# Holographic RG flows and \(AdS_5\) black strings from 5D half-maximal gauged supergravity

## Abstract

We study five-dimensional \(N=4\) gauged supergravity coupled to five vector multiplets with compact and non-compact gauge groups \(U(1)\times SU(2)\times SU(2)\) and \(U(1)\times SO(3,1)\). For \(U(1)\times SU(2)\times SU(2)\) gauge group, we identify \(N=4\) \(AdS_5\) vacua with \(U(1)\times SU(2)\times SU(2)\) and \(U(1)\times SU(2)_{\text {diag}}\) symmetries and analytically construct the corresponding holographic RG flow interpolating between these critical points. The flow describes a deformation of the dual \(N=2\) SCFT driven by vacuum expectation values of dimension-two operators. In addition, we study \(AdS_3\times \Sigma _2\) geometries, for \(\Sigma _2\) being a two-sphere \(S^2\) or a two-dimensional hyperbolic space \(H^2\), dual to twisted compactifications of \(N=2\) SCFTs with flavor symmetry *SU*(2). We find a number of \(AdS_3\times H^2\) solutions preserving eight supercharges for different twists from \(U(1)\times U(1)\times U(1)\) and \(U(1)\times U(1)_{\text {diag}}\) gauge fields. We numerically construct various RG flow solutions interpolating between \(N=4\) \(AdS_5\) critical points and these \(AdS_3\times H^2\) geometries in the IR. The solutions can also be interpreted as supersymmetric black strings in asymptotically \(AdS_5\) space. These types of holographic solutions are also studied in non-compact \(U(1)\times SO(3,1)\) gauge group. In this case, only one \(N=4\) \(AdS_5\) vacuum exists, and we give an RG flow solution from this \(AdS_5\) to a singular geometry in the IR corresponding to an \(N=2\) non-conformal field theory. An \(AdS_3\times H^2\) solution together with an RG flow between this vacuum and the \(N=4\) \(AdS_5\) are also given.

## 1 Introduction

\(\hbox {AdS}_5\)/\(\hbox {CFT}_4\) correspondence has attracted much attention since the first proposal of the AdS/CFT correspondence in [1]. Various aspects of the very well-understood duality between type IIB theory on \(AdS_5\times S^5\) and \(N=4\) Super Yang-Mills (SYM) theory in four dimensions are captured by \(N=8\) *SO*(6) gauged supergravity in five dimensions which is a consistent truncation of type IIB supergravity on \(S^5\) [2]. One aspect of the AdS/CFT correspondence that has been extensively studied is holographic RG flows. There are many previous works considering these solutions both in \(N=8\) five-dimensional gauged supergravity and directly in type IIB string theory, see for example [3, 4, 5, 6, 7].

Results along this direction with less supersymmetry have also appeared in [8, 9, 10, 11]. In this case, gauged supergravities in five dimensions with \(N<8\) supersymmetry provide a very useful framework. In this paper, we consider holographic RG flows in half-maximal \(N=4\) gauged supergravity coupled to vector multiplets. This gauged supergravity has global symmetry \(SO(1,1)\times SO(5,n)\), *n* being the number of vector multiplets. Gaugings of a subgroup \(G_0\subset SO(1,1)\times SO(5,n)\) have been constructed in an \(SO(1,1)\times SO(5,n)\) covariant manner using the embedding tensor formalism in [12], see also [13]. The resulting solutions should describe RG flows arising from perturbing \(N=2\) superconformal field theories (SCFTs) by turning on some operators or their expectation values. Holographic solutions describing these \(N=2\) SCFTs and their deformations are less known compared to the \(N=4\) SYM. The results of this paper will give more examples of supersymmetric RG flow solutions and should hopefully shed some light on strongly coupled dynamics of \(N=2\) SCFTs.

