# Holographic RG flows in \(N=4\) SCFTs from half-maximal gauged supergravity

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## Abstract

We study four-dimensional \(N=4\) gauged supergravity coupled to six vector multiplets with semisimple gauge groups \(SO(4)\times SO(4)\), \(SO(3,1)\times SO(3,1)\) and \(SO(4)\times SO(3,1)\). All of these gauge groups are dyonically embedded in the global symmetry group *SO*(6, 6) via its maximal subgroup \(SO(3,3)\times SO(3,3)\). For \(SO(4)\times SO(4)\) gauge group, there are four \(N=4\) supersymmetric \(AdS_4\) vacua with \(SO(4)\times SO(4)\), \(SO(4)\times SO(3)\), \(SO(3)\times SO(4)\) and \(SO(3)\times SO(3)\) symmetries, respectively. These \(AdS_4\) vacua correspond to \(N=4\) SCFTs in three dimensions with *SO*(4) R-symmetry and different flavor symmetries. We explicitly compute the full scalar mass spectra at all these vacua. Holographic RG flows interpolating between these conformal fixed points are also given. The solutions describe supersymmetric deformations of \(N=4\) SCFTs by relevant operators of dimensions \(\Delta =1,2\). A number of these solutions can be found analytically although some of them can only be obtained numerically. These results provide a rich and novel class of \(N=4\) fixed points in three-dimensional Chern–Simons-Matter theories and possible RG flows between them in the framework of \(N=4\) gauged supergravity in four dimensions. Similar studies are carried out for non-compact gauge groups, but the \(SO(4)\times SO(4)\) gauge group exhibits a much richer structure.

## 1 Introduction

The study of holographic RG flows is one of the most interesting results from the celebrated AdS/CFT correspondence since its original proposal in [1]. The solutions take the form of domain walls interpolating between *AdS* vacua, for RG flows between two conformal fixed points, or between an *AdS* vacuum and a singular geometry, for RG flows from a conformal fixed point to a non-conformal field theory. Many of these fixed points are described by superconformal field theories (SCFTs) which are also believed to give some insight into the dynamics of various branes in string/M-theory.

Rather than finding holographic RG flow solutions directly in string/M-theory, a convenient and effective way to find these solutions is to look for domain wall solutions in lower dimensional gauged supergravities. In some cases, the resulting solutions can be uplifted to interesting brane configurations within string/M-theory, see for example [2, 3, 4]. Apart from rendering the computation simpler, working in lower dimensional gauged supergravities also has an advantage of being independent of higher dimensional embedding. Results obtained within this framework are applicable in any models described by the same effective gauged supergravity regardless of their higher dimensional origins.

Many results along this direction have been found in maximally gauged supergravities, see for examples [5, 6, 7, 8, 9, 10, 11, 12, 13]. A number of RG flow solutions in half-maximally gauged supergravities in various dimensions have, on the other hand, been studied only recently in [14, 15, 16, 17, 18, 19, 20, 21], see also [22, 23] for earlier results. In this paper, we will give holographic RG flow solutions within \(N=4\) gauged supergravity in four dimensions. Solutions in the case of non-semisimple gauge groups with known higher dimensional origins have already been considered in [19, 20]. This non-semisimple gauging, however, turns out to have a very restricted number of supersymmetric \(AdS_4\) vacua. In this work, we will consider semisimple gauging of \(N=4\) supergravity similar to the study in other dimensions. Although some general properties of \(AdS_4\) vacua and RG flows have been pointed out recently in [18], to the best of our knowledge, a detailed analysis and explicit RG flow solutions in \(N=4\) gauged supergravity have not previously appeared.

Gaugings of \(N=4\) supergravity coupled to an arbitrary number *n* of vector multiplets have been studied and classified for a long time [24, 25, 26], and the embedding tensor formalism which includes all possible deformations of \(N=4\) supergravity has been given in [27]. For the case of \(n<6\), the relation between the resulting \(N=4\) supergravity and ten-dimensional supergravity is not known. Therefore, we will consider only the case of \(n\ge 6\) which is capable of embedding in ten dimensions. Furthermore, we are particularly interested in \(N=4\) gauged supergravity coupled to six vector multiplets to simplify the computation. In this case, possible gauge groups are embedded in the global symmetry group \(SL(2,\mathbb {R})\times SO(6,6)\). From a general result of [28], the existence of \(N=4\) supersymmetric \(AdS_4\) vacua requires that the gauge group is purely embedded in the *SO*(6, 6) factor. Furthermore, the gauge group must contain an \(SO(3)\times SO(3)\) subgroup with one of the *SO*(3) factors embedded electrically and the other one embedded magnetically.

