Finite Temperature Holography in Higher Spin Theory/Vector Model

Transcription

1 Finite Temperature Holography in Higher Spin Theory/Vector Model Junggi Yoon Department of Physics Brown University A dissertation submitted for the degree of Doctor of Philosophy May 06

2 Copyright 06 by Junggi Yoon

3 This dissertation by Junggi Yoon is accepted in its present form by the Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Professor Dr. Antal Jevicki, Advisor Recommended to the Graduate Council Date Professor Dr. David Lowe, Reader Date Professor Dr. Anastasia Volovich, Reader Approved by the Graduate Council Date Professor Dr. Peter M. Weber, Dean of the Graduate School

4 Acknowledgements First and foremost, I would like to express my sincere appreciation to my advisor Professor Dr. Antal Jevicki for his dedicated help, encouragement and continuous support as well as for invaluable guidance, profound insight and immense knowledge throughout my Ph.D. I always enjoy all discussions with him, which are the most precious time in my life. I could not have imagined having a better advisor and mentor for my Ph.D study. I would also like to thank my committee members: Professor Dr. David Lowe and Professor Dr. Anastasia Volovich, for their interest and insightful comments to widen my research from various perspectives. I am grateful to my collaborators: Prof. João P. Rodrigues, Prof. Jean Avan for exciting experience of collaboration. Especially, I would like to express my heart-felt gratitude to Prof. Robert de Mello Koch. To me, he is not only an excellent collaborator but also an outstanding teacher and a great mentor. I would like to thank Dr. Kewang Jin and Dr. Qibin Ye for their help and guidance when I joined the HET group. Also, I would also like to thank Dr. Kenta Suzuki for insightful comments, constructive criticisms and enjoyable collaboration. I would like to thank Mary Ann Rotondo for all her supports, time and pleasant conversations. I gratefully acknowledge the funding sources that made my Ph.D. work possible. I was also honored to be a Doctoral Study Abroad Program Fellow from Korea Foundation for Advanced Studies (KFAS) for 5 years. I would also like to thank Mr. Warren Galkin for Galkin Foundation Fellowship award in my 6th year. My work was also supported by the U.S. Department of Energy. I would like to thank my wife, Hee Jung Chung for all her love and encouragement. None of this would have been possible without you, Hee Jung. I am also grateful to my little angel, Amy Yoon who came to us during my Ph.D. study. Both of you are the meaning of my life. Junggi Yoon Brown University April 06

5 Abstract Vector-like models with O(N) and U(N) symmetries at their critical points were seen to exhibit duality with higher spin gravitational theories of Vasiliev. In the dissertation, we study the finite temperature Vector Model/Higher Spin Duality in Large N. For CFTs given by 3d O(N) (or U(N)) vector models, I evaluate the leading and one-loop partition functions in a variety of geometries. This calculations are performed in the scheme of collective field theory, which was seen in earlier studies to represent a bulk description of Vasiliev higher spin theory. The calculations presented provide data for comparison of small fluctuation determinants, giving further evidence for the one-to-one bulk identification between the bi-local and the AdS picture. They also o er insight into the identification of coupling constants G and /N of the two descriptions for models based on O(N) symmetry. I also consider the canonical structure of the collective formulation of Vector Model/Higher Spin Duality in AdS 4. This formulation involves a construction of bulk AdS Higher Spin fields through a time-like bi-local Map, with a Hamiltonian and canonical structure that are established to all orders in /N. Finally, I study the Large N dynamics of the O(N) field theory in the Thermo field dynamics approach. The question of recovering the high temperature phase and the corresponding O(N) gauging is clarified. Through the associated bi-local representation, we discuss the emergent bulk space-time and construction of (Higher spin) fields. We note the presence of evanescent modes in this construction and also the mixing of spins at finite temperature.

6 Contents Introduction Collective approach to the loop corrections 3. Partition function on S R 5. Partition function on S Interpretation of the results 8 3 One loop partition functions in thermal AdS The Heat Kernel method 0 3. The Hamiltonian method 3.3 The collective field theory approach 3 4 Hamiltonian Formulation : Light-cone Frame 5 5 Hamiltonian Formulation : Time-like Frame 0 5. Linearization 5 5. Projection to Currents Reduction 8 6 Thermofield Dynamics Bi-local Collective Field Representation Thermal background 34 7 Collective modes and Bulk Bulk Interpretation 39 8 Conclusions 45 A Derivation of the Measure 47 B Algebra of Bi-local Operators 48 C Inverse Transformation and Polarization Vector 5 D Representation of SO (, 3) 54 E Constraints 59 F Green s Function and Commutators 60

7 Introduction Recently, vector-like models with O(N) and U(N) symmetries at their critical points were seen to exhibit duality [ 3] with higher spin gravitational theories of Vasiliev [4 6]. Typically in 3d vector field theory, there are two conformally invariant fixed points: the free UV fixed point and the interacting IR fixed point. The higher spin duals to these two fixed points are given by the same Vasiliev theory but with di erent boundary conditions in the quantization of the bulk scalar field. This Vector Model/Higher Spin correspondence was also extended to the supersymmetric case [7, 8], Chern-Simons theories [9, 0] and de Sitter space [, ]. Furthermore, one also has the very rich and nontrivial lower dimensional dualities involving d Minimal Model CFTs and 3d Higher Spin Gravities [3 5]. All these dualities have received definite support based on evaluation of three-point correlation functions, finite temperature partition functions and study higher conservation laws. A construction of Higher Spin/vector model Duality based on collective fields was proposed in [6]. A one-to-one Map was explicitly given in [7, 8] in the light-cone gauge [9] (where HS gravity is the simplest). This approach provides a framework for a one-to-one reconstruction of AdS spacetime, higher spin fields in the bulk and their /N interactions. Higher order calculations that were performed concerned the one-loop correction to the free energy [6], correlation functions [0], and an investigation of the (non)triviality of the theory [] based on free fields. A similar identification of AdS space in light-cone QCD was developed in [, 3]. A renormalization group method for bi-local observables is being developed in [4 7]. The collective method is easily formulated in any time-like frame and has been employed explicitly in [, ]. A covariant version is also possible with a more general first principle understanding of the Duality [8]. This concrete AdS/CFT Duality provides insight and allows constructions and studies of issues that otherwise are fairly di cult in HS Gravity and String Theory. One such issue is the procedure for quantization of the theory. In Higher Spin Gravity, Vasiliev s equations of motion are known explicitly, but the canonical description of the equation has been started only in [9]. In String theory, this has not been possible at all except in very low dimension. On the other hand, the collective construction naturally contains a canonical picture that, through Duality, also gives the canonical picture of Higher Spin Gravity. For that reason, in this thesis I discuss in detail the canonical structure brought in by the Collective/HS Gravity identification. This formulation concerns the form of bulk Higher Spin fields and observables, their exact commutation relations and AdS locality, which will be established systematically in the thesis. I stress that the present work deals with the canonical construction in a time-like gauge of the theory. This gauge is appropriate for Hamiltonian description and is central to the unitarity of the theory. It is also of definite interest to have the bi-local map formulated in the covariant gauge framework. Some elements of map to covariant (Fronsdal-type) gauges were recently given in [8]. This construction uses the world line spinning particle framework and, as such, could be related to the bi-local higher spin holographic construction of [30] given in terms spinor variables. This connection is currently being considered.

