Vacua and RG flows in N=9 three dimensional gauged supergravity
aa r X i v : . [ h e p - t h ] O c t Preprint typeset in JHEP style - HYPER VERSION arXiv:1007.5438
Vacua and RG flows in N = 9 threedimensional gauged supergravity Auttakit Chatrabhuti a, b and Parinya Karndumri c, da
Theoretical High-Energy Physics and Cosmology Group, Department of Physics,Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand b Thailand Center of Excellence in Physics, CHE, Ministry of Education, Bangkok10400, Thailand c INFN, Sezione di Trieste, Italy d International School for Advanced Studies (SISSA), via Bonomea 265, 34136Trieste, ItalyE-mail: [email protected] , karndumr @ sissa.it Abstract:
We study some vacua of N = 9 three dimensional gauged supergravity.The theory contains sixteen scalar fields parametrizing the exceptional coset space F − SO (9) . Various supersymmetric and some non-supersymmetric AdS vacua are foundin both compact and non-compact gaugings with gauge groups SO ( p ) × SO (9 − p ) for p = 0 , , , , G − × SL (2) and Sp (1 , × SU (2). We also study many RG flowsolutions, both analytic and numerical, interpolating between supersymmetric AdS critical points in this theory. All the flows considered here are driven by a relevantoperator of dimension ∆ = . This operator breaks conformal symmetry as well assupersymmetry and drives the CFT in the UV to another CFT in the IR with lowersupersymmetries. Keywords:
AdS-CFT Correspondence, Gauge-gravity correspondence, SupergravityModels. . Introduction
Three dimensional Chern-Simons gauged supergravity has a very rich structure. Thetheory admits gauge groups of various types namely compact, noncompact, non semisim-ple and complex gauge groups [1, 2, 3]. This stems from the fact that there is no re-striction on the number of gauge fields. The gauge fields are introduced to the gaugedtheory by a Chern-Simons kinetic term which results in their non propagating naturein the theory. This peculiar feature comes from the duality between vectors and scalarsin three dimensions. All the bosonic propagating degrees of freedom are carried by thescalars because pure supergravity in three dimensions is also topological.Maximal gauged supergravity in three dimensions has been constructed in [1, 2, 3].The construction of the N = 8 theory can be found in [4]. All extended three dimen-sional gauged supergravities with N ≤
16 have been given in a unique formulation in[5]. This is a gauged version of the ungauged theory constructed in [6]. Vacua of thesetheories have been studied in some details e.g. see [7, 8] for N = 16 theory and [9, 10]for N = 4 and N = 8 theories.In three dimensional supergravities with N >
4, the scalar target space manifoldis a symmetric space and can be written as a coset space GH , where G is a global sym-metry group, and H is its maximal compact subgroup. For the theories with N > N = 9 gauged theory in which the scalar manifold is givenby the exceptional coset F − SO (9) . We will study some vacua of this theory with gaugegroups SO ( p ) × SO (9 − p ) for p = 0 , , , , G − × SL (2) and Sp (1 , × SU (2).All these gauge groups have been shown to be consistent gaugings in [5]. We will studysome vacua of these gaugings and give relevant superconformal groups for maximallysupersymmetric vacua with all scalars being zero.The possibility to study holographic renormalization group flows is one of the in-teresting consequences of the AdS/CFT correspondence [11]. We will also study somesupersymmetric RG flow solutions interpolating between supersymmetric AdS vacua.In gauged supergravity, these solutions are domain walls interpolating between criticalpoints of the scalar potential. They have an interpretation in the dual field theory asan RG flow driving the UV CFT to the IR fixed point corresponding to the CFT in theIR. In this paper, we will find flow solutions in three dimensional gauged supergravity.The solutions describe the RG flows in two dimensional field theories. Some super-symmetric flow solutions in three dimensional gauged supergravity have been studiedin [10, 9] for N = 8 and N = 4 theories, respectively. Some flow solutions of N = 2models have been studied in [12]. In this paper, we give the analogous analysis in the N = 9 theory. The is the largest amount of supersymmetry ever studied so far in the– 1 –ontext of RG flows in three dimensional gauged supergravities.The paper is organized as follows. In section 2, we review some useful ingredientsto construct the N = 9 gauged theory. We also give the explicit construction of thegauged theory with symmetric scalar manifold F − SO (9) in detail. The procedure can beapplied to other theories with different values of N as well. Various vacua are foundin section 3. We then find some flow solutions in section 4. Finally, we give someconclusions and comments in section 5. N = 9 three dimensional gauged supergravity In this section, we construct N = 9 three dimensional gauged supergravity using theformulation given in [5]. The N = 9, SO (9) gauged three dimensional supergravity hasalso been constructed in [13], but we will follow the construction of [5] because thisformulation can be easily extended to other gauge groups. We start by reviewing someformulae and all the ingredients needed in this paper.In symmetric spaces G/H , we have the following decompositions of the G genera-tors t M into { X IJ , X α , Y A } . The maximal compact subgroup H of G , consists of the SO ( N ) R-symmetry and an additional factor H ′ such that H = SO ( N ) × H ′ . Thescalar fields parametrizing the target space are encoded in the coset representative L .This transforms under global G and local H symmetries by multiplications from theleft and right, respectively. The latter can be used to eliminate the spurious degreesof freedom such that L is parametrized by dim ( G/H ) physical scalars. In our case,the maximal compact subgroup of G = F − is SO (9), so there is no factor H ′ .Generators X IJ , I, J = 1 , , . . . N generate SO ( N ), and Y A , transforming in a spinorrepresentation of SO ( N ), are non-compact