We will consider \(N=4\) gauged supergravity coupled to five vector multiplets. This \(N=4\) gauged supergravity has a possibility of embedding in ten dimensions since the ungauged supergravity can be obtained via a \(T^5\) reduction of \(N=1\) supergravity in ten dimensions similar to \(N=4\) supergravity in four dimensions coupled to six vector multiplets that descends from \(N=1\) ten-dimensional supergravity compactified on a \(T^6\). However, it should be emphasized that the gaugings considered here have no known higher dimensional origin todate. We mainly focus on domain wall solutions interpolating between \(N=4\) \(AdS_5\) vacua or between an \(AdS_5\) vacuum and a singular domain wall corresponding to a non-conformal field theory. These types of solutions have been extensively studied in half-maximal gauged supergravities in various space-time dimensions, see [10, 11, 14, 15, 16, 17, 18, 19, 20, 21] for an incomplete list. The solutions involve only the metric and scalar fields.

We will also study solutions with some vector fields non-vanishing. These solutions interpolate between the above mentioned supersymmetric \(AdS_5\) vacua and \(AdS_3\times \Sigma _2\) geometries in the IR in which \(\Sigma _2\) is a two-sphere (\(S^2\)) or a two-dimensional hyperbolic space (\(H^2\)). Holographically, the resulting solutions describe twisted compactifications of the dual \(N=2\) SCFTs to two-dimensional SCFTs as first studied in [22]. A number of these flows across dimensions have been found within \(N=8\) gauged supergravity and its truncations in [23, 24, 25, 26, 27], see also a universal result in [28] and solutions obtained directly from type IIB theory in [29]. To the best of our knowledge, solutions of this type have not appeared before in the framework of \(N=4\) gauged supergravity coupled to vector multiplets, see however [30] for similar solutions in pure \(N=4\) gauged supergravity. Our results should give a generalization of the universal RG flows across dimensions in [28] by turning on the twists from flavor symmetries.

In addition, \(AdS_3\times \Sigma _2\) geometries can arise as near horizon limits of black strings. Therefore, flow solutions interpolating between \(AdS_5\) and \(AdS_3\times \Sigma _2\) should describe black strings in asymptotically \(AdS_5\) space. Similar solutions in \(N=2\) gauged supergravity have been considered in [31, 32, 33, 34, 35]. We will give solutions of this type in \(N=4\) gauged supergravity. The solutions presented here will provide further examples of supersymmetric \(AdS_5\) black strings and might be useful for both holographic studies of twisted \(N=2\) SCFTs on \(\Sigma _2\) and certain dynamical aspects of black strings.

The paper is organized as follow. In Sect. 2, we review \(N=4\) gauged supergravity in five dimensions coupled to vector multiplets using the embedding tensor formalism. In Sect. 3, a compact \(U(1)\times SU(2)\times SU(2)\) gauge group is considered. Supersymmetric \(AdS_5\) vacua and RG flows interpolating between them are given. A number of \(AdS_3\times H^2\) solutions will also be given along with numerical RG flows interpolating between the previously identified \(AdS_5\) vacua and these \(AdS_3\times H^2\) geometries. In Sect. 4, we repeat the analysis for a non-compact \(U(1)\times SO(3,1)\) gauge group. An RG flow from \(N=2\) SCFT dual to a supersymmetric \(AdS_5\) vacuum to a singular geometry dual to a non-conformal field theory is considered. A supersymmetric \(AdS_3\times H^2\) geometry and an RG flow between this vacuum and the \(AdS_5\) critical point will also be given. We end the paper with some conclusions and comments in Sect. 5.

## 2 Five dimensional \(N=4\) gauged supergravity coupled to vector multiplets

In this section, we review the structure of five dimensional \(N=4\) gauged supergravity coupled to vector multiplets. We mainly focus on relevant formulae to find supersymmetric solutions. More details on the construction of \(N=4\) gauged supergravity can be found in [12] and [13].

In five dimensions, \(N=4\) gravity multiplet consists of the graviton \(e^{{\hat{\mu }}}_\mu \), four gravitini \(\psi _{\mu i}\), six vectors \(A^0\) and \(A_\mu ^m\), four spin-\(\frac{1}{2}\) fields \(\chi _i\) and one real scalar \(\Sigma \), the dilaton. Space-time and tangent space indices are denoted respectively by \(\mu ,\nu ,\ldots =0,1,2,3,4\) and \({\hat{\mu }},{\hat{\nu }},\ldots =0,1,2,3,4\). The \(SO(5)\sim USp(4)\) R-symmetry indices are described by \(m,n=1,\ldots , 5\) for the *SO*(5) vector representation and \(i,j=1,2,3,4\) for the *SO*(5) spinor or *USp*(4) fundamental representation.

