New Massive Gravity Holography
aa r X i v : . [ h e p - t h ] M a r New Massive Gravity Holography
G.M.Sotkov ∗ , C.P. Constantinidis ∗ and U. Camara dS ∗ ∗ Departamento de F´ısica - CCEUniversidade Federal de Espirito Santo29075-900, Vitoria - ES, Brazil
ABSTRACT
We investigate the holographic renormalization group flows and the classical phase transitionsthat occur in two dimensional QFT model dual to the New Massive 3D Gravity coupled to scalarmatter. Specific matter self-interactions generated by quadratic superpotential are considered. Theoff-critical
AdS /CF T correspondence determines the exact form of the QF T ’s β -function andthe singular part of the reduced free energy. The corresponding scaling laws and critical exponentscharacterizing the RG fixed points as well as the values of the mass gaps in the massive phases areobtained.KEYWORDS: new massive gravity, AdS/CFT correspondence, phase transitions. e-mail: [email protected], [email protected] e-mail: [email protected] e-mail: [email protected] ontents The
AdS d +1 /CF T d correspondence [1] provides holographic description of the d = 4 SU ( N ) super-symmetric large N gauge theories and its off-critical ( a ) AdS /QCD version is expected to solvethe problem of the strong coupling regime of QCD [2]. In this context the two dimensional caserepresents rather “non-physical” problem, which however is known to be of conceptual importance.Since two dimensional (super) conformal group is infinite, the specific features of its unitary repre-sentations [3] allow to exactly calculate all the anomalous dimensions and the n -points correlationfunctions of all the primary and composite fields. Another important fact of purely 2 D nature isthe existence of a vast variety of integrable perturbations of the corresponding CF T ’s [4], as forexample (super)sine-Gordon and the abelian affine (super) Toda models [5], [6], whose S -matrices,mass spectra, form-factors and some correlation functions are known exactly [7]. Apart from thepractical use of all these 2 D models in the description of real condensed matter systems [8], thehuge amount of available exact results also permits to realize non-trivial self-consistency checks ofthe (eventual) validity of the off-critical AdS /CF T correspondence even out of its original super-string/supergravity/SUSY gauge theories frameworks.In what concerns the lessons one can learn about the corresponding realistic higher dimensional d = 4 models, we should mention however one serious disadvantage when 3D Einstein gravity of neg-ative cosmological constant is used as 3D “bulk” gravity. Since it has no local degrees of freedom itsproperties as well as the ones of its 2D dual are rather different from the properties of corresponding d + 1 = 5 versions. It is therefore interesting to study examples of the off-critical AdS /CF T corre-spondence based on appropriate extensions of the Einstein 3D gravity, that have features similar tothe ones of 4D and 5D Einstein gravity such as “propagating gravitons”, non-trivial vacua solutions,etc. The simplest model of such extended 3D gravity is given by the following “higher derivatives”1ction, called New Massive Gravity (NMG) [9]: S NMG ( g µν , σ ; κ, Λ) = 1 κ Z d x √− g n ǫR + 1 m K − κ (cid:16) | ~ ∇ σ | + V ( σ ) (cid:17)o (1.1) K = R µν R µν − R , κ = 16 πG, ǫ = ± K terms above as one loop counter-terms appearing in the perturbativequantization of 3D Einstein gravity. As it was recently shown by Bergshoeff, Hohm and Townsend(BHT) [9] the above model in the absence of matter , i.e. σ = const , unlike the case of higherdimensional D = 4 and D = 5 Einstein gravities with one loop counter-terms added, turns outto be unitary consistent (ghost free) and super-renormalizable for the both choices ǫ = ± R -term, under certain restrictions on the values of the cosmologicalconstant Λ = − κ V ( σ ∗ ) and of the new mass scale m .The problem we are interested in concerns the classical critical phenomena that take place in the(euclidean) QF T ’s dual to NMG model (1.1) and more precisely the phase transitions that occur in2D classical statistical mechanics models in infinite volume, whose thermodynamical limits representmodels dual to NMG. According to the off-critical AdS /CF T correspondence [10] the domain wallsolutions (DW’s) of 3D gravity models of negative cosmological constant provide an alternative“dual” description of the renormalization group (RG) flows in specific 2D deformed conformal fieldtheories CF T ’s, which can be described as as appropriate CF T ’s (called pCF T ’s) perturbed bymarginal or/and relevant operators [2] : S renpCF T ( σ ) = S UVCF T + σ ( L ∗ ) Z d x Φ σ ( x i ) (1.2)In the holographic RG approach [11,12] the “running” coupling constant σ ( L ∗ ) of pCF T is identifiedwith the scalar field σ ( z ) and the RG scale L ∗ – with the scale factor e ϕ ( z ) : ds = dz + e ϕ ( z ) ( dx − dt ) , σ ( x i , z ) ≡ σ ( z ) , L ∗ = l pl e − ϕ/ (1.3)of the DW’s solutions of the NMG-matter model (1.1). Once the pair of dual theories is established,the set of “holographic rules” [2, 11, 12] allows to deduce many of the important features of the quantum pCF T - as anomalous dimensions, fields expectation values, etc. - from the classical DW’ssolutions of the corresponding “bulk” gravity [9, 13–15, 17]. It should be mentioned however thatthe explicit construction of DW’s solutions, even in the simplest case of Einstein gravity, is a ratherdifficult problem. It requires the knowledge of an auxiliary function W ( σ ) called superpotential thatallows to reduce the corresponding DW’s gravity-matter equations to specific BPS-like I st ordersystem. The generalization of the superpotential method [17] to the case of NMG model (1.1) wasrecently introduced in refs. [14, 15] : κ V ( σ ) = 2( W ′ ) (cid:16) − κ W ǫm (cid:17) − ǫκ W (cid:16) − κ W ǫm (cid:17) , ˙ ϕ = − ǫκW, ˙ σ = 2 κ W ′ (cid:16) − κ W ǫm (cid:17) (1.4)2here W ′ ( σ ) = dWdσ , ˙ σ = dσdz etc.The present paper is devoted to the complete description of the holographic RG flows and of theclassical phase transitions that occur in the pCF T dual to the NMG model with quadratic mattersuperpotential W ( σ ) = Bσ + D . The critical exponents characterizing all the RG fixed points aswell as the values of the mass gap in the massive phases are calculated. One of the main statements of the Holographic renormalization group [11, 12] is that the abovescale-radial identifications (1.3) determine the form of the RG equations: dσdl = − β ( σ ) = 2 ǫκ W ′ ( σ ) W ( σ ) (cid:18) − W ( σ ) κ ǫm (cid:19) , l = ln L ∗ (2.1)as well as the explicit β -function of the dual pCF T model in terms of the superpotential W ( σ )only. Let us remember that all the (critical) CF T data is given by the asymptotic behaviour of theNMG’s domain wall solutions [15]. The two types of real zeros of this β − function : (a) W ′ ( σ ∗ a ) = 0and (b) W ( σ ∗ b ) = ǫm κ indeed coincide with (part of) the extrema i.e. V ′ ( σ ∗ A ) = 0 for A = a, b ofthe matter potential V ( σ ). Hence new purely NMG i.e. type (b) critical points exist only in the casewhen ǫm >
0. By construction both -(a) and (b) critical points- describe
AdS vacua ( σ ∗ A , Λ Aeff ) ofthe NMG model ds = dz + e − ǫ q | Λ Aeff | z ( dx − dt ) , A = a, b where the effective cosmological constants Λ Aeff are defined by the vacuum values of the correspondingscalar 3D curvature: R = − ϕ −
32 ˙ ϕ ≡ ǫ ( W ′ ) (cid:18) − κ W ǫm (cid:19) − κ W (2.2) i.e. we have R vac = − κ W ( σ ∗ A ) = 6Λ Aeff . These critical points are known to correspond to II nd order phase transitions occurring in pCF T where it becomes conformal invariant. Therefore thecritical behaviour of this 2D model is described by a set of CF T ’s of central charges : c A = 3 ǫL A l pl L gr L A ! , L gr = 12 ǫm ≫ l pl , κ W ( σ ∗ A ) = 1 L A (2.3)calculated in the approximation of small cosmological constants, i.e. l pl ≪ L gr < L A by the Brown-Henneaux asymptotic method [18] appropriately adapted to the case of NMG coupled to scalarmatter [17], [15].It is natural to consider the quantum (euclidean) pCF T in discussion as describing the univer-sality class of the thermodynamical (TD) limit of certain 2D classical statistical models. We areinterested in studying the infinite volume critical properties of these statistical models by using the3ilson RG methods. As it well known (see for example [19], [5]) they are characterized by the scalinglaws and the critical exponents of their TD potentials as for example the ones y A = ν A related tothe singular part (s.p.) of the reduced free energy (per 2D volume) F As , to correlation length ξ A andto Φ σ ( x i )’s correlation functions: F As ( σ ) ≈ ( σ − σ ∗ A ) yA , ξ A ≈ ( σ − σ ∗ A ) − yA ,G A Φ ( x , σ ) = < Φ σ ( x )Φ σ ( x ) > A ≈ e − | x | ξA | x | − y A ) , (2.4)at the neighbourhood of each critical point σ ∗ A . Once the β − function (2.1) is given, it completelydetermines the scaling properties of TD potentials, correlation functions, etc. under infinitesimalRG transformations as follows [19]: β ( σ ) dF s ( σ ) dσ + 2 F s ( σ ) = 0 , β ( σ ) dξ ( σ ) dσ = ξ ( σ ) , | x | dG Φ ( x , σ ) d | x | + β ( σ ) dG Φ ( x , σ ) dσ + 2(2 + dβ ( σ ) dσ ) G Φ ( x , σ ) = 0 (2.5)One can easily verify for example that the above critical exponents (related to the Φ σ field scalingdimensions ∆ A Φ ) are given by the values of the β − functions derivatives: y ( σ ∗ A ) = 2 − ∆ Φ ( σ ∗ A ) = − dβ ( σ ) dσ (cid:12)(cid:12)(cid:12) σ = σ ∗ A (2.6)In our case (2.1) they have the following explicit form (for W = 0) : y a = y ( σ ∗ a ) = 2 ǫW ′′ a κ W a (cid:16) − κ W a ǫm (cid:17) , y b = y ( σ ∗ b ) = − ǫ ( W ′ b ) κ W b , W b = 2 ǫm κ . (2.7)Their 3D-geometry counterparts appear in the asymptotics of the matter field σ ( z ) of correspondingDW’s solutions of NMG model (see ref. [15]) : σ ( z ) z →±∞ ≈ σ ∗ A − σ A e ∓ A q | Λ Aeff | z , ∆ A = 1 + s − m σ ( A )Λ Aeff , m σ = V ′′ ( σ ∗ A ) , (2.8)thus confirming the basic rule of AdS/CF T correspondence [2] : the scaling dimensions of 2D fieldsare determined by the 3D effective cosmological constants Λ Aeff and by the asymptotic σ − vacuumstates masses m σ ( σ ∗ A ) as follows: m σ ( σ ∗ A ) = − Λ Aeff y A ( y A −