We will consider gauge groups in the form of a simple product \(G_1\times G_2\) in which one of the two factors is embedded electrically in \(SO(3,3)\subset SO(6,6)\) while the other is embedded magnetically in the other *SO*(3, 3) subgroup of *SO*(6, 6). Taking the above criterions for having supersymmetric \(AdS_4\) vacua into account, we will study the case of \(G_1,G_2=SO(4)\) and *SO*(3, 1). There are then three different product gauge groups to be considered, \(SO(4)\times SO(4)\), \(SO(3,1)\times SO(3,1)\) and \(SO(4)\times SO(3,1)\). We will identify possible supersymmetric \(AdS_4\) vacua and supersymmetric RG flows interpolating between these vacua. These solutions should describe RG flows in the dual \(N=4\) Chern–Simons-Matter (CSM) theories driven by relevant operators dual to the scalar fields of the \(N=4\) gauged supergravity. As shown in [29], some of the \(N=4\) CSM theories can be obtained from a non-chiral orbifold of the ABJM theory [30]. Other classes of \(N=4\) CSM theories are also known, see [31, 32] for example. These theories play an important role in describing the dynamics of M2-branes on various backgrounds. The solutions obtained in this paper should also be useful in this context via the AdS/CFT correspondence. It should also be emphasized that all gauge groups considered here are currently not known to have higher dimensional origins. Therefore, the corresponding holographic duality in this case is still not firmly established.

The paper is organized as follow. In Sect. 2, we review \(N=4\) gauged supergravity coupled to vector multiplets in the embedding tensor formalism. This sets up the framework we will use throughout the paper and collects relevant formulae and notations used in subsequent sections. In Sect. 3, \(N=4\) gauged supergravity with \(SO(4)\times SO(4)\) gauge group is constructed, and the scalar potential for scalars which are singlets under \(SO(4)_{\text {inv}}\subset SO(4)\times SO(4)\) is computed. We will identify possible supersymmetric \(AdS_4\) vacua and compute the full scalar mass spectra at these vacua. The section ends with supersymmetric RG flow solutions interpolating between \(AdS_4\) vacua and RG flows to non-conformal field theories. A similar study is performed in Sects. 4 and 5 for non-compact \(SO(3,1)\times SO(3,1)\) and \(SO(4)\times SO(3,1)\) gauge groups. Conclusions and comments on the results will be given in Sect. 6. An appendix containing the convention on ’t Hooft matrices is included at the end of the paper.

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

To set up our framework, we give a brief review of four-dimensional \(N=4\) gauged supergravity. We mainly give relevant information and necessary formulae to find supersymmetric \(AdS_4\) vacua and domain wall solutions. More details on the construction can be found in [27].

\(N=4\) supergravity can couple to an arbitrary number *n* of vector multiplets. The supergravity multiplet consists of the graviton \(e^{\hat{\mu }}_\mu \), four gravitini \(\psi ^i_\mu \), six vectors \(A_\mu ^m\), four spin-\(\frac{1}{2}\) fields \(\chi ^i\) and one complex scalar \(\tau \) containing the dilaton \(\phi \) and the axion \(\chi \). The complex scalar can be parametrized by \(SL(2,\mathbb {R})/SO(2)\) coset. Each vector multiplet contains a vector field \(A_\mu \), four gaugini \(\lambda ^i\) and six scalars \(\phi ^m\). Similar to the dilaton and the axion in the gravity multiplet, the 6*n* scalar fields can be parametrized by \(SO(6,n)/SO(6)\times SO(n)\) coset.

