1/N and Loop Corrections in Higher Spin AdS_4/CFT_3 Duality
PPrepared for submission to JHEP
BROWN-HET-1653 /N and Loop Corrections in Higher Spin AdS /CFT Duality
Antal Jevicki, a Kewang Jin, b Junggi Yoon a a Department of Physics, Brown University,Providence, RI 02912, USA b Department of Physics, University of Illinois,Urbana-Champaign, IL 61801, USA
E-mail: antal [email protected] , [email protected] , jung-gi [email protected] Abstract:
We consider the question of loop corrections (i.e. 1 /N ) in the vector model/higherspin duality following the recent work of Giombi and Klebanov [1]. The purpose of this paperis to gain further more precise comparison between the two sides of the duality. For CFTsgiven by 3d O ( N ) or U ( N ) vector models we evaluate the leading and one loop partitionfunctions in a variety of geometries. Our calculations are performed in the scheme of collectivefield theory which was seen in earlier studies to represent a bulk description of Vasiliev higherspin theory. The calculations presented provide data for comparison of small fluctuationdeterminants giving further evidence for the one-to-one bulk identification between the bi-local and the AdS picture. They also offer insight into the identification of coupling constants G and 1 /N of the two descriptions for models based on O ( N ) symmetry. a r X i v : . [ h e p - t h ] J a n ontents S × R S Recently, vector-like models with O ( N ) and U ( N ) symmetries at their critical points wereseen to exhibit duality [2–4] with higher spin gravitational theories of Vasiliev [5–7]. Typicallyin 3d vector field theory, there are two conformally invariant fixed points, the free UV fixedpoint and the interacting IR fixed point. The higher spin duals to these two fixed points aregiven by the same Vasiliev theory but with different boundary conditions in the quantizationof the bulk scalar field. This Vector Model/Higher Spin correspondence was also extended tothe supersymmetric case [8, 9], Chern-Simons theories [10, 11] and de Sitter space [12, 13].Furthermore one also has the very rich and nontrivial lower dimensional dualities involving 2dMinimal Model CFTs and 3d Higher Spin Gravities [14–16]. All these dualities have receiveddefinite support based on evaluation of three-point correlation functions, finite temperaturepartition functions and study higher conservation laws.In a series of papers [17–19] an explicit operator construction of the (dual) Higher SpinAdS theory in terms of collective fields was developed. This approach provides a frameworkfor a one-to-one reconstruction of AdS spacetime, higher spin fields in the bulk and their 1 /N interactions. Higher order calculations that were performed concerned the 1-loop correctionto the free energy [17], correlation functions [20], and an investigation of the (non)trivialityof the theory [21] based on free fields.The purpose of this paper is to study further the question of loop corrections (i.e. 1 /N ) inthe higher spin duality. We follow up the earlier work of [17] and the recent work of Giombi– 1 –nd Klebanov [1]. These calculations concern the evaluation of partition functions at oneloop in the collective and also in the AdS version of the theory. In both cases, the one-loopcorrections follow from the quadratic LaplaciansTr log (cid:3) bi-local and Tr log (cid:3) hs (cid:3) gh . (1.1)In the light-cone gauge the Laplacians can be shown to be equal (cid:3) bi-local = 2 ∂ + ∂ − − (cid:18) p p +1 + p p +2 (cid:19) (cid:0) p +1 + p +2 (cid:1) = 2 ∂ + ∂ − − (cid:0) ∂ x + ∂ z (cid:1) = ∇ hs (1.2)as a consequence of the spacetime mapping established in [18]. Due to gauge invariance onecould then expect identical results for their one-loop contributions in general. However, sinceone considers backgrounds which do not always easily fit into the light-cone gauge, explicitcalculations are nevertheless worthwhile. They also serve as the purpose for understandingmore completely the nature of loop corrections in higher spin duality. In particular in theheat-kernel AdS calculation the suggestion was made in [1] that the identification of thegravitational coupling constant should be taken as G = 1 / ( N −