2) (2.9)Depending on the values of y A (or equivalently of m σ ( A )) we can have three qualitatively differentnear-critical behaviours of the coupling constant σ ( l ) and therefore different type of critical points the singular points σ s such that W ( σ s ) = 0 (where β − function diverges) divide the coupling space in few indepen-dent regions. in the case of self-interactions the effective masses are defined around each of the extrema σ ∗ A of V ( σ ), i.e. σ ∗ ± A = σ ( z → ±∞ ) and therefore we have m σ ( σ ∗ A ) = V ′′ ( σ ∗ A ) σ . As is well known when ∆ Φ < relevant operator gives rise to an increasing RG flow away the (unstable) UV critical point, whilefor ∆ Φ > irrelevant and we observe decreasing RG flow towardsthe (stable) IR fixed point :( U V ) 0 < y A < , m σ ( A ) < , L ∗ → , ξ → ∞ , e ϕ → ∞ , ( IR ) y A < , m σ ( A ) > , L ∗ → ∞ , ξ → , e ϕ → y A = 0, i.e. of (asymptotically) massless matter m σ ( A ) = 0, is known todescribe marginal operators with ∆ Φ = 2. Such critical points correspond to infinite order phasetransitions, characterized by an essential singularity F As ( σ ) ≈ exp (cid:16) µ A σ − σ ∗ A (cid:17) and ξ A ≈ exp (cid:16) ρ A σ − σ ∗ A (cid:17) instead of the power-like scaling laws (2.4) for thermodynamic potentials in the case of II nd orderphase transitions. Negative m σ ( A ) (tachyons) for scalar fields in AdS backgrounds do not causeproblems when the Breitenlohner-Freedman (BF) condition [21] :Λ Aeff ≤ m σ ( A ) (2.11)is satisfied. The unitarity of the purely gravitational sector of NMG model (1.1) requires that thefollowing Bergshoeff-Hohm-Townsend (BHT) conditions [9]: m (cid:0) Λ Aeff − ǫm (cid:1) > , Λ Aeff ≤ M gr ( A ) = − ǫm + 12 Λ Aeff (2.12)to take place. They impose further restrictions on the values of the cosmological constant Λ aeff = − κ W a = − L a of type (a) critical points (i.e. on NMG vacua) :0 ≤ κ W a ǫm ≤ , ǫm > pCF T depends on the sign factor only : UV - for ǫ = − y b > ǫ = 1 case. The properties of the type (a) critical points ( UV or IR ) do depend on both - the signof ǫ and on the particular form of the matter superpotential, as one can see from eq. (2.7). Let us consider the vacuum structure and related
CF T data of NMG model (1.1) with quadraticsuperpotential W ( σ ) = Bσ + D, D = 0 (3.1)introduced in ref. [15] , where its DW’s solutions have been found. It represents the simplest exampleof extended 3D gravity, whose holographically dual pCF T model still permits rather explicit descrip-tion and as we shall see it exhibits a rich spectrum of different critical phenomena. Its β − function52.1) is parametrized by five parameters (B, D, m , ǫ , κ = 16 πl pl ) - the same that determine theshape of the matter potential V ( σ ) according to eq.(1.4). It is important to remember that the clas-sification of the qualitatively different solutions of the RG eq. (2.1) that describe different criticalbehaviours of the corresponding 2D dual models requires the complete specification of the qualita-tively different regions of the above mentioned parameter space, namely the number and the type ofthe RG critical points in function of the values of the superpotential’s parameters. Independently onthe values of the parameters B and D we always have one type ( a ) vacuum σ ∗ a = 0 represented by AdS of cosmological constant Λ aeff ( σ ∗ a ) = − κ D . The CF T ( a ) describing this critical point hascentral charge given by eq.(2.3) with L a = κ D .We choose to further investigate the particular case of ǫm > B >
0. Then the available type ( b ) RG fixedpoints, determined by the real roots of equation W ( σ ∗ b ) = ǫm κ are given by:( σ ∗± ) = ± √ ǫm κB − DB , ( σ ∗− ) ≤ ( σ ∗ + ) (3.2)Note that there exist two critical values of D: D ± cr = ± D cr , D cr = √ ǫm κ = 1 κL gr (3.3)for which two of the (b) vacua : ±| σ ∗ + | or ±| σ ∗− | coincide with the (a) one σ a = 0, giving rise to aninflection (i.e.massless) point V ′′ ( σ a ) = 0 of the matter potential. It is then clear that depending onthe values of D we have to distinguish the following three regions in the parameters space : • Region (1): no type ( b ) vacuum for D > D cr ; • Region (2): two type ( b ) vacua {±| σ ∗ + } for − D cr < D < D cr ; • Region (3): four type ( b ) vacua {±| σ ∗ + | , ±| σ ∗− |} for D < − D cr .Remembering the definitions of the two NMG scales L a = κ D and L gr , we realize that the abovedivision of the parameters space in regions of different number of critical points is in fact determinedby the relations between these scales: L a > L gr for region (2); L a = L cra = L gr on the borders (2)-(1)and (2)-(3); L a < L gr for the both regions (1)and (3).The corresponding critical exponents y A , defining the scaling dimension ∆ A = 2 − y A of Φ ( A ) σ ,can be obtained from eqs. (2.7) for D = 0 : y a = 4 ǫBL a κ − L gr L a ! , y ± b = 16 ǫBL gr κ (cid:18) L gr L a ∓ (cid:19) At D = ± D cr , i.e. on the borders between the regions we have y a ( ± D cr ) = 0 and therefore suchcritical point describes an infinite order phase transition, as we have mentioned in section 2. The We exclude here the particular case D = 0, that corresponds to a flat E NMG vacuum, i.e. Λ a ( D = 0) = 0, and CF T of c a ( D = 0) = ∞ . σ s = − DB (that exist for D < σ ∈ R in few intervals to be recognized as different phases of QFT model. Due to the Z symmetry σ → − σ of the NMG model with quadratic superpotential the phasestructure in all of the regions remains invariant under σ reflections, i.e. it is enough to study onlythe phases corresponding to σ ≥
0. For example, in the region (2) we have to consider separately thecase (2+) for D ∈ (0 , D cr ) of two critical points only : (0 IR , | σ ∗ + | UV ) , ( | σ ∗ + | UV , ∞ ) and exhibitingonly two phases; from the (2 − ) one for D ∈ ( − D cr ,
0) , where due to the presence of two singularpoints we find the following three phases: (0 UV , | σ s | ) , ( | σ s | , | σ ∗ + | UV ) , ( | σ ∗ + | UV , ∞ ). We note thatthe index UV or IR above ( say 0 IR ) marks the type of the RG fixed points (related to the sign of y A ) for the case ǫ = − m <
0. In the case of ǫ = 1 and m > ± ) − phasestructure is identical to the above one, but now with U V and IR interchanged . The region (3) hasthe richest phase structure formed by the following four phases:(3) ǫ = − IR , | σ ∗− | UV ) ( | σ ∗− | UV , | σ s | ) ( | σ s | , | σ ∗ + | UV ) ( | σ ∗ + | UV , ∞ ) . (3.4)while the region (1) - the simplest one: (0 UV , ∞ ) only.Few comments about the dependence of the properties of the solutions of the RG eq.(2.1) on thevalues of the parameter B are in order. Let us remind that the above described “phase” structure ofthe parameters space was derived under the condition B > negative values of B can be obtained from the ones of positive values of B (as above) by applying the followingformal rules B → − B : σ ∗ + → σ ∗− , reg. (3) → reg. (1) , reg. (2+) → reg. (2 − ) , ǫ → − ǫ, (3.5)and without changing the index UV or IR of the critical points. Consider an arbitrary domain wall solution DW − + of eqs. (1.4) relating two consecutive vacua( σ ± , L ± , y ± ), with σ − < σ + . It has the form (1.3), with the following b.c.’s at z → ±∞ : σ ( ±∞ ) = σ ± ˙ ϕ ± = ˙ ϕ ( ±∞ ) = − ǫκW ± , W ( σ ± ) = W ± . (4.1)Depending on the specific shape of the NMG superpotential, we can have a few different combinationsof couples of b.c.’s defining qualitatively different DW’s geometries [15] . We next consider thedefinitions of the four types of admissible DW’s, and present some simple representative examplesfor most of them.
1. Standard boundary/horizon AdS /AdS DW’s.
Let us first consider a superpotential W ( σ )which has no zeros for σ ∈ ( σ − , σ + ), thus both W ± have the same sign and we also assume that σ + is a critical point of UV -type, i.e. 0 < y + <
2. Then we must take ǫκ W ± = − /L ± in order to7nsure that the “UV-vacuum” ( σ + , L + , y + ) represents, asymptotically, an AdS boundary geometry,and the “deep bulk IR-region ” (i.e. the space around the IR vacuum ( σ − , L − , y − )), of vanishingscale factor, corresponds to a null Cauchy horizon : e ϕ z →∞ ≈ e − ǫκW + z → ∞ , e ϕ z →−∞ ≈ e − ǫκW − z → .
2. Janus two-boundaries
AdS /AdS DW’s.
In the case when W ( σ ) has one zero σ s ∈ ( σ − , σ + ),and consequently W ± have opposite signs, we must introduce another identification ǫκW ± = ∓ /L ± .It is then evident from eqs. (4.1) that the scale factor is now divergent (or zero) at both vacua, whichare both of the same UV or IR type. Such DW’s define a particular ( a ) AdS geometry of Janustype [20] [15] presenting, for ǫ = − two different boundaries (or two horizons, for ǫ = 1). Thecorresponding geometries are rather different from the standard ( a ) AdS ones with one boundary andone horizon , introduced above (see refs. [15] [16] for more details). The simplest explicit exampleof a Janus DW is provided by the linear superpotential W = Aσ kink-like solutions, with ǫm < A < | m | and σ ± = ± /κL gr A , σ s = 0: σ ( z ) = 1 κL gr A tanh (cid:16) A L gr ( z − z ) (cid:17) ,e ϕ ( z ) − ϕ = h cosh (cid:16) A L gr ( z − z ) (cid:17)i y + , y + = y − = 4 A L gr . (4.2)They have as asymptotics at z → ±∞ two very special NMG unitary vacua with λ = − Λ /m = − spaces of equal cosmological constants Λ ± eff = − /L gr . Note that L + = L − isa particular property of the Janus DW’s relating two UV type (b) vacua, while for the correspondingJanus DW’s between one type (a) and one type (b) vacua, we have indeed L − = L + . In both casesthe scale factor reaches its (finite) minimum at the point σ s = 0.