Throughout the paper, space-time and tangent space indices are denoted respectively by \(\mu ,\nu ,\ldots =0,1,2,3\) and \(\hat{\mu },\hat{\nu },\ldots =0,1,2,3\). The \(SO(6)\sim SU(4)\) R-symmetry indices will be described by \(m,n=1,\ldots , 6\) for the *SO*(6) vector representation and \(i,j=1,2,3,4\) for the *SO*(6) spinor or *SU*(4) fundamental representation. The *n* vector multiplets will be labeled by indices \(a,b=1,\ldots , n\). All fields in the vector multiplets accordingly carry an additional index in the form of \((A^a_\mu ,\lambda ^{ia},\phi ^{ma})\).

*SO*(6,

*n*), respectively. Under the full \(SL(2,\mathbb {R})\times SO(6,n)\) duality symmetry, the electric vector fields \(A^{+M}=(A^m_\mu ,A^a_\mu )\), appearing in the ungauged Lagrangian, together with their magnetic dual \(A^{-M}\) form a doublet under \(SL(2,\mathbb {R})\) denoted by \(A^{\alpha M}\). A general gauge group is embedded in both \(SL(2,\mathbb {R})\) and

*SO*(6,

*n*), and the magnetic vector fields can also participate in the gauging. However, each magnetic vector field must be accompanied by an auxiliary two-form field in order to remove the extra degrees of freedom.

From the analysis of supersymmetric \(AdS_4\) vacua in [28], see also [25, 26], purely electric gaugings do not admit \(AdS_4\) vacua unless an \(SL(2,\mathbb {R})\) phase is included [25]. The latter is however incorporated in the magnetic component \(f_{-MNP}\) [27]. Therefore, only gaugings involving both electric and magnetic vector fields, or dyonic gaugings, lead to \(AdS_4\) vacua. Furthermore, the existence of maximally supersymmetric \(AdS_4\) vacua requires \(\xi ^{\alpha M}=0\). Accordingly, we will from now on restrict ourselves to the case of dyonic gaugings and \(\xi ^{\alpha M}=0\).

*SO*(6,

*n*) generators in the fundamental representation and can be chosen as

*SO*(6,

*n*) invariant tensor, and

*g*is the gauge coupling constant that can be absorbed in the embedding tensor \(f_{\alpha MNP}\). For a product gauge group consisting of many simple subgroups, there can be as many independent coupling constants as the simple groups within the product. Note also that with the component \(\xi ^{\alpha M}=0\), the gauge group is embedded solely in

*SO*(6,

*n*).

*SO*(6,

*n*) and local \(SO(6)\times SO(n)\) by left and right multiplications, respectively. By splitting the index \(A=(m,a)\), we can write the coset representative as

*SO*(6,

*n*), the matrix \(\mathcal {V}_M^{\phantom {M}A}\) satisfies the relation

*e*is the vielbein determinant. The scalar potential is given in terms of the scalar coset representative and the embedding tensor by

*m*in the vector representation of

*SO*(6) into an anti-symmetric pair of indices [

*ij*] in the

*SU*(4) fundamental representation. They satisfy the relations

## 3 Supersymmetric \(AdS_4\) vacua and holographic RG flows in \(SO(4)\times SO(4)\) gauged supergravity

We are interested in gauge groups that can be embedded in \(SO(3,3)\times SO(3,3)\subset SO(6,6)\). These gauge groups take the form of a product \(G_1\times G_2\) with \(G_1,G_2\subset SO(3,3)\) being six-dimensional. Semisimple groups of dimension six that can be embedded in *SO*(3, 3) are *SO*(4), *SO*(3, 1) and *SO*(2, 2). The embedding tensors for these gauge groups are given in [33]. Since gauge groups involving *SO*(2, 2) factors do not give rise to \(AdS_4\) vacua, we will not consider these gauge groups in this paper. In this section, we will study \(N=4\) gauged supergravity with compact \(SO(4)\times SO(4)\sim SO(3)\times SO(3)\times SO(3)\times SO(3)\) gauge group.

### 3.1 \(AdS_4\) vacua

*SO*(4) factors are embedded electrically and magnetically, respectively.

