EAdS-Presentation of n-Point Functions

Vertices

We have sketched in the last section how 3-point functions of bilinear twist-2 currents in the free UV fixpoint theory may be given an EAdS-presentation; in short, we almost completed an equality which can be written formally

G^s_(l¹₁^,s_),(l^2,s₂³_),(l3)(x₁, x₂, x₃) =

Z d^dz dz⁰

(z⁰)^d+1 V˜^{s1,s2,s3(µ1),(µ2),(µ3)}(D^z1,D^z2,D^z3) (3.5-137) G^{ft UV}_{bubo (µ}^s1_1),(l1)(z1, x₁)G^{ft UV}_{bubo (µ}^s2_2),(l2)(z2, x₂)G^{ft UV}_{bubo (µ}^s3_3),(l3)(z3, x₃)

zi=z, where ˜V acts as a differential operator on the three propagators and is of order N¹. The propagators are the bulk-to-boundary propagators (3.4-111); and we have agreed to choose them from the field-theoretic prescription, in order to have a definite normalisation with respect to the boundary-to-boundary propagators. They are of order N⁰, in the normalisation of section 3.4.4.

We have thus served the case n = 3 of definition 3.4 on page 74. In this section, we are going to continue the examination to n ≥ 4; based on the results for the three-valent vertex, it will turn out that higher-order vertices are easier to access:

They are given in terms of the three-vertices.

According to proposition 2.4 on page 41 and the following remark 2.5, a correla-tion funccorrela-tion hJ^s1(x₁). . .J^sn(x_n)iconn is equivalent to the inverse Catalan number C_n⁻₋¹₂ times a sum of all amplitudes generated by all possible tree graphs (in the context, they appeared as “cyclic commutative non-associative structures”, in short CCNA’s, cf. page 38) containing symmetric, traceless quasi-primary tensor currents of all even spins s bilinear in the fields φ, using the (EAdS-presented) three-point functions G^s,t,u(x, y, z) as vertices and integrating out the coordinates x with the inverse propagatorD^s(∂_x) defined by equation (2.7-36) as kernel whenever two such correlations have a common midpoint

X

s

Z

d^dx G^s,s1,s2(x, y₁, y₂)D^s(∂x)G^s,t1,t2(x, z₁, z₂)

(suppressing tensor indices). The external currents J^sj(x_j) are inserted at the tips of the tree.

Whenever there appears such a linking term, in the EAdS-presentation this will lead to an effective bulk-to-bulk propagator of orderN⁻¹ (since D^s ∼N⁻¹)

G^eff_{bu (µ}^s _1),(µ2)(z₁, z₂) = Z

d^dx G^{ft UV}_{bubo (µ}^s_1),(l1)(z₁, x)D^s(l1),(l2)(∂_x)G^{ft UV}_{bubo (µ}^s_2),(l2)(z₂, x).

(3.5-138) It is tempting to interpret this as the bulk-to-bulk propagator; however, by construc-tion G^eff_bu^s(z1, z2) should obey the equations of motion w.r.t. z1 and z2 on all EAdS;

there is no singularity on the diagonalz1 =z2 as one expects for any decent propa-gator. By the general arguments of section3.2.2(cf. equation (3.2-38)), the effective propagator G^eff_bu^s(z1, z2) is the difference of the bulk-to-bulk-propagators of different boundary scaling dimensions; it is a completely regular solution of the equation of motion, and not a Green’s function. With the dimension ∆^UV_s = d−2 +s,

G^eff_bu^s(z1, z2) =G^∆_bu^{UVs s}(z1, z2)−G^(d_bu^{−∆UVs )s}(z1, z2) (3.5-139) (G^(d_bu^{−∆UVs )s} never appears as independent propagator because it violates the unitar-ity bound). The na¨ıve EAdS-presentation of the twist-2 CPWE does not yield the correct propagators for an interpretation as effective (classical) Lagrangian field the-ory in the bulk, involving the higher spin tensor fields as basic fields. In addition, we could not get the combinatorics right because of the inverse Catalan number C_n⁻₋¹₂ appearing in the prefactor demanded by the twist-2 CPWE.