generators of G . The target space has ametric g ij , i, j = 1 , , . . . d = dim ( G/H ) given by g ij = e Ai e Bj δ AB . (2.1)Extended supersymmetries are described by N − f P ij , P = 2 , . . . , N . We can construct SO ( N ) generators from these f P ij ’s by forming tensors f IJij via [5] f P Q = f [ P f Q ] , f P = − f P = f P . (2.2)The tensors f IJij , being generators of SO ( N ) in the spinor representation, are given interms of SO ( N ) gamma matrices by f IJij = − Γ IJAB e Ai e Bj . (2.3)– 2 –he indices A and B are tangent space or “flat” indices on the scalar target space. Thevielbein of the target space is encoded in the expansion L − ∂ i L = 12 Q IJi X IJ + Q αi X α + e Ai Y A . (2.4) Q IJi and Q αi are composite connections for SO ( N ) and H ′ , respectively.The gaugings are described by the gauge invariant embedding tensor Θ MN . FromΘ MN , we can compute the A and A tensors as well as the scalar potential via theso-called T-tensor using A IJ = − N − T IM,JM + 2( N − N − δ IJ T MN,MN ,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 ,V = − N g ( A IJ A IJ − N g ij A IJ i A IJ j ) . (2.5)The T-tensors are defined by T AB = V MA Θ MN V NB . (2.6)All the V ’s are given by L − t M L = 12 V M IJ X IJ + V M α X α + V M A Y A . (2.7)Using these, we can now construct the N = 9 theory. We give the procedure in detailbut leave some formulae to the appendix. We begin with the F − SO (9) coset. The 52generators of the compact F have been explicitly constructed by realizing F as anautomorphism group of the Jordan algebra J in [14]. There are 16 non-compact and36 compact generators in F − . Under SO (9), the 52 generators decompose as → + where and are adjoint and spinor representations of SO (9), respectively. Thenon-compact F − can be obtained from the compact F by using “Weyl unitaritytrick”, see [15] for an example with G . This is achieved by introducing a factorof i to each generator corresponding to the non-compact generators. From [14], thecompact subgroup SO (9) is generated by, in the notation of [14], c , . . . , c , c , . . . , c , c , . . . , c . We have chosen the same SO (9) subgroup as in [14] among the threepossibilities, see [14] for a discussion. The remaining 16 generators are our non-compactones which we will define by Y A = ( ic A +21 for A = 1 , . . . , ic A +28 for A = 9 , . . . , . (2.8)– 3 –ote that the SO (9) generators c i in [14], are labeled by the F adjoint index. In orderto apply the SO (9) covariant formulation of N = 9 theory, we need to relabel them byusing the SO (9) antisymmetric tensor indices i.e. X IJ . To do this, we first note therelevant algebra from [5][ t IJ , t KL ] = − δ [ I [ K t L ] J ] , [ t IJ , t A ] = − f IJ,AB t B , [ t A , t B ] = 14 f ABIJ t IJ (2.9)where we have used the flat target space indices in f IJAB and the non-compact generators, t A . Using the first commutator in (2.9), we can map all c i ’s forming SO (9) to the desiredform X IJ . The detail of this is given in the appendix. The next step is to find the f IJ . In order to be compatible with the F algebra given in [14], we need to use thesecond and the third commutators in (2.9) to extract the component of f IJAB ratherthan putting the explicit forms of gamma matrices from another basis. There are eightindependent f IJ from which all other components follow from (2.2). We will not giveall of the f IJ here due to their complicated form.We now come to various gaugings characterized by the embedding tensors Θ. Theembedding tensors for the compact gaugings with gauge groups SO ( p ) × SO (9 − p ), p = 0 , . . . , IJ,KL = θδ KLIJ + δ [ I [ K Ξ L ] J ] (2.10)where Ξ IJ = ( (cid:0) − p (cid:1) δ IJ for I ≤ p − p δ IJ for I > p , θ = 2 p − . (2.11)There is only one independent coupling constant, g . The gauge generators can beeasily obtained from SO (9) generators X IJ by choosing appropriate values for theindices I, J . For example, in the case of SO (2) × SO (7) gauging, we have the followinggauge generators SO (7) : T ab = X ab , a, b = 1 , . . . ,SO (2) : T = X . (2.12)We then move to non-compact gaugings with gauge groups G − × SL (2) and Sp (1 , × SU (2). We find the following embedding tensors G − × SL (2) : Θ MN = η G MN − η SL (2) MN , (2.13) Sp (1 , × SU (2) : Θ MN = η Sp (1 , MN − η SU (2) MN (2.14)where η G is the Cartan Killing form of the gauge group G . The gauge generators inthese two gaugings are given in the appendix.– 4 –sing these embedding tensors and equation (2.7), we can find all the V ’s andT-tensors. With the help of the computer algebra system Mathematica [16], it is thenstraightforward to compute A and A tensors and finally the scalar potential for eachgauge group. In the next section, we will give all of these potentials but refer thereaders to the appendix for V ’s and T-tensors. For completeness, we also give herethe condition for finding stationary points of the potential. We are most interested insupersymmetric AdS vacua, so we mainly work with the condition for supersymmetricstationary points. As given in [5], see also [2] for N = 16, the supersymmetric stationarypoints satisfy the two equivalent conditions A JI i ǫ J = 0and A IK A KJ ǫ J = − V g ǫ I = 1 N ( A IJ A IJ − N g ij A IJ i A IJ i ) ǫ I , (2.15)where V is the value of the potential at the critical point i.e. the cosmological con-stant. ǫ I are the Killing spinors corresponding to the residual supersymmetries at thestationary point. The second condition simply says that ǫ I is an eigenvector of A withan eigenvalue q − V g or − q − V g . In addition, these two conditions are indeed equivalentas shown in [5].The condition for any stationary points, not necessarily supersymmetric, is [5]3 A IK A KJ j + N g kl A IK k A KJ lj = 0 (2.16)where A KL lj is defined by A IJ ij = 1 N (cid:20) − 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 ] (cid:21) . (2.17)For supersymmetric critical points, we will mostly work with the two equivalent con-ditions given by (2.15). However, for non-supersymmetric points, the condition (2.16)is necessary to ensure that all the points are indeed stationary points.
