AdS/CFT Group Theory

Question

I have a two part question about AdS/CFT:

1. Is the only necessary ingredient that the isometry group of AdS matches the conformal group in one dimension less or are there other prerequisites to build a holographic connection?

2. How does one demonstrate that the isometry group of $AdS_{d+1}$ is $SO(d,2)$? I cannot find any references that do this explicitly which is what I need. Is this because it takes a long time to show this?

Answer 1

1) That's not the only ingredient -- it's a prerequisite for holography. In reality, with holographic duality one always means a precise mapping from observables in a gravity theory in AdS to observables in a CFT that lives on the boundary. So holography is much richer: it prescribes for example how you can compute a Wilson loop on the boundary CFT in terms of gravity.

2) No, it's in fact almost trivial. $AdS_{d+1}$ with radius $R$ can be defined as the solution to

$$ \eta_{\mu \nu} X^\mu X^\nu = R^2 $$

with $\eta_{\mu \nu} = (1,1,-1,\ldots,-1)$ and where $X^\mu$ lives in $\mathbb{R}^{d+2}$. But $SO(2,d)$ is precisely the group that leaves the quadratic form $\eta_{\mu \nu} X^\mu X^\nu$ invariant.

If this is too abstract, think of the sphere $S^2$. It can be defined as the set of points $X^\mu \in \mathbb{R}^3$ that obey

$$ \delta_{\mu \nu} X^\mu X^\nu = R^2.$$

Its isometry group is $SO(3)$ because this leaves $X^2$ invariant.

Answer 2

In addition to what was already mentioned, I want to add the following points:

1. From the requirement of matching physical observables on both sides of the correspondence one can deduce a more general principle behind matching symmetries: global symmetries corresponding to Noether currents that are in principle observable have to match. This does not only include spacetime symmetries like the conformal group, but for instance extance applies to supersymmetric R-symmetry. The latter happens to correspond to the isometry group of the compact manifold ($S^5$ in the case of $AdS_5\times S^5$).
2. To determine the isometry group of of a given spacetime, one may solve the Killing equation and determine the Killing vectors. This provides a straightforward algorithm for identifying the isometry group, as it is generated by said vectors.