8 The AdS/CFT correspondence with emergent Gravity from the boundary theory o ers a framework for understanding deep quantum aspects of black holes [3]. Recently, issues concerning the physics at the horizon and applicability of quantum mechanics have been vigorously debated [3]. Of central significance is the understanding of emergent Gravity [33] and its space-time [34]. A particular CFT scheme for understanding the space-time of eternal AdS black holes [35] is the so-called Thermo field dynamics (TFD) where identical copies of the CFT are suggested for right and left regions of a Penrose space-time [36]. In this scenario, one might question at the outset if these (decoupled CFTs) are capable of producing a connected [37] space-time characteristic of a black hole [38, 39]. A further very relevant issue concerns the reconstruction of local bulk fields [40, 4] from the two boundaries. The ability to accomplish this goal is central for a possible reconstruction of behind-the-horizon physics [4] for black holes. BT-type black holes dual to CFT have been investigated in detail [43 46]. Also, a generalization of the 4D black hole solution in Vasiliev Higher spin theory was investigated in [47, 48]. The reconstruction of Higher Spins and AdS space-time through bi-local [6, 7, 5, 8, 49 5] fields was accomplished in a systematic /N expansion scheme. In this thesis we study the Thermo field dynamics [5] of O(N) vector models with the intent of understanding their dual bulk degrees of freedom [53]. The outline of this thesis is as follows. In Section, we consider first the finite temperature case of the CFT reviewing an earlier work of [6]. This example already contains some of the basic e ects that will be observable in the rest of the calculations. We then present details of the bi-local calculation in the case of S 3 (the example of [54, 55]) and point out the role of the measure. In Section 3, we proceed to the other phase of the theory discussing the evaluation of the partition function in thermal AdS both by the heat-kernel method and in the bi-local collective field framework. In Section 4, we review the Hamiltonian formulation of the collect O(N) vector model in light-cone frame. The time-like formulation of the Collective/HS Duality will be established in Section 5. In Section 6, we develop the collective description of the TFD for (free) vector models. Fluctuations of bi-locals and their bulk interpretation is given systematically in Section 7. We give our conclusions in Section 8. Collective approach to the loop corrections The purpose of the first part of the thesis is to study further the question of loop corrections (i.e. /N ) in the higher spin duality. We follow up the earlier work of [6] and the recent work of Giombi and Klebanov [54]. These calculations concern the evaluation of partition functions at one loop in the collective and also in the AdS version of the theory. In both cases, the one-loop corrections follow from the quadratic Laplacians Tr log bi-local and Tr log hs gh. (.) 3

9 In the light-cone gauge, there is a strong operator equality [7] between the respective Laplacians: bi-local =@ r hs x z. (.) p p + + p p + p + + p+ = r hs (.3) where on the right hand side one has the higher spin laplacian given by Metsaeev [9]. r hs x z. (.4) This operator equality with the bi-local Laplacian is a direct consequence of the spacetime mapping established in [7]. Because of gauge invariance one could then expect identical results for general one-loop contributions. However, since one considers backgrounds which do not always easily fit into the light-cone gauge, explicit calculations are nevertheless worthwhile. They also serve as the purpose for understanding more completely the nature of loop corrections in higher spin duality. In particular in the heat-kernel AdS calculation the suggestion was made in [54] that the identification of the gravitational coupling constant should be taken as G =/(N ) for the dualities based on the O(N) symmetry group (no such change was found for the U(N) case). Our results shed some light on this identification. First of all collective theory shows that in addition to the determinant there is one further contribution of O() associated with the measure appearing in the functional integration. The measure does provide the needed cancellation at one loop (as noticed originally in [6]) allowing the standard identification of G =/N. However collective field theory also indicates a freedom of a finite (re)normalization of G into /(N ) as we discuss in the text. These two expansion schemes are compatible, as one can re-expand results of one into another. The collective theory describes the large N dynamics of bi-local collective fields. These fields have the property that they close under the Schwinger-Dyson equations. They represent a more general set than the conformal currents and contain an additional dimension. As such they are natural candidates for representing the bulk AdS 4 theory. This is supported by the fact that an e ective collective field action has a property that the associated functional integral exactly evaluates the O(N) singlet partition function and correlation functions of bi-local operators. The diagramatics accomplished by this reformulation is that of Witten diagrams. The exact partition function of the free vector model with N components in terms of the bi-local field (x, y) is given by [6] = = D (x, y) J(x, y)e S[ (x,y)] D (x, y) µe S col[ (x,y)] (.5) A very similar identification of AdS space in light-cone QCD was found in [, 56]. 4