1) for the dualities basedon the O ( N ) symmetry group (no such change was found for the U ( N ) case). Our resultsshed some light on this identification. First of all collective theory shows that in additionto the determinant there is one further contribution of O (1) associated with the measureappearing in the functional integration. The measure does provide the needed cancellationat one loop (as noticed originally in [17]) allowing the standard identification of G = 1 /N .However collective field theory also indicates a freedom of a finite (re)normalization of G into1 / ( N −
1) as we discuss in the text. These two expansion schemes are compatible, as one canre-expand results of one into another.The content of this paper goes as follows. In section 2 we consider first the finite temper-ature case of the CFT reviewing an earlier work of [17]. This example already contains someof the basic effects that will be observable in the rest of the calculations. We then presentdetails of the bi-local calculation in the case of S (the example of [1, 24]) and point out therole of the measure. In section 3 we proceed to the other phase of the theory discussing theevaluation of the partition function in thermal AdS both by the heat-kernel method and inthe bi-local collective field framework. Some conclusions are given in section 4. The collective theory describes the large N dynamics of bi-local collective fields. These fieldshave the property that they close under the Schwinger-Dyson equations. They represent amore general set than the conformal currents and contain an additional dimension. As suchthey are natural candidates for representing the bulk AdS theory. This is supported by thefact that an effective, collective field action with the property that the associated functional A very similar identification of AdS space in light-cone QCD was found in [22, 23]. – 2 –ntegral exactly evaluates the O ( N ) singlet partition function and correlation functions ofbi-local operators. The diagramatics accomplished by this reformulation is that of Wittendiagrams.The exact partition function of the free vector model with N components in terms of thebi-local field Φ ( x, y ) is given by [17] Z = (cid:90) D Φ ( x, y ) J ( x, y ) e − S [Φ( x,y )] = (cid:90) D Φ ( x, y ) µ e − S col [Φ( x,y )] (2.1)where J ( x, y ) is the Jacobian (generated from the change of variables from the fundamentalvector fields to the bi-local fields), and the collective action reads S col = N (cid:90) d x (cid:16) − ∆ x Φ ( x, y ) | x = y (cid:17) − N Tr log Φ . (2.2)The (integration) measure µ in (2.1) is computed to be µ = (det Φ) − κ (2.3)where the power κ depends on the underlying symmetry of the vector fields. For the O ( N )case, the bi-local field Φ ( x, y ) = N (cid:126)φ ( x ) · (cid:126)φ ( y ) is symmetric and κ = ( K + 1) with K = (cid:80) k U ( N ) case, the bi-local field Φ ( x, y ) = N (cid:126)φ ∗ ( x ) · (cid:126)φ ( y ) is Hermitian and we have κ = K . The details of this derivation can be found inthe appendix A. In the Riemann zeta-function regularization (employed in [1] and also here),we have set K = 0 so that the measure is simplified to be µ = (cid:40) U ( N )(det Φ) − / for O ( N ) . (2.4)We mention that the action on this representation scales with N and the interactionsgenerated are consequently given in powers of 1 /N and the measure would contribute in thesubleading orders. It is also relevant to point out at the outset that the measure in thiscollective representation leads to a contribution of the same form as the Tr log term in theaction (2.2) (which sets the coupling constant). Consequently one can equivalently includethe measure term into the action obtaining an effective coupling constant. We will return tothis issue of interpretation in section 2.3 after presenting the one loop calculations. S × R We start by reviewing first the one-loop calculation performed in [17] for the S × R partitionfunction. This case already demonstrates some of the features of the one loop determinantthat will be general and central to the issues raised in the Introduction.One develops the expansion as usual by shifting the background bi-local fieldΦ ( x , x ) = Φ ( x , x ) + 1 √ N η ( x , x ) (2.5)– 3 –here Φ ( x , x ) represents the stationary point of the collective action (2.2). In the momen-tum space representation one has the Fourier transformed field ˜Φ ( k , k ) with the momenta k , = ( ν n , (cid:126)k , ) where the Matsubara frequency is given by ν n = πnβ and β as the inverse oftemperature T .The zeroth-order collective action is now given by S (0)col = N (cid:88) k k ˜Φ ( k, − k ) − N Tr log ˜Φ . (2.6)Translation invariance implies ˜Φ ( k , k ) = ξ ( k ) δ k , − k , and we get S (0)col = N (cid:88) k k ξ ( k ) − N (cid:88) k log [ ξ ( k )] . (2.7)By the saddle point method one determines ξ ( k ) = k , and the background field isΦ ( x, y ) = 1(2 π ) β (cid:88) n (cid:90) d (cid:126)k (cid:18) (cid:126)k + (cid:16) πnβ (cid:17) (cid:19) e ik · ( x − y ) (2.8)which is nothing but the free two point function (cid:104) φ i ( x ) φ i ( y ) (cid:105) of the bi-local operators. Evalu-ating the action at the background value produces the leading contribution to the free energy F (0) = S (0)col = N (cid:88) n (cid:88) (cid:126)k log (cid:34) (cid:126)k + (cid:18) πnβ (cid:19) (cid:35) (2.9)which is precisely the free energy of N free bosons F (0) = N Tr log ∂ . At high temperature,the free energy scales as F (0) ∼ − N ζ (3) T , producing the lower phase of [25].To evaluate the 1-loop contribution, one expands the collective action S col to the quadraticorder in the fluctuations η : S (2)col = 14 Tr (cid:0) η Φ − η Φ − (cid:1) ≡ Tr ( η (cid:3) η )= 2 (cid:88) k >k k k η k , − k η k , − k + (cid:88) k (cid:0) k (cid:1) η k, − k η k, − k . (2.10)Then the one-loop free energy comes as the determinant of the generalized (bi-local) Lapla-cian. Because of the product form the determinant factorizes and one obtains F (1) = 12 Tr log ( (cid:3) ) = (cid:88) k >k
12 log (cid:0) k k (cid:1) + (cid:88) k log( k )= 12 ( K + 1) (cid:88) k log( k ) (2.11)with a surprising finding that the bi-local determinant produces the local field contribution(with a factor K + 1). This pre-factor is most significant as it is associated with the counting– 4 –f 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 = 1 single field expression. This is aprototype of the result that was also observed in [1], namely the evaluation of the AdS higherspin determinant in the heat-kernel method using the zeta-function regularization gave the N = 1 CFT result.The collective representation however contains one other contribution of order O (cid:0) N (cid:1) .It comes from the measure µ evaluated at the stationary point∆ F (1) = 12 ( K + 1) Tr log Φ = −
12 ( K + 1) (cid:88) k log (cid:0) k (cid:1) . (2.12)Thus the total one-loop correction to the free energy is found to be F (1)total = F (1) + ∆ F (1) = 0 . (2.13)This complete cancellation between the determinant and the measure contribution thereforeassures the required result 0.To recapitulate, the one loop determinant of fluctuations produces an answer identical tothat of N free scalars in d = 3 but with N replaced by K + 1. If K (which is infinite) is set to0 by regularization the result then corresponds to N = 1, i.e. to that of a single scalar field.This is what was also found in [1] 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. Inparticular the appearance of K + 1 in O ( N ) theories (and K in U ( N ) theories) is associatedwith the fact that the fields can be encoded into a symmetric matrix appearing naturallyin 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 (1)total = 0 at one loop. S We now consider the partition function on S , the example that was considered in [1]. Onefollows the same procedure described in detail as in the previous section, the only differencebeing the explicit expressions for the eigenfunctions and eigenvalues.Using spherical harmonics of S , the Fourier transformation of the bi-local field isΦ ( x , x ) = (cid:88) (cid:126)k ,(cid:126)k Φ (cid:126)k ,(cid:126)k Y (cid:126)k ( x ) Y (cid:126)k ( x ) (2.14)where (cid:126)k denotes a full set of quantum numbers (cid:126)k ≡ ( l, n, m ) and l = 0 , , , · · · ,n = 0 , , , · · · , lm = − n, − ( n − , · · · , n − , n . – 5 –enoting the conjugate label of (cid:126)k as (cid:126)k ∗ ≡ ( l, n, − m ), the classical background field is nowΦ ( x , x ) = (cid:88) (cid:126)k ( − m λ ( (cid:126)k ) Y (cid:126)k ( x ) Y (cid:126)k ∗ ( x ) (2.15)where λ ( (cid:126)k ) are the eigenvalues of the Laplacian on S : λ ( (cid:126)k ) = (cid:18) l + 32 (cid:19) (cid:18) l + 12 (cid:19) . (2.16)From the background field, one can calculate the leading free energy F (0) = S (0)col (Φ ) = N (cid:88) (cid:126)k log λ ( (cid:126)k ) = N (cid:18)