*SO*(6, 6) are given by

Scalar masses at the \(N=4\) supersymmetric \(AdS_4\) critical point with \(SO(3)_+\times SO(4)_-\) symmetry and the corresponding dimensions of the dual operators

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

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

\(({\mathbf {1}},{\mathbf {1}},{\mathbf {1}})\) | 4 | 4 |

\(({\mathbf {3}},{\mathbf {1}},{\mathbf {1}})\) | \(0_{\times 3}\) | 3 |

\(({\mathbf {1}},{\mathbf {3}},{\mathbf {3}})\) | \(0_{\times 9}\) | 3 |

\(({\mathbf {5}},{\mathbf {1}},{\mathbf {1}})\) | \(-2_{\times 5}\) | 1, 2 |

\(({\mathbf {3}},{\mathbf {1}},{\mathbf {3}})+({\mathbf {3}},{\mathbf {3}},{\mathbf {1}})\) | \(-2_{\times 18}\) | 1, 2 |

- II. This critical point has \(SO(3)_+\times SO(4)_-\) symmetry with$$\begin{aligned}&\phi =\ln \left[ \frac{2\sqrt{g_1\tilde{g}_1}}{g_1+\tilde{g}_1}\right] ,\quad \phi _1=\frac{1}{2}\ln \left[ \frac{g_1}{\tilde{g}_1}\right] ,\quad \phi _2=0,\nonumber \\&V_0=-\frac{3(g_1+\tilde{g}_1)(g_1-\tilde{g}_1)^2}{\sqrt{g_1\tilde{g}_1}},\nonumber \\&\quad L=\frac{(g_1\tilde{g}_1)^{\frac{1}{4}}}{(\tilde{g}_1-g_1)\sqrt{g_1+\tilde{g}_1}}. \end{aligned}$$(33)
- III. This critical point has \(SO(4)_+\times SO(3)_-\) symmetry with$$\begin{aligned} \phi= & {} -\ln \left[ \frac{2\sqrt{g_2\tilde{g}_2}}{g_2+\tilde{g}_2}\right] ,\quad \phi _2=\frac{1}{2}\ln \left[ \frac{g_2}{\tilde{g}_2}\right] ,\quad \phi _1=0,\nonumber \\ V_0= & {} -\frac{3(g_2+\tilde{g}_2)(g_1-\tilde{g}_1)^2}{\sqrt{g_2\tilde{g}_2}},\nonumber \\ L= & {} \frac{(g_2\tilde{g}_2)^{\frac{1}{4}}}{(\tilde{g}_1-g_1)\sqrt{g_2+\tilde{g}_2}}. \end{aligned}$$(34)
- IV. This critical point is invariant under a smaller symmetry \(SO(4)_{\text {inv}}\) with$$\begin{aligned}&\phi =\ln \left[ \sqrt{\frac{g_1\tilde{g}_1}{g_2\tilde{g}_2}}\frac{g_2+\tilde{g}_2}{g_1+\tilde{g}_1}\right] ,\quad \phi _1=\frac{1}{2}\ln \left[ \frac{g_1}{\tilde{g}_1}\right] ,\nonumber \\&\quad \phi _2=\frac{1}{2}\ln \left[ \frac{g_2}{\tilde{g}_2}\right] ,\nonumber \\&V_0=-\frac{3(g_2+\tilde{g}_2)^2(g_1-\tilde{g}_1)^2}{2\sqrt{g_1\tilde{g}_1g_2\tilde{g}_2}},\nonumber \\&L=\frac{\sqrt{2}(g_1\tilde{g}_1g_2\tilde{g}_2)^{\frac{1}{4}}}{(\tilde{g}_1-g_1)\sqrt{(g_1+\tilde{g}_1)(g_2+\tilde{g}_2)}}. \end{aligned}$$(35)

Scalar masses at the \(N=4\) supersymmetric \(AdS_4\) critical point with \(SO(4)_+\times SO(3)_-\) symmetry and the corresponding dimensions of the dual operators

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

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

\(({\mathbf {1}},{\mathbf {1}},{\mathbf {1}})\) | 4 | 4 |

\(({\mathbf {1}},{\mathbf {1}},{\mathbf {3}})\) | \(0_{\times 3}\) | 3 |

\(({\mathbf {3}},{\mathbf {3}},{\mathbf {1}})\) | \(0_{\times 9}\) | 3 |

\(({\mathbf {1}},{\mathbf {1}},{\mathbf {5}})\) | \(-2_{\times 5}\) | 1, 2 |

\(({\mathbf {1}},{\mathbf {3}},{\mathbf {3}})+({\mathbf {3}},{\mathbf {1}},{\mathbf {3}})\) | \(-2_{\times 18}\) | 1, 2 |

