AdS_4/CFT_3 Construction from Collective Fields
Robert de Mello Koch, Antal Jevicki, Kewang Jin, João P. Rodrigues
aa r X i v : . [ h e p - t h ] N ov BROWN-HET-1606, WITS-CTP-56
AdS /CFT Construction from Collective Fields
Robert de Mello Koch , ∗ Antal Jevicki , † Kewang Jin , ‡ and Jo˜ao P. Rodrigues § National Institute for Theoretical Physics,School of Physics and Centre for Theoretical Physics,University of the Witwatersrand,Wits, 2050, South Africa Department of Physics, Brown University,Providence, RI 02912, USA
We pursue the construction of higher-spin theory in AdS from CFT of the O(N) vector model interms of canonical collective fields. In null-plane quantization an exact map is established betweenthe two spaces. The coordinates of the AdS space-time are generated from the collective coordinatesof the bi-local field. This, in the light-cone gauge, provides an exact one-to-one reconstruction ofbulk AdS space-time and higher-spin fields. PACS numbers: 04.62.+v,11.15.Pg
I. INTRODUCTION
The AdS/CFT correspondence [1–3] represents a veryimportant tool in gauge and string theories. It givesa concrete, analytical procedure for the more generalGauge/String(Gravity) duality. The correspondence ischaracterized by conjectured emerging dimensions ofspace-time (in N = 4 Super Yang-Mills theory the D = 10 of the string in AdS × S background emerges).While the main understanding of the duality itself is pro-vided by ’t Hooft’s large N expansion (which establishes1 /N as the string coupling constant) the origin of theextra spatial dimension is less clearly understood, onespeaks of them as being holographic and have a relation-ship (in the case of radial AdS dimension) with renor-malization group scaling parameters.One framework for analytical understanding of theLarge N limit in general introduced several decades ago[4] is based on the notion of collective fields. Theycapture the relevant degrees of freedom and a generalmethod for describing their effective dynamics both atthe Hamiltonian and Lagrangian level was given. Thisapproach has been successful in analytical treatment aswell as in exhibiting the relevant physics in various modeltheories. In the c = 1 matrix model collective dynamicsnaturally led to (one) extra dimension relevant in estab-lishing the model as a 2D non-critical string theory [5].It has re-emerged in the sub-dynamics of the N = 4Yang-Mills problem in the 1/2 BPS sub-sector. Throughcertain matrix model truncations (of N = 4 Yang-Millstheory) the construction [6] of dual string theory Hamil-tonian was attempted.For further understanding of this mechanism it is use-ful to concentrate on exactly solvable theories. The sim-plest field theory model for which one can build the ∗ Email: [email protected] † Email: antal [email protected] ‡ Email: kewang [email protected] § Email: [email protected]
AdS/CFT correspondence is that of N -component vec-tor theory. It was originally pointed out by Klebanov andPolyakov [7] that the conformal fixed points of the theoryare naturally described in four dimensional AdS space-time. More specifically it was established that it is aparticular higher-spin theory of Vasiliev [8] that emergesin the large N limit. (In a series of works dating back tothe 80’s Vasiliev and collaborators have succeeded in con-structing a remarkable theory providing interactions of asequence of higher-spins in AdS. This (gauge) theory suc-cessfully extends a free theory [9] obtained by Fronsdal[10].) An impressive comparison of three-point bound-ary correlators was performed recently by Giombi andYin [11]. For other relevant work, see [12–15].The relevance of collective fields for higher-spin