3. Vacua of N = 9 gauged supergravity In this section, we give some vacua of the N = 9 gauged theory with the gaugings men-tioned in the previous section. We will discuss the isometry groups of the backgroundwith maximal supersymmetries at L = I . This is a supersymmetric extension of the SO (2 , ∼ SO (1 , × SO (1 ,
2) isometry group of AdS . The superconformal groupcan be identified by finding its bosonic subgroup and representations of superchargesunder this group. A similar study has been done in [8] for models with N = 16 super-symmetry. The full list of superconformal groups in two dimensions can be found in[17]. We first start with compact gaugings.– 5 – .1 Vacua of compact gaugings It has been shown in [18] that the critical points obtained from the potential restrictedon a scalar manifold which is invariant under some subgroups of the gauge group arecritical points of the full potential. This invariant manifold is parametrized by all scalarswhich are singlets under the chosen symmetry. To make things more manageable,we will not study the scalar potential with more than four scalars. We choose toparametrize the scalars by using the coset representative L = e a Y e a Y e a Y e a Y . (3.1)For any invariant manifold with the certain residual symmetry, our choice for L in (3.1)certainly does not cover the whole invariant manifold. Therefore, the critical pointson this submanifold may not be critical points of the potential on the whole scalarmanifold. Nevertheless, we can use the argument of [18] as a guideline to find criticalpoints. After identifying the critical points, we then use the stationarity condition (2.16)to check whether our critical points are truly critical points of the scalar potential.Let us identify some residual symmetries of (3.1). In SO (9) gauging, with only a = 0, L has SO (7) symmetry. For a , a = 0, L preserves SO (6) symmetry. With a , a , a = 0 and a , a , a , a = 0, L preserves SU (3) and SU (2), respectively. In othergauge groups, L will have different residual symmetry. We will discuss the residualgauge symmetry of each critical point, separately. We find that in all cases, non trivialsupersymmetric critical points arise with at most two non zero scalars. With all fourscalar fields turned on, the conditions A JI i ǫ J = 0 are satisfied if and only if two of thescalars vanish. So, we give below only potentials with two scalars.In (3.1), we have used the basis elements of Y ’s to parametrize each scalar field.We also find that, in this parametrization, all the sixteen scalars are on equal footingin the sense that any four of the Y ’s among sixteen of them give the same structureof the potential. As a consequence, any two non zero scalars in (3.1) give rise to thesame critical points with the same location and cosmological constant. Notice that thisis not the case if we use different parametrization of L . For example, by using linearcombinations of Y i ’s as basis for the four scalars in (3.1), different choices of Y i ’s ineach basis may give rise to different structures of the scalar potential.We use the same notation as in [9] namely V is the cosmological constant, and( n − , n + ) refers to the number of supersymmetries in the dual two dimensional filedtheory. On the other hand, the n + ( n − ) corresponds to the number of positive (negative)eigenvalues of A IJ . For definiteness, we will keep a and a non zero. Furthermore, wegive the values of scalar fields up to a trivial sign change.– 6 – SO (9) gauging:The scalar potential is V = 132 g ( − −
232 cosh(2 a ) + 6 cosh(4 a ) + 4 cosh[2( a − a )]+4 cosh(4 a − a ) −
112 cosh[2( a − a )] + cosh[4( a − a )] −
232 cosh(2 a ) + 6 cosh(4 a ) −
112 cosh[2( a + a )]+ cosh[4( a + a )] + 4 cosh[2(2 a + a )] + 4 cosh[2( a + 2 a )]) . (3.2)This is the case in which the full R-symmetry group SO (9) is gauged. There isno non trivial critical point with two scalars. For a = 0, there are two criticalpoints, but only the L = I solution has any supersymmetry.Critical points a V Preserved supersymmetry1 0 − g (9,0)2 cosh − − g -The corresponding A tensor at the supersymmetric point is A (1)1 = diag( − , − , − , − , − , − , − , − , − . (3.3)The notation A (1)1 means that this is the value of the A tensor evaluated at thecritical point number 1 in the table. For L = I , the background isometry is givenby Osp (9 | , R ) × SO (1 , SO (7) gauge symmetry. This point is closely related to the non-supersymmetric SO (7) × SO (7) point found in N = 16 SO (8) × SO (8) gauged supergravity [8].Both the location and the value of the cosmological constants compared to the L = I point are very similar to that in [8]. • SO (8) gauging:The potential is V = − g [(26 + 2 cosh(2 a ) + cosh[2( a − a )] + 2 cosh(2 a )+ cosh[2( a + a )]) − a sinh (2 a )+ cosh a sinh (2 a ))] . (3.4)This case is very similar to the SO (9) gauging. There are two critical points witha single scalar. – 7 –ritical points a V Preserved supersymmetry1 0 − g (8,1)2 cosh − − g -The A tensor is A (1)1 = diag( − , − , − , − , − , − , − , − , . (3.5)For L = I , the background isometry is given by Osp (8 | , R ) × Osp (1 | , R ). Thecritical point 2 is invariant under G subgroup of SO (8). Apart from the splittingof supercharges and residual gauge symmetry, the critical points in this gaugingare the same as the SO (9) gauging. • SO (7) × SO (2) gauging:In this gauging, the potential is V = − g [9(342 + 40 cosh a + 18 cosh(2 a ) − a − a )+16 cosh( a − a ) + 3 cosh[2( a − a )] + 12 cosh(2 a − a )+8 cosh a + 50 cosh(2 a ) + 16 cosh( a + a ) + 3 cosh[2( a + a )]+12 cosh(2 a + a ) − a + 2 a )) + 8( −
