# On *hvLif* -like solutions in gauged Supergravity

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## Abstract

We perform a thorough investigation of Lifshitz-like metrics with hyperscaling violation (*hvLif*) in four-dimensional theories of gravity coupled to an arbitrary number of scalars and vector fields, obtaining new solutions, electric, magnetic, and dyonic, that include the known ones as particular cases. After establishing some general results on the properties of purely *hvLif* solutions, we apply the previous formalism to the case of \({\mathcal {N}}=2,~d=4\) supergravity in the presence of Fayet–Iliopoulos terms, obtaining particular solutions to the \(t^3\)-model, and explicitly embedding some of them in Type-IIB string theory.

## 1 Introduction

*aDS*). On the other hand, in many physical systems critical points are dictated by dynamical scalings in which, even though the system exhibits a scaling symmetry, space and time scale differently under this symmetry. A prototype example of such critical points is a hyperscaling-violating Lifshitz fixed point where the system is spatially isotropic and scale covariant, though there is an anisotropic scaling in the time direction characterized by a dynamical exponent, \(z\), and hyperscaling violation characterized by the exponent \(\theta .\) More precisely the system is covariant under the following scale symmetry [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]:

*aDS*background. For other values of \(z\) and \(\theta ,\) the \(d+2\)-dimensional gravitational backgrounds are supported by metrics of the form

*hvLif*) metrics. The Lifshitz-type spacetimes are known to be singular in the IR. They suffer from a null singularity with diverging tidal forces [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. Just like Lifshitz spacetimes,

*hvLif*metrics are zero-temperature gravity solutions thought to represent a class of quantum critical points characterized by the two parameters \(z\) and \(\theta \) [4, 6]. For holographic related applications, see [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54].

It is interesting to obtain new gravitational solutions that may be used as duals of the corresponding field theories, if any. A first step on this direction, for \({\mathcal {N}}=2,~ d=4\) ungauged supergravity was taken in [55], where a complete analysis on the existence of such kind of solutions was performed. In this note we extend the systematic study to a general class of gravity theories coupled to scalars and vectors, up to two derivatives, in the presence of a scalar potential, in principle arbitrary, focusing later on \({\mathcal {N}}=2,~d=4\) supergravity in the presence of Fayet–Iliopoulos terms.

The structure of the paper goes as follows: in Sect. 2 we dimensionally reduce the general action of gravity coupled to an arbitrary number of scalars and vectors in the presence of a scalar potential assuming a general static background which naturally fits the anisotropic scaling properties which correspond to *hvLif*-like solutions. In Sect. 3 we adapt the general formalism to the Einstein–Maxwell–Dilaton system. In Sect. 4 we focus on \({\mathcal {N}}=2,~d=4\) supergravity in the presence of Fayet–Iliopoulos terms (which correspond to include a scalar potential in ungauged supergravity), were we exploit the symplectic structure of the theory in order to obtain further results. We also embed a particular truncation of the \(t^3\)-model in Type-IIB string theory compactified on a Sasaki–Einstein manifold times \(S^1\). In Sect. 5 we perform an analysis of the properties of purely *hvLif* solutions for the general class of theories considered. In addition, we provide a general recip to obtain *hvLif*-like solutions of a particular class of Einstein–Maxwell–Dilaton systems, reducing the problem to the resolution of an algebraic equation. We apply the procedure to obtain explicit solutions, some of them embedded in string theory. In Sect. 6 we conclude.

## 2 The general theory

*hvLif*

^{1}) of the four-dimensional action

*hvLif*solutions are in particular static, a first step is to constrain the form of the metric to be

*hvLif*solution, is given by

^{2}, by performing a dimensional reduction over the radial coordinate.

^{3}and their coupling to the scalars gets thus encoded in the so-called

*black-hole potential*\(V_\mathrm{bh}\), which is defined as in the asymptotically flat case by

^{4}[56, 58, 63]

^{5}

*Hamiltonian constraint*, associated to the lack of explicit \(r\)-dependence of the Lagrangian:

### 2.1 Constant scalars: generalities

*generalized*squared charge, magnetic and electric. In the case of constant scalars, Eq. (2.13) is not identically satisfied, but it becomes the following constraint:

**Constant scalars as**

**double****critical points:**\(\partial _{i}V_\mathrm{bh}=0, \partial _{i}V = 0\). Of course, the system of equations given by

*i.e.*, the values of the scalars are fixed in terms of the electric and magnetic charges, and we have included a dependence on \(\phi _{\infty }\) to formally consider the existence of flat directions. We will see later on that, in fact, Eq. (2.16) occurs in \({\mathcal {N}}=2,~d=4\) supergravity. The equations of motion reduce to

