Holographic RG flows and Janus solutions from matter-coupled N=4 gauged supergravity
HHolographic RG flows and Janussolutions from matter-coupled N = 4 gauged supergravity Parinya Karndumri
String Theory and Supergravity Group, Department of Physics, Faculty ofScience, Chulalongkorn University, 254 Phayathai Road, Pathumwan, Bangkok10330, ThailandE-mail: [email protected]
Abstract
We study an SO (2) × SO (2) × SO (2) × SO (2) truncation of four-dimensional N = 4 gauged supergravity coupled to six vector multipletswith SO (4) × SO (4) gauge group and find a new class of holographic RGflows and supersymmetric Janus solutions. In this truncation, there is aunique N = 4 supersymmetric AdS vacuum dual to an N = 4 SCFT inthree dimensions. In the presence of the axion, the RG flows generallypreserve N = 2 supersymmetry while the supersymmetry is enhanced to N = 4 for vanishing axion. We find solutions interpolating between the AdS vacuum and singular geometries with different residual symmetries.We also show that all the singularities are physically acceptable within theframework of four-dimensional gauged supergravity. Accordingly, the so-lutions are holographically dual to RG flows from the N = 4 SCFT to anumber of non-conformal phases in the IR. We also find N = 4 and N = 2Janus solutions with SO (4) × SO (4) and SO (2) × SO (2) × SO (3) × SO (2)symmetries, respectively. The former is obtained from a truncation of allscalars from vector multiplets and can be regarded as a solution of pure N = 4 gauged supergravity. On the other hand, the latter is a genuinesolution of the full matter-coupled theory. These solutions describe con-formal interfaces in the N = 4 SCFT with N = (4 ,
0) and N = (2 , a r X i v : . [ h e p - t h ] F e b Introduction
Over the past twenty years, solutions of gauged supergravities in different space-time dimensions have provided a large number of useful holographic descriptionsfor strongly coupled dual field theories. Among various types of these solutions,holographic RG flows and Janus solutions are of particular interest since theoriginal proposal of the AdS/CFT correspondence [1, 2, 3]. These explain defor-mations of the dual conformal field theories (CFT) and the presence of conformalinterfaces or defects within the parent CFTs. The most useful results along thisdirection arise when there is some amount of unbroken supersymmetry in whichmany properties at strong coupling are controllable.Although there are AdS/CFT dualities in many space-time dimensions,AdS /CFT correspondence attracts much attention due to the relevance indescribing world-volume dynamics of M2-branes, the fundamental degrees offreedom in M-theory. In this case, four-dimensional gauged supergravities arevery useful in obtaining holographic solutions of interest which in some casescan be uplifted to ten or eleven dimensions. A number of holographic RGflows and Janus configurations have previously been found in gauged super-gravities with various gauge groups and different numbers of supersymmetries[4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26], forsimilar solutions with other space-time dimensions, see [27, 28, 29, 30, 31, 32, 33,34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44] for an incomplete list.In this paper, we will give a new class of holographic RG flows and su-persymmetric Janus solutions within matter-coupled N = 4 gauged supergravitywith SO (4) × SO (4) gauge group. This gauged supergravity can be constructedby coupling pure N = 4 gauged supergravity to n ≥ n = 6 here. In this gauged supergravity, a numberof supersymmetric AdS vacua and RG flows between them have recently ap-peared in [17]. In that case, the truncation to SO (3) × SO (3) singlet scalars havebeen considered and solutions with only the dilaton and the axion in the gravitymultiplet or solutions with vanishing axion are consistent with supersymmetry.In addition, all solutions given in [17] preserve N = 4 supersymmetry. In thepresent work, we will find more general solutions namely solutions with both theaxion and vector multiplet scalars non-vanishing and breaking the original N = 4supersymmetry to N = 2.We will also find a new class of supersymmetric Janus solutions in N = 4gauged supergravity which has not been considered in [17]. To the best of theauthor’s knowledge, only Janus solutions found in [15] and [16] are obtainedwithin the framework of N = 4 gauged supergravity. All the solutions given in[15] are the so-called singular Janus in the sense that they interpolate betweennon-conformal phases rather than between conformal phases, so these are holo-graphically expected to describe conformal interfaces within non-conformal fieldtheories in three dimensions. On the other hand, regular Janus solutions given in216] only involve the dilaton and axion from the gravity multiplet. There is also asingular Janus solution found in [16] that involve only scalars from vector multi-plets. Therefore, we believe that the N = 2 Janus solution obtained in this paperis the first regular Janus solution in the context of matter-coupled N = 4 gaugedsupergravity that involves scalars from both gravity and vector multiplets.The paper is organized as follows. In section 2, we review the generalstructure of N = 4 gauged supergravity coupled to vector multiplets in the embed-ding tensor formalism and at the end focus on the case of six vector multiplets with SO (4) × SO (4) gauge group. The truncation to SO (2) × SO (2) × SO (2) × SO (2)singlet scalars, AdS vacua and general structure of relevant BPS equations inboth RG flow and Janus solutions are also given in detail. This sets up theframework for finding solutions in subsequent sections. In sections 3 and 4, anumber of N = 4 and N = 2 holographic RG flow and Janus solutions are given,respectively. We end the paper by giving some conclusions and comments on theresults in section 5. N = 4 gauged supergravity In four dimensions, N = 4 supersymmetry allows two types of