holog-raphy was discussed by Das and one of the present au-thors [16]. The framework of covariant bi-local collectivefields was employed and it was shown that they decom-pose into an infinite sequence of integer spin fields in oneextra dimension. The present paper sharpens this pic-ture concentrating on the canonical formulation with thegoal of establishing the correspondence directly at theHamiltonian level. It will be advantageous to work innull-plane quantization, since it gives a physical descrip-tion of higher-spin gauge theory. In this framework, wewill produce an exact one-to-one map between (collec-tive) coordinates of the large N field and the AdS co-ordinates of the higher-spin theory. It is shown how col-lective fields provide a construction of bulk (rather thanboundary) fields of the AdS theory. In particular it isdemonstrated that all the bulk AdS space-time transfor-mation symmetries are recovered from transformationsof the bi-local collective field. Outline of an exact mapof the full interacting theory is given.This paper is organized as follows. In section II, wediscuss the difference between collective fields and con-formal currents which have been the main tool of earlierAdS/CFT comparisons. In section III, we summarize theform of the exact collective field Hamiltonian, discuss itsexpansion in 1 /N as a coupling constant. Realization ofthe conformal symmetries and the quadratic approxima-tion is studied in section IV. We establish a one-to-onemap with the transformations of higher-spin theory inAdS background in section V. Discussion of results andof further topics is done in the Conclusion. II. COLLECTIVE VS CONFORMAL FIELDS
The basis of the holographic map is in a (complete)set of primary operators of the SO(2,d) group. They arebuilt as composite operators from the basic fields of thetheory and obey current conservation once the field equa-tions are used. They are used as sources at the boundaryand their correlators are then shown to be in agreementwith the AdS amplitudes projected to the boundary ofAdS space. The N -component vector model field theorywith the Lagrangian L = Z d d x
12 ( ∂ µ φ a )( ∂ µ φ a ) + v ( φ · φ ) , a = 1 , ..., N (1)possesses two critical points: the UV fixed point at zerovalue of the coupling and an IR fixed point at nonzerocoupling. For the UV case corresponding to the free the-ory where the potential v = 0, a full set of conformalcurrents is explicitly given by [11] O ( ~x,~ǫ ) = φ a ( x − ǫ ) ∞ X n =0 n )! (cid:0) ǫ ←− ∂ x · −→ ∂ x − ǫ · ←− ∂ x )( ǫ · −→ ∂ x ) (cid:1) n φ a ( x + ǫ ) (2)where ~ǫ is a null polarization vector ~ǫ = 0. These cur-rents are conserved and in the holographic scheme ofGKP-W [2, 3] their correlators are compared with theAdS boundary amplitudes.Collective fields for large N theories are introduced ina very different manner. They are to represent a (com-plete) set of invariants under the O ( N ) or U ( N ) (gauge)symmetry group. The meaning of completeness is es-tablished in two not unrelated ways. First one has com-pleteness in group theoretic terms, namely that any otherinvariant can be expressed in terms of them. Second isthe requirement of closure under (quantum) equations ofmotion. This leads to the most important fact, namelythat they provide a complete dynamical description [4]of the large N theory where 1 /N is seen to emerge as thenatural expansion parameter.In the O ( N ) vector model one simply has the bi-localcollective fieldΨ( x µ , y µ ) = N X a =1 φ a ( x ) · φ a ( y ) (3)in the covariant formalism [16]. It is the case for the O ( N ) model, and also more generally that the set of col-lective fields is actually over-complete. This