576 cosh a −
3+ cosh a − a (1 + cosh a )) sinh a − − − a ( − a ) + 47 cosh a + 3 cosh(2 a )(1 + cosh a )+6 cosh a (1 + cosh a )) sinh a )] . (3.6)We find one supersymmetric critical point with V = − g , a = cosh − , a = cosh − A tensor A = −
10 0 0 0 0 0 0 0 00 −
10 0 0 0 0 0 0 00 0 − − √ −
10 0 0 0 0 00 0 0 0 −
10 0 0 0 00 0 0 0 0 −
10 0 0 00 0 0 0 0 0 −
10 0 00 0 0 0 0 0 0 6 00 0 − √ . (3.8)– 8 –fter diagonalization, we find A = diag( − , − , − , − , − , − , − , , . (3.9)This is a (1,2) point with SU (2) symmetry. With a = 0, we find the followingcritical pointsCritical points a V Preserved supersymmetry1 0 − g (7,2)2 cosh − − g (0,1) .The corresponding values of the A tensor are A (1)1 = diag ( − , − , − , − , − , − , − , , A (2)1 = diag (cid:18) − , − , − , − , − , − , − , , (cid:19) . (3.10)For L = I , the background isometry is given by Osp (7 | , R ) × Osp (2 | , R ). Thecritical point 2 preserves SU (3) symmetry. The location and value of the cosmo-logical constant relative to the L = I point are similar to the G × G point in SO (8) × SO (8) gauged N = 16 supergravity. In our result, the residual gaugesymmetry is the SU (3) subgroup of G which is in turn a subgroup of SO (7). • SO (6) × SO (3) gauging:We find the potential V = 1128 g ( − −
424 cosh(2 a ) + 6 cosh(4 a ) + 4 cosh[2( a − a )]+4 cosh(4 a − a ) − a − a ) −
208 cosh[2( a − a )]+ cosh[4( a − a )] −
424 cosh(2 a ) + 6 cosh(4 a ) − a + a ) −
208 cosh[2( a + a )] + cosh[4( a + a )]+4 cosh[2(2 a + a )] + 4 cosh[2( a + 2 a )]) . (3.11)One supersymmetric critical point is V = − g , a = cosh − , a = cosh − . (3.12)– 9 –ith the value of the A tensor A = −
16 0 0 0 0 0 0 0 00 −
16 0 0 0 0 0 0 00 0 − − √
30 0 0 −
16 0 0 0 0 00 0 0 0 −
16 0 0 0 00 0 0 0 0 −
16 0 0 00 0 0 0 0 0 8 0 00 0 0 0 0 0 0 8 00 0 − √ . (3.13)This can be diagonalized to A = diag ( − , − , − , − , − , − , , , . (3.14)This is a (1,3) point and has SO (3) ⊂ SO (6) symmetry. With a = 0, we findthe following critical pointsCritical points a V Preserved supersymmetry1 0 − g (6,3)2 cosh − − g (0,2) .The corresponding values of the A tensor are A (1)1 = diag ( − , − , − , − , − , − , , , A (2)1 = diag ( − , − , − , − , − , − , , , . (3.15)For L = I , the background isometry is given by Osp (6 | , R ) × Osp (3 | , R ). Thecritical point 2 is also invariant under SO (3) subgroup of SO (6). • SO (5) × SO (4) gauging:The potential for this gauging is V = 132 g (3 + cosh a cosh a ) ( −
86 + 2 cosh(2 a ) −
24 cosh( a − a )+ cosh[2( a − a )] + 2 cosh(2 a ) −
24 cosh( a + a )+ cosh[2( a + a )]) . (3.16)There is no critical point with two non zero scalars. With a = 0, we find thefollowing critical points: – 10 –ritical points a V Preserved supersymmetry1 0 − g (5,4)2 cosh − − g (0,3) .The corresponding values of the A tensor are A (1)1 = diag ( − , − , − , − , − , , , , A (2)1 = diag ( − , − , − , − , − , , , , . (3.17)For L = I , the background isometry is given by Osp (5 | , R ) × Osp (4 | , R ). Thecritical point 2 preserves SO (4) diag symmetry which is the diagonal subgroup of SO (4) × SO (4) with the first SO (4) being a subgroup of SO (5). We now give some critical points of the non-compact gaugings. The isometry groupof the background with L = I consists of the maximal compact subgroup of the gaugegroup and SO (2 ,
2) as the bosonic subgroup. Using the generators given in the ap-pendix, we can compute the scalar potentials for these two gaugings. Notice that inthe non-compact gaugings, all sixteen scalars are not equivalent. At the maximallysymmetric vacua, the gauge group is broken down to its maximal compact subgroup,and some of the scalars become Goldstone bosons making some of the vector fieldsmassive. This “Higgs-mechanism” results in the propagating n ng massive vector fieldswhere n ng denotes the number of non compact generators which are broken at the crit-ical point. The total number of degrees of freedom remains the same because of thedisappearance of the n ng scalars, Goldstone bosons. For further detail, see [8] in thecontext of N = 16 models. • G − × SL (2) gauging:The coset representative is chosen to be L = e a Y e a Y . (3.18)This parametrization has residual gauge symmetry SU (2) which is a subgroup of G − . With one of the scalars vanishing, L has SU (3) symmetry. The potential– 11 –ith two scalars is given by V = 14608 g [ − − a ) + 70 cosh(4 a ) + 8 cosh(4 a − a )+28 cosh[2( a − a )] + 28 cosh(4 a − a ) −
560 cosh[2( a − a )]+ cosh[4( a − a )] − a − a ) + 56 cosh(4 a − a )+3472 cosh( a ) − a ) −
16 cosh(3 a ) + 198 cosh(4 a ) −
560 cosh[2( a + a )] + cosh[4( a + a )] − a + a )+28 cosh[2(2 a + a )] + 56 cosh(4 a + a ) + 28 cosh[2( a + 2 a )]+8 cosh(4 a + 3 a )] . (3.19)We find the following critical points:critical point a a V preservedsupersymmetries1 0 0 − g (7,2)2 0 cosh − q √ − √ g -3 cosh − − g (0,1)4 cosh − cosh − √ − g (1,2)The corresponding values of the A tensor are A (1)1 = diag (cid:18) − , − , − , − , − , − , − , , (cid:19) ,A (3)1 = − − − − − − −
