Sasaki Spaces Research Articles

Hidden symmetries in Sasaki–Einstein geometries

We describe a method for constructing Killing–Yano tensors on Sasaki spaces using their geometrical properties, without the need of solving intricate generalized Killing equations. We obtain the Killing–Yano tensors on toric Sasaki–Einstein spaces using the fact that the metric cones of these spaces are Calabi–Yau manifolds which in turn are described in terms of toric data. We extend the search of Killing–Yano tensors on mixed 3-Sasakian manifolds. We illustrate the method by explicit construction of Killing forms on some spaces of current interest.

Supersymmetry on curved spaces and holography

We study superconformal and supersymmetric theories on Euclidean four- and three-manifolds with a view toward holographic applications. Preserved supersymmetry for asymptotically locally AdS solutions implies the existence of a (charged) “conformal Killing spinor” on the boundary. We study the geometry behind the existence of such spinors. We show in particular that, in dimension four, they exist on any complex manifold. This implies that a superconformal theory has at least one supercharge on any such space, if we allow for a background field (in general complex) for the R-symmetry. We also show that this is actually true for any supersymmetric theory with an R-symmetry. We also analyze the three-dimensional case and provide examples of supersymmetric theories on Sasaki spaces.

Constructing Calabi–Yau metrics from hyperkähler spaces

Recently, a metric construction for Calabi–Yau threefolds from a four-dimensional hyperkähler space by adding a complex line bundle was proposed. We extend the construction by adding a U(1) factor to the holomorphic (3, 0)-form and obtain the explicit formalism for a generic hyperkähler base. We find that a discrete choice arises: the U(1) factor can either depend solely on the fiber coordinates or vanish. In each case, the metric is determined by a differential equation for the modified Kähler potential. As explicit examples, we obtain the generalized resolutions (up to orbifold singularity) of the cone of the Einstein–Sasaki spaces Yp, q. We also obtain a large class of new singular CY3 metrics with SU(2) × U(1) or SU(2) × U(1)2 isometries.

On the supersymmetric limit of Kerr-NUT-AdS metrics

Generalizing the scaling limit of Martelli and Sparks [D. Martelli, J. Sparks, Phys. Lett. B 621 (2005) 208, hep-th/0505027] into an arbitrary number of spacetime dimensions we re-obtain the (most general explicitly known) Einstein–Sasaki spaces constructed by Chen et al. [W. Chen, H. Lü, C.N. Pope, Class. Quantum Grav. 23 (2006) 5323, hep-th/0604125]. We demonstrate that this limit has a well-defined geometrical meaning which links together the principal conformal Killing–Yano tensor of the original Kerr-NUT-(A)dS spacetime, the Kähler 2-form of the resulting Einstein–Kähler base, and the Sasakian 1-form of the final Einstein–Sasaki space. The obtained Einstein–Sasaki space possesses the tower of Killing–Yano tensors of increasing rank—underlined by the existence of Killing spinors. A similar tower of hidden symmetries is observed in the original (odd-dimensional) Kerr-NUT-(A)dS spacetime. This rises an interesting question whether also these symmetries can be related to the existence of some ‘generalized’ Killing spinor.

Resolutions of cones over Einstein–Sasaki spaces

Recently an explicit resolution of the Calabi–Yau cone over the inhomogeneous five-dimensional Einstein–Sasaki space Y 2 , 1 was obtained. It was constructed by specialising the parameters in the BPS limit of recently-discovered Kerr–NUT–AdS metrics in higher dimensions. We study the occurrence of such non-singular resolutions of Calabi–Yau cones in a more general context. Although no further six-dimensional examples arise as resolutions of cones over the L p q r Einstein–Sasaki spaces, we find general classes of non-singular cohomogeneity-2 resolutions of higher-dimensional Einstein–Sasaki spaces. The topologies of the resolved spaces are of the form of an R 2 bundle over a base manifold that is itself an S 2 bundle over an Einstein–Kähler manifold.

On a class of 4D Kähler bases and AdS5supersymmetric black holes

We construct a class of toric Kahler manifolds, M_4, of real dimension four, a subset of which corresponds to the Kahler bases of all known 5D asymptotically AdS_5 supersymmetric black-holes. In a certain limit, these Kahler spaces take the form of cones over Sasaki spaces, which, in turn, are fibrations over toric manifolds of real dimension two. The metric on M_4 is completely determined by a single function H(x), which is the conformal factor of the two dimensional space. We study the solutions of minimal five dimensional gauged supergravity having this class of Kahler spaces as base and show that in order to generate a five dimensional solution H(x) must obey a simple sixth order differential equation. We discuss the solutions in detail, which include all known asymptotically AdS_5 black holes as well as other spacetimes with non-compact horizons. Moreover we find an infinite number of supersymmetric deformations of these spacetimes with less spatial isometries than the base space. These deformations vanish at the horizon, but become relevant asymptotically.

Supersymmetric solutions based on [formula omitted] and [formula omitted

We explicitly realize supersymmetric cones based on the five-dimensional Y p , q and L p , q , r Einstein–Sasaki spaces. We use them to construct supersymmetric type IIB supergravity solutions representing a stack of D3- and D5-branes as warped products of the six-dimensional cones and R 1 , 3 .

A note on Einstein–Sasaki metrics in D ⩾ 7

In this paper, we obtain new non-singular Einstein–Sasaki spaces in dimensions D ⩾ 7. The local construction involves taking a circle bundle over a (D − 1)-dimensional Einstein–Kähler metric that is itself constructed as a complex line bundle over a product of Einstein–Kähler spaces. In general, the resulting Einstein–Sasaki spaces are singular, but if parameters in the local solutions satisfy appropriate rationality conditions, the metrics extend smoothly onto complete and non-singular compact manifolds. The seven-dimensional space, whose base is a complex line bundle over S2 × S2, is discussed in detail since it has relevance in terms of the AdS/CFT correspondence.

Imaginary Killing spinors in Lorentzian geometry

We study the geometric structure of Lorentzian spin manifolds, which admit imaginary Killing spinors. The discussion is based on the cone construction and a normal form classification of skew-adjoint operators in signature (2,n−2). Derived geometries include Brinkmann spaces, Lorentzian Einstein–Sasaki spaces and certain warped product structures. Exceptional cases with decomposable holonomy of the cone are possible.