supermultiplets,the gravity and vector multiplets. The former contains the following field content( e ˆ µµ , ψ iµ , A mµ , χ i , τ ) (1)which are given by the graviton e ˆ µµ , four gravitini ψ iµ , six vectors A mµ , four spin- fields χ i and one complex scalar τ . The scalar consists of the dilaton φ and theaxion χ and can be parametrized by SL (2 , R ) /SO (2) coset. Indices µ, ν, . . . =0 , , , µ, ˆ ν, . . . = 0 , , , m, n = 1 , . . . , i, j = 1 , , , SO (6) R and SU (4) R R-symmetry.The component fields in a vector multiplet are given by a vector field A µ ,four gaugini λ i and six scalars φ m . In general, the gravity multiplet can coupleto an arbitrary number n of vector multiplets. These vector multiplets will belabeled by indices a, b = 1 , . . . , n , so all fields in the vector multiplets will carryan additional index in the form ( A aµ , λ ia , φ ma ) . (2)Similar to the dilaton and the axion in the gravity multiplet, the 6 n scalar fields φ ma can be parametrized by SO (6 , n ) /SO (6) × SO ( n ) coset.Fermionic fields and supersymmetry parameters transforming in funda-mental representation of SU (4) R ∼ SO (6) R are subject to the chirality projec-tions γ ψ iµ = ψ iµ , γ χ i = − χ i , γ λ i = λ i (3)3hile those transforming in anti-fundamental representation of SU (4) R satisfy γ ψ µi = − ψ µi , γ χ i = χ i , γ λ i = − λ i . (4)In general, all possible gaugings of a supergravity theory can be describedby the embedding tensor. For the matter-coupled N = 4 supergravity, supersym-metry requires that the embedding tensor can have only two non-vanishing com-ponents denoted by ξ αM and f αMNP . Indices α = (+ , − ) and M, N = ( m, a ) =1 , . . . , n + 6 describe fundamental representations of the global SL (2 , R ) and SO (6 , n ) symmetries, respectively. The electric vector fields A + M = ( A mµ , A aµ ),appearing in the ungauged Lagrangian with the usual Yang-Mills kinetic term,together with their magnetic dual A − M form a doublet under SL (2 , R ) denotedby A αM . In addition, the embedding tensor needs to satisfy the quadratic con-straint in order to define a consistent gauging. This ensures that the resultinggauge generators form a closed subalgebra of SL (2 , R ) × SO (6 , n ). We will notdiscuss all these details here but refer to the original construction in [45], see also[46, 47, 48] for an earlier construction.Consistent gauge groups can generally be embedded in both SL (2 , R ) and SO (6 , n ) factors, and the magnetic vector fields can also participate in the gaug-ing. However, each magnetic vector field must be accompanied by an auxiliarytwo-form field in order to remove the extra degrees of freedom. The analysis ofthe existence of maximally supersymmetric AdS vacua given in [49], see also[47, 48] for an earlier result, requires the gaugings to involve both electric andmagnetic vector fields with the corresponding gauge groups embedded solely in SO (6 , n ). This implies that both electric and magnetic components of f αMNP must be non-vanishing and ξ αM = 0. We also note that the magnetic compo-nents f − MNP are related to the duality phases first introduced in [47, 48].In this paper, we are interested in supersymmetric solutions that areasymptotic to
AdS vacua and involve only the metric and scalar fields. Accord-ingly, from now on, we will set all the other fields including ξ αM to zero. Thescalar coset SL (2 , R ) /SO (2) × SO (6 , n ) /SO (6) × SO ( n ) can be described by thecoset representative V α and V AM , respectively. With the definition τ = χ + ie φ , (5)we will choose the explicit form of V α to be V α = e φ (cid:18) χ + ie φ (cid:19) . (6)The coset representative V AM transforms under the global SO (6 , n ) and local SO (6) × SO ( n ) by left and right multiplications, respectively. This implies thesplitting of the index A = ( m, a ). We can then write the SO (6 , n ) /SO (6) × SO ( n )coset representative as V AM = ( V mM , V aM ) . (7)4urthermore, as an element of SO (6 , n ), the matrix V AM also satisfies the relation η MN = −V mM V mN + V aM V aN (8)with η MN = diag( − , − , − , − , − , − , , . . . ,
1) being the SO (6 , n ) invarianttensor. With ξ αM = 0 and only the metric and scalars non-vanishing, the bosonicLagrangian can simply be written as e − L = 12 R + 116 ∂ µ M MN ∂ µ M MN − τ ) ∂ µ τ ∂ µ τ ∗ − V (9)where e = √− g is the vielbein determinant. The scalar potential is given interms of the scalar coset representative and the embedding tensor by V = 116 (cid:20) f αMNP f βQRS M αβ (cid:20) M MQ M NR M P S + (cid:18) η MQ − M MQ (cid:19) η NR η P S (cid:21) − f αMNP f βQRS (cid:15) αβ M MNP QRS (cid:21) . (10)At this point, we should note that the components f αMNP of the embeddingtensor include the gauge coupling constants.The SO (6) × SO ( n ) invariant and symmetric matrix M MN is defined by M MN = V mM V mN + V aM V aN (11)with its inverse denoted by M MN . The tensor M MNP QRS is obtained from raisingindices of M MNP QRS given by M MNP QRS = (cid:15) mnpqrs V mM V nN V pP V qQ V rR V sS . (12)Similarly, M αβ is the inverse of the symmetric 2 × M αβ defined by M αβ = Re( V α V ∗ β ) . (13)Fermionic supersymmetry transformations are given by δψ iµ = 2 D µ (cid:15) i − gA ij γ µ (cid:15) j , (14) δχ i = − (cid:15) αβ V α D µ V β γ µ (cid:15) i − igA ij (cid:15) j , (15) δλ ia = 2 i V Ma D µ V ijM γ µ (cid:15) j − igA i aj (cid:15) j . (16)The fermion shift matrices are in turn defined by A ij = (cid:15) αβ ( V α ) ∗ V Mkl V ikN V jlP f NPβM ,A ij = (cid:15) αβ V α V Mkl V ikN V jlP f NPβM ,A j ai = (cid:15) αβ V α V aM V ikN V jkP f PβMN (17)5here V ijM is defined in terms of the ’t Hooft symbols G ijm and V mM as V ijM = 12 V mM G ijm . (18)Similarly, the inverse elements V ijM can be written as V ijM = − V mM ( G ijm ) ∗ . (19) G ijm satisfy the relations G mij = ( G ijm ) ∗ = 12 (cid:15) ijkl G klm . (20)An explicit representation of these matrices can be chosen as G ij = − − , G ij = − − ,G ij = − − , G ij = i − i − i i ,G ij = i
