Domain Wall Holography for Finite Temperature Scaling Solutions
OOctober 22, 2018
Domain Wall Holography for Finite Temperature Scaling Solutions
Eric Perlmutter Department of Physics and AstronomyUniversity of California, Los Angeles, CA 90095, USA
Abstract
We investigate a class of near-extremal solutions of Einstein-Maxwell-scalar theorywith electric charge and power law scaling, dual to charged IR phases of relativisticfield theories at low temperature. These are exact solutions of theories with domainwall vacua; hence, we use nonconformal holography to relate the bulk and boundarytheories. We numerically construct a global interpolating solution between the IRcharged solutions and the UV domain wall vacua for arbitrary physical choices ofLagrangian parameters. By passing to a conformal frame in which the domain wallmetric becomes that of AdS, we uncover a generalized scale invariance of the IR scalingsolution, indicating a connection to the physics of Lifshitz fixed points. Finally, guidedby effective field theoretic principles and the physics of nonconformal D-branes, weargue for the applicability of domain wall holography even in theories with AdS criticalpoints, namely those theories for which a scalar potential is dominated by a singleexponential term over a large range. [email protected] a r X i v : . [ h e p - t h ] J un Introduction and summary 22 Construction of the action and scaling solution 7
The form and function of the AdS/CFT correspondence [1] have evolved since the idea’sbirth. Initially, the correspondence was borne from consideration of the decoupling limitof branes in string and M-theory with conformal near-horizon supersymmetries, includingthe paradigmatic case of duality between type IIB supergravity on AdS × S and stronglycoupled N = 4 super-Yang-Mills theory in the large N limit. Subsequent work relaxed thecorrespondence to include bulk spacetimes that are asymptotically AdS and to those withnon-maximal or no supersymmetry. The last few years have also seen a broadening of thescope of correspondence to include bulk actions that have no origin in string or M-theory: onesimply writes down a “bottom-up” Lagrangian that admits an AdS vacuum, and asks whatsort of field theories it can describe, while tabling questions regarding the quantum existenceof the bulk theory. This has led to a productive interface between AdS/CFT and condensedmatter physics, including constructions of gravity duals to holographic superconductors [5,6, 7], non-relativistic theories with anisotropic Lifshitz scaling [8, 9, 10], and theories withGalilean invariance [11, 12], among much other work.There has also been work on theories with no microscopic scaling symmetry at all, butrather merely relativistic symmetry. This correspondence [13] models strongly coupled, For foundational work and a review, see [2, 3, 4]. ds = ( Ar ) γ (cid:48) η µν dx µ dx ν + dr ( Ar ) γ (cid:48) , (1.1)along with some number of rolling scalar fields. This goes by the name “Domain Wall/QFTcorrespondence”[14]. The fact that the metric (1.1) is conformal to AdS can be phrasedas crucial to the duality, which was established by decoupling the near-horizon dynamicsof non-conformal D-branes: passage to a conformal, so-called “dual” frame in which theD-brane near-horizon metric becomes AdS p +2 × S − p reveals a manifest ( p + 2)-dimensionalgravitational description, in which we identify the radial direction with the energy scale ofthe dual field theory [15]. Because the scalars vary and the curvature is somewhere singular,the duality holds at intermediate energies, away from large curvature and order one effectivestring coupling.This picture can be summed up by the statement that these nonconformal D-branes havea “generalized conformal structure”[16, 17]: upon passage to the dual frame, the conformalsymmetry of the metric is broken by the nonconstant scalar field, implying that the radialscale transformation leaves the solution invariant if one shifts the scalar field simultaneously.This follows from the conformal structure of M-theory branes and their relation to ten-dimensional type II branes.In [16, 18] a precise holographic dictionary was established in this case, both for choicesof γ (cid:48) that derive from branes and those that do not – that is, the program of holographicrenormalization was extended to these non-asymptotically AdS spacetimes. It seems naturalthen to extend the phenomenological philosophy described above in pursuit of the question,“what are all IR spacetimes that can be patched onto a domain wall solution?” This dualizesto the question, “what long-range behavior is possible for systems with merely Poincar´e sym-metry?” This approach was utilized in [19] in the context of hydrodynamics of nonconformalbranes, for example, but we would like to continue to study phases of field theories to whichDW/QFT applies.In a separate development, we recall that gravity duals of condensed matter systemsoften have the undesirable feature of nonzero entropy at extremality. This runs counter tothe empirical Nernst’s “Theorem”, which says that a generic physical system cooled to zerotemperature should lose all its entropy as it occupies a unique ground state. Gauge/gravityduality translates this to the statement that we should study finite temperature black holeswhich have a degenerating horizon as the temperature is tuned to zero, unlike the standardexample of the charged Reissner-Nordstrom black brane. Much work has succeeded in Henceforth labeled “DW/QFT” for short. Of course, this near-horizon AdS region of the Reissner-Nordstrom black hole has also been used S = − πG D (cid:90) d D x √− g (cid:16) R + f ( φ ) F µν F µν + 12 ( ∂φ ) + V ( φ ) (cid:17) , (1.2)which admits a general scaling solution for the metric, ds ∼ − r β f ( r ) dt + dr r β f ( r ) + r γ dx i · dx i (1.3)where f ( r ) is a near-extremal “emblackening” factor that vanishes at the near-extremalhorizon, r = r h , and equals one in the extremal solution ( r h = 0). The field equationsdemand β, γ >
0, so this metric, by design, has zero extremal entropy. Hence, its extremallimit can be viewed as approximating the leading near-horizon behavior of a solution withvanishing horizon area; that is, as the far infrared gravity dual to a system with a uniquezero temperature ground state.If one insists that this class of metrics is an exact solution of the theory (1.2), it can onlybe supported with rolling scalars of the form φ ( r ) ∼ ln r + φ (1.4)Moreover, the field equations demand that the scalar potential and gauge coupling are singleexponentials in φ , V ( φ ) = − V e ηφ , f ( φ ) = e αφ . (1.5)An action of the form (1.2) with this exponential potential clearly does not admit AdS;in fact, its flux-less vacuum is a domain wall. Therefore, if the scaling solution (1.3) ispatched onto an asymptotically domain wall spacetime, then DW/QFT holography dualizes constructively, for instance in modeling behavior of IR CFTs in 0+1 dimensions. See e.g. [20, 21, 22, 23]. (cid:101) ds IR = − r l f ( r ) dt + l r dr f ( r ) + (cid:16) rl (cid:17) ˜ γ dx i · dx i , (1.6)modified by a logarithmically rolling scalar (1.4).The low-energy physics of our dual field theory, therefore, is controlled by a “generalizedscale invariance,” despite the solution not exhibiting scale invariance in Einstein frame. Ofcourse, the invariance is broken by the finite temperature, but it is nonetheless an interestingresult, natural in the context of DW/QFT: just as the UV domain wall spacetime possessesa generalized conformal structure, the charged, Lorentz-symmetry breaking phase to whichthe theory flows in the IR retains the part of this structure unbroken by the presence of thatcharge.Four-dimensional holography for the scaling solution (1.3) was studied for the scale-invariant β = 2 case in [26], both numerically and analytically, and extended to arbitraryspacetime dimension in [27]. We refer to this as the “modified Lifshitz solution.” The five-dimensional β (cid:54) = 2 solution was presented in [28], where the authors studied a charged dilatonAdS black hole with a ten-dimensional uplift to spinning D3 branes. That full dilatonicblack hole solution approaches this one, for certain values of η and α , in the limit that thescalar field is large and one can neglect an exponential term in its potential. We noted earlierthat the single-power scaling of the metric is characteristic of the leading small r behaviorof some solution near an extremal horizon of vanishing area. Restricting our attention toonly that term, and demanding that it be part of an exact solution to the action (1.2), wefound that the potential must be a single exponential in φ , namely the term in some fullpotential which dominates near the horizon, of course; this was indeed the motivation of [28]to consider the scaling solution at all.It was also suggested in [28] that a deformation of the potential V ( φ ) which generates anAdS critical point would allow one to study the Einstein frame solution (1.3) holographicallyvia AdS/CFT. That would necessarily turn the solution into an IR phase of a conformal, notmerely quantum, field theory. We take a different approach, asking as we did earlier aboutIR behavior of theories with relativistic symmetry instead.5ven in the conformal case, one should be able to use DW/QFT at intermediate energiesbelow the scale at which conformal symmetry is manifest, where a single term of a full (evenstringy) potential dominates. We argue for this role of DW/QFT in the body of the paper,guided by the philosophy of effective field theory. Consider, for instance, a scalar potentialof the form V ( φ ) = − V ( e − bφ + be φ ) , < b + 1 < (cid:115) D − V , (1.7)and V >
0. This admits an AdS vacuum at the origin, stable with respect to the Breitenlohner-Freedman bound. When φ is large, the potential is dominated by the second term. So fora solution in which φ goes to + ∞ at the horizon, then decreases monotonically outwardsand settles at its φ = 0 AdS critical point, the fields behave as though near a domain wallboundary over a large range of r . This suggests that domain wall holography can act as aneffective holographic tool at intermediate energies where a theory is only relativistic, and theterms in the full potential that become large at larger radii (higher energies) are not needed.The paper is organized as follows. We first show, in section 2, that such scaling be-havior for the metric is only compatible with single exponential scalar potentials and gaugecoupling functions. In the process, we come to bear on why there is attractor behavior inthe extremal modified Lifshitz case, despite the presence of a scalar field that breaks theisometry: the functional form of the scalar is fixed once the metric is given. We constructthe scaling solution in terms of a fixed Lagrangian and explore its parameter space. Sec-tion 3 reviews the relevant aspects of domain wall geometries and introduces the DW/QFTcorrespondence itself. We argue that only a subclass of all domain walls can be treated holo-graphically, namely those with a boundary. The meat of the paper begins in section 4, wherewe describe and show evidence of the numerical construction of the global solution, whichis of scaling form in the IR and domain wall form in the UV. We put this to use in section5, where we expose a generalized scale invariance of the scaling solution that descends fromthe generalized conformal structure of the domain wall. In section 6, we argue for the widerapplicability of domain wall holography, and classify when it is safe to use, by drawing oneffective field theory principles and lessons from the case of nonconformal D-branes. Section7 concludes with a discussion of the results and prospects for future work. Note:
As this work was being finalized, two papers appeared which give different treat-ments of the same solutions.The authors of [29] characterize a wide class of theories that includes ours and providea thorough analytic investigation. They make the assumption that the theories have AdScritical points in the UV, which we do not, as discussed earlier. Additionally, we provide thenumerical construction of the global solution which explicitly permits holographic analysis.The paper [30] complements our work by calculating conductivities of the scaling solution.6
Construction of the action and scaling solution
We reproduce the action of our D-dimensional Einstein-Maxwell-scalar bulk system as S = − πG D (cid:90) d D x √− g (cid:16) R + f ( φ ) F µν F µν + 12 ( ∂φ ) + V ( φ ) (cid:17) , (2.1)with no reference to any string or M-theory origin as yet. f ( φ ) is a positive definite function,to ensure the correct sign for the gauge kinetic term. We will consider solutions with electriccharge only, so we do not write any Chern-Simons terms; we also do not show the boundaryterms required for the usual construction of a well-defined variational problem.Our electric, non-relativistic, planar-symmetric ansatz is A t = A t ( r ) , A r = (cid:126)A = 0 φ = φ ( r ) ds = − U ( r ) dt + dr U ( r ) + V ( r ) dx i · dx i , (2.2)where i indexes the D − D ,AdS × R D − , and the modified Lifshitz geometry as limiting cases of the most generalsolution.Extracting the field equations, the t component of Maxwell’s equations can be integratedto give the field strength, F rt = A (cid:48) t ( r ) = ρf ( φ ) V D − ; (2.3)the integration constant ρ acts as the charge density of the black hole. Writing the rest ofthe field equations in these terms, we have the scalar equation, S : U φ (cid:48)(cid:48) + (cid:16) D − (cid:17) U V (cid:48)
V φ (cid:48) + U (cid:48) φ (cid:48) = d V ( φ ) dφ − ρ f ( φ ) V D − df ( φ ) dφ (2.4) Throughout the paper, we work in D ≥
4, avoiding the peculiarities of lower-dimensional gravity.
7s well as three Einstein equations, E (cid:16) D − (cid:17)(cid:16) V (cid:48)(cid:48) V − V (cid:48) V (cid:17) = − ( φ (cid:48) ) CON : (cid:16) D − (cid:17) U (cid:48) V (cid:48) V + (cid:16) ( D − D − (cid:17) U V (cid:48) V − U ( φ (cid:48) ) V ( φ ) + 2 ρ f ( φ ) V D − = 0 (cid:110) E D − U (cid:48)(cid:48) + (cid:16) D − (cid:17) U (cid:48) V (cid:48) V + V ( φ ) − D − ρ f ( φ ) V D − = 0 (cid:111) (2.5)The bracketed Einstein equation, ( E CON ) and substitution from theothers.As we plug in the scaling behavior U ( r ) ∼ r β , V ( r ) ∼ r γ , (2.6)we note that will only consider solutions with β >
1, so that our solutions obey the usualdefinition of extremality, namely T = r h = 0, ensuring smooth connection to the finitetemperature solutions.We also point out that the field equations dictate that β ≤ β = 2 – and 0 ≤ γ ≤
2. Upon fixing the form of the La-grangian consistent with admission of the scaling solution, these bounds become clear; so letus proceed.Returning to the field equations, then, the first Einstein equation (E1) tells us that thescalar field must take the form φ ( r ) = C ln r + φ . (2.7)The remaining undetermined functions are those of the scalar field, f ( φ ) and V ( φ ). But fora neutral scalar, we can form a linear combination of the Einstein equations in which thescalar only appears in f ( φ ): taking ( CON ) − ( E
2) + U · ( E U (cid:48) V (cid:48) V (cid:16) D − − (cid:16) D − (cid:17) (cid:17) + U V (cid:48) V (cid:16) ( D − D − (cid:17) + (cid:16) U V (cid:48)(cid:48) V − U (cid:48)(cid:48) (cid:17) D − − ρ (2 − D ) f ( φ ) V D − = 0 (2.8)The gauge coupling function f ( φ ) is fully determined by the metric and therefore, by the first-order equation ( CON ), so is V ( φ ). Specifically, they are both constrained to be exponentialin φ , that is, power law in r : plugging in the scaling form of the metric reveals, up to positiveconstants, f ( φ ) ∼ ρ F ( β, γ ) r − β − γ ( D − , (2.9)8here we have defined F ( β, γ ) = (cid:16) D − (cid:17) γ (cid:16) β − γ (cid:17) + β ( β − − γ ( γ − . (2.10)Demanding reality of the flux, ρ ≥
0, implies F ( β, γ ) >
0: allowed combinations of β and γ are bounded by the lines γ = β and γ = D − (1 − β ). For β >
1, the metric must have β ≥ γ , (2.11)where saturation occurs for vanishing flux, e.g. in AdS D where β = γ = 2. This is the cousinof the fact that, for example, the Lifshitz geometry (1.6) sourced by real two- and three-formfluxes, as in [8], can only act as a gravity dual to Lifshitz fixed points with z >
1. Such asimilarity in the causal structure of our scaling solution to the Lifshitz solution is our firsthint that the two may have some connection.Furthermore, by plugging this form for f ( φ ) back into the Einstein equations, one seesthat the potential is also a power law in r : V ( φ ) = − ρ G ( β, γ ) r β − , (2.12)where G ( β, γ ) ≡ (cid:16) D − γF ( β, γ ) (cid:16) γ (cid:16) D − (cid:17) + ( β − (cid:17)(cid:17) . (2.13) β, γ > G ( β, γ ) > D >