00 0 0 0 0 0 0 0 (3.20)– 12 –nd A (4)1 = − − − − − − q − − − −
00 0 0 − q . (3.21) A (3)1 and A (4)1 can be diagonalized to A (3)1 = diag (cid:18) − , − , − , − , − , − , − , , (cid:19) ,A (4)1 = diag (cid:18) − , − , − , − , − , − , − , , (cid:19) . (3.22)For L = I , the gauge group is broken down to its compact subgroup G × SO (2).The background isometry is given by G (3) × Osp (2 | , R ). There are two SU (3)points with completely broken supersymmetry (point 2) and (0,1) supersymmetry(point 3). Point 4 has SU (2) symmetry. • Sp (1 , × SU (2) gauging:We choose the coset representative L = e a ( Y − Y ) e a ( Y + Y ) . (3.23)This has symmetry SO (3) × SO (3) if any one of the scalars vanishes. Thisis the case in which our critical points lie. This symmetry is a subgroup ofthe SO (5) × SO (3) compact subgroup of Sp (1 ,
2) with the first SO (3) being asubgroup of SO (5). We find the potential V = 132 g [ − −
232 cosh(2 √ a ) + 6 cosh(4 √ a )+4 cosh[2 √ a − a )] −
112 cosh[2 √ a − a )]+ cosh[4 √ a − a )] + 4 cosh[2 √ a − a )] −
232 cosh(2 √ a )+6 cosh(4 √ a ) −
112 cosh[2 √ a + a )] + cosh[4 √ a + a )]+4 cosh[2 √ a + a )] + 4 cosh[2 √ a + 2 a )]] . (3.24)– 13 –ome of the critical points are given bycritical point a a V preserved supersymmetries1 0 0 − g (5,4)2 0 cosh − √ − g -3 ln(2 −√ √ − g -4 ln(2+ √ √ − g -with the corresponding A tensor A (1)1 = diag ( − , − , − , − , − , , , ,
4) (3.25)for the critical point 1. For L = I , the gauge group is broken down to its compactsubgroup Sp (1) × Sp (2) × SU (2) ∼ SU (2) × SO (5) × SU (2). The two SU (2)’sfactors combine to SO (4) under which the right handed supercharges transformas . So, the background isometry is given by Osp (5 | , R ) × Osp (4 | , R ). Point2, 3, and 4 are SO (3) × SO (3) points with completely broken supersymmetry.We have checked that all critical points given above are truely critical points of the cor-responding potential. In the next section, we will find RG flow solutions interpolatingbetween some of these vacua.
4. RG flow solutions
In this section, we study RG flow solutions in the N = 9 theory whose vacua areobtained in the previous section. We start by giving the general formulae we will usein various gaugings. The strategy to find supersymmetric flow solutions is to find thesolutions to the BPS equations coming from the supersymmetry transformations offermions which in this case, are δχ iI and δψ Iµ .We start by giving an ansatz for the metric ds = e A ( r ) dx , + dr . (4.1)The relevant spin connection is ω ˆ ν ˆ r ˆ µ = A ′ δ νµ (4.2)where hatted indices denote the tangent space indices, ˆ µ, ˆ ν = 0 ,
1. We use the notation ′ ≡ ddr from now on. We then recall the supersymmetry transformations from [5] δψ Iµ = D µ ǫ I + gA IJ γ µ ǫ J ,δχ iI = 12 ( δ IJ − f IJ ) i j D /φ j ǫ J − gN A JIi ǫ J . (4.3)– 14 –e will not repeat the meaning of all the notations here but refer the readers to [5] forthe detailed explanation. Using (2.4), we find, in our normalization, dφ A dr = 16 Tr( Y A L − L ′ ) . (4.4)With this information, we are now in a position to set up the BPS equations which areour flow equations. The δχ Ii = 0 equation gives flow equations for the scalars whilethe δψ Iµ = 0 is used to determine A ( r ) in the metric. In order to obtain the equationfor A ( r ), we impose γ r ǫ I = ǫ I , so the solution preserves half of the original supersym-metries. We now apply this result to various gaugings. In the gauging that admits asupersymmetric flow solution, there must exist at least two AdS supersymmetric crit-ical points with different cosmological constants. The latter is related to the centralcharge of the dual CFT as c ∼ √− V . (4.5)According to the c-theorem, the c-function interpolating between the central charges inthe UV and IR fixed points is a monotonically decreasing function along the flow fromthe UV to the IR. From the previous section, there is no flow solution in the SO (9), SO (8) and Sp (1 , × SU (2) gaugings because there is only one supersymmetric criticalpoint. We start by finding flow solutions in the compact gaugings. SO (7) × SO (2) gauging With a single scalar, the flow equation is given by da dr = g sinh a (3 cosh a − . (4.6)Changing the variable to b = cosh a , we find the solution r = 120 g ln(1 + b ) − g ln( b −
1) + 340 g ln(3 b − . (4.7)The supersymmetry transformation of the gravitino gives dAdr = − g ( b − b ) . (4.8)We can solve this equation to obtain A as a function of b using the equation for dbdr .The solution is A = − ln( b − −