00 0 0 i − i − i , G ij = i − i i − i . (21)Finally, we also note that the scalar potential can be written in terms ofthe fermion shift matrices A and A as V = − A ij A ij + 19 A ij A ij + 12 A j ai A i a j . (22)It is useful to remark here that upper and lower i, j, . . . indices are related bycomplex conjugation. SO (4) × SO (4) gauge group and SO (2) × SO (2) × SO (2) × SO (2) truncation We now consider the case of n = 6 vector multiplets with SO (4) × SO (4) gaugegroup as constructed in [50]. With a slightly different notation for various gaugecoupling constants, the corresponding embedding tensor has the following non-vanishing components f + ˆ m ˆ n ˆ p = g (cid:15) ˆ m ˆ n ˆ p , f +ˆ a ˆ b ˆ c = ˜ g (cid:15) ˆ a ˆ b ˆ c ,f − ˜ m ˜ n ˜ p = g (cid:15) ˜ m ˜ n ˜ p , f − ˜ a ˜ b ˜ c = ˜ g (cid:15) ˜ a ˜ b ˜ c , (23)6n which we have used the convention on the SO (6 ,
6) fundamental index as M = ( m, a ) = ( ˆ m, ˜ m, ˆ a, ˜ a ) with ˆ m = 1 , ,
3, ˜ m = 4 , ,
6, ˆ a = 7 , , a = 10 , ,
12. We also note that the two SO (4) factors are electrically andmagnetically embedded in SO (6 ,
6) and will be denoted by SO (4) + × SO (4) − .In terms of the SO (3) factors given by the embedding tensor in (23), we willwrite the gauge group as SO (3) + × SO (3) − × SO (3) + × SO (3) − with the firsttwo factors embedded in the SU (4) R ∼ SO (6) R .To parametrize the SO (6 , /SO (6) × SO (6) coset representative, we firstdefine SO (6 , n ) generators in the fundamental representation by( t MN ) QP = 2 δ Q [ M η N ] P . (24)We then identify the SO (6 ,
6) non-compact generators as Y ma = t m,a +6 . (25)The truncation of this gauged supergravity to SO (4) diag ∼ SO (3) diag × SO (3) diag singlet scalars has already been studied in [17] in which a number of supersym-metric AdS critical points and domain walls interpolating between them havebeen given.In the present case, we are interested in another truncation to SO (2) × SO (2) × SO (2) × SO (2) invariant scalars. For later convenience, we also notethat the gauge generators are given by X αM = f αM NP t NP . (26)Each SO (2) factor is embedded in the SO (3) factor of SO (3) + × SO (3) − × SO (3) + × SO (3) − ∼ SO (4) + × SO (4) − as → + . To identify these singlets,we first consider the transformation of the scalars under the SO (2) × SO (2) × SO (2) × SO (2) subgroup. The 36 scalars transform under the compact subgroup SO (6) × SO (6) as ( , ). The fundamental representation of SO (6) is in turndecomposed as ( , ) + ( , ) under SO (3) × SO (3). With the aforementionedembedding of SO (2) in SO (3), we find the transformation of all scalars under SO (2) × SO (2) × SO (2) × SO (2)( , ) → ( + , + , , )+( + , , , + )+( , + , + , )+( , , + , + )(27)from which we can easily see that there are four singlets corresponding to therepresentation ( , , , ). With the gauge generators obtained from the embed-ding tensor given in (23), we can choose the SO (2) × SO (2) × SO (2) × SO (2)generators to be X +3 , X − , X +9 and X − . We then identify non-compact gen-erators corresponding to these singlets as Y , Y , Y and Y in term of whichthe coset representative can be written as V = e φ Y e φ Y e φ Y e φ Y . (28)7ogether with the dilaton and axion from the gravity multiplet which are SO (4) × SO (4) singlets, there are six scalars in the SO (2) × SO (2) × SO (2) × SO (2) sector.By a straightforward computation, we find a very simple scalar potential V = − e − φ ( g + e φ g + g χ ) − g g cosh φ cosh φ cosh φ cosh φ . (29)It can be readily verified that this potential admits a unique AdS critical pointat φ = ln (cid:20) g g (cid:21) and φ = φ = φ = φ = χ = 0 . (30)The cosmological constant and AdS radius are given by V = − g g and L = (cid:114) − V = 1 √ g g . (31)This is the maximally N = 4 supersymmetric AdS vacuum preserving the full SO (4) × SO (4) gauge symmetry. By shifting the dilaton φ by a constant, we canchoose g = g = g which make the dilaton vanish at the AdS critical point. It isuseful to note that all scalars have mass m L = − , N = 4 SCFT, see more detail in[17]. We are interested in two types of supersymmetric solutions in the forms of holo-graphic RG flows and Janus configurations. The former is described by the stan-dard (flat) domain wall with the metric ds = e A ( r ) dx , + dr (32)while the latter takes the form of a curved domain wall with an AdS -slicedworld-volume corresponding to the metric ds = e A ( r ) ( e ξ(cid:96) dx , + dξ ) + dr . (33)It should be noted that the Janus ansatz (33) is obtained from the domain wallmetric (32) by replacing the three-dimensional Minkowski metric dx , by themetric on AdS with radius (cid:96) . The flat domain wall is in turn recovered from theJanus metric (33) in the limit (cid:96) → ∞ .To preserve the isometry of dx , or the AdS metric, scalar fields candepend only on the radial coordinate r . In the SO (2) × SO (2) × SO (2) × SO (2)truncation, the kinetic term for all the six scalars Φ r = ( φ, χ, φ , φ , φ , φ ), with8 , s = 1 , , . . . ,
6, takes the form L kin = 12 G rs Φ r (cid:48) Φ s (cid:48) = −
14 ( φ (cid:48) + e − φ χ (cid:48) ) −
116 [6 + cosh 2( φ − φ )+ cosh 2( φ + φ ) + 2 cosh 2 φ (cosh 2 φ cosh 2 φ − φ (cid:48) − cosh φ cosh φ sinh φ sinh φ φ (cid:48) φ (cid:48) − cosh φ cosh φ sinh φ sinh φ φ (cid:48) φ (cid:48) + sinh φ sinh φ φ (cid:48) φ (cid:48) −
12 cosh φ φ (cid:48) −
