0912/0912.0512.md

Quantum Criticality and Holographic Superconductors in M-theory

Jerome P. Gauntlett, Julian Sonner and Toby Wiseman

Theoretical Physics Group, Blackett Laboratory,
Imperial College, Prince Consort Rd, London SW7 2AZ, U.K.

Abstract

We present a consistent Kaluza-Klein truncation of $D = 11$ supergravity on an arbitrary seven-dimensional Sasaki-Einstein space ( $SE_7$ ) to a $D = 4$ theory containing a metric, a gauge-field, a complex scalar field and a real scalar field. We use this $D = 4$ theory to construct various black hole solutions that describe the thermodynamics of the $d = 3$ CFTs dual to skew-whiffed $AdS_4 \times SE_7$ solutions. We show that these CFTs have a rich phase diagram, including holographic superconductivity with, generically, broken parity and time reversal invariance. At zero temperature the superconducting solutions are charged domain walls with a universal emergent conformal symmetry in the far infrared.# Contents

1	Introduction	2
2	The consistent KK truncation of [7]	8
3	$AdS\_4$ vacuum solutions	9
3.1	$AdS\_4\; \backslash times\; SE\_7$ solutions: supersymmetric and skew-whiffed . . . . .	10
3.2	Pope-Warner solutions . . . . .	11
3.3	Englert solutions . . . . .	12
3.4	Flux quantisation and central charges . . . . .	12
4	Further consistent KK truncation	13
5	Ansatz for black hole and domain wall solutions	15
5.1	Skew-whiffed to Pope-Warner uncharged and charged domain walls . .	17
5.2	Skew-whiffed to $AdS\_2\; \backslash times\; \backslash mathbb\{R\}^2$ solutions . . . . .	18
5.3	Black hole solutions . . . . .	18
5.4	Counter terms and black hole thermodynamics . . . . .	19
6	Uncharged domain wall solutions: holographic RG flows	22
7	Interpolating solutions with $T\; =\; 0,\; \backslash mu\; \backslash neq\; 0$	25
7.1	Pope-Warner IR: zero temperature superconductors . . . . .	25
7.2	$AdS\_2\; \backslash times\; \backslash mathbb\{R\}^2$ IR: zero temperature normal phase . . . . .	28
8	Uncharged black hole solutions	29
9	Charged black hole solutions and superconductivity	30
9.1	Black hole solutions with $\backslash mu\; \backslash neq\; 0$ and no scalar hair: the unbroken phase	32
9.2	Black hole solutions with $\backslash mu\; \backslash neq\; 0$ and scalar hair: the superconducting phase . . . . .	34
9.3	Conductivity . . . . .	35
10	Discussion	38
A	The differential equations	43

## 1 Introduction

Black holes with charged hair in anti-de-Sitter space provide a holographic description of superconductivity via the AdS/CFT correspondence. This idea was first discussed in [1][2] and some of the subsequent developments have been nicely reviewed in [3][4]. In more detail, one considers a CFT with an AdS dual in a theory of gravity with matter fields which include a Maxwell gauge field and additional charged fields. The CFT is studied at finite temperature $T$ and at fixed chemical potential $\mu$ (or at fixed charge density) by studying electrically charged black holes in the dual gravity theory. Charged black holes with vanishing charged matter fields describe the high temperature, normal phase of the superconductor. Below some critical temperature, one requires a new branch of charged black hole solutions to emerge that carry charged hair and are thermodynamically favoured. The charged hair spontaneously breaks the $U(1)$ gauge symmetry in the bulk which corresponds to a spontaneous breaking of a global $U(1)$ symmetry in the boundary CFT, signalling the superconductivity. More precisely, this signals superfluidity, but for certain phenomena one expects that the difference between the two is not significant [5].

Most work has focussed on $D = 4$ theories of gravity, corresponding to superconductors in $d = 3$ spacetime dimensions, as this is likely to be the most promising arena to make contact with real materials. The black holes are usually taken to have flat $\mathbb{R}^2$ horizons, and hence are also called black branes, corresponding to considering the CFTs in $d = 3$ Minkowski spacetime. The conformal invariance then implies that the critical temperature for the onset of superconductivity is fixed by the scale set by $\mu$ .

Most studies have been carried out within the context of “phenomenological” models of gravity without any obvious embedding into string/M-theory. This bottom up approach has the virtue of simplicity but has the drawback that one is not guaranteed that there is a well defined underlying conformal field theory. Furthermore, if the models are posited to just provide approximate supergravity solutions to string/M-theory, the approximations can obscure important physical features, such as the low-temperature behaviour of the superconductors. Many investigations have also used a “probe approximation” within these phenomenological models, in which the back reaction onthe gravitational field is ignored. This approximation again makes the analysis more tractable and while it should capture some important features it cannot be used to study, for example, low-temperature phenomena. The incorporation of back reaction for the most studied class of phenomenological models with a single charged scalar field was initiated in the foundational work [5].

There has also been recent progress in a top down approach, where one aims to find exact solutions of $D = 11$ and type IIB supergravity. The constructions have been based on new consistent KK truncations of the supergravity theories. Building on the work of [6][7], fully back reacted solutions of $D = 11$ supergravity that describe holographic superconductors in $d = 3$ spacetime dimensions were constructed in [8]. In the zero temperature limit, $T \rightarrow 0$ , the event horizon of the superconducting black holes disappears implying that the entropy of the dual superconductors vanishes in this limit. More precisely, it was shown in [8] (and further studied in [9]) that as $T \rightarrow 0$ the superconducting black hole solutions approach charged domain wall solutions that interpolate between (perturbed) $AdS_4$ solutions in the UV that preserve the $U(1)$ gauge symmetry and different $AdS_4$ solutions in the IR that break the gauge symmetry. These domain wall solutions demonstrate that when the dual CFT is held at zero temperature and finite chemical potential there is an emergent conformal symmetry at low energies, exactly as in a class of phenomenological models¹ studied in [11]. The $D = 11$ superconducting black hole solutions of [8] were constructed using a $D = 4$ theory of gravity, to be discussed momentarily. This $D = 4$ theory has some similarities with the $D = 4$ phenomenological model of [5] after fixing some parameters. However, there are also important differences. In particular, it was shown in [8] that the superconducting black hole solutions in the model of [5], for these parameters², become singular as $T \rightarrow 0$ .

Analogous constructions of solutions of type IIB supergravity dual to superconductors in $d = 4$ spacetime dimensions have been carried out in [13][9], but not yet in the same level of detail. It has been shown that superconducting black hole solutions should exist using a probe analysis in [13]. Furthermore, zero temperature domain wall solutions with emergent conformal symmetry have been constructed [9]; while it is expected that they arise as the zero temperature limit of the superconducting black hole solutions at finite temperature, this has not yet been shown. These superconducting black hole solutions in $D = 11$ and type IIB carry abelian charge of “ $R$ -symmetry

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¹Note that models with an interesting emergent Lifshitz scaling in the IR were also studied in [10].

²The $T \rightarrow 0$ limit for other values of the parameters in the model of [5] was discussed in [12].type” (it is an $R$ -symmetry for supersymmetric vacua); the construction of type IIB solutions carrying a baryonic abelian charge were recently presented in [14].

In this paper we will expand upon and extend the analysis of $d = 3$ holographic superconductivity arising in $D = 11$ supergravity that was initiated in [8]. The solutions found in [8] were obtained using the consistent KK truncation [7] of $D = 11$ supergravity on a seven-dimensional Sasaki-Einstein space down to a $D = 4$ theory of gravity that includes six real scalar fields and two gauge fields. The consistency of this KK reduction means that given an arbitrary Sasaki-Einstein metric, any solution of the $D = 4$ theory can be uplifted to obtain an exact solution of $D = 11$ supergravity. It was shown in [8] that it is possible to further truncate the $D = 4$ theory to a theory with a metric, $g$ , a gauge field, $A$ , with field strength $F$ , and a charged scalar field, $\chi$ , provided that one restricts to solutions satisfying $F \wedge F = 0$ (such as electrically charged black holes). One result of this paper is that there is a consistent truncation, with no additional restrictions, that is obtained by also keeping an additional neutral scalar field $h$ . As we now discuss this truncated $D = 4$ theory with fields $(g, A, \chi, h)$ , with action given in (4.3), has a rich structure to explore different aspects of superconductivity. A particularly interesting feature of the $D = 4$ theory is that the neutral scalar field $h$ couples to $F \wedge F$ and this coupling implies that the solutions we discuss with $h \neq 0$ break $d = 3$ parity and time reversal invariance.

The truncated theory has three $AdS_4$ vacua which uplift to $AdS_4 \times SE_7$ solutions in $D = 11$ . One of them uplifts to the skew-whiffed $AdS_4$ solutions of Freund-Rubin type [15], as shown in [7], the second uplifts to the $AdS_4$ solutions of Pope and Warner type [16][17], as shown in [8], and the third, which is the only one with $h \neq 0$ , uplifts to the $AdS_4$ solutions of Englert type [18][19], as we will show here. Recall that the skew-whiffed solutions do not preserve any supersymmetry, except in the special case that the $SE_7$ space is the round seven-sphere in which case it preserves all supersymmetry. All skew-whiffed solutions are known to be perturbatively stable [20]. Some discussion on the possibility of $1/N$ effects destabilising the non-supersymmetric skew-whiffed solutions has been discussed in [21][22]. The Pope-Warner and the Englert solutions do not preserve any supersymmetry for any choice of $SE_7$ . A stability analysis for the Pope-Warner solutions has not been carried out; in light of the results presented in [8] and here, we feel that this would now be a worthwhile investigation. On the other hand the Englert solutions are known to be unstable [23]: indeed we will show that an unstable mode is already present in our truncated Lagrangian.

