The quest for a conifold conformal order

The holographic duality between cascading gauge theory and type IIB supergravity on warped deformed conifold with fluxes reveals exotic thermal phases with nonzero expectation values of certain operators, persistent to high temperatures. These phases, in the limit of vanishing the strong coupling scale of the cascading gauge theory, would realize thermal ordered conformal phases in relativistic QFTs. We find that the dual Klebanov-Strassler and Klebanov-Tseytlin black branes in this limit are outside the regime of the supergravity approximation, rendering the construction of such conformal ordered states unreliable. While we have been able to construct conformal order in phenomenologically deformed effective theory of type IIB supergravity reduced on warped deformed conifold with fluxes, the removal of the deformation parameter causes the destruction of the thermal conformal ordered phases. Once again, we find that the holographic models with the conformal ordered phases are in the String Theory swampland.


Introduction
Conformal order stands for exotic thermal phases of conformal field theories (CFTs), characterized with nonzero one-point correlation function(s) of certain operator(s) [1][2][3][4][5][6][7][8][9][10]. For a CFT d+1 in Minkowski space-time R d,1 the existence of the ordered phases implies that there are at least two distinct thermal phases: where F is the free energy density, T is the temperature, C is a positive constant proportional to the central charge of the theory, and {O ∆ i } is the set of the order parameters with the conformal dimension spectrum {∆ i }. The parameters κ and {γ i } characterizing the thermodynamics of the ordered phase are necessarily constants.
Conformal order can realize spontaneous breaking of discrete [1][2][3] of continuous [10] global symmetries; but it does not have to be the case: in the model we discussed in [9], the conformal order parameter is not associated with spontaneous breaking of any global symmetry 1 .
From (1.1), note that when κ > 1 (κ < 1), the symmetry broken phase dominates (is subdominant) both in the canonical and the microcanonical ensembles. Irrespectively of the value, provided κ > 0, the symmetry broken phase is thermodynamically stable.
It is difficult to compute directly in a CFT the values {κ, γ i }, thus establishing the presence and the (in)stability of the ordered phase. Rather, the authors of [1,[6][7][8] established the instability of the disordered thermal phases in discussed CFTs. The condensation of the identified unstable mode then leads to O ∆ i = 0 for the new equilibrium thermal state -the conformal order.
Conformal order states are very interesting in the context of holography [11,12], as they imply the existence of the dual black branes in a Poincare patch of asymptotically AdS d+2 bulk geometry that violate the no-hair theorem. In [9] we proved a theorem that the disordered conformal thermal states are always stable in dual holographic models of Einstein gravity with multiple scalars. Thus, the mechanism for the conformal order presented in [1] is not viable in these holographic models.
The first holographic model of the conformal order was discovered (though not appreciated in this context prior to the QFT construction [1]) purely by accident in [2].
The general framework for constructing holographic conformal order, the phenomenologically deformed effective theory, was presented in [5] -we now review those arguments as they will be utilized in this paper 2 . Consider a top-down holographic model, dual to 3 a CFT 4 with a single operator O ∆ of a dimension ∆. The five-dimensional gravitational bulk effective action takes the form where c is the central charge of the boundary CFT, φ is the gravitational bulk scalar dual to the order parameter O ∆ , and the scalar potential P[φ] is The O(φ 0 ) term in (1.3) is a negative cosmological constant, setting the radius of the asymptotically AdS 5 bulk geometry to L; the mass term,

4)
2 All the known constructions of the holographic conformal order can be understood within this framework. 3 Extensions to AdS d+2 /CFT d+1 models with multiple order parameters O ∆i is trivial -such models will be studied in section 3. represents the standard encoding of the dimension of the order parameter O ∆ [13,14].
In what follows we set L = 1. It is vital that in real holographic models (contrary to the phenomenological toys), the full scalar potential P[φ] is nonlinear. Unfortunately, holography is not understood at the level where given a boundary CFT, with a spectrum of gauge invariant operators, we can engineer/compute the scalar potential. In specific holographic examples, like the N = 2 * correspondence [15][16][17], the cascading gauge theory duality [18,19] or the Maldacena-Nunez model [20], the scalar potential is computed from the realization of the duality correspondence in type IIB supergravity.