**Metric functions identified**: \(e^{U}=\beta e^{W},~\beta \in {\mathbb {R}}^{+}\)

**and**\(2\beta ^4\partial _{i}V_\mathrm{bh}=\partial _{i}V(\phi )\). In this case, the equations of motion imply

## 3 The Einstein–Maxwell–Dilaton model

*i.e.*, we consider a single vector field and a single scalar field. Moreover, the coupling given by \(R\) is taken to be zero, which greatly simplifies the black-hole potential \(V_\mathrm{bh}(\phi ,{\mathcal {Q}})\), which is therefore given by

**Another coordinate system:** \(A-B-f\) **coordinates.**

*hvLif*asymptotics when \(f(\tilde{r})\) is a function of \(\tilde{r}\) that obeys

*hvLif*limit is, thus, assumed to be at \(\tilde{r}_0\), whereas the horizon is at \(\tilde{r}_h\). The equations of motion (3.4), (3.5) and (3.6) can be rewritten accordingly as

^{6}

^{7}

## 4 \({\mathcal {N}}=2\) Supergravity with FI terms

^{8}

^{9}:

*central charge*of the theory.

**Constant scalars and supersymmetric attractors.**In Sect. (2) we studied the case of constant scalars in the general theory (2.1). We found that, besides the solution \(aDS_2\times {\mathbb {R}}^2\), there was another possible solution, if Eq. (2.16) holds. We will see now how this is always possible in \({\mathcal {N}}=2 ~\mathrm FI\). The general theory of the attractor mechanism in ungauged \(d=4\) supergravity proves that, for extremal black holes, the value of the scalars at the horizon is fixed in terms of the charges \({\mathcal {Q}}^M\), and given by the so-called

*critical points*or

*attractors*,

*i.e.*, solutions to the system

### 4.1 The \(t^3\)-model

**Embedding the**\(t^3\)

**-model system in the EMD.**As it can be trivally verified, we have just obtained the action (3.2) with

## 5 *hvLif* solutions

In this section we are going to construct (purely and asymptotically) *hvLif* solutions to Eq. (2.1). After establishing some results on the properties of the solutions corresponding to the pure *hvLif* case in the general set-up of Eq. (2.14), we focus on the EMD system, obtaining the *hvLif* solutions allowed by the embedding of our *axion-free* supergravity model in this system. Then, we provide a recipe to construct asymptotically *hvLif* solutions to these theories in the presence of constant and non-constant dilaton fields, recovering and extending some of the results already present in the literature.

### 5.1 Purely *hvLif* solutions: general remarks

*hvLif*metric in four dimensions, given by

*hvLif*solutions, the equations of motion can be further simplified by direct substitution of (5.2)

*hvLif*case. We find

*hvLif*solution to any theory that belongs to the class defined by Eq. (2.1). Likewise, Eq. (5.7) provides the behavior of the black-hole potential and the scalar potential, in terms of the variable \(r\), for any

*hvLif*solution consistent with the equations of motion. \({\mathcal {G}}_{ij}\) is positive-definite, therefore

*hvLif*solution of any theory describable by Eq. (2.1) are constant

*iff*\(\theta =2\), or \(z=1+\theta /2\). In addition, \(V_\mathrm{bh}\) is, in our conventions, a negative definite function, hence \(V_\mathrm{bh}\le 0\Leftrightarrow {\mathcal {Y}}_{(\theta ,z)}\le 0\). These two conditions on the sign of \({\mathcal {W}}_{(\theta ,z)}\) and \( {\mathcal {Y}}_{(\theta ,z)}\) are equivalent to imposing the null-energy condition (NEC) to our purely

*hvLif*solutions, as we commented before, and they define a region of acceptable solutions in the \((\theta ,z)\)-plane, as we shall see.

*hvLif*solutions of any theory describable by Eq. (2.1) attending to the vanishing of \(V\), \(V_\mathrm{bh}\), and/or \({\mathcal {G}}_{ij} \dot{\phi }^i\dot{\phi }^j\). Let us proceed.