propertyhas significant implications on the emerging space-time,when implemented it naturally leads to space-time cut-offs and ultimately non-commutativity. As far as the relationship between the conformal andcollective fields we have the following. Clearly any con-formal field is contained in the collective (bi-local) field,one has a prescription with derivatives given above. Butthe converse is not true, collective fields represent a moregeneral set. This property will have important implica-tions on the bulk vs boundary description of the theory.It has already seen in approximate manner [16] that therelative coordinate in the bi-local field into angles gener-ating a sequence of spins and the radial part which playsthe role of an extra dimension. What prevented a pre-cise identification however was the fact that higher-spinis a gauge theory, whose dynamical form depends on thegauge chosen. Consequently for establishing a preciseone-to-one map, one has to bring both theories to thesame gauge. This will be accomplished in the presentwork in a canonical description.The canonical formalism for collective fields is based(in equal-time quantization) on the observablesΨ( t ; ~x, ~y ) = X a φ a ( t, ~x ) · φ a ( t, ~y ) ≡ Ψ xy (4)which are local in time but bi-local in d − O ( N ) invariant canonical variables (obtained throughscalar product). To deduce the dynamics obeyed by thesefields, one performs an operator change of variables [4]from φ a ( t, ~x ) to the bi-local field Ψ( t ; ~x, ~y ) using the chainrule δδφ ( ~x ) = δ Ψ( ~y, ~z ) δφ ( ~x ) δδ Ψ( ~y, ~z ) . (5)Starting from the canonical Hamiltonian H = Z (cid:16) − δδφ a ( ~x ) δδφ a ( ~x ) + 12 ▽ x φ a ▽ x φ a + v ( φ · φ ) (cid:17) d~x, one deduces an equivalent representation in terms collec-tive variables H = 2Tr(ΠΨΠ) + N − + Z d~xv (Ψ(˜ x, ˜ y ) | ˜ x =˜ y )+ 12 Z d~x [ − ▽ x Ψ(˜ x, ˜ y ) | ˜ x =˜ y ] + ∆ V (6)where we have the conjugate momentum denoted byΠ( ~x, ~y ) = − i δδ Ψ( ~x, ~y ) (7)and ∆ V summarizes ordering terms which are lower orderin 1 /N ∆ V = − N (cid:16)Z dxδ (0) (cid:17) TrΨ − + 12 (cid:16)Z dxδ (0) (cid:17) TrΨ − . The product of two bi-local fields is defined by AB = Z d~yA ( ~x, ~y ) B ( ~y, ~z ) (8)and the trace of a bi-local field meansTr( A ) = Z d~xA ( ~x, ~x ) . (9)For more details on this representation, including the factthat it generates correctly the large N Schwinger-Dysonequations, the reader should consult Refs. [4, 17].
III. EXPANSION
The main feature of the collective representation interms of the Hamiltonian (6) is that it can be expandedin series of 1 /N with an infinite number of polynomialvertices to generate systematically the 1 /N expansion.This is seen by a simple rescaling of field variables:Ψ → N Ψ , Π → Π /N whereby N factorizes in front of theaction. The terms in ∆ V are seen to be of lower order,consequently they provide counter-terms in the system-atic 1 /N expansion.To generate the expansion, one first evaluates the staticlarge N background ψ ( ~x, ~y ) obtained from the time-independent equations of motion ∂V∂ Ψ( ~x, ~y ) = 0 , (10)where we have set v = 0 and the effective potential reads V = 18 TrΨ − + 12 Z d~x [ − ▽ x Ψ(˜ x, ˜ y ) | ˜ x =˜ y ] . (11)One performs a shiftΨ = ψ + 1 √ N η,