18 log 2 − ζ (3)16 π (cid:19) , (2.17)where we have used the Riemann zeta function regularization as in [24].For the one-loop contribution, following the same procedure which leads to (2.11), wehave the result F (1) = 12 ( K + 1) (cid:88) (cid:126)k log λ ( (cid:126)k ) . (2.18)In the zeta function regularization, the constant K gives K = (cid:88) (cid:126)k ∞ (cid:88) l =0 l (cid:88) n =0 n (cid:88) m = − n ζ ( −
2) = 0 . (2.19)Therefore, the one-loop contribution to free energy is F (1) = 18 log 2 − ζ (3)16 π (2.20)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 (1) U ( N ) = K (cid:80) (cid:126)k log λ ( (cid:126)k ). After regularization, the free energy of U ( N ) vector model vanishes F (1) U ( N ) = 0 as a result of K = 0. This also agrees with [1].Remember there is another correction to the one-loop free energy from the measure µ byplugging in the background bi-local field∆ F (1) = 12 ( K + 1) Tr log Φ = −
12 ( K + 1) (cid:88) (cid:126)k log λ ( (cid:126)k ) . (2.21)The total one-loop free energy is therefore F (1)total = F (1) + ∆ F (1) = 0 . (2.22)The cancellation of one-loop free energy by the contribution of the measure also occurs in thecase of U ( N ). – 6 – .3 Interpretation of the results Collective higher spin field theory based on bi-local fields realizes AdS/CFT duality in thebulk through the path integral Z = (cid:90) d Φ ( x, y ) µ [Φ] e − S col [Φ] = Z (cid:18) G = 1 N (cid:19) (2.23)where the action is given by S col = S − N . (2.24)Compared with the original CFT action S , we have an extra O ( N ) term given by the Tr logterm in (2.24) responsible for the G = 1 /N expansion, and a O (cid:0) N (cid:1) measure term µ = (det Φ ( x, y )) − κ (2.25)with κ = (cid:40) K for U ( N ) ( K + 1) for O ( N ) . (2.26)These two terms both represent the quantum effects, they specifically come from the Jacobianarising in the change of variables from N -component scalar fields φ i ( x ) to the singlet bi-localfields Φ( x, y ) = φ i ( x ) φ i ( y ) : log J = 12 ( N − κ ) Tr log Φ . (2.27)Altogether the action (expandable in 1 /N ) and the measure of lower order define the system-atic 1 /N expansion of the theory.But the collective field representation offers another possibility. One notices the fact themeasure term and the additional term contributing to the action have the same functionalform. This then allows an alternative splitting for example with the whole log J added to theaction Z = (cid:90) d Φ ( x, y ) e ∂ Φ+ ( N − κ )Tr log Φ = Z ( G ∗ ) . (2.28)This leads to a formulation without any measure and an effective coupling constant given by G ∗ = 1 N − κ . (2.29)One can be worried about this scheme considering the fact that this represents an infiniterenormalization of the coupling constant. But in the case of O ( N ) models where 2 κ = K + 1(and K is infinite), we can include the 2 κ = 1 part into the coupling resulting in Z = (cid:90) d Φ( x, y ) µ (cid:48) [Φ] e − ( N − S col = Z (cid:18) G (cid:48) = 1 N − (cid:19) (2.30)– 7 –nd an expansion based on the new coupling constant G (cid:48) = 1 N − . (2.31)In this case, the measure is µ (cid:48) = (det Φ) − K . Employing a regularization which sets K = 0we have the expansion parameter G (cid:48) = 1 / ( N −
1) and no extra measure. This would be inagreement with the identification suggested in [1].In general, gravitational theories come with a nonzero measure [26]. For example, thefunctional measure in (quantized) general relativity was computed in [27, 28] to be µ = (cid:89) x (cid:104) g / ( x ) g ( x ) (cid:89) σ ≤ λ dg σλ ( x ) (cid:105) , (2.32)where g ≡ det g µν . It contributes infinite δ (4) (0) terms in perturbation theory cancelinganalogous 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 requirethe use of an action). But, the existence of a collective representation for this theory wouldindicate that there will be an analogous measure. If what we have learned in the collectiverepresentation is telling, then in a regularization where such a measure is removed, one coulddefine an effective coupling constant so that expansion would naturally become G (cid:48) = 1 / ( N − O ( N ) theories as compared to G = 1 /N for U ( N ) duals. We mention however that fornon-perturbative studies involving the Hilbert space (and entropy) it might not be appropriateto use a regularization which removes the measure. Such is for example the case of dS/CFT[13]. 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 [29]. On the field theory side, anexplicit symmetry breaking from O ( N ) → O ( N −
