Sasaki-Einstein Metrics Research Articles

Toric geometry of convex quadrilaterals

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kahler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kahler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kahler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.

Lorentzian Sasaki-Einstein metrics on connected sums of S 2 × S 3

We settle completely an open problem formulated by Boyer and Galicki in [5] which asks whether or not #kS 2 × S 3 are negative Sasakian manifolds for all k. As a consequence of the affirmative answer to this problem, there exists so-called Sasaki η-Einstein and Lorentzian Sasaki-Einstein metrics on these five-manifolds for all k and moreover all of these can be realized as links of isolated hypersurface singularities defined by weighted homogenous polynomials. The key step is to construct infinitely many hypersurfaces in weighted projective space that contain branch divisors \({\Delta=\sum_{i}\left(1-\frac{1}{m_{i}}\right)D_i}\) such that the D i are rational curves.

Kähler–Sasaki geometry of toric symplectic cones in action-angle coordinates

In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kähler cone. Kähler–Sasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kähler geometry and how it was recently generalized, by Burns–Guillemin–Lerman and Martelli–Sparks–Yau, to toric Kähler–Sasaki geometry. It also describes, as an application, how this approach can be used to relate a recent new family of Sasaki–Einstein metrics constructed by Gauntlett–Martelli–Sparks–Waldram in 2004, to an old family of extremal Kähler metrics constructed by Calabi in 1982.

Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant $f1$ for the first Chern class case becomes an obstruction to the existence of transverse Kähler metric of constant scalar curvature. We prove the existence of transverse Kähler-Ricci solitons (or Sasaki-Ricci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if $S$ is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on S. As an application we obtain Sasaki-Einstein metrics on the $U(1)$-bundles associated with the canonical line bundles of toric Fano manifolds, including as a special case an irregular toric Sasaki-Einstein metrics on the unit circle bundle associated with the canonical bundle of the two-point blow-up of the complex projective plane.

On Sasaki-Einstein manifolds in dimension five

We prove the existence of Sasaki-Einstein metrics on certain simply connected 5-manifolds where until now existence was unknown. All of these manifolds have non-trivial torsion classes. On several of these we show that there are a countable infinity of deformation classes of Sasaki-Einstein structures.

Uniqueness and Examples of Compact Toric Sasaki-Einstein Metrics

In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on \(S^5 \sharp k(S^2 \times S^3)\) for each positive integer k.

Obstructions to the Existence of Sasaki–Einstein Metrics

We describe two simple obstructions to the existence of Ricci-flat Kahler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kahler-Einstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3-fold and 4-fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the R-charge of a gauge invariant chiral primary operator violates the unitarity bound.

Semiclassical strings in Sasaki-Einstein manifolds and long operators in Script N = 1 gauge theories

We study the AdS/CFT relation between an infinite class of 5-d Ypq Sasaki-Einstein metrics and the corresponding quiver theories. The long BPS operators of the field theories are matched to massless geodesics in the geometries, providing a test of AdS/CFT for these cases. Certain small fluctuations (in the BMN sense) can also be successfully compared. We then go further and find, using an appropriate limit, a reduced action, first order in time derivatives, which describes strings with large R-charge. In the field theory we consider holomorphic operators with large winding numbers around the quiver and find, interestingly, that, after certain simplifying assumptions, they can be described effectively as strings moving in a particular metric. Although not equal, the metric is similar to the one in the bulk. We find it encouraging that a string picture emerges directly from the field theory and discuss possible ways to improve the agreement.

Toric Geometry, Sasaki–Einstein Manifolds and a New Infinite Class of AdS/CFT Duals

Recently an infinite family of explicit Sasaki–Einstein metrics Y p,q on S 2×S 3 has been discovered, where p and q are two coprime positive integers, with q<p. These give rise to a corresponding family of Calabi–Yau cones, which moreover are toric. Aided by several recent results in toric geometry, we show that these are Kähler quotients namely the vacua of gauged linear sigma models with charges (p,p,−p+q,−p−q), thereby generalising the conifold, which is p=1,q=0. We present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold for all q<p with fixed p. We hence find that the Y p,q manifolds are AdS/CFT dual to an infinite class of superconformal field theories arising as IR fixed points of toric quiver gauge theories with gauge group SU(N)2 p . As a non–trivial example, we show that Y 2,1 is an explicit irregular Sasaki–Einstein metric on the horizon of the complex cone over the first del Pezzo surface. The dual quiver gauge theory has already been constructed for this case and hence we can predict the exact central charge of this theory at its IR fixed point using the AdS/CFT correspondence. The value we obtain is a quadratic irrational number and, remarkably, agrees with a recent purely field theoretic calculation using a-maximisation.

Toric Sasaki–Einstein metrics on [formula omitted

We show that by taking a certain scaling limit of a Euclideanised form of the Plebanski–Demianski metrics one obtains a family of local toric Kähler–Einstein metrics. These can be used to construct local Sasaki–Einstein metrics in five dimensions which are generalisations of the Yp,q manifolds. In fact, we find that these metrics are diffeomorphic to those recently found by Cvetic, Lu, Page and Pope. We argue that the corresponding family of smooth Sasaki–Einstein manifolds all have topology S2×S3. We conclude by setting up the equations describing the warped version of the Calabi–Yau cones, supporting (2,1) three-form flux.

Sasaki-Einstein Metrics on S^2\times S^3

We present a countably infinite number of new explicit co-homogeneity one Sasaki-Einstein metrics on $S^2\times S^3$ of both quasi-regular and irregular type. These give rise to new solutions of type IIB supergravity which are expected to be dual to $\mathcal{N}=1$ superconformal field theories in four dimensions with compact or non-compact $R$-symmetry and rational or irrational central charges, respectively.