10 where J(x, y) is the Jacobian (generated from the change of variables from the fundamental vector fields to the bi-local fields), and the collective action reads S col = N d 3 x x (x, y) x=y NTr log. (.6) The (integration) measure µ in (.5) is computed to be µ =(det ) apple (.7) where the power apple depends on the underlying symmetry of the vector fields. For the O(N) case, the bi-local field (x, y) = ~ N (x) ~ (y) is symmetric and apple = (K + ) with K = P k the volume of the momentum space. While for the U (N) case, the bi-local field (x, y) = N ~ (x) ~ (y) is Hermitian and one has apple = K. The details of this derivation can be found in the Appendix A. In the Riemann zeta-function regularization (employed in [54] and also here), I have set K = 0 so that the measure is simplified to be ( for U(N) µ = (det ) / for O(N). (.8) I mention that the action on this representation scales with N and the interactions generated are consequently given in powers of /N and the measure would contribute in the subleading orders. It is also relevant to point out at the outset that the measure in this collective representation leads to a contribution of the same form as the Tr log term in the action (.6) (which sets the coupling constant). Consequently one can equivalently include the measure term into the action obtaining an e ective coupling constant. I will return to this issue of interpretation in Section.3 after presenting the one loop calculations.. Partition function on S R We start by reviewing first the one-loop calculation performed in [6] for the S R partition function. This case already demonstrates some of the features of the one loop determinant that will be general and central to the issues raised in the Introduction. One develops the expansion as usual by shifting the background bi-local field (x,x )= 0 (x,x )+ p N (x,x ) (.9) where 0 (x,x ) represents the stationary point of the collective action (.6). For finite temperature T, the fundamental vector field ~ (x) is periodic in Euclidean time with periodicity, the inverse of temperature T. Hence, in the momentum space representation one has the Fourier transformed field 0(k,k ) with the momenta k, =( n,, ~ k, )wherethe Matsubara frequency is given by n, = n,. The zeroth-order collective action is now given by S (0) col = N X k k 0(k, k) NTr log 0. (.0) 5

11 Translation invariance implies 0(k,k )= (k ) k, S (0) col = N X k k (k) N X k k, and one gets log [ (k)]. (.) By the saddle point method one determines (k) =, and the background field is k X 0 (x, y) = ( ) d ~ k e ik (x y) (.) n ~ k + n which is nothing but the free two point function h i (x) i (y)i of the bi-local operators. Evaluating the action at the background value produces the leading contribution to the free energy " F (0) = S (0) col = N X X # log ~ n k + (.3) n ~ k which is precisely the free energy of N free bosons F (0) = N Tr At high temperature, the free energy scales as F (0) N (3)T, producing the lower phase of [57]. To evaluate the -loop contribution, one expands the collective action S col to the quadratic order in the fluctuations : S () col = 4 Tr 0 0 Tr ( ) (.4) = X X kk k, k k, k + k k, k k, k. k >k k Then the one-loop free energy comes as the determinant of the generalized (bi-local) Laplacian. Because of the product form the determinant factorizes and one obtains F () = X Tr log ( ) = log k k + X log(k ) k >k k = (K + ) X k log(k ) (.5) with a surprising finding that the bi-local determinant produces the local field contribution (with a factor K + ). This pre-factor is most significant as it is associated with the counting of bi-local degrees of freedom. With a zeta-function regularization the infinite volume K would be set to 0 and the result corresponds to the N = single field expression. This is a prototype of the result that was also observed in [54], namely the evaluation of the AdS higher spin determinant in the heat-kernel method using the zeta-function regularization gave the N =CFTresult. The collective representation however contains one other contribution of order O N 0. It comes from the measure µ evaluated at the stationary point F () = (K + ) Tr log 0 = (K + ) X k log k. (.6) 6

12 Thus the total one-loop correction to the free energy is found to be F () total = F () + F () =0. (.7) This complete cancellation between the determinant and the measure contribution therefore assures the required result 0. To recapitulate, the one loop determinant of fluctuations produces an answer identical to that of N free scalars in d =3butwithN replaced by K +. If K (which is infinite) is set to 0 by regularization the result then corresponds to N =, i.e. to that of a single scalar field. This is what was also found in [54] and will be the case in all the other examples that follow. One can trace its origin of this to the bi-local nature of degrees of freedom in this theory. In particular the appearance of K +ino(n) theories (and K in U(N) theories) is associated with the fact that the fields can be encoded into a symmetric matrix appearing naturally in the bi-local description. Equally importantly in the collective higher spin representation, one also has a measure in the functional integral which leads to cancellation and the result F () total = 0 at one loop.. Partition function on S 3 We now consider the partition function on S 3, the example that was considered in [54]. One follows the same procedure described in detail as in the previous section, the only di erence being the explicit expressions for the eigenfunctions and eigenvalues. Using spherical harmonics of S 3, the Fourier transformation of the bi-local field is (x,x )= X ~ k, ~ k ~ k, ~ k Y ~k (x ) Y ~k (x ) (.8) where ~ k denotes a full set of quantum numbers ~ k (l, n, m) and l =0,,,, n =0,,,,l m = n, (n ),,n,n. Denoting the conjugate label of ~ k as ~ k (l, n, 0 (x,x )= X ~ k ( ) m m), the classical background field is now ( ~ k) Y ~ k (x ) Y ~k (x ) (.9) where ( ~ k) are the eigenvalues of the Laplacian on S 3 : ( ~ k)= l + 3 l +. (.0) From the background field, one can calculate the leading free energy F (0) = S (0) col ( 0)= N X ~ k log ( ~ k) 7

13 = N 8 log 3 (3) 6, (.) where I have used the Riemann zeta function regularization as in [55]. For the one-loop contribution, following the same procedure which leads to (.5), one has the result F () = (K + ) X ~ k log ( ~ k). (.) In the zeta function regularization, the constant K gives K = X ~ k = X lx nx = ( ) = 0. (.3) l=0 n=0 m= n Therefore, the one-loop contribution to free energy is F () = 8 log 3 (3) 6 (.4) which is exactly the contribution from a single scalar field. Notice that a bi-local field in the U (N) vector model is not symmetric, but Hermitian, the one-loop free energy of U (N) is F () U(N) = K P ~ k log ( ~ k). After regularization, the free energy of U (N) vector model vanishes F () U(N) = 0 as a result of K = 0. This also agrees with [54]. Remember there is another correction to the one-loop free energy from the measure µ by plugging in the background bi-local field F () = (K + ) Tr log 0 = (K + ) X ~ k log ( ~ k). (.5) The total one-loop free energy is therefore F () total = F () + F () =0. (.6) The cancellation of one-loop free energy by the contribution of the measure also occurs in the case of U (N)..3 Interpretation of the results Collective higher spin field theory based on bi-local fields realizes AdS/CFT duality in the bulk through the path integral = d (x, y) µ [ ] e S col[ ] = G = (.7) N where the action is given by S col = S 0 N Tr log. (.8) 8