In the skew-whiffed $AdS_4$ vacuum the operators $\mathcal{O}_\chi, \mathcal{O}_h$ dual to the fields $\chi, h$ , re-spectively, are both relevant operators with scaling dimension taken³ to be $\Delta = 2$ . Before investigating superconductivity, we first construct uncharged domain wall solutions (i.e. the gauge-field is identically zero) that describe ordinary RG flows between the different $AdS$ vacua. We will show that there is a one parameter family of such domain wall solutions which describe RG flows between the skew-whiffed vacuum in the UV, perturbed by $\mathcal{O}_\chi$ and $\mathcal{O}_h$ , and the Pope-Warner vacuum in the IR. As is usual for such RG flows the operators have non-zero vevs, $\langle \mathcal{O}_\chi \rangle, \langle \mathcal{O}_h \rangle \neq 0$ . We will also show that there is a single domain wall solution that flows between the skew-whiffed vacuum in the UV to the Englert vacuum in the IR and also a domain wall solution that flows from the Englert vacuum in the UV to the Pope-Warner vacuum in the IR. Of course, given the instability of the Englert vacuum, the physical relevance of these latter domain wall solutions is not clear.

We then turn our attention to superconductivity. We will be interested in the CFT dual to the skew whiffed vacuum that has been deformed by $\mathcal{O}_h$ , generalising the analysis of [8], and the solutions we construct imply that we also have $\langle \mathcal{O}_h \rangle \neq 0$ . The superconductivity is signalled by a spontaneous breaking of the $U(1)$ symmetry, so we demand that the CFT is not perturbed by $\mathcal{O}_\chi$ (in contrast to the RG flow solutions discussed in the last paragraph) and look for solutions with $\langle \mathcal{O}_\chi \rangle \neq 0$ .

We first construct charged domain walls that describe the deformed skew-whiffed CFT at zero temperature and non-zero chemical potential. After observing that in the Pope-Warner vacuum the gauge-field is dual to an irrelevant operator a simple parameter count suggests that, given the one parameter family of uncharged domain wall RG flow solutions mentioned above, there should also be a one parameter family of charged domain wall solutions that interpolate between the skew-whiffed vacuum in the UV and the Pope-Warner vacuum in the IR. This is in accord with the conjecture made in [9]. In the special case that $h = 0$ such a solution was found in [8] (and further studied in [9]) and here we will construct a more general one parameter family of solutions with $h \neq 0$ . It is interesting to note that these solutions only exist for a certain range of deformations by $\mathcal{O}_h$ (at fixed $\mu$ ). We then show that this one parameter family of charged domain wall solutions arises as the zero temperature limit of a more general class of superconducting black holes, once again generalising what was found in [8] for $h = 0$ . We first construct Reissner-Nordström-like charged black

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³This arises because of the boundary conditions we shall impose on the skew-whiffed $AdS_4$ solution; different boundary conditions would lead to $\Delta = 1$ .holes⁴ with $\chi = 0$ , but with $h \neq 0$ , which describe the high temperature normal phase of these superconductors. At finite temperature these solutions have a regular horizon with $h$ going to zero at the horizon. Furthermore, in the zero temperature limit, for a certain range of deformations $\mathcal{O}_h \neq 0$ , they approach $AdS_2 \times \mathbb{R}^2$ , exactly as for the Reissner-Nordström black hole with $h = 0$ . For larger deformations, in the zero temperature limit they have vanishing entropy and become singular. We then show that for a certain range of deformations $\mathcal{O}_h \neq 0$ a new branch of black holes carrying charged scalar hair appear and that they are thermodynamically favoured, thus demonstrating that we do indeed have holographic superconductors. Our numerics indicate that the superconducting black holes exist for exactly the same class of deformations where the Reissner-Nordström like black holes have an $AdS_2 \times \mathbb{R}^2$ limit at zero temperature and hence at zero temperature the $AdS_2 \times \mathbb{R}^2$ solutions are thermodynamically disfavoured. Furthermore, we show that the solutions with charged hair smoothly map on the zero temperature charged domain wall solutions with the Pope-Warner $AdS_4$ region in the IR, demonstrating the emergent $d = 3$ conformal symmetry of these holographic superconductors. It is worth highlighting that in the far IR all of these superconductors (for a given $SE_7$ space), when held at zero temperature and finite chemical potential, are described by exactly the same universal CFT i.e. the CFT dual to the Pope-Warner $AdS_4$ vacuum.

If we assume that our analysis has captured all of the relevant instabilities in M-theory, the phase diagram for our new holographic superconductors, for $\mu \neq 0$ , is summarised in Figure 1. The vertical axis is the temperature, the horizontal axis is the value of $h_1$ which determines the isotropic deformation by the operator $\mathcal{O}_h$ . The most striking feature is the superconducting dome that appears for $-h_1^c \leq h_1 \leq h_1^c$ . Under this dome at zero temperature, and in the far IR, the superconductors are all described by the same Pope-Warner $AdS_4$ solution. Outside of the dome at zero temperature the system is described by singular solutions which require further investigation. We will also construct black hole solutions with $\mu = 0$ and this part of the phase diagram is incorporated in Figure 14 in the discussion section.

Finally, we calculate the electrical conductivity of our black holes using linear response theory. We find that the electrical conductivity contains both longitudinal and, when $h \neq 0$ , transverse (Hall) components for both the superconducting and normal phase black holes, the latter arising from the broken parity and time reversal invariance

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⁴It is interesting to compare this class of charged black holes with those that were very recently constructed in a top down model in [24] and a phenomenological model in [25].Figure 1: The phase diagram for the holographic superconductors. The vertical axis is temperature, the horizontal axis determines the deformation of the skew-whiffed CFT by the operator $\mathcal{O}_h$ and the chemical potential $\mu$ is non-zero.

in the boundary theory.

The plan of the rest of the paper is as follows. In section 2 we briefly review the consistent KK truncation on an arbitrary Sasaki-Einstein seven-manifold that was presented in [7]. In section 3 we discuss the three $AdS_4$ vacua of the $D = 4$ theory and their uplifts to $D = 11$ solutions. Section 4 presents the new additional consistent KK truncation of the $D = 4$ theory. In section 5 we present the ansatz for the $D = 4$ fields that we shall use to construct black hole and domain wall solutions. We also discuss boundary counter terms in the action and some aspects of thermodynamics. As somewhat of an aside, section 6 discusses uncharged domain walls corresponding to the RG flows between the $AdS_4$ vacua. Section 7 discusses the zero temperature limit solutions of the charged black hole solutions that we construct in section 9. We construct the charged domain walls interpolating between the deformed skew-whiffed $AdS_4$ vacuum and the Pope-Warner $AdS_4$ vacuum and also the solutions interpolating between the skew-whiffed $AdS_4$ vacuum and the $AdS^2 \times \mathbb{R}^2$ . Section 8 briefly discusses uncharged black hole solutions (i.e. $\mu = 0$ ) with $h \neq 0$ . Section 9 discusses both the normal and superconducting phase charged black hole solutions for general $h$ and presents some results on the conductivity of the black holes. We briefly conclude in section 10 and we have three appendices.## 2 The consistent KK truncation of [7]

We start by summarising the consistent KK truncation of $D = 11$ supergravity on an arbitrary Sasaki-Einstein space $SE_7$ found in [7] (extending [26][27]). First recall that any Sasaki-Einstein metric can, locally, be written as a fibration over a six-dimensional Kähler-Einstein space

$$d{s}^2(S{E}_7)\equiv d{s}^2(K{E}_6)+\eta \otimes \eta (2.1)ds^2(SE\_7)\; \backslash equiv\; ds^2(KE\_6)\; +\; \backslash eta\; \backslash otimes\; \backslash eta\; \backslash quad\; (2.1)$$

Here $\eta$ is the one-form dual to the Reeb Killing vector satisfying $d\eta = 2J$ where $J$ is the Kähler form of $KE_6$ . We denote the $(3, 0)$ form defined on $KE_6$ by $\Omega$ and $d\Omega = 4i\eta \wedge \Omega$ . The volume form is taken to be $vol(SE_7) = \eta \wedge J^3/3! = (i/8)\eta \wedge \Omega \wedge \Omega^*$ .

For a regular or quasi-regular Sasaki-Einstein manifold, the orbits of the Reeb vector all close, corresponding to compact $U(1)$ isometry, and the $KE_6$ is a globally defined manifold or orbifold, respectively. For an irregular Sasaki-Einstein manifold, the Reeb-vector generates a non-compact $\mathbb{R}$ isometry and the $KE_6$ is only locally defined. For applications to holographic superconductivity one is most interested in cases with $U(1)$ isometry.

In the KK ansatz the $D = 11$ metric is written as

$$\frac{1}{(2L{)}^2}d{s}^2=e^{-6U-V}d{s}_4^2+e^{2U}d{s}^2(K{E}_6)+e^{2V}(\eta +A_1)\otimes (\eta +A_1),(2.2)\backslash frac\{1\}\{(2L)^2\}\; ds^2\; =\; e^\{-6U-V\}\; ds\_4^2\; +\; e^\{2U\}\; ds^2(KE\_6)\; +\; e^\{2V\}\; (\backslash eta\; +\; A\_1)\; \backslash otimes\; (\backslash eta\; +\; A\_1)\; ,\; \backslash quad\; (2.2)$$

while the four-form is written

where $ds_4^2$ is a four-dimensional metric⁵, $U, V, h$ are real scalars, $\chi$ is a complex scalar defined on the four-dimensional space. Furthermore, also defined on this four-dimensional space are $A_1$ a one-form potential, with field strength $F_2 \equiv dA_1$ , two-form and three-form field strengths $H_2$ and $H_3$ , related to one-form and two-form potentials via

Note that the scalar $\chi$ is charged with respect to $A_1$ and in particular we have $D\chi \equiv d\chi - 4iA_1\chi$ . Also note that $\epsilon$ appearing in the four-form flux is a constant:

$$ϵ=\pm 1(2.5)\backslash epsilon\; =\; \backslash pm\; 1\; \backslash quad\; (2.5)$$

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⁵Note that we are using the four-dimensional Einstein frame metric.whose significance will be explained below. Our conventions for $D = 11$ supergravity are as in [28]; in particular we note that the $D = 11$ volume form is given by $vol_4 \wedge vol(SE_7)$ . Finally, $L$ is an arbitrary length scale which we will later set to $1/2$ .