310 ln(1 + b ) + 45 ln(3 b − . (4.9)– 15 –n all these solutions, we have neglected all the additive constants to A and r becausewe can always shift A and r to absorb them. From (4.7), we see that as a = 0, r → ∞ and r → −∞ when a = cosh − . The UV point corresponds to a = 0, and the IRpoint is at a = cosh − . The ratio of the central charges is given by c UV c IR = s V V = 43 . (4.10)At the UV point, the AdS radius is L = g . Near this point, the scalar fluctuationbehaves as δa ∼ e − gr = e − r L . (4.11)Using the argument in [19, 20], we find that the flow is driven by a relevant operatorof dimension ∆ = . In the IR, we find δa ∼ e r L , L = 332 g . (4.12)The corresponding operator is irrelevant with dimension ∆ = . The UV and IRpoints have supersymmetries (7,2) and (0,1), respectively. Our scalars are canonicallynormalized as can be easily checked by looking at the scalar kinetic terms, so we candirectly read off the value of m from the potential. Near the UV point, we find V = − g − g a . (4.13)The mass squared in unit of L is m L = − . The mass-dimension formula ∆(∆ −
2) = m L gives ∆ = in agreement with what we have found from the behavior of thescalar near the critical point. At the IR point, we find V = − g + 20809 g a . (4.14)The mass squared is m L = which gives precisely ∆ = .We now consider a flow solution with two non zero scalars. Unfortunately, we arenot able to find an analytic solution in this case. We do find a numerical solutioninterpolating between maximal supersymmetric point at L = I and the non trivialcritical point with two scalars given in the previous section. We start by giving flowequations da dr = g e a cosh a sinh a e a [3 cosh a (1 + cosh a ) − cosh a − , (4.15) da dr = g
16 ( −
65 + 9 cosh a (1 + cosh a ) − a (7 + cosh a )+3 sinh a + cosh a (47 + 3 sinh a )) sinh a . (4.16)– 16 –hanging the variables to a = cosh − b , a = cosh − b , (4.17)we can rewrite (4.15) and (4.16) as b ′ = g b − b (1 + b ) − b − , (4.18) b ′ = g b − b −
17 + 3 b (1 + b ) − b (7 + b )] . (4.19)It can be easily checked that b = , b = 2 is a fixed point of these equations. In orderto find a numerical solution, we set g = 1 and b = z . Taking b as a function of z , wecan write the two equations as a single equation db dz = 2( − − z + 3(1 + z ) b ) ( − b )( − z ) ( −
17 + 11 z − z ) b + 3(1 + z ) b ) . (4.20)The numerical solution to this equation is shown in Figure 1. The gravitino variationgives dAdr = − g [3 − z + 11 z − z + z ) b + 3(1 + z ) b ] (4.21)or dAdz = (3 − z + 11 z − z + z ) b + 3(1 + z ) b )2( − z )( −
17 + 11 z − z ) b + 3(1 + z ) b ) . (4.22)The numerical solution for A is shown in Figure 2. The UV point is at r → ∞ andhas (7,2) supersymmetries. The IR point has (1,2) supersymmetries and correspondsto r → −∞ . The ratio of the central charges is c UV c IR = 32 . (4.23)The behavior of the fluctuations of a and a near the fixed point can be found bylinearizing (4.15) and (4.16). We find δa ∼ e − r L , δa ∼ e − r L , (4.24)near the UV point with L = g . We see that the flow is driven by a relevant operatorof dimension . Near the IR point with L = g , we find δa ∼ δa ∼ e gr = e r L . (4.25)So, the operator becomes irrelevant at the IR and has dimension ∆ = . We canalso check this by computing the scalar masses from the potential although it is more– 17 –omplicated in this case because we will need to diagonalize the mass matrix. We onlygive the analysis at the UV point. The potential is fortunately diagonal and given by V = − g − g ( a + a ) . (4.26)We find m L = − which gives ∆ = .In all other gaugings studied here, the same pattern appears, and the analysis isthe same. So, we will quickly go through these cases and give only the main resultswithout giving all the details. In particular, we will not give the scalar masses. Thesecan be worked out as above. z Figure 1:
Solution for b ( z ) in SO (7) × SO (2) gauging. z - - A Figure 2:
Solution for A ( z ) in SO (7) × SO (2) gauging. – 18 – .1.2 SO (6) × SO (3) gauging We begin with a flow with one scalar. The flow equation is a ′ = g (2 cosh a −
6) sinh a . (4.27)With a = cosh − b , we find b ′ = 2 g (3 − b − b + b ) . (4.28)This can be solved directly and gives r = 116 ln(9 + 6 b − b ) −
18 ln( b − . (4.29)The gravitino variation gives A ′ = − g a ) −
12 cosh a − . (4.30)The solution is given by A = 34 ln( b − − ln( b − −
14 ln(1 + b ) . (4.31)Near the UV point with (6,3) supersymmetries, the fluctuation behaves as δa ∼ e − r L , L = 18 g . (4.32)The flow is driven by a relevant operator of dimension . At the IR (0,2) point, we find δa ∼ e r L , L = 112 g . (4.33)The operator becomes irrelevant with dimension ∆ = . The ratio of the centralcharges is c UV c IR = 32 . (4.34)We then move to a flow with two scalars. With a = cosh − b , a = cosh − b , (4.35)the flow equations are given by b ′ = g ( b − b − − b + b b ) , (4.36) b ′ = g b − b (1 + b ) − b − b (3 + b )] . (4.37)– 19 –aking b as a function of z = b , we find db dz = 2( − − z + (1 + z ) b )( − b )( − z )( − z − z ) b + (1 + z ) b ) . (4.38)The numerical solution is given in Figure 3. The metric function A can be determinedby using the equation dAdz = − − − z + 5 z − z + z ) b + (1 + z ) b − z )( − z − z ) b + (1 + z ) b ) . (4.39)The numerical solution is given in Figure 4. The linearized equations give δa ∼ δa ∼ e − r L , L = 18 g (4.40)near the UV point. The flow is driven by a relevant operator of dimension andinterpolates between (6,3) and (1,3) critical points. Near the IR, we find δa ∼ δa ∼ e r L , L = 116 g . (4.41)So, in the IR the operator has dimension . The ratio of the central charges is c UV c IR = 2 . (4.42) z Figure 3:
Solution for b ( z ) in SO (6) × SO (3) gauging. – 20 – .5 2.0 2.5 3.0 z - - A Figure 4:
Solution for A ( z ) in SO (6) × SO (3) gauging. SO (5) × SO (4) gauging In this gauging, there is no critical point with two non zero scalars, so there is no flowwith two scalars. The flow equation with one scalar is a ′ = g sinh a (cosh a − . (4.43)The solution for r as a function of b = cosh a is r = 124 g ln( b − − g ln( b −
1) + 112 g ln(1 + b ) . (4.44)The gravitino variation gives A ′ = − g a ) −
20 cosh a − . (4.45)The solution for A as a function of b is A = 23 ln( b − − ln( b − −
16 ln[20(1 + b )] . (4.46)We find the scalar fluctuations near the UV and IR point asUV : δa ∼ e − r L , L UV = 18 g , (4.47)IR : δa ∼ e r L , L IR = 116 g . (4.48)From these, we find that the flow is driven by a relevant operator of dimension . Theoperator has dimension in the IR. The ratio of the central charges is c UV c IR = 2 . (4.49)– 21 – .2 RG flows in non-compact gaugings4.2.1 G − × SL (2) gauging Remarkably, there exist flow solutions in this non-compact exceptional gauging. Westart with a single scalar giving rise to the flow equation a ′ = − g a (cosh a − . (4.50)The solution to this equation is r = 34 g (cid:2)
13 ln( b − −
12 ln( b −
1) + 16 ln(1 + b ) (cid:3) (4.51)where as usual b = cosh a . The equation for A and its solution are given by A ′ = 8 g a − g a (4.52)and A = 5 ln( b − − b − − b )6 . (4.53)The solution interpolates between (7,2) and (0,1) critical points with the ratio of thecentral charges c UV c IR = 54 . (4.54)The linearized equation givesUV : δa ∼ e − r L , L UV = 38 g , (4.55)IR : δa ∼ e r L , L IR = 310 g . (4.56)The flow is driven by a relevant operator of dimension . In the IR, the operator hasdimension .We now move to a flow solution with two scalars. The flow equations are a ′ = − g (cid:2) a − cosh a a ) (cid:3) , (4.57) a ′ = − g
12 [2(9 + 8 cosh a − cosh(2 a )) sinh a − (7 + cosh(2 a )) sinh (2 a )] . (4.58)Using a = cosh − a, a = cosh − b, (4.59)– 22 –e can combine the two equations into one with a being independent variable dbda = ( b − b (3 + a ) + a − a − a − ab + a − . (4.60)We give a numerical solution to this equation in Figure 5. The equation for A is dAda = − − a + a + 2( a − a − b + (3 + a ) b a − ab + a −
4) (4.61)whose solution is shown in Figure 6. The scalar fluctuations are given byUV : δa ∼ δa ∼ e − r L , L UV = 38 g , (4.62)IR : δa ∼ δa ∼ e r L , L IR = 932 g . (4.63)The flow interpolates between the (7,2) and (1,2) points with the ratio of the centralcharges c UV c IR = 43 . (4.64)The flow is driven by a relevant operator of dimension . a b Figure 5:
Solution for b ( a ) in G − × SL (2) gauging.
5. Conclusions
We have studied N = 9 three dimensional gauged supergravity with compact and non-compact gaugings. We have found some supersymmetric AdS vacua corresponding to– 23 – .1 1.2 1.3 1.4 1.5 a - - A Figure 6:
Solution for A ( a ) in G − × SL (2) gauging. some two dimensional CFT’s. We have identified the superconformal groups from theisometry of the AdS backgrounds with L = I . These backgrounds have dual conformalfield theories at their boundaries. We then studied RG flow solutions describing adeformation of the CFT in the UV to the CFT in the IR. In the scalar sector studiedhere, only SO (7) × SO (2), SO (6) × SO (3), SO (5) × SO (4) and G − × SL (2) gaugingsadmit supersymmetric flow solutions. This is because there is only one supersymmetriccritical point in SO (9), SO (8) and Sp (1 , × SU (2) gaugings. This is not unexpected,the bigger gauge groups give rise to a simpler structure of vacua in general. We havefound analytic flow solutions with one active scalar and numerical solutions for theflows with two active scalars. All the flows are operator flows driven by a relevantoperator of dimension . It is interesting to identify the CFT’s dual to these gravitysolutions. Because two dimensional field theories are more controllable and the gravitysolutions correspond to strong coupling limits of the dual field theories, we hope tounderstand many aspects of the AdS/CFT correspondence in the case of AdS /CFT .The higher dimensional origin of many three dimensional gauged supergravities isstill mysterious. Only the case of non semisimple gaugings is known to be related todimensional reductions of higher dimensional theories [23]. It is interesting to studythe non semisimple gaugings in this N = 9 theory although there is another subtletywith the theories with odd N . This is because we cannot obtain these theories directlyfrom dimensional reductions due to the mismatch in the number of supercharges. Thereduced theory, always having even N in three dimensions, needs to be truncated inorder to give odd values of N . The models with compact and non-compact gauge groupsstudied in this paper and elsewhere are not obtainable from dimensional reductions,so it is very interesting to study whether there exist any higher dimensional origin for– 24 –hese models. This will provide an interpretation of our flow solutions and that studiedin [10] in terms of higher dimensional geometries. Acknowledgments
We would like to thank Henning Samtleben and K. S. Narain for valuable discussionsand Ahpisit Ungkitchanukit for reading the manuscript.