12 cosh φ φ (cid:48) − φ (cid:48) (34)in which we have introduced a symmetric matrix G rs . Throughout the paper, wewill denote the r -derivative by (cid:48) .We now analyze the BPS equations obtained by setting fermionic super-symmetry transformations to zero. We use Majorana representation for space-time gamma matrices with all γ ˆ µ real and γ purely imaginary. This implies thatleft and right chiralities of fermions are simply related to each other by complexconjugation. To proceed, we first note that for both types of solutions, the varia-tions δχ i and δλ ia contain γ ˆ r matrix due to the r -dependence of scalar fields. Wethen impose the following projector γ ˆ r (cid:15) i = e i Λ (cid:15) i (35)with an r -dependent phase Λ. This projector relates the two chiralities of (cid:15) i andbreaks half of the supersymmetry.To complete the analysis, we consider the variation of the gravitini δψ iµ .With the coset representative (28), we obtain the A ij tensor of the form A ij = diag( α + , α − , α − , α + ) (36)with α ± = 34 e − φ (cid:2) ( g e φ cosh φ ± ig sinh φ sinh φ ) cosh φ + g cosh φ (cosh φ ± i sinh φ sinh φ ) ++ ig χ cosh φ cosh φ ] . (37)For χ = 0, α + and α − are complex conjugate of each other. The eigenvalue of A ij tensor corresponding to unbroken supersymmetry will give a superpotential interm of which the scalar potential can be written. From the structure of A ij givenabove, we find that in general, solutions within the SO (2) × SO (2) × SO (2) × SO (2)truncation will preserve at most N = 2 supersymmetry corresponding to (cid:15) , or (cid:15) , depending on which eigenvalue among α + and α − is chosen to be thesuperpotential.By defining the superpotential W as W = 23 α (38)9nd taking into account the projector (35), we can write the variations δψ iµ for µ = 0 , , A (cid:48) e i Λ − W = 0 . (39)This condition is satisfied along (cid:15) , and (cid:15) , for W = W ± = α ± , respectively.In the following analysis, we will consider only W = W + . The final results showthat the other choice W = W − is related to this by a sign change in χ , so thetwo options are equivalent. Equation (39) implies that A (cid:48) = ± W = ±|W| and e i Λ = ± W W . (40)Using the explicit phase factor e i Λ in term of W in δχ i and δλ ia equations, weobtain the BPS equations for scalar fields. In what follows, we will choose theupper sign choice in order to relate the AdS vacuum to the limit r → ∞ .Finally, the variation δψ ir leads to2 (cid:15) ˆ i (cid:48) − W γ ˆ r (cid:15) ˆ i = 0 , ˆ i = 1 , (cid:15) ˆ i = e A (cid:15) ˆ i (42)for constant spinors (cid:15) ˆ i satisfying (35).For Janus solutions, we can perform a similar analysis with some modifi-cations. The variations δψ ˆ i , and δψ ˆ iξ lead to the following conditions (cid:18) A (cid:48) γ ˆ r + e − A (cid:96) γ ˆ ξ (cid:19) (cid:15) ˆ i − W (cid:15) ˆ i = 0 , (43)2 ∂ ξ (cid:15) ˆ i + A (cid:48) e A γ ˆ ξ γ ˆ r (cid:15) ˆ i − e A W γ ˆ ξ (cid:15) ˆ i = 0 . (44)The analysis closely follows that given in [19] to which we refer for more detail.Equation (43) implies the integrability condition A (cid:48) + e − A (cid:96) = W . (45)Using equation (43) in (44), we find2 ∂ ξ (cid:15) ˆ i = 1 (cid:96) (cid:15) ˆ i (46)which gives (cid:15) ˆ i = e ξ (cid:96) ˜ (cid:15) ˆ i (47)with ˜ (cid:15) ˆ i being ξ -independent spinors.It should be noted that condition (43) takes the form of a projector on10 ˆ i which must be compatible with the projector (35) used in the scalar flowequations. Following [19], we impose the projector γ ˆ ξ (cid:15) ˆ i = iκe i Λ (cid:15) ˆ i (48)with κ = 1. Using equations (35) and (48) in (43), we arrive at (cid:18) A (cid:48) + iκ(cid:96) e − A (cid:19) e i Λ = W (49)which can be used to determine the phase e i Λ in terms of W and A . We noteagain that in the limit (cid:96) → ∞ , the phase factor reduces to that given in (40).Finally, with all the previous results, the condition δψ ˆ ir = 0 gives2 ∂ r (cid:15) ˆ i = A (cid:48) (cid:15) ˆ i + iκ(cid:96) e − A (cid:15) ˆ i . (50)By redefining the Killing spinors with the phase factor e i Λ2 , this condition impliesthe following form of the Killing spinors (cid:15) ˆ i = e A + r (cid:96) + i Λ2 (cid:15) ˆ i (51)in which (cid:15) ˆ i can include an r -dependent phase and satisfy the γ ˆ r and γ ˆ ξ projectorswithout the phase e i Λ γ ˆ r (cid:15) ˆ i = (cid:15) i and γ ˆ ξ (cid:15) ˆ i = iκ(cid:15) i . (52)We also note that the constant κ = ± AdS slice.With all these results, we will look for supersymmetric RG flow and Janussolutions within the SO (2) × SO (2) × SO (2) × SO (2) truncation of the SO (4) × SO (4) N = 4 gauged supergravity. We are now in a position to present an explicit form of the BPS equations givenin the previous section and look for possible solutions. In this section, we considerRG flow solutions. Since there is only one supersymmetric
AdS critical pointdual to a single conformal fixed point, the solutions will describe RG flows fromthe dual N = 4 SCFT to non-conformal phases in the IR. This type of solutionsin the case of pure N = 4 gauged supergravity, with all scalars but the dilatonand axion vanishing, has already been studied in [17]. In this paper, we will findsolutions in the matter-coupled N = 4 gauged supergravity.As previously mentioned, the flow equations for scalars can be obtained11y using the projector (35) and the phase e i Λ given in (40). Within the SO (2) × SO (2) × SO (2) × SO (2) truncation, the variation δχ i reduces to a single complexequation which can be straightforwardly solved for φ (cid:48) and χ (cid:48) . However, it turnsout that consistency of the equations from the variation δλ ia requires some scalarsto vanish. This is similar to the SO (3) × SO (3) truncation studied in [17] in whichthe BPS equations from δλ ia imply χ = 0, and non-vanishing χ is possible onlywhen all scalars from the vector multiplets are set to zero. In that case, the axionand vector multiplet scalars cannot be turned on simultaneously.In the present case, the constraint is somewhat weaker. We find that aconsistent set of BPS equations that are compatible with the second-order fieldequations can be obtained only for two of the scalars φ i , i = 1 , , , N = 2 supersymmetric solutions. In addition, if the axion χ vanishes, supersymmetry is enhanced to N = 4. As expected, truncating outall scalars from the vector multiplets also leads to N = 4 supersymmetry. Wethen see that in the presence of both axion and vector multiplet scalars, thesolutions break some supersymmetry. We also point out that this is not possiblein the SO (3) × SO (3) truncation considered in [17] since in that case the A ij