This provides a consistent KK truncation of $D = 11$ supergravity in the sense that if one finds a solution to the equations of motion for the $D = 4$ metric $g$ and matter fields $U, V, A_1, B_1, B_2, h, \chi$ , as given in [7], then one has found a solution to the $D = 11$ supergravity equations of motion. The $D = 4$ equations of motion can be derived from an action given in [7]. In this paper we will find it convenient to work with an action that is obtained after dualising the one-form $B_1$ to another one form $\tilde{B}_1$ and the two-form $B_2$ to a scalar $a$ as explained in section 2.3 of [7]. The dual action is given by

where $\tilde{H}_2 \equiv d\tilde{B}_1$ . The dual fields are related to the original fields via

Note that the factors of $L$ appear in a more conventional way if one uses the rescaled metric $\tilde{g} = (2L)^2 g$ .

3 $AdS_4$ vacuum solutions

The $D = 4$ equations of motion arising from (2.6) admit various $AdS_4$ solutions with $F_2 = \tilde{H}_2 = a = 0$ and for various values of the scalar fields $U, V, h$ and $\chi$ . In the following $ds^2(AdS_4)$ will always denote the standard unit radius $AdS_4$ metric and $Vol(AdS_4)$ the corresponding volume-form as given in appendix A of [7]. We will determine the masses of the other fields considered as perturbations around each $AdS_4$ solution to obtain the scaling dimensions of the dual operators in the boundary CFT. For scalarfields with mass $m$ the scaling dimensions are given by

$$\mathrm{\Delta }=\frac{3}{2}\pm \frac{1}{2}[9+4m^2{R}_{AdS}^2{]}^{1\mathrm{/}2}(3.1)\backslash Delta\; =\; \backslash frac\{3\}\{2\}\; \backslash pm\; \backslash frac\{1\}\{2\}[9\; +\; 4m^2\; R\_\{AdS\}^2]^\{1/2\}\; \backslash quad\; (3.1)$$

while those for vector fields are given by

$$\mathrm{\Delta }=\frac{3}{2}\pm \frac{1}{2}[1+4m^2{R}_{AdS}^2{]}^{1\mathrm{/}2}(3.2)\backslash Delta\; =\; \backslash frac\{3\}\{2\}\; \backslash pm\; \backslash frac\{1\}\{2\}[1\; +\; 4m^2\; R\_\{AdS\}^2]^\{1/2\}\; \backslash quad\; (3.2)$$

where $R_{AdS}^2$ is the radius squared of the Einstein frame $AdS$ metric $g$ . Note that if we used the metric $\tilde{g} = (2L)^2 g$ then $R_{AdS}^2 \rightarrow (2L)^2 R_{AdS}^2$ and $m^2 \rightarrow m^2/(2L)^2$ .

3.1 $AdS_4 \times SE_7$ solutions: supersymmetric and skew-whiffed

The simplest $AdS_4$ vacua have $\epsilon = \pm 1$ ,

$$U=0,V=0,\chi =0,h=0(3.3)U\; =\; 0,\; \backslash quad\; V\; =\; 0,\; \backslash quad\; \backslash chi\; =\; 0,\; \backslash quad\; h\; =\; 0\; \backslash quad\; (3.3)$$

and the radius squared of the Einstein $AdS_4$ metric is given by

$$R_{AdS}^2=\frac{1}{4}\mathrm{.}(3.4)R\_\{AdS\}^2\; =\; \backslash frac\{1\}\{4\}.\; \backslash quad\; (3.4)$$

These uplift to the $D = 11$ solutions:

When $\epsilon = +1$ , these $AdS_4 \times SE_7$ solutions are supersymmetric and are dual to $d = 3$ SCFTs with, generically, $\mathcal{N} = 2$ supersymmetry. For these solutions the Killing vector dual to the one-form $\eta$ in the $SE_7$ metric (2.1) is dual to an $R$ -symmetry. On the other hand when $\epsilon = -1$ the solutions are skew-whiffed $AdS_4 \times SE_7$ solutions and generically do not preserve any supersymmetry at all. An important exception is when $SE_7$ is the round seven-sphere in which case both $AdS_4$ solutions preserve maximal supersymmetry. The skew-whiffed solutions have been shown to be perturbatively stable [20] and thus should be dual to well defined CFTs at least in the supergravity approximation. Some discussion on the possibility of $1/N$ effects destabilising the non-supersymmetric skew-whiffed solutions has been discussed in [21][22] and it would be interesting to explore this further. Note that for the skew-whiffed $AdS_4 \times SE_7$ solutions the Killing vector dual to the one-form $\eta$ in the $SE_7$ metric (2.1) is dual to a global symmetry in the dual CFT and is an $R$ -symmetry just for the case of $SE_7 = S^7$ .The spectrum of the $D = 4$ theory in these backgrounds was discussed in [7]. For $\epsilon = +1$ , $m_h^2 = 40$ , $m_\chi^2 = 40$ and $U, V$ mix to give $m^2 = 16, 72$ . These give scaling dimensions $\Delta = 5, 5$ and $\Delta = 4, 6$ respectively. There is also a massless gauge field and a massive gauge field with $m^2 = 48$ corresponding to $\Delta = 2$ and $\Delta = 5$ , respectively.

When $\epsilon = -1$ , $U, V$ mix to again give $m^2 = 16, 72$ , corresponding to scaling dimensions $\Delta = 4, 6$ respectively. On the other hand now $m_h^2 = m_\chi^2 = -8$ with each corresponding to scaling dimension $\Delta_\pm = 1, 2$ . The masses of the gauge fields are unchanged. Note that in both cases the field $a$ becomes the longitudinal mode of the massive gauge field.

3.2 Pope-Warner solutions

Another $AdS_4$ vacuum is obtained when $\epsilon = -1$ ,

$$e^U=2^{-1\mathrm{/}6},e^V=2^{1\mathrm{/}3},{\chi }^2=2\mathrm{/}3,h=0(3.6)e^U\; =\; 2^\{-1/6\},\; \backslash quad\; e^V\; =\; 2^\{1/3\},\; \backslash quad\; \backslash chi^2\; =\; 2/3,\; \backslash quad\; h\; =\; 0\; \backslash quad\; (3.6)$$

and the radius squared of the Einstein $AdS_4$ metric is given by

$$R_{AdS}^2=\frac{3}{16}(3.7)R\_\{AdS\}^2\; =\; \backslash frac\{3\}\{16\}\; \backslash quad\; (3.7)$$

Choosing $\chi = +(2/3)^{1/2}$ for definiteness we find that this uplifts to the $D = 11$ class solutions

which were first constructed in [16][17]. It has been shown that they do not preserve any supersymmetry but a stability analysis has not yet been performed. Note that these solutions are topologically $AdS_4 \times SE_7$ : the fibre of the $SE_7$ metric being stretched by a factor of $\sqrt{2}$ compared to the $SE_7$ metric in (2.1).

In this background, the $D = 4$ theory gives two massive scalars with $m^2 = 32$ and two with $m^2 = 96$ corresponding to scaling dimensions $\Delta = 3/2 + \sqrt{33}/2$ and $\Delta = 6$ , respectively. There are also two massive gauge fields with $m^2 = 32$ and $m^2 = 96$ corresponding to scaling dimensions $\Delta = 4$ and $\Delta = 3/2 + \sqrt{73}/2$ , respectively. Note that the phase of $\chi$ and the field $a$ become longitudinal modes of the massive gauge-fields.### 3.3 Englert solutions

Another $AdS_4$ vacuum is obtained when $\epsilon = -1$ ,

$$e^U=(4\mathrm{/}5{)}^{1\mathrm{/}6},e^V=(4\mathrm{/}5{)}^{1\mathrm{/}6},{\chi }^2=4\mathrm{/}15,h^2=1\mathrm{/}5(3.9)e^U\; =\; (4/5)^\{1/6\},\; \backslash quad\; e^V\; =\; (4/5)^\{1/6\},\; \backslash quad\; \backslash chi^2\; =\; 4/15,\; \backslash quad\; h^2\; =\; 1/5\; \backslash quad\; (3.9)$$

and the radius squared of the Einstein $AdS_4$ metric is given by

$$R_{AdS}^2=\frac{12}{25\sqrt{5}}(3.10)R\_\{AdS\}^2\; =\; \backslash frac\{12\}\{25\backslash sqrt\{5\}\}\; \backslash quad\; (3.10)$$

Choosing $\chi = +(2/15)^{1/2}$ , $h = +1/\sqrt{5}$ for definiteness we find that this uplifts to the $D = 11$ class of solutions

This an Englert-type solution and note that the metric on $SE_7$ is exactly the same as in (2.1). For the special case when $SE_7 = S^7$ it was first constructed in [18] and the generalisation was suggested in [19]. This solution is known not to preserve any supersymmetry (for the $S^7$ case this was shown in [29] and the results of [23] show this more generally) and to be unstable [23].