The construction of the holographic conformal order proposed in [5] relies on (and is applicable to) models where the leading nonlinear correction is unbounded from below 4 along certain directions on the scalar manifold. To be specific, we assume that with the constant deformation parameter b being positive. The claim of [5], see also appendix A, is that the thermal conformal order always exists in the limit b → +∞, when the thermal ordered phase is holographically realized as AdS-Schwarzschild black brane, with a perturbatively small "scalar hair" (1.7) The existence of the thermal conformal order in real holography then boils to the question whether this perturbative constructions survives as b decreases from +∞ to In this paper we continue the quest for constructing the thermal conformal order in String Theory holography. Our focus is on top-down holographic dualities between regular and fractional D3-branes on a conifold, the simplest non-compact Calabi-Yau threefold [21], and N = 1 supersymmetric gauge theories -the Klebanov-Witten (KW) [22] and the Klebanov-Strassler (KS) [18] models. There are two reasons for this choice: • Analysis of the phenomenological model [2] revealed for the first time the exotic thermal phases in a holographic system, which are associated with the spontaneous breaking of a discrete symmetry, and persist to arbitrary high temperature.
As T → ∞, the fact that the model of [2] was non-conformal becomes irrelevant since for any fixed mass scale m/T → 0. In this way one can obtain a holographic thermal conformal order, as emphasized in [3]. Holographic duality between type IIB supergravity on warped deformed conifold with fluxes and the KS cascading gauge theory also reveals the exotic thermal phase [23] -the deconfined phase with spontaneously broken U(1) R chiral symmetry, that exists only above certain critical temperature T χSB . Like the model in [2], the cascading gauge theory is non-conformal, and has a strong coupling scale Λ. It is natural to explore this exotic phase in the limit Λ/T → 0, and potentially obtain a top-down holographic model of the thermal conformal order. We discuss this in section 2.
• The limit of Λ/T → 0 in the above example effectively removes the fractional D3branes from the holographic model. On the gravity side one ends with type IIB supergravity on warped deformed conifold with the self-dual five-form flux. The corresponding boundary gauge theory is N = 1 superconformal KW model. The gravitational bulk scalars encode the gauge invariant operators (potential conformal order parameters). As we review in section 3, the resulting holographic model (when the conifold is not deformed) is a universal example of AdS 5 /CFT 4 duality on warped and squashed Sasaki-Einstein manifolds [24]. Thus, analysis of the conformal order on the conifold will cover all such cases, see sections 3.1 and 3.2. In the case of the deformed conifold, we will have potentially the first example of the holographic conformal order with spontaneously broken continuous symmetry, see section 3.3.
We summarized our results in section 4.
The theory is not conformal, and the gauge couplings g 1 and g 2 , of the gauge group factors SU(N + M) and SU(N) correspondingly, run with the renormalization group scaleμ, where γ is the anomalous dimension of operators Tr A i B j and Λ is the strong coupling scale of the cascading gauge theory. To leading order in M/N, γ = − 1 2 [18,22], so that 8π 2 g 2 while the sum of the gauge couplings is constant along the RG flow The thermal phase diagram of the theory is rich: At large temperatures, T ≫ Λ, its thermal equation of state is that of a conformal theory with the effective temperature-dependent central charge [25], e.g, the free energy density F takes the form the theory undergoes the first-order confinement/deconfinement phase transition [29]; precisely at the transition point the free energy density vanishes, where the first equality introduces dimensionless quantityF we use to describe cascading gauge theory thermodynamics in the canonical ensemble.
