From $\beta$ to $\eta$: a new cohomology for deformed Sasaki-Einstein manifolds

DOI10.1007/JHEP04(2022)075arXiv2112.09167MaRDI QIDQ6385936

Author name not available (Why is that?)

Publication date: 16 December 2021

Abstract: 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

graded by their charge under the Reeb vector. We then introduce a new cohomology,

e t a

-cohomology, which is defined by a CR structure and a holomorphic function

f

with non-vanishing

e t a e q u i v m a t h r m 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

m a t h c a l 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

e t a

-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

e t a

-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

m a t h b b P^{2}

,

m a t h b b P^{1} i m e s m a t h b b P^{1}

and the del Pezzo surfaces.

Mathematics Subject Classification ID

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