Π = √ Nπ (12)generating an infinite sequence of verticesTrΨ − = Tr ψ − + ∞ X n =1 ( − n N n Tr( ψ ( ηψ ) n ) . (13)The quadratic and cubic terms in the Hamiltonian areseen to be given by H (2) = 2Tr( πψ π ) + 18 Tr( ψ ηψ ηψ ) , (14) H (3) = 2 √ N Tr( πηπ ) − √ N Tr( ψ ηψ ηψ ηψ ) . (15)The higher order vertices are obtained directly from theexpansion (13).We now discuss the evaluation of the spectrum whichfollows from diagonalization of H (2) . In doing this wefollow closely [17]. Using a Fourier transform ψ xy = Z d~ke i~k · ( ~x − ~y ) ψ k , (16)with ψ k = 12 p ~k , (17) and for the fields η xy ≡ Z d~k d~k e − i~k · ~x e + i~k · ~y η k k , (18) π xy ≡ Z d~k d~k e + i~k · ~x e − i~k · ~y π k k , (19)the quadratic Hamiltonian now becomes H (2) = 2 Z d~k d~k ψ k π k k π k k + 116 Z d~k d~k η k k ( ψ − k ψ − k + ψ − k ψ − k ) η k k . Redefining π k k → ψ − / k π k k η k k → ψ / k η k k (20)one has the quadratic Hamiltonian H (2) = 12 Z d~k d~k π k k π k k + 18 Z d~k d~k η k k ( ψ − k + ψ − k ) η k k (21)from which one reads off the frequencies ω k k = 12 ψ − k + 12 ψ − k = q ~k + q ~k . (22)To summarize, the quadratic Hamiltonian and momen-tum can be written in use of bi-local fields as H (2) = Z d~xd~y Ψ † ( ~x, ~y ) (cid:16)p −▽ x + q −▽ y (cid:17) Ψ( ~x, ~y ) ,P (2) = Z d~xd~y Ψ † ( ~x, ~y )( ▽ x + ▽ y )Ψ( ~x, ~y ) . (23)In the light-cone quantization, we have the quadraticHamiltonian P − (2) = H (2) + P (2) = Z dx − dx − d~x d~x Ψ † (cid:16) − ▽ p +1 − ▽ p +2 (cid:17) Ψ . (24)Here Ψ( x + ; x − , x − ; ~x , ~x ) is a bi-local field where 1 , IV. CONFORMAL TRANSFORMATIONS OFTHE COLLECTIVE FIELDS
Our goal is to demonstrate that the collective field con-tains all the necessary information and is in a one-to-onemap with the physical fields of the higher-spin theory inAdS . For this comparison to be done it is advantageousto work in the light-cone gauge, where the physical de-grees of freedom of a gauge theory are most transparent[18, 19]. Our strategy is to compare directly the action ofthe conformal group of the d = 3 field theory with thatof the Anti de Sitter higher spin field. This comparisonis similar to the study in D-brane case and N = 4 Su-per Yang-Mills theory performed in [20]. In this directcomparison we will see that as expected we have verydifferent set of space-time variables and a different real-ization of SO(2,3). The number of canonical variableshowever will be shown to be identical and one can searchfor a (canonical) transformation to establish a one-to-onerelation between the two representations.One can work out the conformal transformations inlight-cone notation ( x + = t ) for any dimension d . As forthe linear momenta, we have P − = H = Z d~x (cid:16) −
12 ( ∂ i φ ) (cid:17) ,P + = Z d~x (cid:16) π (cid:17) ,P i = Z d~x (cid:16) π∂ i φ (cid:17) , (25)where π = ∂ + φ is the conjugate momentum and i is thetransverse index (for the specific case when d = 3, theindex i runs over a single value). Similarly, for Lorentztransformations, the conserved charges are M + − = tH − Z d~x (cid:16) x − π (cid:17) ,M + i = Z d~x (cid:16) tπ∂ i φ − x i π (cid:17) ,M − i = Z d~x (cid:16) x − π∂ i φ − x i H (cid:17) ,M ij = Z d~x (cid:16) x i π∂ j φ − x j π∂ i φ (cid:17) . (26)The Dilatation operator takes the form D = tH + Z d~x (cid:16) π ( d φ + x i ∂ i ) φ + x − π (cid:17) , (27)where d φ = d − is the scaling dimension of the φ field.The special conformal generators are K − = Z d~x (cid:16) x − D −
12 (2 tx − + x j x j ) H − d φ φ (cid:17) ,K + = tD − Z d~x (cid:16)
12 (2 tx − + x j x j ) π (cid:17) ,K i = Z d~x (cid:16) x i D −