In this background, the $D = 4$ theory gives four massive scalars with $m^2 = -5\sqrt{5}$ , $25\sqrt{5}/2$ , $20\sqrt{5}$ , $75\sqrt{5}/2$ . Note that the first mode has complex scaling dimension and thus violates the BF bound. This unstable mode, as well as the other three modes, are precisely the same modes considered in [23]. The scaling dimensions of the three other scalars are $\Delta = 3/2 + \sqrt{33}/2$ , $3/2 + (237/20)^{1/2}$ , $6$ , respectively. There are also two massive gauge fields with $m^2 = 5\sqrt{5}$ and $m^2 = 30\sqrt{5}$ corresponding to scaling dimensions $\Delta = 3/2 + (53/20)^{1/2}$ and $\Delta = 3/2 + (293/20)^{1/2}$ , respectively.

3.4 Flux quantisation and central charges

The $D = 11$ equation of motion for the four-form is $d * G_4 + \frac{1}{2} G_4 \wedge G_4 = 0$ and the quantised membrane charge is given by

$$N=\frac{1}{(2\pi l{)}^6}\int (\ast G_4+\frac{1}{2}{C}_3\wedge G_4)(3.12)N\; =\; \backslash frac\{1\}\{(2\backslash pi\; l)^6\}\; \backslash int\; (*G\_4\; +\; \backslash frac\{1\}\{2\}\; C\_3\; \backslash wedge\; G\_4)\; \backslash quad\; (3.12)$$

where $dC_3 = G_4$ and $l$ is the $D = 11$ Planck length. For each of the above solutions with $\epsilon = -1$ , we find

$$N=6{\left(\frac{(2L)}{2\pi l}\right)}^{6}vol(S{E}_7)(3.13)N\; =\; 6\; \backslash left(\; \backslash frac\{(2L)\}\{2\backslash pi\; l\}\; \backslash right)^6\; vol(SE\_7)\; \backslash quad\; (3.13)$$Note that in our conventions this is counting the number of anti-membranes. We can also calculate the central charge using the formula [30]

$$c\equiv 384\frac{(2L{)}^2{R}_{AdS}^2}{16\pi G}(3.14)c\; \backslash equiv\; 384\; \backslash frac\{(2L)^2\; R\_\{AdS\}^2\}\{16\backslash pi\; G\}\; \backslash quad\; (3.14)$$

In our conventions the D=11 action is

$$S=\frac{1}{(2\pi {)}^8{l}^9}\int d^{11}x\sqrt{-g_{11}}[R_{11}+\dots ](3.15)S\; =\; \backslash frac\{1\}\{(2\backslash pi)^8\; l^9\}\; \backslash int\; d^\{11\}x\; \backslash sqrt\{-g\_\{11\}\}\; [R\_\{11\}\; +\; \backslash dots]\; \backslash quad\; (3.15)$$

and hence

$$\frac{1}{16\pi G}=\frac{(2L{)}^7\text{vol}(S{E}_7)}{(2\pi {)}^8{l}^9}(3.16)\backslash frac\{1\}\{16\backslash pi\; G\}\; =\; \backslash frac\{(2L)^7\; \backslash text\{vol\}(SE\_7)\}\{(2\backslash pi)^8\; l^9\}\; \backslash quad\; (3.16)$$

We thus deduce

$$c=\frac{128\pi N^{3\mathrm{/}2}}{6^{1\mathrm{/}2}\text{vol}(S{E}_7{)}^{1\mathrm{/}2}}{R}_{AdS}^2(3.17)c\; =\; \backslash frac\{128\backslash pi\; N^\{3/2\}\}\{6^\{1/2\}\; \backslash text\{vol\}(SE\_7)^\{1/2\}\}\; R\_\{AdS\}^2\; \backslash quad\; (3.17)$$

By comparing $R_{AdS}^2$ of the different $AdS_4$ vacua, one concludes that it might be possible to find a domain wall solution that interpolates between a perturbed skew-whiffed vacuum in the UV and the Pope-Warner vacuum in the IR, which would be dual to an RG flow between the corresponding dual CFTs. We shall see later that this is indeed the case. In fact we will see that there is a one parameter family of domain walls that interpolates between these vacua. We will also find a domain wall solution that interpolates from the skew-whiffed vacuum in the UV to the Englert vacuum in the IR and another domain wall solution that interpolates between the Englert vacuum in the UV and the Pope-Warner in the IR. The instability of the Englert vacuum makes these latter solutions less interesting as far as RG flows are concerned.

4 Further consistent KK truncation

When $\epsilon = -1$ there is a further consistent truncation of the $D = 4$ theory described by (2.6) that is obtained by setting

Note that this implies

After substituting this into the equations of motion derived from (2.6) (see appendix B of [7]) we obtain equations that can be derived from the following action

This action is also obtained by substituting the ansatz (4.1) directly into (2.6). From now on we have set $L = 1/2$ . Observe that we must have $|\chi| < \frac{2}{\sqrt{3}}$ and $|h| < 1$ . We also observe that all terms in the Lagrangian except for the coupling of $h$ to $F \wedge F$ are invariant under $h \rightarrow -h$ . As we shall see later this coupling leads to breaking of parity and time reversal invariance in the dual CFT.

It is worth pointing out that there are further consistent truncations that one can consider. For example, one can consistently set $\chi = 0$ to obtain a theory of gravity coupled to a gauge field $A_1$ and a neutral scalar $h$ , and we will find black hole solutions of this theory (it is interesting to compare and contrast with the phenomenological models studied in [24][25]). It is also possible to then further consistently set $A_1 = 0$ . Alternatively, we can set $A_1 = 0$ provided that we restrict $\chi$ to be real: $\chi = \chi_R$ . This theory with two real scalars can then be further consistently truncated⁶ by either setting $\chi_R = (2/\sqrt{3})h$ or by setting $h = 0$ . Finally, observe that we can set $h = 0$ provided that we impose $F \wedge F = 0$ by hand [8] (and hence setting $h = 0$ is not a consistent KK truncation).

Observe that all of the $AdS_4$ vacua discussed in the last subsection are solutions of the consistent truncation (4.3). For the skew-whiffed $AdS_4$ solution the perturbed fields $\delta\chi$ and $\delta h$ have masses given by $m_\chi^2 = m_h^2 = -8$ , as before, and the gauge field is massless. For the Pope-Warner $AdS_4$ solution we have $m_\chi^2 = m_h^2 = 32$ and the

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⁶Observe that this latter theory of a single real scalar field can be obtained from the truncation considered in equation (3.3) of [7]. In particular, we obtain the same theory after setting $e^{2v} = 1 - h^2$ in (3.3) of [7], after flipping the sign of 3 in the last term in the potential which is needed for the skew-whiffed case with $\epsilon = -1$ .gauge field is massive with $m^2 = 32$ . For the Englert solution we find that the scalar perturbations mix and that $\delta h - (\sqrt{3}/2)\delta\chi$ and $\delta\chi + (\sqrt{3}/2)\delta h$ have mass squared $-5\sqrt{5}$ and $25\sqrt{5}/2$ , respectively. In particular, the unstable, BF violating mode about the Englert solution is contained within the truncation (4.3). The gauge field is massive in the Englert solution with $m^2 = 5\sqrt{5}$ .

In the skew-whiffed vacuum $h, \chi$ are dual to operators $\mathcal{O}_h, \mathcal{O}_\chi$ each with conformal dimension $\Delta_\pm = 1, 2$ . It is worth emphasising that in the solutions of the truncated theory (4.3) with $h, \chi \neq 0$ , in addition to activating these operators in the dual CFT we are also activating the operators with $\Delta = 4, 6$ that are dual to linear combinations of the fields $U, V$ , via (4.1). More precisely, in the solutions we shall consider that asymptotically approach the skew-whiffed $AdS_4$ vacuum, the asymptotic falloffs of $h, \chi$ will include cases where the skew-whiffed CFT is deformed by $\mathcal{O}_h$ and, for the RG flows in section 6 only, by $\mathcal{O}_\chi$ , and also where these operators acquire vevs. This means that the operators dual to $U, V$ are not deforming the skew-whiffed CFT but they are acquiring vevs which can easily be worked out for our solutions. However, we will not include the details.

Finally, we observe that if we set $h = 0$ in the truncated action (4.3) (which is only possible for configurations with $F \wedge F = 0$ ), and then linearize in $|\chi|$ , we make contact with the model with a simple mass term considered in [5] (after rescaling their gauge field by a factor of 2 and setting their $q = 2$ ). Also when $h = 0$ the action (4.3) is in the class considered in [31].

5 Ansatz for black hole and domain wall solutions

For the remainder of the paper we will consider the following ansatz for the $D = 4$ fields in the truncated theory described by the action (4.3). For the metric we take

$$d{s}^2=-g{e}^{-\beta }d{t}^2+g^{-1}d{r}^2+r^2(d{x}^2+d{y}^2)(5.1)ds^2\; =\; -ge^\{-\backslash beta\}dt^2\; +\; g^\{-1\}dr^2\; +\; r^2(dx^2\; +\; dy^2)\; \backslash quad\; (5.1)$$

where $g, \beta$ are functions of $r$ only. The gauge-field $A_1$ is taken to be purely electric

$$A_1=\varphi (r)dt,(5.2)A\_1\; =\; \backslash phi(r)dt,\; \backslash quad\; (5.2)$$

and we will also impose

$$\chi \equiv \xi (r)\in \mathbb{R},h=h(r)(5.3)\backslash chi\; \backslash equiv\; \backslash xi(r)\; \backslash in\; \backslash mathbb\{R\},\; \backslash quad\; h\; =\; h(r)\; \backslash quad\; (5.3)$$