*n*of vector multiplets. Each vector multiplet contains a vector field \(A_\mu \), four gaugini \(\lambda _i\) and five scalars \(\phi ^m\). The

*n*vector multiplets will be labeled by indices \(a,b=1,\ldots , n\). Components fields in the

*n*vector multiplets are accordingly denoted by \((A^a_\mu ,\lambda ^{a}_i,\phi ^{ma})\). The 5

*n*scalar fields parametrized the \(SO(5,n)/SO(5)\times SO(n)\) coset. Combining the gravity and vector multiplets, we have \(6+n\) vector fields denoted by \(A^{\mathcal {M}}_\mu =(A^0_\mu ,A^m_\mu ,A^a_\mu )\) and \(5n+1\) scalars. All fermionic fields are symplectic Majorana spinors subject to the condition

*C*and \(\Omega _{ij}\) being the charge conjugation matrix and

*USp*(4) symplectic form, respectively.

*H*indices \(A,B,\ldots \) can be split into \(A=(m,a)\). We can then write the coset representative as

*SO*(5,

*n*), satisfies the relation

*SO*(5,

*n*) invariant tensor. In addition, the \(SO(5,n)/SO(5)\times SO(n)\) coset can also be described in term of a symmetric matrix

The full global symmetry of \(N=4\) supergravity coupled to *n* vector multiplets is \(SO(1,1)\times SO(5,n)\). The \(SO(1,1)\sim {\mathbb {R}}^+\) factor is identified with the coset space described by the dilaton \(\Sigma \). Gaugings can be efficiently described, in an \(SO(1,1)\times SO(5,n)\) covariant manner, by using the embedding tensor formalism. \(N=4\) supersymmetry allows three components of the embedding tensor \(\xi ^{M}\), \(\xi ^{MN}=\xi ^{[MN]}\) and \(f_{MNP}=f_{[MNP]}\). The existence of supersymmetric \(AdS_5\) vacua requires \(\xi ^M=0\), see [36] for more detail. Since, in this paper, we are only interested in supersymmetric \(AdS_5\) vacua and solutions interpolating between these vacua or solutions asymptotically approaching \(AdS_5\), we will restrict ourselves to the gaugings with \(\xi ^{M}=0\).

*SO*(5,

*n*). The gauge generators in the fundamental representation of

*SO*(5,

*n*) can be written in terms of the

*SO*(5,

*n*) generators \({(t_{MN})_P}^Q=\delta ^Q_{[M}\eta _{N]P}\) as

*SO*(5,

*n*) indices \(M,N,\ldots \) are lowered and raised by \(\eta _{MN}\) and its inverse \(\eta ^{MN}\).

*SO*(5,

*n*). This requires \(\xi ^{MN}\) and \(f_{MNP}\) to satisfy the quadratic constraints

*U*(1) is gauged by \(A^0_\mu \) while \(H\subset SO(n+3-\text {dim}\, H_0)\) is a compact group gauged by vector fields in the vector multiplets. \(H_0\) is a non-compact group gauged by three of the graviphotons and \(\text {dim}\, H_0-3\) vectors from the vector multiplets. In addition, \(H_0\) must contain an

*SU*(2) subgroup. For simple groups, \(H_0\) can be \(SU(2)\sim SO(3)\),

*SO*(3, 1) and \(SL(3,{\mathbb {R}})\).

*n*vector multiplets can be written as

*e*is the vielbein determinant. \(\mathcal {L}_{\text {top}}\) is the topological term which we will not give the explicit form here due to its complexity. The covariant gauge field strength tensors read

*U*(1) part of the gauge group.

*SO*(5) gamma matrices. Similarly, the inverse \({\mathcal {V}_{ij}}^M\) can be written as

## 3 Supersymmetric RG flows in \(U(1)\times SU(2)\times SU(2)\) gauge group

*SO*(5,

*n*) non-compact generators are given by

### 3.1 RG flows between \(N=4\) supersymmetric \(AdS_5\) critical points

*SO*(5, 5), the scalars transform under \(U(1)\times SU(2)\times SU(2)\) gauge group as

*U*(1) charges. According to this decomposition, there is one singlet corresponding to the following

*SO*(5, 5) non-compact generator

*SO*(5, 5) global symmetry. Furthermore, we can rescale \(\Sigma \), or equivalently set \(g_2=-\sqrt{2}g_1\) to bring this critical point located at \(\Sigma =1\). The cosmological constant, the value of the scalar potential at the critical point, is

*SU*(2) dual to a flavor symmetry of the dual \(N=2\) SCFT is present.

