Prepared for submission to JHEP

Lecture notes on Sasaki-Einstein and G2 manifolds

Dan Xie CMSA, Harvard, MA 02138, USA

Abstract: Lecture notes on Math 273. Contents

1 Introduction

Theorem 1 (Berger): Suppose M is a simply-connected of dimension n, and that g is a Riemannian metric on M, that is irreducible and nonsymmetric. Then exactly one of the following seven cases holds: (i): Hol(g) = SO(n), generic case. (ii): n = 2m with m ≥ 2, and Hol(g) = U(m) in SO(2m); They are called Kahler metrics. (iii): n = 2m with m ≥ 2, and Hol(g) = SU(m) in SO(2m); They are called Calabi- Yau metrics. Typical examples are T 2,K3 and quintic, etc. (iv): n = 4m with m ≥ 2, and Hol(g) = Sp(m) in SO(4m); They are called Hyper- kahler metrics. (v): m = 4m with m ≥ 2, and Hol(g) = Sp(m)Sp(1) in SO(4m); They are called quaternionic Kahler metrics (they are actually not Kahler).

(vi): n = 7 and Hol(g) = G2 in SO(7); (vii) n = 8 and Hol(g) = Spin(7) in SO(8).

The list iii, iv, vi, vii admits Ricci-flat metric, and it admits covariant constant spinor i.e. solution for equation ∆Ψ = 0 exists. The number of covariant constant spinors are described by the following theorem:

Theorem 2 Let M be an orientable, simply-connected spin n-manifold for n ≥ 3, and g an irreducible Riemannian metric on M. Define N to be the dimension of the space of parallel spinors on M. If n is even, define N± to be the dimensions of the space of parallel spinors ∞ in C (S±), so that N = N+ + N−. Suppose N ≥ 1. Then, after making an appropriate choice of orientation for M, exactly one of the following holds:

i) n=4m for m ≥ 1 and Hol(g) = SU(2m), with N+ = 2, N− = 0. ii) n=4m for m ≥ 2 and Hol(g) = Sp(m), with N+ = m + 1, N− = 0. iii) n=4m+2 for m ≥ 1 and Hol(g) = SU(2m + 1), with N+ = 1, N− = 1. iv) n=7 and Hol(g) = G2 with N = 1, and v) n=8 and Hol(g) = Spin(7), with N+ = 1 and N− = 0. These manifolds are important in as we can have unbroken supersym- metry if we compactify string/M theory on the corresponding compact manifold. A lot of interesting things have been discovered for these manifolds such as Mirror symmetry for Calabi-Yau manifold, and Homological mirror symmetry involving calibrated sub-manifolds of CY manifolds, etc.

– 1 – The list (ii, iii, iv, v) can be studied using complex geometry (for quaternionic Kahler manifold, the corresponding twistor space (4n+2) dimensional space is complex). The existence of Calabi-Yau metric on compact manifold is known for many examples due to

Yau’s theorem. Much less is known for G2 and Spin(7) manifolds as there is no analog of Yau’s theorem. G2 manifolds are simpler and more tractable than Spin(7) as it admits more . So G2 is one of the subject which will be covered in this course, and our focus will be on constructing compact and non-compact examples and determining some basic properties.

Remark 1: M theory on CY3 gives five dimensional N = 1 theory (eight supersymme- tries); On G2 manifold gives four dimensional N = 1 theory (four super symmetries); On Spin(7) manifold gives three dimensional N = 1 theory (two ). Usually the larger the unbroken super symmetries, the easier the dynamics of the corresponding theories. The dynamics of the theories is related to the geometric property of the corre- sponding manifold, so we can see that Calabi-Yau is the simplest, while G2 is the next simplest case to consider. A second list of special metrics consists of those with Killing spinors, spinor fields satisfies the equation ∆X Ψ = λXΨ. These are

• Spheres;

• Sasaki-Einstein manifolds in dimension 2n + 1;

• 3-Sasakian manifolds in dimension 4n + 3;

• Nearly Kahler manifolds in dimension 6;

• Weak G2 manifolds in dimension 7;

They are Einstein metric with Rij = Λgij, Λ > 0. The link between this class of example with the above class is noticed by Bar (1993): consider the following metric on the metric cone of above manifold X = R+ × M:

2 2 2 dsX = dr + r dgL, (1.1) and it is Ricci-flat! We have the following correspondence:

• Spheres → flat metric

• Sasaki-Einstein manifolds in dimension 2n + 1 → Calabi-Yau metric in dimension 2n + 2.

