Laplace operators on Sasaki-Einstein manifolds

Abstract

We decompose the de Rham Laplacian on Sasaki-Einstein manifolds as a sum over mostly positive definite terms. An immediate consequence are lower bounds on its spectrum. These bounds constitute a supergravity equivalent of the unitarity bounds in dual superconformal field theories. The proof uses a generalisation of Kähler identities to the Sasaki-Einstein case.

Localisation on Sasaki-Einstein manifolds from holomorphic functions on the cone

“…The latter follows from the inequality for the Laplacian, ∆ ≥ −£ 2 ξ − 4ı£ ξ proved in [33,35]. Alternatively one can simply follow the considerations in [43] in the context of the discussion of the Lichnerowicz obstruction.…”

“…Previously, the H p,q ∂ b (Y ) have appeared in the context of holographic calculations of superconformal indices of three-and four-dimensional SCFTs [33][34][35]. Together with the isomorphism H 0,0 ∂ b (Y ) ∼ = H 0 O C(Y ) , our result allows for a easy evaluation of the perturbative partition function, as the whole calculation reduces to the counting of holomorphic functions on the Calabi-Yau cone C(Y ), weighted by their charge along the Reeb.…”

“…For a more detailed introduction, we refer to the review articles [37,38]; further material on the tangential Cauchy-Riemann operator and Kohn-Rossi cohomology groups can be found in [33][34][35] and references therein.…”

“…We review some notation from [35] that is quite useful when manipulating expressions involving the Hodge star operator. The material is a straightforward generalization of identical ideas on Kähler manifolds to the Sasaki-Einstein case.…”

Gauged supergravities from M-theory reductions

“…For twisted Dolbeault cohomology, the argument is a little different, but similar in spirit. We will use the following result 8 for the Laplacian acting on primitive p-forms on a Sasaki-Einstein space [30]…”