*SO*(5) gamma matrices \({\Gamma _{mi}}^j\)

*r*.

*r*-derivative. Multiply this equation by \(A'\gamma _r\) and iterate, we find

*W*will be identified with the superpotential. When substitute this result in Eq. (41), we find

*r*-dependent Killing spinors of the form \(\epsilon _i=e^{\frac{A}{2}}\epsilon _{0i}\) for constant spinors \(\epsilon _{0i}\) satisfying (44). Using the projector (44) in conditions \(\delta \chi _i=0\) and \(\delta \lambda ^a_i=0\), we can derive the first order flow equations for \(\Sigma \) and \(\phi \).

*W*and solve Eqs. (50) and (51). These critical points are then \(N=4\) supersymmetric. Together with the \(A'\) equation

*A*. The constant

*C*will be chosen in such a way that \(\Sigma \) approaches the second \(AdS_5\) vacuum. This requires \(C=-\frac{g_1(g_3+g_2)^2}{\sqrt{2}g_2g_3}\) leading to the final form of the solution for \(\Sigma \)

Scalar masses at the \(N=4\) supersymmetric \(AdS_5\) critical point with \(U(1)\times SU(2)\times SU(2)\) symmetry and the corresponding dimensions of the dual operators

Scalar field representations | \(m^2L^2\phantom {\frac{1}{2}}\) | \(\Delta \) |
---|---|---|

\(({\mathbf {1}},{\mathbf {1}})_0\) | \(-4\) | 2 |

\(({\mathbf {1}},{\mathbf {1}})_{\pm 2}\) | \(-3_{\times 4}\) | 3 |

\(({\mathbf {1}},{\mathbf {3}})_{\pm 2}\) | \(-3_{\times 6}\) | 3 |

\(({\mathbf {3}},{\mathbf {1}})_0\) | \(-4_{\times 6}\) | 2 |

\(({\mathbf {3}},{\mathbf {3}})_0\) | \(-4_{\times 9}\) | 2 |

Scalar masses at the \(N=4\) supersymmetric \(AdS_5\) critical point with \(U(1)\times SU(2)_{\text {diag}}\) symmetry and the corresponding dimensions of the dual operators

Scalar field representations | \(m^2L^2\phantom {\frac{1}{2}}\) | \(\Delta \) |
---|---|---|

\({\mathbf {1}}_0\) | \(-4\) | 2 |

\({\mathbf {1}}_0\) | 12 | 6 |

\({\mathbf {1}}_{\pm 2}\) | \(-3_{\times 4}\) | 3 |

\({\mathbf {3}}_{\pm 2}\) | \(5_{\times 6}\) | 5 |

\({\mathbf {3}}_0\) | \(-4_{\times 6}\) | 2 |

\({\mathbf {3}}_0\) | \(0_{\times 3}\) | 4 |

\({\mathbf {5}}_0\) | \(0_{\times 5}\) | 4 |

### 3.2 Supersymmetric RG flows from \(N=2\) SCFTs to two dimensional SCFTs

We now consider another type of solutions namely solutions interpolating between supersymmetric \(AdS_5\) vacua identified previously and \(AdS_3\times \Sigma _2\) geometries. In the present consideration, \(\Sigma _2\) is a two-sphere (\(S^2\)) or a two-dimensional hyperbolic space (\(H^2\)).

To preserve some amount of supersymmetry, we impose a twist condition by cancelling the spin connection on \(S^2\) with some gauge connections. We will consider abelian twists from \(U(1)\times U(1)\times U(1)\subset U(1)\times SU(2)\times SU(2)\) and its \(U(1)\times U(1)_{\text {diag}}\) subgroup. The corresponding gauge fields are denoted by \((A^0,A^5,A^8)\). Note that turning on \(A^0\) and \(A^5\) correspond to a twist along the R-symmetry \(U(1)\times SU(2)\) of the dual \(N=2\) SCFTs while a non-vanishing \(A^8\) is related to turning on the gauge field of *SU*(2) flavor symmetry. The latter cannot be used as a twist since the Killing spinor is neutral under this symmetry.

