We present a countably infinite number of new explicit co-homogeneity one Sasaki-Einstein metrics on S 2 × 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 N = 1 superconformal field theories in four dimensions with compact or non-compact R-symmetry and rational or irrational central charges, respectively.

We show that the Reeb vector, and hence in particular the volume, of a Sasaki-Einstein metric on the base of a toric Calabi-Yau cone of complex dimension n may be computed by minimising a function Z on R n which depends only on the toric data that defines the singularity. In this way one can extract certain geometric information for a toric Sasaki-Einstein manifold without finding the metric explicitly. For complex dimension n = 3 the Reeb vector and the volume correspond to the R-symmetry and the a central charge of the AdS/CFT dual superconformal field theory, respectively. We therefore interpret this extremal problem as the geometric dual of a-maximisation. We illustrate our results with some examples, including the Y p,q singularities and the complex cone over the second del Pezzo surface.

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the DuistermaatHeckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kähler-Einstein metric.

We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi-Yau singularity. Our method combines information from the geometry and topology of Sasaki-Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3-branes probing a toric Calabi-Yau singularity can be deduced. The infinite family of toric singularities with known horizon Sasaki-Einstein manifolds L a,b,c is used to illustrate these ideas. We construct the corresponding quiver gauge theories, which may be fully specified by giving a tiling of the plane by hexagons with certain gluing rules. As checks of this construction, we perform a-maximisation as well as Z-minimisation to compute the exact R-charges of an arbitrary such quiver. We also examine a number of examples in detail, including the infinite subfamily L a,b,a , whose smallest member is the Suspended Pinch Point.

We describe an infinite family of quiver gauge theories that are AdS/CFT dual to a corresponding class of explicit horizon Sasaki-Einstein manifolds. The quivers may be obtained from a family of orbifold theories by a simple iterative procedure. A key aspect in their construction relies on the global symmetry which is dual to the isometry of the manifolds. For an arbitrary such quiver we compute the exact R-charges of the fields in the IR by applying a-maximization. The values we obtain are generically quadratic irrational numbers and agree perfectly with the central charges and baryon charges computed from the family of metrics using the AdS/CFT correspondence. These results open the way for a systematic study of the quiver gauge theories and their dual geometries.1 The main results of [21] were computed independently by some of us, unpublished.

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 /Z p+1 × Z p+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 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 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.the metric Y 7 is built using any positive curvature Kähler-Einstein metric in real dimension four [15]. These have been classified [18,19]. For the case when the Kähler-Einstein manifold is toric, one has only three cases: CP 2 , CP 1 × CP 1 , and dP 3 , where the latter is the third del Pezzo surface. Using the techniques developed in this paper, one can show that for the first two cases the metric cones over Y 7 are given by Kähler quotients C 5 //U(1), and C 6 //U(1) 2 , respectively, where the various U(1) charges are, with appropriate definitions 5 of the Chern numbers p and k, Q = (p, p, p, −3p+k, −k)and Q 1 = (p, p, 0, 0, −2p + k, −k), Q 2 = (0, 0, p, p, −2p + k, −k), respectively.

For every positively curved Kähler-Einstein manifold in four dimensions we construct an infinite family of supersymmetric solutions of type IIB supergravity. The solutions are warped products of AdS3 with a compact seven-dimensional manifold and have non-vanishing five-form flux. Via the AdS/CFT correspondence, the solutions are dual to two-dimensional conformal field theories with (0, 2) supersymmetry. The corresponding central charges are rational numbers. Dedicated to Rafael Sorkin in celebration of his 60th birthday. The AdS/CFT correspondence [1] states that any solution of string or M-theory with an AdS d+1 factor should be equivalent to a conformal field theory (CFT) in d spacetime dimensions. This correspondence, and its gen-eralisations, has provided profound insight into the non-perturbative structure of string theory, the structure of quantum field theory and the quantum properties of black holes. Backgrounds with AdS 3 factors are of particular interest because, unlike in higher dimensions, the confor-mal group in two-dimensions is infinite dimensional. As a consequence two-dimensional conformal field theories are much more tractable than their higher dimensional cousins; for instance, many models are exactly solvable, and there is a considerable literature on the subject. It would be a significant development if, via the AdS/CFT correspondence, string or M-theory can make contact with this large body of work. However, until now there were only a few known explicit AdS 3 × M solutions, with compact M. The most well studied class of examples are the AdS 3 × S 3 × X backgrounds of type IIB supergravity, where X = T 4 or K3. These are dual to N = (4, 4) conformal field theories that are deformations of the sigma model based on the orbifold Sym(X) n /S n. From a string theory perspective, these backgrounds describe the backreaction of a D-brane configuration that can be related to a black hole in five dimensions. It is a remarkable fact that the entropy of this black hole can be precisely derived from the central charge of the dual conformal field theory [2]. There have also been recent investigations into the con-formal field theory dual to the AdS 3 × S 3 × S 3 × S 1 background of type II string theory [3] (see [4]-[8] for earlier discussions). Despite the fact that the field theory has a larger version of N = (4, 4) superconformal symmetry than those dual to the AdS 3 × S 3 × X solutions, it has proved more difficult to identify them as a number of subtleties arise. The purpose of this letter is to present a new infinite class of supersymmetric AdS 3 backgrounds of type IIB string theory, which are dual to two-dimensional confor-mal field theories with N = (0, 2) supersymmetry. It will be very interesting if these conformal field theories can be explicitly identified. It will also be very interesting to know whether or not our solutions can be related to black holes. The new solutions are warped products of AdS 3 with a compact seven-dimensional manifold M 7 and have non-trivial self-dual five-form. T...

We show that for every positive curvature Kähler-Einstein manifold in dimension 2n there is a countably infinite class of associated SasakiEinstein manifolds X 2n+3 in dimension 2n + 3. When n = 1 we recover a recently discovered family of supersymmetric AdS 5 × X 5 solutions of type IIB string theory, while when n = 2 we obtain new supersymmetric AdS 4 ×X 7 solutions of D = 11 supergravity. Both are expected to provide new supergravity duals of superconformal field theories. e-print archive: