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  • UM - TH - 00 - 13

    A Variational Principle for Radial Flows in

    Holographic Theories.

    R. Akhoury

    The Randall Laboratory of Physics

    University of Michigan

    Ann Arbor, MI 48109-1120


    We develop furthur the correspondence between a d + 1 dimensional theory and a d dimensional one with the ”radial” (d + 1)th coordinate ρ playing the role of an evolution parameter. We discuss the evolution of an effective action defined on a d dimensional surface charactarized by ρ by means of a new variational principle. The conditions under which the flow equations are valid are discussed in detail as is the choice of boundary conditions. It is explained how domain walls may be incorporated in the framework and some generalized junction relations are obtained. The gen- eral principles are illustrated on the example of a supergravity theory on AdSd+1.


  • 1 Introduction

    The holographic principle [1, 2, 3, 4], motivated by Black Hole considerations, is an extension

    of the Bekenstein bound [5] which limits the number of degrees of freedom that generate

    entropy in a theory including gravitation. It is a dynamical statement postulating that

    the evolution of gravitational and matter fields in bulk space-time is specified by the data

    stored in its boundary. In the conventional quantum field theoretic treatment of gravity,

    this property is not apparent, however, recent conjectures in string theory concerning the

    AdS/CFT duality [6, 7, 8, 9] provide an example of this principle. If future investigations

    validate it, the holographic principle could well turn out to be one of the most important

    physical ideas in recent times.

    There are many forms that a correspondence between a d + 1 dimensional and a d

    dimensional theory can take. Perhaps the simplest one is that between a classical theory in

    the higher dimensional space and a quantum theory at the boundary. An example is provided

    by the conjectured duality between the classical supergravity theory in 5 dimensions and a

    quantum gauge theory in 4 dimensions [6]. More specifically, let S(φ, g) denote the classical

    action for a supergravity solution where the boundary values of the fields are φ and of the

    metric, gµν , then the presciption [7, 8] for the conjectured duality is:

    1√ g


    δφi1 ......

    1√ g


    δφin S = < Oi1......Oin > (1)

    where, O denote certain gauge invariant operators in the boundary theory. In order for

    the classical description in the bulk of the AdS5 space to be valid one considers the limit

    of large N and large ’t Hooft coupling on the gauge theory side. A key ingredient of this

    correspondence is the UV/IR connection [10, 11] : the infrared regulator of the bulk theory

    is equivalent to the ultraviolet regulator of the boundary theory. This indicates that one

    may interpret the d+1th coordinate, ρ as a renormalization group parameter of the 4 dimen-

    sional theory. Indeed, the evolution of bulk fields with ρ has been studied in a fundamental

    paper [12] within the Hamilton-Jacobi framework where it was shown that the resulting flow


  • equations can asymptotically be cast in the form of a Callan-Symanzik equation, with the

    bulk scalar field representing the 4 dimensional gauge coupling. In ref. [10] the boundary

    AdS/CFT conjectured correspondence has been extended to a finite radius AdS foliation.

    This is of relevance to the connection with the Randall-Sundrum type scenarios [13]. A more

    generalized correspondence was originally proposed [6], i.e., between the type II B string the-

    ory in the bulk (of AdS5 times S5) and a CFT on the boundary. In the limit of large N

    and large ’t Hooft coupling, the string coupling g → 0 and one may limit to the classical supergravity description, otherwise string loops need to be considered.

