Non-semisimple gauging of a magical N=4 supergravity in three dimensions
aa r X i v : . [ h e p - t h ] D ec Preprint typeset in JHEP style - HYPER VERSION arXiv: 1509.07431
Non-semisimple gauging of a magical N = 4 supergravity in three dimensions Parinya Karndumri
String Theory and Supergravity Group, Department of Physics, Faculty of Science,Chulalongkorn University, 254 Phayathai Road, Pathumwan, Bangkok 10330,ThailandE-mail: [email protected]
Abstract:
We construct a new N = 4 non-semisimple gauged supergravity in threedimensions with E /SU (6) × SU (2) scalar manifold and SO (4) ⋉ T gauge group.Depending on the values of the gauge coupling constants, the theory admits boththe maximally supersymmetric AdS vacuum preserving SO (4) gauge symmetry andhalf-supersymmetric domain walls with unbroken SO (4) symmetry. We give all scalarmasses at the supersymmetric AdS critical point corresponding to an N = (4 ,
0) su-perconformal field theory (SCFT) in two dimensions. The scalar potential also admitstwo flat directions corresponding to marginal deformations that preserve full super-symmetry and conformal symmetry. This SO (4) ⋉ T gauged supergravity is expectedto arise from a dimensional reduction on a three-sphere of the minimal N = (1 , Keywords:
AdS-CFT correspondence, Gauge/Gravity Correspondence andSupergravity Models, Supersymmetric Effective Theories. . Introduction
Gauged supergravities in various dimensions play an important role in many aspects ofstring/M theory in particular the AdS/CFT correspondence [1]. Unlike higher dimen-sional analogues, gauged supergravity in three dimensions has a much richer structuredue to the duality between vectors and scalars. Since three-dimensional supergravityfields are topological, the propagating bosonic degrees of freedom of the matter-coupledsupergravity can be described entirely in terms of scalar fields. The resulting theory isa supersymmetric non-linear sigma model coupled to supergravity. All of these theorieswith different numbers ( N ) of supersymmetries have been classified in [2]. For N >
G/H in which G and H are global and local symmetries, respectively. The latter is the maximal compactsubgroup of G and takes the form SO ( N ) × H ′ with SO ( N ) being the R-symmetrygroup.Gauged supergravity can be constructed by introducing gauge vector fields viaChern-Simons (CS) terms which are topological in nature. This CS formulation makesthe connection to usual Yang-Mills (YM) gauged supergravities, in which gauge fieldsappear via YM kinetic terms, obscure. Furthermore, since conventional dimensionalreductions result in YM gauged supergravities, the embedding of these CS gaugedsupergravities in higher dimensions is then a non-trivial task. However, it has beenshown that CS gauged supergravity with a non-semisimple gauge group G ⋉ T dim G ison-shell equivalent to YM gauged supergravity with G gauge group [3]. The T dim G factor is a nilpotent translational group transforming in the adjoint representation of asemisimple group G . Therefore, at least some of these particular gauge groups mighthave higher dimensional origins in terms of known dimensional reductions. Since theembedding in higher dimensions is necessary for interpreting three-dimensional solu-tions in string/M-theory context, CS gauged supergravities with non-semisimple gaugegroups are of particular interest in the AdS/CFT correspondence.Until now, a number of works considering CS gauged supergravity with non-semisimple gauge groups and their application in the AdS/CFT correspondence haveappeared [4, 5, 6, 7, 8, 9, 10, 11, 12]. All of these gauged supergravities are expectedto arise from dimensional reductions of higher dimensional theories, and explicit reduc-tion ansatze for obtaining three-dimensional gauged supergravities from a three-spherereduction, both the full S reduction and SU (2) group manifold reduction, have beenconstructed in [13, 14, 15, 16]. Furthermore, a number of N = 2 gauged supergravitieswith abelian gauge groups have also been obtained from wrapped branes in type IIBand M-theory [17, 18, 19].In this paper, we construct new N = 4 gauged supergravity with E /SU (6) × – 1 – U (2) scalar manifold and SO (4) ⋉ T gauge group. According to the result of [3],the resulting gauged supergravity is equivalent to SO (4) YM gauged supergravity. Anumber of possible semisimple gaugings of this theory have already been classified in[8]. In the present work, we will study possible non-semisimple gauge groups and alsoclassify supersymmetric vacua of the gauged supergravity both maximally supersym-metric and half-supersymmetric with the full SO (4) symmetry unbroken. These wouldbe useful in AdS /CFT correspondence.The theory considered here is one of the magical supergravities in three dimen-sions, see [20, 21, 22] for higher dimensional magical supergravities. In [8], another SO (4) ⋉ T gauged magical supergravity with F /U Sp (6) × SU (2) scalar manifoldhas been studied. This theory is expected to arise from an S reduction of N = (1 , SO (4) ⋉ T gauged theory with E /SU (6) × SU (2)scalar manifold to arise from an S reduction of this N = (1 ,