3; thus, V ( φ ) must be negative. And because β ≤ V ( φ ) must diverge at small r or be constant everywhere. As expected, a scale-invariantsolution can only solve a theory with the latter: as we rescale r , φ picks up a constant,which must not affect the energy of the theory because we are simply executing a symmetrytransformation.What we have shown, in the end, is that the scalar rolls down the exponential potential asit nears the horizon, presumably signaling the dive toward zero entropy at zero temperature: φ is “looking” for a critical point of the potential, as exists for the unique finite entropyextremal AdS × R D − geometry [31], but cannot find it.Even in the case where the potential V ( φ ) is constant, the exponentiality of the gaugecoupling function f ( φ ) explains the zero extremal entropy in terms of the attractor mech-anism [26]: the diverging scalar drives the system toward the runaway minimum of theeffective attractor potential, V eff = ρ f − ( φ ). The fact that there is attractor behavior atall despite the lack of true SO(2,1) isometry is accounted for by our analysis above: the formof the attractor potential is fixed once the metric’s SO(2,1) isometry is given. Thus we havea case of an attractor in which the full functional form of the massless scalar is fixed nearthe extremal horizon. 9et us note that if the scalar is charged, the form of f ( φ ) and V ( φ ) is not fixed as above,indicating that a charged interaction between φ and A µ is compatible with extremal scalingbehavior (1.3) for a range of gauge couplings and potentials.In anticipation of a possible embedding of this solution into a consistent truncation ofsome higher-dimensional supergravity, we end this subsection with the observation that amulti-scalar version of this solution can also support the metric (2.6). Writing a schematicaction S = − πG D (cid:90) d D x √− g (cid:16) R + f ( φ i ) F µν F µν + 12 ( ∂φ i ) + V ( φ i ) (cid:17) , (2.14)field equation (E1) is satisfied for all scalars logarithmic in r . Then the power law behaviorof f ( φ i ) and V ( φ i ) means that both are products of exponentials, V ( φ ) = − V e η i φ i , f ( φ ) = e α i φ i . (2.15)The space of solutions is then dictated by which of the { η i , α i } is nonzero. Having shown that such exact scaling solutions only exist in theories with an exponentialscalar potential and gauge coupling function, we rewrite the action and establish parametricdefinitions in terms of Lagrangian parameters. Our action is S = − πG D (cid:90) d D x √− g (cid:16) R + e αφ F µν F µν + 12 ( ∂φ ) − V e ηφ (cid:17) , (2.16)Withou the flux term, this theory describes a consistent sphere truncation of a higher-dimensional supergravity to gravity coupled to a single scalar [32]. Our ansatz, once more,is ds = − C r β dt + dr C r β + C r γ dx i · dx i φ ( r ) = C ln r + φ A (cid:48) t ( r ) = ρr αC + γ D − (2.17)As φ and C can be eliminated by rescaling r and x i , respectively, we will set φ = 0and C = 1. Then the parameters of our ansatz, { C , C , ρ, β, γ } are given in terms of the10hysical parameters of the theory, { V , η, α } as follows: β = 2 − D − α + η )( α + η ) + 2( D − ηγ = 2( α + η ) ( α + η ) + 2( D − C = − ( D − α + η γρ = V − η − αη α + αηC = V (( α + η ) + 2( D − ( D − α + αη )(2( D −
2) + α ( D − − η ( D −
3) + 2 αη ) (2.18)As we have no scale invariance, C receives a constant rescaling as we rescale r . For order oneradii, the validity of our classical analysis demands that C is small. This is essentially thephenomenological version of taking the large N limit: for an action without fluxes descendantfrom string theory, there is no clear concept of what N is, and instead we just insist thatthe gravity theory is classical. On the field theory side, this guarantees that the density ofdegrees of freedom is large.Without loss of generality we restrict η >
0, and equation (1.5) then implies that V > φ must diverge to positive infinity at the horizon.We elucidate the content of these expressions by noting the following: • When η = 0, β = 2: the scalar potential is constant, and our solutions become scaleinvariant. This framework enables us to consider AdS D ( α → + ∞ , γ = 2, no flux, φ constant), AdS × R D − ( α → , γ = 0, flux through R D − , φ constant), and themodified Lifshitz solution ( α arbitrary, 0 ≤ γ ≤
2, flux through R D − , φ ∼ ln r ) asformal limits. • In order for φ ( r ) → + ∞ for small r and the flux to be real, the bound on α in termsof some fixed η is − η < α < η − η (2.19)The lower bound says that β ≤
2, where saturation occurs only if the potential isconstant ( η = 0).To be certain that this scaling solution is within the domain of validity of a classical grav-itational treatment, we study the singularity structure of the spacetime. Calculation of theRicci scalar, squared Ricci tensor and Kretschmann invariant reveal a curvature singularity11t r = 0 for the general scaling solution: R = A r β − R µν R µν = A ( r β − ) R µνλσ R µνλσ = A ( r β − ) , (2.20)where A i = f i ( β, γ ) that can never simultaneously vanish. This reproduces the result thatthe extremal Lifshitz metric ( β = 2) has constant, finite curvature everywhere, but alsotells us that spacetimes with β < r , and in fact have asymptoticallyvanishing curvature invariants. We will return to this important fact in the next section,where we begin to discuss domain walls as holographic spacetimes. As for the singularity at r = 0, we will show the near-extremal generalization of this solution presently, shielding thesingularity in the usual manner. This theory also admits a finite temperature generalization of our scaling solution, wherebyone adds an emblackening factor to the metric that protects the singularity at the origin:now, U ( r ) = C r β (cid:16) − (cid:16) r h r (cid:17) ω (cid:17) , (2.21)where ω = β − γ D − , and all other fields and parametric definitions remain unchanged.Preservation of the correct metric signature is ensured for β > T = 14 π C ωr β − h . (2.22)The entropy per unit volume of the planar horizon is s ≡ SV R D − = 14 G D r γ ( D − ) h , (2.23)yielding an entropy density-temperature scaling relation, s ∼ T χ , χ = ( D − α + η ) D −
2) + ( α + η )( α − (2 D − η ) . (2.24) We also note that our solution, and its near-extremal generalization, have a pp singularity at the horizonwhich indicates geodesic incompleteness due to diverging tidal forces as measured by a freely falling observer.The Lifshitz spacetime is known to suffer from such a feature at its horizon as well; as there, one can acceptsuch singularities in hopes that they have some stringy resolution [33]. r h as the entropy density.By definition, χ > β > T = s = r h = 0extremal limit is obtained smoothly as we lower the temperature. One can arrange for aninfinite range of χ by changing the physical parameters of the theory.Let us quickly note two facts about this scaling:1. χ = D −
2, its free field value, when α = η − η , a D -independent result.2. For a given η , there is a one-parameter family of theories which have linear specificheat, χ = 1, for which α = 1 D − (cid:16) η (5 − D ) + (cid:112) η ( D − + 2( D − D − (cid:17) (2.25)This α is consistent with the bound (2.19). Generically, this value of α appears to haveno relation to any AdS geometry of the gravity dual.How do we use gauge/gravity duality in this situation? In the usual AdS/CFT corre-spondence, the behavior of the bulk fields maps onto properties of the dual field theory livingon the boundary. The area of the black hole horizon would correspond to the low temper-ature entropy of the theory whose IR physics is captured by the bulk dynamics at small r . When β = 2, AdS is the natural vacuum of the theory, and so such an interpretation ispossible, presuming an interpolating solution exists that patches the IR geometry onto anasymptotic AdS in the UV. This was found numerically in [26] for the extremal modifiedLifshitz solution. But in the context of a theory which does not admit an AdS vacuum, this frameworkobviously cannot apply. A global solution, if it exists, patches the IR dynamics onto someother UV spacetime: one must ask what this spacetime is, and whether a correspondencecan be set up which permits us to make such thermodynamic identifications of geometricquantities.This latter question is answered affirmatively as we turn to the solution which is theanalog of AdS in this theory, namely the domain wall spacetime.
