A. O. Barvinsky*

l.i In ih r Physical Institute, Russian Academy of Sciences

119991, Moscow, Russia

Theory Division, CERN CH-1211, Geneva 23, Switzerland

Received October 20, 2014

We suggest that the principle of holographic duality can be extended beyond conformal invariance and AdS isometry. Such an extension is based on a special relation between functional determinants of the operators acting in the bulk and on its boundary, provided that the boundary operator represents the inverse propagators of the theory induced on the boundary by the Dirichlet boundary value problem in the bulk spacetime. This relation holds for operators of a general spin-tensor structure on generic manifolds with boundaries irrespective of their background geometry and conformal invariance, and it apparently underlies numerous 0(N") tests of the AdS/CFT correspondence, based on direct calculation of the bulk and boundary partition functions, Casimir energies, and conformal anomalies. The generalized holographic duality is discussed within the concept of the "double-trace" deformation of the boundary theory, which is responsible in the case of large-A' CFT coupled to the tower of higher-spin gauge fields for the renormalization group flow between infrared and ultraviolet fixed points. Potential extension of this method beyond the one-loop order is also briefly discussed.

Contribution for the JETP special issue in honor of V. A. Rubakov's 60th birthday

DOI: 10.7868/S0044451015030131


It is a groat pleasure to write this paper dedicated to Valery Rubakov on the occasion of his sixties birthday. Our scientific careers have started simultaneously when we were students at the Moscow University and shared common interests in physics classical and quantum gravity and invariably pursued these interests, in our own ways and styles, throughout the years to come. In particular, the results of this work were conceived in the course of discussions, when Valery suggested to work out a covariant method for calculating radiative corrections in brano gravity models fl] as a means of establishing applicability limits of the perturbation theory. By the time this method has become ready for use, the peak of interest in brano models was basically-over, and interests of scientific community have shifted to other areas, not the least of those being the idea of holographic duality and the AdS/CFT correspondence.

E-mail: barvin'ffltd.lpi.ru

Interestingly, that old method now seems to find application in this field, and, I hope, Valery will be amused to see how his suggestions are realized in this nonper-turbative concept of high-energy physics.

The idea of holographic duality between a ii-di-mensional conformal field theory (CFT) and a theory in the (d+ 1 )-dimensional anti-de Sitter (AdS) spacetime that initially began with supersymmotric models of AT x Ar-matrix valued fields [2 4] was later formulated for much simpler "vectorial" models without the need in supersymmetry [5]. These models have an infinite tower of nearly conserved higher-spin currents and in this way naturally lead to a corresponding tower of massless higher-spin gauge fields. Therefore, the holography concept implies that the dual theory should contain these fields in AdS spacetime, thus forming the Vasiliev theory of nonlinear higher-spin gauge fields [6, 7], which necessarily imply an infinite set of those, because the principle of gauge invariance for spins a > 2 cannot be realized for a finite tower of spins. In contrast to the original supersymmotric models in which the AdS/ CFT correspondence was checked

for suporsynimotry-protoctod correlators, holographic duality in vectorial models underwent verification by numerous nontrivial calculations that go beyond simple kinematical or group-theoretical reasoning and extend from the tree level 0(Arl) to the "one-loop" order 0( №).

In particular, the calculation of the U(N) singlet scalar CFT partition function on 5'1 x S2 was shown to agree with the corresponding higher-spin partition function calculation in AdSi [8], a result extended to the 0(N) singlet sector of a scalar CFT [9]. Then these results were confirmed and extended to arbitrary dimensions in ("10], including the comparison of thermal and Casimir energy parts of partition functions in CFTd and AdSd+i in [11]. The vanishing Casimir energy in odd-dimensional theory (associated with the absence of the conformal anomaly) implies the same on the AdS side, which is nontrivial because it implies an infinite summation over the tower of higher-spin gauge fields the property that was observed in d = 4 on the AdS.5 side [12] and confirmed by an explicit summation of conformal anomaly coefficients us for conformal higher-spin fields on the 5'4 side [13]. The list of similar results agreeing on both sides of the AdSd+i/CFTd correspondence was extended in [11].

A special class of holographic dualities is associated with the so-called double-trace deformations of the scalar CFT [14], which generates its rcnormalization group (RG) flow from the IR fixed point (free CFT) to the UV fixed point [15]. The associated holographic dual of this RG flow in the AdS spacetime is the transition between two different boundary conditions on the dual massless gauge fields of higher spins at the AdS boundary [12,15].