- 1.\( {\theta =2}\) In this situation \({\mathcal {G}}_{ij} \dot{\phi }^i\dot{\phi }^j=0\), and$$\begin{aligned}&V= -\ell ^{-2} z(z-1) r^{-2},\end{aligned}$$(5.12)The NEC imposes \({z\in (-\infty ,0]\cup [1,+\infty )}\), and we have the two special cases: \({\theta =2,~z=0}\) (which corresponds to Rindler spacetime) and \({\theta =2,~z=1}\) (which is Minkowski spacetime) for which \(V=V_\mathrm{bh}=0\) as well.$$\begin{aligned}&V_\mathrm{bh}= -\frac{1}{2}\ell ^2 z(z-1)r^{-2}. \end{aligned}$$(5.13)
- 2.\({z=1+\frac{\theta }{2}}\) We have again \({\mathcal {G}}_{ij} \dot{\phi }^i\dot{\phi }^j=0\), and$$\begin{aligned}&V= -\ell ^{-2} \left( \frac{\theta }{2}-3\right) \left( \frac{\theta }{2}-2\right) r^{-\theta },\end{aligned}$$(5.14)The NEC translates into \({\theta \in [0,6]}\), and we have three more special cases: the Ricci flat one: \({\theta =6,~z=4}\) corresponding to \(V=V_\mathrm{bh}=0\) (this is a particular case of the general formalism developed in [55] for ungauged \({\mathcal {N}}=2,~d=4\) supergravity); \({\theta =4,~z=3}\), which corresponds to \(V=0\), \(V_\mathrm{bh}=-\ell ^2\) (also in agreement with the results of [55]); and \({\theta =0,~z=1}\), which is nothing but the \(aDS_{4}\) spacetime in a conformally flat representation, and the only solution with vanishing black-hole potential, and constant (non-zero) scalar potential compatible with the equations: \(V_\mathrm{bh}=0\), \(V\equiv \varLambda =-\ell ^{-2} 6\).$$\begin{aligned}&V_\mathrm{bh}= -\frac{1}{2}\ell ^2 \left( \frac{\theta }{2}-3\right) \frac{\theta }{2}r^{\theta -4}. \end{aligned}$$(5.15)
- 3.\( {z=1,~\theta \ne 2,~z\ne 1+\frac{\theta }{2}}\) We have \(V_\mathrm{bh}=0\), whereas$$\begin{aligned}&V= -\ell ^{-2} \left( \theta -3\right) \left( \theta -2\right) r^{-\theta },\end{aligned}$$(5.16)The NEC becomes now \({\theta \in (-\infty ,0]\cup [2,\infty )}\), and we have the limit case \({\theta =3,~z=1}\), which will be a particular case of the family considered in the next paragraph.$$\begin{aligned}&{\mathcal {G}}_{ij}\dot{\phi }^i\dot{\phi }^j=\frac{1}{2}(\theta -2)\theta ~ r^{-2}. \end{aligned}$$(5.17)
- 4.\( {z=\theta -2,~\theta \ne 2,~z\ne 1+\frac{\theta }{2}}\) This situation imposes \(V=V_\mathrm{bh}=0\), whereasThe NEC reads \({\theta \in [2,6]}\). These will be solutions of the Einstein–Dilaton system for \({\mathcal {G}}_{ij}=\frac{1}{2}\delta _{ij},~i=1\), and$$\begin{aligned} {\mathcal {G}}_{ij}\dot{\phi }^i\dot{\phi }^j&= \frac{1}{2}(\theta -2)(6-\theta ) ~ r^{-2}. \end{aligned}$$(5.18)$$\begin{aligned} \phi =\phi _0+\sqrt{(\theta -2)(6-\theta )}\log r. \end{aligned}$$(5.19)
- 5.\({z=\theta -1,~\theta \ne 2,~z\ne 1+\frac{\theta }{2}}\) We have now \(V=0\), while$$\begin{aligned}&V_\mathrm{bh}= -\frac{1}{2}\ell ^2 \left( \theta -2\right) \frac{\theta }{2}r^{\theta -4}.\end{aligned}$$(5.20)and the NEC becomes \({\theta \in [2,4]}\).$$\begin{aligned}&{\mathcal {G}}_{ij}\dot{\phi }^i\dot{\phi }^j=\frac{1}{2}(\theta -2)(4-\theta ) ~ r^{-2}, \end{aligned}$$(5.21)

*hvLif*solution (for non-vanishing vector fields) for such model.

#### 5.1.1 Purely *hvLif* in the EMD

*hvLif*solutions has a scalar potential which depends on \(\phi \) through one single exponential [becoming a constant when \(\theta =0\), \(\theta =2\) or \(z=1+\theta /2\) (\(\phi =\phi _0\) in the last two cases)] [17]. On the other hand, the gauge coupling function is constant for \(\theta =4\), and again if \(\phi =\phi _0\).