Consider the parametric relations (2.18). Suppose our theory was such that α = η − η , andhence the scaling solution would have zero flux: then the metric would have β = γ ≡ γ (cid:48) .When γ (cid:48) = 2, this is of course AdS. When γ (cid:48) (cid:54) = 2, this is none other than a domain In Appendix A, we find it for the near-extremal Lifshitz solution. { α, η } .Ignoring supersymmetries, this spacetime can have, at most, Poincar´e symmetry. Whereasasymptotically AdS solutions are gravity duals to conformal field theories, asymptoticallydomain wall solutions are gravity duals to theories with only a relativistic symmetry. Thedecrease in elegance in establishing a correspondence in this case is countered by the increasein the number of real-world systems to which this treatment is applicable.We first review the details of domain walls with an eye toward establishing the holographiccorrespondence. In the process, we show why domain walls with γ (cid:48) < The most general single-scalar domain wall solution can be written in the form ds = ( Ar ) γ (cid:48) η µν dx µ dx ν + dr ( Ar ) γ (cid:48) φ ( r ) = C (cid:48) ln Ar (3.1)In our model, the parametric definitions are γ (cid:48) = 42 + η ( D − C (cid:48) = − ηγ (cid:48) D − A = 4 V D − γ (cid:48) ( Dγ (cid:48) − . (3.2)The forms of γ (cid:48) and C (cid:48) can be obtained from the scaling solution parameters γ and C ,respectively, by substituting α = η − η (i.e. the ρ = 0 condition) into the expressions (2.18).The form of A has a similar relation to its counterpart, C , which is only manifest in agauge in which the constant part of the scalar is chosen to vanish: defining a new radialcoordinate R = Ar (3.3)and shifting the boundary coordinates as { t, x i } → A { t, x i } (3.4)14e have the solution ds = A R γ (cid:48) η µν dx µ dx ν + dR A R γ (cid:48) φ ( r ) = C (cid:48) ln R (3.5)Indeed, A = C ( α → η − η ).Let us stress that the aforementioned substitution for α is only a parametric manipulationdesigned to derive the form of the vacuum solution; the solution exists for any values of { α, η } consistent with other physical principles mentioned elsewhere.One can perform a perturbative analysis in the parameter η by considering η (cid:28)
1, whichmeans a potential that is effectively constant over a large range of φ . Because η entersquadratically in the exponent γ (cid:48) but linearly in C (cid:48) , the solution (3.1), to first order in η ,looks like AdS with a slowly rolling scalar field . The scalar Klein-Gordon equation expandedabout the AdS background to first order in η , (cid:3) φ = − V η + O ( η ) , (3.6)reveals logarithmic behavior for φ near the boundary, to leading order in r .While this domain wall solution makes clear its relation to our scaling solutions andto Dp-brane geometries, we wish to briefly make contact with other literature on solitonicsupergravity domain walls by rewriting this metric in new, conformally flat coordinates: aharmonic function on the (one-dimensional) transverse space multiplying a Minkowski lineelement plus a “radial” term, describing a solitonic D − y , we can write the metric as ds = H ( y ) x ( η µν dx µ dx ν + dy ) , (3.7)which defines the harmonic function and radial coordinate as( Ar ) γ (cid:48) = H ( y ) x , ± dy = dr ( Ar ) γ (cid:48) . (3.8)Assuming that γ (cid:48) (cid:54) = 1, integrating gives H ( y ) = (1 ± my ) , x = γ (cid:48) − γ (cid:48) (3.9)where we define m = A (1 − γ (cid:48) ) (3.10)and have chosen the integration constant such that the constant in H ( y ) is equal to one.The full solution, with the scalar field, is thus ds = H ( y ) γ (cid:48) − γ (cid:48) ( η µν dx µ dx ν + dy ) e φ = H ( y ) − η D − γ (cid:48) − γ (cid:48) (3.11)15he likeness of this solution to the noncompact part of Dp-brane metrics along with therunning scalar is no coincidence, because various dimensional reductions of 10- and 11-dimensional maximal supergravities give rise to a panoply of such solitons in the corre-sponding lower-dimensional supergravities [34]. The interpolating structure is evident: themetric asymptotes to flat space as y →
0, and to near-horizon form with the asymptoticallyflat part decoupled as y → ±∞ . Note the ± sign in H ( y ), which enables one to constructdomain walls with Z symmetry.Further defining the constant ∆ = η − D − D − ds = H ( y ) D − ( η µν dx µ dx ν + dy ) e φ = H ( y ) − η ∆+2 m = − V (∆ + 2)
2∆ (3.13)This matches the solution in [32], for example.When γ (cid:48) = 1, we instead have the relation H ( y ) x = Ar = e ± Ay , (3.14)again with an integration constant chosen for simplification. Now, η = 2 D − , A = 2 D − (cid:112) V , ∆ = − ds = e ± D − √ V y ( η µν dx µ dx ν + dy ) ,φ = ∓ (cid:114) V D − y (3.16)Returning to the form (3.1), one sees that, at least for γ (cid:48) >
1, the domain wall spacetimeis well-suited for holographic correspondence. In accord with the discussion in section 2,it is well-behaved at large r , and more importantly, it has a boundary in the AdS sense.Specifically, the coordinate time to r = ∞ along a null (timelike) geodesic is finite (infinite),despite an infinite proper radial distance to the boundary from any point in the interior: (cid:90) t f t i dt = (cid:90) ∞ r i drr γ (cid:48) = finite (cid:90) ds = (cid:90) ∞ r i drr γ (cid:48) = ∞ (3.17)16rucially, the domain wall spacetime is conformal to AdS: ds AdS = r γ (cid:48) − ds DW = Ar γ (cid:48) − η µν dx µ dx ν + dr Ar (3.18)One should think of this metric as AdS in “interpolating coordinates”, where the value of γ (cid:48) determines whether the boundary lies at r → ∞ or r = 0. That is to say, γ (cid:48) = 0 gives AdSin conformally flat coordinates with a boundary at r = 0, γ (cid:48) = 2 gives AdS in coordinateswith the boundary at r → ∞ , and other values on either side of γ (cid:48) = 1 simply stretch oneof these limiting spacetimes. These interpolating coordinates provide a convenient heuristic to understand why domainwalls with γ (cid:48) < r ) = r − γ (cid:48) . This vanishesat r = 0. Therefore, if we start from an AdS metric with the boundary at r = 0, the map tothe domain wall spacetime will not be faithful because the infinite volume near the boundaryis cancelled by the degenerating conformal factor.In addition to not having a boundary, the extremal γ (cid:48) < ds = − ( Ar ) γ (cid:48) (1 − ( r h r ) ω (cid:48) ) dt + dr ( Ar ) γ (cid:48) (1 − ( r h r ) ω (cid:48) ) + ( Ar ) γ (cid:48) dx i · dx i , (3.19)where ω (cid:48) = D γ (cid:48) − , (3.20)we see that as γ (cid:48) → D , A diverges and the emblackening factor becomes identically one forall temperatures. When γ (cid:48) < D , A becomes imaginary, so there is clearly some region of γ (cid:48) < γ (cid:48) < Just as the AdS/CFT correspondence was fundamentally built on the physics of D3-branesand subsequently extended to apply to a priori unrelated asymptotically AdS spaces, the The value γ (cid:48) = 1 is once again special, as the metric is not AdS; we recognize this from the cases of thenear-horizon metrics of NS5 and D5-branes of type II supergravity, a connection we discuss more later. p <
6, there is a formally well-defined limitin which gravity decouples at intermediate energies, and therefore one can describe theworldvolume theory using the strong field dynamics of the supergravity solution. This theory,the toroidal reduction of D = 10, N = 1 super-Yang-Mills to p + 2 dimensions, has adimensionful coupling, g Y M , that runs with scale. This maps to a variable bulk dilaton.Consider the type II supergravity bosonic string frame action with the NS 2-form set tozero, S = 1(2 π ) l s (cid:90) d x √− g (cid:18) e − φ ( R + 4( ∂φ ) ) − p + 2)! F p +2 (cid:19) (3.21)A Dp-brane is electrically charged under the RR field strength as F p +2 = dA p +1 . The D-branesolution is ds = 1 H p ( r ) / ds M p, + H p ( r ) / ( dr + r d Ω − p ) e φ = g s H p ( r ) (3 − p ) / A p +1 = g − s ( H p ( r ) − −
1) (3.22)with H a harmonic function on the transverse (9 − p )-dimensional space, chosen as H p ( r ) = 1 + c p g s N l − ps r − p (3.23)where c p is a p -dependent constant chosen to satisfy Maxwell’s equation, d (cid:63) F p +2 = 0. N isthe quantized RR flux through S − p .We now take the low energy, l s → g Y M N , where g Y M = g s l p − s . We also take g s →
0, hence working at tree level in the closed loop string perturbationexpansion. Doing so reveals that, as for p = 3, we decouple the asymptotically flat regionand end up with a near-horizon geometry with metric ds = (cid:18) r − p c p g s N l − ps (cid:19) / ds M p, + (cid:18) c p g s N l − ps r − p (cid:19) / ( dr + r d Ω − p ) (3.24)This geometry is a warped product of a p + 2-dimensional domain wall with an (8 − p )-spherewith an r -dependent radius. For p <
3, the curvature is well-behaved at small r , but theeffective string coupling e φ blows up; for p >