0) supergravity coupledto four vector and three tensor multiplets. Both of these six-dimensional supergravitiesare also magical supergravities, and the possible gaugings have been systematicallyconsidered in [23] using the embedding tensor formalism.The paper is organized as follow. In section 2, we review N = 4 three-dimensionalgauged supergravity with symmetric scalar target spaces. We mainly focus on a specificcase of exceptional coset E /SU (6) × SU (2). In section 3, the embedding tensor of SO (4) ⋉ T is given. We then study the resulting scalar potential for the SO (4) singletscalars and identify some of their possible supersymmetric vacua. We end the paper bygiving some conclusions and comments on a possible higher dimensional origin of thegauged supergravity constructed here in section 4. Two appendices with some usefulformulae are also included. N = 4 gauged supergravity in three dimensions with E /SU (6) × SU (2) scalar manifold Three dimensional matter-coupled supergravity is given by a nonlinear sigma modelcoupled to supergravity. Scalar fields will be denoted by φ i , i = 1 , . . . , d with d beingthe dimension of the scalar target space. Throughout this paper, we will work in the SO ( N ) covariant formulation of [24].Coupling the non-linear sigma model to N extended supergravity requires N − f P i j , P = 2 , . . . N , on the target space of thesigma model. The R-symmetry in three dimensions is given by SO ( N ) under whichscalars transform in a spinor representation. The tensor f IJ , I, J = 1 , . . . N , generating– 2 – O ( N ) R-symmetry in this spinor representation, can be constructed from f P via f P = − f P = f P , f P Q = f [ P f Q ] . (2.1)The f IJij have a symmetry property f IJij = − f JIij = − f IJji .For N = 4 theory, the scalar target space must be a quaternionic manifold. Specialto N = 4 supersymmetry, there exists a tensor J = ǫ P QR f P f Q f R that commutes withthe almost complex structures and is covariantly constant. This implies the productstructure of the target space. Therefore, a general N = 4 matter-coupled supergravityadmits a scalar manifold of the form M = M − × M + with a total dimension d = d + + d − . The full SO (4) ∼ SO (3) − × SO (3) + R-symmetry will split into each factorof the full target space. In this paper, we are however only interested in the so-calleddegenerate case with only one factor of M present. For definiteness, we will consider anon-vanishing M + by setting d − = 0. Furthermore, we will restrict ourselves to only asymmetric target space of the form G/H although this is not in general a requirementfrom N = 4 supersymmetry.The symmetric target space of the form G/H has a global symmetry G and a localsymmetry H given by its maximal compact subgroup. In general, the local symmetry H contains the SO ( N ) R-symmetry and takes the form H = SO ( N ) × H ′ . But, for the N = 4 theory, we have the compact group H ± = SO (3) ± × H ′± ∼ SU (2) ± × H ′± for eachsubspace M ± . In the present case, the scalar target space is given by E /SU (6) × SU (2) of dimension 40.We now decompose the E generators t M = ( T IJ + , X α , Y A ) into those of thecompact SO (3) + × SU (6) and the non-compact generators Y A , A = 1 , . . . ,
40. The SO (3) + generators denoted by T IJ + are given by the self-dual part of the full SO (4)R-symmetry generators T IJ : T IJ + = T IJ + 12 ǫ IJKL T KL . (2.2) X α , α = 1 , . . . ,
35 are SU (6) generators.The E /SU (6) × SU (2) manifold can be described by the coset representative L transforming under E and SU (6) × SU (2) by left and right multiplications, respec-tively. In particular, L can be used to find the SU (6) × SU (2) composite connections, Q IJ + i and Q αi , and the vielbein on E /SU (6) × SU (2), e Ai , by the relation L − ∂ i L = 12 Q IJ + i T IJ + + Q αi X α + e Ai Y A . (2.3)The metric on the target space g ij can be computed from the vielbein e Ai by the usualrelation g ij = e Ai e Bj δ AB . Here, indices A, B = 1 , . . .