14 Compared with the original CFT action S 0, one has an extra O(N) term given by the Tr log term in (.8) responsible for the G =/N expansion, and a O N 0 measure term with µ =(det (x, y)) apple (.9) ( K for U (N) apple =. (.30) (K + ) for O (N) These two terms both represent the quantum e ects, they specifically come from the Jacobian arising in the change of variables from N-component scalar fields i (x) to the singlet bi-local fields (x, y) = i (x) i (y) : log J = (N apple) Tr log. (.3) Altogether the action (expandable in /N ) and the measure of lower order define the systematic /N expansion of the theory. But the collective field representation o ers another possibility. One notices the fact the measure term and the additional term contributing to the action have the same functional form. This then allows an alternative splitting for example with the whole log J added to the action = d (x, y) + (N apple)tr log = (G ). (.3) This leads to a formulation without any measure and an e ective coupling constant given by G = N apple. (.33) One can be worried about this scheme considering the fact that this represents an infinite renormalization of the coupling constant. But in the case of O (N) modelswhereapple = K + (and K is infinite), one can include the apple = part into the coupling resulting in = d (x, y) µ 0 [ ] e (N )S col = G 0 = (.34) N and an expansion based on the new coupling constant G 0 = N. (.35) In this case, the measure is µ 0 =(det ) K. Employing a regularization which sets K =0 one has the expansion parameter G 0 =/(N ) and no extra measure. This would be in agreement with the identification suggested in [54]. In general, gravitational theories come with a nonzero measure [58]. For example, the functional measure in (quantized) general relativity was computed in [59, 60] tobe µ = Y h g 7/ (x)g 00 (x) Y i dg (x), (.36) x apple 9

15 where g det g µ. It contributes infinite (4) (0) terms in perturbation theory canceling analogous divergences of Feynman diagrams. In dimensional or zeta function regularization, such terms are set to 0. In Vasiliev theory, one has not yet worked out the measure (evaluating it would require the use of an action). But, the existence of a collective representation for this theory would indicate that there will be an analogous measure. If what one has learned in the collective representation is telling, then in a regularization where such a measure is removed, one could define an e ective coupling constant so that expansion would naturally become G 0 =/(N ) for O(N) theories as compared to G =/N for U(N) duals. I mention however that for nonperturbative studies involving the Hilbert space (and entropy) it might not be appropriate to use a regularization which removes the measure. Such is for example the case of ds/cft []. In any case it is of interest to evaluate the one loop measure of higher spin theories. Another possibility was suggested by Leigh and Petkou [6]. On the field theory side, an explicit symmetry breaking from O(N)! O(N ) can be triggered by adding a singleton deformation. Such deformation, in the bulk, can be absorbed by the higher spin fields with a shift of the parameter N! N +. Therefore, the singleton deformation breaks higher-spin symmetry and generates a /N correction to the free energy. 3 One loop partition functions in thermal AdS 4 I now proceed to the study (and evaluation) of the free energy in the case of another geometry (thermal AdS 4 ). This actually represents a di erent phase of the theory, involving the phase transition described in [57]. In this case, one performs calculations both in the AdS heatkernel version and the bi-local collective version. The purpose is first of all to observe an agreement between the two calculations and also to see that the phenomena put forward in Section persist in the case of a di erent background. This will happen even though the physics of the two phases (as emphasized in [57]) is very di erent. 3. The Heat Kernel method Thermal AdS 4 is defined by periodicity conditions on the Euclidean time variable [0, ]. One expands the metric g around the AdS background which is taken the same (static) solution as the AdS vacuum g = g AdS +. In [6, 63], the partition functions of higher spin theories in odd dimensional AdS spaces are explicitly calculated using the heat kernel method. One can follow exactly the same method in performing the calculations in AdS 4. 0

16 The partition function of massless spin-s field is then " (s) =exp Tr log r + s s ( r + s ) # = = " Y q s+m+ s m= Y m= ( q s+m ) s+ # m(m+) ( q s+m ) m(m+s) (3.) where q = e. The partition function of the massless scalar field is log (0) = = = X m= log det m ( X m + m= 4+M(0) q 3 ± 9 4 +M (0) q m ) 3 qm log q (0) +m (3.) X m= q m (0) m ( q m ) 3 resulting in (0) = Y m= q (0) +m m(m+) (3.3) where (0) is the scaling dimension of the bulk scalar field. For the UV fixed point, which corresponds to (0) =, one has the partition function for the scalar field as (0) = Y m= ( q m ) m(m+). (3.4) Multiplying with all the higher spin contributions, the total one loop partition function of higher spin gravity (which corresponds to the U(N) vector model on the boundary) is = Y (s) = s=0 ( q)( q ) 3 Y m= ( q m ) (m+ ) ( q m+ ) 3(m+ )+4( m+ 3 ). (3.5) Therefore, the associated free energy is F = log = X m= 3 m3 + 3 m log ( q m )= X k= q k +q k k ( q k ) 4 (3.6) which agrees with eq. (0) of [57].

