3.4 EAdS-Presentation of Three-Point Functions
3.4.1 Conformal Invariants in the Embedding Space
Embedding Geometry. If we want to obtain a truly covariant expression on EAdS resembling a Feynman graph, we should be introducing a “vertex”-like object situated at (z0, z). This will include covariant derivatives acting on the boundary-to-bulk propagators. Since these propagators will certainly propagate tensor fields, the Christoffel symbols which appear in the covariant derivative whenever it acts on these tensors will have to be taken care of. These are rather inconvenient to handle.
A possible way out is to perform the calculation in the embedding space Rd+1,1 which is a flat space (see section 3.1), and only in the end restrict the expressions to the embedded hyperboloid defining Euclidean Anti-de-Sitter space. The resulting tensors and vectors will have indices running from 0 to d + 1, however. A similar use of tensor representations on de-Sitter space obtained by restriction of tensors from the embedding space has been suggested already by Dirac [27, 28]. Einstein and Mayer [34, 35] have ventured to obtain a sort of “holographic” description of Kaluza-Klein theory on a 4-dimensional submanifold embedded in a 5-dimensional
spacetime necessitated by Kaluza-Klein theory, yielding similar tensor fields with one “odd” orthogonal direction 8. Ultimately, these orthogonal components should not to contribute at all to the correlations.
SinceRd+1,1 is a flat space, the covariant derivative is replaced by the partial deriva-tive (as long as we stick to the Euclidean coordinates for Rd+1,1). The vertex coor-dinatez = (z0, z) must then be replaced by its embedded coordinate
˜
z0 =1−(z)2
2z0 , z˜i =zi
z0, z˜d+1 =1 + (z)2
2z0 , (3.4-68) with (z)2 = (z0)2+z2. The scalar product of two points on the embedded hyperboloid which are specified in Poincar´e coordinates is according to section 3.1
(˜y,z) =˜ −(y0)2+ (z0)2+ (y−z)2
2y0z0 ≤ −1. (3.4-69)
In what follows, we will frequently use different geometric objects; while the same letter denotes always the same object, a tilde ˜z means that we consider the object in Rd+1,1, an underscore x means that this object lives on the conformal boundary (say, a boundary point or a tangent vector on the boundary) and letters without any decoration point to an object in EAdS. What are the coordinates of these objects in Rd+1,1 depends on their type, and we will for every single object give a coordinatisation in the Euclidean embedding space.
A boundary point x ∈ Rd can be represented by a lightlike ray asymptotically tangential to the hyperboloid; these rays are characterised by vectors
˜
xµ˜ =s(x)
1−x2
2 , x, 1 +x2 2
µ˜
(3.4-70) (we decorate indices in the Euclidean coordinate system for Rd+1,1 with a tilde).
Here, s(x) is an arbitrary scale factor on the ray which will change under conformal transformations; since every bulk-to-boundary propagator is thought to be connected to a boundary operator with a certain scaling dimension, the appearance of the factor s(x) has to be expected. The scale factor will play a role, however, when pushing forward the tangent vectors of the boundary into EAdS.
For scalar products of mixed boundary/bulk vectors z ∈EAdS,x, y ∈Rd, we get (˜x,z) =˜ −s(x)(z0)2+ (z−x)2
2z0 , (˜x,y) =˜ −s(x)s(y)(x−y)2
2 , (3.4-71) and obviously (˜x,x) = 0. On the boundary space, let˜ v ∈ Tx be a tangent vector at x; it can be taken into EAdS be the relation
v· ∂
∂x =v· ∂x˜µ˜
∂x
∂
∂x˜µ˜
= ˜! v˜µ ∂
∂x˜µ˜, (3.4-72)
8The author is indebted to Prof. H. G¨onner for pointing out this connection.
whence its components in the embedding space Rd+1,1 are
˜
vµ˜ =s(x) (−v ·x, v, v·x)µ˜+ (v· ∇s(x)) ˜xµ˜. (3.4-73) For v, w∈Tx tangent vectors atx, we have
(˜v,x) = 0,˜ (˜v,w) =˜ s(x)2v·w. (3.4-74) This implies that vectors v ∈ Tx are tangent to the light cone at ˜x in EAdS; from (3.4-73), one can see that boundary tangent vectors properly correspond to the equiv-alence class ˜v+Rx˜ in the bulk.