The next critical temperature is [31] T χSB = 0.541(9)Λ ,  symmetry, the solid magenta curves in fig. 1, is exotic: it exists for T > T χSB and is realized holographically as the Klebanov-Strassler black brane [23]. If we could extend this phase for T Λ → ∞, by analogy with [3], we would have realized the conformal order on the conifold. Alas, our numerics allowed the construction of this exotic phase in the There is a practical and a conceptual reason for this: • from the practical perspective, certain normalizable components of the scalar fields near the boundary become too large for a reliable numerics 5 ; • the conceptual reason causing the above growth is the following: as the temperature increases, the curvature of the dual Klebanov-Strassler black brane evaluated at the horizon grows, and the construction becomes unreliable in the supergravity approximation. Specifically, in the right panel of fig. 1 we present the value of the Kretschmann scalar of the KS black brane horizon, for the exotic phase (the magenta curve) and the deconfined chirally symmetric phase (the black curve), relative to the value of the Kretschmann scalar K χSB evaluated at T = T χSB .
Over the range (2.9), the Kretschmann scalar corresponding to the exotic phase changes as Thus, we conclude that there can not be a reliable conformal order on the warped deformed conifold with fluxes, arising from the T → ∞ limit of the Klebanov-Strassler black branes -such black branes are outside the validity of the supergravity approximation.
• Lastly, there is a terminal temperature [30] T fail to extend it as T → ∞. We constructed the latter states for fairly narrow temperature range: As the temperature of the exotic deconfined chirally symmetric phase increases, the curvature of the corresponding dual Klebanov-Tseytlin black brane evaluated at the horizon rapidly grows, see the right panel of fig. 2. Over the range (2.12), the Kretschmann scalar corresponding to this exotic phase (the brown curve) changes as On the contrary, the deconfined chirally symmetric phase with the positive specific heat (the black curves in fig. 2) can be extended as T Λ → ∞ -however it is not exotic, as the thermal expectation values of O ∆={4,6,8} operators vanish [29] in this limit, and we end up with the log-dressed conformal equation of state (2.5). Thus, we conclude that there can not be a reliable conformal order on the warped squashed conifold with fluxes, arising from the T → ∞ limit of the Klebanov-Tseytlin black branes with the negative specific heat -such black branes are outside the validity of the supergravity approximation.
It is a functional of the a five-dimensional metricĝ µν on M 5 , scalars Ω i=1···3 describing the warping and the deformation of the conifold, a dilaton Φ, scalars h i=1···3 parameterizing the 3-form fluxes, a constant parameter Ω 0 (necessary to define the self-dual 5-form flux), and a topological parameter P , related to the number of fractional D3 branes on the conifold. Finally, R 10 is the Ricci scalar of the ten-dimensional type IIB metric, obtained from uplifting (3.2), andR 5 is the five-dimensional Ricci scalar of the metric (3.2).
We find it convenient to rewrite the action (3.1) in five-dimensional Einstein frame.
The latter is achieved with the following rescalinĝ Further introducing the five-dimensional effective action becomes Consistent truncation of (3.8), i.e., produces effective action of SU(2) × SU(2) × U(1) invariant sector of type IIB supergravity on warped squashed conifold with fluxes derived in [34].
The boundary holographic dual represented by (3.8) is the N = 1 supersymmetric SU(N + M) ×SU(N) cascading gauge theory, often referred to as a Klebanov-Strassler gauge theory [18]. This theory is not conformal, and has a strong coupling scale Λ, where g s is the asymptotic value of the string coupling constant 7 , and K 0 is a parameter that can always be adjusted (using the scaling symmetries of the holographic radial coordinate) to a fixed positive value. The precise definition of K 0 can be found in [32,35]. We are interested here in the conformal holographic model, thus we must take the limit Λ → 0, which is equivalent to sending P → 0. Finding the conformal order in the model with 7 scalars is a daunting task -so 8 we will follow the framework developed in [5], and reviewed in appendix A.
As a first step, we truncate the general potential P f lux +P scalar in (3.8) to scalars that enter nonlinearly, and have "unbounded" directions on the field space manifold, at least close to the origin. Notice that h i , encoding the 3-form fluxes on the conifold, enter quadratically and generate always nonnegative contribution to the scalar potential, i.e., P f lux ≥ 0. Thus, in addition to P = 0, we set h i ≡ 0. Then, where in the second equality we set a constant parameter Ω 0 = 1 108 to ensure that the radius of the asymptotically AdS 5 geometry is L = 1.