*SO*(5, 5) non-compact generators \(Y_{53}\), \(Y_{54}\) and \(Y_{55}\). The coset representative is then given by

*SO*(5, 5)

For the \(H^2\) case, we simply change \(\sin \theta \) to \(\sinh \theta \) in the metric (60) and take the gauge fields to be \(A^{\mathcal {M}}=a_{\mathcal {M}}\cosh \theta d\phi \). The twist procedure works as in the \(S^2\) case. However, due to the opposite sign in the covariant field strengths \(\mathcal {H}^{\mathcal {M}}=dA^{\mathcal {M}}\), the resulting BPS equations for the two cases are related to each other by a sign change in the twist parameters \(a_{\mathcal {M}}\).

#### 3.2.1 Flow solutions with \(U(1)\times U(1)\times U(1)\) symmetry

*SU*(2) or

*U*(1) flavor symmetry depending on the value of \(\varphi _0\). An asymptotic analysis near the \(AdS_3\times H^2\) critical point shows that \(\varphi \) is dual to a marginal operator in the two-dimensional SCFT. The central charge of the dual SCFT can also be computed by [38]

*g*(

*r*) at the \(AdS_3\) critical point. For \(H_2\) being a genus \({\mathfrak {g}}>1\) Riemann surface, we have \(\text {vol}(H^2)=4\pi ({\mathfrak {g}}-1)\).

Examples of numerical flow solutions interpolating between \(N=4\) supersymmetric \(AdS_5\) and \(N=4\) supersymmetric \(AdS_3\times H^2\) with different values of \(\varphi _0\) are given in Fig. 1. The solution with \(\varphi _0=0\) is effectively the same as that studied in [28] which is in turn obtained from the solutions in [30] by turning off the *U*(1) gauge field. In this case, the matter multiplets can be decoupled. Solutions with \(\varphi _0\ne 0\) are only possible in the matter-coupled gauged supergravity and have not previously appeared.

*r*

*C*. This has also been pointed out in [28].

#### 3.2.2 Flow solutions with \(U(1)\times U(1)_{\text {diag}}\) symmetry

We firstly consider the twist from \(A^0\) gauge field. For \(a_5=0\), the \(U(1)_{\text {diag}}\) symmetry also demands \(a_8=0\). The BPS equations for \(\phi _1\), \(\phi _2\) and \(\phi _3\) will not depend on the twist parameter \(a_0\) since they are not charged under \(A^0\). Therefore, the only possibility to have \(AdS_3\) vacua is to set these scalars to their values at the two \(AdS_5\) critical points. Setting all \(\phi _i=0\) for \(i=1,2,3\) dose not lead to any \(AdS_3\) solutions as in the previous case. The other choice namely \(\phi _3=0\) and \(\phi _1=\phi _2=\frac{1}{2}\ln \left[ \frac{g_3-g_2}{g_3+g_2}\right] \) does not give rise to any \(AdS_3\) vacua either. Therefore, we will not give the explicit form of the BPS equations in this case.