1) can be triggered by adding a singletondeformation. Such deformation, in the bulk, can be absorbed by the higher spin fields with ashift of the parameter N → N + 1. Therefore, the singleton deformation breaks higher-spinsymmetry and generates a 1 /N correction to the free energy. We now proceed to the study (and evaluation) of the free energy in the case of anothergeometry (thermal AdS ). This actually represents a different phase of the theory, involvingthe phase transition described in [25]. In this case we perform calculations both in the AdSheat-kernel version and the bi-local collective version. The purpose is first of all to observean agreement between the two calculations and also to see that the phenomena put forwardin section 2 persist in the case of a different background. This will happen even though thephysics of the two phases (as emphasized in [25]) is very different.– 8 – .1 The Heat Kernel method Thermal AdS is defined by periodicity conditions on the Euclidean time variable τ ∈ [0 , β ].One expands the metric g around the AdS background which is taken the same (static)solution as the AdS vacuum g = g AdS + η . In [30, 31], the partition functions of higherspin theories in odd dimensional AdS spaces are explicitly calculated using the heat kernelmethod. One can follow exactly the same method in performing the calculations in AdS .The partition function of massless spin- s field is then Z ( s ) = exp (cid:34) −
12 Tr log (cid:0) −∇ + s − s − (cid:1) ( −∇ + s − (cid:35) = ∞ (cid:89) m =1 (cid:34) (cid:0) − q s + m +1 (cid:1) s − (1 − q s + m ) s +1 (cid:35) m ( m +1)2 = ∞ (cid:89) m =1 − q s + m ) m ( m +2 s ) . (3.1)The partition function of the massless scalar field islog Z (0) = −
12 log det (cid:16) − (cid:52) + M (cid:17) = ∞ (cid:88) m =1 m (1 − q m ) q m (cid:16) ± (cid:113) + M (cid:17) ≡ ∞ (cid:88) m =1 q m ∆ (0) m (1 − q m ) = ∞ (cid:88) m =1 (cid:18) m + 12 (cid:19) log 11 − q ∆ (0) + m − (3.2)resulting in Z (0) = ∞ (cid:89) m =1 (cid:0) − q ∆ (0) + m − (cid:1) m ( m +1)2 (3.3)where ∆ (0) is the scaling dimension of the bulk scalar field.For the UV fixed point, which corresponds to ∆ (0) = 1, we have the partition functionfor the scalar field as Z (0) = ∞ (cid:89) m =1 − q m ) m ( m +1)2 . (3.4)Multiplying with all the higher spin contributions, the total one loop partition function ofhigher spin gravity (which corresponds to the U ( N ) vector model on the boundary) is Z = ∞ (cid:89) s =0 Z ( s ) = 1(1 − q ) (1 − q ) ∞ (cid:89) m =2 − q m )( m +12 ) (1 − q m +1 ) ( m +12 ) +4 ( m +13 ) . (3.5)Therefore, the associated free energy is F = − log Z = ∞ (cid:88) m =1 (cid:18) m + 13 m (cid:19) log (1 − q m ) = − ∞ (cid:88) k =1 k q k (cid:0) q k (cid:1) (1 − q k ) (3.6)which agrees with eq. (10) of [25]. – 9 –lso, for the minimal higher spin theory which includes only the even higher spin fields,the free energy is then F min = − log Z min = ∞ (cid:88) m =1 m ( m + 1)2 log (1 − q m ) + ∞ (cid:88) m,s =1 m ( m + 4 s ) log (cid:0) − q s + m (cid:1) = ∞ (cid:88) m =1 m ( m + 1)2 log (1 − q m ) + ∞ (cid:88) m =3 [ m − ] (cid:88) s =1 (cid:0) m − s (cid:1) log (1 − q m )= − ∞ (cid:88) k =1 q k k (cid:0) q k + 4 q k + q k + q k (cid:1) ( q k + 1) ( q k − = − ∞ (cid:88) k =1 q k k (cid:34) (cid:0) q k (cid:1) (1 − q k ) + 1 + q k (1 − q k ) (cid:35) . (3.7)This result will be seen to agree with the singlet O ( N ) model case using the collective fieldmethod. For completeness, let us describe how the heat kernel evaluations can be equivalently obtainedby a Hamiltonian method as described in [32]. The one-particle partition function of amassless field in AdS as a function of the temperature T = β − and the chemical potentialΩ is written as Y ( β, Ω) = (cid:88)
E,j e − ( βE + αj ) . (3.8)For the representations which are relevant for the UV fixed point, we have Y (1 , ( β, Ω = 0) = e β ( e β − (3.9) Y ( s +1 ,s ) ( β, Ω = 0) = e (1 − s ) β (cid:2) (2 s + 1) e β + 1 − s (cid:3) ( e β − ( s ≥ . (3.10)From the single-particle partition function, one deduces energy spectrum and the degeneracies D (1 ,
0) : E n = n, d n = 12 n ( n + 1) , ( n ≥
1) (3.11) D ( s + 1 , s ) : E n = n, d n = n − s , ( n ≥ s ≥ . (3.12)Therefore, through the formula F = (cid:88) n d n log (cid:16) − e − βE n (cid:17) (3.13)one can obtain free energies of massless particles in AdS as F (0) = ∞ (cid:88) m =1 m ( m + 1)2 log (1 − q m ) (3.14) F ( s ) = ∞ (cid:88) m = s (cid:0) m − s (cid:1) log (1 − q m ) ( s ≥
