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We obviously cannot talk about AdS /CFT without knowing the geometry of Anti De Sitter (AdS ) space. First of all a little background on Maximally symmetric spaces in general is presented.

Infinitesimal symmetries in space time can be described via the technology of Killing vectors. Suppose the metric is symmetric under the transformation

x^µ→ x^0µ= x^µ−  ξ^µ,   1 . (2.3.1) This means that

g⁰_µν(x⁰) − gµν(x⁰) =  gµν,λξ^λ+  gµσξ^σ_,ν+  gρνξ^ρ_,µ+ O(²) ≡ £ξgµν = 0 . (2.3.2) Notice that the two arguments are the same in both terms: we need to compare the new and old metric in the same point. The symbol £_ξ goes under the name of Lie derivative.

If on the space is defined a connection ∇ (such that ∇_µξ^ρ= ∂_µξ^ρ+Γ^ρ_µνξ^ν) we can rewrite the last expression as

∇_µξ_ν+ ∇_νξ_µ= £_ξg_µν = 0 . (2.3.3) This is called the Killing equation and its solution are the Killing vectors. To each Killing vector is associated a symmetry of space time. This equation gives a very strong statement about multiple derivatives of ξ; in fact it is known that the commutator of two derivatives is just the Riemann curvature

[∇µ, ∇ν]ξρ = R^λ_ρµνξλ. (2.3.4) The Killing equation combined with this and the ciclicity property of the Riemann tensor R^λ_[µνρ]= 0 allows us to write

∇ρ∇νξµ= R^λ_ρµνξλ. (2.3.5) Since the second derivatives of ξ are just linear combinations of ξ, via a Taylor expansion we can reconstruct ξ^µ(x) in each point in space just by knowing ξ^µ(x0) and ξ^µ_,ν(x0) for some fixed point x0. In terms of some universal functions A(x) and B(x) we can write:

ξµ(x) = A^λ_µ(x)ξλ(x0) + B^ρλ_µ (x)∇λξρ(x0) , (2.3.6) so the maximal number of linearly independent Killing vectors (with constant coeffi-cients!) is the total number of independent vectors and derivatives of vectors. The Killing equation reduces the number to n + n(n − 1)/2 = n(n + 1)/2 where n is the dimension of space. A Maximally symmetric spaces is defined as a space whose num-ber of Killing vectors is maximal. When the numnum-ber of Killing vectors is maximal we have many interesting properties, in fact it can be shown that any Maximally symmetric space is isotropic and homogeneous, which means that the space enjoys translational and rotational invariance, hence the general coordinate scalars should not depend on the point. The scalar curvature R(x) = R is always a constant in these class of spaces and the Riemann tensor can be cast in the following form

Rλρσν= R

n(n − 1)(gνρgλσ− gσρgλν) . (2.3.7) A fundamental theorem of unicity about Maximally symmetric spaces will be enounced.

Theorem A Maximally symmetric space is uniquely determined by 3 characteristics:

the dimension n, the scalar curvature R and the signature of the metric.

Since Maximally symmetric spaces are unique, we can build them in a

stan-(a) AdS or dS space (b) Hyperbolic space (c) Sphere

Figure 2.3: Three different kinds of hyperspheres. The first has undefined signature, the last two have positive definite signature with curvature respectively negative and positive.

dard way. Suppose we need a space in p + q dimensions with signature (p+, q−);

then we can go in a space of p + q + 1 dimensions and embed an hypersphere: the metric restricted to the hypersphere will be that of the space we are looking for.

To be more concrete let us distinguish between positive and negative curvature.

 Negative curvature with signature (p+, q−), Anti de Sitter AdSp,q

Ambient metric: ds² =

p

 Positive curvature with signature (p+, q−), de Sitter dSp,q

Ambient metric: ds² =

p+1

The special cases q = 0 with positive and negative curvature are, respectively, the p–Sphere S^p and the Hyperbolic space H^p. In Figure 2.3 there is a depiction of these spaces in p+q = 2. Strictly speaking Anti de Sitter has Lorentian signature, so in our notation is AdSp+1 ≡ AdS_p,1: it originates from the metric induced over

Ω

τ

0 θ ^π2

∂

Ω

τ

Figure 2.4: Half of R×S^pis conformally equivalent to AdSp+1. The figure shows p = 2.