- I. The simplest solution is obtained by setting \(\phi _i=0\), \(i=1,2,3\) and$$\begin{aligned} \Sigma= & {} -\left( \frac{\sqrt{2}g_2}{g_1}\right) ^{\frac{1}{3}},\quad g=\frac{1}{6}\ln \left[ \frac{2a_5}{g_1^2g_2}\right] ,\nonumber \\ L_{AdS_3}= & {} \left( \frac{\sqrt{2}}{g_1g_2^2}\right) ^{\frac{1}{3}}\, . \end{aligned}$$(97)
- II. One of the \(AdS_3\times H^2\) solutions with vector multiplet scalars non-vanishing is given by$$\begin{aligned} \phi _1= & {} \phi _2=\frac{1}{2}\ln \left[ \frac{g_3-g_2}{g_3+g_2}\right] ,\quad \phi _3=0,\nonumber \\ \Sigma ^3= & {} -\frac{\sqrt{2}g_2g_3}{g_1\sqrt{g_3^2-g_2^2}},\nonumber \\ g= & {} \frac{1}{6}\ln \left[ \frac{a_5^3(g_3^2-g_2^2)^2}{g_1^2g_2g_3^4}\right] ,\nonumber \\ L_{AdS_3}= & {} \left( \frac{\sqrt{2}(g_3^2-g_2^2)}{g_1g_2^2g_3^2}\right) ^{\frac{1}{3}}\, . \end{aligned}$$(98)
- III. There is another \(AdS_3\times H^2\) solution located at$$\begin{aligned} \phi _1= & {} 0,\quad \phi _2=\phi _3=\frac{1}{2}\ln \left[ \frac{g_3-g_2}{g_3+g_2}\right] ,\nonumber \\ \Sigma ^3= & {} -\frac{\sqrt{2}g_2g_3}{g_1\sqrt{g_3^2-g_2^2}},\nonumber \\ g= & {} \frac{1}{6}\ln \left[ \frac{2a_5^3(g_3^2-g_2^2)^2}{g_1^2g_2g_3^4}\right] ,\nonumber \\ L_{AdS_3}= & {} \left( \frac{\sqrt{2}(g_3^2-g_2^2)}{g_1g_2^2g_3^2}\right) ^{\frac{1}{3}}\, . \end{aligned}$$(99)

*g*and \(\Sigma \) of the form

\(AdS_3\times H^2\) critical point II is more interesting in the sense that it can be connected to both of the \(N=4\) \(AdS_5\) vacua. In order to obtain RG flow solutions, we set \(\phi _3=0\) which is a consistent truncation. An example of flows from \(AdS_5\) with \(U(1)\times SU(2)\times SU(2)\) symmetry to \(AdS_3\times H^2\) critical point II is given in Fig. 3. With suitable boundary conditions, we can find a solution that flows from \(AdS_5\) with \(U(1)\times SU(2)\times SU(2)\) symmetry and approaches \(AdS_5\) with \(U(1)\times SU(2)_{\text {diag}}\) symmetry before reaching the \(AdS_3\times H^2\) critical point II. A solution of this type is shown in Fig. 4.

## 4 Supersymmetric RG flows in \(U(1)\times SO(3,1)\) gauge group

Scalar masses at the \(N=4\) supersymmetric \(AdS_5\) critical point with \(U(1)\times SO(3)\) symmetry and the corresponding dimensions of the dual operators for the non-compact \(U(1)\times SO(3,1)\) gauge group

Scalar field representations | \(m^2L^2\phantom {\frac{1}{2}}\) | \(\Delta \) |
---|---|---|

\({\mathbf {1}}_0\) | \(-4\) | 2 |

\({\mathbf {1}}_0\) | 12 | 6 |

\({\mathbf {1}}_{\pm 2}\) | \(-3_{\times 4}\) | 3 |

\({\mathbf {3}}_{\pm 2}\) | \(5_{\times 6}\) | 5 |

\({\mathbf {3}}_0\) | \(-4_{\times 6}\) | 2 |

\({\mathbf {3}}_0\) | \(0_{\times 3}\) | 4 |

\({\mathbf {5}}_0\) | \(0_{\times 5}\) | 4 |

### 4.1 BPS equations and holographic RG flow solutions

Since there is only one supersymmetric \(AdS_5\) critical point, supersymmetric RG flows between \(AdS_5\) critical points do not exist. We will look for solution describing a domain wall with one limit being the \(AdS_5\) critical point identified above and another limit being a singular geometry dual to an \(N=2\) non-conformal field theory.

*W*has only one critical point. The potential can be written in term of the superpotential as

*A*and the dilaton \(\Sigma \) can be written as

*A*can be straightforwardly obtained. The result is

*r*to \(\rho \) via \(\frac{d\rho }{dr} = \Sigma ^{-1}\), we find the solution for \(\phi (\rho )\)

*C*. In both cases, \(\Sigma \rightarrow 0\) and \(V\rightarrow \infty \). As a result, these singularities are unphysical by the criterion of [39].

### 4.2 RG flows to \(AdS_3\times \Sigma _2\) geometries

which admits only a single supersymmetric critical point at which all vector multiplet scalars vanish.