Scalar masses at the \(N=4\) supersymmetric \(AdS_4\) critical point with \(SO(3)\times SO(3)\sim SO(4)_{\text {inv}}\) symmetry and the corresponding dimensions of the dual operators

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

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

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

\(({\mathbf {1}},{\mathbf {5}})+({\mathbf {5}},{\mathbf {1}})\) | \(-2_{\times 10}\) | 1, 2 |

\(({\mathbf {1}},{\mathbf {3}})+({\mathbf {3}},{\mathbf {1}})\) | \(0_{\times 6}\) | 3 |

\(({\mathbf {3}},{\mathbf {3}})\) | \(0_{\times 18}\) | 3 |

### 3.2 Holographic RG flows between \(N=4\) SCFTs

*r*in order to preserve Poincaré symmetry in three dimensions. The BPS conditions coming from setting \(\delta \chi ^i=0\) and \(\delta \lambda ^i_a=0\) require the following projection

*r*-derivative. The superpotential \(\mathcal {W}\) is defined by

*W*as

*A*as functions of \(\phi _1\), we can combine the above BPS equations into

*r*-dependent of \(\phi _1\) and \(\phi _2\) can be obtained in the same way as Eqs. (58) and (62).

### 3.3 RG flows to \(N=4\) non-conformal theory

A consistent truncation of the above \(N=4\) \(SO(4)\times SO(4)\) gauged supergravity is obtained by setting \(\phi _1=\phi _2=0\). In this case, only scalars in the gravity multiplet are present. As previously mentioned, the axion \(\chi \) cannot be turned on simultaneously with \(\phi _1\) and \(\phi _2\).

*A*as functions of \(\chi \), we can combine the BPS equations into

*A*has been neglected. It should also be noted that we must keep the constant \(C\ne 0\) in order to obtain the correct behavior near the \(AdS_4\) critical point as given in Eq. (74).

*r*. However, we are not able to solve for \(\chi \) analytically. We then look for numerical solutions. From Eq. (7778), we see that \(\phi \rightarrow 0\) as \(\chi \rightarrow 0\). This limit, as usual, corresponds to the \(AdS_4\) critical point. We can also see that \(\phi \) is singular at \(\chi _0\) for which \(1-2C\chi _0 -\chi _0^2=0\) or

*SO*(4) flavor symmetry and \(N=4\) Poincaré supersymmetry in three dimensions.

## 4 \(N=4\) \(SO(3,1)\times SO(3,1)\) gauged supergravity

### 4.1 Supersymmetric \(AdS_4\) vacuum

*SO*(3, 1) in

*SO*(3, 3), there are two \(SO(3)\times SO(3)\) singlets corresponding to the non-compact generators

Scalar masses at the \(N=4\) supersymmetric \(AdS_4\) critical point with \(SO(3)\times SO(3)\) symmetry and the corresponding dimensions of the dual operators for \(SO(3,1)\times SO(3,1)\) gauge group

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

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

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

\(({\mathbf {1}},{\mathbf {5}})+({\mathbf {5}},{\mathbf {1}})\) | \(-2_{\times 10}\) | 1, 2 |

\(({\mathbf {1}},{\mathbf {3}})+({\mathbf {3}},{\mathbf {1}})\) | \(0_{\times 6}\) | 3 |

\(({\mathbf {3}},{\mathbf {3}})\) | \(0_{\times 18}\) | 3 |

### 4.2 RG flows without vector multiplet scalars

### 4.3 RG flows with vector multiplet scalars

*r*.

*C*. It can be verified that, in this limit, the scalar potential blows up as \(V\rightarrow \infty \). Therefore, the singularity is unphysical.

*A*depends on the value of

*C*.

A similar analysis shows that the truncation with \(\phi _1=0\) also leads to unphysical singularities. It would be interesting to uplift these solutions to ten or eleven dimensions and determine whether these singularities are resolved.