We now substitute into the equations of motion arising from the action (4.3). After some calculation we are led to five ordinary differential equations, which we havepresented in (A.1), (A.2), (A.3), (A.5) and (A.6), for five real functions $\phi, \xi, h, g$ and $\beta$ . These equations can also be obtained from an action obtained by substituting the above ansatz directly in to the action (4.3):

$$S=c_0\int dr{r}^2{e}^{-\beta \mathrm{/}2}[-g^{\mathrm{\prime }\mathrm{\prime }}+g^{\mathrm{\prime }}(\frac{3}{2}{\beta }^{\mathrm{\prime }}-\frac{4}{r})+g({\beta }^{\mathrm{\prime }\mathrm{\prime }}-\frac{1}{2}({\beta }^{\mathrm{\prime }}{)}^2+2\frac{{\beta }^{\mathrm{\prime }}}{r}-\frac{2}{r^2})-\frac{3(g{\xi }^{\mathrm{\prime }2}-16g^{-1}{e}^{\beta }{\varphi }^2{\xi }^2)}{2(1-\frac{3}{4}{\xi }^2{)}^2}-\frac{3g(h^{\mathrm{\prime }}{)}^2}{2(1-h^2{)}^2}+\frac{2(1-h^2{)}^{3\mathrm{/}2}{e}^{\beta }{\varphi }^{\mathrm{\prime }2}}{1+3h^2}-\frac{24(-1+h^2+{\xi }^2)}{(1-\frac{3}{4}{\xi }^2{)}^2(1-h^2{)}^{3\mathrm{/}2}}](5.4)S\; =\; c\_0\; \backslash int\; dr\; r^2\; e^\{-\backslash beta/2\}\; \backslash left[\; -g\text{'}\text{'}\; +\; g\text{'}\; \backslash left(\; \backslash frac\{3\}\{2\}\; \backslash beta\text{'}\; -\; \backslash frac\{4\}\{r\}\; \backslash right)\; +\; g\; \backslash left(\; \backslash beta\text{'}\text{'}\; -\; \backslash frac\{1\}\{2\}\; (\backslash beta\text{'})^2\; +\; 2\; \backslash frac\{\backslash beta\text{'}\}\{r\}\; -\; \backslash frac\{2\}\{r^2\}\; \backslash right)\; -\; \backslash frac\{3(g\backslash xi\text{'}^2\; -\; 16g^\{-1\}e^\backslash beta\; \backslash phi^2\; \backslash xi^2)\}\{2(1\; -\; \backslash frac\{3\}\{4\}\backslash xi^2)^2\}\; -\; \backslash frac\{3g(h\text{'})^2\}\{2(1\; -\; h^2)^2\}\; +\; \backslash frac\{2(1\; -\; h^2)^\{3/2\}\; e^\backslash beta\; \backslash phi\text{'}^2\}\{1\; +\; 3h^2\}\; -\; \backslash frac\{24(-1\; +\; h^2\; +\; \backslash xi^2)\}\{(1\; -\; \backslash frac\{3\}\{4\}\backslash xi^2)^2\; (1\; -\; h^2)^\{3/2\}\}\; \backslash right]\; \backslash quad\; (5.4)$$

where

$$c_0\equiv \frac{1}{16\pi G}\int dtdxdy(5.5)c\_0\; \backslash equiv\; \backslash frac\{1\}\{16\backslash pi\; G\}\; \backslash int\; dt\; dx\; dy\; \backslash quad\; (5.5)$$

It will be helpful to note the following scaling symmetries:

$$r\to ar,(t,x,y)\to a^{-1}(t,x,y),g\to a^{2}g,\varphi \to a\varphi ,(5.6)r\; \backslash rightarrow\; ar,\; \backslash quad\; (t,\; x,\; y)\; \backslash rightarrow\; a^\{-1\}(t,\; x,\; y),\; \backslash quad\; g\; \backslash rightarrow\; a^2\; g,\; \backslash quad\; \backslash phi\; \backslash rightarrow\; a\backslash phi,\; \backslash quad\; (5.6)$$

and

$$e^{\beta }\to a^2{e}^{\beta },t\to at,\varphi \to a^{-1}\varphi (5.7)e^\backslash beta\; \backslash rightarrow\; a^2\; e^\backslash beta,\; \backslash quad\; t\; \backslash rightarrow\; at,\; \backslash quad\; \backslash phi\; \backslash rightarrow\; a^\{-1\}\backslash phi\; \backslash quad\; (5.7)$$

which leave the metric, $A_1$ , and all equations of motion invariant. Notice that this ansatz has the symmetry $h \rightarrow -h$ . We also have the $\mathbb{Z}_2$ symmetries $\xi \rightarrow -\xi$ and $\phi \rightarrow -\phi$ . Also notice that it is consistent to separately set $\xi = 0$ , $\phi = 0$ or $h = 0$ .

We will be almost exclusively interested in solutions that asymptote to a perturbation of the skew-whiffed $AdS_4$ vacuum. We recall that a scalar field dual to an operator in the CFT with scaling dimension $\Delta$ has the two asymptotic behaviours

$$r^{\mathrm{\Delta }-3}\text{and}{r}^{-\mathrm{\Delta }}(5.8)r^\{\backslash Delta-3\}\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; r^\{-\backslash Delta\}\; \backslash quad\; (5.8)$$

whereas a vector field behaves as

$$r^{\mathrm{\Delta }-2}\text{and}{r}^{1-\mathrm{\Delta }}(5.9)r^\{\backslash Delta-2\}\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; r^\{1-\backslash Delta\}\; \backslash quad\; (5.9)$$We thus focus on the asymptotic expansion ⁷

It is worthwhile noting that

$$\begin{array}{cccc}& e^{-\beta }g=e^{-{\beta }_a}[4r^2-\frac{8\pi G[\epsilon +\frac{4}{3}{\xi }_1{\xi }_2+\frac{4}{3}{h}_1{h}_2]}{r}+\dots ]& & \text{(5.11)}\end{array}e^\{-\backslash beta\}g\; =\; e^\{-\backslash beta\_a\}\; \backslash left[\; 4r^2\; -\; \backslash frac\{8\backslash pi\; G[\backslash varepsilon\; +\; \backslash frac\{4\}\{3\}\backslash xi\_1\backslash xi\_2\; +\; \backslash frac\{4\}\{3\}h\_1h\_2]\}\{r\}\; +\; \backslash dots\; \backslash right]\; \backslash tag\{5.11\}$$

5.1 Skew-whiffed to Pope-Warner uncharged and charged domain walls

As we will discuss in more detail later, we will be interested in both charged and uncharged domain wall solutions that asymptote in the IR to the Pope-Warner vacuum as $r \rightarrow 0$ . Hence we will consider the following expansion

where $\Delta_{\text{PW}}^\xi = \Delta_{\text{PW}}^h = (3 + \sqrt{33})/2$ and $\Delta_{\text{PW}}^\phi = 4$ . Here ${a_\xi, a_h, a_\beta, a_\phi}$ parametrises all possible marginal and irrelevant operators of the Pope-Warner vacuum within our truncation.

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⁷To compare with [8] we should identify $\varepsilon, \mu, q, \xi_i$ with $m, \hat{\mu}, \hat{q}, \sigma_i$ , respectively.## 5.2 Skew-whiffed to $AdS_2 \times \mathbb{R}^2$ solutions

The Reissner-Nordström electrically charged black hole solution (with planar horizon) is given by,

$$g=4r^2-\frac{1}{r}(4r_{+}^3+\frac{{\alpha }^2}{r_{+}})+\frac{{\alpha }^2}{r^2},\varphi =\alpha (\frac{1}{r_{+}}-\frac{1}{r})(5.13)g\; =\; 4r^2\; -\; \backslash frac\{1\}\{r\}(4r\_+^3\; +\; \backslash frac\{\backslash alpha^2\}\{r\_+\})\; +\; \backslash frac\{\backslash alpha^2\}\{r^2\},\; \backslash quad\; \backslash phi\; =\; \backslash alpha(\backslash frac\{1\}\{r\_+\}\; -\; \backslash frac\{1\}\{r\})\; \backslash quad\; (5.13)$$

with $h = \xi = \beta = 0$ , where $\alpha = \sqrt{4\pi G}q$ , $r_+ = q/\mu$ . At zero temperature, when $\alpha^2 = 12r_+^4$ and $g = 4(r - r_+)^2(r^2 + 2r_+r + 3r_+^2)/r^2$ , the near horizon geometry of this solution is given by $AdS_2 \times \mathbb{R}^2$ with the $AdS_2$ having radius squared $1/24$ . In other words, at zero temperature this charged solution interpolates between skew-whiffed $AdS_4$ in the UV and $AdS_2 \times \mathbb{R}^2$ in the IR.

Later we will consider deformations of this interpolating solution by marginal and irrelevant operators with respect to the $AdS_2$ factor in the IR which allow us to find zero temperature charged solutions with $h \neq 0$ and $\chi = 0$ which again interpolate from a deformed skew-whiffed $AdS_4$ vacuum in the UV to an $AdS_2 \times \mathbb{R}^2$ geometry in the IR. Specifically we find the $AdS_2 \times \mathbb{R}^2$ IR geometry is asymptotically given by

where $\beta_+$ parameterizes the only marginal deformation, and $h_+$ the only irrelevant deformation in our ansatz.