The metric ansatz is still given by (60). We will consider the twists obtained from turning on \(U(1)\times U(1)\subset U(1)\times SO(3,1)\) gauge fields along \(\Sigma _2\). These gauge fields will be denoted by \(A^0\) and \(A^5\). As in the previous section, the twists from \(A^0\) and \(A^5\) cannot be turned on simultaneously. Furthermore, the \(A^0\) twist does not lead to \(AdS_3\times \Sigma _2\) solutions. We will therefore consider only the twist from \(A^5\). It turns out that the two-form fields can also be consistently set to zero provided that we set the gauge fields \(A^1=A^2=0\).

## 5 Conclusions and discussions

We have studied gauged \(N=4\) supergravity in five dimensions coupled to five vector multiplets with compact and non-compact gauge groups \(U(1)\times SU(2)\times SU(2)\) and \(U(1)\times SO(3,1)\). For \(U(1)\times SU(2)\times SU(2)\) gauge group, we have recovered two supersymmetric \(N=4\) \(AdS_5\) vacua with \(U(1)\times SU(2)\times SU(2)\) and \(U(1)\times SU(2)_{\text {diag}}\) symmetries together with the RG flow interpolating between them found in [11]. However, we have also given the full mass spectra for scalar fields at both critical points which have not been studied in [11]. These should be useful in the holographic context since it provides information about dimensions of operators dual to the supergravity scalars. For \(U(1)\times SO(3,1)\) gauge group, there is only one \(N=4\) supersymmetric \(AdS_5\) critical point with vanishing vector multiplet scalars. We have given an RG flow solution from an \(N=2\) SCFT dual to this vacuum to a non-conformal field theory dual to a singular geometry. However, this singularity is unphysical within the framework of \(N=4\) gauged supergravity. It would be interesting to embed this solution in ten or eleven dimensions and further investigate whether this singularity is resolved or has any physical interpretation in the context of string/M-theory.

We have also considered \(AdS_3\times \Sigma _2\) solutions by turning on gauge fields along \(\Sigma _2\). We have found that in order to preserve eight supercharges, the twists from the *U*(1) factor in the gauge group and the Cartan \(U(1)\subset SU(2)\), denoted by the parameters \(a_0\) and \(a_5\), cannot be performed simultaneously. It should also be noted that for less supersymmetric solutions, both \(a_0\) and \(a_5\) can be non-vanishing such as \(\frac{1}{4}\)-BPS solutions found in [30] for pure \(N=4\) gauged supergravity with \(U(1)\times SU(2)\) gauge group. It would also be interesting to look for more general solutions of this type.

For \(U(1)\times SU(2)\times SU(2)\) gauge group, we have identified a number of \(AdS_3\times H^2\) solutions preserving eight supercharges. We have given numerical RG flow solutions from the two \(AdS_5\) vacua to these \(AdS_3\times H^2\) geometries. For \(U(1)\times SO(3,1)\) gauge group, there is one \(AdS_3\times H^2\) solution when all scalars from vector multiplets vanish. The solution preserves eight supercharges similar to the solutions in the compact gauge group. A numerical RG flow between this solution and the \(N=4\) \(AdS_5\) vacuum has also been given. All of these solutions describe twisted compactifications of \(N=2\) SCFTs on \(H^2\) and should be of interest in holographic studies of \(N=2\) SCFTs in four dimensions and in the context of supersymmetric black strings. It is noteworthy that the space of \(AdS_5\) and \(AdS_3\) solutions in the compact gauge group is much richer than that of the non-compact gauge group. This is in line with similar studies of half-maximal gauged supergravities in other dimensions.

There are a number of future works extending our results presented here. It is interesting to consider flow solutions with non-vanishing two-form fields similar to the recently found solutions in seven and six dimensions in [40, 41, 42]. These solutions will also give a description of conformal defects in the dual \(N=2\) SCFTs. Furthermore, finding Janus solution within this \(N=4\) gauged supergravity is also of particular interest. This can be done by an analysis similar to that initiated in [43] and [44]. Up to now, this type of solutions has only appeared in \(N=8\) and \(N=2\) gauged supergravities, see for example [45, 46].

## Notes

### Acknowledgements

P. K. is supported by The Thailand Research Fund (TRF) under Grant RSA5980037.

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