3, the situation is opposite. The domain wall and sphere parts of the metric contribute to the curvature divergence in concert, sothat one need only know where the sphere becomes small to know where the curvature blows up. p +2 × S − p : (cid:101) g µν = ( N e φ ) p − g µν ∼ ( g Y M N ) − r − p g µν (3.25)and so, defining a new so-called “horospherical” radial coordinate u ∼ ( g Y M N ) − r − p (3.26)this metric reads as (cid:101) ds ∼ u ds M p, + du u + d Ω − p (3.27)This suggests identification of the radial coordinate u with the energy scale of the worldvol-ume SYM theory, naturally incorporating the energy-distance relation of Dp-brane super-gravity probes [15]. This frame allows a simple sphere reduction ansatz, in which the fluxis through the sphere and the D=( p + 2)-dimensional action consists only of the universalsector with gravity and the scalar alone. Passage back to the ( p + 2)-dimensional Einsteinframe reveals a domain wall metric.One can calculate the near-extremal entropy of the ( p + 2)-dimensional domain wallsolution compactified on a torus with S radii L , S = A G D ∼ L p N ( g Y M N ) p − − p u − p − p , (3.28)which is the correct result [13]. Written in terms of an effective dimensionless Yang-Millscoupling g eff ( E ) = g Y M
N E p − , (3.29)and identifying E ∼ u , the entropy is S ∼ L p N E p ( g eff ( E )) p − − p . (3.30)In this form, the departure from conformality when p (cid:54) = 3 is clearest.We have only sketched the derivation, having been cavalier about factors of l s and left outthe conformal frame definitions of φ and F p +2 . What we wish to emphasize are the followingthree things:1. The scalar field is nonconstant in the dual frame, an obvious fact since the conformaltransformation on the metric leaves the scalar alone. The one exception to this is the case p = 5, for which the variable u is ill-defined and the conformalgeometry is not AdS × S , but rather M , × R × S .
19. These brane solutions are simply special (and -supersymmetric) cases of the domainwalls we considered in section 3, where the value of η is given in terms of p = D − γ (cid:48) = 2(9 − p ) p − p + 18 (3.31)Alternatively, they are the vacua of theories with η = 2( p − p (9 − p ) (3.32)As a check, we note that for p = 3, γ (cid:48) = 2 and η = 0 as required. We also pointout that as anticipated earlier, the case p = 5 gives a 7-dimensional domain wall with γ (cid:48) = 1, which is not conformal to AdS .We present the fourth reason why domain walls with γ (cid:48) < p >
5, for which γ (cid:48) <
1, there is not a well-defined brane decouplinglimit. All branes for which there is such a limit have γ (cid:48) ≥
1, by inspection of (3.31).Therefore, when γ (cid:48) < ds = 1 H p ( r ) / (cid:0) − f ( r ) dt + dx i · dx i (cid:1) + H p ( r ) / (cid:18) dr f ( r ) + r d Ω − p (cid:19) (3.33)where f ( r ) = 1 − (cid:16) r h r (cid:17) − p (3.34)Performing the above conformal transformation (3.25), and transforming once againto the u coordinate (3.26), yields (cid:101) ds ∼ u ( − f ( u ) dt + dx i · dx i ) + du u f ( u ) + d Ω − p (3.35) The case of γ (cid:48) = 1 itself – that is, fivebranes – does admit a decoupling limit, though it requires separateanalysis. D0 branes also have γ (cid:48) = 1, but our treatment of domain walls does not apply to two dimensionsin which several of our formulae break down; from the standpoint of the D0-brane theory itself, the dualityis not problematic. f ( u ) = 1 − (cid:16) u h u (cid:17) − p )5 − p (3.36)The noncompact part of this geometry is at finite temperature, but is not AdS-Schwarschild. Upon reduction to D dimensions and passage back to the Einstein frame,one obtains the D-dimensional finite temperature domain wall solution (3.19) with ω (cid:48) = 2 p (7 − p ) p − p + 18 (3.37)This is positive for all 0 < p <
7. As a check, we note that for p = 3, ω = 4,corresponding to AdS -Schwarzchild.Having elaborated on the relevant properties of domain walls, we return to the goal athand: construction of interpolating solutions between the finite temperature scaling solutionand the domain wall. The interpolation we construct connects the two spacetimes (2.17) and (3.1), along with theirassociated scalar and gauge fields. This is done numerically, using a shooting technique.Let us summarize the method. The equations of motion fix the relation between fields atthe horizon, r = r h , save for some number of free parameters whose horizon values determinethe full set of initial data. One must also use symmetries of the horizon metric to eliminategauge degrees of freedom. Thusly, we can vary the temperature of the black hole and,ultimately, the boundary conditions at large r by varying the horizon values of fields.As a numerical matter, one must start integration at some small distance outside thehorizon to seed the perturbations, so we develop the fields in a power series about thehorizon. Integrating out to infinity, one finds that for sufficiently low temperatures, themetric looks like the finite temperature scaling solution over a large but finite range of r ,after which the growing perturbations become large enough to backreact upon the metricand induce its asymptotic form.Consider the solution at the horizon. By definition of the horizon, and by separaterescalings of t and the boundary coordinates { x i } , we can fix U ( r h ) = 0 , U (cid:48) ( r h ) = V ( r h ) = 1 . (4.1)(This gauge is only appropriate for nonzero temperatures.) We expand the fields to second-21rder in the expansion parameter (cid:15) , U ( r h + (cid:15) ) ≈ (cid:15) + u (cid:15) + . . .V ( r h + (cid:15) ) ≈ v (cid:15) + v (cid:15) + . . .φ ( r h + (cid:15) ) ≈ φ + φ (cid:15) + φ (cid:15) + . . . (4.2)with Maxwell’s equation determining the field strength to arbitrary order in terms of theseexpansions, A (cid:48) t = Qe αφ V D − (4.3)Then the field equations give the following relations in terms of two free parameters, { φ , Q } : v = (cid:16) D − (cid:17)(cid:16) V e ηφ − Q e − αφ (cid:17) φ = − (cid:0) V ηe ηφ + 2 Q αe − αφ (cid:1) v = 12 v − (cid:16) D − (cid:17) φ φ = (cid:16) D − (cid:17)(cid:16) v − v (cid:17) · v φ u = (cid:16) − V η φ e ηφ + 2 Q αe − αφ ( αφ + ( D − v ) − D − v − φ (cid:17) · φ (4.4)At large r , the field equations near the domain wall boundary dictate the falloff of the fieldsas U ≈ U c r γ (cid:48) + . . .V ≈ V c r γ (cid:48) + . . .φ ≈ C (cid:48) ln r + φ c + . . .A t ≈ µ + ρr αC (cid:48) + γ (cid:48) D − + . . . (4.5)where we have kept only a handful of leading and subleading terms. We are free to normalizeour horizon coordinates such that the asymptotic domain wall is Poincar´e symmetric, U c = V c . (We do so with the understanding that the entropy density scales accordingly – seeAppendix A for a deeper discussion of this issue). It is our task, then, to show this falloffnumerically.The constants { U c , φ c } in the numerical large r asymptotic region will not necessarily beequal to those in (3.1): we need to present a more general domain wall solution which allowsus to account for differences in gauge between the horizon and infinity. Taking r → rc and22ormalizing the boundary coordinates appropriately gives a solution ds = c − γ (cid:48) ( Ar ) γ (cid:48) η µν dx µ dx ν + dr c − γ (cid:48) ( Ar ) γ (cid:48) φ ( r ) = C (cid:48) ln Ar − C (cid:48) ln c (4.6) c is an unconstrained gauge parameter which relates asymptotic constants in the simulation.Solving for c yields the relation U c = A e − (cid:18) − γ (cid:48) C (cid:48) (cid:19) φ c (4.7)If we set η = 0, then we will interpolate between the modified Lifshitz solution and AdS.This was done in [26] at zero temperature by solving for the exact form of the perturbationto linear order. In appendix A, we construct the analogous numerical solutions for the finitetemperature modified Lifshitz solution, showing that at successively lower temperatures, thepower law relation (2.24) is increasingly obeyed. The lack of Poincar´e invariance in thezero temperature solution implies the presence of an analytically unknown coefficient in theentropy due to the stretching of the spacetime between the horizon and infinity; but as thedimensionless temperature ˆ T →
0, this coefficient stabilizes to its zero temperature value,and power law behavior is observed.Here we present the results of one simulation with η (cid:54) = 0. Specifically, we choose the setof parameters D = 4 , η = 12 , α = 1 , V = 6 , r h = 1 . (4.8)This translates to small r scaling solution parameters β = 3825 , γ = 1825 C = − , ρ = (cid:114) , C = 1875434 (4.9)and large r domain wall parameters γ (cid:48) = 85 , C (cid:48) = − , A = 7544 . (4.10)The interpolation is shown in figure 1. The upper plot shows the metric turnover: fora large but finite range in r , V ( r ) scales as r γ as seen in the slope of the plot. (Thoughwe did not show it, one can of course directly extract the IR power law relation for U ( r ) aswell.) At large r , both metric components scale with domain wall exponent γ (cid:48) = , wherethe turnover on the V ( r ) plot happens around ln r ∼ r based on the asymptotic scalings described above. In particular,in the large r domain wall region, we have A (cid:48) t ( r ) ∼ r − ∼ e φ (4.11)23y plugging the values (4.10) into the falloff (4.5).One can also extract the value of φ c from the scalar plot and compare it to the value U c (not shown); the numerical values for this trial, to six significant figures, are U c = 19 . , φ c = 4 . log(r) U(r)/V(r)V(r) r A t ’ r r e φ r Figure 1: Behavior of fields in the interpolating solution. a.
