From $\beta$ to $\eta$: 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 $H_{\bar\partial}^{(p,0)}(k)$ 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 $\eta\equiv \mathrm{d}f$. 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 $\mathcal{N}=1$ 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 $\mathbb{P}^2$, $\mathbb{P}^1\times\mathbb{P}^1$ and the del Pezzo surfaces.