Holographic RG flows for four-dimensional \( \mathcal{N}=2 \) SCFTs

Journal of High Energy Physics, Jun 2018

Abstract We study holographic renormalization group flows from four-dimensional \( \mathcal{N}=2 \) SCFTs to either \( \mathcal{N}=2 \) or \( \mathcal{N}=1 \) SCFTs. Our approach is based on the framework of five-dimensional half-maximal supergravity with general gauging, which we use to study domain wall solutions interpolating between different supersymmetric AdS5 vacua. We show that a holographic RG flow connecting two \( \mathcal{N}=2 \) SCFTs is only possible if the flavor symmetry of the UV theory admits an SO(3) subgroup. In this case the ratio of the IR and UV central charges satisfies a universal relation which we also establish in field theory. In addition we provide several general examples of holographic flows from \( \mathcal{N}=2 \) to \( \mathcal{N}=1 \) SCFTs and relate the ratio of the UV and IR central charges to the conformal dimension of the operator triggering the flow. Instrumental to our analysis is a derivation of the general conditions for AdS vacua preserving eight supercharges as well as for domain wall solutions preserving eight Poincaré supercharges in half-maximal supergravity.

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Holographic RG flows for four-dimensional \( \mathcal{N}=2 \) SCFTs

Accepted: June Holographic RG ows for four-dimensional SCFTs Nikolay Bobev 0 1 4 Davide Cassani 0 1 2 Hagen Triendl 0 1 3 Prince Consort Road 0 1 London SW 0 1 0 Via Marzolo 8 , 35131 Padova , Italy 1 Celestijnenlaan 200D , B-3001 Leuven , Belgium 2 INFN , Sezione di Padova 3 Department of Physics, Imperial College London 4 Instituut voor Theoretische Fysica, KU Leuven We study holographic renormalization group ows from four-dimensional N = 2 SCFTs to either N = 2 or N = 1 SCFTs. Our approach is based on the framework of ve-dimensional half-maximal supergravity with general gauging, which we use to study domain wall solutions interpolating between di erent supersymmetric AdS5 vacua. We AdS-CFT Correspondence; Supergravity Models; Extended Supersymmetry - show that a holographic RG ow connecting two N = 2 SCFTs is only possible if the avor symmetry of the UV theory admits an SO(3) subgroup. In this case the ratio of the IR and UV central charges satis es a universal relation which we also establish in eld theory. In addition we provide several general examples of holographic ows from N = 2 to N = 1 SCFTs and relate the ratio of the UV and IR central charges to the conformal dimension of the operator triggering the ow. Instrumental to our analysis is a derivation of the general conditions for AdS vacua preserving eight supercharges as well as for domain wall solutions preserving eight Poincare supercharges in half-maximal supergravity. 1 Introduction 2 Gauged half-maximal supergravity 2.1 Supersymmetric domain walls 3 Holographic ows between N = 2 SCFTs 3.1 3.2 3.3 3.4 3.5 4.1 5.1 5.2 Review of conditions for N = 4 AdS5 vacua Uniqueness in the absence of avor symmetries Two distinct N = 4 AdS5 vacua Conditions for ows with eight Poincare supercharges Flow between two N = 4 AdS5 vacua and its holographic dual 4 Field theory derivation of ratio between central charges Anomalies in four-dimensional N = 2 SCFTs 4.2 RG ow between N = 2 SCFTs 5 Holographic ows from N = 2 to N = 1 SCFTs Conditions for N = 2 AdS5 vacua Review of conditions for minimally supersymmetric ows 5.3 A model with one N = 2 vacuum 5.4 A model with two N = 2 vacua 6 Discussion A Uniqueness of half-maximal AdS solutions in various dimensions A.1 Four dimensions A.2 Six dimensions A.3 Seven dimensions B The generator of the IR U(1)R symmetry was used to obtain a holographic proof of the a-theorem. There have been numerous deformations on the other hand are less accessible with eld theory tools and thus the supergravity results derived here should teach us important general lessons for the structure of supersymmetric RG ows. To understand the general constraints for the existence of distinct supersymmetric AdS5 vacua and the ow connecting them we present a detailed analysis of the supersymmetry conditions in half-maximal gauged supergravity. The results depend on the number, n, of vector multiplets in the theory and on the type of gauging performed. The existence of at least one AdS5 vacuum with 16 supercharges implies that an U(1) SU(2) Hc subgroup of the SO(5; n) global symmetry of the supergravity theory should be gauged [8]. The U(1) SU(2) gauge eld is dual to the R-symmetry of the four-dimensional N = 2 SCFT dual to this AdS5 while Hc represents the continuous avor symmetry. If Hc is trivial we nd that there is a unique AdS5 vacuum with 16 supercharges in the supergravity theory.1 However when Hc is non-trivial then it must contain an SO(3) subgroup and there can be another AdS5 vacuum in the supergravity theory with a di erent value of the cosmological constant. Moreover these two distinct AdS5 vacua are connected by a regular supersymmetric domain wall solution in the gauged supergravity theory which we construct analytically. In addition we establish that the RG ow in the dual QFT should be triggered by vacuum expectation values (vevs) for two scalar operators of dimension = 2 and the ratio of these vevs has to be a xed constant. One of the two scalar operators belongs to the energy momentum multiplet of the SCFT and the other one sits in the SO(3) Hc avor current multiplet. The di erent values of the cosmological constants of the two AdS5 vacua translate into di erent values for the conformal anomalies of the dual UV and IR N = 2 SCFTs. We compute this ratio of central charges using our supergravity results and are able to reproduce it by an anomaly calculation in the dual SCFT. The result is a universal expression for the IR conformal anomalies in terms of the UV conformal anomalies as well as the central charges of the SO(3) avor current. We also nd that these anomalies are related to the constant that controls the relation between the scalar vevs triggering the ow. 1This result can also be established for AdS vacua with 16 supercharges in four-, six-, and sevendimensional half-maximal gauged supergravity. { 2 { HJEP06(218) Having described the conditions for the existence of N = 4 AdS vacua in vedimensional gauged supergravity it is natural to ask whether there are other AdS vacua which preserve less supersymmetry. To answer this we analyze the general conditions for N = 2 AdS5 vacua and then we focus on theories that admit both an N = 4 and one or more N = 2 vacua. Perhaps not surprisingly we nd that as we increase the number of vector multiplets in the supergravity theory we can have an increasing number of distinct N = 2 AdS5 vacua. The details of this structure depend on the matter content and the choice of gauging in the supergravity theory. To illustrate our general approach we focus on two particular examples. We rst establish a holographic analog of the QFT result in [14] in which it was shown that every four-dimensional N = 2 SCFT with an exactly marginal ow was in fact rst constructed and discussed in some particular holographic examples, see [2, 15, 16], but here we o er a more general treatment. Our general setup should capture the RG ow relating the N = 2 and N = 1 Maldacena-Nun~ez SCFTs [17] arising from M5-branes wrapped on a Riemann surface. While it is widely believed that this RG ow exists, and is of the class discussed in [14], its explicit holographic construction is still elusive. Our results should o er some insight into this problem. Moreover, if our setup can be embedded in eleven-dimensional supergravity it can potentially capture holographic RG ows connecting the N = 2 Maldacena-Nun~ez SCFT [17] and one of the N = 1 SCFTs studied in [18, 19]. In addition to this we study a setup with one N = 4 and two distinct N = 2 AdS5 vacua and discuss the supersymmetric domain wall solutions which interpolate between them. This may capture holographic RG ows which relate the N = 2 Maldacena-Nun~ez SCFT and two of the N = 1 SCFTs of [18, 19]. Finally we would like to note that we do not study a speci c embedding of the gauged supergravity theories we work with in string or M-theory. Thus our results are universal and apply to all supersymmetric AdS vacua which admit a lower-dimensional e ective description in terms of half-maximal supergravity. This universality is somewhat similar in spirit to the results for holographic RG ows across dimensions discussed in [20]. We begin our presentation in the next section with a brief general introduction to vedimensional N = 4 gauged supergravity. In section 3 we identify under what conditions there can be two distinct AdS vacua of such a supergravity theory which preserve all 16 supercharges and construct gravitational domain wall solutions interpolating between these vacua. Whenever such a ow is possible it exhibits a universal relation between the UV and IR central charges which we establish by eld theory methods in section 4. We continue in section 5 with a study of the conditions for the existence of AdS5 vacua with 8 supercharges and a discussion on domain wall solutions connecting such vacua. Section 6 is devoted to a short discussion on our results and their implications for holography. In appendix A we present the extension of some of the results in section 3 to half-maximal gauged supergravity in four, six and seven dimensions. In appendix B we give some more details on the ow in section 5. { 3 { Gauged half-maximal supergravity In this section we review the basic properties of ve-dimensional gauged N = 4 (halfmaximal) supergravity [21{24] that are relevant for our analysis, mainly following [24]. Ungauged N = 4 supergravity has USp(4) R-symmetry and consists of a gravity multiplet and n vector multiplets. The gravity multiplet contains the metric g , four gravitini i ; i = 1; : : : ; 4 transforming in the 4 of USp(4), six vectors (dubbed the graviphotons) A0 ; Am, with Am, m = 1; : : : ; 5 transforming in the 5 of USp(4) and A0 being neutral, four spin-1/2 fermions i in the 4 of USp(4), and one neutral real scalar . We will label the vector multiplets with the index a = 1; : : : ; n. Each vector multiplet contains a vector Aa , four spin-1/2 gaugini ai, and parametrize the coset space ve real scalar elds. All together the scalar elds Mscal = SO(1; 1) SO(5; n) SO(5) SO(n) ; ds2(Mscal) = 3 2 d 2 dMMN dM MN : 1 8 The isometry group of the scalar manifold, SO(1; 1) SO(5; n), is the global symmetry group of the ungauged supergravity action. In addition, the scalar eld space admits a local invariance under SO(5) SO(n). The group SO(5) is promoted to Spin(5) ' USp(4) when discussing the couplings to the fermions. It is then convenient to convert the SO(5) index m of the scalar vielbeine VM m into USp(4) indices i; j via SO(5) gamma matrices, This satis es VM ij = VM [ij] and ij VM ij = 0 and hence transforms in the 5 of USp(4). Here ij is a 4 4 real symplectic matrix. where the rst factor is spanned by while the second factor is spanned by the scalars in the vector multiplet, which we denote by x, x = 1 : : : ; 5n. We indicate the coset representative of the second factor by V = (VM m; VM a), where M = 1; : : : ; n + 5 labels the fundamental representation of SO(5; n). Being an element of SO(5; n) this obeys MN = VM m VN m + VM aVN a ; where MN = diag( 1; 1; 1; 1; 1; +1; : : : ; +1) is the at SO(5; n) metric, which is also used to raise and lower the M; N indices (while the m; n and a; b indices are contracted with the SO(5) and SO(n) Kronecker delta, respectively). Alternatively, the coset can be represented by the positive de nite scalar metric MMN = VM m VN m + VM aVN a ; which also plays the role of the gauge kinetic matrix for the (5 + n) vector elds AM = (Am; Aa ). The metric on the scalar manifold, which determines the scalar kinetic terms, is 1 2 VM ij = VM m imj : { 4 { (2.1) (2.2) (2.3) (2.4) (2.5) In gauged supergravity a subgroup of the global symmetry group SO(1; 1) SO(5; n) is promoted to a local gauge symmetry by introducing minimal couplings to the gauge elds and their supersymmetric counterparts. In this way part of the global symmetry group is broken. When some vector elds transform in non-trivial non-adjoint representations of the gauge group, additional Stuckelberg-like couplings to antisymmetric rank-two tensor elds may be required in order to ensure closure of the gauge symmetry algebra. Such vector