    Though much work has gone into the development of these ideas, there are many chal-

    lenges ahead [9]. Of particular relevance for this paper is the work of [12] concerning the

    evolution of bulk fields with ρ. We will formulate this evolution quite generally by means of a

    variational principle. The advantages of doing so are many : a variational principle replaces

    many mathematical expressions by a single general principle and moreover the formulation

    is coordinate independant. The variational principle which we derive in section 2 involves a

    multi-stage optimization procedure -a minimization condition which leads to a flow equation

    for the effective action S. The effective action is a function of the fields on a d dimensional

    surface charactarized by a particular value of ρ. As one changes ρ the action changes and if we

    still require a minimum of the action integral, then the flow equations must be satisfied. This

    procedure is discussed in detail in section 2 where its connection with the Hamilton-Jacobi

    formalism is also shown. The problem is first discussed in a general setting and then the

    generalized flow equations are applied to the example of the supergravity theory on AdSd+1

    where the flow equations of ref.[12] are reproduced. In addition to the above, a variational

    formulation provides us with a new way to formulate the boundary conditions which the

    flow equations must satisfy. This is done in section 3 together with a discussion of the Weyl

    anomaly. Another decided advantage of the variational formulation is discussed in section

    4. There we discuss in a quantitative manner, the conditions under which the ”classical”

    approximation for the flow equations is valid. In particular, we seek the conditions under


  • which terms involving the second partial derivatives of the effective action which are ignored

    in the derivation of the flow equations in section 2 can become unbounded. This allows us

    to formulate the applicability criterion quite generally in terms of (the solutions of) a set of

    partial differential equations. The example of a single scalar field in AdS is then considered

    where the differential equation reduces to a Jacobi type equation. The question of how the

    second partial derivatives of the action may be incorporated into the flow equations in this

    framework will be dealt with in a subsequent publication [14]. Our formulation, as with

    ref.[12], allows one to consider the effective action for values of ρ away from the boundary.

    In the AdS case, for example, we can consider any of its foliations. This allows us to study

    the scenario when a domain wall is introduced into the d + 1 dimensional space-time. This

    is done quite generally in section 5 where starting from the results of section 2, generalized

    junction conditions [15] are derived. The example of the Randall-Sundrum [13] scenario and

    its connection with the supergravity theory on AdS [16] is also discussed in some detail.

    Finally, in section 6 we conclude with a discussion of the results.

    2 The Variational Principle

    In this section we will consider the dynamical evolution of fields in a d + 1 dimensional

    spacetime with a d dimensional (time-like) boundary. We will denote the d dimensional

    coordinates by xµ and the ”radial” coordinate is ρ. We consider a d dimensional hypersurface

    defined by ρ = constant after making appropriate gauge choices for the metric. In the AdS

    case, for example, this hypersurface would be a particular foliation. The boundary of the

    d+1 dimensional space is located at ρ = ρ0. We will be interested in fields φi(x, ρ) [17] whose

    dynamics is governed by a lagrangian L. In particular, we would like to understand how the effective action changes as we move from one hypersurface to a neighbouring one. The

    trajectory that one follows in moving from one hypersurface to the other is not arbitrary but

    is determined by an optimization condition. We propose below that this optimal trajectory


  • is determined by a variational principle which will be seen to imply a (functional) partial

    differential equation satisfied by the effective action S(φi) of the boundary values of the


    The optimal trajectory arises as a result of a minimization problem and the solution

    suggests itself as a consequence of a multi-stage procedure. To see how this works, consider

    first the integral:

    J ′[φi] = ∫ ρ2

    ρ1 dρF (ρ, φi, φ̇i) (2)

    In the above, φ̇i = dφi/dρ. In general, F (ρ, φi, φ̇i) will be of the form, ∫ d

    √ GL(ρ, φi, φ̇i), with

    L the Lagrangian, and Gµν is the d + 1 dimensional metric. In this and the following we suppress any dependance on the d dimensional coordinates. We will call the critical trajec-

    tory, the one which is obtained by minimizing J ′. Consider a trajectory, and let us choose a

    point (ρ3, φi3) [18] in between the initial and final points, and follow the curve from (ρ1, φi1)

    to (ρ3, φi3). For the rest of the curve ((ρ3, φi3) to (ρ2, φi2) to be also critical we must mini-

    mize ∫ ρ2 ρ3

    dρF (ρ, φi, φ̇i). This must be true for all such inbetween points ρ3. Thus whatever

    the initial point (ρn, φin), or the initial arc of the trajectory, the remaining transformations

    must constitute an optimal sequence for the rem