In the absence of 3-form fluxes the dilaton becomes an exact modulus, and we set (3.14) We now arrive at the effective action we use to analyze conformal order on the warped 7 It can always be fixed to g s = 1. 8 While it is conceivable that other general constructions of the conformal order might exist in the model, finding them without any guidances is a lost cause.
deformed conifold: is the central charge of N = 1 superconformal SU(N) × SU(N) Klebanov-Witten model [22], and From (3.15) we obtain the following equations of motion In the rest of this section we consider three conformal models, obtained by consistent truncations of the effective action S 5 = S 5 [f, w, λ] (3.15): • Model I: (3.23) • Model III: can be truncated to our Model I [24]. In the latter case, the scalar f represents the breathing mode of Y 5 .
We follow appendix A to construct conformal order in Model I. From (3.15) we find i.e., the leading nonlinear term is n = 3 (see (1.5)) and the unbounded direction is along f → +∞. The b-deformed scalar potential takes the form, compare with (1.6), Using the radial coordinate as in (A.10), and introducing we obtain the following equations of motion ( ′ ≡ d dx ) Note that in total we have four parameters {f 2 , f h 0 , h h 1 , g h 0 }, along with an arbitrary constant A. They determine the thermal conformal order of Model I, specifically, for the entropy density s, the free energy density F , the temperature T , and the thermal order parameter O 8 . Thus, see (1.1), where the subscript I refers to Model I.
In the limit b → +∞ we find, see appendix A, ing [4] we expect that this conformal order is perturbatively unstable.

Conformal order in Model II
Conformal Model II realizes holographic dual to the KW gauge theory with a pair of order parameters: a dimension ∆ = 8 operator O 8 and a dimension ∆ = 6 operator O 6 , represented by the bulk scalars f and w correspondingly. Since these scalars represent the warping of the T 1,1 base, along with the squashing of the U(1) fibration of the Kähler-Einstein base of T 1,1 , this model is universal as well: any holographic duality between a CFT 4 and a type IIB supergravity on AdS 5 ×Ỹ 5 , whereỸ 5 is a fivedimensional squashed Sasaki-Einstein manifold, can be truncated to our Model II [24].
In the latter case, the scalars f and w represent the breathing and the squashing modes i.e., the leading nonlinear terms are n = 3 (see (1.5)) and the unbounded direction is along {f, w} → +∞. The b-deformed scalar potential takes the form, Repeating the analysis as in section 3.1, in the limit b → +∞, we find where Notice that we modified the leading nonlinear interactions in (3.37) as , at the horizon   below the BF bound; while they would be expected to become large-negative for the effective mass of λ to dip below its effective BF bound near the horizon. As we shortly demonstrate, conformal order in deformed Model III exists when both scalars {f, w} are negative at the horizon. The b-deformed scalar potential takes the form,

Conformal order in Model III
(3.47) Using the radial coordinate as in (A.10), and introducing we obtain the following equations of motion ( ′ ≡ d dx ) as y ≡ (1 − x) → 0 + , i.e., at the horizon (3.55) Note that in total we have eight parameters , along with an arbitrary constant A. They determine the thermal conformal order of Model III, specifically, where the subscript III refers to Model III.
In the limit b → +∞ we find, see appendix A, with from above. We take closeness of b crit,III to 1 as the strong indication that the actual value of b crit,III is precisely 1. We thus conclude that while there is a conformal order in b-deformed Model III, it disappears once this model becomes a realization of the topdown holography, much like in top-down holographic models discussed in [5,9]. Since κ III (b) < 1, the conformal order in deformed Model III is subdominant in canonical and microcanonical ensembles. Following [4] we expect that this conformal order is perturbatively unstable.