17 Also, for the minimal higher spin theory which includes only the even higher spin fields, the free energy is then F min = log min = = X k= q k k X m= " +q k m (m + ) ( q k ) 4 + +qk ( q k ) log ( q m )+ # X m,s= m (m +4s) log q s+m. (3.7) This result will be seen to agree with the singlet O(N) model case using the collective field method. 3. The Hamiltonian method For completeness, I will describe how the heat kernel evaluations can be equivalently obtained by a Hamiltonian method as described in [64]. The one-particle partition function of a massless field in AdS 4 as a function of the temperature T = and the chemical potential is written as Y (, ) = X (E e j). (3.8) E,j For the representations which are relevant for the UV fixed point, one has e Y (,0) (, = 0) = (e ) 3 (3.9) Y (s+,s) (, = 0) = e( s) (s + ) e + s (e ) 3 (3.0) for s. From the single-particle partition function, one deduces energy spectrum and the degeneracies D (, 0) : E n = n, d n = n (n + ), (n ) (3.) D (s +,s) : E n = n, d n = n s, (n s ) (3.) Therefore, through the formula F = X n d n log e E n (3.3) one can obtain free energies of massless particles in AdS 4 as F (0) = F (s) = X m= X m=s m (m + ) log ( q m ) (3.4) m s log ( q m ) (s ). (3.5)

18 Thus, the total free energy is X X X F = F (s) = F (0) + s=0 s= m=s m s log ( q m )= X k= q k +q k k ( q k ) 4 (3.6) which agrees with (3.6). Also, one can add up only the even spin fields and the scalar field contributions to get " X F min = F (s) = X q k +q k # k ( q k ) 4 + +qk ( q k ) (3.7) which agrees with (3.7). s=0 k= 3.3 The collective field theory approach I will now describe the evaluation of the partition function in the bi-local picture. Since the background is given by the ground state solution it is appropriate to use the Hamiltonian (single-time) representation of the bi-local theory [7]. The full nonlinear collective Hamiltonian for the equal-time bi-local field (and its canonical conjugate) reads H = d~xd~yd~z (~x, ~y) (~y,~z) (~z,~x)+ d~xd~y (~x, ~y) (~y,~x) (~x, ~x) + d~xd~y (~x, ~x) (~x, ~y) (~y,~x)+ d~x (~x, ~x) (~x, ~x) (~x, ~x) + d~x ~x (~x, ~y) ~y=~x + N 8 Tr + V (3.8) where ~x is the Laplacian on S and the counterterms (which are lower orders in /N ) are N V = 4 (K + ) + (K + ) Tr. (3.9) 8 The first five (integral) terms on the RHS of (3.8) comes from a direct rewriting of the original Hamiltonian (of the vector fields) in terms of the bi-local fields (after a repeated use of the chain rule) (see [65] for details). The rest terms in (3.8) (including the interaction term Tr and the counterterm V ) arises from a similarity transformation to make the Hamiltonian Hermitian. This is in the same spirit as the Jacobian present in the action approach, and the counter-term is related to the lower order measure. The collective Hamiltonian (3.8) is well suited to perform a /N expansion after the rescaling! N and! /N. By expanding (~x, ~y) around the background field = 0 + p N, and similarly for the conjugate momenta = p N, one can show that the leading Hamiltonian H (0) of order O (N) is E (0) =H (0) = N h i d x r~x 0 (~x, ~y) ~y=~x + N 8 Tr 0 = N X (l + ) (3.0) 4 l=0 3

19 which is exactly the ground state energy of N free bosons. The one-loop calculation follows similarly as the covariant formulation used in the previous section. The quadratic Hamiltonian of order O N 0 is H () = X ~ k apple ~ k h i ~k, ~ k ~k, ~ + k ~k, ~! k ~ k, ~ k ~k, ~ k (3.) where ~ k =(l, m) and ~ k =(l, m). The frequencies are! ~k, ~ k = l + l + on S, so that the free energy of the singlet sector can be easily calculated as F 0 min =E () + =E () X (l,m ),(l,m ) X n= e n n h log e (l +l +) i + X (l,m) " # +e n +e n ( e n 4 + ) ( e n ) h log e (l+) i (3.) where the first factor (on the first line) are necessary for avoiding double-counting. Furthermore, the one-loop correction to the ground state energy is E () = P ~ k apple ~ k! ~k, ~ k = (K + ) P ~ k ( ~ k) which precisely cancels the O(N 0 ) contribution from the counterterm V : E () = 4 (K +)Tr 0 = (K +) P ~ k ( ~ k). This ensures the vanishing of the total one loop correction to the ground state energy: E () total = E() + E () = 0. However, the one-loop correction to the free energy is non-vanishing and one has " # F min = X e n +e n +e n n ( e n 4 + ) ( e n ) (3.3) n= which agrees with (3.7) after the identification q = e. In a similar way, one can calculate the free energy of singlet sector of U (N) vector theory. In this case, the bi-local field (~x, ~y) is not symmetric hence there will be no potential doublecounting, the final result is F = X (l,m ),(l,m ) h log e (l +l +) i = X n= e n n +e n ( e n ) 4 (3.4) which agrees with (3.6). At high temperature, the free energy scales as F 4 (5) T 4, showing the higher phase of [57]. What one has seen in the present series of calculations is that in this case the free energies do not vanish at one loop. But the ground state energy is indeed much like the free energy of the previous section: one obtains the exact result in the leading evaluation while the one loop contribution cancels with the contribution from the counter-term. The picture regarding the redefinition of the coupling constant in the O(N) case therefore appears in this background too. That is satisfactory as there should not be a change in the identification of the coupling constant just by changing the background. 4

20 4 Hamiltonian Formulation : Light-cone Frame The basis of the AdS/CFT lies in the di erent manifestation of the theory when seen through the Large N expansion. Collective field theory is built to implement this picture to all orders in /N with a collective representation [66] of the Hamiltonian: which is systematically given in powers of /N. H = H CFT = H col ( c, /N ) (4.) H col = NH 0 + H + p N H 3 + N H 4 + (4.) in terms of collective fields. The main property of collective fields is that they are canonical. i.e. [ c, c 0]= c,c 0 (4.3) Here c labels the collective degrees of freedom, with kinematics that depends on the theory. For the case of O(N) vector models, one has the bi-local collective fields (t; ~x, ~x )defined as NX (t; ~x, ~x ) ' i (t, ~x ) ' i (t, ~x ) (4.4) i= In this case, the collective Hamiltonian is given by H col = N Tr ( )+N 8 Tr + N h i d~x r x (~x, ~y) ~y=~x + V [ ]+ V (4.5) Here V [ ] represents the original interaction potential and V represents (known) lower order counter-terms. (A detailed expression can be seen in Section ) The/N series is generated systematically as follows: One first determines the Large N background field 0(~x, ~x ) through minimization of the collective Hamiltonian. Expanding the bi-local field (t, ~x, ~x ) around the background field 0(~x, ~x ) (t; ~x, ~x )= 0 (~x, ~x )+ p N 0 (t; ~x, ~x ) (4.6) (4.5) gives the series of higher vertices interaction vertices: H n =Tr( 0? 0? 0 0?? {z } n? 0 ) (4.7) with a natural star product defined as A?B R d~x A(~x, ~x )B(~x, ~x 3 ) representing a matrix product in bi-local space. Here I assumed the scalar field interaction V to be at most quartic in ', otherwise there is an additional interaction term generated from V. To summarize, the two main properties of the collective representation are: 5