40 can be considered as “flat” target– 3 –pace indices.Gaugings are implemented by a symmetric and gauge invariant tensor called theembedding tensor Θ MN [25, 26]. A viable gauging consistent with supersymmetry ischaracterized by the embedding tensor that satisfies two consistency conditions. Thefirst condition, called the quadratic constraint, requires that the gauge group is a propersubgroup of the global symmetry G . This constraint is explicitly given byΘ PL f KL ( M Θ N ) K = 0 (2.4)where f KLM are the G -structure constants. Furthermore, supersymmetry requires thatthe T-tensor defined by the image of the embedding tensor under a map V T AB = V MA Θ MN V NB (2.5)satisfies the constraint T IJ,KL = T [ IJ,KL ] − N − δ [ I [ K T L ] M,MJ ] − N − N − δ I [ K δ L ] J T MN,MN . (2.6)The T-tensor transforms under the local H symmetry with the index A = { IJ, α, A } .The above constraint implies that the ⊞ representation of SO ( N ) is absent from the T IJ,KL component of the T-tensor. Therefore, we can equivalently write this constraintas P ⊞ T IJ,KL = 0 . (2.7)In the case of symmetric target spaces, the condition (2.7) can be expressed as aconsistency condition on the embedding tensor which lives in a symmetric product of theadjoint representation of G . It turns out that under the decomposition of this productinto irreducible representations of G , there is a unique representation of G , called R ,which gives rise to the SO ( N ) representation ⊞ under the branching G → SO ( N ).Therefore, in this case, the constraint (2.7) can be written in a G -covariant way by P R Θ MN = 0 . (2.8)In the case of G = E , the representation R is given by [24]. In addition, forsymmetric spaces in the form of a coset space, the map V will be an isomorphism, andits components are given by the relation L − t M L = 12 V M IJ T IJ + + V M α X α + V M A Y A . (2.9)Various components of the T-tensor can be straightforwardly computed from theembedding tensor by using the definition (2.5) and the map V from (2.9). Combinations– 4 –f these T-tensor components are used to construct the A , A and A tensors which willappear as fermion mass-like terms and the scalar potential in the gauged Lagrangian.They are defined by A IJ = − N − T IM,JM + 2( N − N − δ IJ T MN,MN , (2.10) A IJ j = 2 N T
IJj + 4 N ( N − f M ( Imj T J ) Mm + 2 N ( N − N − δ IJ f KL mj T KLm , (2.11) A IJ ij = 1 N h − D ( i D j ) A IJ + g ij A IJ + A K [ I f J ] Kij + 2 T ij δ IJ − D [ i T IJj ] − T k [ i f IJkj ] i (2.12)where D i is the covariant derivative with respect to φ i . In terms of these tensors, thescalar potential can be written as V = − N (cid:18) A IJ A IJ − N g ij A IJ i A IJ j (cid:19) . (2.13)As a final ingredient, we give the supersymmetry transformations of the gravitini ψ Iµ and the spinor fields χ iI , involving only bosonic fields, δψ Iµ = D µ ǫ I + gA IJ γ µ ǫ J , (2.14) δχ iI = 12 ( δ IJ − f IJ ) i j D /φ j ǫ J − gN A JIi ǫ J (2.15)where the covariant derivative of ǫ I is given by D µ ǫ I = ∂ µ ǫ I + 14 ω abµ γ ab ǫ I + ∂ µ φ i Q IJi ǫ J + g Θ MN A M µ V N IJ ǫ J . (2.16)The covariant derivative on scalars φ i is defined by D µ φ i = ∂ µ φ i + Θ MN A M µ X N i (2.17)with X N i being the Killing vectors associated to the isometries of G/H .Note that there are only d physical χ i fields in consistent with d scalar fields φ i asrequired by supersymmetry. In order to work with the SO ( N ) covariant formulation of[24], in which the explicit dependence on f P is absent, the χ i fields have been writtenin terms of the constrained fields χ iI satisfying χ iI = 1 N ( δ IJ δ ij − f IJij ) χ jJ . (2.18)– 5 –e finally note here that, for maximally symmetric vacua, the unbroken supersymmetrycorresponds to Killing spinors ǫ I satisfying the relation A IK A KJ ǫ J = − V ǫ I . (2.19)This relation can be derived by solving the BPS conditions δψ Iµ = 0 and δχ iI = 0 atconstant scalars, see the relevant detail in [24]. N = 4 , SO (4) ⋉ T gauged supergravity and some supersym-metric vacua We first give an explicit construction of the E /SU (6) × SU (2) coset space. Genera-tors of the compact H = SU (6) × SU (2) subgroup and the 40 non-compact generatorsare given in appendix A.We first describe the embedding of SO (4) ⋉ T gauge group in the global symmetrygroup E . In order to do this, we will decompose E into its maximal subgroup SO (6 , × U (1) generated by ˜ X ˆ i ˆ j and ˆ X given explicitly in appendix A. The full gaugegroup SO (4) ⋉ T can be embedded in SO (6 ,
