Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals
The authors consider an infinite family of Sasaki-Einstein metrics \(Y^{p,q}\) on \(S^2 \times S^3\), where \(p\) and \(q\) are coprime positive integers and \(q < p\). A family of toric Calabi-Yau cones is associated to it, which are shown to be Kähler quotients \({\mathbb C}^4 /U(1)\) (the vacua of gauged linear sigma models, generalizing the conifold). It is proved that the corresponding toric diagrams may be embedded in that of the orbifold \({\mathbb C}^3 / {\mathbb Z}_{p+1} \times {\mathbb Z}_{p+1}\). It follows that the \(Y^{p,q}\) manifolds are AdS/CFT dual to an infinite class of superconformal field theories. As a particular application, it is shown that \(Y^{2,1}\) is an irregular Sasaki-Einstein metric on the horizon of the complex cone over the first del Pezzo surface; the exact central charge at its \(\mathbb{R}\) fixed point, predicted for this theory, is a quadratic irrational number and agrees with an independent recent purely field theoretic calculation. The paper contains also some reviews on Sasaki-Einstein geometry, symplectic toric geometry and the moment map.
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