The quest for a conifold conformal order

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 ${\mathbb{R}}^{d,1}$ the existence of the ordered phases implies that there are at least two distinct thermal phases:

where ${\cal F}$ is the free energy density, $T$ is the temperature, ${\cal C}$ is a positive constant proportional to the central charge of the theory, and $\{{\cal O}_{\Delta_{i}}\}$ is the set of the order parameters with the conformal dimension spectrum $\{\Delta_{i}\}$. The parameters $\kappa$ and $\{\gamma_{i}\}$ characterizing the thermodynamics of the ordered phase are necessarily constants. Conformal order can realize spontaneous breaking of discrete [2, 1, 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 symmetry¹¹1This is strictly true in the context of the effective Kaluza-Klein reduced holographic models. The nonvanishing scalar expectation value would signal the spontaneous breaking of the global symmetry of the compactification manifold upon uplift to 10d supergravity. Unfortunately, no model with a thermal conformal order has been constructed in top-down holography yet..

From (1.1), note that when $\kappa>1$ ($\kappa<1$), the symmetry broken phase dominates (is subdominant) both in the canonical and the microcanonical ensembles. Irrespectively of the value, provided $\kappa>0$, the symmetry broken phase is thermodynamically stable. It is difficult to compute directly in a CFT the values $\{\kappa,\gamma_{i}\}$, 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 $\langle{\cal O}_{\Delta_{i}}\rangle\neq 0$ 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 paper²²2All the known constructions of the holographic conformal order can be understood within this framework.. Consider a top-down holographic model, dual to³³3Extensions to AdS_d+2/CFT_d+1 models with multiple order parameters ${\cal O}_{\Delta_{i}}$ is trivial — such models will be studied in section 3. a CFT₄ with a single operator ${\cal O}_{\Delta}$ of a dimension $\Delta$. The five-dimensional gravitational bulk effective action takes the form

where $c$ is the central charge of the boundary CFT, $\phi$ is the gravitational bulk scalar dual to the order parameter ${\cal O}_{\Delta}$, and the scalar potential ${\cal P}[\phi]$ is

The ${\cal O}(\phi^{0})$ term in (1.3) is a negative cosmological constant, setting the radius of the asymptotically AdS₅ bulk geometry to $L$; the mass term,

represents the standard encoding of the dimension of the order parameter ${\cal O}_{\Delta}$ [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 ${\cal P}[\phi]$ 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 ${\cal N}=2^{*}$ 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 below⁴⁴4This feature is ubiquitous in top-down holographic examples. along certain directions on the scalar manifold. To be specific, we assume that

for some constant $s$ and an integer $n\geq 3$. Note that ${\cal O}(\phi^{n})$-truncated scalar potential is unbounded as $\phi\to+\infty$. Of course, the full scalar potential ${\cal P}[\phi]$ might be bounded, but this is not important for the perturbative construction of the thermal conformal order in the phenomenologically deformed model. The phenomenologically deformed model is defined as a holographic correspondence where the top-down scalar potential ${\cal P}[\phi]$ is deformed as

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 $b\to+\infty$, when the thermal ordered phase is holographically realized as AdS-Schwarzschild black brane, with a perturbatively small “scalar hair”

The existence of the thermal conformal order in real holography then boils to the question whether this perturbative constructions survives as $b$ decreases from $+\infty$ to $1$, since

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 ${\cal N}=1$ 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 $T\to\infty$, the fact that the model of [2] was non-conformal becomes irrelevant since for any fixed mass scale $m/T\to 0$. 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 $T_{\chi\rm{SB}}$. Like the model in [2], the cascading gauge theory is non-conformal, and has a strong coupling scale $\Lambda$. It is natural to explore this exotic phase in the limit $\Lambda/T\to 0$, and potentially obtain a top-down holographic model of the thermal conformal order. We discuss this in section 2.

• •

  The limit of $\Lambda/T\to 0$ in the above example effectively removes the fractional D3-branes 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 ${\cal N}=1$ 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${}_{5}/$CFT₄ 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.

Thermodynamics of the Klebanov-Strassler cascading gauge theory [18] has by now a long history [25, 26, 27, 28, 29, 30, 31, 23, 32]. We refer the reader to a recent comprehensive review [32], and focus here on the results only.

Cascading gauge theory is ${\cal N}=1$ supersymmetric $SU(N+M)\times SU(N)$ gauge theory with pairs of chiral multiplets $A_{k}$ and $B_{\ell}$, $k,\ell=1,2,$ in the bifundamental $(N+M,\overline{N})$ and $(\overline{N+M},{N})$ representations, and the superpotential

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 scale $\hat{\mu}$,

where $\gamma$ is the anomalous dimension of operators $\mathop{\rm Tr}A_{i}B_{j}$ and $\Lambda$ is the strong coupling scale of the cascading gauge theory. To leading order in $M/N$, $\gamma=-\frac{1}{2}$ [22, 18], so that

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, $T\gg\Lambda$, 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 ${\cal F}$ takes the form

   At

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 quantity $\hat{\cal F}$ we use to describe cascading gauge theory thermodynamics in the canonical ensemble.
   The next critical temperature is [31]

represented by the vertical dashed red lines in figs. 1 and 2. For $T>T_{{\chi\rm{SB}}}$ the deconfined phase, represented by the solid black curves in figs. 1 and 2, is perturbatively stable to $U(1)_{R}\to{\mathbb{Z}}_{2}$ chiral symmetry breaking fluctuations. The deconfined states of the cascading gauge theory plasma unstable to chiral symmetry breaking fluctuations are represented by the dashed solid black curves in figs. 1 and 2.
   Similar to the model of [2], the deconfined phase with spontaneously broken chiral symmetry, the solid magenta curves in fig. 1, is exotic: it exists for $T>T_{\chi\rm{SB}}$ and is realized holographically as the Klebanov-Strassler black brane [23]. If we could extend this phase for $\frac{T}{\Lambda}\to\infty$, 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 range