12 (2 tx − + x j x j ) π∂ i φ (cid:17) , (28)where D and H are the densities of these two operators.The dynamical variables in the light-cone formulationare ( x − , x i ). The momentum conjugate to x − is p + . Inthe massless case, the energy can be expressed as p − = − p i p i p + . (29)To define the mode expansion, we perform a Fouriertransform of the fields φ ( x − , x i ) and π ( x − , x i ) along the x − direction. The creation and annihilation operatorsare defined in terms of φ ( x − , x i ) = Z ∞ dp + √ π p p + (cid:16) a ( p + , x i ) e ip + x − + a † ( p + , x i ) e − ip + x − (cid:17) , (30) π ( x − , x i ) = − i Z ∞ dp + √ π r p + (cid:16) a ( p + , x i ) e ip + x − − a † ( p + , x i ) e − ip + x − (cid:17) . (31)The actions of linear momenta now take the form P − : δa ( p + , x i ) = ∂ i p + a ( p + , x i ) ,P + : δa ( p + , x i ) = p + a ( p + , x i ) ,P i : δa ( p + , x i ) = i∂ i a ( p + , x i ) . (32)For the Lorentz generators, one has M + − : δa ( p + , x i ) = (cid:16) t ∂ i p + − i p p + ∂∂p + p p + (cid:17) a ( p + , x i ) ,M + i : δa ( p + , x i ) = (cid:16) it∂ i − x i p + (cid:17) a ( p + , x i ) ,M − i : δa ( p + , x i ) = (cid:16) − ∂ i ∂∂p + − ∂ j x i ∂ j p + (cid:17) a ( p + , x i ) ,M ij : δa ( p + , x i ) = (cid:16) ix i ∂ j − ix j ∂ i (cid:17) a ( p + , x i ) . (33)and the Dilatation operator D : δa ( p + , x i ) = (cid:16) t ∂ i p + + i h d φ + x i ∂ i + p p + ∂∂p + p p + i(cid:17) a ( p + , x i ) . (34)Finally, for the special conformal generators K − : δa ( p + , x i ) = n − ∂ j x i x i ∂ j p + − p p + ∂∂p + ∂∂p + p p + − x i ∂ i ∂∂p + − d φ p p + ∂∂p + p p + o a ( p + , x i ) ,K + : δa ( p + , x i ) = n t ∂ i p + + it ( d φ + x i ∂ i ) − x i x i p + o a ( p + , x i ) ,K i : δa ( p + , x i ) = n t ∂ j x i ∂ j p + + t∂ i ∂∂p + − i x j x j ∂ i + ix i h d φ + x j ∂ j + p p + ∂∂p + p p + io a ( p + , x i ) . (35)We next deduce the transformation for the collectivefields. In creation-annihilation form A ( x − , x − , ~x , ~x ) = a ( x − , ~x ) a ( x − , ~x ), we have δA (1 ,
2) = δa (1) a (2) + a (1) δa (2) and any conformal generator G = Z dx − dx − d~x d~x A † ˆ gA = Z dx − dx − d~x d~x A † (ˆ g + ˆ g ) A. (36)Denoting the conjugate momenta as ( p +1 , p +2 , p i , p i ), wecan write down the following generatorsˆ p − = p − + p − = − (cid:16) p i p i p +1 + p i p i p +2 (cid:17) , (37)ˆ p + = p +1 + p +2 , (38)ˆ p i = p i + p i , (39)ˆ m + − = t ˆ p − − x − p +1 − x − p +2 , (40)ˆ m + i = t ˆ p i − x i p +1 − x i p +2 , (41)ˆ m − i = x − p i + x − p i + x i p j p j p +1 + x i p j p j p +2 , (42)ˆ d = t ˆ p − + x − p +1 + x − p +2 + x i p i + x i p i + 2 d φ , (43)ˆ k − = x i x i p j p j p +1 + x i x i p j p j p +2 + x − ( x − p +1 + x i p i + d φ )+ x − ( x − p +2 + x i p i + d φ ) , (44)ˆ k + = t ˆ p − + t ( x i p i + x i p i + 2 d φ ) − x i x i p +1 − x i x i p +2 , (45)ˆ k i = − t (cid:16) x i p j p j p +1 + x i p j p j p +2 + x − p i + x − p i (cid:17) − x j x j p i − x j x j p i + x i ( x − p +1 + x j p j + d φ )+ x i ( x − p +2 + x j p j + d φ ) . (46) V. MAPPING TO ADS The correspondence introduced in [7] is specific for
CF T ↔ AdS . We will from now on consider the caseof d = 3 for the vector model. In the light-cone nota-tion, there is only one transverse dimension x i = x and x µ = ( x + , x − , x ).The AdS spacetime coordinates in the light-cone no-tation ( x + = t ) are denoted with the Poincar´e metric ds = 2 dtdx − + dx + dz z . (47)The lowercase transverse index i = 1 denotes x only,while the uppercase transverse index I = (1 ,
2) denotes( x, z ). In AdS higher-spin theory, the generators wereworked out by Metsaev in [18] which we now summarize. A. Conformal generators from higher-spin theory
The four-dimensional case has the unique propertythat, after fixing light-cone gauge [21], the only phys-ical states are the ± s helicity states [22]. Let us nowexplain how to fix the light-cone gauge. Starting fromthe covariant notation | Φ i = ∞ X s =1 Φ µ ...µ s a † µ ...a † µ s | i . (48)where µ = (0 , , z,
