11/09/2008, 11:00 — 12:00 — Amphitheatre Pa1, Mathematics Building
James Sparks, Mathematical Institute, Oxford
Sasaki-Einstein Geometry and the AdS/CFT Correspondence - II
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.