21 I. The representation features /N as a coupling constant, leading to a natural Witten type expansion. II. Collective fields are exactly canonical, through the commutation relations (4.3). This property follows from the definition of the conjugate For the bi-local case one therefore has [ (x,x ), (x 0,x 0 )] = (x x 0 ) (x x 0 )+ (x x 0 ) (x x 0 ) (4.8) for all order in /N. The only nontrivial issue with respect to exact duality is the interpretation of the collective space c. This represents a kinematical problem. When interpreted in physical terms, the collective space leads to extra emerging coordinates (as in [67]) and emerging gravitational and string degrees of freedom ([68]). In some relatively simple cases, this de-coding of collective degrees can be completely done. Such is the case of N-component vector theories with dual Higher Spin fields and AdS d+. For these one has a one-to-one map between bi-local (collective) degrees of freedom and Higher Spin fields in AdS space-time. The simplest form of the Map is found in light-cone quantization where it is given explicitly for AdS 4 /CFT 3 in [7]. I give a short summary of the light-cone case since the construction in any other frame will be related. Denoting a sequence of higher spin fields by H s (x + ; x,x,z), a Fourier transformation with respect to spin s leads to H(x + ; x,x,z, ) = X s=0 : even cos(s ) H s (x + ; x,x,z) (4.9) representing a field on AdS 4 S. It was established by Metsaev that a representation [9] of SO(, 3) with all spins can be built on this space-time. To construct a one-to-one map between bi-local and bulk fields, one first builds a transformation [7] from bi-local momentum space to the momentum space of AdS 4 S. This is given by the following transformations: p + =p + + p+ (4.0) p =p + p (4.) p z =p sp + p + = arctan p sp + s p + p + This transformation induces the Map between fields through: eh p +,p,p z, = dp + dp dp + dp K p +,p,p z, ; p +,p,p +,p p + (4.) (4.3) e p +,p,p +,p (4.4) Here, the conjugate to spin s, is a coordinate for S is treated as a momentum. 6

22 where the kernel K is simply K p +,p,p z, ; p +,p,p +,p ( + cos ) p +sin p = p z ( cos ) p sin p p z p + p + sin p + p + cos (4.5) This is built through the inverses of the above (momentum space) transformations. Consequently, this map between fields is one-to-one and invertible. The coordinate space map can be determined from this kernel through the the chain rule. In particular, z H z dp + dp dp + dp K e p +,p,p +,p (4.6) establishes the following identification between the extra AdS 4 coordinate and the relative bi-local space: q z = (x x ) p + p+ p + + (4.7) p+ By construction, the transformations between coordinates/momenta corresponds to a canonical transformation. This fact has a number of consequences. One can show that the SO(, 3) generators of AdS 4 higher spin fields [9] correspond to those of the bi-local CFT 3 [7], namely: L AdS H(p e +,p,p z, ) = dp + dp dp + dp K L bi-local e (p +,p,p +,p ) (4.8) for any generator of SO(, 3). Furthermore, the canonical commutators of collective fields are seen to imply canonical commutation relations in the constructed AdS 4 space. Canonical quantization of scalar field in light-cone time introduces the notation e' i (p +,x) which is Fourier transformed with respect to x. For p + > 0, e' i (p +,x) and its conjugate e' i ( p +,x)playsa role of creation operator and annihilation operator, respectively. One defines bi-local field to be And, one can represent its conjugate as e (p +,x,p +,x )= p N e' i (p +,x ) e' i (p +,x ) (4.9) e (p +,x,p +,x )= p N e' i ( p +,x ) e' i ( p +,x )= p+ p+ @ e' i (p +,x e' i (p +,x ) (4.0) Using chain rule, one can easily express e (p +,x,p +,x ) in terms of e (p +,x,p +,x ) and e (p + I,x I,p + J,x J ). e (p +,x,p +,x )=p e (; ) + p+ p+ N e (J, e (,I) I,J (4.) 7

23 up to contact terms and where I used compact notation, I =(p + I,x I). Note that the contact terms in (4.) appear because one has to take independent collective fields e into account when using chain rule. The algebraic structure and /N expansion of these O(N) bi-local operators was studied in []. I also present this algebraic structure in the U(N) example in B. The derivative defines a canonical conjugate of the bi-local field e (p +,p,p +,p ), and one denotes it by e (p +,p,p +,p ) with the canonical (bi-local) commutation relation: h e(p +,p,p +,p ), (k e +,k,k )i +,k = ip+ p+ (p + k + ) (p k ) (p + k + ) (p k ) +((k +,k )! (k +,k )) (4.) It follows that the Higher Spin fields H(p e +,p,p z, ) and W(p f +,p,p z, ) constructed from e and e through the kernel K(p +,p,p z, ; p +,p,p +,p ): eh(p +,p,p z, ) = dp + dp dp + dp K e (4.3) fw(p +,p,p z, ) = dp + dp dp + dp K e (4.4) obey canonical commutation relations: h eh(p +,p,p z, ), f W(k +,k,k z, )i = ip + (p + k + ) (p k) (p z k z ) ( ) (4.5) Continuing with the description of the construction, I emphasize the feature that as constructed it is o -shell. Equations of motion and time evolution of the dual theory will be dictated by those of bi-local collective fields e. They are given to all orders in /N by the Hamiltonian in (6.9). In leading expansion, the (linearized) equations for the bi-local field + + p p + + p e (x + p + ; p +,p,p +,p ) = 0 (4.6) coming from the leading term in the /N expansion of the Hamiltonian (6.9). This gives the time evolution: e (x + ; p +,p,p +,p )=e ix+ p p + p p + e (p +,p,p +,p ) (4.7) and after the momentum space map, the following leading time evolution for the Higher Spin field e H(x + ; p +,p,p z, ): eh(x + ; p +,p,p z, ) =e ix+ p +(p z ) p + e H(p +,p,p z, ) (4.8) and the associated linearized equation of motion: p + + p +(p z ) eh(x + ; p +,p,p z, ) = 0 (4.9) 8