3. AdS /flat DW’s. Within the family of the standard
AdS ( L − ) /AdS ( L + ) “boundary/horizon”type of DW’s, we have to separately consider the limiting case of say L − → ∞ (i.e. Λ eff − → E (or M , ) vacuum, cf. eqs.(1.3) and (2.2), with e ϕ z →−∞ → constant = 0.These vacua do not correspond to critical points at all. As we have shown (cf. eq.(2.1)), the β -functionis divergent at such points and furthermore the E metric is not scale invariant, since its isometrygroup — the 3D Poincar´e group — is different from the conformal one, SO (3 , space. In fact, they have a rather different 2D statistical mechanicsor/and QFT interpretation. It turns out that such “flat” vacua play an important role in thedescription of certain massive RG flows that occur in the dual pCFT . The particular DW’s ofAdS /E type (interpolating between one AdS vacuum of UV type and one such E vacuum in thedeep IR) represent the NMG geometrical counterpart of specific massive phases in the corresponding2D model . In order to make transparent the properties of this kind of flat vacua, we present here In the euclidean case, the corresponding ( a ) H geometry is given by t → iτ . Then this limit is just one point thatshould be added to the 2 − d boundary to complete the 2-sphere S representing the conformal compactification of the2 d -euclidean plane E . To be compared with the AdS /AdS type of DW’s [15] relating two different AdS vacua, corresponding both tocritical points (i.e. to certain CFT ’s), and thus describing a massless RG flow. W = Bσ + D with D = 0 (we fix B > ǫm < σ ∈ (0 , σ + ), it has a flat vacuum at σ − = 0 with W − = W (0) = 0: σ ( z ) = 1 p κL gr B (cid:16) e − Bκ ( z − z ) (cid:17) − / , e ϕ ( z ) − ϕ = − κL gr Bσ ( z )1 + κL gr Bσ ( z ) ! κǫ/ BL gr . (4.3)This example represents an interesting self-dual pCFT model with two massive phases, whosefeatures require further investigations.
4. AdS /n.s. singular DW’s. Let us complete the list of the qualitatively different DW’s solutionsof the NMG-matter model (1.1) by adding the case of “singular” DW’s relating one UV-type vacuumwith a naked singularity (n.s.) at z = z , i.e. R (3) ( z ) = −∞ . They are defined in the interval z ∈ ( z , ∞ ) and correspond to the following specific b.c.’s: σ ( z → ∞ ) = σ + , σ ( z ) = ∞ and e ϕ z → z → constant = 0. An example of such DW’s is the solution for quadratic superpotential with D = 0, but now within the interval σ ∈ ( σ + , ∞ ): σ ( z ) = 1 p κL gr B (cid:16) − e − Bκ ( z − z ) (cid:17) − / . Let us mention the important fact that both DW’s for the quadratic D = 0 superpotential –flat/ AdS ( L + ) and AdS ( L + )/n.s. – we have considered above share a common boundary at z → ∞ given by the UV vacua σ + . This is quite a general property, valid for a generic superpotential withvarious different NMG vacua σ a k , σ b k and zeros at σ s ( k = 1 , , ...N ) which divide the entire couplingspace σ ∈ R into intervals p k,k +1 = ( σ A k , σ A k +1 ) of a few different kinds, each one corresponding toa DW of one the above considered four types: (1) n.s./AdS or AdS /n.s.; (2) AdS /AdS of Janustwo boundaries; (3) AdS /AdS of standard boundary/horizon type and (4) flat/AdS or AdS /flattype. Notice that the change of the IR-type of b.c.’s that occurs at each one of the boundariesdescribes in fact the transition between two different types of ( a ) AdS geometries depending on theinterval p k,k +1 = ( σ A k , σ A k +1 ) to which the initial “RG” condition σ ( l = 0) = σ belongs to. The
CF T ’s data ( σ ∗ n , c n , y n ) UV/IR specific for each parameters space region, provide the boundaryconditions necessary for the derivation of the solutions of RG eqs. (2.1) and (2.5) characterizing eachphase p nk = ( σ ∗ n , σ ∗ k ). The RG flows by definition represent the way the coupling constant σ ( l, D ) isrunning between two neighbour critical points when the RG scale L ∗ increases from L UV ∗ = 0 (i.e. l UV = ∞ ) to L IR ∗ = ∞ (i.e. l IR = −∞ ). Depending on the behaviour of the correlation length ξ ( σ ),the s.p. of the free energy F s ( σ ) (and its derivatives) and of the correlation functions G Φ ( x , σ ) wedistinguish in the non-degenerate case D = ± D cr the following three types of phases:(1) massless ( U V /IR ) : 0 < L ∗ ≤ ∞ ξ ( σ ∗ UV ) ≈ ∞ , ξ ( σ ∗ IR ) ≈ σ ( −∞ ) = σ ∗ UV , σ ( ∞ ) = σ ∗ IR , (2) massive ( U V / ∞ ) : 0 < L ∗ ≤ L ms ∗ ξ ( σ ∗ UV ) ≈ ∞ , ξ ( σ ≈ ∞ ) ≈ L ms ∗ , (3) J anus ( U V + /σ s /U V − ) : 0 < L ∗ ≤ L max ∗ ξ ( σ ± UV ) ≈ ∞ , ξ ( σ s ) ≈ L max ∗ (5.1)9he simplest example is provided by the phase structure of pCF T model in region (2+) [15] . For σ > ǫ = − p ml = (0 R , | σ ∗ + | UV ) and p ms = ( | σ ∗ + | UV , ∞ ), characterizedby the singularities and asymptotic behaviour of the solutions of eqs. (2.1), (2.5): ξ (2+) ( σ, σ ) ≈ e − l = (cid:18) σ σ (cid:19) − y (cid:18) ( σ ∗ + ) − σ ( σ ∗ + ) − σ (cid:19) − y + (cid:18) ( | σ ∗− ) | + σ | ( σ ∗− ) | + σ (cid:19) − y − , (5.2)where σ = σ ( l = 0) represents the “initial” condition of RG rescalings, i.e. L (0) ∗ ≈ F (2+) s ( σ ) = e l and the properties of the correlationlength within p ml = (0 IR , | σ ∗ + | UV ) – starting at the UV critical point as ξ (2+) ( σ ∗ + | UV , σ ) ≈ ∞ andterminating at the the IR on 0 IR , where we have ξ (2+) (0 IR , σ ) ≈ IR , | σ ∗ + | UV )as a massless phase. In this case (i.e. for ǫ = − B > L a > L gr ) the massless RG flow is drivenby the (relevant) operator Φ UVσ of dimension ∆ UV Φ = 2 − ǫBL gr ( L gr − L a ) /κL a < CF T of central charge c UV = 3 L gr /l pl . According to the discussion in section 2 (see also [15]) thecorresponding IR- CF T has central charge c IR = ǫL a l pl (cid:0) L gr /L a (cid:1) and due to the renormalizationof the coupling constant during the flow, the renormalized operator Φ σ becomes irrelevant at theIR limit of dimension ∆ IR = 2 − y a = 2 − ǫBL a (1 − L gr /L a ) /κ >