1) (3.15)– 10 –here q = e − β . Thus, the total free energy is F = ∞ (cid:88) s =0 F ( s ) = F (0) + ∞ (cid:88) s =1 ∞ (cid:88) m = s (cid:0) m − s (cid:1) log (1 − q m )= F (0) + ∞ (cid:88) m =1 m (cid:88) s =1 (cid:0) m − s (cid:1) log (1 − q m ) = ∞ (cid:88) m =1 m (cid:0) m + 1 (cid:1) log (1 − q m )= − ∞ (cid:88) m =1 ∞ (cid:88) k =1 m (cid:0) m + 1 (cid:1) q mk k = − ∞ (cid:88) k =1 k q k (cid:0) q k (cid:1) (1 − q k ) (3.16)which agrees with (3.6). Also, one can add up only the even spin fields and the scalar fieldcontributions to get F min = ∞ (cid:88) s =0 F (2 s ) = − ∞ (cid:88) k =1 q k k (cid:34) (cid:0) q k (cid:1) (1 − q k ) + 1 + q k (1 − q k ) (cid:35) (3.17)which agrees with (3.7). We will now describe the evaluation of the partition function in the bi-local picture. Sincethe background is given by the ground state solution it is appropriate to use the Hamil-tonian (single-time) representation of the bi-local theory [18]. The full nonlinear collectiveHamiltonian for the equal-time bilocal field (and its canonical conjugate) reads H = 12 (cid:90) d(cid:126)xd(cid:126)yd(cid:126)z Π ( (cid:126)x, (cid:126)y ) Ψ ( (cid:126)y, (cid:126)z ) Π ( (cid:126)z, (cid:126)x ) + 12 (cid:90) d(cid:126)xd(cid:126)y
Π ( (cid:126)x, (cid:126)y ) Ψ ( (cid:126)y, (cid:126)x ) Π ( (cid:126)x, (cid:126)x )+ 12 (cid:90) d(cid:126)xd(cid:126)y
Π ( (cid:126)x, (cid:126)x ) Ψ ( (cid:126)x, (cid:126)y ) Π ( (cid:126)y, (cid:126)x ) + 12 (cid:90) d(cid:126)x
Π ( (cid:126)x, (cid:126)x ) Ψ ( (cid:126)x, (cid:126)x ) Π ( (cid:126)x, (cid:126)x )+ 12 (cid:90) d(cid:126)x (cid:16) − (cid:3) (cid:126)x Ψ ( (cid:126)x, (cid:126)y ) | (cid:126)y = (cid:126)x (cid:17) + N − + ∆ V (3.18)where (cid:3) (cid:126)x is the Laplacian on S and the counterterms (which are lower orders in 1 /N ) are∆ V = (cid:18) − N K + 1) + 18 ( K + 1) (cid:19) TrΨ − . (3.19)The first five (integral) terms on the RHS of (3.18) comes from a direct rewriting of theoriginal Hamiltonian (of the vector fields) in terms of the bi-local fields (after a repeated useof the chain rule) (see [33] for details). The rest terms in (3.18) (including the interactionterm TrΨ − and the counterterm ∆ V ) arises from a similarity transformation to make theHamiltonian Hermitian. This is in the same spirit as the Jacobian present in the actionapproach, and the counterterm is related to the lower order measure.The collective Hamiltonian (3.18) is well suited to perform a 1 /N expansion after therescaling Ψ → N Ψ and Π → Π /N . By expanding Ψ ( (cid:126)x, (cid:126)y ) around the background field– 11 – = Ψ + √ N η , and similarly for the conjugate momenta Π = √ N π , one can show that theleading Hamiltonian H (0) of order O ( N ) is E (0) = H (0) = N (cid:90) d x (cid:104) − ∇ (cid:126)x Ψ ( (cid:126)x, (cid:126)y ) (cid:12)(cid:12) (cid:126)y = (cid:126)x (cid:105) + N − = N (cid:88) (cid:126)k (cid:18) l + 12 (cid:19) = N (cid:88) l =0 (2 l + 1) (3.20)which is exactly the ground state energy of N free bosons.The one-loop calculation follows similarly as the covariant formulation used in the pre-vious section. The quadratic Hamiltonian of order O (cid:0) N (cid:1) is H (2) = 12 (cid:88) (cid:126)k ≤ (cid:126)k (cid:104) π (cid:126)k ,(cid:126)k ∗ π (cid:126)k ,(cid:126)k ∗ + η (cid:126)k ,(cid:126)k ∗ ω (cid:126)k ,(cid:126)k η (cid:126)k ,(cid:126)k ∗ (cid:105) (3.21)where (cid:126)k = ( l, m ) and (cid:126)k ∗ = ( l, − m ). The frequencies are ω (cid:126)k ,(cid:126)k = l + l + 1 on S , so that thefree energy of the singlet sector can be easily calculated as F (cid:48) min = E (1) β + 12 (cid:88) ( l ,m ) , ( l ,m ) log (cid:104) − e − β ( l + l +1) (cid:105) + 12 (cid:88) ( l,m ) log (cid:104) − e − β (2 l +1) (cid:105) = E (1) β − ∞ (cid:88) n =1 e − nβ n (cid:34) (cid:0) e − nβ (cid:1) (1 − e − nβ ) + 1 + e − nβ (1 − e − nβ ) (cid:35) (3.22)where the factor in the second term are necessary for avoiding double-counting. Fur-thermore, the 1-loop correction to the ground state energy is E (1) = (cid:80) (cid:126)k ≤ (cid:126)k ω (cid:126)k ,(cid:126)k = ( K + 1) (cid:80) (cid:126)k λ ( (cid:126)k ) which precisely cancels the O ( N ) contribution from the counterterm ∆ V :∆ E (1) = − ( K + 1)TrΨ − = − ( K + 1) (cid:80) (cid:126)k λ ( (cid:126)k ). This ensures the vanishing of the totalone loop correction: E (1)total = E (1) + ∆ E (1) = 0. Therefore, the leftover correction to the freeenergy is F min = − ∞ (cid:88) n =1 e − nβ n (cid:34) (cid:0) e − nβ (cid:1) (1 − e − nβ ) + 1 + e − nβ (1 − e − nβ ) (cid:35) (3.23)which agrees with (3.7) after the identification q = e − β .