algebras with a deformed superpotential. We show how to compute the eta-cohomology in terms of the transverse Dolbeault cohomology of the undeformed Sasaki-Einstein space. The gauge-gravity correspondence implies a direct relation between the cyclic homologies of the Calabi-Yau algebra, or equivalently the counting of short multiplets in the deformed SCFT, and the eta-cohomology groups. We verify that this relation is satisfied in the case of S-5, and use it to predict the reduced cyclic homology groups in the case of deformations of regular Sasaki-Einstein spaces. The corresponding Calabi-Yau algebras describe non-commutative deformations of P-2, P-1 x P-1 and the del Pezzo surfaces.