2. We have to note thatalthough for ǫ = − c ( | σ ∗ + | UV ) > c (0 IR ), i.e. the central charge is decreasing during the massless flow as required fromthe c-theorem [22] . In the case ( ǫ = 1 , m >
0) the direction of the flow is inverted, since now wehave that y (0) = y a > IR becomes a critical point of UV type, while y ( | σ ∗ + | ) < | σ ∗ + | UV is of IR-type. Nevertheless, the central charge is again decreasing during the flow, since nowboth the central charges are positive and we realize that indeed c (0 IR ) > c ( | σ ∗ + | UV ). Although wehave no characteristic (mass) scale in this interval σ ∈ p ml = (0 R , | σ ∗ + | UV ), the considered pCF T model is not conformal invariant.The (2+)-phase corresponding to the coupling space interval σ ∈ p ms = ( | σ ∗ + | UV , ∞ ) is charac-terized by the finite value of correlation length for σ → ∞ : ξ (2+) ( σ → ∞ , σ > | σ ∗ + | UV ) ≈ e − l ms = (cid:0) σ (cid:1) y (cid:0) σ − ( σ ∗ + ) (cid:1) y + (cid:0) | ( σ ∗− ) | + σ (cid:1) y − (5.3)as one can easily verify from the limit of eq.(5.2) taking into account the remarkable ”resonance”property y + y + + y − = 0, specific for our quadratic superpotential . We observe that in thisphase the coupling constant runs to infinity while the RG scale is running in the finite interval L ∗ ∈ (0 , L ( ms ) ∗ ) thus defining particular mass gap M ( ms ) ≈ L ( ms ) ∗ = (cid:0) σ (cid:1) − y (cid:0) σ − ( σ ∗ + ) (cid:1) − y + (cid:0) | ( σ ∗− ) | + σ (cid:1) − y − (5.4) we are further requiring B < κL a L gr ( L a − L gr ) in order to ensure that 0 < y UV = y b + <
2, which is the condition forrelevance 0 < ∆ UV < Which in fact assures that the “naked singularity” σ = ∞ is not a critical point.
10s a consequence the corresponding Φ σ correlation function (2.5) changes its behaviour includingnow at the leading order specific exponential decay term e − M ms | x | that determines the massiveproperties of this pCF T -phase. We have therefore an example of phase transition from massless tothe massive phase that occurs at the UV critical point | σ ∗ + | in the (2+)-phase of pCF T model. The3d gravity description of such phase transition involves two different NMG solutions having coincidingboundary conditions ( | σ ∗ + | , Λ + eff , ∆ + ) at their common boundary z → ∞ , i.e. at σ ( ∞ ) = | σ ∗ + | . Themassive phase is “holographically” described by singular DW metrics giving rise to ( a ) AdS space-time with naked singularity [15] , while the massless one corresponds to the regular DW (constructedin ref. [15]) interpolating between the two NMG vacua | σ ∗ + | UV and 0 R .The above analysis of the critical phenomena in pCF T model (and their 3D geometrical counter-parts) based on the standard statistical mechanical and RG methods, allows us to establish the basicrule of the off-critical ( a ) AdS /CF T correspondence, namely: the NMG-geometrical description ofthe phase transitions in its dual pCF T model is given by the analytic properties - poles, zeros, cutsand essential singularities - of the scale factor e ϕ of 3D DW’s metrics of the ( a ) H (euclidean) type: F (2+) s ( σ, σ ) ≈ e l ≈ ξ − ≈ e − ϕ ( σ ) (5.5)as a function of the matter field σ obtained by excluding the radial variable z from the correspondingDW’s solutions [15] . Another important ingredient of the off-critical holography is the so calledZamolodchikov’s central function for NMG model introduced in refs. [23] [15] : C ( σ ) = − GκW ( σ ) (cid:18) κ W ( σ )2 ǫm (cid:19) (5.6)which at the critical points σ ∗ A ± takes the values (2.3). Remember that according to the I st ordereqs.(1.4) we have W ( σ ) = − ˙ ϕ ǫκ and therefore the central charges c A and the central function aswell are geometrically described by the log-derivative ˙ ϕ of the scale factor. As a consequence of itsdefinition (5.6) and of the RG eqs. (2.1) we conclude that [15] : dC ( σ ) dl = − GW ( σ ) (cid:18) dσdl (cid:19) (5.7)Hence for W ( σ ) positive (as in our example) the central function is decreasing during the masslessflow, i.e. we have c ( | σ ∗ + | UV ) > c (0 IR ) for ǫ = − CF T model are in order. All the properties of the massless-to-massive phase transition observed in the dual QFT in the region (2+) obtained from the NMG-induced exact β -function turns out to be almost identical to those calculated perturbatively inpCFT , with action (1.2), where the CF T in the UV critical point and the corresponding relevantoperator are chosen to coincide with those we have found above. It is important to also mention thatthe operator Φ UVσ of dimension ∆ UV Φ has OPE of Φ adj (or Φ ) type [3] with structure constant given It