#### 5.1.2 \(t^3\)-model

Let us see now what the situation is for the truncation of the \(t^3\)-model considered in the previous section. In this case, \(V^{I,II}=c_1 e^{-\phi /\sqrt{3}}+c_2 e^{\phi /\sqrt{3}}\) with \(c_2=0~\Rightarrow ~c_1=0\) in the case I, and \(c_1=0~\Rightarrow ~c_2=0\) in the case II. Since we can only keep one of the exponentials [in order to match \(V\) with Eq. (5.25)], the only possibility is setting \(g^0=0\) (\(c_1=0\)) in the case I (which leaves us with the string theory embedded model), and \(g_0\) (\(c_2=0\)) in the case II. In both situations, \(Z(\phi )=-2~e^{-\sqrt{3}\phi }\). In I there exists one only solution, which is magnetic, and corresponds to \({\theta =-2,~z=3/2}\), \(g_1^2=297/(4 \ell ^2)\) and \(p^2=11\ell ^2/4\). On the other hand, case II admits one only solution (magnetic as well) for \({\theta =1,~z=3}\), \({g^1}^2=4/ \ell ^2\), and \(p^2=8\ell ^2\). Both solutions satisfy the NEC, as was desirable, and have a running dilaton given by Eq. (5.24) with \({\mathcal {Z}}=12\) and \({\mathcal {Z}}=3\), respectively.

### 5.2 Asymptotically *hvLif* in the EMD

#### 5.2.1 Non-constant scalar field

*hvLif*asymptotics, we switch now to \(A\)–\(B\)–\(f\) variables. The required form for \(A\) and \(B\) is

^{10}

*hvLif*metrics, and taking into account the form of \(V(\phi )\) and \(Z(\phi )\) for our

*axion-free*model (and others present in the literature), we can start by considering these functions to have the generic form

*axion-free*\(t^3\) model, as well as in other string theory truncations present in the literature (see, e.g., [71, 72]). On the other hand, additional terms to the single-exponential gauge coupling have been introduced to mimic the quantum corrections appearing from string theory (see, e.g., [73]), in an attempt to cure the logarithmic behavior of the dilaton, which blows up in the deep IR, pointing out the non-negligibility of quantum corrections in this regime. The expressions for \(V(\phi )\) and \(Z(\phi )\) can be introduced in Eqs. (5.29) and (3.10), (3.11) or (3.12) (depending on whether we are searching for electric, magnetic or dyonic solutions) using Eq. (5.30). Once this is done, we are left with two second-order differential equations for \(f(r)\) which can in general be converted into a first order equation plus a constraint that remains to be fulfilled. Obtaining the general solution in the presence of so many arbitrary parameters (\(c_1\), \(c_2\), \(c_3\), \(d_1\), \(d_2\), \(d_3\), \(s_1\), \(s_2\), \(t_1\), \(t_2\), \(z\), and \(\theta \)) seems not to be possible and therefore we are forced to consider further simplifications, keeping in mind that the procedure does work for other set-ups in which \(Z(\phi )\) and \(V(\phi )\) are given by a different choice of the parameters in (5.31) and (5.32). Taking into account the form of the potentials obtained in the

*axion-free*\(t^3\) model, let us assume \(s_1=s_2\), \(d_2=d_3=0\) (we allow \(t_1\) to be positive or negative) and we have