4) as follow.The semisimple part SO (4) is given by a diagonal subgroup of SO (4) × SO (4) ⊂ SO (6) × SO (4) which is in turn the maximal compact subgroup of SO (6 , SO (4) is accordingly generated by J ab = ˜ X ab + ˜ X a +6 ,b +6 , a, b = 1 , . . . , . (3.1)The other combination ˜ X ab − ˜ X a +6 ,b +6 together with a suitable set of non-compactgenerators will give rise to the translational generators T transforming as an adjointrepresentation of the above mentioned SO (4). All of the T generators also commutewith each other.In order to identify the appropriate non-compact generators constituting T , wedecompose the Y A generators under the SO (4) part of the gauge group. Under SU (6) × SU (2), the 40 generators Y A transform as ( , ). Under SU (6) × SU (2) → SU (4) × SU (2) × U (1) × SU (2), we find( , ) → ( , , ) + (¯ − , − , ) + ( , , ) . (3.2)From now on, we will neglect all the U (1) charges since they will not play any importantrole in the following analysis. We now decompose SU (4) ∼ SO (6) → SO (4) × SO (2)by the embedding → + + . With the SO (2) ∼ U (1) charges neglected, further– 6 –ecompositions to SO (4) × SO (4) and finally to SO (4) diag respectively give SO (4) × SO (4) : 2 × ( , ; , ) + 2 × ( , ; , ) + ( , ; , ) + ( , ; , ) SO (4) diag : 3 × ( , ) + 2 × ( , ) + 4 × ( , ) + ( , )+( , ) + ( , ) (3.3)where we have denoted the SO (4) ∼ SU (2) × SU (2) representations by ( + , + )with j , j corresponding to the spins of the two SU (2)’s.The last two representations in (3.3) are the adjoint representation of SO (4) diag .These will be part of the gauge generators T . Explicitly, we find that these generatorsare given by t ab = ˜ X ab − ˜ X a +6 ,b +6 + ˜ X a,b +6 + ˜ X a +6 ,b , a, b = 1 , . . . , . (3.4)Note that ˜ X a,b +6 and ˜ X a +6 ,b are non-compact generators of E . It can be verifiedthat ( J ab , t ab ) generators satisfy the SO (4) ⋉ T algebra (cid:2) J ab , J cd (cid:3) = − δ [ a [ c J d ] b ] , (cid:2) J ab , t cd (cid:3) = − δ [ a [ c t d ] b ] , (cid:2) t ab , t cd (cid:3) = 0 . (3.5)The non-vanishing components of the embedding tensor corresponding to the fullgauge group are denoted by Θ ab and Θ bb with a and b associated to the J ab and t ab parts, respectively [3]. It turns out that the embedding tensor satisfying the linear andquadratic constraints given in (2.4) and (2.8) is given byΘ = g Θ ab + g Θ bb (3.6)where both Θ ab ab,cd and Θ bb ab,cd are given by ǫ abcd . This is much similar to the N = 8 and N = 4 theories studied in [6] and [8]. It should be noted that supersymmetry allowsfor two independent coupling constants. SO (4) symmetry We now consider some vacua of the N = 4 gauged supergravity constructed previously.From equation (3.3), we see that there are three scalars which are singlets under SO (4).These singlets correspond to the following non-compact generatorsˆ Y = Y − Y + Y − Y − Y − Y + Y + Y , ˆ Y = Y + Y − Y + Y + Y − Y − Y − Y , ˆ Y = Y + Y + Y + Y + Y − Y + Y − Y . (3.7)The coset representative L can be parametrized by L = e Φ ˆ Y e Φ ˆ Y e Φ ˆ Y . (3.8)– 7 –y using the formulae in section 2 and appendix B, we obtain the scalar potential for(Φ , Φ , Φ ) given by V = 32 g [4 cosh(2Φ ) cosh(2Φ ) cosh(2Φ ) − )] × [4 g cosh(2Φ ) cosh(2Φ ) cosh(2Φ ) − g sinh(2Φ ) − g ] . (3.9)From this potential, it can be readily verified that the SO (4) ⋉ T gauged super-gravity admits a maximally supersymmetric AdS critical point. By shifting the valuesof scalar fields at the vacuum, we can bring the critical point to L = I × at whichΦ = Φ = Φ = 0. This can also be achieved by setting g = g = g . The cosmologicalconstant at the critical point is given by V = − g . In our convention, the AdS radius is given by L = r − V = 132 g (3.10)where we have taken g > N = (4 ,
0) superconformal symmetry in the dual two-dimensional SCFT.All of the scalar masses at this critical point are given in table 1. In the table, we havealso given the dimensions of the dual operators in the dual SCFT according to the rela-tion m L = ∆(∆ − AdS critical points. SO (4) representations m L ∆( , ) 0 ( × , 3 2 , , ) + ( , ) 0 ( × , ) −
89 ( ×
6) 23 , ( , ) −
34 ( ×
16) 12 , ( , ) − ( × Table 1:
