The aim of these lectures will be to describe some recent developments in Sasaki-Einstein geometry, and also to explain, in a way that is hopefully accessible to geometers, how these results are related to the AdS/CFT correspondence in string theory. I will begin with a general introduction to Sasakian geometry, which is the odd-dimensional cousin of Kahler geometry. I will then introduce Sasaki-Einstein geometry, and describe a number of different constructions of Sasaki-Einstein manifolds. In particular, I will develop the theory of toric Sasakian manifolds, culminating with the recent theorem of Futaki-Ono-Wang on the existence of toric Sasaki-Einstein metrics. Next I will describe a number of different obstructions to the existence of Sasaki-Einstein metrics, together with some simple examples. Finally, I will outline how Sasaki-Einstein manifolds arise as solutions to supergravity, and describe their role in the AdS/CFT correspondence. The latter conjectures that for each Sasaki-Einstein \(5\)-manifold there exists a corresponding conformal field theory on \(\mathbb{R}^4\) This map is only really understood in certain examples, and for concreteness I will focus mainly on the toric case. The conformal field theory is then (conjecturally) described by a gauge theory on \(\mathbb{R}^4\) that is determined from the algebraic geometry of the cone over the Sasaki-Einstein manifold. Mathematically, this data is encoded by a bipartite graph on a two-torus. I will conclude by explaining how AdS/CFT relates some of the properties of Sasaki-Einstein manifolds described earlier to this combinatorial structure.

References

The article arXiv:math/0701518 [math.DG] reviews much of the material that I will cover, and also contains references to the original literature.