#### 5.2.2 \(t^3\)-model

- 1.
**Magnetic solutions.**As we saw, a consistent truncation of the \(t^3\)-model can be embedded in the EMD system for \(s_1=1/\sqrt{3}\), \(t_1=\sqrt{3}\), \(c_3=0\). It turns out that setting \(q=0\), it is possible to construct two families of solutions which, in the apropriate cases, asymptote to the purely*hvLif*ones constructed in the previous subsection. The first one is determined bywhere \(A\) is a constant depending on \(z\) and \(\theta \). The blackening factor is given by$$\begin{aligned} c_1=0,~\theta =2\left( 1-\frac{\varDelta }{\sqrt{3}}\right) ,~c_2=Ap^2, \end{aligned}$$(5.47)where \(C\) is another \(z,~\theta \)-dependent constant. Needless to say, the metric will not, in general, asymptote to a$$\begin{aligned} f(r)=Cp^2r^{\left( 2-\frac{\varDelta }{\sqrt{3}}\right) }+K r^{\left( \frac{2\varDelta }{\sqrt{3}}+z\right) }, \end{aligned}$$(5.48)*hvLif*(with exponents \(z,~\theta \)) as \(r\rightarrow 0\) except for particular values of \(\theta \) and \(z\). However, if we choose \({\theta =-2,~z=3/2}\), \(c_2=-9p^2\), we findThe second family is characterized by$$\begin{aligned} f(r)=\frac{4p^2}{11}\left[ 1-K r^{\frac{11}{2}} \right] . \end{aligned}$$(5.49)where \(A\) is another constant, and the blackening factor reads$$\begin{aligned} c_2=0,~\theta =\left( 2-\frac{\varDelta }{\sqrt{3}}\right) ,~c1=Ap^2, \end{aligned}$$(5.50)If we set \({\theta =1,~z=3}\), it becomes$$\begin{aligned} f(r)=C p^2 r^{\left( 2-\frac{2\varDelta }{\sqrt{3}}\right) }+K r^{\left( \frac{\varDelta }{\sqrt{3}}+z\right) }. \end{aligned}$$(5.51)which, as we will see in a moment, is a particular a case of a dyonic solution admitted by the model.$$\begin{aligned} f(r)=\frac{p^2}{8}\left[ 1-K r^4 \right] \end{aligned}$$(5.52) - 2.
**Electric solutions.**Similarly, we can construct two families of electric solutions. The first one is characterized bywhere, once more, \(A\) is a constant depending on \(z\) and \(\theta \). The blackening factor is given by$$\begin{aligned} c_1=0,~\theta =\left( 2+\frac{\varDelta }{\sqrt{3}}\right) ,~c_2=Aq^2, \end{aligned}$$(5.53)whereas for the second$$\begin{aligned} f(r)=Cq^2r^{\left( 2+\frac{2\varDelta }{\sqrt{3}}\right) }+K r^{\left( -\frac{\varDelta }{\sqrt{3}}+z\right) }, \end{aligned}$$(5.54)$$\begin{aligned}&c_1=Aq^2,~\theta =2\left( 1+\frac{\varDelta }{\sqrt{3}}\right) ,~c_2=0,\end{aligned}$$(5.55)In contradistinction to the magnetic cases, for no values of \((\theta ,z)\) the above solutions take the form of Eq. (5.44). This is obviously connected to the fact that no purely$$\begin{aligned}&f(r)=Cq^2r^{\left( 2+\frac{\varDelta }{\sqrt{3}}\right) }+K r^{\left( -\frac{2\varDelta }{\sqrt{3}}+z\right) }. \end{aligned}$$(5.56)*hvLif*electric solutions exist in this model for non-constant dilaton and scalar potential, as we saw before. - 3.
**Dyonic solutions.**It is possible to show that a dyonic solution does exist for \(\theta =1\), \(z=3\), \(c_2=0\), and \(c_1=-3p^2/2\), with a blackening factor given byThe corresponding metric Eq. (3.16) reads (after the redefinitions \(\mathrm{d}R^2=8\mathrm{d}r^2/p^2\), \(\mathrm{d}T^2=8\mathrm{d}t^2/p^2\))$$\begin{aligned} f(r)=\frac{p^2}{8}\left[ 1-Kr^4+\frac{q^2}{p^2}r^6\right] . \end{aligned}$$(5.57)It asymptotes to a$$\begin{aligned} \mathrm{d}s^2_f&= \frac{L^2}{R}\left\{ \left[ 1-K R^4+\frac{p^4 q^2}{512} R^6 \right] \frac{\mathrm{d}T^2}{R^4}\right. \nonumber \\&\left. -\frac{\mathrm{d}R^2}{\left[ 1-K R^4+\frac{p^4 q^2}{512} R^6 \right] } -\mathrm{d}\vec {x}^2\right\} . \end{aligned}$$(5.58)*hvLif*as \(R\rightarrow 0\) with \(\theta =1,~ z=3\), and to a different one as \(R\rightarrow \infty \) with \(\theta =5/2\), \(z=3/2\) as can be seen by taking the limit in the previous expression, and defining \(\rho \sim R^{-2}\)$$\begin{aligned}&\mathrm{d}s^2_f\overset{R\rightarrow +\infty }{\sim }\frac{L^2}{R}\left[ R^2 \mathrm{d}T^2 - \frac{\mathrm{d}R^2}{R^6} -\mathrm{d}\vec {x}^2\right] ,\end{aligned}$$(5.59)which corresponds to \(\theta =5/2,~z=3/2\). The value of \(K\) can be fixed in a way such that \(\exists ~R_h\in {\mathbb {R}}^{+}/f(R_h)=0\), or chosen to get a positive-definite metric in the whole spacetime.$$\begin{aligned}&\mathrm{d}s^2_f \overset{\left[ R\rightarrow +\infty ,~R^{-2}=\rho \right] }{\sim } L^2\rho ^{1/2}\left[ \frac{\mathrm{d}T^2}{\rho } - \mathrm{d}\rho ^2 -\mathrm{d}\vec {x}^2\right] , \end{aligned}$$(5.60)

*hvLif*with \(\theta =3,~ z=1\). On the other hand, restoring \(q\) and setting \(p=0\), imposes the vanishing of \(V_\mathrm{fi}\), and the solution is \(\theta =3,~z=1\) as \(R\rightarrow 0\), and again \(\theta =5/2,~ z=3/2\) as \(R\rightarrow + \infty \).

^{11}. This is somehow “dual” to the previous one, as it presents the same IR and UV behavior but with both regimes interchanged. It is characterized by \(c_1=0\), \(c_2=-\frac{3q^2}{8}\), and

*hvLif*with \(\theta =1,~z=3\).