Scalar masses at the N = 4 supersymmetric AdS critical point and the corre-sponding dimensions of the dual operators From the table, we see the presence of six massless scalars in the adjoint repre-sentations of SO (4), ( , ) + ( , ). These are Goldstone bosons corresponding to thesymmetry breaking SO (4) ⋉ T → SO (4) at the vacuum. Furthermore, there areadditional massless scalars which are singlets under SO (4). These are expected tocorrespond to marginal deformations in the dual SCFT. The deformations preserve allsupersymmetry as well as the full SO (4) symmetry. These deformations can be given– 8 –xplicitly by the relationcosh(2Φ ) cosh(2Φ ) cosh(2Φ ) = 1 + sinh(2Φ ) . (3.11)When Φ = Φ = 0, there is only one solution Φ = 0. Non-vanishing values of Φ andΦ give rise to the same value of V , unbroken SO (4) symmetry and the same numberof supersymmetry. Therefore, Φ and Φ correspond to flat directions of the potential.There is another class of vacua given by the relationcosh(2Φ ) cosh(2Φ ) = tanh(2Φ ) . (3.12)This gives supersymmetric Minkowski vacua in three dimensions with V = 0. We now move to half-supersymmetric vacuum solutions. To find these solutions, we setup the corresponding BPS equations from the supersymmetry transformations (2.14)and (2.15). The three-dimensional metric is taken to be the standard domain wallansatz ds = e A ( r ) dx , + dr . (3.13)With the projection γ r ǫ I = − ǫ I corresponding to N = (4 ,
0) supersymmetry in twodimensions, equations δχ iI = 0 and δψ Iµ = 0 for µ = 0 , ′ = 16 sinh(2Φ )cosh(2Φ ) cosh(2Φ ) [ g − g cosh(2Φ ) cosh(2Φ ) cosh(2Φ )+ g sinh(2Φ )] , (3.14)Φ ′ = 16 sinh(2Φ ) cosh(2Φ )cosh(2Φ ) [ g − g cosh(2Φ ) cosh(2Φ ) cosh(2Φ )+ g sinh(2Φ )] , (3.15)Φ ′ = 16 [ g − g cosh(2Φ ) cosh(2Φ ) cosh(2Φ ) + g sinh(2Φ )] × [cosh(2Φ ) cosh(2Φ ) sinh(2Φ ) − cosh(2Φ )] , (3.16) A ′ = 32 [2 g − g cosh(2Φ ) cosh(2Φ ) cosh(2Φ ) + g sinh(2Φ )] × [cosh(2Φ ) cosh(2Φ ) cosh(2Φ ) − sinh(2Φ )] (3.17)where ′ denotes the r -derivative. All of these equations satisfy the second-order fieldequations. Some details of this verification is given in appendix C.We first consider the case Φ = Φ = 0. It can be readily seen that the first twoequations are identically satisfied. We are left with two equationsΦ ′ = 16 e − ( g − g e ) , (3.18) A ′ = 32 e − (2 g e − g ) . (3.19)– 9 –or g = 0, the Φ ′ equation has a critical point at Φ = ln h g g i while the A ′ equationgives A = g g r + C . This is the maximally supersymmetric AdS critical point iden-tified previously.Equations (3.18) and (3.19) can be solved explicitly with the corresponding solution A = − − ln( e g − g ) + C , g r = − g e − g ln( e g − g ) + C . (3.20)It should be noted that the integration constants C and C can be removed by shiftingthe coordinate r and rescaling the x and x coordinates, respectively. The solutioninterpolates between N = 4 AdS critical point at r → ∞ to a half-supersymmetricdomain wall at a finite value of r . At large | Φ | , we findΦ ∼
12 ln( C − g r ) , A ∼ − ln( C − g r ) (3.21)with C being a constant. We have set g = g for simplicity. The metric at r ∼ C g becomes ds = ( C − g r ) − dx , + dr (3.22)which takes the form of a domain wall. However, the scalar potential becomes un-bounded when Φ → −∞ . The singularity of the above metric is then unphysical bythe criterion of [27]. Therefore, the holographic interpretation of the solution as anRG flow between an N = (4 ,
0) SCFT and an N = (4 ,
0) non-conformal field theory intwo dimensions cannot be given at least in the three-dimensional framework. It wouldbe interesting to further investigate the singularity in the context of higher dimensionsin which this solution is embedded. Note also that since the operator dual to Φ isirrelevant, we would expect the N = (4 ,
0) SCFT to appear in the IR.There is another simple exact solution to equations (3.14), (3.15), (3.16) and (3.17)namely Φ = 0 , g cosh(2Φ ) cosh(2Φ ) = g , A = 32 g g r + C . (3.23)This solution corresponds to a marginal deformation of the supersymmetric
AdS crit-ical point.We now move to another type of domain walls which is a half-supersymmetricvacuum of the theory without any limit with enhanced supersymmetry. Recall that su-persymmetry allows for two independent gauge couplings g and g , by setting g = 0,we also have a consistent gauged supergravity. In this case, the resulting gauged super-gravity possesses a half-supersymmetric domain wall vacuum. We will present a simple– 10 –olution of this type. With Φ = Φ = g = 0, the BPS equations becomeΦ ′ = − g e − ,A ′ = 64 g e − (3.24)which admit a solutionΦ = 12 ln( C ′ − g r ) , A = − C ′ − g r ) (3.25)with an integration constant C ′ . This solution gives a domain wall metric ds = ( C ′ − g r ) − dx , + dr . (3.26)For Φ = 0 which satisfies equation (3.15), a more general solution can be found bytreating Φ as an independent variable. After combing equations (3.16) and (3.17) with(3.14), we can solve for Φ and A as functions of Φ . The result is then substituted inequation (3.14) to find the solution for Φ ( r ). The resulting solution is given byΦ = 14 ln (cid:20) C − coth Φ tanh Φ − C (cid:21) ,A = ln(1 − e ) −
12 ln (cid:2) (1 + 4 C ) − (1 − C ) e (cid:3) −