24 agree with the Higher Spin equations of Metsaev [9]. Let us now come to another relevant property of this bulk construction, which is the behavior of the constructed AdS fields near the z = 0 boundary. For this, one can proceed to be on-shell and in linearized approximation, and consider the map for the spin s field in particular: H s (x + ; x,x,z)= d H(x + ; x,x,z, ) cos(s ) = d 4 p (p + p + p +(p z ) )e ixµ p µ dp + dp dp + dp J (p +,p+ )! sp (p + + p+ p + + ) (p + p p) p p z where J z, +,p,p +,p ) s! s + = p + p + p sp + p + P, p + p + e s (p + p + +,p,p + p+,p ) (4.30) + p + is Jacobian of the transformation in (4.0) (4.3) and P, s (x) is Jacobi polynomial. Also note that when put on-shell through linearized approximation the Map produces the low spin formulae of [40, 69 7]. Using the symmetry of the collective bi-local field e (p +,p,p +,p )= e (p +,p,p +,p ) which corresponds to a symmetry of the higher spin field H(p e +,p,p z, ) = H(p e +,p, p z, ) by the kernel K and performing a change of variables from p z to p = p +(p z ) (coming p + from the delta function), one obtains, after short calculation, the relation: H s (x +,x,x,z)= dp dp + dp e ix+ p +ix p + +ixp J z p p + p p p + p p >0 s z p p + p p s! s + (p+ ) s e O s (p,p +,p) (4.3) where O e s (p,p +,p) is recognized to represent Fourier modes of spin-s primary operators O (x) of the O(N) model[73, 74] which in terms of the bi-local collective read: eo s (p,p +,p)= dp + dp dp + dp J (p +,p+ ) (p + + p+ p + ) p p p (p + p p) p + + p+ s P, s p + p + p + p + e (p + p + +,p + p+,p,p ) (4.3) Now one registers the following boundary behavior of H s (x +,x,x,z): H s (x +,x,x,z) z!0 s!! s + dp dp + dp e ix+ p +ix p + +ixp Os e (x,x +,x) p + p p >0 9

25 The fact that the collective field map to AdS fields leads, at z = 0, to the conserved primary operators of the CFT is a clear verification of the above bulk construction. This represents a significant consistency check. 5 Hamiltonian Formulation : Time-like Frame One can follow the basics of the light-cone construction and give an analogous construction in any other frame. Here I give the details of quantization performed in the time-like frame. Some parts of this construction have appeared before 3, while some parallel the light-cone case. Nevertheless, the full construction is su ciently nontrivial so that I finds it worthwhile to present it. Regarding the change of (phase space) coordinates from bi-local to AdS, first it is natural to identify the center of momentum of the bi-local space with the momentum of AdS 4 space. ~p =~p + ~p (5.) p 0 = ~p + ~p (5.) Second, based on light-cone case, one can make use of the same on-shell condition adopted for time-like kinematics. This gives the following identification of p z with bi-local momenta: p z = ± p (p 0 ) ~p = p ' ' ~p ~p sin (5.3) where ~p =( ~p cos ', ~p sin ' ), ~p =( ~p cos ', ~p sin ' ) (5.4) There is an ambiguity in choosing the sign of p z.onedeterminedp z in a way that the sign of p z is changed when one exchanges ~p and ~p. Also, note that p z = p ~p ~p ~p ~p. Next, for further identification it is useful to use the second-order Casimir which, for the unitary irreducible representations D(E 0,s) of SO(, 3), equals: C SO(,3) = E 0 (E 0 3) + s(s + ) (5.5) where E 0 is the lowest energy and s is the spin. The massless representations are characterized by E 0 = s +. At semiclassical level, ignoring the ordering term (e.g. s + s) the secondorder Casimir for the massless representation is then C SO(,3) =s = (p ) (5.6) where p plays the same role as in the light-cone case, namely labeling the internal spin degree of freedom. Comparing this the second-order Casimir with the one expressed in terms of bi-local SO(, 3) generators, one can express p in terms of the bi-local variables. 4 p = p ~p ~p cos ' + ' (x x )+ p ~p ~p sin ' + ' (x x ) (5.7) 3 In particular, [] where higher order calculations were done in the time-like collective method. 4 Also, I chose the sign of each term to give correct Poisson brackets. 0

26 Finally, consider the following ansatz for conjugate to p. ~p ~p = arctan ( ~p ~p )p z (5.8) where ~p ~p p p p p. One can confirm that and p satisfy canonical Poisson bracket n o p, = (5.9) and the Poisson brackets with others vanish. One can also obtain this identification of from a polarization vector and a primary operator. (See 5.3) In sum, the identification between the bi-local (momentum) space and AdS 4 S (momentum) space is ~p =~p + ~p (5.0) p z = p ' ' ~p ~p sin (5.) ~p ~p = arctan ( ~p ~p )p z (5.) One can consider this identification as a point transformation in momentum space. 5 Jacobian of the transformation is J (~p, ~p )= ~p + ~p The (5.3) Moreover, inverting this transformation one can express bi-local momentum in terms of momenta of AdS 4 S space. ~p a = ~ a(~p, p z, ) (a =, ) (5.4) where ~ a(~p, p z, ) is given in C. Using the momentum transformation, one can construct a higher spin field H(~p, e p z, ) from a bi-local field e (~p, ~p ). eh(~p, p z, ) = d~p d~p K(~p, p z, ; ~p, ~p ) e (~p, ~p ) (5.5) where a kernel K(~p, p z, ; ~p, ~p )isdefinedtobe K(~p, p z, ; ~p, ~p )=J(~p, ~p ) () (~p + ~p ~p) p ' ' ~p ~p sin ~p ~p arctan ( ~p ~p )p z p z = () (~p ~ (~p, p z, )) () (~p ~ (~p, p z, )) (5.6) 5 Again, though is a coordinate for S, I treat it like momentum.