Next to it also the boundary is shown.

an hyperboloid of negative radius with two times. The isometry group for AdSp+1

is SO(2, p); the construction showed here renders this symmetry evident because this is the symmetry of the hyperboloid. The following coordinate chart covers the whole surface only once (if ρ > 0)

T₁ = L cosh ρ cos τ , T2 = L cosh ρ sin τ , X_i = L sinh ρ Ωi , 

i= 1, . . . p , X

i

Ω²_i = 1

. (2.3.10)

Actually one subtlety must be fixed: the coordinate τ is periodic so this coor-dinate chart admits closed time like curves. In order to avoid this paradoxical consequence it is preferable to cut the hyperboloid at τ = 2π and analytically continue the metric for τ ∈]−∞, ∞[. After the change of variables tan θ = sinh ρ, θ ∈[0,^π₂], we get the following induced metric on the surface

ds² = L²

cos²θ −dτ²+ dθ²+ sin²θdΩ²_p−1

(2.3.11) If we are interested in studying the causal structure of the space we are allowed to make a conformal rescaling (Weyl transformation), which allows us to get rid of the factor 1/ cos²θ in front, leaving us with R × S^p. Actually this is not quite true, since θ ∈ [0,^π₂] and we are covering half of the S^p, which is a B^p (p dimensional ball). In Figure 2.4 the case AdS3 is shown. The space R × S^p is conformally equivalent to R^1,p (this can be proved in a similar way). This is crucial in the AdS /CFT correspondence because we just discovered that AdSp+1

has a conformal boundary which is R × S^p−1 (simply from ∂B^p = S^p−1), which is conformally equivalent to R^1,p−1. The causal structure of the boundary is of the Lorentzian type, so it is suitable to describe causal dynamics. When we refer to the “boundary” of AdS we mean conformal boundary in the sense exposed here.

t u = const.

θ = −^π₂ θ = ^π₂

u = ∞ u=0

u= 0 τ

Figure 2.5: The strip AdS2 and the Poincar´e patch (the triangular region). The Poincar´e patch covers only half of the strip if is thought to be periodically continued for τ > 2π, τ < 0. The lines at u = const. are drawn, those are also the lines along which t grows.

We now show a change of variables to express the metric of AdS in a more familiar form. This new chart of coordinates is called the “Poincar´e patch” and it does not cover the whole hyperboloid but only half of it (u ≥ 0, ~x ∈ R^p−1):

T1 = 1

2u 1 + u²(L² + ~x²− t²)

, T2 = Lut , Xⁱ = Luxⁱ , (i = 1, . . . p − 1) ,

X^p = 1

2u 1 − u²(L²− ~x²+ t²)
.

(2.3.12)

The metric reads

ds² = L² du²

u² + u²(−dt²+ d~x²)



= dz²− dt²+ d~x²

z² .

(2.3.13)

The second line is also a familiar form of the metric, where the substitution Lu = 1/z has been made. In Figure 2.5 is shown the part of the hyperboloid covered by the Poincar´e patch in the case p = 1, where AdS is just a strip R × I:

we can see that indeed only half of it is covered.

It turns out that (A)dS is the Maximally symmetric solution of Einstein equa-tions with (negative)positive cosmological constant. The Einstein–Hilbert action in n dimensions with a cosmological constant term is

SEH = − 1 2κ²₀

Z

dⁿx√

g(R − 2Λ) . (2.3.14)

The equation in the vacuum is just R_µν− 1

2g_µνR+ Λgµν = 0 . (2.3.15) The AdS metric (2.3.13) is a solution of the above equation where the parameter L (which in the second line appears implicitly in the units of the adimensional coordinate z) is related to the cosmological constant:

R_µν = −n −1

L² g_µν = 2Λ

n −2g_µν , Λ = −(n − 2)(n − 1)

2L² . (2.3.16)

As mentioned before the isometry group of AdSn+1 is the orthogonal group SO(2, n). We prefer to postpone the discussion of this group to the next sec-tion because, as we shall see, it will be related to conformal symmetry.