Bulk-to-bulk Propagators of Tensor Fields

While we have discussed the bulk-to-boundary propagators, a full theory in the bulk must certainly contain bulk-to-bulk propagators. We will, without giving them explicitly, shortly discuss how they may be evaluated.

As starting point, consider the “propagator” (3.5-138) which features in the EAdS-presentation of the twist-2 CPWE and which is given by the difference of the field-theoretic and dual prescription for the boundary current in the intermediate channel.

We denoted this effective propagator of order ∼N⁻¹ by G^eff_bu^s(˜z,u)[˜˜ a,˜b] =

Z

d^dx G^{ft UV}_{bubo (l}^s₁₎(˜z, x)[˜a]D^s(l1),(l2)(∂_x)G^{ft UV}_{bubo (l}^s₂₎(˜u, x)[˜b]; (3.6-140) it is independent of the specific boundary prescriptions for the spin-s tensor field in the bulk (up to a sign). In this notation, the bulk ends of the propagators are con-tracted with the ˜aresp. ˜b-vectors, whereas the boundary ends are contracted directly with the inverse propagator D^s. We have found the bulk-to-boundary propagators (3.4-111); the difficulty now lies in the application of the inverse propagatorD^s. This very technical operation has been performed in [60,62], resulting in a representation of the AdS space propagator in terms of Legendre functions of the second kind.

Note that, by the structure of equation (3.6-140), the propagator G^eff_bu^s fulfills the equations (3.4-113a) to (3.4-113e) for each end separately (ie, for the pairs of points and tangent vectors ˜z ∈ R^d+1,1,˜a ∈ Tz˜ and ˜u ∈ R^d+1,1,˜b ∈ T˜u separately), which hold for the bulk end of the bulk-to-boundary propagators. A true propagator G^∆_bu^s with boundary behaviour (z⁰)^∆ should fulfill these equations as well, up to δ-terms on the diagonal (when the endpoints coincide). We have thus the following set of equations, which we write with an undetermined propagator function G^s_bu:

˜

z^µ˜∂˜a^µ˜G^s_bu =0, z˜G^s_bu =0 + diag. terms, ˜aG^s_bu =0, ∂z˜^µ˜∂˜aµ˜G^s_bu =0 + diag. terms

˜

u^µ˜∂˜b^µ˜G^s_bu =0, u˜G^s_bu =0 + diag. terms, ^˜bG^s_bu =0, ∂_u˜^µ˜∂˜bµ˜G^s_bu =0 + diag. terms;

(3.6-141) and the homogeneity relation

G^s_bu(˜z,u)[˜˜ a,˜b] = (|z˜||u˜|)^−∆G^s_bu z˜

|z˜|, u˜

|u˜|

[˜a,˜b]. (3.6-142) Note that all these equations hold in the embedding space. Intrinsic EAdS equations could be formulated using the material of section3.4.4to relate the embedding space-and the covariant EAdS-derivative, using the homogeneity of the propagator in the

embedding space Euclidean coordinates. We know furthermore that G^s_bu is of order

∼ N⁻¹, homogeneous of degree s in ˜a and ˜b; moreover, G^s_bu(˜z,u)[˜˜ a,˜b] should be symmetric in the argument pairs (˜z,˜a) and (˜u,˜b) by construction.

By (3.4-114), we expect that this system of equations has two linearly independent propagator solutions, with either the boundary behaviour (z⁰)^∆ or the conjugate (z⁰)^d−∆; the generic solution is

G^s_bu(˜z,u)[˜˜ a,˜b] =αG^∆_bu^s(˜z,u)[˜˜ a,˜b]

+ (1−α)(|z˜||u˜|)^d−2∆G^d_bu^−∆s(˜z,u)[˜˜ a,˜b]. (3.6-143) By way of its definition, there may no other vectors appear inG^s_bu, so that it can be written as a function of scalar products of its arguments,

G^s_bu(˜z,u)[˜˜ a,˜b] = (|z˜||u˜|)^−∆g^s (˜z,u)˜

|z˜||u˜|,(˜a,z)˜

|z˜| ,(˜b,z)˜

|z˜| ,(˜a,u)˜

|u˜| ,(˜b,u)˜

|u˜| ,˜a²,˜b²,(˜a,˜b)

! . (3.6-144) The orthogonality conditions ˜z^µ˜∂_˜a^µ˜G^s_bu = 0 and resp. for (˜u,˜b) are the easiest to fulfill; we use the scheme which is by now well-known and introduce the invariants

Kab =(˜a,˜b)−(˜a,u)(˜˜ z,˜b)

(˜z,u)˜ Kaa =(˜a,a)˜ − (˜a,z)˜ ²

(˜z,z)˜ Kbb =(˜b,˜b)− (˜b,u)˜ ² (˜u,u)˜ J_a =(˜a,u)˜

(˜z,u)˜ −(˜a,z)˜

(˜z,z)˜ J_b =(˜b,z)˜

(˜z,u)˜ − (˜b,u)˜

(˜u,u)˜ . (3.6-145)

Furthermore, define the scale invariant bifunction X = (˜z,u)˜

|z˜||u˜| ≤ −1. (3.6-146) We find that the most general form possible for the solution is

G^s_bu(˜z,u)[˜˜ a,˜b] = (|z˜||u˜|)^−∆g^s(X, Kab, Kaa, Kbb,|z˜|Ja,|u˜|Jb), (3.6-147) with the usual homogeneity and symmetry requirements for (˜z,˜a) and (˜u,˜b). One can see that this must be the most general ansatz since it is a function of six argu-ments fulfilling two differential equations, whereas the functions (3.6-144) has eight arguments and does not fulfill any differential equation ¹². We have six equations left, and this should be just sufficient to determine G^s_bu, up to a multiple.