tensor is proportional to the identity matrix. Therefore, partial supersymmetrybreaking is not possible there. N = 4 holographic RG flow with SO (2) × SO (3) × SO (2) × SO (2) symmetry We first consider N = 4 solutions with χ = 0 and SO (2) × SO (3) × SO (2) × SO (2)symmetry. In this case, the SO (4) subgroup of the SO (6) R-symmetry is brokento SO (2) × SO (3) while the remaining SO (2) × SO (2) factor is a subgroup of SO (3) × SO (3) ∼ SO (4) symmetry of the vector multiplets. There are twopossibilities namely setting φ = φ = 0 or φ = φ = 0. The two choices arerelated to each other by interchanging e φ with e − φ factors. Therefore, we willconsider only the solution with φ = φ = 0.The superpotential is real and given by W = W + = W − = 12 ( g e φ + g e − φ cosh φ cosh φ ) . (53)The resulting BPS equations read φ (cid:48) = − ∂W∂φ = g e − φ cosh φ cosh φ − g e φ , (54) φ (cid:48) = − φ ∂W∂φ = − g e − φ sech φ sinh φ , (55) φ (cid:48) = − ∂W∂φ = − g e − φ cosh φ sinh φ , (56) A (cid:48) = W = 12 ( g e φ + g e − φ cosh φ cosh φ ) . (57)12o find the solution, we first take the following combination dφ dφ = csch φ sech φ tanh φ (58)with the solution sinh φ = tan ϕ tanh φ (59)for a constant ϕ . Similarly, taking the combination between φ (cid:48) and φ (cid:48) equationstogether with this result leads to dφdφ = g e φ csch φ g (cid:112) ϕ tanh φ − coth φ . (60)The solution to this equation is given by φ = ln (cid:34) √ g cos ϕ csch φ (cos ϕ csch φ √ cos 2 ϕ + cosh 2 φ − C ) g [1 − C + cos 2 ϕ + 2 cos ϕ (coth φ + cos 2 ϕ csch φ )] (cid:35) . (61)Using these results and redefining the radial coordinate to ρ given by dρdr = e − φ ,we obtain the solution for φ ( ρ ) of the formcosh 2 φ = 2 cos ϕ tanh [ g sec ϕ ( ρ − ρ )] − cos 2 ϕ (62)with an integration constant ρ .Finally, taking a linear combination 2 A (cid:48) + φ (cid:48) , we find2 dAdφ + dφdφ = − e φ + 1) e φ − A = φ − φ − ln(1 − e φ ) . (64)We have neglected an additive integration constant in A since this can be removedby rescaling coordinates on dx , .We now look at asymptotic behaviors of the solution. For convenience,we first set g = g to bring the AdS critical point to the origin of the scalarmanifold. As r → ∞ , we find φ ∼ φ ∼ φ ∼ e − g r ∼ e − rL , A ∼ g r ∼ rL . (65)This gives the supersymmetric AdS vacuum and indicates that all scalars aredual to relevant operators of dimensions ∆ = 1 , ρ . We find that, for C <
0, the above solution is singular at ρ = ρ + ρ ∗ with ρ ∗ = 1 g coth − sec ϕ (cid:118)(cid:117)(cid:117)(cid:116) cos 2 ϕ + cosh (cid:16) − (cid:113) − sec ϕ + C sec ϕ (cid:17) . (66)Near the singularity, we find φ = φ ∗ = coth − (cid:114) − sec ϕ + 12 C sec ϕ ,φ = sinh − (tan ϕ tanh φ ∗ ) , φ ∼ − ln( φ − φ ∗ ) ,A ∼ − φ ∼
12 ln( φ − φ ∗ ) (67)and V ∼ − g e φ → −∞ . (68)We see that near the singularity both φ and φ are constant, and the singularityis physical by the criterion given in [51]. Therefore, the solution describes anRG flow from N = 4 SCFT to a non-conformal phase in the IR with the N = 4superconformal symmetry broken to N = 4 Poincare supersymmetry in threedimensions. The SO (4) × SO (4) symmetry is also broken to SO (2) × SO (3) × SO (2) × SO (2) along the flows. N = 4 holographic RG flow with SO (2) × SO (2) × SO (2) × SO (2) symmetry Unlike the previous case, if we set φ = φ = 0 or φ = φ = 0 along with χ = 0,we obtain N = 4 solutions with only SO (2) × SO (2) × SO (2) × SO (2) symmetry.The two choices give rise to equivalent results with scalars ( φ , φ ) and ( φ , φ )interchanged. We will consider only the φ = φ = 0 case for definiteness.Setting χ = φ = φ = 0, we again find a real superpotential W = 12 ( g e − φ cosh φ + g e φ cosh φ ) . (69)14y repeating the same procedure as in the previous case, we find the followingBPS equations A (cid:48) = W = 12 ( g e − φ cosh φ + g e φ cosh φ ) , (70) φ (cid:48) = − ∂W∂φ = g e − φ cosh φ − g e φ cosh φ , (71) φ (cid:48) = − ∂W∂φ = − g e − φ sinh φ , (72) φ (cid:48) = − ∂W∂φ = − g e φ sinh φ . (73)Unlike the previous case, solving these equations is slightly more complicated dueto the different dilaton prefactors in φ (cid:48) and φ (cid:48) equations. However, with someeffort, we can find an analytic solution to these equations.We first treat φ as an independent variable and take a combination dφ dφ = g g e − φ csch φ sinh φ . (74)By changing to a new variable ϕ defined by φ = ln sinh ϕ − ln cosh ϕ , (75)we can rewrite the above equation as dϕdφ = g g e − φ csch φ . (76)We then combine φ (cid:48) and φ (cid:48) equations with the e − φ factor obtained from (76) andarrive at dφdφ = coth φ + coth ϕ dϕdφ (77)with the solution φ = φ + ln sinh φ + ln sinh ϕ . (78)We can now use this solution in (76) and find the solutioncosh ϕ = g g e − φ coth φ . (79)Taking a linear combination 2 A (cid:48) − φ (cid:48) gives2 dAdφ − dφdφ = − φ (80)with the solution A = 12 φ − ln sinh φ . (81)15inally, changing r to a new coordinate ρ given by dρdr = e φ , we can readilyfind φ ( ρ ) solution φ = ln(1 + e − g ( ρ − ρ ) ) − ln(1 − e − g ( ρ − ρ ) ) . (82)As in the previous case, for r → ∞ , we find φ ∼ φ ∼ φ ∼ e − g r ∼ e − rL (83)in which we have taken g = g for convenience. This is an asymptotic AdS geometry.As ρ → ρ , the solution is singular with φ ∼ constant , φ ∼ ln( ρ − ρ ) ,φ ∼ − ln( ρ − ρ ) , A ∼