Upper : The metric component V ( r ) = g ii , as well as the ratio U ( r ) V ( r ) = − g tt g ii , rescaled to fit on the same graph. The turnoverin the slope of the former curve, and the flatness of the latter, indicate the domain wallasymptotics. b. Lower left : The field strength, multiplied by its asymptotic domain wallpower of r . c. Lower right : The exponentiated scalar field, multiplied by its asymptoticdomain wall power of r . 24 Generalized scale invariance
Having shown that the scaling solutions (2.17) can indeed asymptote to domain walls (3.1) toform a global geometry, we ask what domain wall holography can tell us about this solution.Our scaling solution is not scale-invariant for η (cid:54) = 0, but its thermodynamics are governedby power law relations. One can ask what aspects of this IR scaling behavior can be explainedin connection to the symmetries of the domain wall. More specifically, the UV domain wallspacetime and hence the theory at hand has Poincar´e symmetry, but also the generalizedconformal structure explained earlier. It is worthwhile to consider if the IR scaling solutionhas any such “hidden” symmetry as well: although the scaling solution is not scale invariant,one might expect it to possess a “generalized scale invariance,” manifest in the conformalframe defined by the domain wall.To motivate this prospect, one may think of the flow from the domain wall to the scalingsolution as a symmetry-breaking flow: the IR flux breaks the Lorentz symmetry of theasymptotic domain wall. But in Einstein frame, one cannot easily tell whether this particularsolution leaves the domain wall’s generalized scale invariance intact in the IR. That is tosay, the presence of the IR flux might be expected to preserve that part of the generalizedconformal structure which is independent of the Lorentz symmetry-breaking.We now show that this is indeed the case, by passage to the conformal frame.The DW/QFT correspondence says that we should use the UV domain wall geometryto determine the conformal factor, as discussed in section 3. This is easily done, writingthe factor in terms of r and converting it to an expression in terms of the field φ . Withthe conformal factor in hand, we note that φ has a different r -dependence in the IR scalingsolution, and therefore, the IR metric receives a different multiplicative factor in terms of r .Altogether, we have the new global conformal frame metric which interpolates between twoscale-invariant metrics: a modified Lifshitz solution in the IR and an AdS plus rolling scalarsolution in the UV.For clarity, we reproduce the interpolating solution in Einstein frame as follows. Themetric is ds IR = − C r β f ( r ) dt + dr C r β f ( r ) + r γ dx i · dx i ds UV = c − γ (cid:48) ( Ar ) γ (cid:48) η µν dx µ dx ν + dr c − γ (cid:48) ( Ar ) γ (cid:48) (5.1)and f ( r ) is defined in (2.21). The scalar field is φ ( r ) = (cid:40) C ln r r → C (cid:48) ln Ac r r → ∞ (5.2)25he field strength is A (cid:48) t ( r ) = ρe αφ V D − ∼ (cid:40) ρr − ( αC + γ D − ) r → ρr − ( αC (cid:48) + γ (cid:48) D − ) r → ∞ (5.3)We set c = 1 for simplicity, remembering that a generic choice of horizon coordinate nor-malizations will change this value.From ds UV , define (cid:101) ds UV = Ω( r ) UV ds UV (5.4)where Ω( r ) UV = ( Ar ) γ (cid:48) − (5.5)This will give us a UV AdS spacetime. In terms of φ , this is proportional to the potential,Ω( φ ) = e ηφ (5.6)On the global solution, therefore, Ω( r ) takes the following limiting forms:Ω( r ) = (cid:40) r β − r → Ar ) γ (cid:48) − r → ∞ (5.7)where we have made use of algebraic relations ηC = β − ηC (cid:48) = γ (cid:48) −
2, by inspectionof (2.18) and (3.2), respectively. Hence, the conformal frame metric is (cid:101) ds IR = − C r β − f ( r ) dt + dr C r f ( r ) + r β + γ − dx i · dx i (cid:101) ds UV = ( Ar ) γ (cid:48) − η µν dx µ dx ν + dr ( Ar ) (5.8)Both the IR and UV metrics now have scale invariance, anisotropic in the former andisotropic in the latter. In other words, the general scaling solutions have a generalized scaleinvariance, manifest only in this conformal frame. Actually, the IR metric has its scaleinvariance broken by nonzero temperature, but the emergence of finite temperature Lifshitzbehavior is still non-generic.Because these solutions lie in the IR of a global solution asymptoting to a domain wall,and because one uses the domain wall alone to determine the Weyl factor Ω( φ ), this gener-alized scale invariance follows from the generalized conformal structure of the UV domainwall, itself grounded in the identical structure of nonconformal D-branes.To clarify the solution, we can define separate new radial coordinates in the IR and theUV which will restore the gauge choice g tt = − g rr (cid:54) = g ii , thereby reinstituting horospherical26oordinates in the UV AdS spacetime. (Note that as it stands, the UV metric is AdS inpreviously defined interpolating coordinates, (3.18).) Define the IR coordinate u as( β − u = r β − (5.9)and the UV coordinate R as A ( γ (cid:48) − R = ( Ar ) γ (cid:48) − (5.10)Then the metric is (cid:101) ds IR = − (cid:101) C u f ( u ) dt + du (cid:101) C u f ( u ) + (cid:101) C u (cid:101) γ dx i · dx i (cid:101) ds UV = − ( (cid:101) AR ) dt + dR ( (cid:101) AR ) + ( (cid:101) AR ) dx i · dx i (5.11)with conformal frame parameters (cid:101) γ = β + γ − β − , (cid:101) C = C ( β − , (cid:101) C = ( β − (cid:101) γ , (cid:101) A = A ( γ (cid:48) −
1) (5.12)and an emblackening function f ( u ) = 1 − (cid:16) u h u (cid:17) (cid:101) ω , (cid:101) ω = ωβ − φ = (cid:40) (cid:101) C ln ( u ( β − r → (cid:101) C (cid:48) ln (cid:101) AR r → ∞ (5.14)where (cid:101) C = C β − , (cid:101) C (cid:48) = C (cid:48) γ (cid:48) − F = F µν dx µ ∧ dx ν , not just the tensor components. Doing so reveals an IR two-form F IR ∼ u D − γβ − du ∧ dt (5.16)and UV two-form F UV ∼ R − D − (cid:101) C (cid:48) ( α − η ( D − dR ∧ dt (5.17)The full conformal frame solution, then, is (5.11), (5.14), (5.16), and (5.17), with theassociated parametric definitions. Note that when β = γ (cid:48) = 2 – which would be the case27or the theory with constant potential, giving rise to an interpolation between the modifiedLifshitz solution and AdS – the conformal frame and the Einstein frame are identical, andall of these formulae reduce to tautology.The two radial coordinates u (in the IR) and R (in the UV) dualize to the energy scaleof the field theory, each in its respective regime. If we are to consistently identify thebulk radial direction with the energy scale, then it must be true that the two coordinatessmoothly interface at intermediate energies: when u decreases, so must R , and both musthave a strictly positive range. The coordinates share the relation u = A R ( β − / ( γ (cid:48) − , (5.18)where A = β − ( A − γ (cid:48) ( γ (cid:48) − β − γ (cid:48)− . When β > γ (cid:48) >
1, the relationship is as required.By definition, the conformal frame solutions are those of the conformal frame action,with Lagrangian density˜ L = (cid:112) − ˜ ge − η ( D − ) φ (cid:16) ˜ R + (cid:16) η D − D − (cid:17) ∂ µ φ ˜ ∂ µ φ e ( α + η ) φ F µν ˜ F µν − V (cid:17) The tilde’s in this action, as usual, represent quantities calculated using the conformal framemetric, ˜ g µν . One can easily confirm that the gauge field scalings (5.16) and (5.17), forexample, satisfy Maxwell’s equation, ∂ µ (cid:16)(cid:112) − ˜ ge ( α − η ( D − φ F µν (cid:17) = 0 (5.19)To summarize, the fact that we recover the finite temperature modified Lifshitz solution inthe IR conformal metric (5.11) indicates that the dynamics of the dual relativistic field theoryat low energies are determined in some fashion by those of scale-invariant fixed points. Because this connection is a direct result of the inherent structure of the domain wallmetric, it would be interesting to answer the question, “what kinds of matter Lagrangianscoupled to the universal Einstein-scalar sector can support solutions without generalizedscale invariance?” We elaborate on this in the conclusion.
The solution we have constructed interpolates between two exact solutions of our single-exponential, domain wall gravity theory. We found ourselves restricted to such a theory We note, without clear understanding of its meaning, that for β > γ (cid:48) > γ = z <