Conclusion
In this paper we searched for the holographic conformal order [3] on warped deformed conifold with fluxes [32] -representing the supergravity dual to Klebanov-Strassler cascading gauge theory [18]. We focused on two exotic phases [2] of the black branes on the conifold: • the Klebanov-Strassler black branes [23], realizing the spontaneous breaking of the continuous R-symmetry; • the branch of Klebanov-Tseytlin black branes with the negative specific heat [30]. the conformal order one needs to construct these black branes in the high temperature limit T Λ → 0. We showed that as one increases the temperature, these exotic black branes on the conifold become ever more stringy. As a result, the conformal order can not be reached within the controllable supergravity approximation.
In the second part of the paper we changed the strategy: instead of starting with the non-conformal exotic black branes and taking the high-temperature limit, we took the conformal limit in the effective action of type IIB supergravity on warped deformed conifold with fluxes [31]. The latter limit effectively removes the fractional D3 branes from the model. The resulting five-dimensional effective action has seven bulk scalars; but only three of these scalars have the nonlinear potential. The nonlinearity of the scalar potential appears to be crucial in holographic examples of the conformal order constructed so far [2,3,5,9]. Taking this fact as a hint, we consistently truncated the "conformal" effective action to these three scalars. Within this action, we further identified two additional universal consistent truncations: • type IIB supergravity on warped Sasaki-Einstein manifolds with the self-dual five-form flux [24,30]; • type IIB supergravity on warped and squashed Sasaki-Einstein manifolds with the self-dual five-form flux [24,30].
We showed that in both truncations one can construct the conformal order, provided one deforms the bulk scalar potential obtained from the Kaluza-Klein reduction on the Sasaki-Einstein manifold. However, before the deformation parameter is removed, the conformal order disappears. The same story repeats in the full effective action on the warped deformed conifold with the three bulk scalars -the conformal order exists, as long as the scalar potential is (arbitrarily small) deformed from the one obtained from the type IIB supergravity Kaluza-Klein reduction. Along with the previous results [5,9] we believe this sends a powerful message: there is a 'real' holography and a 'toy'

A Perturbative holographic thermal conformal order
Consider thermal conformal order in the phenomenological model with the scalar potential P b [φ] given by (1.6), i.e., where p n > 0, and the following next nonzero coefficient p k is for k = n + p ≥ n + 1.
We assume ∆ ≥ 2. To this end, we assume the black brane metric ansatz along with φ = φ(r), all depending on the radial coordinate r ∈ [r 0 , +∞). There is the smooth Schwarzschild horizon as r → r 0 + , i.e., Once the perturbative (as b → +∞) solution (A.11) is constructed, the finite-b conformal order can be analyzed numerically, incrementally decreasing the deformation parameter b [3,5,9].
The reason why we expect the specific large-b scaling of the conformal order as in (A.11) was given in [5]: • Recall the story of the holographic superconductor [37,38]. A scalar field in asymptotically AdS geometry must have a mass above the (space-time dependent) Breitenlohner-Freedman (BF) bound to avoid the condensation. In the vicinity of the Schwarzschild horizon, the BF bound can be modified either by changing the effective dimensionality of the space-time (as in the extremal limit of a Reissner-Nordstrom black brane) [38], or via nonlinear scalar interactions, leading to large negative contribution of the effective mass [37]. Thus, it is possible for a scalar field to be above the BF bound close to the AdS boundary, and below the effective BF bound close to the horizon. This scenario triggers the condensation of the scalar -the black brane develops the scalar hair. Close to the transition point, the condensation can be studied in the probe approximationneglecting the backreaction.
• The conformal order realizes the black brane horizon scalarization mechanism of [37]. It becomes a probe approximation in the limit b → +∞. Indeed, with the scaling φ ∝ 1 b Thus, any nonlinear term in the scalar potential with k > n is subleading in the b → +∞ limit compare to the quadratic scalar terms in the effective action (A.1), while the leading nonlinear k = n term scales precisely as the former. and the scalar field φ 1 satisfies (A.12). Notice from (A. 19) that a large positive value of φ 1 at the horizon, see (A.14), can make m 2 ef f sufficiently negative, and trigger the scalarization as in [37].