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 Ψ ( (cid:126)x, (cid:126)y ) is not symmetric hence there will be no potential double-counting, the final result is F = (cid:88) ( l ,m ) , ( l ,m ) log (cid:104) − e − β ( l + l +1) (cid:105) = − ∞ (cid:88) n =1 e − nβ n (cid:0) e − nβ (cid:1) (1 − e − nβ ) (3.24)which agrees with (3.6). At high temperature, the free energy scales as F ∼ − ζ (5) T ,showing the higher phase of [25]. – 12 –hat we have seen in the present series of calculations is that in this case the free energiesdo not vanish at one loop. But the ground state energy is indeed much like the free energy ofthe previous section: one obtains the exact result in the leading evaluation while the one loopcontribution cancels with the contribution from the counterterm. The picture regarding theredefinition of the coupling constant in the O ( N ) case therefore appears in this backgroundtoo. That is satisfactory as there should not be a change in the identification of the couplingconstant just by changing the background. The purpose of the calculations presented in this paper is two-fold. First, the calculations(essentially of the determinants) in the bi-local collective picture and the AdS picture wereseen to give identical results. This can serve as a further confirmation of the exact equivalenceof the two pictures at the level of Laplacians representing small fluctuations. This agreementwas seen for a variety of backgrounds. We note that the bi-local picture provides an explana-tion of the curious observation of [1] that the evaluation of one-loop AdS determinants (aftersumming over all spins and zeta function regularization) gives a result identical to that of asingle local field.Second, the calculations performed offer further data for a specification of the dictionarybetween higher spin gravities and vector field theories (both U ( N ) and O ( N )). The issue(addressed in [1]) is the possible difference between the identification of G (the couplingconstant in the higher spin theory) with the parameter 1 /N or 1 / ( N −
1) of the vector model.The collective field representation shows that the 1 /N expansion can be maintained due toa presence of a measure factor in the functional integration. Due to the specific form ofthe measure we have also observed that for O ( N ) based theories one can define an effectivecoupling constant given by G (cid:48) = 1 / ( N −
1) as proposed in [1]. We caution however that thismight be a gauge dependent phenomenon and might not be a feature of an arbitrary gauge.This problem (of measure) is well understood in bi-local version of higher spin theory butremains to be understood in Vasiliev type gravities. For this, one requires the knowledge ofthe action [34], which then through its (nonlinear) Poisson structure determines the measureappearing in the functional integral. Work on this is in progress.Finally let us mention that the collective approach can be extended to the interactingIR fixed point as well. For the AdS computations including the heat-kernel method andthe Hamiltonian method, adapting to the IR fixed point is relatively straightforward just bychanging the conformal dimension of the bulk scalar field from ∆ (0) = 1 to ∆ (0) = 2. Onthe field theory side, the IR fixed point can be reached either by turning on a double tracedeformation or using the nonlinear sigma model. The collective approach can be applied toeither method and we have checked that the free energy decreases along the RG flow (theF-theorem) with the difference F IR − F UV = − ζ (3)8 π in the case of S [24]. Most recently, anextension of the results in [1] to other dimensions d was performed in [35]. It is clear thatsome of our conclusions from d = 3 indeed hold more generally for other dimensions. Most– 13 –mportant among them is the reduction of the small fluctuation determinant to a single localfield one. Acknowledgments
We would like to thank S. Das, M. Gaberdiel, S. Giombi, I. Klebanov, R. Leigh, A. Petkouand especially Sumit Das for helpful discussions. The work of AJ and JY is supported by theDepartment of Energy under contract DE-FG02-91ER40688. The work of KJ is supportedby the DOE grant DE-FG02-13ER42001.