The tangent vectors to EAdS space can be represented in a similar fashion: Iftµ, µ = 0. . .d, is a tangent vector to the point z living on the Poincar´e patch, then we have
tµ ∂
∂zµ =tµ∂z˜ν
∂zµ
∂
∂z˜ν
= ˜! tν ∂
∂z˜ν, (3.4-75)
whence its components in the embedding space Rd+1,1 are
˜tν ≡tµ∂z˜ν
∂zµ =
−t01−z2+ (z0)2
2(z0)2 − t·z
z0 , tz0−zt0
(z0)2 , −t01 +z2−(z0)2
2(z0)2 +t·z z0
ν
, with the usual EAdS scalar product
(˜t,˜t) = tµtµ= (t0)2+t2
(z0)2 . (3.4-76)
We list possible expressions which can be constructed from the scalar product: if v ∈Tx, we can form
vll(˜x,z)˜ ≡v·∂x(˜x,z) = (˜˜ v,z) =˜ s(x)v·(z−x)
z0 −(v· ∇s(x))(z0)2+ (z−x)2
2z0 .
We indicate the derivative by simply appending the relevant index. The other deriva-tive µ˜(˜z,˜x) = ˜xµ˜ is trivial. Two scalar products may be contracted by
(˜x,z)˜ µ˜ µ˜(˜z,y) = (˜˜ x,y) =˜ −s(x)s(y) (x−y)2
2 ,
and thereby
(˜y,z)˜ µ˜ µ˜(˜z,v) = (˜˜ y,˜v) =s(x)s(y)v ·(y−x)−(v· ∇s(x))s(y)(x−y)2 2 and, for w∈Ty,
( ˜w,z)˜ µ˜ µ˜(˜z,v) = ( ˜˜ w,v˜) =s(y)s(x)w·v+ (v· ∇s(x))s(y)w·(x−y)
+s(x)(w· ∇s(y))v·(y−x)−(v· ∇s(x))(w· ∇s(y))(x−y)2
2 .
We finally mention the boundary point∞(the point needed for conformal compact-ification of the boundary of EAdS); in the embedding space, it can be represented by the ray in direction
∞˜µ˜ = lim
x→∞
˜ xµ˜ x2 =
−1 2, 0, 1
2 µ˜
. (3.4-77)
The expressions involving boundary points and their tangent vectors are in general not invariants on EAdS, since they do not have the required form; a conformal symmetry transformation will change the scale factor s(x) locally, and therefore, the push-forward (3.4-73) of the boundary tangent vectors v will contain factors of the form (v · ∇s(x)) ˜x. While the scale factor s(x) whenever it appears can be attributed to a physical scale dependence of the underlying physical quantities (eg the boundary operators whose correlations we are computing), the derivative ∇s(x) does not have such an interpretation. The logical consequence is that any expression linear in the boundary tangent vector ˜v should be invariant under the transformation
˜
v 7→ v˜+r·x,˜ r ∈ R. In other words, if T˜x is the tangent space at ˜x, then there should be an equivalence relation T˜x ∋ x˜ ∼ 0 efficient. For generic functions G(˜v) containing several instances of ˜v, conformal covariance means that they vanish under the application of the differential operator
˜
xµ˜∂v˜˜µG(˜v) = 0. (3.4-78) Note that this notion may be even transferred to points situated within EAdS: If z ∈EAdS, then the tangent vector ˜z ∈Tz˜ points in a direction orthogonal to EAdS should not have a physical meaning, and therefore, covariant expressions should not depend on this component of the tangent vectors.
For ˜v ∈T˜x and ˜a ∈Tz˜, a typical invariant expression is (˜v,a)˜ − (˜v,z)(˜˜ x,a)˜
(˜x,z)˜ . (3.4-79)
For ˜v ∼x˜ or ˜a ∼z, this will vanish. Or, for˜ x a boundary point and ˜z,u˜∈Rd+1, (˜v,u)˜
(˜x,u)˜ − (˜v,z)˜
(˜x,z)˜ (3.4-80)
fulfills the same purpose. Forone additional point (without a tangent vector) besides v ∈Tx, the construction of such invariants is not possible.
We will use the scalar product as the basic building block for the EAdS-presentation of the correlations. While in the following, we will set s(x) = 1 and ignore the scale factor, we have to take care that all expression which we obtain ultimately are formed of invariants like (3.4-79), independent of derivatives ∇s(x) of the boundary scale factor s(x).