A. Essential formulae
In this appendix, we give all necessary formulae in order to obtain the scalar potentialand flow equations. We use the 52 generators of F from [14]. The generators arenormalized by Tr( c i c j ) = − δ ij . (A.1)With this normalization, we find that V αIJ = −
16 Tr( L − T αG LX IJ ) (A.2) V αA = 16 Tr( L − T αG LY A ) (A.3)where we have introduced the symbol T αG for gauge group generators. T αG will bereplaced by some appropriate generators of the gauge group being considered in eachgauging.The following mapping provides the relation between c i and X IJ , generators of SO (9), 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 = 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 = − c , X = c , X = − c , X = − c ,X = − c , X = − c . (A.4)All the f IJ ’s components can be obtained from the structure constants of the [ X IJ , Y A ]given in [14], but we will not repeat them here.In the non-compact G − × SL (2) gauging, we use the following generators. The– 25 –enerators of G − are obtained by using the embedding of G − in SO (7) generatedby X IJ , I, J = 1 , . . . ,
7. The adjoint representation of SO (7) decomposes under G − as → + . (A.5)The generators of G − can be explicitly found by combinations of SO (7) generators[22] T = 1 √ X + X ) , T = 1 √ X − X ) ,T = 1 √ X − X ) , T = 1 √ X − X ) ,T = − √ X + X ) , T = − √ X + X ) ,T = 1 √ X − X ) , T = 1 √ X + X − X ) ,T = − √ X − X + 2 X ) , T = − √ X + X − X ) ,T = 1 √ X + X + 2 X ) , T = − √ X − X + 2 X ) ,T = 1 √ X − X + 2 X ) , T = 1 √ X + X − X ) . (A.6)We have verified that these generators satisfy G algebra given in [21]. The SL (2)generators are J = i √ c + c ) , J = i √ c + c ) , J = 2 c (A.7)which can be easily checked that they commute with all T ’s and form SL (2) algebra.The generators of non-compact Sp (1 ,
2) can be constructed by first finding itscompact subgroup generators Sp (1) × Sp (2) ∼ SO (3) × SO (5). The latter can beobtained by taking SO (8) with generators X IJ , I, J = 1 , . . . ,
8. We then identify the SO (3) generators with X IJ for I, J = 1 , . . . , SO (5) with X IJ for I, J = 4 , . . . , Sp (1 ,
2) can be obtained by taking combinationsof Y A ’s which commute with the SU (2) gauge group. The latter has three generatorsobtained by looking for the combinations of SO (9) generators that commute with SO (3) × SO (5) mentioned above. We find the following gauge generators:– 26 – Sp(1,2): Q = √ c , Q = −√ c , Q = √ c , Q = √ c , Q = −√ c ,Q = √ c , Q = √ c , Q = −√ c , Q = √ c , Q = −√ c ,Q = − c − c , Q = c − c , Q = c + c ,Q = Y + Y , Q = Y − Y , Q = Y + Y ,Q = Y + Y , Q = Y − Y , Q = Y − Y ,Q = Y + Y , Q = Y − Y . (A.8) • SU(2): K = 12 ( c − c ) , K = −
12 ( c + c ) , K = 12 ( c − c ) . (A.9)With these generators and (A.3), we can compute the T-tensors T IJ,KL = V IJ,α V KL,β δ SO ( p ) αβ − V IJ,α V KL,β δ SO (9 − p ) αβ , (A.10) T IJ,A = V IJ,α V A,β δ SO ( p ) αβ − V IJ,α V A,β δ SO (9 − p ) αβ (A.11)for compact gaugings and T IJ,KL = V IJ,α V KL,β η G αβ − K V IJ,α V KL,β η G αβ , (A.12) T IJ,A = V IJ,α V A,β η G αβ − K V IJ,α V A,β η G αβ (A.13)for non-compact gaugings with K being and 12 for G × G = G − × SL (2) and Sp (1 , × SU (2), respectively. We have used summation convention over gaugeindices α , β with the notation δ G and η G meaning that the summation is restrictedto the G generators. References [1] H. Nicolai and H. Samtleben, “Maximal gauged supergravity in three dimensions”,Phys. Rev. Lett. (2001) 1686-1689, arXiv: hep-th/0010076.[2] H. Nicolai and H. Samtleben, “Compact and noncompact gauged maximalsupergravities in three dimensions”, JHEP 0104 (2001) , arXiv: hep-th/0103032.[3] T. Fischbacher, H. Nicolai and H. Samtleben, “Non-semisimple and Complex Gaugingsof N = 16 Supergravity”, Commun.Math.Phys. (2004) 475-496, arXiv:hep-th/0306276. – 27 –
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