12 ln h(cid:2) (1 − C ) e − (1 + 4 C ) (cid:3) g − C ( e − g − C g g e q sinh(2Φ ) [8 C cosh(2Φ ) − (1 + 16 C ) sinh(2Φ )] (cid:21) , C g r = g q C coth(2Φ ) − C − C g ln sinh(2Φ ) − C g ln (cid:2) ( g + ( g + g )16 C ) sinh(2Φ ) − C g cosh(2Φ ) (cid:3) − C g tanh − " g p C coth(2Φ ) − C − C g . (3.27)In the above equations, we have neglected additive integration constants in Φ ( r ) and A (Φ ) solutions.Apart from these solutions, we are not able to completely solve the BPS equationswith all scalars non-vanishing in an analytic form. We will however give a partial resulton this solution since it might be useful for further investigation. It turns out thatcombining equations (3.14) and (3.15) gives an equation for Φ as a function of Φ witha solution Φ = 14 ln (cid:20) − e C sinh(2Φ )1 + e C sinh(2Φ ) (cid:21) . (3.28)– 11 –f all of the integration constants are set to zero, a simple solution for Φ can also befound Φ = cosh − (cid:20) q − csch(2Φ ) p cosh(4Φ ) − (cid:21) . (3.29)With these relations, equation (3.17) would in principle give a solution for A (Φ ) whileequation (3.14) would give a solution for Φ ( r ). We have not succeeded in obtainingan analytic form for these solutions.
4. Conclusions
In this paper, we have constructed N = 4 gauged supergravity in three dimensions with SO (4) ⋉ T gauge group and E /SU (6) × SU (2) scalar manifold. We have studiedsome of the maximally supersymmetric and half-supersymmetric vacua of this gaugedsupergravity. The N = 4 AdS critical point with SO (4) symmetry corresponds to an N = (4 ,
0) SCFT in two dimensions. We have given the full scalar mass spectrum atthis critical point which might be useful for other holographic applications. In addi-tion, some half-supersymmetric domain wall solutions have also been explicitly givenin an analytic form. According to the DW/QFT correspondence [28], we expect thesesolutions to be dual to N = (4 ,
0) non-conformal field theories in two dimensions.We have also identified two flat directions of the scalar potential corresponding toexactly marginal deformations of the N = (4 ,
0) SCFT that preserve all supersymme-tries and SO (4) symmetry. Remarkably, these flat directions are not Goldstone bosons.This is in contrast to the four-dimensional N = 4 gauged supergravity studied in [29].In that case, all flat directions correspond to Goldstone bosons. It would be interestingto identify the N = (4 ,
0) SCFT and two-dimensional non-conformal field theories dualto the vacua identified here. Further investigations of the scalar potential in other scalarsectors invariant under smaller residual gauge symmetry could be useful for the studyof other deformations of the dual N = (4 ,
0) SCFT in particular relevant deformationsgiven by scalars in ( , ), ( , ) and ( , ) representations.Due to the equivalence between the gauged supergravity constructed here and theYang-Mills gauged supergravity with SO (4) gauge group, it is possible that this theorymight be obtained from higher dimensions. The ungauged N = 4 supergravity with E /SU (6) × SU (2) scalar manifold can be obtained from a reduction on a 3-torus( T ) of the minimal supergravity in six dimensions coupled to three tensor and fourvector multiplets. The three tensor multiplets consist of three scalars parametrized bythe SO (3 , /SO (3) coset manifold. After dimensional reduction and dualization ofthe vector fields coming from the six-dimensional metric and the (anti) self-dual tensorfields, the resulting three-dimensional supergravity consists of 40 scalars as required by– 12 –he dimension of E /SU (6) × SU (2) coset manifold.We then expect that the SO (4) ⋉ T gauged supergravity constructed here shouldarise from a dimensional reduction of this six-dimensional supergravity on a 3-shpere( S ). Along the line of this uplifting, it could be useful to compute vector and fermionmasses and match with the AdS × S spectrum of N = (1 ,
0) six-dimensional su-pergravity carried out in [32]. It should also be remarked here that, when coupledto hypermultiplets with hyper-scalars described by M − manifold, the six-dimensionalsupergravity could give rise to three-dimensional gauged supergravity with two scalartarget manifolds M + × M − . It would be interesting to construct an explicit reductionansatz similar to the recent result in [16]. The result will be very useful in uplifting thethree-dimensional solutions to higher dimensions. We leave this issue and related onesfor future works. Acknowledgments
This work is supported by Chulalongkorn University through Ratchadapisek SompochEndowment Fund under grant GF-58-08-23-01 (Sci-Super II).