5.3 Black hole solutions

For the general black hole solutions we demand that there is a regular finite temperature horizon located at $r = r_+$ . Specifically we impose that

$$g(r_{+})=\varphi (r_{+})=0(5.15)g(r\_+)\; =\; \backslash phi(r\_+)\; =\; 0\; \backslash quad\; (5.15)$$

and so the solution is specified by five parameters at the horizon

$$r_{+},{\beta }_{+}=\beta (r_{+}),{\varphi }_{+}={\varphi }^{\mathrm{\prime }}(r_{+}),{\xi }_{+}=\xi (r_{+}),h_{+}=h(r_{+})(5.16)r\_+,\; \backslash quad\; \backslash beta\_+\; =\; \backslash beta(r\_+),\; \backslash quad\; \backslash phi\_+\; =\; \backslash phi\text{'}(r\_+),\; \backslash quad\; \backslash xi\_+\; =\; \backslash xi(r\_+),\; \backslash quad\; h\_+\; =\; h(r\_+)\; \backslash quad\; (5.16)$$(we will return to this counting later). We have the expansion as $r \rightarrow r_+$

where for example,

$$\begin{array}{cccc}& g_{+}^{(1)}=\frac{r_{+}[12(1+3h_{+}^2)(-1+h_{+}^2+{\xi }_{+}^2)+e^{{\beta }_{+}}{\varphi }_{+}^2(1-h_{+}^2{)}^3{(1-\frac{3}{4}{\xi }_{+}^2)}^2]}{\sqrt{1-h_{+}^2}{(1-\frac{3}{4}{\xi }_{+}^2)}^2(-1-2h_{+}^2+3h_{+}^4)}& & \text{(5.18)}\end{array}g\_+^\{(1)\}\; =\; \backslash frac\{r\_+\; \backslash left[\; 12(1\; +\; 3h\_+^2)(-1\; +\; h\_+^2\; +\; \backslash xi\_+^2)\; +\; e^\{\backslash beta\_+\}\; \backslash phi\_+^2\; (1\; -\; h\_+^2)^3\; \backslash left(1\; -\; \backslash frac\{3\}\{4\}\backslash xi\_+^2\backslash right)^2\; \backslash right]\}\{\backslash sqrt\{1\; -\; h\_+^2\}\; \backslash left(1\; -\; \backslash frac\{3\}\{4\}\backslash xi\_+^2\backslash right)^2\; (-1\; -\; 2h\_+^2\; +\; 3h\_+^4)\}\; \backslash tag\{5.18\}$$

and analogous expressions can be obtained for $g_+^{(2)}, \xi_+^{(1)}, h_+^{(1)}, \beta_+^{(1)}, \phi_+^{(2)}, \dots$ in terms of the data given in (5.16).

5.4 Counter terms and black hole thermodynamics

To calculate thermodynamic quantities for the black hole solutions we would like to calculate the on-shell Euclidean action $I$ . As usual this will require adding counter terms which are also relevant for the domain wall solutions. Interesting early work analysing the thermodynamics of $R$ -charged Reissner-Nordström black holes using AdS/CFT techniques was carried out in [32][33].

We analytically continue by writing

$$\begin{array}{cccc}& I=-iS,t=-i\tau & & \text{(5.19)}\end{array}I\; =\; -iS,\; \backslash quad\; t\; =\; -i\backslash tau\; \backslash tag\{5.19\}$$

The temperature of the black hole is $T = e^{\beta_a/2}/\Delta\tau$ where $\Delta\tau$ is fixed by demanding regularity of the Euclidean metric at $r = r_+$ . We find using (A.5) and (5.17) that

As explained in appendix B, we find that the on-shell action can be expressed as

$$\begin{array}{cccc}& I_{OS}=\frac{\mathrm{\Delta }\tau {\text{vol}}_2}{16\pi G}{\int }_{r_{+}}^{\mathrm{\infty }}dr[2rg{e}^{-\beta \mathrm{/}2}{]}^{\mathrm{\prime }}& & \text{(5.21)}\end{array}I\_\{OS\}\; =\; \backslash frac\{\backslash Delta\backslash tau\; \backslash text\{vol\}\_2\}\{16\backslash pi\; G\}\; \backslash int\_\{r\_+\}^\{\backslash infty\}\; dr\; [2r\; g\; e^\{-\backslash beta/2\}]\text{'}\; \backslash tag\{5.21\}$$where $\text{vol}2 \equiv \int dx dy$ . Since $g(r+) = 0$ , this expression only gets contributions from the on-shell functions at $r = \infty$ . An alternative expression for the on-shell action is given by

$$I_{OS}=\frac{\mathrm{\Delta }\tau {\text{vol}}_2}{16\pi G}{\int }_{r_{+}}^{\mathrm{\infty }}dr{[r^2{e}^{-\beta \mathrm{/}2}(g^{\mathrm{\prime }}-g{\beta }^{\mathrm{\prime }}-4e^{\beta }\frac{(1-h^2{)}^{3\mathrm{/}2}}{1+3h^2}\varphi {\varphi }^{\mathrm{\prime }})]}^{\mathrm{\prime }}(5.22)I\_\{OS\}\; =\; \backslash frac\{\backslash Delta\backslash tau\; \backslash text\{vol\}\_2\}\{16\backslash pi\; G\}\; \backslash int\_\{r\_+\}^\{\backslash infty\}\; dr\; \backslash left[\; r^2\; e^\{-\backslash beta/2\}\; (g\text{'}\; -\; g\backslash beta\text{'}\; -\; 4e^\{\backslash beta\}\; \backslash frac\{(1-h^2)^\{3/2\}\}\{1+3h^2\}\; \backslash phi\backslash phi\text{'})\; \backslash right]\text{'}\; \backslash quad\; (5.22)$$

which gets contributions from both $r = r_+$ and $r = \infty$ .

The on-shell action diverges and we need to regulate by adding appropriate counter terms. By examining the asymptotic expansion of the fields given in (5.10), we find that the following counter-term action renders the total action finite:

$$I_{ct}=\frac{1}{16\pi G}\int d\tau d^{2}x\sqrt{g_{\mathrm{\infty }}}[-2K+8+3{\xi }^2+3h^2](5.23)I\_\{ct\}\; =\; \backslash frac\{1\}\{16\backslash pi\; G\}\; \backslash int\; d\backslash tau\; d^2x\; \backslash sqrt\{g\_\{\backslash infty\}\}\; [-2K\; +\; 8\; +\; 3\backslash xi^2\; +\; 3h^2]\; \backslash quad\; (5.23)$$

where $\sqrt{g_{\infty}} = \lim_{r \rightarrow \infty} g^{1/2} r^2 e^{-\beta/2}$ and $K = \lim_{r \rightarrow \infty} g^{\mu\nu} \nabla_{\mu} n_{\nu}$ is the trace of the extrinsic curvature. For the class of solutions under consideration we find

$$I_{ct}=\frac{\mathrm{\Delta }\tau {\text{vol}}_2}{16\pi G}\underset{r\to \mathrm{\infty }}{\lim }{e}^{-\beta \mathrm{/}2}[-r^2{e}^{\beta }(g{e}^{-\beta }{)}^{\mathrm{\prime }}-4rg+r^2{g}^{1\mathrm{/}2}(8+3{\xi }^2+3h^2)](5.24)I\_\{ct\}\; =\; \backslash frac\{\backslash Delta\backslash tau\; \backslash text\{vol\}\_2\}\{16\backslash pi\; G\}\; \backslash lim\_\{r\; \backslash rightarrow\; \backslash infty\}\; e^\{-\backslash beta/2\}\; [-r^2\; e^\{\backslash beta\}\; (ge^\{-\backslash beta\})\text{'}\; -\; 4rg\; +\; r^2\; g^\{1/2\}\; (8\; +\; 3\backslash xi^2\; +\; 3h^2)]\; \backslash quad\; (5.24)$$

Defining

$$I_{Tot}=I+I_{ct}(5.25)I\_\{Tot\}\; =\; I\; +\; I\_\{ct\}\; \backslash quad\; (5.25)$$

we find that corresponding to the two expressions (5.21), (5.22), the on-shell total action can be written as

respectively, where we have defined the entropy density $s$ to be

$$s=\frac{(r_{+}^2)}{4G}(5.27)s\; =\; \backslash frac\{(r\_+^2)\}\{4G\}\; \backslash quad\; (5.27)$$

(the total entropy is $s\text{vol}_2$ ). The equality of these two expressions imply the Smarr-type relation

$$\frac{3}{2}\epsilon =\mu q+Ts-2{\xi }_1{\xi }_2-2h_1{h}_2(5.28)\backslash frac\{3\}\{2\}\backslash varepsilon\; =\; \backslash mu\; q\; +\; Ts\; -\; 2\backslash xi\_1\backslash xi\_2\; -\; 2h\_1h\_2\; \backslash quad\; (5.28)$$

A variation of the action $I$ yields the equations of motion together with surfaceterms. For an on-shell variation the only terms remaining are the surface terms

In the Euclidean black hole solution the only boundary is the conformal boundary $r \rightarrow \infty$ and hence this integral only gets contributions there. In addition one must also add the variation of the counter terms,

Combining these expressions we deduce that

$$[\delta I_{Tot}{]}_{OS}=\mathrm{\Delta }\tau vo{l}_2{e}^{-{\beta }_a\mathrm{/}2}[(-\frac{1}{2}\epsilon +\frac{1}{2}\mu q)\delta {\beta }_a-q\delta \mu -4{\xi }_2\delta {\xi }_1-4h_2\delta h_1](5.31)[\backslash delta\; I\_\{Tot\}]\_\{OS\}\; =\; \backslash Delta\backslash tau\; vol\_2\; e^\{-\backslash beta\_a/2\}\; \backslash left[\; \backslash left(-\backslash frac\{1\}\{2\}\backslash varepsilon\; +\; \backslash frac\{1\}\{2\}\backslash mu\; q\backslash right)\; \backslash delta\backslash beta\_a\; -\; q\backslash delta\backslash mu\; -\; 4\backslash xi\_2\; \backslash delta\backslash xi\_1\; -\; 4h\_2\; \backslash delta\; h\_1\; \backslash right]\; \backslash quad\; (5.31)$$

(which corrects equation (17) of [8] by a factor of $16\pi$ ). Note that we are keeping $\Delta\tau$ fixed in this variation, and hence $\delta\beta_a = 2\delta T/T$ . Hence we see that $I_{Tot}$ is stationary for fixed temperature and chemical potential (i.e. $\delta\beta_a = \delta\mu = 0$ ) and for either $\xi_2 = 0$ or fixed $\xi_1$ and similarly either $h_2 = 0$ or fixed $h_1$ . In our applications we will always fix $\xi_1$ and $h_1$ .