#### 5.2.3 Constant scalar field

- 1.\({z=1+\frac{\theta }{2},~\theta \ne 2}\). In this situation, it is possible to find a solution which imposes no further constraints on \(V\) and \(V_\mathrm{bh}\). This readsThe case \(z=1,\theta =0\), in which we expect to recover \(aDS_4\) asymptotically is a particularization of this. The blackening factor then reads$$\begin{aligned} f(r)=-K r^{3-\theta /2}+\frac{\left[ 12Z_0^2 r^{4-\theta }-2\varLambda r^{\theta } \right] }{3(\theta -2)^2}. \end{aligned}$$(5.70)Assuming a negative cosmological constant, \(\varLambda =-|\varLambda |\), this can be rewritten as$$\begin{aligned} f(r)=-\frac{\varLambda }{6}-K r^3+Z_0^2 r^4, \end{aligned}$$(5.71)If we define \(\mathrm{d}T^2=|\varLambda |\mathrm{d}t^2/6\), \(\mathrm{d}R^2=6\mathrm{d}r^2/|\varLambda |\), the metric Eq. (3.16) becomes$$\begin{aligned} f(r)=\frac{|\varLambda |}{6}\left[ 1-K r^3+\frac{6Z_0^2}{|\varLambda |}r^4 \right] . \end{aligned}$$(5.72)which, of course, asymptotes to \(aDS_4\) as \(R\rightarrow 0\), and is such that \(\exists ~R_h\in {\mathbb {R}}^{+}/f(R_h)=0\) for \(K>0\). Similarly, the metric blows up as \(R\rightarrow \infty \), behaving as a$$\begin{aligned} \mathrm{d}s_f^2&= \frac{L^2}{R^2}\left\{ \left[ 1-K R^3+\frac{|\varLambda |Z_0^2}{6} R^4 \right] \mathrm{d}T^2\right. \nonumber \\&\quad \left. -\frac{\mathrm{d}R^2}{\left[ 1-K R^3+\frac{|\varLambda |Z_0^2}{6} R^4 \right] }-\mathrm{d}\vec {x}^2\right\} , \end{aligned}$$(5.73)
*hvLif*with \(\theta =4,~z=3\). Indeed,up to constants; if we make now the change \(\rho \sim 1/R\)$$\begin{aligned} \mathrm{d}s_f^2\overset{R\rightarrow \infty }{\sim }\frac{L^2}{R^2}\left[ R^4 \mathrm{d}T^2-\frac{\mathrm{d}R^2}{R^4}-\mathrm{d}\vec {x}^2\right] \end{aligned}$$(5.74)we find a$$\begin{aligned} \mathrm{d}s_f^2\overset{\rho \rightarrow 0}{\sim }L^{\prime 2}\rho ^2\left[ \frac{\mathrm{d}T^2}{\rho ^4}-\mathrm{d}\rho ^2-\mathrm{d}\vec {x}^2\right] , \end{aligned}$$(5.75)*hvLif*metric with \(\theta =4,~z=3\) as we have said. If we plug these values \(\theta =4,~z=3\) in Eq. (5.70) we find a new solution, which behaves asymptotically as this one (with the IR and UV regions interchanged). Indeed, its blackening factor readsand with the redefinitions \(\mathrm{d}R^2=\mathrm{d}r^2/Z_0^2,~\mathrm{d}T^2=\mathrm{d}t^2/Z_0^2\)$$\begin{aligned} f(r)=Z_0^2\left[ 1-K r +\frac{|\varLambda |}{6Z_0^2}r^4 \right] , \end{aligned}$$(5.76)As \(R\rightarrow 0\), it becomes a$$\begin{aligned} \mathrm{d}s_f^2&= L^2 R^2\left\{ \left[ 1-K R+\frac{|\varLambda |Z_0^2}{6}R^4 \right] \frac{\mathrm{d}T^2}{R^4}\right. \nonumber \\&\quad \left. -\frac{\mathrm{d}R^2}{\left[ 1-K R+\frac{|\varLambda |Z_0^2}{6}R^4 \right] }-\mathrm{d}\vec {x}^2 \right\} . \end{aligned}$$(5.77)*hvLif*with \(\theta =4,~z=3\), and as \(R\rightarrow \infty \),which we can rewrite as (\(\rho =1/R\))$$\begin{aligned} \mathrm{d}s^2_f=L^2 R^2 \left[ \mathrm{d}T^2-\frac{\mathrm{d}R^2}{R^4}-\mathrm{d}\vec {x}^2 \right] , \end{aligned}$$(5.78)which is \(aDS_4\).$$\begin{aligned} \mathrm{d}s^2_f=\frac{{L^{\prime }}^2 }{\rho ^2} \left[ \mathrm{d}T^2-\mathrm{d}\rho ^2-\mathrm{d}\vec {x}^2 \right] , \end{aligned}$$(5.79) - 2.\({\theta =2}\). This case imposes the constraint \(Z_0^2=-\frac{\varLambda }{2}\), and can be solved for any value of \(z\). The general form of \(f(r)\), which applies for \(z\ne 2\) is nowwhereas for \(z=2\) we have$$\begin{aligned} f(r)=\frac{2Z_0^2 r^2}{(z-2)^2}+r^{z}K_1+r^{2(z-1)}K_2, \end{aligned}$$(5.80)For example, if we consider the case \(\theta =2\), \(z=1\), we immediately find the asymptotically flat metric (as \(r\rightarrow 0\))$$\begin{aligned} f(r)=2 r^2\log (r) \left[ K_2+Z_0^2 \log (r) \right] +K_1 r^2. \end{aligned}$$(5.81)$$\begin{aligned} f(r)&= 1-Kr+2Z_0^2r^2,\end{aligned}$$(5.82)for which, once more \(\exists ~r_h\in {\mathbb {R}}^{+}/f(r_h)=0\) for \(K>0\). As \(r\rightarrow \infty \), up to constants, it behaves as$$\begin{aligned} \mathrm{d}s^2_f&= l^2\left\{ \mathrm{d}t^2\left[ 1-Kr+2Z_0^2r^2 \right] \right. \nonumber \\&\quad \left. -\frac{\mathrm{d}r^2}{\left[ 1-Kr+2Z_0^2 r^2 \right] }-\mathrm{d}\vec {x}^2 \right\} , \end{aligned}$$(5.83)where we defined \(R=\log r\). This is nothing but \(aDS_2\times {\mathbb {R}}_2\). On the other hand, if we set \(\theta =2\), \(z=2\), from Eq. (5.81) we find$$\begin{aligned} \mathrm{d}s^2_f\overset{\left[ R\rightarrow +\infty \right] }{\sim }{l^{\prime }}^2 \left[ e^{2R}\mathrm{d}t^2-\mathrm{d}R^2-\mathrm{d}\vec {x}^2\right] , \end{aligned}$$(5.84)$$\begin{aligned} f(r)&= 2 r^2\log (r) \left[ -K+Z_0^2 \log (r) \right] \nonumber \\&\quad +r^2\overset{\left[ R=\log r\right] }{=}e^{2R}\left[ 1-KR+2Z_0^2R^2 \right] ,\end{aligned}$$(5.85)which is nothing but Eq. (5.83).$$\begin{aligned} \mathrm{d}s^2_f&= l^2\left\{ \mathrm{d}t^2\left[ 1-KR+2Z_0^2R^2 \right] \right. \nonumber \\&\quad \left. -\frac{\mathrm{d}R^2}{\left[ 1-KR+2Z_0^2R^2 \right] }-\mathrm{d}\vec {x}^2 \right\} \end{aligned}$$(5.86)