## 5 \(N=4\) \(SO(4)\times SO(3,1)\) gauged supergravity

*SO*(4) and

*SO*(3, 1) are electrically and magnetically embedded in \(SO(3,3)\times SO(3,3)\), respectively. The corresponding embedding tensor is given by

### 5.1 Supersymmetric \(AdS_4\) vacua

- The first critical point is a trivial one with \(SO(4)\times SO(3)\) symmetry at$$\begin{aligned} \phi =\chi =\phi _1=\phi _2=0,\quad V_0=-6g_2^2. \end{aligned}$$(121)
- A non-trivial supersymmetric critical point is given byThis critical point is invariant under a smaller symmetry \(SO(3)\times SO(3)\).$$\begin{aligned} \phi _2= & {} \chi =0,\quad \phi _1=\frac{1}{2}\ln \left[ \frac{g_1}{g_1+g_2}\right] ,\nonumber \\ \phi= & {} \frac{1}{2}\ln \left[ \frac{4g_1(g_1+g_2)}{(2g_1+g_2)^2}\right] ,\quad V_0=-\frac{3g_2^2(2g_1+g_2)}{\sqrt{g_1(g_1+g_2)}}.\nonumber \\ \end{aligned}$$(122)

Scalar masses at the \(N=4\) supersymmetric \(AdS_4\) critical point with \(SO(4)\times SO(3)\) symmetry and the corresponding dimensions of the dual operators for \(SO(4)\times SO(3,1)\) gauge group

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

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

\(({\mathbf {1}},{\mathbf {1}},{\mathbf {1}})\) | 4 | 4 |

\(({\mathbf {1}},{\mathbf {1}},{\mathbf {3}})\) | \(0_{\times 3}\) | 3 |

\(({\mathbf {3}},{\mathbf {3}},{\mathbf {1}})\) | \(0_{\times 9}\) | 3 |

\(({\mathbf {1}},{\mathbf {1}},{\mathbf {5}})\) | \(-2_{\times 5}\) | 1, 2 |

\(({\mathbf {1}},{\mathbf {3}},{\mathbf {3}})+({\mathbf {3}},{\mathbf {1}},{\mathbf {3}})\) | \(-2_{\times 18}\) | 1, 2 |

Scalar masses at the \(N=4\) supersymmetric \(AdS_4\) critical point with \(SO(3)\times SO(3)\) symmetry and the corresponding dimensions of the dual operators for \(SO(4)\times SO(3,1)\) gauge group

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

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

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

\(({\mathbf {1}},{\mathbf {5}})+({\mathbf {5}},{\mathbf {1}})\) | \(-2_{\times 10}\) | 1, 2 |

\(({\mathbf {1}},{\mathbf {3}})+({\mathbf {3}},{\mathbf {1}})\) | \(0_{\times 6}\) | 3 |

\(({\mathbf {3}},{\mathbf {3}})\) | \(0_{\times 18}\) | 3 |

### 5.2 Holographic RG flow

In this section, we will give a supersymmetric RG flow solution interpolating between the two \(AdS_4\) vacua identified above. As in the previous cases, turning on vector multiplet scalars requires the vanishing of the axion \(\chi \). Since we are only interested in the solution interpolating between two \(AdS_4\) vacua, we will accordingly set \(\chi =0\) from now on.

*r*by the relation \(\frac{d\tilde{r}}{dr}=e^{\frac{\phi }{2}}\).

This solution preserves \(N=4\) supersymmetry in three dimensions and describes an RG flow from \(N=4\) SCFT in the UV with \(SO(4)\times SO(3)\) symmetry to another \(N=4\) SCFT in the IR with \(SO(3)\times SO(3)\) symmetry at which the operator dual to \(\phi _1\) is irrelevant. Although the number of supersymmetry is unchanged, the flavor symmetry *SO*(3) in the UV is broken by the operator dual to \(\phi _1\). We can also truncate out the vector multiplet scalars and study supersymmetric RG flows to non-conformal field theories as in the previous cases. However, we will not consider this truncation since it leads to similar structure as in the previous two gauge groups.