• 3-Sasakian manifolds in dimension 4n+3 → Hyperkahler metric in dimension 4n+4.

• Nearly Kahler manifolds in dimension 6 → G2 metric in dimension 7.

• Weak holonomy G2 manifolds in dimension 7 → Spin(7) metric in dimension 8.

– 2 – Using this correspondence, we can put the study of this class of metrics into the study of Ricci flat metric on the non-compact manifold. The simplest non-trivial case is the Sasaki-Einstein metric or Ricci-flat conic metric on its metric cone. Unlike the compact case, there is no analog of Yau’s theorem on Ricci-flat conic metric, and it is an important question to tell whether a 2n + 1 dimensional manifold admits Sasaki-Einstein manifolds. Such manifolds are interesting due to following reasons:

• Einstein metrics was thought to be rare, but the studies in last twenty years on Sasaki- Einstein metrics show that Einstein metric is abundant. Studying them should tell us many things about the Einstein metrics.

• Sasaki manifold is agolog of Kahler manifold in odd dimension, but it is actually easier than Kahler case as the corresponding metric cone is algebraic.

• It covers the case of Kahler-Einstein metric on the Fano manifolds or orbifolds. The existence of Kahler-Einstein metric on Fano manifolds has been studied intensively recently, and it is proven that it is equivalent to K stability of the manifold. One can also have a notion of K stability for Sasaki manifold.

• The metric cone seems closed related to the degeneration of the Calabi-Yau manifold, and it is closed related to the singularity theory and the minimal model program.

• It is very useful for studying string theory and AdS/CFT. In fact, any Sasaki-Einstein five or seven manifold will produce a AdS/CFT pair. In certain SE five manifold cases, the CFT dual admits a quiver gauge theory description which is related to non-commutative crepant resolution of the singularity. The SE geometry should be closed related to the CY algebra and resolutions.

Let me illustrate the last point using a simple example. Consider the following singu- larity x2 + y2 + z2 + w2 = 0. (1.2) The link of this singularity admits a Sasaki-Einstein metric (The link is a symmetric space SU(2)×SU(2)/U(1)). If we use D3 brane probing this singularity, the quantum field theory on the link has the quiver gauge theory description. The gauge theory is a supersymmetric field theory (four dimensional N = 1 theory). It is shown by Witten and Klebanov (1999) that the superconformal field theory in the large N limit is due to type IIB string theory on the following background

AdS5 × L5 (1.3)

One of the purpose of this lecture is to produce a large number of generalizations of this duality using the recent mathematical results. Let’s discuss what will be covered in this lecture note:

• Definitions of Sasakian manifold. Examples involving toric manifolds, link of isolated singularities.

– 3 – D3

A A 2 1

X N N

B B 1 2 W=A B A B −A B A B L 1 1 2 2 1 2 2 1

Figure 1. tt

• Topological obstruction on positive Sasakian structure.

• Sasaki-Einstein manifold. Sufficient conditions: alpha invariant and toric classes. Generally, we need to use K stability. Various property of Sasaki-Einstein manifold such as Homology groups, volume, automorphism group, etc will be discussed.

• 3-Sasaki-Einstein manifold.

• AdS/CFT correspondence will be covered with emphasis on the correspondence be- tween the geometry and the quantum field theory.

• G2 manifolds including the basic properties and various constructions. Calibrated submanfolds will be discussed shortly.

2 Background material: complex and Kahler geometry

3 Sasakian geometry

4 Sasaki-Einstein manifold

5 AdS/CFT correspondence

6 G2 manifold

– 4 –