## 6 Conclusions

We have studied purely *hvLif* and *hvLif*-like solutions of the general class of theories defined by the Lagrangian (2.1), which covers any theory of gravity coupled to an arbitrary number of scalars and vector fields up to two derivatives. We have obtained the general effective one-dimensional equations of motion that need to be solved in order to obtain *hvLif*-like solutions.

The general analysis is intended to complete the case-by-case results present in the literature in a unified framework: given a particular kinetic matrix \(\left( I_{\varLambda \Sigma }(\phi ),~R_{\varLambda \Sigma }(\phi )\right) \), a scalar metric \({\mathcal {G}}_{ij}(\phi )\) and a scalar potential \(V(\phi )\), the equations of motion of the theory follow trivially by plugging them into (2.11)–(2.13) and the Hamiltonian constraint (2.10).

For this broad family of theories, we have discussed the existence and properties of purely *hvLif* solutions attending to the presence (or absence) of non-constant scalar fields and non-vanishing black-hole and scalar potentials.

In the context of \({\mathcal {N}}=2\) FI supergravity, we have studied the \(t^3\)-model, for which we have explicitly constructed two consistent *axion-free* embeddings in the EMD system, one of which is, in turn, embedded in Type-IIB string theory for a particular choice of embedding tensor \(\theta _{M}\).

In addition, we obtained the general form of the \(f(r)\) function (for the set of metrics determined by Eqs. (5.27) and (3.16)), up to a constraint, for a rather general family of (supergravity-inspired) scalar and black-hole potentials, and explicitly constructed some dyonic solutions for the \(t^3\) truncations considered. We have provided a straightforward procedure to construct asymptotically *hvLif* solutions covered by Eqs. (5.27) and (3.16) for the family of theories specified by Eq. (5.33). This reduces the task to solving a single algebraic constraint, given by Eq. (5.41).

