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 S2 × S3 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 C4//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/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 constructed for this case and hence we can give 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 purely field theoretic calculation using a–maximisation.