represents a natural generalization [23] of the well known result for m → ∞ limit [12], [11] C ΦΦΦ ( | σ ∗ + | UV ) = 8 ǫB L gr | σ + | (1 + 2 L gr L a ), similar to the conformal OPE’s we have in the Virasoroand Liouville models [22] [24] . In order to make this equivalence exact one need to identify theNMG -matter model parameters L gr , L a with the one (denoted usually by b ) of the corresponding CF T ’s central charges c ± ( b ) = 1 ± Q b with Q b = b ± b . Further investigations are needed in orderto find an appropriate large c UV (i.e. large L gr /l pl ) limits consistent with the conformal perturbationtheory.The RG flows in region (2 − ) are rather different from the ones of (2+) due to the fact that allthe critical points are now of UV-type and to the presence of singular points as well. The massivephase ( | σ ∗ + | UV , ∞ ) coincides with the corresponding one in reg.(2+) and the mass gap is given againby M ms of eq. (5.4) except that the values of the exponents y ± > y > D < UV / | σ s | / | σ ∗ + | UV both provided with relevant operators. As one can see from the scalefactor and from the correlation length ξ (2 − ) behaviours (5.10) the RG scale is now start running from L ∗ = 0 at the both 0 UV and | σ ∗ + | UV critical points and it gets its maximal value L ( max ) ∗ at the “endpoint” | σ s | = q | D | B . Note the important difference with the normal massive phase where the L ( ms ) ∗ was reached for σ ≈ ∞ . The proper existence of L ( max ) ∗ however introduces mass scale: M (2 − ) J ( σ s , σ ) ≈ e l max = (cid:18) σ s σ (cid:19) y (cid:18) ( σ ∗ + ) − σ s ( σ ∗ + ) − σ (cid:19) y + (cid:18) ( | σ ∗− ) | + σ s | ( σ ∗− ) | + σ (cid:19) y − (5.8)specific for the new Janus-massive phase. Hence in this case we have two different massive phases thatstart from the same critical point | σ ∗ + | UV . This massive-to-massive phase transition is characterizedby the ratio of the two mass gaps: M (2 − ) J M (2 − )( ms ) = (cid:18) L a L gr (cid:19) y + (cid:18) − L a L gr (cid:19) y − , (5.9)which differently from the corresponding ξ ’s and mass gaps is completely determined by the super-potential data and turns out to be an important characteristics of corresponding QFT model. TheNMG description of (2 − ) − phase diagram is therefore given by one Janus-type DW and one singularsolution representing naked-singularity.The phase structure of the pCF T model corresponding to region (3) turns out to combine allthe critical phenomena observed in regions (2 ± ). Consider again the σ > IR , | σ ∗− | UV ) , ( | σ ∗− | UV , | σ s | ) , ( | σ s | , | σ ∗ + | UV ) , ( | σ ∗ + | UV , ∞ )containing three critical and one singular points. As one can verify from the behaviour of thecorresponding correlation length: ξ (3) ( σ, σ ) ≈ (cid:18) σ σ (cid:19) − y (cid:18) ( σ ∗ + ) − σ ( σ ∗ + ) − σ (cid:19) − y + (cid:18) ( | σ ∗− ) | − σ | ( σ ∗− ) | − σ (cid:19) − y − (5.10)12he phase (0 IR , | σ ∗− | UV ) is describing massless RG flow similar to the one in the region (2+) butinvolving the new critical point | σ ∗− | UV . The next two phases ( | σ ∗− | UV , | σ s | ) and ( | σ s | , | σ ∗ + | UV ) areboth representing Janus-massive phases, while the last one ( | σ ∗ + | UV , ∞ ) is identical to the mass phaseof region (2+) except that the exponents y A with A = ± , D < − D cr , i.e. L a < L gr . It is evident that the holographicdescription of the reg. (3) phase structure in terms of NMG’s DW solutions consists of three differentDW’s of common boundaries: one of UV-IR type interpolating between two different NMG vacuaof cosmological constants Λ eff and Λ eff − , one of Janus type and the last one that involves nakedsingularity as defined in Sect.4 above.The nature of the phase transitions in QFT occurring at the borders of the parameter space D = ± D cr (i.e. for L a = L gr ) where y a ( ± D cr ) = 0 is rather different from the ones of II nd orderwe have described above. As one can see by comparing the forms of the corresponding solutions σ = σ ( l, D ) of RG equations (2.1), (2.5): ξ ( D = ± D cr ) ≈ e − l ≈ e ϕ = ± √ ǫm κBσ ! y ∓ ( cr ) e ρ σ − y ∓ ( cr ) , y ∓ ( cr ) = ± ǫBκ √ ǫm ξ ( D = ± D cr ) ≈ e − l ≈ e ϕ = (cid:0) σ (cid:1) − y (cid:0) ( σ ∗ + ) − σ (cid:1) − y + (cid:0) ( σ ∗− ) − σ (cid:1) − y − , ρ = − m B (5.11)the specific power-like singularities for D = ± D cr are replaced now by an essential singularity atthe (here triple) zero of the β − function. According to definitions of section 2, the presence of suchsingularity in ξ ( σ, D cr ) and F crs ( σ ) at the critical point σ = 0 mar in the case D = D cr is an indicationof infinite order phase transition. The corresponding massive phase (0 mar , ∞ ) is characterized bythe mass gap M crms ≈ e /y ∓ ( cr ) obtained from the σ → ∞ limit of eq. (5.11). The phase structurein the case D = − D cr is richer: for σ > mar , one singularpoint | σ crs | and one UV critical point | σ cr + | UV giving rise to four massive phases. The first two weak-coupling massive phases (0 mar , | σ crs | ) and ( | σ crs | , | σ cr + | UV ) are of Janus type with mass gap given by M crJ ≈ e /y ∓ ( cr ) = ( M crms ) and the last one ( | σ cr + | UV , ∞ ) is the standard strong coupling massivephase ( | σ cr + | UV , ∞ ). The phase