The variety of these miraculous coincidences and the gradually extending area of validity of duality relations (from supersymmetric models to nonsupcr-syninietric ones, from lower spacetime dimensions and lower spins to higher ones, from divergent and Casimir energy parts of partition functions to their thermal parts, from bosons to fermions, etc) imply that there should be some deep functional reasons underlying all this and perhaps even allowing one to extend holographic duality beyond AdS isometry and conformal invariance. The goal of this paper is to show that this is indeed possible. Within the class of holographic dualities associated with the double-trace deformation of CFT, there exist universal relations for one-loop functional determinants of local and nonlocal operators on generic (d + 1 )-diniensional spacetime and its il-dimen-sional boundary [16] that guarantee this duality irrespective of the background geometry and conformal in-

variance. The only condition that relates (d+ ^-dimensional and (/-dimensional theories is that at the tree level, the boundary theory be induced from the bulk by a Dirichlet boundary value problem; then their one-loop quantum corrections dutifully match. The proof of this statement is based on linear algebra of (pseudo)differential operators and a sequence of Gaussian functional integrations. When the theory has a small parameter 1/Лг playing the role of a semiclassi-cal Planck constant, this sequence of integrations might apparently be extended to holographic duality beyond the one-loop order 0(AT°).


The double-trace deformation [14] of the large-Ar CFT of scalar fields Фг(.г), i = 1,... , Л\ by the square of the O(N) invariant single-trace scalar operator

,J(x) = <!>'(>• )<!>'(>•).

Scft($) Scft($) - jj j dx,J'2{x),

leads to the rcnormalization group flow between the IR fixed point of the free CFT and its UV fixed point. In the limit of large N, this was clearly demonstrated by using the Habbard Stratonovich transformation as follows [15].

We consider the generating functional Zcprif) of the correlators of J for the perturbed theory with sources if,

ZcftW) = j (1Ф x

x exp (-SCft($) + i J($)f_1 J($) + , with

= (cxp №))'срт, (2.1)

Jf~1J= / dx dy ,J (x) f_ 1 (x, у )J(y),

ip,J = / dxip(x)J(x).

For the sake of generality of our formalism we write the operator f = f(x.y) in what follows in a rather



general form even though it is ultralocal in CFT models, f(x.tj) = f6(x,y), and we also use the condensed notation omitting the sign of integration over (/-dimensional coordinates. A functional dependence in the d-dimensional space is denoted by round brackets, like Scft(^) = Scft($(z))i and the operators acting in this space, like f, are boldfaced.

Representing the part of the exponential in (2.1) quadratic in J as a Gaussian integral over an auxiliary-field <f> (the Habbard Stratonovich transform), we have


f f exp(<PJ)) = (detf)1/'2 / (1ф x

x (exp (~<t>f<t> + (ф+ ç)J



where det f denotes the functional determinant of the operator f(x. tj) on the space of functions of (I-dimon-sional coordinates.

As usual in largo-Ar CFT, we assume the vanishing expectation value of J, (.J) = 0, and the smallnoss of higher-order correlators {JJ ... J) as AT —¥ oc,

exp(wJ)i «exp







where ^F is the notation for the undoformod two-point correlator of J. From the new Gaussian integration in (2.3), we then have

' rap if J) ]


= (detf)1/2 (detF/F17'2 x


x exp


2 ^F-1

F + f.





Therefore, the correlator {JJ){:FT in the double-trace deformed CFT interpolates between the UV and IR fixed points of the theory:

{JJ){ 1

)cFT — p_i

-f+ f(f"1F)-

f^F -С 1, f^F > 1.


For an ultralocal f = f6(x,tj) in the CFT with a sing-lo-traco scalar operator J of dimension A, the correlator {JJ)CFT = ^F in the coordinate and momentum representations behaves as

F - У

2 A


-2A '

Thus, the above two limits indeed correspond to the respective UV,


and IR.

r1 F



■c 1,

> 1,


fixed points. In the IR limit, the correlator (modulo the contact term f = f6(x. tj)) is dominated by the second term j

f^Fr1 ~ k _ у11/Ы-2А)

in the long-distance regime |.i: — y\ |/|1/'d_2A) [15]. The ronormalization group flow interpolates between two phases in which the operator J(x) has different dimensions, Д = Д+ in IR and d/2 — A = Д_ in UV.

This double-trace deformation picture also applies in the context of the dual descr

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