3) in the case of AdS , one fixes thelight-cone gauge in two steps. First, we drop the oscilla-tors a ± = a ± a and keep only the transverse oscillators a I , a † J including the z component. The oscillators satisfythe commutators[ a I , a † J ] = δ IJ , [ a I , a J ] = [ a † I , a † J ] = 0 . (49)The spin matrix of the Lorentz algebra now takes theform M IJ = a † I a J − a † J a I . (50)The next step is to impose a further constraint T | Φ i = 0 , T = a I a I (51)so that only two components will survive. With the com-plex oscillators α = 1 √ a + ia ) , α † = 1 √ a † + ia † ) , (52)¯ α = 1 √ a − ia ) , ¯ α † = 1 √ a † − ia † ) , (53)we find the simple expansion for | Φ i| Φ i = ∞ X λ =1 (cid:16) Φ ( λ ) (¯ α † ) λ + ¯Φ ( λ ) ( α † ) λ (cid:17) | i . (54)This expansion obviously satisfies the constraint T | Φ i = 0 , T = ¯ αα. (55)The spin matrix M = α † ¯ α − ¯ α † α (56)also reduces to (50).In four dimensions, the only non-vanishing spin matrixis M xz . One can represent α = e iθ , ¯ α = e − iθ . In a coher-ent basis, the operator M xz becomes ∂∂θ . Then we haveΦ( x µ , z, θ ) or in light-cone notation Φ( x + , x − , x, z ; θ ).The generators can be written as G = Z dx − dxdzdθ ¯Φˆ g Φ . (57)Denoting the conjugate momenta as ( p + , p x , p z , p θ ), onehas [18]ˆ p − = − p x p x + p z p z p + , (58)ˆ p + = p + , (59)ˆ p x = p x , (60)ˆ m + − = t ˆ p − − x − p + , (61)ˆ m + x = tp x − xp + , (62)ˆ m − x = x − p x − x ˆ p − + p θ p z p + , (63)ˆ d = t ˆ p − + x − p + + xp x + zp z + d a , (64)ˆ k − = −
12 ( x + z )ˆ p − + x − ( x − p + + xp x + zp z + d a )+ 1 p + (cid:0) ( xp z − zp x ) p θ + ( p θ ) (cid:1) , (65)ˆ k + = t ˆ p − + t ( xp x + zp z + d a ) −
12 ( x + z ) p + , (66)ˆ k x = t ( x ˆ p − − x − p x − p θ p z p + ) + 12 ( x − z ) p x + x ( x − p + + zp z + d a ) + zp θ , (67)where the scaling dimension d a = 1 in the case of AdS . B. The map: canonical transformation
We will now show how the two pictures are relatedby a canonical transformation. At this point, we willgive the classical transformation (it can be specified inits full quantum version also). So in what follows we donot compare terms with d φ which will receive quantumcorrections (due to ordering).By relating (38-41) to (59-62), one can easily solve for x − = x − p +1 + x − p +2 p +1 + p +2 , (68) p + = p +1 + p +2 , (69) x = x p +1 + x p +2 p +1 + p +2 , (70) p x = p + p . (71)From (58,63,64,66), we get z = ( x − x ) p +1 p +2 ( p +1 + p +2 ) , (72) p z p z = ( p p +2 − p p +1 ) p +1 p +2 , (73) zp z = ( x − x )( p p +2 − p p +1 )( p +1 + p +2 ) , (74) p θ p z = ( x − − x − )( p p +2 − p p +1 )+( x − x ) (cid:16) p +2 ( p ) p +1 − p +1 ( p ) p +2 (cid:17) . (75)The solution to (72-75) can be written as z = ( x − x ) q p +1 p +2 p +1 + p +2 , (76) p z = s p +2 p +1 p − s p +1 p +2 p , (77) p θ = q p +1 p +2 ( x − − x − )+ x − x (cid:16)s p +2 p +1 p + s p +1 p +2 p (cid:17) . (78)A nontrivial check of the consistency is given by compar-ing (65,67) with (44,46).We now turn to the construction of θ . The conditionthat θ Poisson commutes with p x implies θ is a function of x − x and the condition that θ Poisson commutes with p + implies that θ is a function of x − − x − . Requiringthat θ Poisson commutes with x − , x , z and p z as well as θ and p θ Poisson commute to give 1 we obtain θ = 2 arctan s p +2 p +1 . (79)An important consistency check on the correctness ofthe map that we have constructed is that all the Pois-sion brackets of the derived variables (like z and p z etc.)take the canonical form with distinct canonical sets com-muting with each other. One can confirm the Poissonbrackets { x − , p + } = { x, p x } = { z, p z } = 1 (80)and others vanish.Finally, as a consequence of the above map it followsthat the wave equation in the collective picture has amap [23] to the wave equation of higher-spin gravity infour-dimensional