Induced Representations of the Conformal Group

Since the symmetry group SO0(d + 1,1) of Euclidean AdS and the conformal sym-metry group of its conformal boundary, the compactified R^d, are identical, it is suggestive to compare the representations on both spaces and find intertwiners be-tween these representations. The construction of the representations proceeds by the technique of induced representations; we summarise the results here with the utmost brevity and urge the reader to consult the original publications quoted at

the beginning of the section.

In the sequel, we will need the following subgroups of the Euclidean conformal group:

The Euclidean Lorentz (rotation) groupM =SO(d), the one-dimensional dilatation group A, the Abelian group of Euclidean translations N^tr ∼= R^d, and the Abelian group of special conformal transformations N^sc∼=R^d.

Representations on the boundary. The representations used in the conformal boundary theory are called elementary representations (ER) after [29]; we give them in the noncompact picture, which is relevant for physics. They are obtained by induction from the so-called parabolic subgroup P =MAN^sc. This is natural since N^tr is locally isomorphic to G/MAN^sc, so we identify the conformal boundary with N^tr. The representation space of the representation Tχ is, following Dobrev,

Cχ={f ∈C^∞(R^d, Vµ)},

whereχ= [µ,∆], ∆ is the conformal weight,µis a unitary irreducible representation of the Euclidean Lorentz groupSO(d), andVµis the finite-dimensional representation space ofµ. We also have to demand a special asymptotic behaviour of these functions as → ∞, with the leading termf(x)∼(x²)^−∆.

The irreducible representation Tχ acts like

(Tχ(g)f)(x) =|a|^−∆ D^µ(m)f(x^′), (3.3-58) whereais the associated scale factor, and mis the rotation obtained from nonglobal Bruhat decomposition g =n^trman^sc, more precisely g⁻¹n^tr_x =n^tr_x′m⁻¹a⁻¹(n^sc)⁻¹ (see [31]). The matrix D^µ(m) is the representation matrix of m in the representation µ.

The “coordinates” x and x^′ are related by a geometric point transformx^′ =g⁻¹x.

Representations in the Bulk. The bulk EAdS can be identified with N^trA ≃ G/K, where K =SO(d + 1) is the maximal compact subgroup; we have to discuss the representations induced by the maximal compact subgroup K. This is shown by Iwasawa decomposition G=N^trAK. The representation spaces are

Cˆ^τ ={φ∈C^∞(R^d×R₊,Vˆτ)}, (3.3-59) where τ is an irreducible representation of K, and ˆVτ is its finite-dimensional repre-sentation space. The action of the reprerepre-sentation is

( ˆT^τ(g)φ)(x,|a|) = ˆD^τ(k)φ(x^′,|a^′|), (3.3-60) where the group elements are related by the Iwasawa decomposition g⁻¹n^tr_xa = n^tr_x′a^′k⁻¹, and ˆD^τ(k) is the representation matrix ofkin ˆVτ; again, we obtain the geo-metric point transformation (|a|, x) =g⁻¹(|a^′|, x^′). While we may choose any irrep τ of the maximal compact subgroupK, the bulk representations which are usually dis-cussed in the perturbative approach to AdS/CFT mostly are selected from so-called

“minimal representations” τ =τ(µ), uniquely specified by a choice of the boundary representationµ.

However, ˆT^τ is not irreducible; so an ER on the boundary can in general only be equivalent to a subrepresentation of ˆT^τ. A good method to single out such subre-presentations is to select the eigenspaces of the Casimir operators of the conformal group. The Casimir operators are in general differential operators acting on the func-tions in the representation space. The selection of an irreducible subrepresentation amounts therefore to solving the corresponding differential equations; this is not a simple task.

Pairs of dual representations. Since we have established the content of these representations, we will now ask in which sense they are equivalent. To begin, we find that the boundary representations always arrive in pairs: For the representation χ = [µ,∆], we find that χ^∗ = [µ^∗,d−∆] is the representation conjugated by Weyl reflection. Here, µ^∗ is the “mirror image” ofµ.

The conformal two-point function Gχ on the boundary is given by Gχ(x)∼ 1

(x²)^∆D^µ(m(x))

with m(x)ij = _x²2xixj −δij ∈ M a rotation. It serves as intertwiner between these representations:

Gχ:Cχ^∗ →Cχ, Tχ(g)◦Gχ =Gχ◦Tχ^∗(g), ∀g ∈G where Gχ is the convolution operator with kernel Gχ(x) defined by

(Gχf)(x₁) = Z

d^dx2Gχ(x₁−x₂)f(x₂) f ∈Cχ^∗.

Therefore, we have partial equivalence Cχ ≃ Cχ^∗. In particular, the values of all the Casimirs coincide. At generic points, the representations are even equivalent (so GχGχ^∗ =¹χ and Gχ^∗Gχ=¹χ^∗).

The dual representationsχ= [µ,∆] andχ^∗ = [µ^∗,d−∆] can be contracted naturally.