A. Generators of E and relevant subgroups E generators in the fundamental representation, used throughout this paper, have beenconstructed in [30] and [31]. All of these 78 generators are denoted by c i , i = 1 , . . . c i c j ) = − δ ij . (A.1)In order to construct the non-compact form E from the compact E , we first identifythe maximal compact subgroup H = SU (6) × SU (2) with the SU (2) factor correspond-ing to the SU (2) + subgroup of the full SO (4) R-symmetry. We then apply the “Weylunitarity trick” to the remaining 40 generators to make them non-compact. A.1 SU (6) × SU (2) subgroup and non-compact generators The R-Symmetry group SU (2) + ∼ SO (3) + is generated by T = −
12 ( c + c ) , T = 12 ( c − c ) , T = 12 ( c + ˜ c ) . (A.2)– 13 –he generators of the group H ′ = SU (6) are given by X i = c i , i = 1 , . . . , ,X = 1 √ c + c ) , X = 1 √ c − c ) , X = 1 √ c − c ) ,X = 1 √ c + c ) , X = 1 √ c − c ) , X = 1 √ c + c ) ,X = 1 √ c − c ) , X = 1 √ c + c ) , X = 1 √ c + c ) ,X = 1 √ c − c ) , X = 1 √ c − c ) , X = 1 √ c + c ) ,X = 1 √ c − c ) , X = 1 √ c − c ) , X = 1 √ c + c ) ,X = 1 √ c + c ) , X = 1 √ c − c ) , X = 1 √ c − c ) ,X = 1 √ c + c ) , X = ˜ c (A.3)where ˜ c and ˜ c generators are defined by˜ c = 12 c + √ c , ˜ c = − √ c + 12 c . (A.4)It is useful to note that the SU (4) × SU (2) × U (1) ∼ SO (6) × SO (3) × U (1) subgroupof SU (6) is generated by X i , i = 1 , . . . ,
15, ( X , X , X ) and X , respectively.With the compact H generators defined above, the 40 non-compact generators are– 14 –ccordingly given by Y = i c − c ) , Y = i c + c ) , Y = i c − c ) ,Y = i c + c ) , Y = i c − c ) , Y = i c − c ) ,Y = i c + c ) , Y = i c + c ) , Y = i c − c ) ,Y = i c + c ) , Y = i c + c ) , Y = i c − c ) ,Y = i c − c ) , Y = i c + c ) , Y = i c + c ) ,Y = i c − c ) , Y = i c + c ) , Y = − i c + c ) ,Y = i c + c ) , Y = − i c + c ) , Y = − i c + c ) ,Y = − i c + c ) , Y = i c − c ) , Y = i c − c ) ,Y = i c − c ) , Y = i c − c ) , Y = i c − c ) ,Y = i c − c ) , Y = i c − c ) , Y = i c − c ) ,Y = i c − c ) , Y = i c − c ) , Y = i c − c ) ,Y = i c − c ) , Y = i c + c ) , Y = − i c + c ) ,Y = i c + c ) , Y = − i c + c ) , Y = − i c + c ) ,Y = − i c + c ) . (A.5)For advantages of future investigations, we give the non-compact generators corre-sponding to scalar fields that are singlets under various subgroups of the SO (4) gaugesymmetry. The following results can be obtained by further decompositions of the SO (4) representations given in (3.3). • SO (3) diag ⊂ SO (3) × SO (3) ∼ SO (4) singlets:ˆ Y = Y + Y + Y + Y − Y + Y , ˆ Y = Y + Y + Y − Y , ˆ Y = Y − Y , ˆ Y = Y − Y + Y + Y , ˆ Y = Y − Y , ˆ Y = Y + Y + Y − Y , ˆ Y = Y − Y , ˆ Y = Y + Y − Y + Y (A.6)– 15 – SU (2) × SO (2) s singlets:ˆ Y = Y + Y + Y + Y + Y − Y + Y − Y , ˆ Y = Y + Y − Y − Y + Y − Y − Y + Y , ˆ Y = Y − Y + Y − Y , ˆ Y = Y + Y − Y + Y , ˆ Y = Y + Y − Y − Y , ˆ Y = Y − Y − Y − Y (A.7) • SU (2) s singlets:ˆ Y = Y − Y + Y − Y − Y − Y + Y + Y , ˆ Y = Y + Y − Y + Y + Y − Y − Y − Y , ˆ Y = Y + Y − Y − Y , ˆ Y = Y − Y + Y − Y , ˆ Y = Y + Y + Y + Y , ˆ Y = Y − Y − Y + Y (A.8)The SU (2) s denotes the SU (2) subgroup of SO (4) generated by self-dual SO (4) gen-erators with SO (2) s ⊂ SU (2) s . A.2 SO (6 , × U (1) subgroup The U (1) is generated by ˆ X = ˜ c while the SO (6 ,