12 ln( ρ − ρ ) . (84)Unlike the previous case, the scalars from vector multiplets are not all constantnear the singularity. Using the scalar potential given in (29), we find that, as ρ → ρ , V ∼ − g ( ρ − ρ ) → −∞ . (85)The singularity is again physically acceptable, and the solution could be inter-preted as an RG flow from the N = 4 SCFT to a non-conformal phase in the IRdriven by relevant operators of dimensions ∆ = 1 , N = 2 holographic RG flow with SO (2) × SO (2) × SO (3) × SO (2) symmetry We now consider the most general solutions in the SO (2) × SO (2) × SO (2) × SO (2)truncation with non-vanishing χ and φ = φ = 0. In this case, the SO (4) ⊂ SO (6) R is broken to SO (2) × SO (2) while the SO (4) subgroup of the SO (6)symmetry of the vector multiplets is broken to SO (3) × SO (2), or SO (2) × SO (3)if we choose φ = φ = 0 rather than φ = φ = 0.With φ = φ = 0, the superpotential is given by W = W + = 12 (cid:104) g e − φ (cosh φ + i sinh φ sinh φ ) + g e φ cosh φ + ig e − φ χ cosh φ (cid:105) (86)in which for definiteness, we have chosen W + to be the superpotential with (cid:15) and (cid:15) being the corresponding Killing spinors. If we choose W = W − , the unbrokensupersymmetry will correspond to (cid:15) and (cid:15) .With the same analysis as in the previous cases, we find the following16PS equations A (cid:48) = W = 12 (cid:113) ( g e − φ cosh φ + g e φ cosh φ ) + e − φ ( g sinh φ sinh φ + g cosh φ χ ) , (87) φ (cid:48) = − ∂W∂φ = 18 W e − φ (cid:2) g (3 + cosh 2 φ + 2 cosh 2 φ sinh φ ) − g e φ cosh φ +4 g χ ( g sinh φ sinh 2 φ + g χ cosh φ ) (cid:3) , (88) χ (cid:48) = − e φ ∂W∂χ = − e φ W g cosh φ ( g sinh φ sinh φ + g cosh φ χ ) , (89) φ (cid:48) = − φ ∂W∂φ = − W e − φ g (cid:2) sinh φ ( g cosh φ + g e φ sech φ ) + g χ cosh φ tanh φ (cid:3) , (90) φ (cid:48) = − ∂W∂φ = − W e − φ (cid:2) sinh 2 φ ( g e φ + g χ + g sinh φ )+2 g g e φ cosh φ sinh φ + 2 g g χ cosh 2 φ sinh φ (cid:3) . (91)In this case, we are not able to find analytic solutions, so we will insteadperform a numerical analysis for solving these equations. By choosing the nu-merical values of the coupling constants g = g = √ g , we obtain examples ofnumerical RG flow solutions for suitable boundary conditions and different valuesof the parameter g as shown in figures 1, 2 and 3. From these figures, we see somepatterns namely, in all of these solutions, φ is constant near the singularities asin the previous cases, and all the singularities are physically acceptable. Theformer indeed seems to be a general feature at least among all the solutions wehave found. The latter follows directly from the scalar potential (29) for g = g , V = − g e − φ (1 + e φ + χ + 4 e φ cosh φ cosh φ ) (92)from which we immediately see that V → −∞ for any diverging behaviors ofall the scalar fields. This implies that all possible singularities in this SO (2) × SO (2) × SO (3) × SO (2) truncation are physical. Accordingly, the correspondingsolutions describe RG flows from the dual N = 4 SCFT to various non-conformalphases in the IR. 17 r ϕ ( r ) (a) φ ( r ) solution - r χ ( r ) (b) χ ( r ) solution - r ϕ ( r ) (c) φ ( r ) solution - r ϕ ( r ) (d) φ ( r ) solution - r - - - - - A ( r ) (e) A ( r ) solution - r - - - - - (f) Scalar potential Figure 1: An N = 2 RG flow from the N = 4 SCFT in the UV to a non-conformalphase in the IR for g = g = √ . 18 r ϕ ( r ) (a) φ ( r ) solution r - χ ( r ) (b) χ ( r ) solution r - - - - - ϕ ( r ) (c) φ ( r ) solution r - - - - - ϕ ( r ) (d) φ ( r ) solution r A ( r ) (e) A ( r ) solution r - - - - (f) Scalar potential Figure 2: An N = 2 RG flow from the N = 4 SCFT in the UV to a non-conformalphase in the IR for g = g = √ . 19 .0 7.5 8.0 8.5 9.0 9.5 10.0 r ϕ ( r ) (a) φ ( r ) solution r χ ( r ) (b) χ ( r ) solution r - - - - ϕ ( r ) (c) φ ( r ) solution r ϕ ( r ) (d) φ ( r ) solution r A ( r ) (e) A ( r ) solution r - - - (f) Scalar potential Figure 3: An N = 2 RG flow from the N = 4 SCFT in the UV to a non-conformalphase in the IR for g = g = √ . 20 Supersymmetric Janus solutions
In this section, we look at supersymmetric Janus solutions with N = 4 and N = 2 supersymmetries. In this case, the resulting BPS equations are muchmore complicated than those in the RG flow case, and we are not able to obtainany analytic solutions. N = 4 Janus solutions with SO (4) × SO (4) symmetry We first consider solutions with all φ i = 0 that can be regarded as solutions ofpure N = 4 gauged supergravity with SO (4) ∼ SO (3) × SO (3) gauge group. Inthis case, we find a simple superpotential W = 12 e − φ [ g + g ( e φ + iχ )] . (93)Using the prescription given in section 2, we obtain the following BPS equations φ (cid:48) = − A (cid:48) W ∂W∂φ − κ e − A (cid:96)W e φ ∂W∂χ = 2 (cid:96)A (cid:48) ( g − g e φ + g χ ) − κg χe φ − A (cid:96) [( g + g e φ ) + g χ ] , (94) χ (cid:48) = − e φ A (cid:48) W ∂W∂χ + 4 κ e − A (cid:96)W e φ ∂W∂φ = − (cid:96)g A (cid:48) χe φ − κe φ − A ( g − g e φ + g χ ) (cid:96) [( g + g e φ ) + g χ ] , (95) A (cid:48) = 14 e − φ [( g + g e φ ) + g χ ] − e − A (cid:96) . (96)As expected, in the limit (cid:96) → ∞ , these equations reduce to those of the RG flowsstudied in [17] after a redefinition of the scalars and coupling constants. It shouldalso be noted that setting χ = 0 is not consistent unless (cid:96) → ∞ . Therefore, Janussolutions are possible only for non-vanishing pseudoscalars needed to support thecurvature of the AdS -sliced world-volume. This has also been pointed out in[19]. We now perform a numerical analysis and give some examples of super-symmetric Janus solutions as shown in figure 4. These solutions preserve halfof the original N = 4 supersymmetry, or eight supercharges, with all (cid:15) i non-vanishing and subject to the projector (35). On the conformal interfaces, theKilling spinors can have left or right chiralities depending on whether the valuesof κ = 1 or κ = −
1. The two-dimensional interfaces then preserve (4 ,