elds can then be gauged away, leaving just massive tensor elds together with the other vectors [21, 23, 24]. The possible gaugings are classi ed by the embedding tensor formalism [25{27]. This introduces the gauge couplings via a spurionic object | the embedding tensor | and elds that consist of a tensor eld for each of the original vector elds. In N = 4 supergravity, the embedding tensor splits into three di erent representations of SO(1; 1) SO(5; n), denoted by M ; MN = [MN] and fMNP = f[MNP ]. Their transformation under SO(5; n) follows from the indicated index structure. With respect to SO(1; 1), M and fMNP carry charge 1=2, while MN has charge 1. Supersymmetry of the Lagrangian imposes a set of quadratic constraints on the embedding tensor, whose possible solutions parametrize the di erent consistent gauged N = 4 supergravity theories. In this paper we are interested in theories admitting at least one fully supersymmetric AdS5 vacuum. In [6] it was shown that a necessary condition for this is M = 0. This means that the SO(1; 1) part of the global symmetry is not involved in the gauging and the gauge group is entirely contained in SO(5; n). We therefore take M = 0 from now on. In this case, the quadratic constraints are simply given by fR[MN fP Q] R = 0 ; M QfQNP = 0 : The fMNP correspond to structure constants for a (non-Abelian) subgroup of SO(5; n), while the MN assign the charges under the U(1) gauge eld A0 . The embedding tensor determines the gauge covariant derivatives, D = r AM fM NP tNP A 0 NP tNP ; where tMN = t[MN] generate so(5; n). It also determines the shift matrices that appear in the fermion supersymmetry variations and specify the scalar potential. In the following we abbreviate the contraction of the embedding tensor components f MNP and MN with the coset representatives VM m and VM a by f^mnp = f MNP f^mna = f MNP f^mab = f MNP f^abc = f MNP VM VM VM m m m VN nVP p ; VN nVP a ; VN aVP b ; VM aVN bVP c : ^mn = ^ma = ^ab = MN MN MN VM VM m m VN n ; VN a ; VM aVN b ; These \dressed" embedding tensor components will always be denoted by a hat symbol. Since they depend on the scalars, generically they vary along domain wall solutions. Also, they appear in the conditions for supersymmetric AdS vacua. { 5 { (2.6) (2.7) (2.8) where A(r) is the warp factor which depends only on the radial coordinate r. The oneand two-form supergravity elds vanish, while the scalars have a radial pro le, = (r), x = x(r). In particular, when the solution is AdS5, the scalars are constant and we have A = r=`, where ` is the AdS radius. The latter is related to the cosmological constant, which in our conventions is the same as the critical value of the scalar potential, V = 6=`2. The supersymmetry conditions for solutions of this form (and with M = 0) read [28] i x0vxa ij 5 j 2P a ij j = 0 ; SO(5;n) the SO(5) SO(n) scalar manifold, de ned as where i are the supersymmetry parameters, satisfying the symplectic-Majorana condition i = ij C( j )T . A prime means derivative with respect to r and vxam are the vielbeins on d xvam = x ( V 1 dV)am : Moreover we introduced the shift matrices where "mnpqr is the totally antisymmetric symbol, and P ij = P mn mnij ; with P mn = 2 ^mn + 1"mnpqrf^pqr ; 1 36 P aij = 1 p 1f^amn mn ij : 1 p The shift matrices also determine the scalar potential as 8 2 3 P ij Pij : The supersymmetry conditions above are obtained by setting to zero the fermion variations given in [24].2 Eqs. (2.10), (2.11) arise from the gravitino variation, (2.12) arises from the variation of the spin 1/2 fermion in the N = 4 gravity multiplet, while (2.13) q 38 @ P ij, A2a ij = p12 P a ji. For the scalar manifold geometry and the Cli ord algebra we use the same conventions as in [28]. We have reabsorbed the gauge coupling constant g appearing in [24] into the embedding tensor. comes from the gaugino variation. The derivation of (2.10), (2.11) assumes that the supersymmetry parameters depend on the coordinate r but are constant on R1;3; this means that we are only describing the Poincare supersymmetries. For generic domain walls these are all the supersymmetries allowed, however in the special case of AdS solutions one also has the conformal supersymmetries, which depend on the coordinates on R1;3. For this reason, the case of AdS solutions will be analyzed separately in the next sections. As we discuss in detail later, the domain wall supersymmetry conditions imply the existence of a real superpotential function W constructed out of the shift matrix P mn, which drives the ow of the warp factor and the scalar elds. Introducing an index X = (0; x), we can denote the scalars as X = ( ; x) and the scalar kinetic matrix as Then the ow equations read gXY = 3 0 However, this is not the full information encoded into supersymmetry. Indeed, one also nds a set of algebraic constraints restricting the scalar elds that can possibly ow. After these constraints are satis ed, the scalar potential (2.17) can be expressed in terms of the superpotential as This is su cient to ensure that the Einstein and scalar equations of motion are satis ed [29, 30]. When in particular the superpotential is extremized, @X W = 0, we obtain an AdS solution with radius ` 1 = W . The speci c form of the superpotential and of the constraints depends on the amount of supersymmetry being preserved and will be discussed in the next sections. 3 Holographic ows between N = 2 SCFTs In this section, we rst review the conditions for fully supersymmetric AdS5 vacua in halfmaximal gauged supergravity. Then we show that if there is one such vacuum and the gauge group does not contain any compact part in addition to the U(1) SU(2) R-symmetry of the vacuum, then the latter is unique, up to moduli. If on the other hand there is one N = 4 vacuum preserving an SO(3) in addition to the R-symmetry and a certain condition on the gauge coupling constants is satis ed, then we show that there exists a second N = 4 AdS vacuum and we construct an explicit ow connecting the two. 3.1 Review of conditions for N = 4 AdS5 vacua It was shown in [8] that the necessary and su cient conditions for ve-dimensional halfmaximal supergravity to admit a fully supersymmetric AdS5 solution amount to a simple { 7 { set of constraints on the dressed components of the embedding tensor. In addition to the aforementioned M = 0, these conditions read: ^[mn ^pq] = 0 ; ^ma = 0 ; f^mna = 0 ; p where necessarily ^mn and f^mnp are not identically zero.3 The rst condition arises from the gravitino equation while (3.2){(3.4) are equivalent to P a ij = @ P ij = 0. The AdS cosmological constant is read from the scalar potential (2.17) and is The conditions above imply [8] that the theory has gauge group G = U(1) Hnc Hc SO(5; n) ; where Hc SO(n) is a compact semi-simple subgroup, while Hnc is a generically noncompact group admitting SO(3) as maximal compact subgroup. If Hnc is simple, it can be either SO(3), SO(3; 1), or SL(3; R). When ^ab = 0, the product of the U(1) factor in G with the SO(3) subgroup of Hnc embeds block-diagonally as SO(2) SO(3) in SO(5). 6 If ^ab = 0, the U(1) factor is a diagonal subgroup of SO(2) In the vacuum, the gauge vectors of U(1) and of SO(3) SO(5) and SO(2) SO(n). Hnc are graviphotons, with U(1) being always gauged by the vector A0, while the gauge vectors of Hc and of the noncompact generators of Hnc belong to vector multiplets. The non-compact part of Hnc is spontaneously broken and the corresponding gauge vectors are massive. Finally, the vectors that are charged under the U(1) factor of the gauge group are eaten up by antisymmetric rank-two tensor elds via the Stuckelberg mechanism. In total, the AdS vacuum is invariant under U(1) SU(2) Hc. The U(1) SU(2) corresponds to the R-symmetry of the dual N = 2 SCFT, while Hc represents the avor group of that SCFT. These properties are most easily seen if we perform a global SO(1; 1) SO(5; n) transformation sending the N = 4 critical point to the origin of the scalar manifold, so that and (VM m; VM a) is the identity element of SO(5; n). By further making an SO(5) = 1 SO(n) transformation, we can choose AB, where A; B; C = 6; 7; : : : ; n + 5. Then f 123 are SU(2) structure constants, while the 3Condition (3.4) di ers by a factor of 2 from the one given in [8] because we are including a factor of 1=2 in the map (2.5) and when evaluating the shift matrices of [24] we are taking VP m = P QVQm. See footnote 5 in [8]. { 8 { (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) non-vanishing 45 implies that the vectors A4 , A5 are eaten up by tensor elds. Moreover, f 1AB; f 2AB; f 3AB complete the SU(2) structure constants to those of Hnc, while f ABC are the Hc structure constants. From (3.5) we nd that the cosmological constant is VM 5 such that 45 is invariant, i.e. The N = 4 vacuum may admit a set of moduli, namely at directions of the scalar potential along which full supersymmetry is preserved. These are deformations of VM 4 and It was proven in [8] that these moduli span the space U(1) SU(m) for some m. SU(1;m) 3.2 Uniqueness in the absence of avor symmetries In the absence of any avor symmetries Hc we can prove that there cannot be two N = 4 AdS5 solutions with di erent values of the cosmological constant. We arrive at this result by showing that in any two such solutions the contractions ^mn ^mn and f^mnpf^mnp must take the same value. From (3.4) we infer that the SO(1; 1) scalar is also unchanged. Then from (3.5) we conclude that the cosmological constant takes the same value in the two solutions. ^ V T V = We rst consider the MN components of the embedding tensor, in their dressed version ^mn ^mb ^an ^ab . The supersymmetry conditions (3.2), (3.1) and the spectral theory of real, antisymmetric matrices imply that by a local SO(5) evaluated on the solution can be put in the canonical block-diagonal form: SO(n) transformation, ^ ^ = diag ( 0; 0; 0; ; 1 ; 2 ; : : : ; p ; 0; : : : ; 0 ) ; i are the only non-vanishing eigenvalues of ^mn and where = 0110 , while ^ab = 0 there are no i 2; : : : ; i p are the non-vanishing eigenvalues of ^ab. It is understood that when eigenvalues. Let us now assume there are two di erent eld (3.8) (3.9) (3.10) i 1, con gurations corresponding to N = 4 AdS5 solutions. The two corresponding vielbeins V are related by an SO(5; n) transformation. However the latter cannot change the eigenvalues of ^, neither can it reshu e the eigenvalue with the 's, because the former lives in the timelike eigenspace while the latter live in the spacelike eigenspace. It follows that ^ is the same in the two vacua up to SO(5) is the same in the two vacua. SO(n) transformations. In particular, ^mn ^mn = 2 2 We now turn to the f MNP components of the embedding tensor. We can assume with admit an SU(2) no loss of generality that one of the N = 4 AdS5 solutions sits at the origin of the scalar manifold. In an SO(5) gauge such that (3.7) is true, the other N = 4 AdS vacuum must Hnc gauge group with structure constants f^123 = VM 1VN 2VP 3f MNP : The choice of an SU(2) subgroup inside Hnc is described by the coset Hnc=SU(2). Hence { 9 { the rst three components of the coset representative in the two vacua are related as (3.11) (3.12) HJEP06(218) VM VM VM where (fc)mb are the non-compact generators of Hnc and c are free real parameters. These transformations have been identi ed in [6, 8] as the Goldstone bosons for the spontaneous breaking Hnc ! SU(2). Since the V's in the two AdS5 vacua are related by a gauge transformation, the structure constants f^mnp should be preserved. This can easily be seen at rst order in c recalling that (3.3) holds for the vacuum at the origin: f^123 = VM 1VN 2VP 3f MNP = M 1 N 2 P 3f MNP = f 123 + 3 cfca[1f 23]a + O( 2) = f 123 + O( 2) : (3.13) In particular, f^mnpf^mnp takes the same value in the two vacua. This concludes our proof. We remark that a similar argument of uniqueness for fully supersymmetric AdS vacua when Hc is trivial can be derived in N = 4 supergravity in dimensions four, six and seven. We provide this in appendix A. Two distinct N = 4 AdS5 vacua 3.3 Hc Now let us assume that the Hc part of the gauge group is non-trivial. Since by de nition SO(n) and does not contain any U(1) factor, a non-trivial Hc must contain an SO(3)c subgroup. As we are going to show, in this case one may have multiple fully supersymmetric vacua by modifying the choice of the SO(3) subgroup corresponding to the SU(2) U(1) vacuum R-symmetry within the full gauge group G given in (3.6). We will assume in the following that the rst vacuum is set at the origin of the scalar manifold and is invariant under Hc (hence the dual SCFT has Hc avor symmetry). In the second vacuum, the U(1) part of the R-symmetry must also be a diagonal subgroup of SO(2) SO(5) and SO(2) SO(n). Since A0 is the gauge vector of that U(1) globally over scalar eld space, this can only be if VM 4 and VM 5 di er from their values in the rst vacuum by moduli, that is ^45 = 45, as discussed in section 3.1. Therefore the two vacua are only distinguished by the values of VM then means that in the second vacuum we m for m = 1; 2; 3. The condition (3.3) nd an SO(3)2 subgroup of G that is gauged by A^m = AM SO(3)1 SO(3)c VM m; m = 1; 2; 3. Most generally this subgroup can be a subgroup SO(3)2 SO(3)c, where SO(3)1 is part of the R-symmetry in the original vacuum, while Hc. We can use SO(5; n) rotations to choose this SO(3)c group to be in the M = 6; 7; 8 directions at the origin. We will denote the SO(3)c structure constants by (3.14) where is a real constant, while as before we will take . . . cosh 2 sinh 2 0 0 0 0 0 0 . . . 0 0 0 0 0 0 . . . cosh 3 0 0 sinh 3 0 0 With the choice above for the embedding tensor and for the scalar elds, the only non-trivial N = 4 condition on the scalars m is given by (3.3), which leads to tanh m tanh n = tanh p ; with (m; n; p) cyclic permutations of (1; 2; 3). Apart for the trivial solution m = 0, these equations have the solution 1 = 2 = 3 = (or 1 = 2 = 3 = , etc.), with This implies that a second vacuum can only exist for for the gauge coupling constant of SO(3)1 U(1). The gauge elds of SO(3)1 are thus A1;2;3, those of SO(3)c are A6;7;8, while A4;5 are eaten up by tensor elds since they are charged under the U(1) gauged by A0. As already seen before, the embedding tensor above leads to an N = 4 vacuum at the origin of the scalar manifold, with cosmological constant V = assume that in the second vacuum the coset representative VM 2 3 g2. We can also m has for m = 1; 2; 3 only More explicitly, its non-trivial part is N]P are the generators of so(5; n) in the fundamental representation. In that vacuum, we nd that the coupling constant of SO(3)2 is Using this and the fact that ^45 = p12 g, we nd from (3.4) that the scalar is tanh = : j j < 1 : In order to identify which gauge symmetries are spontaneously broken we study the covariant derivative of the scalar elds around the second vacuum. Starting from (2.7), one can see that in general the scalar covariant derivative reads D am = d am ^amA0 + f^amnA^n f^ambA^b ; where A^n = AP VP n, A^a = AP VP a are dressed vectors and we have de ned d am We expand the covariant derivative at rst order in the eld uctuations around the second vacuum. In particular the constants Then (3.5) gives for the cosmological constant are non-zero and lead to D( 17 26) = d( 17 while 17 + 26 remains uncharged (here 6; 7; 8 denote the values taken by the a index). One also has similar expressions for simultaneous cyclic permutations of the indices m = (123) and a; b = (678). It follows that A^a = 1 2 1=2 Aa Aa 5 , with a = 6; 7; 8, are all massive, and the gauge group SO(3)1 SO(3)c is indeed broken to the diagonal subgroup SO(3)2 with structure constant (3.21), gauged by A^m = (1 2) 1=2(Am A5+m), for m = 1; 2; 3. If moreover SO(3)c is part of a larger gauge group Hc, and there are other generators of Hc that do not commute with SO(3)c, then the constants f MNP ; M = 6; 7; 8 and N; P > 8 are non-zero. In the second vacuum this leads to non-vanishing structure constants given by f^mab = sinh f (M=m+5)(a=N)(b=P ) that give a mass to the gauge vectors corresponding to those symmetries. That means that SO(3)1 Hc is spontaneously broken to the product of SO(3)2 with the maximal commutant of SO(3)c in Hc . We emphasize that by the procedure above we nd a possible second N = 4 vacuum for every inequivalent embedding of SO(3)c into Hc such that the condition (3.20) holds. In section 3.5 we present a domain wall solution between the two N = 4 vacua above and discuss its holographic interpretation. 3.4 Conditions for ows with eight Poincare supercharges Domain wall solutions preserving eight of the sixteen supercharges were only partially discussed in [28]. Here we provide their complete characterization (when M = 0), which to the best of our knowledge has not appeared in the literature before.4 Starting from the gravitino shift matrix P de ned in (2.15), we introduce the superpotential W = p2 PmnP mn : (3.27) 4The analysis is also similar in spirit to the one performed in N = 2 supergravity in [31]. Then the supersymmetry conditions are equivalent to the ow equations together with the constraints A 0 = W ; P [mnP pq] = 0 ; W 1P mn = 0 ; f^amnPmn = 0 ; 3"mnpqrP pq ^ra = Pp[mf^n]pa : (3.28) (3.29) (3.30) (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) (3.37) (3.38) (3.39) When these constraints are satis ed, the scalar potential (2.17) can be written in terms of the superpotential as 3 2 Clearly, the ow equations and the form of the potential agree with (2.19), (2.20). One can show that if the constraints (3.31){(3.34) are satis ed and the superpotential W is extremized, then the N = 4 AdS conditions of section 3.1 are recovered. In other words, the xed points of ows preserving eight supercharges are N = 4 AdS solutions. The converse implication is of course also true, as an N = 4 AdS5 solution can be seen as a domain wall preserving eight Poincare supercharges and having constant scalars. Proof. Let us prove the supersymmetric ow equations above. We start from the gravitino equation (2.10). Multiplying by P we obtain P mnP pq( mnpq)ij j = h2PmnP mn A In order to solve this equation while preserving eight degrees of freedom in the supersymmetry parameter i, we need the two sides to vanish separately [28]. In this way we obtain the constraint (3.31) and the evolution equation (3.28) for the warp factor, where W is de ned as in (3.27). Since now we can write so that I2 = the form of the projector which precisely reduces the number of independent components in i by half. 1 is an almost complex structure. Then the gravitino equation (2.10) takes PikPkj = W 2 ij ; P = W I ; I j i j + i 5 i = 0 ; Using the relations just