2, which is the ˜ z > S = − πG D (cid:90) d D x √− g (cid:16) R + f ( φ ) F µν F µν + 12 ( ∂φ ) + V ( φ ) (cid:17) , (6.1)As noted in the introduction, this is because the scaling ansatz, which models generic near-horizon behavior of zero entropy extremal solutions, singles out the term in some general V ( φ ) that dominates at small r . Including subleading terms in r in our ansatz – essentiallynear-horizon corrections – would turn on other contributions to the potential.For these reasons, our numerical work has broader suggestive power, specifically to the-ories with scalar potentials that are dominated by a single exponential term over a largerange of field space but have an AdS critical point. We give a schematic example in theintroduction, equation (1.7); more generally, we refer to the behavior V ( φ ) ∼ (cid:40) − V e ηφ r → − V (cid:48) r → ∞ (6.2)Suppose that our theory was UV-completed with a full string/M-theory-derived potentialwith this behavior; the scaling solution of this paper would then be only approximate. If thescaling solution at small r could be patched onto an approximate domain wall solution atintermediate r whose only nonzero matter field is the scalar, then the interpolation betweenthe domain wall and the AdS critical point at large r would be guaranteed: the scalar wouldjust roll toward its AdS critical point, in the manner of an RG flow. All other fields areturned off.Whether the interpolation could actually be constructed in this imaginary theory is amatter of numerics, not argument: one cannot assume the existence of an interpolating so-lution in a given theory without actually constructing it. But it seems reasonable that givena stringy effective action, a more complicated extremal solution to that theory which ap-proaches our scaling solution in the near-horizon limit would interpolate to the approximatedomain wall solution of that theory at intermediate energies. This should also be true foran ad hoc action with a scalar potential as in (6.2).In fact, this sort of analysis leads one to consider DW/QFT as an effective holographictool, applicable in settings far more general than domain wall supergravities, where we use“effective” in the field theory sense. Our assertion is that even when the UV completion ofsome bulk theory is unknown, if that theory admits an approximate domain wall solution atsome intermediate value of r then one can use DW/QFT to develop a holographic map.Let us explore this idea.The central tenet of effective field theory, informed by the philosophy of the renormaliza-tion group, is that physics at low energies should not be sensitive to physics at high energies.29aively, AdS/CFT seems to violate this idea: given an effective field theory, one needs toknow something about its UV physics, namely whether the theory is conformal, in order toknow whether it has a well-defined IR gravity dual. Of course this is an incorrect mode ofthought, because gauge/gravity duality, it can be said, is not a right, but a privilege of con-formal field theories (and their deformations and subsequent holographic extensions). Thereis no a priori reason why every field theory should have a gravity dual. From what we knowso far, one could argue that classifying the types of field theories which might have gravityduals boils down to finding bulk spacetimes with boundaries, in which case the attendantisometries dictate the field theory symmetries.Actually, we can consider an elementary case in which the bulk has no AdS solution inthe UV and we know how to treat it holographically: the case of a positive mass scalarfield. As a bulk field dual to an irrelevant CFT operator, the scalar blows up at large r .At the radius at which its backreaction ruins the AdS asymptotics, we work with a radialcutoff, dual to working with an effective field theory below the corresponding energy scale.To incorporate the backreaction of the scalar would be to find the full UV completion of thetheory; in its absence, one works with the effective AdS boundary and proceeds essentiallyas usual.Analogously, one is entitled to use DW/QFT in settings beyond those in which thedomain wall geometry is a true vacuum of the theory – one can work with a cutoff boundaryat finite r , at which the bulk fields will have falloffs characteristic of a domain wall boundary.This is exactly what one does in the case of nonconformal D-branes, fundamental strings,and NS5 branes: the supergravity approximation cannot be trusted for large (or small) r ,i.e. high (or low) energies. Of course, for some of these branes, the UV completion is givenin terms of M-theory solutions in one greater spacetime dimension, which is interesting inand of itself: even in cases like the D4-brane where we know that the solution uplifts to anAdS × S vacuum in the UV, we can still define an effective ten-dimensional nonconformalholography. This M-theory resolution of strong effective ten-dimensional string coupling isobviously a deep and special case, but it alludes to the general possibility that even when apotential (borne from string/M-theory compactification or otherwise) has no AdS vacuum,DW/QFT can be used at intermediate energies.To rephrase, accepting the validity of nonconformal holography demands that it shouldhave the same role as AdS holography in situations where the bulk must have a UV radialcutoff. In the absence of a continuum limit, even in AdS/CFT, one cannot be sure thatholography is describing a theory that is not sick; this is no different in the nonconformalcase and permits us to extend its regime likewise.We can summarize when it is safe to use domain wall holography in theories that admit(at least approximate) domain wall vacua.1. When a phenomenological theory admits an exact domain wall solution, like our theory302.16), domain wall holography is on firm footing. Without a string/M-theory embed-ding, the theory is not quantumly well-defined; when an exact domain wall solution does arise in a consistent truncation of string/M-theory, domain wall holography isvalid even on the level of quantum corrections.In fact, our action (2.16) can indeed be obtained in such a manner, where α and η arefixed by the compactification. Specifically, [37] showed that starting with the canonical S , S and S sphere compactifications of string and M-theory, one can deform thesespheres by taking certain moduli to a limit in which the compactification on S n becomesone on S a × R b , where a + b = n . The resulting D-dimensional effective Lagrangian,suitably (and consistently) truncated to a single scalar, reads S = − πG D (cid:90) d D x √− g (cid:16) R + e − √ D − φ F µν F µν + 12 ( ∂φ ) − V e √ D − φ (cid:17) ; (6.3)that is, − α = η = (cid:113) D − . Unfortunately, our scaling solution does not solve theseactions – it reduces to the AdS × R D − solution – and so we make no use of them inthis instance.2. When a bulk action has a potential V ( φ ) which has a large r AdS vacuum, we canstill use domain wall holography at low and intermediate energies if V ( φ ) admits anapproximate domain wall solution there, as exemplified in (1.7) and (6.2). (Again,issues of UV completion come to bear on whether this is a microscopically allowedmap.) The bulk fields that run with scale in the domain wall region eventually stabilizein AdS. For instance, in nonconformal ( p + 1)-dimensional super-Yang-Mills, the two-point function of a ∆ = p + 1 operator dual to a massless scalar field, as shown in [16],has scale-dependence determined by generalized conformal invariance as (cid:104)O ( x ) O (0) (cid:105) ∼ N ( g eff ( | x | )) p − − p | x | (6.4)where g eff ( | x | ) = g Y M N | x | − p , essentially as defined in (3.29). In the domain wallregion, this function runs with scale; at the eventual AdS critical point, g eff runssmoothly to a constant and the conformal structure of the two-point function is recov-ered.In connection to the scaling solution studied in this paper, these arguments serve tosuggest that we do not need to know whether the potential V ( φ ) has an AdS vacuum atlarge r in order to know something about the universal behavior of the dual field theory.As a final point, the relation between higher-dimensional AdS solutions of M-theory andlower-dimensional domain wall solutions of string theory may be a more general aspect of31he existence of DW/QFT. In [19], the authors show that any D-dimensional domain wallgravity theory with action S = − πG D (cid:90) d D x √− g (cid:16) R + 12 ( ∂φ ) − V e ηφ (cid:17) , (6.5)can be derived by toroidal dimensional reduction of a pure AdS gravity, where the toroidaldimension is related to the value of the potential parameter η . This leads one to speculatethat all holography is intimately connected to the existence of some AdS vacuum, be it inthe same spacetime dimension as the theory one is considering or otherwise. This would bean interesting conclusion that would generalize the way the strong coupling singularities oftype IIA solutions are resolved.