0) or (0 , κ = 1.21 - r - - - ϕ ( r ) (a) φ ( r ) solutions - - r - - χ ( r ) (b) χ ( r ) solutions - - r A ( r ) (c) A ( r ) solutions Figure 4: N = 4 supersymmetric Janus solutions from N = 4 SO (4) × SO (4)gauged supergravity for g = g = 2 √ (cid:96) = κ = 1 with different boundaryconditions. 22 .2 N = 2 Janus solutions with SO (2) × SO (2) × SO (3) × SO (2) symmetry We now look at more complicated solutions in matter-coupled N = 4 gaugedsupergravity. We first note that Janus solutions are not possible for all the sub-truncations with χ = 0. We then consider the case of N = 2 supersymmetry and SO (2) × SO (2) × SO (3) × SO (2) symmetry with φ = φ = 0 and (cid:15) = (cid:15) = 0.The other equivalent cases can be obtained by replacing ( φ , φ ) and/or ( (cid:15) , (cid:15) )by ( φ , φ ) and/or ( (cid:15) , (cid:15) ).With the superpotential (86) and the same procedure as in the previouscase, we find the following BPS equations φ (cid:48) = − φ A (cid:48) W ∂W∂φ − φ κe − A (cid:96)W ∂W∂φ , (97) φ (cid:48) = − A (cid:48) W ∂W∂φ + 2sech φ κe − A (cid:96)W ∂W∂φ , (98) φ (cid:48) = − A (cid:48) W ∂W∂φ − e φ κe − A (cid:96)W ∂W∂χ , (99) χ (cid:48) = − e φ A (cid:48) W ∂W∂χ + 4 e φ κe − A (cid:96)W ∂W∂φ , (100) A (cid:48) = W − e − A (cid:96) (101)with 1 W ∂W∂φ = 14 W g e − φ cosh φ (cid:2) sinh φ ( g e φ + g cosh φ cosh φ )+ g χ cosh φ sinh φ ] , (102)1 W ∂W∂φ = 14 W (cid:2) e − φ ( g sinh φ sinh φ + g χ cosh φ )( g cosh φ sinh φ + g χ sinh φ ) g sinh φ ( g cosh φ + g e φ cosh φ ) (cid:3) , (103)1 W ∂W∂φ = 18 W (cid:2) g cosh φ ( g cosh φ + g e φ cosh φ ) − e − φ ( g sinh φ sinh φ + g χ cosh φ ) − e − φ ( g cosh φ + g e φ cosh φ ) (cid:3) , (104)1 W ∂W∂χ = 14 W g e − φ cosh φ ( g sinh φ sinh φ + g χ cosh φ ) . (105)Examples of numerical solutions are given in figure 5. In the figure, thesolutions for the warped factor A are very close to each other. In this case, thesolutions describe conformal interfaces with N = (2 ,
0) or N = (0 ,
2) supersym-metries corresponding respectively to κ = 1 or κ = − N = 4 SCFT. In this paper, we have studied a number of holographic RG flows from matter-coupled N = 4 gauged supergravity with SO (4) × SO (4) gauge group by truncat-ing to SO (2) × SO (2) × SO (2) × SO (2) singlet scalars. For vanishing axion, thereexist N = 4 supersymmetric flow solutions with SO (2) × SO (3) × SO (2) × SO (2)and SO (2) × SO (2) × SO (2) × SO (2) symmetries. In the presence of axion, the so-lutions preserve only N = 2 supersymmetry and SO (2) × SO (2) × SO (3) × SO (2)symmetry. All of these solutions are singular at finite values of the radial coor-dinate, but all the singularities turn out to be physically acceptable according tothe criterion given in [51]. The solutions can be interpreted as RG flows from the N = 4 SCFT, dual to the unique N = 4 supersymmetric AdS vacuum withinthis truncation, to non-conformal phases in the IR with different global symme-tries. All of these solutions could hopefully be useful in studying various massdeformations of strongly coupled N = 4 SCFT via AdS/CFT holography.We have also found N = 4 and N = 2 supersymmetric Janus solutionswith SO (4) × SO (4) and SO (2) × SO (2) × SO (3) × SO (2) symmetries. The for-mer is obtained by truncating out all scalars from the vector multiplets resultingin the solutions of pure N = 4 gauged supergravity while the latter is a genuinesolution of the matter-coupled theory. These solutions holographically describetwo-dimensional conformal interfaces in the dual N = 4 SCFT with unbroken N = (4 ,
0) or N = (2 ,
0) supersymmetries on the interfaces. Generally, theseconfigurations are identified with deformations of the dual SCFT by operatorsor vacuum expectation values that depend on the coordinate transverse to theinterfaces.It would be interesting to explicitly identify the dual N = 4 SCFT to-gether with relevant deformations dual to the solutions found here. Uplifting the SO (4) × SO (4) gauged supergravity considered here to higher dimensions willallow to embed these solutions in ten or eleven dimensions giving rise to new ex-amples of AdS/CFT duality in the context of string/M-theory. In particular, itcould be interesting to check whether the singularities allowed by the criterion of[51] are physical in string/M-theory by using the criterion of [52]. If this is indeedthe case, identifying relevant M-brane or D-brane configurations related to thesefour-dimensional solutions also deserves further study. Finally, finding more gen-eral holographic solutions of N = 4 gauged supergravities in other truncations orin other gauge groups could be interesting as well. These solutions might includeholographic solutions describing spatially varying mass terms given in [53] and[54]. We leave all these issues for future investigation.24 - r ϕ ( r ) (a) φ ( r ) solutions - - r - χ ( r ) (b) χ ( r ) solutions - - r ϕ ( r ) (c) φ ( r ) solutions - - r - ϕ ( r ) (d) φ ( r ) solutions - - r A ( r ) (e) A ( r ) solutions Figure 5: N = 2 supersymmetric Janus solutions from N = 4 SO (4) × SO (4)gauged supergravity for g = g = 2 √ (cid:96) = 2 and κ = 1 with different boundaryconditions. 25 cknowledgement This work is supported by The Thailand Research Fund (TRF) under grantRSA6280022. It is a pleasure to thank Patharadanai Nuchino for reading themanuscript.
References [1] J. M. Maldacena, “The large N limit of superconformal field theories andsupergravity”, Adv. Theor. Math. Phys. (1998) 231-252, arXiv: hep-th/9711200.[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge Theory Correlatorsfrom Non-Critical String Theory”, Phys. Lett. B428 (1998) 105-114, arXiv:hep-th/9802109.[3] E. Witten, “Anti De Sitter Space and holography”, Adv. Theor. Math. Phys. (1998) 253-291, arXiv: 9802150.[4] R. Corrado, K. Pilch and N. P. Warner, “An N = 2 supersymmetric mem-brane flow”, Nucl. Phys. B629 (2002) 74-96, arXiv: hep-th/0107220.[5] C. N. Gowdigere and N. P. Warner, “Flowing with Eight Supersymmetriesin M-Theory and F-theory”, JHEP 12 (2003) , arXiv: hep-th/0212190.[6] K. Pilch, A. Tyukov and N. P. Warner, “Flowing to Higher Dimensions: ANew Strongly-Coupled Phase on M2 Branes”, JHEP 11 (2015) , arXiv:1506.01045.[7] C. Ahn and K. Woo, “Supersymmetric Domain Wall and RG Flow from 4-Dimensional Gauged N = 8 Supergravity”, Nucl. Phys. B599 (2001) 83-118,arXiv: hep-th/0011121.[8] C. Ahn and T. Itoh, “An N = 1 Supersymmetric G -invariant Flow in M-theory”, Nucl. Phys. B627 (2002) 45-65, arXiv: hep-th/0112010.[9] N. Bobev, N. Halmagyi, K. Pilch and N. P. Warner, “Holographic, N = 1Supersymmetric RG Flows on M2 Branes”, JHEP 09 (2009) , arXiv:0901.2376.[10] A. Guarino, “On new maximal supergravity and its BPS domain-walls”,JHEP 02 (2014) , arXiv: 1311.0785.