obtained, the di erential equation (2.11) for the spinor is solved by i = eA=2^i ; where ^i is a covariantly constant spinor on R1;3 (with the covariant derivative including the USp(4) connection). We now pass to the supersymmetry condition (2.12). Since it has to hold for any spinor satisfying the projector (3.39), it must be that x0vxa ij = 2P a ikIkj : P a ij Iij = 0 ; P a(ikIj)k = 0 ; 21 vya ij P a ikIkj = gyx x0 ; which is equivalent to 0 ik 2 because the terms linear in 5 cannot compensate the others and thus have to vanish separately. Using (3.38) and noting that I2 = 1 implies Tr(I@ I) = 0, gives the ow equation (3.29) for , together with constraint (3.32). It remains to discuss the supersymmetry equation (2.13). The same argument used to manipulate equation (2.12) allows to infer that (2.13) together with the projection (3.39) is equivalent to Separating the terms transforming in di erent irreducible representations of USp(4), we get (3.40) (3.41) (3.42) (3.43) (3.44) (3.45) (3.46) where to obtain the last equation we used vxa ij vyaij = 4gxy. Recalling the de nition of the gaugino shift matrix (2.16), the rst and the second are easily seen to correspond to constraints (3.33) and (3.34), respectively. The third instead gives the ow equation (3.30), because 21 vya ij P a ikIkj = This can be seen by an explicit computation: evaluating the derivative of (3.27) one nds p 2 2 ^anInm + 1"mnpqrInpf^qra ; 1 2 where we used DxVM m = VM avxam. evaluating 12 vya ij P a ikIkj . This concludes our proof. Exactly the same expression is obtained by 3.5 Flow between two N = 4 AdS5 vacua and its holographic dual We now construct a ow connecting the two N = 4 AdS5 vacua discussed in section 3.3. This should correspond to a holographic RG ow connecting two N = 2 four-dimensional SCFTs. We preserve all the eight Poincare supersymmetries along this ow and these get enhanced to sixteen at the AdS xed points by the eight additional conformal supercharges. We again use the local symmetry on the scalar manifold to choose the relevant components of the embedding tensor as in (3.14), (3.15). We see from the solution for the second N = 4 vacuum that besides the only owing scalars should be 1 parametrization (3.17) of the coset representative. Since we do not want to break the diagonal SO(3)2 symmetry along the ow, we set the three scalars equal to each other, 1 = 2 = . We can then construct the shift matrix (2.15) and the superpotential (3.27). We obtain: with the superpotential being P mn = W 4[m 5n] ; (3.47) where we are assuming g > 0 for simplicity.5 It is easy to check that the constraints (3.31){ (3.34) are satis ed with no further assumptions. The metric on the subspace spanned by the two scalars is computed from (2.4) and reads V = The scalar potential is 2) : The superpotential and the scalar potential are related as in (2.20), namely 4 1 cosh3 sinh3 + 2 sinh4 (cosh(2 ) + 2) fully supersymmetric AdS vacua, that is the one at the origin, and the one at non-trivial values of the scalar elds, = 2 1=6 ; = = log ; W = V = We recall that we should impose < 1 in order to have a well-de ned vacuum. It is easy to compute the masses of the scalar elds at these two vacua. They are given by the eigenvalues of the matrix gXY @X @Y V where gXY is the inverse of the scalar metric. 5Strictly speaking, formula (3.27) for the superpotential yields the absolute value of the right hand side of (3.48). However assuming g > 0 we see that both in the rst vacuum ( = 1; = 0) and in the second vacuum the right hand side of (3.48) is positive; we can thus remove the absolute value. It is useful to compute the dimensionless scalar mass, i.e. the combination m2`2 where ` is the scale of AdS. At the UV vacuum one nds At the IR vacuum one has We can now employ the holographic identity m2`2 = ( 4) to extract the conformal dimensions of the operators dual to the two scalars at the UV and IR AdS vacua. At the UV vacuum we nd that both scalars are dual to operators of dimension = 2. In the IR vacuum is still dual to an operator of dimension = 2 and is thus relevant, however the operator dual to is irrelevant and has dimension = 6. Notice that in an N = 2 SCFT the energy-momentum multiplet contains the SU(2) U(1) R-current as well as a real operator of dimension 2 (see for example page 18 in [32]). We thus nd that the conformal dimensions computed in (3.54) and (3.55) are consistent with identifying the scalar as the gravitational dual to the operator of dimension 2 in the energy-momentum multiplet. This is also consistent with the supergravity analysis since sits in the gravity multiplet of ve-dimensional half-maximal supergravity. Through similar reasoning one nds that the operator dual to the scalar is the bottom component in the UV SO(3) avor current multiplet. This operator is sometimes referred to as momentum map operator. It transforms as a triplet of both the R-symmetry and the avor SO(3)'s and we are giving a vev to the component invariant under the diagonal SO(3) subgroup. The value of the cosmological constants at the two AdS vacua in (3.54) and (3.55) determines the ratio of the central charges of the dual SCFTs, see for example [2]. We nd cIR = cUV VIR VUV 3=2 2 : Since 2 < 1 this result is compatible with the a-theorem. Notice that this is also the same ratio as (gIR=g) 2, where gIR is the gauge coupling of the IR R-symmetry, given in (3.21). The ow equations generated by the superpotential (3.48) via (3.28){(3.30) read (3.54) (3.55) (3.56) (3.57) (3.58) It is possible to solve analytically for one nds that the solution for is and A as a function of . After a short calculation 0 = 0 = 1 3 g A0 = W : 1 3 g 3 + g cosh3 ; 1 sinh cosh cosh 1 sinh ; cosh 1 sinh 1=3 ( ) = (cosh(2 ) + c1 sinh(2 ))1=3 ; 1.5 1.0 red/blue lines are the values for the scalars at the IR vacuum in (3.53). We have xed g = 1 and 1 = 1:1. Right : numerical solution for A(r) for the same values of g and . The IR/UV is at large negative/positive values of r. The function A(r) is linear in these regions and the scalars attain their xed point values as expected from (3.52) and (3.53). HJEP06(218) in (3.53) we should x c1 = solution for the warp factor, where c1 is an integration constant. In order for the