AdS background. This follows from thelittle generators (37) and (58) coinciding after the canoni-cal transformation. The canonical transformation can beunderstood as a point transformation in the momentumspace (if we interpret θ as momentum (79), the othermomenta are given by (69,71,77)). Consequently, thetransformation between the higher-spin field and bi-localfield is simple in momentum spaceΦ( x − , x, z, θ ) = Z dp + dp x dp z e i ( x − p + + xp x + zp z ) Z dp +1 dp +2 dp dp δ ( p +1 + p +2 − p + ) δ ( p + p − p x ) δ (cid:16) p q p +2 /p +1 − p q p +1 /p +2 − p z (cid:17) δ (cid:0) q p +2 /p +1 − θ (cid:1) ˜Ψ( p +1 , p +2 , p , p ) (81)where ˜Ψ( p +1 , p +2 , p , p ) is the Fourier transform of thebi-local field Ψ( x − , x − , x , x ). VI. CONCLUSION / ORIGIN OF THE EXTRADIMENSION
The main contribution of this paper is an explicit one-to-one map between the collective field (in the case of theO(N) vector model) and the field of higher-spin gravityin 4D AdS space-time. This map is defined by the canon-ical transformation which establishes the relationship be-tween the coordinates of the bi-local collective field andthe coordinates of the AdS space-time plus spin vari-ables. The map is one to one, in particular the mosttelling formula is the one for the extra radial coordinateof AdS space-time z = ( x − x ) q p +1 p +2 p +1 + p +2 . Here we have an explicit expression, in terms of thecollective coordinates contained in the bi-local field. Thephysical picture for this extra dimension is much like the(collective) coordinates of solitons, which are containedin the field itself but are nontrivial to exhibit. Theirorigin is again through a canonical map from the exist-ing field degrees of freedom. Naturally, if the boundaryconditions are too restrictive then these degrees will beabsent. In more recent phenomenological studies of scat-tering processes in QCD, a dipole picture [24] was usedwhich can have a relation to the construction presented.It is interesting to confront this collective mechanism forthe emerging dimension with other viewpoints such asholographic [25], Feynman diagrams [26] and stochasticquantization [27].Returning to future issues we have the following. Thecollective field theory gives a bulk Hamiltonian represen-tation for the higher-spin gravity. It specifies an infi-nite set of bulk interacting vertices, which can be explic-itly evaluated. These can be compared with the higherspin approaches, in particular Vasiliev’s and we expect tofind agreement. This comparison is presently being per- formed. It is also interesting to consider various canonicalgauge fixings of Vasiliev’s theory.
ACKNOWLEDGMENTS
AJ would like to thank J. Avan, S. Das, T. Yoneyaand C. I. Tan for discussions and interest in this work.He is also grateful to Prof. T. Takayanagi for hospitalityat the IPMU, Kashiwa, Tokyo where part of this workwas done. KJ would like to thank X. Yin for interestingdiscussions on this subject, M. A. Vasiliev for clarifyingone of his papers and also I. Messamah for the discus-sion on gravity. The work of AJ and KJ is supportedby the Department of Energy under contract DE-FG-02-91ER40688. The work of KJ is also supported by theGalkin fellowship at Brown University. RdMK is sup-ported by the South African Research Chairs Initiativeof the Department of Science and Technology and Na-tional Research Foundation. [1] J. M. Maldacena, Adv. Theor. Math. Phys. ,231 (1998) [Int. J. Theor. Phys. , 1113 (1999)][arXiv:hep-th/9711200].[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys.Lett. B , 105 (1998) [arXiv:hep-th/9802109].[3] E. Witten, Adv. Theor. Math. 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