This is not fatal to us, because, by the philosophy of section 3.2.3, we have to find a single (probably nonlocal) bulk vertex ˜V^s1...sn(D^z1, . . . ,D^zn) which EAdS-presents then-point correlations. Such a bulk vertex can be obtained from the twist-2 CPWE.

We state the procedure in

Proposition 3.7. The n-valent bulk vertex V˜^s1...sn(D^z1, . . . ,D^zn), necessary for the EAdS-presentation of the n-point function of quasi-primary bilinear tensor currents in the free UV fixpoint theory according to definition 3.4 on page 74, can be con-structed as follows:

(i) EAdS-presenting each three-point function arising in the twist-2 CPWE (sec-tion 2.7) of the n-point function of the currents by (3.5-137) in the bulk, (ii) amputating those propagators which constitute external legs,

(iii) summing over the spins of the internal propagators and integrating out the bulk coordinates of those vertices which are not connected to an external leg.

(iv) summing over all the different CCNA’s which constitute the total twist-2 CPWE, with the combinatorial prefactorC_n⁻₋¹₂.

Since the n-point vertices are generated by EAdS-presentation of φ-loops with n operator insertions, and each such insertion from the boundary UV theory carries

an additional coupling−i, there will be a factor (−i)ⁿ which we have to supplement explicitly.

The truth of this proposition follows by construction. Since the boundary φ-loops and their twist-2 CPWE are all ∼N¹, the resulting n-valent bulk vertices will also be∼N¹. For three-valent vertices, the result is simply given by the EAdS-presented three-valent vertex which we have constructed in section 3.4.

As a more complex example, the quartic scalar bulk vertex is V˜^s1,s2,s3,s4(z₁, z₂, z₃, z₄) =

The explicit computation of the vertices shall not be undertaken here, as we are still missing the precise form of the three-valent vertex. Note that the vertices contain derivatives acting on the external propagators to be engrafted onto the vertex; the way we have written it, these are simply to be connected from the right, and the δ-distributions are to be acknowledged on the final integration of coordinates in the finished bulk graph.

One might argue that the sum over spins should make the internal propagators in the n-valent vertices vanish, by the philosophy of section 3.2.3. However, this is not so: The sum over bulk-to-bulk propagators of different spins was to vanish only for the true propagators G^UV_bu ^s, and possibly an additional (hypothetical) field must be included into the sum to this effect.

The total scalar 4-point function in the UV theory is then given by G^{UV 0,0,0,0}(x₁, x₂, x₃, x₄) =i⁴

Under the hypothesis that the bulk-to-bulk propagation cancels in total, there is a second term which might possibly be contained, but which vanishes completely,

G^{UV 0,0,0,0}_cancel (x₁, x₂, x₃, x₄) =i⁴

where we have suppressed the tensor indices and written the trivalent vertices in the same nonlocal notation as the four-valent vertex. G^UV_bu ^s(y, u) is the bulk-to-bulk propagator for the intermediate tensor field with spin s. The sum over spins s possibly includes any hypothetical field needed to make the second summand vanish in total, by equation (3.2-56).

By their construction, the n-valent vertices are very non-local in nature. Their internal structure reminds one very much of string theory. We can simulate a “tree-level” interaction of several strings by inserting n vertex operators on a worldsheet with the topology of a sphere. The worldsheet in the vicinity of vertex operator j would then be interpreted as “stringj”. There is no saying in which order the strings resp. vertex operators interact with each other; even if the string worldsheet is drawn suggestively as a system of tubes linked by regions connecting three tubes each, then by deforming this worldsheet, we can modify the network in such a manner that the order of interactions is different. In a way, then-valent vertices we have constructed are mirroring this structure since they contain a sum over all different interaction schemes (CCNA’s).

The non-local nature of string vertices is further reflected in the effective propagator G^eff_bu^s which serves as internal propagator inside the vertices: G^eff_bu^s(z1, z2) does not contain singularities on the diagonal z1 = z2, since it is the difference (3.5-138) of two propagators which have the same singularity behaviour on the diagonal (but different boundary conditions).

So it is not totally unrealistic to imagine that the bulk theory we are constructing may be obtained as an infinite-tension limit of an underlying string theory in (E)AdS.