solution to reach the IR AdS vacuum + 1 . In a similar way one can nd the following A( ) = log 1 6 1 cosh )(cosh sinh3(2 ) 1 sinh ) 3 + c2 ; (3.59) where c2 is a trivial integration constant that can be set to any desired value by shifting the radial coordinate r. The asymptotic behavior of A close to the two AdS vacua is AUV 1 2 log ; 1 2 AIR log( ) : This is the expected divergent behavior of the metric function close to the two AdS vacua. One can plug the analytic solution for ( ) back into the second equation in (3.57) and solve for the function (r) in quadratures. Then one can use this solutions to nd also the functions (r) and A(r). We were not able to solve for (r) analytically, however a typical numerical plot for the scalars and metric function is not hard to generate, see gure 1. It is also instructive to analyze the ow close to the UV AdS vacuum in order to understand what drives it. We can linearize the ow equations in (3.57) around the vacuum Using that the AdS scale is ` = 2=g we nd the approximate solution v e 2r=` ; 1 + v e 2r=` ; (3.60) (3.61) (3.62) Since the scalars and are dual to operators of dimension 2 in the SCFT we can conclude that the RG ow is driven by vacuum expectation values for these two operators. If there were explicit sources for the operators the approximate UV solution should have had an re 2r=` term in the asymptotic expansion. This is clearly absent in our setup. -10 -5 5 10 4 2 -2 -4 -6 -8 A g 2 : r ` : Expanding the explicit analytic solution in (3.58) around the UV AdS vacuum at We thus conclude that the constants v and v in (3.62) are related by 1 + 3 + : : : : v = v : 3 It would be interesting to understand eld-theoretically the corresponding relation between the operator vevs. Field theory derivation of ratio between central charges (3.63) (3.64) (4.1) (4.2) (4.3) (4.4) We pause here our supergravity analysis and present a eld theory explanation for the ratio of UV and IR central charges of N = 2 SCFT's found holographically in (3.56). Anomalies in four-dimensional N = 2 SCFTs 't Hooft anomalies are6 The R-symmetry of four-dimensional N = 1 SCFTs is U(1)RN=1 . The cubic and linear Tr(RN3 =1) and Tr(RN =1) : Via N = 1 supersymmetric Ward identities these anomalies are related to the conformal anomalies by the well-known relations [33] a = 9 32 Tr(RN3 =1) 3 32 Tr(RN =1) ; c = 9 32 Tr(RN3 =1) 5 32 Tr(RN =1) : For four-dimensional N = 2 SCFTs the R-symmetry is SU(2)R U(1)RN=2 . The generators of SU(2)R are denoted by Ia, a = 1; 2; 3.7 There is a unique N = 1 superconformal subalgebra of the N = 2 superconformal algebra. This xes how the U(1)RN=1 is embedded into the Cartan of the SU(2)R U(1)RN=2 R-symmetry, see for example [14, 34], RN =1 = 1 N =1 that is used to compute the conformal anomalies via (4.2). Continuous avor symmetries in four-dimensional N = 2 SCFTs are characterized by a avor central charge kF given by the 't Hooft anomaly (see eq. (2.6) of [34]) where Ta are the generators of the avor group. 6The Tr symbol in all equations below should be understood formally. In the presence of a Lagrangian it indicates a trace over the charges of the chiral fermions in the theory. 7The indices a; b used in the present eld theory section are unrelated to the SO(n) indices used in the rest of the paper. 4.2 ow between N = 2 SCFTs We are interested in an RG ow which connects two distinct four-dimensional N = 2 SCFTs. In parallel with the supergravity setup, assume that the UV SCFT has SU(2)R U(1)RN=2 and an SU(2)F avor symmetry.8 The generators of the avor symmetry algebra in the UV will be denoted by Ta. In the IR SCFT the symmetry is S^U(2)R U(1)RN=2 where S^U(2)R is the diagonal subgroup of SU(2)R are computed by (4.2) using the generator SU(2)F . The UV conformal anomalies (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) Tr(RN2 =2Ta) = Tr(T3T3T3) = Tr(Ia) = Tr(Ta) = 0 : With this at hand it is easy to show that Tr[(RNIR=1)3) = Tr[(RNUV=1)3] Tr(RNIR=1) = Tr(RNUV=1) : 8 9 kF ; Using these identities we arrive at the following simple relations between the UV and IR conformal anomalies aIR = aUV 1 4 kF ; cIR = cUV 1 4 kF : In unitary SCFTs one can show that kF > 0 so the result above is in harmony with the a-theorem.9 For theories with a = c, such as the large N theories described by our holographic setup, the result (4.9) can be written as while for the IR conformal anomalies we use the generator RNUV=1 = 1 where we are assuming that the SU(2)R and SU(2)F generators are normalized in the same way, so that the respective structure constants are the same. We now note that the following identities are true due the properties of the generators of SU(2)R and SU(2)F kF = 1 + 418 Tr(R3 Tr(RN =2T3T3) N =2) + Tr(RN =2I3I3) : Now we can use the AdS/CFT dictionary to compare this expression with our supergravity results. The relation between the SCFT symmetry generators and the supergravity vectors gauging that symmetry is RN =2 ! sA0 ; I3 ! g below applies to an SU(2) subgroup of the avor group. and (4.17) of [35]. 8This analysis can be generalized to a more general avor symmetry group. In that case the discussion 9Notice that there are stronger unitarity bounds on the avor central charge given in Equations (4.16) where the 1=g and =g rescalings are introduced because in the conventions of section 3.3 the supergravity vectors A1;2;3 and A6;7;8 are gauging the SO(3)1 and SO(3)c groups with gauge couplings g and g= , respectively, while we have assumed that Ia and Ta have the same structure constants. Moreover, s is a real constant that is taking care of any potential rescaling of the A0 gauge eld in order to match CFT and supergravity conventions. It turns out that the speci c value of this constant is not important for our analysis. Using (4.11), the 't Hoof anomalies translate into coe cients of supergravity topological terms as Tr(RN3 =2) ! s3d000 ; Tr(RN =2I3I3) ! g2 d033 ; s Tr(RN =2T3T3) ! s 2 where we are omitting a possible overall factor that will not play any role in our calculation. Therefore in supergravity language the expression in (4.10) reads VIR VUV 3=2 = 1 + 2 d088 s24g82 d000 + d033 : In ve-dimensional half-maximal supergravity, the coe cients d000, d033, d088 are components of a symmetric tensor