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 numerics⁵⁵5In the notations of [32], e.g., it is the coefficient $f_{c,8,0}$ — see (A.55) there.;

• •

  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 ${\cal K}_{{\chi\rm{SB}}}$ evaluated at $T=T_{\chi\rm{SB}}$. Over the range (2.9), the Kretschmann scalar corresponding to the exotic phase changes as

  $${\cal K}={\cal K}_{{\chi\rm{SB}}}\times\biggl{\{}1\cdots 8890\biggr{\}}\,.$$

  (2.10)

  Thus, we conclude that there can not be a reliable conformal order on the warped deformed conifold with fluxes, arising from the $T\to\infty$ 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]

  $$T_{u}=0.537(3)\Lambda\,,$$

  (2.11)

  represented by the vertical dashed brown lines in fig. 2, for the deconfined chirally symmetric states of the cascading gauge theory plasma — these states exist only for $T\geq T_{u}$. In the vicinity of $T_{u}$, there are two branches of states: the stable branch with respect to the energy density fluctuations (the black curves in fig. 2), and the unstable branch (the brown curves in fig. 2). On the former branch the specific heat is positive, while it is negative on the latter branch, explaining the (in)stability to the energy density fluctuations [33]. The unstable branch of the deconfined chirally symmetric states is exotic: it extends for $T>T_{u}$ with the ever increasing values of the thermal expectation values of ${\cal O}_{\Delta=\{4,6,8\}}$ , however, much like in the case of the exotic phase with the chiral symmetry breaking, we fail to extend it as $T\to\infty$. We constructed the latter states for fairly narrow temperature range:

  $$\frac{T}{\Lambda}=\frac{T_{u}}{\Lambda}\times\biggl{\{}1\cdots 1.0181\biggr{\}}\,.$$

  (2.12)

  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

  $${\cal K}={\cal K}_{{\chi\rm{SB}}}\times\biggl{\{}1.669\cdots 73.1\biggr{\}}\,.$$

  (2.13)

  On the contrary, the deconfined chirally symmetric phase with the positive specific heat (the black curves in fig. 2) can be extended as $\frac{T}{\Lambda}\to\infty$ — however it is not exotic, as the thermal expectation values of ${\cal O}_{\Delta=\{4,6,8\}}$ 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 $T\to\infty$ limit of the Klebanov-Tseytlin black branes with the negative specific heat — such black branes are outside the validity of the supergravity approximation.

Consistent truncation in the $SU(2)\times SU(2)\times{\mathbb{Z}}_{2}$ invariant sector of type IIB supergravity on warped deformed conifold with fluxes to a five dimensional manifold ${\cal M}_{5}$ was derived in [31]⁶⁶6See [32] for a recent comprehensive review.:

It is a functional of the a five-dimensional metric $\hat{g}_{\mu\nu}$ on ${\cal M}_{5}$,

scalars $\Omega_{i=1\cdots 3}$ describing the warping and the deformation of the conifold, a dilaton $\Phi$, scalars $h_{i=1\cdots 3}$ parameterizing the 3-form fluxes, a constant parameter $\Omega_{0}$ (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),

and $\hat{R}_{5}$ 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 rescaling

leading to

Further introducing

the five-dimensional effective action becomes

where

Consistent truncation of (3.8), i.e.,

produces effective action of $SU(2)\times SU(2)\times U(1)$ 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 ${\cal N}=1$ supersymmetric $SU(N+M)\times SU(N)$ cascading gauge theory, often referred to as a Klebanov-Strassler gauge theory [18]. This theory is not conformal, and has a strong coupling scale $\Lambda$,

where $g_{s}$ is the asymptotic value of the string coupling constant⁷⁷7It can always be fixed to $g_{s}=1$., 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 [35, 32]. We are interested here in the conformal holographic model, thus we must take the limit $\Lambda\to 0$, which is equivalent to sending $P\to 0$. Finding the conformal order in the model with 7 scalars is a daunting task — so⁸⁸8While 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. we will follow the framework developed in [5], and reviewed in appendix A.
   As a first step, we truncate the general potential ${\cal P}_{flux}+{\cal P}_{scalar}$ 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., ${\cal P}_{flux}\geq 0$. Thus, in addition to $P=0$, we set $h_{i}\equiv 0$. Then,

where in the second equality we set a constant parameter $\Omega_{0}=\frac{1}{108}$ to ensure that the radius of the asymptotically AdS₅ geometry is $L=1$.
   In the absence of 3-form fluxes the dilaton becomes an exact modulus, and we set

   We now arrive at the effective action we use to analyze conformal order on the warped deformed conifold:

where

is the central charge of ${\cal N}=1$ superconformal $SU(N)\times SU(N)$ 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 $S_{5}=S_{5}[f,w,\lambda]$ (3.15):

• •

  Model I:

  $$S_{5}^{I}[f]\ \equiv\ S_{5}\bigg{|}_{w=\lambda\equiv 0}\,;$$

  (3.22)

• •

  Model II:

  $$S_{5}^{II}[f,w]\ \equiv\ S_{5}\bigg{|}_{\lambda\equiv 0}\,;$$

  (3.23)

• •

  Model III:

  $$S_{5}^{III}[f,w,\lambda]\ \equiv\ S_{5}\,.$$

  (3.24)