To recapitulate, the scaling behavior of low temperature, relativistic quantum field theorieswith finite charge density with IR gravity duals (5.1) can be understood via domain wallholography and an inherited generalized scale invariance (broken by the finite temperature).The power law s-T relation of the bulk ansatz encodes the physics of a system with a uniquezero temperature ground state, though the thermodynamic description breaks down due toa physical singularity in the extremal limit.We have numerically constructed the interpolating solution between the near-extremalscaling solution in the IR and the asymptotic domain wall vacuum in the UV, thus permittingthe mapping of the near-horizon physics to the low-energy dynamics of the dual field theory.The formalism developed in [16, 18] ensures that the holographic relation is faithful andthermodynamically well-defined.We also made some comments on the nature of domain wall holography for bottom-up actions with no connection to type II supergravity, delineating which domain walls areamenable to holography and arguing for an effective role of domain wall holography insettings where an exact domain wall vacuum does not exist.Let us emphasize that the generalized scale invariance of the IR solution followed, viathe construction of the full interpolating solution, directly from the generalized conformalinvariance of domain walls which is itself descendant of M-theory. The D4 brane of 10-dimensional, type IIA supergravity descends from the 11-dimensional solitonic M5-branewrapped on the M-theory circle, for example, and similarly for the IIA fundamental stringfrom the M2-brane. By using S- and T-duality on, say, the D4 brane supergravity solu-tion, one can generate all branes of type II supergravity, including the D-branes of course;therefore, insofar as one defines their generalized conformal structure as the presence of ametric conformal to AdS p +2 × S − p , the entire domain wall holography has a non-perturbative32onnection to M-theory in this way. Domain walls with no necessary relation to D-branenear-horizons should still be considered as a rung on this non-perturbative ladder, just asany asymptotically AdS spacetime can be treated holographically in its own right.It would be interesting to investigate what matter Lagrangians, when coupled to anEinstein-scalar sector with a domain wall vacuum, would break this generalized scale invari-ance. One might phrase this as follows. Suppose our potential is still given as V ( φ ) = − V e ηφ . (7.1)We know that the domain wall metric will be conformal to AdS via the conformal factorΩ( φ ) = e ηφ , so the Einstein frame metric ds = − r e ηφ dt + dr r e ηφ + V ( r ) dx i · dx i (7.2)will have conformal frame scale invariance. What type of matter Lagrangian can supportthis metric?It would also be worthwhile to find a string/M-theory embedding of our scaling solution,or a generalization of it. The string theory-derived effective action (6.3) which we presentedearlier could not accomodate it, but there are large families of similar actions of domain wallgravities, as laid out in [38]. The domain wall would generally be supported by some numberof scalars, which would be no impediment to use of DW/QFT.Lastly, it would clearly be nice to delve deeper into the thermodynamics of this system,though that has largely been done in the recent papers cited earlier [29, 30]. Perhaps addingfermionic degrees of freedom to the theory would be worthwhile as well. Acknowledgments
We thank Per Kraus for frequent guidance, crucial insight and consistent support. Thiswork was supported in part by a Leo P. Delsasso Fellowship from the UCLA Department ofPhysics and Astronomy.
A Numerical study of finite temperature modified Lif-shitz solution
We clarify a lingering issue in the extraction of field theory thermodynamics from interpolat-ing gravity solutions, as we present the numerical construction of the interpolating solutionfor a finite temperature black hole with IR Lifshitz scaling. This work was done with Per Kraus. ds = − U ( r ) dt + dr U ( r ) + V ( r ) dx i · dx i (A.1)For an extremal black hole with Lifshitz scaling, r → rλ , t → λt , x i → λ z x i (A.2)the IR behavior of the metric is determined by this scale invariance up to a constant: ds = − r l dt + l r dr + ξ (cid:16) rl (cid:17) /z dx i · dx i , (A.3)where ξ is a constant. The solution is not Poincar´e invariant. As noted earlier, this interpo-lating solution was numerically constructed in [26] for the D = 4 extremal solution, wherethe authors solved the linearized perturbation equations. The metric goes as U ( r ) = (cid:26) ( r/l ) r → r/L ) r → ∞ (A.4)and V ( r ) = (cid:40) ξ ( r/l ) /z r → r/L ) r → ∞ (A.5)where l and L are the characteristic length scales of the Lifshitz and AdS geometries, re-spectively. ξ is not fixed by any symmetry.In thinking about the finite temperature solutions, one can view them as being “gluedin” to the zero temperature background, which is to say that the near-extremal black holeshave extremal asymptotics: ds = − r l f ( r ) dt + l r dr f ( r ) + ξ (cid:16) rl (cid:17) /z dx i · dx i , (A.6)where f ( r ) = 1 − (cid:16) r h r (cid:17) ω and ω = 1 + D − z . In the context of an interpolating solution, thisis only strictly true in the infinitesimal temperature limit: as one raises the temperature,34he horizon extends outward toward the IR extremal asymptotic region so that in the fullglobal solution, the UV asymptotics will change. The D-dimensional Lifshitz black holes weare considering in this paper are supported by a scalar field and a U(1) gauge field; theirentropy densities, as determined by scaling symmetry, will have the formˆ s = cf ( ˆ φ ) ˆ T D − z (A.7)where hats denote dimensionless quantities, made with appropriate powers of, in our case, thecharge density. What we wish to show numerically, as a consequence of the above, is that asone lowers the dimensionless temperature ˆ T of the black hole while keeping the dimensionlesssource ˆ φ fixed, the combination cf ( ˆ φ ) asymptotes to a fixed value, determined by the valueof ˆ φ and ξ : ˆ T → , ˆ φ fixed ⇒ ˆ s ˆ T D − z → Constant (A.8)We form dimensionless parameters with the asymptotic charge density ρ , defined by theusual AdS asymptotics as A (cid:48) t ( r ) = ρr D − + . . . 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