We now define the thermodynamic potential for a grand canonical ensemble via $W \equiv T[I_{Tot}]_{OS} \equiv w vol_2$ . From the above variations we see that $w = w(T, \mu, \xi_1, h_1)$ and using the second expression in (5.26) we deduce the first law

$$\delta w=-s\delta T-q\delta \mu -4{\xi }_2\delta {\xi }_1-4h_2\delta h_1(5.32)\backslash delta\; w\; =\; -s\backslash delta\; T\; -\; q\backslash delta\backslash mu\; -\; 4\backslash xi\_2\; \backslash delta\backslash xi\_1\; -\; 4h\_2\; \backslash delta\; h\_1\; \backslash quad\; (5.32)$$

Note that from the second expression in (5.26) we can write $w = \varepsilon - Ts - \mu q$ . We therefore have $\varepsilon = \varepsilon(s, q, \xi_1, h_1)$ with

$$\delta \epsilon =T\delta s+\mu \delta q-4{\xi }_2\delta {\xi }_1-4h_2\delta h_1(5.33)\backslash delta\backslash varepsilon\; =\; T\backslash delta\; s\; +\; \backslash mu\backslash delta\; q\; -\; 4\backslash xi\_2\; \backslash delta\backslash xi\_1\; -\; 4h\_2\; \backslash delta\; h\_1\; \backslash quad\; (5.33)$$

and we can identify $\varepsilon$ as the energy density of the thermal system.As a consistency check we can also calculate the energy by calculating the holographic energy-momentum tensor. From [34] we have

$$(8\pi G)T_{ij}=K_{ij}+{\gamma }_{ij}(-K+4+\frac{3}{2}{\xi }^2+\frac{3}{2}{h}^2)(5.34)(8\backslash pi\; G)T\_\{ij\}\; =\; K\_\{ij\}\; +\; \backslash gamma\_\{ij\}(-K\; +\; 4\; +\; \backslash frac\{3\}\{2\}\backslash xi^2\; +\; \backslash frac\{3\}\{2\}h^2)\; \backslash quad\; (5.34)$$

where $\gamma_{ij}$ is the spatial metric at a fixed radius $r$ , $K_{ij}$ is the extrinsic curvature tensor and we note that the $\xi$ and $h$ terms have arisen from the corresponding terms in $I_{ct}$ given in (5.24). We find that the $T_{ij}$ is diagonal with

Using equation (45) of [34] to calculate the total energy we obtain

$$E=e^{{\beta }_a\mathrm{/}2}\epsilon vo{l}_2(5.36)E\; =\; e^\{\backslash beta\_a/2\}\; \backslash varepsilon\; vol\_2\; \backslash quad\; (5.36)$$

which agrees upon setting $\beta_a = 0$ which we will do. It is also worth noting that from the spatial part of the stress tensor we deduce that the pressure is $p = \frac{1}{2}\varepsilon + \frac{3}{2}\xi_1\xi_2 + \frac{3}{2}h_1h_2$ and we thus see that the Smarr-type formula (5.28) can be written in the familiar form $\varepsilon + p = \mu q + Ts$ .

6 Uncharged domain wall solutions: holographic RG flows

Recall that the scalar potential for the truncated action (4.3) with $\chi = \xi \in \mathbb{R}$ is given by

$$V=24\frac{-1+h^2+{\xi }^2}{(1-h^2{)}^{3\mathrm{/}2}(1-\frac{3}{4}{\xi }^2{)}^2}(6.1)V\; =\; 24\; \backslash frac\{-1\; +\; h^2\; +\; \backslash xi^2\}\{(1\; -\; h^2)^\{3/2\}\; (1\; -\; \backslash frac\{3\}\{4\}\backslash xi^2)^2\}\; \backslash quad\; (6.1)$$

and has extrema for $(h, \xi) = (0, 0), (0, \pm\sqrt{2/3}), (\pm 1/\sqrt{5}, \pm 2/\sqrt{15})$ . These correspond to the skew-whiffed, Pope-Warner and Englert $AdS_4$ vacua, respectively, that were discussed in section 3. As before, we will restrict our considerations to vacua with positive values of the fields. Before discussing new solutions related to holographic superconductivity in subsequent sections, in this section we pause to numerically construct interpolating uncharged domain-wall solutions, with vanishing gauge field, $\phi = 0$ , which have the interpretation as ordinary holographic RG flows.

For these solutions, as $r \rightarrow \infty$ we require the asymptotic expansion given in (5.10) (with $\mu = q = 0$ ), corresponding to a perturbed skew-whiffed vacuum in the UV. ForFigure 2: The plot shows the scalar potential of our model in the $(h, \xi)$ plane. The extrema indicated by dots correspond to the skew-whiffed $AdS_4$ vacuum (SW), the Pope-Warner vacuum (PW) and the Englert vacuum (E). The interpolating trajectories are domain wall solutions that describe holographic flows interpolating between a deformed skew-whiffed vacuum in the UV and a Pope-Warner vacuum in the IR.

uncharged domain wall solutions that flow to the Pope-Warner vacuum in the IR, we demand that as $r \rightarrow 0$ we have the expansion (5.12) (with $\phi_{IR} = 0$ ). Note that in the UV $\xi, h$ are both dual to relevant operators and non-zero values of $\xi_1, h_1$ correspond to deforming the skew-whiffed CFT by the corresponding operators, $\mathcal{O}_h$ and $\mathcal{O}_\xi$ , while $\xi_2, h_2$ correspond to giving vevs for these operators in the deformed CFT. By contrast, in the IR $\xi, h$ are both dual to irrelevant operators in the Pope-Warner CFT.

Do such interpolating domain wall solutions exist? A simple counting suggests the following picture. We have four fields, $g, \beta, \xi, h$ , two of which satisfy⁸ first-order equations and the rest second order equations (given in appendix A). Thus we must specify six constants to obtain a unique solution. We can use the scaling symmetries of the theory given in (3.1), (3.2) to set $\beta_a = 0$ as well as $(16\pi G/3)^{1/2}\xi_1 = 1$ . We therefore have seven parameters, $\varepsilon, \xi_2, h_1, h_2, a_\beta, a_\xi, a_h$ left to specify and thus we expect to ob-

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⁸Note that the domain walls we are interested in do not satisfy first order RG flow equations. For example when $h = 0$ , after redefining $\xi = (2/\sqrt{3}) \tanh(s/2)$ , the $D = 4$ Lagrangian can be written in the form

$$16\pi G\mathcal{L}=\sqrt{-g}[R-\frac{1}{2}(\mathrm{\nabla }s{)}^2-V](6.2)16\backslash pi\; G\backslash mathcal\{L\}\; =\; \backslash sqrt\{-g\}[R\; -\; \backslash frac\{1\}\{2\}(\backslash nabla\; s)^2\; -\; V]\; \backslash quad\; (6.2)$$

with the potential $V$ given in terms of a superpotential $W$ via $V = 8(4(W')^2 - 3W^2)$ and $W = 1/2(1 + \cosh(s))$ . This is of the form considered in e.g. [35] and we note that while for the skew-whiffed vacuum $W'(0) = 0$ , by contrast for the Pope-Warner vacuum $W'(1) \neq 0$ .tain a one-parameter family of solutions. Let us take this parameter to be $h_1$ . We will take $h_1 \geq 0$ and solutions with $h_1$ negative can be obtained using the $h \rightarrow -h$ symmetry of the equations of motion.

Using a shooting technique (see Appendix C), we have constructed this one-parameter family of solutions numerically, finding solutions with $h_1$ in the range $(16\pi G/3)^{1/2}h_1 \in [0, \sim .86)$ . Figure 2 shows a contour plot of the scalar potential with domain-wall trajectories superposed. The solution with $h_1 = 0$ has $h = 0$ identically. As $h_1$ approaches the maximum value $(16\pi G/3)^{1/2}h_1 \sim 0.86$ the solution gets closer and closer to the unstable Englert vacuum. In Figure 3 we show the values of $h_2$ and $\xi_2$ , which are fixing the vevs $\langle \mathcal{O}_h \rangle$ and $\langle \mathcal{O}_\xi \rangle$ , as a function of $h_1$ .

Figure 3: Plot showing the behaviour of $h_2$ and $\xi_2$ as a function of $h_1$ in the uncharged domain wall solutions.

Following similar considerations we have also constructed two further domain wall solutions, one that interpolates between the skew-whiffed vacuum in the UV and the Englert vacuum in the IR and another between the Englert vacuum in the UV and the Pope-Warner vacuum in the IR. Since the Englert solution is unstable, the physical significance of such solutions, if any, is not clear.

For the one parameter family of domain wall solutions flowing between the skew-whiffed vacuum to the Pope Warner vacuum to describe sensible RG flows we require that the corresponding $AdS_4$ solutions of $D = 11$ supergravity are stable, at least perturbatively. While perturbative stability has been demonstrated for the skew-whiffed solutions it has not yet been shown for the Pope-Warner solutions. Assuming that they are in fact stable, one might then be concerned that the instability of the Englert vacuum implies a concomitant pathology of the RG flows, especially for values of $h_1$near the maximum value $(16\pi G/3)^{1/2}h_1 \sim 0.86$ for which the domain wall solutions are getting close to the Englert solution. We think that this is unlikely to be a problem. While the solutions do have a region that is approximated by the Englert solution, the unstable mode of the Englert solution will not be localised in that region. Furthermore, if there was a critical value of $h_1$ for the solutions in which they become unstable, one would expect a marginal static mode to appear which we are able to explicitly test for numerically, as described in detail in appendix C, and do not find.