4) generators are given by˜ X = c , ˜ X = − c , ˜ X = c , ˜ X = c , ˜ X = c , ˜ X = − c , ˜ X = c , ˜ X = − c , ˜ X = c , ˜ X = − c , ˜ X = − c , ˜ X = c , ˜ X = − c , ˜ X = − c , ˜ X = c , ˜ X = ic , ˜ X = − ic , ˜ X = − ic , ˜ X = ic , ˜ X = − ic , ˜ X = − ic , ˜ X = − c , ˜ X = ic , ˜ X = − ic , ˜ X = − ic , ˜ X = ic , ˜ X = − ic , ˜ X = − ic , ˜ X = − ic , ˜ X = ic ˜ X = − ic , ˜ X = ic , ˜ X = − ic , ˜ X = − ic , ˜ X = − c , ˜ X = − c , ˜ X , = − ic , ˜ X , = ic , ˜ X , = − ic , ˜ X , = ic , ˜ X , = ic , ˜ X , = ic , ˜ X , = c , ˜ X , = c , ˜ X , = − ˜ c . (A.9)All generators are labelled by SO (6 ,
4) adjoint indices with ˜ X ˆ i ˆ j = − ˜ X ˆ j ˆ i , ˆ i, ˆ j = 1 , . . . , SO (6) × SO (4) is generated by ˜ X ˆ i ˆ j , ˆ i, ˆ j = 1 , . . . , X ˆ i ˆ j ,ˆ i, ˆ j = 7 , . . . ,
10, respectively. This coincides with the SO (6) × SO (3) ⊂ SU (6) togetherwith the SO (3) R-symmetry. The 24 non-compact generators are identified with ˜ X ˆ i, ˆ j +6 for ˆ i = 1 , . . . , j = 1 , . . . ,
4. – 16 – . Essential formulae
For a general symmetric space of the form
G/H with G and H = SO ( N ) × H ′ beingthe global and local symmetry groups, the G algebra is given by (cid:2) T IJ , T KL (cid:3) = − δ [ I [ K T L ] J ] , (cid:2) T IJ , Y A (cid:3) = − f IJ,AB Y B , (cid:2) X α , X β (cid:3) = f αβγ X γ , (cid:2) X α , Y A (cid:3) = h α AB Y B , (cid:2) Y A , Y B (cid:3) = 14 f ABIJ T IJ + 18 C αβ h βAB X α . (B.1) C αβ is an invariant tensor of H ′ , and h αAB are antisymmetric tensors that commute with f IJAB .Using the above algebra, we find that components of f IJ tensor written in flat cosetspace indices are given by f IJAB = −
23 Tr( Y B (cid:2) T IJ , Y A (cid:3) ) . (B.2)In term of the coset representative, various components of the V map can be computedby using the relations V a ab,IJ = −
13 Tr( L − J ab LT IJ ) , V b ab,IJ = −
13 Tr( L − t ab LT IJ ) , V a ab,A = 13 Tr( L − J ab LY A ) , V b ab,A = 13 Tr( L − t ab LY A ) . (B.3)The T-tensors are then obtained from T IJ,KL = g (cid:0) V a ab,IJ V b cd,KL + V b ab,IJ V a cd,KL (cid:1) ǫ abcd + g V b ab,IJ V b cd,KL ǫ abcd ,T IJ,A = g (cid:0) V a ab,IJ V b cd,A + V b ab,IJ V a cd,A (cid:1) ǫ abcd + g V b ab,IJ V b cd,A ǫ abcd . (B.4)From these relations, the tensors A IJ , A IJ i and the scalar potential can be straightfor-wardly computed. C. Field equations for SO (4) singlet scalars and the metric In this appendix, we explicitly verify that the BPS equations given in (3.14), (3.15),(3.16), and (3.17) indeed satisfy the corresponding second-order field equations. With– 17 –nly scalar fields and the metric, the Lagrangian of the gauged supergravity read, inour convention, L = √− g (cid:20) R − P Aµ P Aµ − V (cid:21) (C.1)where the scalar potential is given in (3.9). The scalar kinetic term is written in termof the coset vielbein e Ai as P Aµ = ∂ µ φ i e Ai . (C.2)It should be noted that, with the relation g ij = e Ai e Aj , the scalar kinetic term is thesame as that given in [24] − P Aµ P Aµ = − g ij ∂ µ φ i ∂ µ φ j . (C.3)In the present case, the coset vielbein can be computed from (2.3) P Aµ = 13 Tr( L − ∂ µ LY A ) . (C.4)For completeness, we will explicitly give the result here − P Aµ P Aµ = − ′ − (2Φ ) (cid:2) cosh (2Φ )Φ ′ + Φ ′ (cid:3) . (C.5)Einstein’s equation coming from the above Lagrangian is given by R µν − g µν R = P Aµ P Aν − g µν (cid:20) P Aρ P Aρ + V (cid:21) (C.6)For the metric ansatz (3.13), non-vanishing components of the Ricci tensor and Ricciscalar are the following R µν = − e A η µν ( A ′′ + 2 A ′ ) ,R rr = − A ′′ + A ′ ) ,R = − A ′′ − A ′ (C.7)for µ, ν = 0 ,
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