27 One can also invert the map (5.5). e (~p, ~p )= d~pdp z d Q(~p, ~p ; ~p, p z, ) H(~p, e p z, ) (5.7) where the inverse kernel Q(~p, ~p ; ~p, p z, ) is Q(~p, ~p ; ~p, p z, ) = () (~p + ~p ~p) p ' ' ~p ~p sin ~p ~p arctan ( ~p ~p )p z p z (5.8) The most important property of this map is again that it preserves the canonical commutation relations but now with AdS e in the momentum space satis- Consider the bi-local field e and its conjugate e = fying the canonical commutation relation. h e(~p, ~p ), ( e ~ k, ~ k )i = i ~p ~p () (~p ~ k ) () (~p ~ k ) (5.9) where (~p, ~p ) and ( ~ k, ~ k ) are independent collective degrees of freedom. e.g. p >p or p = p if p = p, and similar for (~ k, ~ k ). Through the kernel K(~p, p z, ; ~p, ~p ), one can construct e H(~p, p z, ) and f W(~p, p z, ) from e and e,respectively. eh(~p, p z, ) = fw(~p, p z, ) = d~p d~p K(~p, p z, ; ~p, ~p ) e (~p, ~p ) (5.0) d~p d~p K(~p, p z, ; ~p, ~p ) e (~p, ~p ) (5.) Then, e H(~p, p z, ) and f W(~p, p z, ) satisfy canonical commutation relation. i.e. h eh(~p, p z, ), W( f ~ k, k z, )i = i p ~p +(p z ) () (~p ~ k) (p z k z ) ( ) (5.) For s = 0, this is the same commutation relation of scalar field in AdS 4 in [69, 75] Using the kernel, one can construct the transformation for the coordinates. The kernel K(~p, p z, ; ~p, ~p ) induces the identification of bi-local coordinates (~x, ~x ) with AdS 4 S space according to the chain rule. For example, x H(~p, e p z, ) e H(~p, p z, ) d~p d~p K(~p, p z, ; ~p, ~x e (~p, ~p ) (5.3)

28 As a result, the map for AdS 4 S coordinates in terms of the bi-local variables is given by x = ~p x + ~p x p (p z ) + ~p p p z p ~p p (p z ) + ~p (5.4) x = ~p x + ~p x p p z p p + (p z ) + ~p ~p p (5.5) (p z ) + ~p z = (~x ~x ) ~p ~p (~x ~x ) ~p ~p p z ( ~p + ~p ) (5.6) p = p ~p ~p cos ' + ' (x x ) + p ~p ~p sin ' + ' (x x ) (5.7) By construction, this transformation, (5.0) (5.) and (5.4) (5.7), is canonical. i.e. p i,x j = ij (i, j =, ) {p z,z} = n o p, = (5.8) and others vanish. The kernel K(~p, p z, ; ~p, ~p ) also maps the bi-local SO(, 3) generators L bi-local to SO(, 3) generators L ads acting on e H(~p, p z, ). d~p d~p K(~p, p z, ; ~p, ~p )L bi-local e (~p, ~p )=L ads e H(~p, p z, ) (5.9) Classically, (5.0) (5.) and (5.4) (5.7) corresponds to a canonical transformation from the bi-local space to AdS 4 S. One can also obtain L ads from L bi-local by this canonical transformation. P µ ads =P µ bi-local (5.30) J µ µ ads =J bi-local (5.3) D ads =D bi-local (5.3) K µ ads =Kµ bi-local (5.33) 3

29 Then, setting t = 0 for simplicity, L ads is given by P 0 ads =p ~p +(p z ) (5.34) P ads =p (5.35) P ads =p (5.36) J 0 ads = x P 0 p p z p ~p (5.37) J ads =x p x p (5.38) J 0 ads =x P 0 p p z p ~p (5.39) D ads =x p + x p + zp z (5.40) Kads 0 = (~x + z )P 0 p z p ~p J P 0 (p ) ~p (5.4) Kads = (~x + z )p + x D + zp P 0 p ~p + p (p ) ~p (5.4) K ads = (~x + z )p + x D zp P 0 p ~p + p (p ) ~p (5.43) There is another way to obtain the map between bi-local momentum space and AdS 4 S momentum space. In [7], the map was found by comparing generators of AdS 4 and CFT 3 in light-cone gauge. This is the inverse procedure of the derivation in this section. Unfortunately, one does not have a representation of SO (, 3) for higher spin field in timelike gauge. However, there is alternative way to obtain the generators. In [9], Metsaev constructed a realization of SO (, 3) for spin-s current in CFT 3. One can show that this agrees with a realization of SO (, 3) for higher spin field in light-cone gauge after manipulating the generators. For the case of time-like gauge, one can repeat the same procedure to get SO(, 3) generators in time-like gauge. (see D, (D.9) (D.38) for t = 0.) And, one can accept this result as a representation of SO (, 3) for higher spin field in time-like gauge. Note that these generators in (D.9) (D.38) are identical to L ads in (5.34) (5.43). Then, one can identify L ads with L bi-local to obtain the map. i.e. see (5.30) (5.33). First of all, (5.30) gives expression for ~p and p z in terms of ~p and ~p in (5.0) and (5.). Moreover, comparing Casimir of SO (, 3) of both representation, one can get p in (5.7). Solving two equations, Jads 0 = J bi-local 0 and J 0 ads = J bi-local 0, for x and x, respectively, one has (5.4) and (5.5). Finally, the identification D ads = D bi-local,(5.4) and (5.5) give the map for z in (5.6). Again, it is di cult to derive the map for by this identification because does not appear in the generators. Instead, using a primary operator and polarization vector, one can find the map for. (see5.3). This result perfectly agrees with the previous result, (5.0) (5.) and (5.4) (5.7). 4