7 Interpolating solutions with $T = 0$ , $\mu \neq 0$

In this section we will study two classes of regular interpolating solutions with non-zero gauge field, $\phi \neq 0$ , that arise as the zero temperature limit of black hole solutions which will be constructed in section 9. The first class is a one parameter family of charged domain walls that interpolate between deformed skew-whiffed $AdS_4$ in the UV and the Pope-Warner $AdS_4$ solution in the IR. These solutions have scalar $\xi$ hair and, as we will show in section 9, are the zero temperature limit of superconducting black holes. In particular, the $AdS_4$ region in the IR corresponds to an emergent $d = 3$ conformal symmetry in the IR. For the special case when $h = 0$ these solutions were found in [8][9]. The second class of solutions is a one parameter family of charged solutions that interpolate between deformed skew-whiffed $AdS_4$ in the UV and $AdS_2 \times \mathbb{R}^2$ in the IR. These solutions have no scalar hair and, as we show later, will give the zero temperature limit of some of the normal phase black holes. For the special case when $h = 0$ these solutions are simply the zero temperature limit of the Reissner-Nordström black hole solution given in (5.13). As we discuss in section 9, our numerical results indicate that the class of interpolating solutions with $AdS_2$ factors in the IR are never thermodynamically favoured while those with $AdS_4$ factors are.

7.1 Pope-Warner IR: zero temperature superconductors

As $r \rightarrow \infty$ we again impose the asymptotic expansion given in (5.10), corresponding to a perturbed skew-whiffed vacuum in the UV. Similarly as $r \rightarrow 0$ we impose the expansion (5.12) in order that we approach the Pope-Warner vacuum in the IR (observe that $\phi$ is dual to an irrelevant operator in the CFT dual to the Pope-Warner vacuum). We now have five fields, $g, \beta, \xi, h, \phi$ , two of which satisfy first-order equations and the rest second order equations (given in appendix A). Thus we must specify eight constantsto obtain a unique solution. We next use the scaling symmetries of the theory given in (3.1), (3.2) to set $(16\pi G)^{1/2}\mu = 1$ as well as $\beta_a = 0$ . This leaves the ten parameters $\varepsilon, q, \xi_1, \xi_2, h_1, h_2, a_\beta, a_\xi, a_h, a_\phi$ and so we expect a two parameter family of solutions. We will fix one of these parameters by choosing to set $\xi_1 = 0$ (as we discuss further below). This then leaves us with a single parameter which we choose to be $h_1$ . Once again we take $h_1 \geq 0$ and we can recover negative $h_1$ by using the symmetry $h \rightarrow -h$ . Using a shooting techniques described in appendix C, we do indeed find a one parameter family of such charged domain wall solutions for $h_1 \leq h_1^c$ with $(16\pi G/3)^{1/2}h_1^c \sim 0.35$ (for $(16\pi G)^{1/2}\mu = 1$ ). In Figure 4 we have plotted the trajectories of the scalar fields and Figure 5 displays the dependence of $h_2, \xi_2$ and $q$ on $h_1$ . Notice that as $h \rightarrow h_1^c$ , the charge carried by the black hole is going to zero since $q \rightarrow 0$ .

Figure 4: Plot in the $(h, \xi)$ plane showing the interpolating trajectories of the charged domain wall solutions interpolating between the skew-whiffed vacuum in the UV and the Pope-Warner vacuum in the IR. Along the trajectories $\phi \neq 0$ .

The special solution with $h_1 = 0$ , which has $h = 0$ identically, has been shown to arise as the zero temperature limit of holographic superconducting black holes with non-zero chemical potential in [8]. We will see in the next section that all of the new charged domain walls arise in a similar way. As in [8] we have imposed $\xi_1 = 0$ because it corresponds to allowing the operator $\mathcal{O}_\chi$ , dual to $\xi$ , to obtain a vev, determined by $\xi_2$ , without being sourced i.e. without adding the operator to the CFT dual to the skew-whiffed vacuum. Equivalently, the abelian symmetry in the dual CFT is thenFigure 5: Plots showing $h_2$ , $\xi_2$ and $q$ for the one-parameter family of charged domain wall solutions labelled by $h_1$ with $(16\pi G)^{1/2}\mu = 1$ .

broken spontaneously and not explicitly⁹. In our new charged domain walls with $h \neq 0$ we necessarily have $h_1 \neq 0$ , $h_2 \neq 0$ corresponding to the dual $d = 3$ CFT having been perturbed by the operator $\mathcal{O}_h$ , dual to $h$ , as well as this operator acquiring a vev. As we will discuss later the coupling of $h$ to $F \wedge F$ in (4.3) implies that when $h \neq 0$ the dual theory breaks parity and time reversal invariance.

In the far IR, all of the new charged domain wall solutions approach the Pope-Warner CFT. Thus, in this limit, all of the $h$ -deformed skew-whiffed CFTs with $T = 0$ and $\mu \neq 0$ are described by the same universal CFT.

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⁹Note that we expect entirely analogous results if we set $\xi_2 = 0$ and $\xi_1 \neq 0$ .## 7.2 $AdS_2 \times \mathbb{R}^2$ IR: zero temperature normal phase

To construct the solutions interpolating between deformed skew-whiffed $AdS_4$ in the UV and $AdS_2 \times \mathbb{R}^2$ in the IR, we proceed in a similar fashion. However, since the scalar $\xi$ has no irrelevant behaviour about $AdS_2 \times \mathbb{R}^2$ (see (5.14)) these solutions have $\xi = 0$ identically (i.e. no scalar hair). The counting of parameters is exactly the same as that for the Pope-Warner domain walls after excluding $\xi$ , and so we again expect a one parameter family of solutions, which we label by $h_1 \geq 0$ .

We do indeed find a one parameter family of interpolating solutions which for $h_1 = 0$ (in fact $h = 0$ identically) includes the $T = 0$ Reissner-Nordström solution (5.13). As we will see later all of these solutions arise as the zero temperature limit of charged black holes with $\xi = 0$ corresponding to the normal phase of the system. In Figure 6 we have plotted the dependence of $h_2$ and $q$ on $h_1$ . Note that as for the Pope-Warner charged domain walls, as $h \rightarrow h_1^c$ it appears that the charge carried by the black hole is vanishing, $q \rightarrow 0$ .

Figure 6: Plots showing $h_2$ and $q$ for the one-parameter family of charged domain wall solutions labelled by $h_1$ with $(16\pi G)^{1/2}\mu = 1$ .

Interestingly we find that these solutions exist for a very similar, if not identical, range of $h_1$ as for the Pope-Warner charged domain wall solutions of the last section, i.e. $h_1 \leq h_1^c$ with $(16\pi G/3)^{1/2}h_1^c \sim 0.35$ (for $(16\pi G)^{1/2}\mu = 1$ ). We will see later that the broken phase superconducting phase solutions are thermodynamically preferred when $h_1 < h_1^c$ . Thus, if the ranges are indeed identical (as we expect), for $h_1 < h_1^c$ the system at zero temperature is described by the interpolating solutions with $AdS_4$ factors in the IR and not those with $AdS_2$ factors. More detailed investigations near $h_1 = h_1^c$ wouldcertainly be worthwhile.

8 Uncharged black hole solutions

In this section we construct uncharged black hole solutions that are asymptotic to the perturbed skew-whiffed $AdS_4$ solution. These describe the skew-whiffed CFT deformed by the relevant operator $\mathcal{O}_h$ at finite temperature $T$ and $\mu = 0$ . These solutions have unbroken gauge symmetry and we will see later how they interface with the superconducting solutions.

The asymptotic behaviour of the uncharged black holes is given in (5.10) and the behaviour at the black hole horizon is given by (5.17). The black hole solutions that we construct have $\xi = \phi = 0$ identically, but can have $h_1 \neq 0$ , which corresponds to a deformation of the skew-whiffed CFT by the operator dual to $h$ , $\mathcal{O}_h$ , as well as having $h_2 \neq 0$ corresponding to giving $\mathcal{O}_h$ a vev. For solutions with $h_1 = 0$ (which have $h = 0$ identically) we have the usual neutral AdS-Schwarzschild black hole,

$$g=4r^2-\frac{8\pi G\epsilon }{r}(8.1)g\; =\; 4r^2\; -\; \backslash frac\{8\backslash pi\; G\backslash varepsilon\}\{r\}\; \backslash quad\; (8.1)$$

As the temperature is taken to zero, one recovers the skew-whiffed $AdS_4$ vacuum in the usual manner. For $h_1 \neq 0$ we have found new solutions for all temperatures. To solve the differential equations we use the scaling symmetries (3.1), (3.2) to set $\beta_a = 0$ and $(16\pi G/3)^{1/2}h_1 = 1$ . Since $\phi = \xi = 0$ we set $\mu = q = \xi_1 = \xi_2 = 0$ . This leaves us with five parameters $\varepsilon, h_2, r_+, \beta(r_+), h(r_+)$ and since a solution to the differential equations for $g, \beta, h$ is specified by four parameters we expect a one parameter family of black hole solutions. We take this parameter to be the temperature of the black hole.

In Figure 7 we have plotted the dependence of $h_2$ , the thermodynamic potential $w = -\varepsilon/2 - 2h_1h_2$ (since $\xi_1 = 0$ ) and also $r_+^2 = 4Gs$ , where $s$ is the entropy density, against temperature. Observe that as the temperature goes to zero the entropy goes to zero. In Figure 8 we have plotted the value of $h$ at the horizon, $h(r_+)$ and we see that as the temperature goes to zero it is approaching the singular value of 1. We have also plotted the Ricci scalar at the horizon as a function of temperature in Figure 8 which confirms that, unlike the solutions with $h_1 = 0$ , as the temperature is decreased to zero, the solutions become singular.

We have verified that as the temperature goes to zero, the solution appears to approach Poincaré invariant behaviour with $ge^{-\beta} = r^2$ . First observe that it is consistent with the equations of motion to set $ge^{-\beta} = r^2$ when $\phi = \xi = 0$ and that the equations