## 6 Conclusions and discussions

We have studied dyonic gaugings of \(N=4\) supergravity coupled to six vector multiplets with compact and non-compact gauge groups \(SO(4)\times SO(4)\), \(SO(3,1)\times SO(3,1)\) and \(SO(4)\times SO(3,1)\). We have identified a number of supersymmetric \(N=4\) \(AdS_4\) vacua within these gauged supergravities and studied several RG flows interpolating between these vacua. The solutions describe supersymmetric deformations of the dual \(N=4\) SCFTs with different flavor symmetries in three dimensions. These deformations are driven by relevant operators of dimensions \(\Delta =1,2\) which deform the UV SCFTs to other SCFTs or to non-conformal field theories in the IR.

For \(SO(4)\times SO(4)\) gauge group, there are four supersymmetric \(AdS_4\) vacua with \(SO(4)\times SO(4)\), \(SO(4)\times SO(3)\), \(SO(3)\times SO(4)\) and *SO*(4) symmetries. These vacua should correspond to \(N=4\) conformal fixed points of \(N=4\) CSM theories with *SO*(4), *SO*(3) and no flavor symmetries, respectively. We have found various RG flows interpolating between these critical points including RG flows connecting three critical points or a cascade of RG flows. These should be useful in holographic studies of \(N=4\) CSM theories.

In the case of non-compact \(SO(3,1)\times SO(3,1)\) gauge group, we have found only one supersymmetric \(AdS_4\) vacuum with \(SO(3)\times SO(3)\) symmetry. We have given a number of RG flow solutions describing supersymmetric deformations of the dual \(N=4\) SCFT to \(N=4\) non-conformal field theories. The solutions with only scalar fields from the gravity multiplet non-vanishing give rise to physical singularities. Flows with vector multiplet scalars turned on, however, lead to physically unacceptable singularities. The mixed gauge group \(SO(4)\times SO(3,1)\) also exhibits similar structure of vacua and RG flows with two supersymmetric \(AdS_4\) critical points.

Given our solutions, it is interesting to identify their higher dimensional origins in ten or eleven dimensions. Along this line, the result of [36, 37] on \(S^3\times S^3\) compactifications could be a useful starting point for the \(SO(4)\times SO(4)\) gauge group. The uplifted solutions would be desirable for a full holographic study of \(N=4\) CSM theories. This should provide an analogue of the recent uplifts of the GPPZ flow describing a massive deformation of \(N=4\) SYM [38, 39]. The embedding of the non-compact gauge groups \(SO(3,1)\times SO(3,1)\) and \(SO(4)\times SO(3,1)\) would also be worth considering.

Another direction is to find interpretations of the solutions given here in the dual \(N=4\) CSM theories with different flavor symmetries similar to the recent study in [18] for AdS\(_5\)/CFT\(_4\) correspondence. The results found here is also in line with [18]. In particular, scalars in the gravity multiplet are dual to relevant operators at all critical points. These operators are in the same multiplet as the energy-momentum tensor. Another result is the exclusion between the operators dual to the axion and vector multiplet scalars which cannot be turned on simultaneously as required by supersymmetry in the gravity solutions. It would be interesting to find an analogous result on the field theory side.

A generalization of the present results to include more active scalars with smaller residual symmetries could provide more general holographic RG flow solutions in particular flows that break some amount of supersymmetry. We have indeed performed a partial analysis for \(SO(3)_{\text {inv}}\) scalars. In this case, there are six singlets. It seems to be possible to have solutions that break supersymmetry from \(N=4\) to \(N=1\), but the scalar potential takes a highly complicated form. Therefore, we refrain from presenting it here. Solutions from other gauge groups more general than those considered here also deserve investigations. Finally, finding other types of solutions such as supersymmetric Janus and flows across dimensions to \(AdS_2\times \Sigma _2\), with \(\Sigma _2\) being a Riemann surface, would also be useful in the holographic study of defect SCFTs and black hole physics. Recent works along this line include [20, 40, 41, 42, 43, 44, 45].

## Notes

### Acknowledgements

This work is supported by The 90th Anniversary of Chulalongkorn University Fund (Ratchadaphiseksomphot Endowment Fund) and the Graduate School, Chulalongkorn University. P. K. is also supported by The Thailand Research Fund (TRF) under Grant RSA5980037.

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