For spin 0 one finds, solving these equations up to the normalisation¹³, G⁰_bu = (|z˜||u˜|)^−∆ X²−1^1−d₄

C1P_∆^1−d2

−^d+1₂ (−X) + C₂

Γ(∆−d + 1)Q_∆^1−d2

−^d+1₂ (−X)

. (3.6-148)

12A general argument for the invariants which are fundamental for propagators on maximally symmetric spaces can be found in [3].

13 For s= 0 and ∆ = d−2, this result can be computed directly from formulas (3.6-140) and (3.5-138), by employing a Schwinger representation not only for the bulk-to-boundary propagator, but also for the kernel of the wave operatorD⁰. This would also give the proper normalisation.

The C1-term is the difference of propagators with different boundary conditions, without the singular terms on the diagonal X =−1, similar to G^{eff 0}_bu obtained from the EAdS-presentation of the twist-2 CPWE. The C₂-term is the scalar bulk-to-bulk propagator with a boundary behaviour (z⁰)^∆; It obeys the differential equation only for X < −1, with a singular contribution at X = −1. This is in accordance with the results (6.1-18) and (6.1-17) which will be found in a later chapter from a direct functional integral approach. The modified Bessel function of the second kindK_∆−^d

2(τ) can be expressed by mod-ified Bessel functions of the first kind,

2

Γ(^d₂ −∆)Γ(1− ^d2 + ∆)K_∆−^d

2(τ) =I_∆−^d

2(τ)−I^d

2−∆(τ). (3.6-150) When the Bessel functions I_∆−^d

2(τ) or I^d

2−∆(τ) are substituted into (3.6-149), we obtain certain linear combinations of theC₁- and theC₂-terms; the necessary integral is given in (C.3-11) in the appendix. Each of these generates one particular boundary behaviour of the propagator. Since we want to reserve the term “propagator” for the objects which feature in the actual bulk theory and have a normalisation which is adapted to the normalisation of the vertices ˜V^s1,...,sn, we will for now be content to give the normalised Green’s functions H_bu^{∆, s} in the cases

H_bu^{eff 0} =− (|z˜||u˜|)^2−d given. The actual propagators will be multiples of these Green’s functions.

For the normalisation of the Green’s functions, we have choosen the following con-ventions: By computing the action of the d’Alembertian, one shows that

and equivalently for the dual prescription. There will be a divergent contribution whenever ˜z and ˜uare lying on the same ray in the embedding space. On the EAdS-hypersurface |z˜|=|u˜|= 1, ˜z^d+1,u˜^d+1 >0, we obtain the usual EAdS scalar Green’s function.

The first nontrivial case is spin 1 (we choose ∆ = d−1). For spin 1, the general form (3.6-147) can be reduced to

G¹_bu=|z˜|^1−d|u˜|^1−dn

gA(X)·(Kab+ (˜z,u)J˜ aJb)−XgB(X)·Kab

o (3.6-153)

in terms of the invariants (3.6-145). We have chosen the specific parametrisation with a view to the solution.

While the second-order equations in ˜a and ˜b are automatically fulfilled for a tensor of order 1 (ie, a vector), we have to take care of the z˜- and u˜- and the mixed equations (3.4-113d) to (3.4-113e). The solution of these equations is complicated, but standard, taking care that they are fulfilled for any ˜z,u,˜ ˜a,˜b, and we obtain the general solution

gA(X) =C1(X²−1)^−d+34 Pd−3^−d+32 2

(−X) +C2(X²−1)^−d+34 Q^d+3d−3² 2

(−X), (3.6-154a) gB(X) =C1

d + 1

2 (X²−1)^−d+34 Pd−1^−d+32 2

(−X)−C2(X²−1)^−d+34 Q^d+3d−1² 2

(−X).

We do not give the correct normalisation in this place, because we are just interested in the mechanism for the solution of these equations. It is in principle possible to obtain an integral representation of the kind (3.6-151a) to (3.6-151c) for these expressions. Note that when these solutions are substituted into (3.6-153) to obtain the EAdS propagators, all radii become |z˜| = |u˜| = 1. The asymptotic behaviour of these two linearly independent solutions reveals that the C₁-term describes the effective propagator G^{eff 1}_bu won by lifting of the twist-2 CPWE (see (3.5-138)); by linear combinations of the C1- and C2-terms, one obtains the true propagators with behaviour (z⁰)^d−1 resp. (z⁰)¹; they diverge at X = −1, ie z = u and z⁰ = u⁰ simultaneously. The propagator with dimension ∆₋= 1 already violates in 2<d<4 the unitarity bound ∆¹_ub= d−1 for spin 1.

Summary. We have found characteristic equations for the propagators in the em-bedding space. The advantage of working in the emem-bedding space is that we can work without having to use the covariant derivative, using the partial derivative of the embedding space exclusively. The mass of the fields is contained in a homogeneity condition; and the system of equations has two propagator solutions, corresponding to the boundary behaviour (z⁰)^∆ and (z⁰)^d−∆. For higher spins, the solution of the propagator equations is involved.

Im Dokument AdS/CFT Holography of the O(N)-symmetric $\phi^4$ Vector Model (Seite 109-113)