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 C 4 //U(1), 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 C 3 /Zp+1 × Zp+1 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 N = 1 superconformal field theories arising as infra–red (IR) fixed points of toric quiver gauge theories with gauge group SU(N) 2p. 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

We describe all almost contact metric, almost hermitian and G2-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇-parallel spinors. In particular, we obtain solutions of the type II string equations in dimension n = 5, 6 and 7.

The purpose of this paper is to study a special kind of Riemannian metrics on Sasakian manifolds. A Sasakian manifold M of dimension 2n + 1 with a Sasakian structure S = (ξ, η,Φ, g) is said to be η-Einstein if the Ricci curvature tensor of the metric g satisfies the equation Ricg = λg + νη ⊗ η for some constants λ, ν ∈ R.

Any sphere S n admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S 4m+3, m> 1 are known to have another Sp(m + 1)-homogeneous Einstein metric discovered by Jensen [Jen73]. In addition,

The purpose of this note is to prove the existence of new Sasakian-Einstein met- rics on S 2 x S a and on (S 2 x Sa)#(S = x Sa). These give the first known examples of non-regular Sasakian-Einstein 5-manifolds. Our method involves describing the Sasakian-Einstein structures as links of certain isolated hypersurface singularities, and makes use of the recent work of Demailly and Kolltr who obtained new examples of Kiihler-Einstein del Pezzo surfaces with quotient singularities.

Abstract: We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres Σ 2n+1 the moduli space of Sasakian structures has infinitely many positive components determined by inequivalent underlying contact structures. We also prove the existence of Sasakian metrics with positive Ricci curvature on each of the 2 2m distinct diffeomorphism types of homotopy real projective spaces RP 4m+1. Milnor’s discovery of exotic spheres [Mil1] presented Riemannian geometry with a very natural question. What kind of special metrics or, more generally, geometric structures can exist on exotic spheres? Perhaps the most intriguing example of such a question concerns the existence of metrics with positive sectional curvature. In 1974 Gromoll and

The AdS/CFT correspondence relates dibaryons in superconformal gauge theories to holomorphic curves in Kähler-Einstein surfaces. The degree of the holomorphic curves is proportional to the gauge theory conformal dimension of the dibaryons. Moreover, the number of holomorphic curves should match, in an appropriately defined sense, the number of dibaryons. Using AdS/CFT backgrounds built from the generalized conifolds of Gubser, Shatashvili, and Nekrasov (1999), we show that the gauge theory prediction for the dimension of dibaryonic operators does indeed match the degree of the corresponding holomorphic curves. For AdS/CFT backgrounds built from cones over del Pezzo surfaces, we are able to match the degree of the curves to the conformal dimension of dibaryons for the nth del Pezzo surface, 1 ≤ n ≤ 6. Also, for the del Pezzos and the Ak type generalized conifolds, for the dibaryons of smallest conformal dimension, we are able to match the number of holomorphic curves with the number of possible dibaryon operators from gauge theory. AdS/CFT correspondence [1, 2, 3] asserts that type IIB string theory on AdS5 × X is

: A Sasakian structure S=(;j;\Phi;g) on a manifold M is called positive if its basic first Chern class c1 (F ) can be represented by a positive (1;1)-form with respect to its transverse holomorphic CR-structure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This allows us by example to give a completely independent proof of a result of Sha and Yang [SY] that for every nonnegative integer k the 5-manifolds k#(S 2 \ThetaS 3 ) admits metrics of positive Ricci curvature.

Abstract: In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2-connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit regular positive Sasakian structures.

Abstract. We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one. From a Riemannian point of view, an Einstein-Sasaki manifold is a Riemannian manifold (M, g) such that the conical metric on M × R + is Kähler and Ricci-flat. In particular, this implies that (M, g) is odd-dimensional, contact and Einstein with positive scalar curvature. The Einstein-Sasaki manifolds that are simplest to describe are the regular ones, which arise as circle bundles over Kähler-Einstein manifolds. In five dimensions, there is a classification of regular Einstein-Sasaki manifolds [11], in which precisely two homogeneous examples appear, namely the sphere S 5 and the Stiefel manifold (1) V2,4 = SO(4)/SO(2) ∼ = S 2 × S 3.