[11] J. Tarrio and O. Varela, “Electric/magnetic duality and RG flows in AdS /CF T ”, JHEP 01 (2014) , arXiv: 1311.2933.2612] Y. Pang, C. N. Pope and J. Rong, “Holographic RG Flow in a New SO (3) × SO (3) Sector of ω -Deformed SO (8) Gauged N = 8 Supergravity”, JHEP 08(2015) , arXiv: 1506.04270.[13] P. Karndumri, “Holographic RG flows in N = 3 Chern-Simons-Matter theoryfrom N = 3 4D gauged supergravity”, Phys. Rev. D94 (2016) 045006, arXiv:1601.05703.[14] P. Karndumri and K. Upathambhakul, “Gaugings of four-dimensional N = 3supergravity and AdS /CFT holography”, Phys. Rev. D93 (2016) 125017arXiv: 1602.02254.[15] P. Karndumri, “Supersymmetric deformations of 3D SCFTs from tri-sasakian truncation”, Eur. Phys. J. C (2017) , 130, arXiv: 1610.07983.[16] P. Karndumri and K. Upathambhakul, “Supersymmetric RG flows and Janusfrom type II orbifold compactification”, Eur. Phys. J. C (2017) , 455,arXiv: 1704.00538.[17] P. Karndumri and K. Upathambhakul, ”Holographic RG flows in N = 4SCFTs from half-maximal gauged supergravity”, Eur. Phys. J. C78 (2018)626, arXiv:hep-th/1806.01819.[18] P. Karndumri and C. Maneerat, “Supersymmetric solutions from N = 5gauged supergravity”, Phys. Rev. D101 (2020) 126015, arXiv: 2003.05889.[19] N. Bobev, K. Pilchand N. P. Warner, “Supersymmetric Janus Solutions inFour Dimensions”, JHEP 1406 (2014) , arXiv: 1311.4883.[20] P. Karndumri, “Supersymmetric Janus solutions in four-dimensional N = 3gauged supergravity”, Phys. Rev. D93 (2016) 125012, arXiv: 1604.06007.[21] M. Suh, “Supersymmetric Janus solutions of dyonic
ISO (7)-gauged N = 8supergravity”, JHEP 04 (2018) , arXiv: 1803.00041.[22] N. Kim and S. J. Kim, “Re-visiting Supersymmetric Janus Solutions: APerturbative Construction”, arXiv: 2001.06789.[23] A. Guarino, “On new maximal supergravity and its BPS domain-walls”,JHEP 02 (2014) , arXiv: 1311.0785.[24] J. Tarrio and O. Varela, “Electric/magnetic duality and RG flows inAdS4/CFT3”, JHEP 10 (2014) , arXiv: 1311.2933.[25] P. Karndumri and J. Seeyangnok, “Supersymmetric solutions from N = 6gauged supergravity”, arXiv: 2012.10978.2726] P. Karndumri and C. Maneerat, “Supersymmetric Janus solutions in ω -deformed N = 8 gauged supergravity”, arXiv: 2012.15763.[27] P. Karndumri, “RG flows in 6D N = (1 ,
0) SCFT from SO (4) half-maximal7D gauged supergravity”, JHEP 06 (2014) , arXiv: 1404.0183.[28] P. Karndumri, “Holographic RG flows in six dimensional F(4) gauged super-gravity”, JHEP 01 (2013) , erratum ibid JHEP 06 (2015) , arXiv:1210.8064.[29] P. Karndumri, “Gravity duals of 5D N = 2 SYM from F(4) gauged super-gravity”, Phys. Rev. D90 (2014) 086009, arXiv: 1403.1150.[30] G. Bruno De Luca, A. Gnecchi, G. Lo Monaco and A. Tomasiello, “Holo-graphic duals of 6d RG flows”, JHEP 03 (2019) , arXiv: 1810.10013.[31] L. Girardello, M. Petrini, M. Porrati, and A. Zaffaroni, “Novel local CFTand exact results on perturbations of N = 4 super Yang Mills from AdSdynamics, JHEP 12 (1998) , arXiv: hep-th/9810126.[32] D. Z. Freedman, S. S. Gubser, K. Pilch, and N. P. Warner, “Renormalizationgroup flows from holography supersymmetry and a c theorem”, Adv. Theor.Math. Phys. (1999) 363–417, arXiv: hep-th/9904017.[33] D. Cassani, G. Dall’Agata and A. F. Faedo, “BPS domain walls in N = 4supergravity and dual flows”, JHEP 03 (2013) , arXiv: 1210.8125.[34] N. Bobev, D. Cassani and H. Triendl, “Holographic RG flows for four-dimensional N = 2 SCFTs”, JHEP 06 (2018) , arXiv: 1804.03276.[35] H. L. Dao and P. Karndumri, “Holographic RG flows and AdS black stringsfrom 5D half-maximal gauged supergravity”, Eur. Phys. J. C79 (2019) 137,arXiv: 1811.01608.[36] C. Bachas, J. de Boer, R. Dijkgraaf, and H. Ooguri, “Permeable conformalwalls and holography”, JHEP 06 (2002) , arXiv:hep-th/0111210.[37] C. Bachas and M. Petropoulos, “Anti-de-Sitter D-branes”, JHEP 02 (2001) , arXiv:hep-th/0012234.[38] D. Bak, M. Gutperle and S. Hirano, “Three dimensional Janus and time-dependent black holes”, JHEP 02 (2007) , arXiv: hep-th/0701108.[39] M. Chiodaroli, M. Gutperle and D. Krym, “Half-BPS Solutions locallyasymptotic to
AdS × S and interface conformal field theories”, JHEP 02(2010) , arXiv: 0910.0466. 2840] M. Chiodaroli, E. D’Hoker, Y, Guo and M. Gutperle, “Exact half-BPS string-junction solutions in six-dimensional supergravity”, JHEP 12 (2011) ,arXiv: 1107.1722.[41] D. Bak and H. Min, “Multi-faced Black Janus and Entanglement”, JHEP03 (2014) , arXiv: 1311.5259.[42] E. D’Hoker, J. Estes, M. Gutperle and D. Krym, “Janus solutions in M-theory”, JHEP 06 (2009) , arXiv: 0904.3313.[43] M. Gutperle, J. Kaidi and H. Raj, “Janus solutions in six-dimensional gaugedsupergravity”, JHEP 12 (2017) , arXiv: 1709.09204.[44] K. Chen and M. Gutperle “Janus solutions in three-dimensional N = 8gauged supergravity”, arXiv: 2011.10154.[45] J. Schon and M. Weidner, “Gauged N = 4 supergravities”, JHEP 05 (2006) , arXiv: hep-th/0602024.[46] E. Bergshoeff, I. G. Koh and E. Sezgin, “Coupling of Yang-Mills to N = 4, d = 4 supergravity”, Phys. Lett. B155 (1985) 71-75.[47] M. de Roo and P. Wagemans, “Gauged matter coupling in N = 4 supergrav-ity”, Nucl. Phys. B262 (1985) 644-660.[48] P. Wagemans, “Breaking of N = 4 supergravity to N = 1, N = 2 at Λ = 0,Phys. Lett. B206 (1988) 241.[49] J. Louis and H. Triendl, “Maximally supersymmetric
AdS vacua in N = 4supergravity”, JHEP 10 (2014) , arXiv:1406.3363.[50] D. Roest and J. Rosseel, “De Sitter in Extended Supergravity”, Phys. Lett. B685 (2010) 201-207, arXiv: 0912.4440.[51] S. S. Gubser, “Curvature singularities: the good, the bad and the naked”,Adv. Theor. Math. Phys. (2000) 679-745, arXiv: hep-th/0002160.[52] J. Maldacena and C. Nunez, “Supergravity description of field theories oncurved manifolds and a no go theorem”, Int. J. Mod. Phys. A16 (2001) 822,arXiv: hep-th/0007018.[53] J. P. Gauntlett and C. Rosen, “Susy Q and spatially modulated deformationsof ABJM theory”, JHEP 10 (2018) , arXiv: 1808.02488.[54] I. Arav, J. P. Gauntlett, M. Roberts, C. Rosen, “Spatially modulated andsupersymmetric deformations of ABJM theory”, JHEP 04 (2019)099