A Derivation of the measure
In the transformation from the vector fields φ i ( x ) to the bi-local fields Φ ( x, y ), the partitionfunction gets a Jacobian (cid:90) D (cid:126)φ ( x ) e − S [ (cid:126)φ ] = (cid:90) D Φ ( x, y ) J ( x, y ) e − S [Φ( x,y )] . (A.1)For the symmetric bi-local field Φ ( x, y ), we can give an ordering to the spacetime coordinates x and y . Then, the independent bi-local fields areΦ ( x, y ) ( x ≤ y ) . (A.2)Denoting a ≡ ( x, y ) and b ≡ ( x (cid:48) , y (cid:48) ) with x ≤ y and x (cid:48) ≤ y (cid:48) , respectively, in general, one canderive a differential equation for the Jacobian [36] (cid:90) dx (cid:48) dy (cid:48) Ω ( a, b ) ∂ log J∂ Φ ( b ) + ω ( a ) + (cid:90) dx (cid:48) dy (cid:48) ∂ Ω ( a, b ) ∂ Φ ( b ) = 0 (A.3)where ω ( a ) and Ω ( a, b ) are O ( N ) and O (cid:0) N (cid:1) contributions to log J , respectively ω ( a ) ≡ − (cid:90) dz ∂ ∂φ i ( z ) ∂φ i ( z ) Φ ( a ) = − N δ ( x − y ) (A.4)Ω ( a, b ) ≡ (cid:90) dz ∂ Φ ( a ) ∂φ i ( z ) ∂ Φ ( b ) ∂φ i ( z ) ∼ O (cid:0) N (cid:1) . (A.5)One can also compute (cid:90) dx (cid:48) dy (cid:48) ∂ Ω ( a, b ) ∂ Φ ( b ) = 4 κ δ ( x − y ) (A.6)where the coefficient κ depends on the type of bi-local collective field theory κ = (cid:40) K for U ( N ) ( K + 1) for O ( N ) (A.7) For example, for x = ( x , x , · · · , x n ) and y = ( y , y , · · · , y n ), x < y means iff x < y or x = y , x < y etc. – 14 –nd K ≡ (cid:82) dx δ (0) = (cid:80) k (cid:90) dx (cid:48) dy (cid:48) Ω ( a, b ) ∂ log J∂ Φ ( b ) − N − κ ) δ ( x − y ) = 0 , (A.8)from which one can solve for the Jacobian for general collective field theory (see also [37]) aslog J = 12 ( N − κ ) Tr log Φ , (A.9)where we have used the explicit form of (A.5).The parameter κ is also related to weight of bi-local space when we express a function ofbi-local space in terms of a function of local space. For example, (cid:88) k ,k (cid:48) [ λ ( k ) + λ ( k )] = 2 κ (cid:88) k λ ( k ) (A.10)where (cid:80) (cid:48) means the summation over the independent bi-local momentum. For the O ( N )vector model, we have (cid:80) (cid:48) k ,k = (cid:80) k ≤ k and κ = ( K + 1). While for the U ( N ) vector model,there is no restriction on the summation, therefore κ = K . References [1] S. Giombi and I. R. Klebanov, “One Loop Tests of Higher Spin AdS/CFT,” JHEP , 068(2013) [arXiv:1308.2337 [hep-th]].[2] I. R. Klebanov and A. M. Polyakov, “AdS dual of the critical O(N) vector model,” Phys. Lett.B , 213 (2002) [hep-th/0210114].[3] E. Sezgin and P. Sundell, “Massless higher spins and holography,” Nucl. Phys. B , 303(2002) [Erratum-ibid. B , 403 (2003)] [hep-th/0205131].[4] S. Giombi and X. Yin, “Higher Spin Gauge Theory and Holography: The Three-PointFunctions,” JHEP , 115 (2010) [arXiv:0912.3462 [hep-th]].[5] M. A. Vasiliev, “Consistent equation for interacting gauge fields of all spins in(3+1)-dimensions,” Phys. Lett. B , 378 (1990).[6] M. A. Vasiliev, “More on equations of motion for interacting massless fields of all spins in(3+1)-dimensions,” Phys. Lett. B , 225 (1992).[7] X. Bekaert, S. Cnockaert, C. Iazeolla and M. A. Vasiliev, “Nonlinear higher spin theories invarious dimensions,” hep-th/0503128.[8] R. G. Leigh and A. C. Petkou, “Holography of the N=1 higher spin theory on AdS(4),” JHEP , 011 (2003) [hep-th/0304217].[9] E. Sezgin and P. Sundell, “Holography in 4D (super) higher spin theories and a test via cubicscalar couplings,” JHEP , 044 (2005) [hep-th/0305040].[10] O. Aharony, G. Gur-Ari and R. Yacoby, “d=3 Bosonic Vector Models Coupled to Chern-SimonsGauge Theories,” JHEP , 037 (2012) [arXiv:1110.4382 [hep-th]]. – 15 –
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