transitions at σ = | σ cr + | UV point is of J-massive-to-massive (weak-to-strong coupling) type, quite similar to the one observed in the region (2 − ) above (i.e. of secondorder), but indeed with different mass ratio M crJ M crms = e /y ∓ ( cr ) . Our investigation of the classical critical phenomena in the pCF T ’s duals to the NMG models withquadratic matter superpotential has revealed many essential features of these 2D non-conformalmodels leaving however still open the problem of their complete identification. It is important to13mphasize that the phase transitions we have described concern the TD limits of certain 2D classical statistical models (s.m.), related to the pCF T in discussion. We have studied the infinite volume critical properties of these statistical models by using the Wilson RG methods. As it well known(see section 4 . finite temperature phase transitions in the classical d = 2 s.m. in infinite volume correspond to zero temperature phase transition in certain equivalentquantum d = 1 s.m. (or its TD 1 + 1 QFT limit) when some other coupling in the quantum modelbecomes critical, say the transverse magnetic field in the case of 1D Ising model. Observe thatthe temperatures used in both models are different: the inverse temperature (i.e. 1 /kT ) in thequantum 1D model corresponds to the period of the extra (time) direction, while the temperaturein 2D classical s.m. is related to the extra coupling constant in 1D model. Hence the descriptionof finite T -phase transitions in the quantum 1D s.m. requires to study the finite-size effects in its2D classical counterpart. This fact explains the successful use of DW solutions in the descriptions ofclassical phase transitions (in 2D s.m.) instead of say black holes and other periodic (or finite time)solutions of NMG model, which are indeed the geometric ingredients required in the investigation offinite T quantum phase transitions.The detailed description of the main features - critical exponents, mass gaps, s.p. of the reducedfree energy - of the variety of second and infinite order classical phase transitions in 2D s.m. modelsthat are conjectured to be dual of the NMG model (1.1), allows us to establish the following impor-tant rule of the off-critical AdS /CF T correspondence: the phase transitions observed in the dual pCF T models are determined by the analytic properties of the scale factor e ϕ of 3D (euclidean)DW’s type metrics of NMG model [15] written as a function of the matter field σ . As we haveshown by using RG methods, the inverse of the scale factor is proportional of the s.p. of the freeenergy . In order to calculate the exact values of the entropy, the specific heat and other importantTD characteristics one need to know the finite part of the free energy as well, which is a rathercomplicated problem even for the simplest 2D s.m. models. It is well known however that the infi-nite 2D conformal symmetry at the critical points offers powerful methods, based on the knowledgeof the characters of the Virasoro algebra representations, which allow to construct the exact formof the corresponding “critical” partition functions. An important step towards the construction ofcorresponding holographic partition functions is the calculation of the spectra of the linear fluctua-tions around the DW’s solutions of the considered NMG models. One has to further consider thedifficult problem of the construction of the off-critical correlation functions of 2D fields dual to 3Dmatter scalar by applying AdS/CF T methods [2] [10] and to next compare with the known resultsof corresponding 2D models [5] [7] as well .Let us also mention the problem with the interpretation of the negative values of the ”classical”central charges (2.3) for ǫ = − m <
0, that are usually considered as non-unitary
CF T ’s.According to the well known exact results for the quantum minimal CF T [3] [19] , based on thedegenerate representations of the Virasoro algebra, in all the cases when c < CF T ’s contain primary fields of negative dimensions and hence they indeed represent non-unitary QF T ’s. We should remember however also the well established results [24] [25] concerning the14lassical and semi-classical limits ~ → unitary series c quant = 1 − Q quant ~ . They lead to large negative centralcharges c quant → c cl ≈ −∞ . Hence the classical and semi-classical large negative central charges arecommon feature for both unitary and non-unitary quantum minimal models of c quant <
1. Furtherinvestigations of the limiting properties (involving the calculation of certain sub-leading terms) ofthe dimensions of the primary fields an of the structure constants their OPE’s are needed in orderto conclude whether the strong-coupling CFT’s data we have extracted from the NMG holographicdual
CF T ’s are representing limits of unitary or non-unitary quantum m.m.’s. Acknowledgements.
We are grateful to A.L.A.Lima for critical reading of the manuscript andfor pointing out to us few mistakes and misprints in the e-Print version arXiv:1009.2665 [hep-th], aswell as for his suggestions for improvements.This work has been partially supported by PRONEX project number 35885149/2006 from FAPES-CNPq (Brazil).
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