Symmetric Spaces

We start off with a brief discussion of the differential geometry of maximally symmetric spaces in two dimensions such as $\mathbb{S}^2$, ${\rm dS}^2$, $\mathbb{H}^2$ and ${\rm AdS}^2$ which play a prominent role in the near-horizon geometry of black holes, in Euclidean quantum field theory, in noncommutative geometry and in the  ${\rm AdS}^{d+1}/{\rm CFT}_d$ correspondence.

 

First we note that maximally symmetric spaces are essential ingredient in quantum gravity theories and cosmological models. These homogeneous and isotropic spaces enjoy the largest possible amount of spactime symmetries (isometries) and in Lorentzian signature they are exhausted with the three maximally symmetric spaces \cite{Bengtsson}:

The de Sitter spacetime ${\rm dS}^d$ (positive scalar curvature, repulsive cosmological constant, topology $\mathbb{S}^{d-1}\times \mathbb{R}$) which is relevant to cosmology.  The de Sitter spacetime ${\rm dS}^d$ as embedded in $\mathbb{M}^{1,d}$ is given by the ambiant metric and the quadric form
\begin{eqnarray}
&&ds^2=-dX_1^2+dX_2^2+...+dX_{d+1}^2\nonumber\\
&&-X_1^2+X_2^2+...+X_{d+1}^2=R^2.
\end{eqnarray}

Minkowski spacetime $\mathbb{M}^d$ (zero scalar curvature, zero cosmological constant, topology $\mathbb{R}^d$) which can be viewed as a zero cosmological constant limit of de Sitter spacetime  $\mathbb{dS}^d$.

The anti-de Sitter spacetime ${\rm AdS}^d$ (negative scalar curvature, attractive cosmological constant,  topology $\mathbb{R}^{d-1}\times \mathbb{S}^1$) which is relevant to quantum gravity. The anti-de Sitter spacetime ${\rm AdS}^d$ as embedded in $\mathbb{M}^{2,d-1}$ is given by the ambiant metric and the quadric form
\begin{eqnarray}
&&ds^2=-dX_1^2-dX_2^2+dX_3^2+...+dX_{d+1}^2\nonumber\\
&&-X_1^2-X_2^2+X_3^2+...+X_{d+1}^2=-R^2.
\end{eqnarray}

However, Wick rotation to Euclidean signature remains crucial to both quantum field theory and noncommutative geometry where quantization of fields and geometries makes strict sense only in Euclidean setting. In Euclidean signature, the maximally symmetric spaces are then given by the three spaces \cite{Bengtsson}:

The sphere $\mathbb{S}^d$ (positive curvature). The sphere $\mathbb{S}^d$ as embedded in $\mathbb{R}^{d+1}$ is given by the ambiant metric and the quadric form
\begin{eqnarray}
&&ds^2=dX_1^2+...+dX_{d+1}^2\nonumber\\
&&X_1^2+...+X_{d+1}^2=R^2.
\end{eqnarray}
The Killing vectors fields which leave both the ambiant metric and the quadric form invariant are
\begin{eqnarray}
J_{\alpha\beta}=X_{\alpha}\partial_{\beta}-X_{\beta}\partial_{\alpha}.\label{isometry}
\end{eqnarray}
These $d(d+1)/2$ isometries generate the group of rotations $SO(d+1)$. This is to be contrasted with the isometry group of de Sitter spacetime is $SO(1,d)$.

Euclidean space $\mathbb{R}^d$ (zero curvature).

The pseudo-sphere $\mathbb{H}^d$ (negative curvature).  The pseudo-sphere (Hyperboloic space) $\mathbb{H}^d$  as embedded in $\mathbb{M}^{1,d}$ is given by the ambiant metric and the quadric form
\begin{eqnarray}
&&ds^2=-dX_1^2+dX_2^2+...+dX_{d+1}^2\nonumber\\
&&-X_1^2+X_2^2+...+X_{d+1}^2=-R^2.
\end{eqnarray}
The Hyperboloic space $\mathbb{H}^d$ is defined as the upper sheet of the two-sheeted hyperboloid $-X_1^2+X_2^2...+X_{d+1}^2=-R^2$. The Killing vectors fields which leave both the ambiant metric and the quadric form invariant are still given by as before but the underlying symmetry group is now given by $SO(1,d)$. This is to be contrasted with the isometry group of anti-de Sitter spacetime which is given by $SO(2,d-1)$.

It is intriguing to note that in two dimensions the spaces $\mathbb{S}^2$, ${\rm dS}^2$, $\mathbb{H}^2$ and ${\rm AdS}^2$ are simply related. For example, we can go from  ${\rm AdS}^2$ (closed timelike curves with isometry group $SO(2,1)$) to  ${\rm dS}^2$ (closed spacelike curves with isometry group $SO(1,2)$) and vice versa by switching the meaning of timelike and spacelike. While both ${\rm dS}^2$ and $\mathbb{H}^2$ share precisely the same isometry group $SO(1,2)$. And we can go from ${\rm dS}^2$ to $\mathbb{S}^2$ by an ordinary Wick rotation. We can also go from ${\rm AdS}^2$ to $\mathbb{H}^2$ by a Wick rotation.

Representation theory of the Lorentz groups $SO(1,2)$ and $SO(2,1)$ can be found for example in \cite{barg,bns}. See also \cite{Mukunda:1974gb,Girelli:2015ija,Basu:1981ju}.


In this article we will focus on the case of two dimensions with Euclidean signature where the positive curvature space is given by a sphere $\mathbb{S}^2$ with isometry group $SO(3)$ and the negative curvature space is given by a pseudo-sphere $\mathbb{H}^2$  with isometry group $SO(1,2)$. We will be mostly interested in the case of the pseudo-sphere $\mathbb{H}^2$ which we will simply denote by ${\rm AdS}^2$.


The quantization of these two spaces yields the fuzzy sphere $\mathbb{S}^2_N$  \cite{Hoppe,Madore:1991bw} and the noncommutative pseudo-sphere ${\rm AdS}^2_{\theta}$ \cite{Ho:2000fy,Ho:2000br,Jurman:2013ota,Pinzul:2017wch} respectively which enjoy the same isometry groups $SO(3)$ and $SO(1,2)$ as their commutative counterparts. The fuzzy sphere is unstable and suffers collapse in a phase transition to Yang-Mills matrix models (topology change or geometric transition) whereas the noncommutative pseudo-sphere can sustain black hole configurations (by including a dilaton field) and also suffers collapse in the form of the information loss process (quantum gravity transition).

In fact, the product space $\mathbb{S}^2\times {\rm AdS}^2$ is the near-horizon geometry of extremal black holes in general relativity and string theory, e.g. the four-dimensional Reissner-Nordstrom black hole. It is then observed that the information loss problem in  four dimensions on  $\mathbb{S}^2_N\times {\rm AdS}^2_{\theta}$ reduces to the information loss problem in two dimensions on  noncommutative ${\rm AdS}^2_{\theta}$.

As we have said we will be mostly interested here in the case of the pseudo-sphere $\mathbb{H}^2$ or the Euclidean ${\rm AdS}^2$. The goal naturally is to construct a consistent ${\rm AdS}^2$/${\rm CFT}_1$ correspondence.