We have avoided, on purpose, the term *black hole* to denote the *hvLif*-like solutions obtained in this paper. The reason is that, although they look like black holes, a complete and rigorous proof (for example by constructing the corresponding Penrose–Carter diagram) is still missing. Therefore, any results obtained from them implicitly assuming that they do represent a black hole must be interpreted carefully, knowing that those would be yet to be proven statements.

As a final remark, we would like to point out that, thanks to the *BH-hvLif-Topological* triality (BHvTriality) discovered in [55], the new fascinating results that are being obtained in the context of static, spherically symmetric black holes in ungauged \({\mathcal {N}}=2,~d=4\) supergravity [74, 75, 76, 77, 78, 79] can also be applied to *hvLif*, giving therefore, the first examples of *hvLif*-like solutions in the presence of quantum corrections induced by Type-IIA string theory Calabi–Yau compatifications.

**Note added:** During the very last stage of this project, the very interesting Ref. [80] appeared, containing a minor overlap with our work.

## Footnotes

- 1.
We will understand for

*hvLif*any non-trivial gravitational solution that presents some kind of Lifshitz limit with hyperscaling violation.*Purely hvLif*stands for metrics that are exactly Lifshitz with hyperscaling violation. - 2.
- 3.The form of the vector fields can be recovered following the dimensional-reduction procedure. The corresponding field strengths \(F^{\varLambda }_{\mu \nu }\) are given bywhere \(\Psi = (\psi ^{\varLambda },\chi _{\varLambda })^{T}\) is a symplectic vector whose components are the time components of the electric and magnetic vector fields, \(A^{\varLambda }\) and \(A_{\varLambda }\). \(\Psi \) is given by$$\begin{aligned} F^{\varLambda }_{\underline{m}t} = -\partial _{\underline{m}}\psi ^{\varLambda },\qquad F^{\varLambda }_{\underline{m}\underline{n}}&= \frac{e^{-2U}}{\sqrt{|\gamma |}}\epsilon _{\underline{m}\underline{n}\tau }\left[ \left( I^{-1}\right) ^{\varLambda \Omega }\partial _{\tau }\chi _{\Omega }\right. \nonumber \\&\quad \left. - \left( I^{-1} R\right) ^{\varLambda }_{\Omega }\partial _{\tau }\psi ^{\Omega }\right] , \end{aligned}$$(2.5)$$\begin{aligned} \Psi = \int \frac{1}{2} e^{2U} {\mathcal {M}}^{MN}{\mathcal {Q}}_{N}\mathrm{d}\tau . \end{aligned}$$(2.6)
- 4.
It is important to stress that, in spite of its name, which is such because the dimensionally reduced actions in which it is often used are intended to build black-hole solutions, the black-hole potential should be understood as a very convenient generalization of a (negative) linear combination of squared electric and magnetic charges associated with the corresponding dimensionally reduced theory, regardless of whether the solution under consideration contains a black-hole spacetime or not.

- 5.
The canonical choice for \(d=4\) is \(\alpha = \frac{1}{2}\).

- 6.
From now on, we will use always the symbol “\(r\)” to denote the “radial” coordinate, independently of which coordinate system we use, which will be specified by other means.

- 7.
- 8.
We assume the conventions of [66].

- 9.
Supergravity gaugings are originally electric, breaking therefore the symplectic covariance present in the ungauged case. The embedding tensor formalism allows to formally keep the theory simplectically covariant by introducing magnetic and electric gaugings.

- 10.
- 11.
One may wonder why we did not find a purely

*hvLif*for these values of the exponents in the previous subsection. The reason is that for \(\theta =5/2,~z=3/2\) we have \({\mathcal {X}}_{(\theta ,z)}=0\), which implies the vanishing of \(V_\mathrm{fi}\) in the purely*hvLif*case. In fact, to recover the pure solution, we have to set \(K=q=c_2=0\), and since we have already set \(c_1=0\), this would make \(V_\mathrm{fi}=0\).

## Notes

### Acknowledgments

The authors thank Tomás Ortín for useful discussions. WC acknowledges the hospitality at IFT where the final part of this work was performed This work has been supported in part by the Spanish Ministry of Science and Education grant FPA2009-07692, the Comunidad de Madrid grant HEPHACOS S2009ESP-1473 and the Spanish Consolider-Ingenio 2010 program CPAN CSD2007-00042. The work of P.B. and C.S.S. has been supported by the JAE-predoc grants JAEPre 2011 00452 and JAEPre 2010 00613. P.B. wishes to thank N. Fuster for her unshakeable support. C.S.S. wishes to thank S. Ruiz for her unswerving support.

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