For consider a tensor operator O in the representation χ. If J is a source function transforming under the representationχ^∗, then the integral

Z

d^dx J(x)·O(x),

where · denotes the natural contraction in the representation µ, is invariant under the action of the conformal group. This is the situation we come across in Witten’s proposal for the implementation of the AdS/CFT correspondence.

Equivalent and partially equivalent representations. Since we are inter-ested in possible equivalences between the bulk and boundary representartions, we further restrict the bulk representations: Namely, givenχ= [µ,∆] with ∆ real, we de-fine that ˆC_χ^τ is the maximal subrepresentation of ˆC^τ which has the same Casimir val-ues asCχ and has matching asymptotic behaviourφ(x,|a|)∼ |a|^−∆ϕ(x) for |a| →0.

It is clear that there must be at least partial equivalence between C_χ and ˜C_χ^τ. It can be established that the two representations are indeed equivalent if ∆ is generic (see [30] for a list of the exceptional values). For these values, it is also established that Cˆ_χ^τ = ˆC_χ^τ∗ ≡Cˆ_χ,χ^τ ∗.

There are two intertwiners: The bulk-to-boundary intertwiner L^τ_χ : ˆC_χ,χ^τ ∗ →Cχ acts like a projection

(L^τ_χφ)(x) = lim

|a|→0|a|^−∆Π^τ_µφ(|a|, x),

where Π^τ_µis the standard projection operator from the K-representation space ˆVτ to the M-representation space Vµ. The intertwining property of L^τ_χ is

L^τ_χ◦Tˆ^τ(g) = Tχ(g)◦L^τ_χ ∀g ∈G.

The boundary-to-bulk intertwiner ˆL^τ_χ :Cχ →Cˆ_χ,χ^{τ ∗}, on the other hand, is constructed as integral convolution operator

( ˆL^τ_χf)(|a|, x) = Z

d^dx^′|a|^∆−dK_χ^τ

x−x^′

|a|

f(x^′),

where K_χ^τ :Vµ→Vˆτ is some linear operator. The intertwining property of ˆL^τ_χ is Tˆ^τ(g)◦Lˆ^τ_χ = ˆL^τ_χ◦Tχ(g) ∀g ∈G.

In particular, ˆL^τ_χL^τ_χ = ˆL^τ_χ∗L^τ_χ∗ = ¹Cˆ_χ,χ^τ ∗ and L^τ_χLˆ^τ_χ = ¹Cχ. It can be shown that at generic points, we can reconstruct the bulk field completely from its boundary values.

Dual intertwiners can be related to each other by the boundary propagator: We have up to a prefactor

L^τ_χ^∗ ∼Gχ^∗◦L^τ_χ. (3.3-61) Bulk-to-bulk Propagators. Different Prescriptions. We are relating the group theoretical results to the discussion of the propagators in section 3.2.2. We found that the propagators have a distinguished boundary behaviour, which for a given mass m² in the bulk is characterised by (z⁰)^∆±, with ∆_± given in (3.2-27).

Can they be characterised by the representation method? The answer is that for generic points, there is indeed a simple characterisation. Both types of propagators act according to

G^∆_χ,χ^±∗ : ˆC_χ,χ^{τ ∗} →Cˆ_χ,χ^{τ ∗}. (3.3-62)

For a crossed contraction of the bulk-to-boundary intertwiners, we have up to a multiple

Lˆ^τ_χ∗◦L^τ_χ ∼G^∆_χ,χ⁻∗−G^∆_χ,χ⁺∗ ∼Lˆ^τ_χ◦L^τ_χ∗ (3.3-63) (this equation will be shown in a functional integral setup in section 6.1.1 below).

The integral kernels of the propagatorsG^∆_χ,χ^±∗(|a|, x; |b|, y) (as function of|a|, x, with

|b|, y fixed, or vice versa) are not elements ofC_χ,χ^τ ∗, since they solve the free equation of motion only up to contact terms δ^(d)(x−y)δ(|a| − |b|), and the free equations of motion are precisely the Casimir differential equations required to hold for ele-ments of C_χ,χ^τ ∗. However, they still can be identified by their asymptotic behaviour, G^∆_χ,χ^−∗(|a|, x; |b|, y) ∼ |a|^∆− ∼ |a|^d−∆ for |a| → 0 and G^∆_χ,χ^+∗(|a|, x;|b|, y) ∼ |a|^∆ for

|a| →0, if we identify the ∆ from this section with ∆₊ from (3.2-27). It is tempting to denote the field-theoretic propagator by G^τ_χ^∗ ≡G^∆_χ,χ^−∗ and say that it propagates the representationχ^∗, whereas the dual propagator might be called G^τ_χ ≡G^∆_χ,χ⁺∗ and said to propagate the “shadow field” in the representation χ [60].

From what has been said before, however, there is really no stringent reason to make a distinction between those two representations in the bulk. The limiting behaviour doesnot imply relations like “L^τ_χ◦G^∆_χ,χ^−∗ = 0” or similar: G^∆_χ,χ^−∗, acting as convolution operator according to (3.3-62), will certainly not equal 0, and therefore will not map into the kernel of neither L^τ_χ not L^τ_χ∗.