d MN P , with M; N ; P = f0; M g = 0; 1; : : : 5 + n, that controls the topological term. In particular, the gauge variation of the topological term contains d MN P HM^HN ^ AP , where HM are covariant eld strengths [24]. Crucially, the only nonzero components of the d MN P tensor are d0MN = dM0N = dMN0 = MN . Plugging d000 = 0 and d088 = d033 into (4.13) we obtain precisely the relation (3.56) we found in supergravity. Thus we nd that the ratio of central charges of the UV and IR N = 2 SCFTs which we found in supergravity is precisely reproduced by the anomaly matching calculation above. The discussion above also provides a eld theory counterpart of the constant entering in the supergravity embedding tensor and controlling the relation between the vevs of the operators triggering the ow. Comparing (3.56) with (4.9), we obtain 2 = kF : 4cUV The existence of the holographic RG ow imposes that 0 < 2 < 1 and it is important to understand whether this constraint can be understood from the dual large N eld theory. Unitarity of the SCFT immediately implies that any eld theory argument for why one should nd 2 > 0, however we are not aware of 2 < 1. It will be most interesting to understand better this condition and for which N = 2 SCFT it is obeyed. 5 In this section we study holographic ows between an N = 4 AdS5 vacuum and an N = 2 AdS5 vacuum with a di erent cosmological constant. First we will provide the conditions for the existence of N = 2 AdS vacua, independently of whether there is also an N = 4 vacuum. Then we consider speci c models allowing for an N = 4 AdS vacuum and study the existence of N = 2 AdS vacua. Finally, we construct domain wall solutions between such AdS vacua and discuss their holographic interpretation. (4.12) (4.13) (4.14) with norm e Xm = mnpqrPnpPqr ; jXe j q XmXem = e q Let us focus on the generic case where this does not vanish (we will comment on the special case Xe = 0 at the end of this section). Then we can introduce a normalized vector Xm = Xe m = jXe j ; Xij j = i : which speci es an SO(4) subgroup of SO(5). On the spinors, this de nes a reduction Xij = Xm USp(4) ! SU(2)+ mij . The supersymmetry preserved by our N = 2 AdS vacuum transforms under either one of these SU(2) factors. Without loss of generality we can choose SU(2)+, meaning that the supersymmetry parameters satisfy the projection SU(2) , where the plus and minus refer to the 1 eigenvalues of We start by providing general conditions for AdS5 solutions of half-maximal gauged supergravity preserving eight supercharges, which have not been discussed in the literature so far. The only assumption we make is M = 0. The supersymmetries of an N = 2 AdS5 solution transform as a doublet of SU(2) ' USp(2), hence we need to identify the relevant USp(4) ! SU(2) breaking of the Rsymmetry of half-maximal supergravity. This was already discussed in [28] and we summarize it here. The gravitino shift matrix (2.15) de nes the SO(5) vector (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) HJEP06(218) Having identi ed the USp(4) ! SU(2) breaking by means of the vector Xm, we nd that the conditions for an N = 2 AdS5 vacuum are: ( pm 2 3 ^ma p2f^mnaXn = 0 ; 2 1 "mnpqrf^pqaXr = 0 : The proof is given below. We observe that (5.6) and (5.8) are self-duality conditions on the four-dimensional space orthogonal to Xm.10 The AdS radius is xed by 10We can derive some other, non-independent, relations. Contracting (5.6) with Xn and using (5.5) we obtain f^[mnpXq] = 0, while contracting (5.7) with Xm we nd ^anXn = 0. ` 1 = W ; where r W = 2 P mnPmn jXe j = 2 P mnPmn As we will discuss in the next section, this expression for W de nes the superpotential driving supersymmetric ows of the scalar elds. This is extremized at the AdS point. It would be interesting to study the moduli space of the conditions above. This would most easily be done by exploiting the symmetry of the scalar manifold to set the undeformed vacuum at the origin and the unit vector Xm to point in a chosen direction. However, this analysis goes beyond the scope of the present paper and we leave it for future work. We can also discuss the spontaneous breaking of the gauge group in the N = 2 vacuum by looking at the scalar covariant derivative (3.24). Working at leading order in the eld uctuations around the vacuum, separating the term along the vector Xm from those transverse to it and using the supersymmetry conditions above, we get XnXm projects on the subspace transverse to Xm. The terms containing the A^a gauge vectors signal that all non-compact generators of the gauge group are spontaneously broken in the N = 2 vacuum and their gauge bosons acquire a mass via the Stuckelberg mechanism. This is analogous to what happens in N = 4 AdS5 vacua. The remaining terms give generically mass to some of the vectors of the form combination A0 + p1 3XmA^m. The U(1) generated by the transformation nmAn and to the 1 2 A 0 ! A0 + p 3 d ; Am ! Am Xmd ; is unbroken and corresponds to the R-symmetry of the N = 2 vacuum. This also corresponds to the R-symmetry of the dual N = 1 SCFT. Proof. Let us derive the N = 2 supersymmetry conditions given above. Using the AdS conditions A0 = 1` and 0 = x0 = 0, the supersymmetry equations (2.10){(2.13) reduce to i Pij j = 0i = P a ij j = 0 : 1 ` 5 i ; 1 2` i ; (5.12) (5.13) (5.14) (5.15) (5.16) Xeij j = 1 `2 2PmnP mn i ; and one can easily see that, as long as Xe does not vanish, and after making a harmless sign choice, this is equivalent to the USp(4) ! SU(2) projection (5.4) together with Eq. (5.14) is trivially solved in terms of a constant spinor ^i as i = e 2` ^i. However we must recall that (5.13), (5.14) were derived from the gravitino variation assuming that the supersymmetry parameter i does not depend on the R1;3 domain wall coordinates, therefore they only capture the Poincare supersymmetry of AdS. When the conformal supersymmetries are also taken into account, one nds that the gravitino equation does not constrain the degrees of freedom in i further than (5.4). For this reason, the analysis from now on di ers from the one in [28], where only the Poincare supersymmetries were considered. The remaining two equations, namely (5.15) and (5.16), constrain the embedding tensor and lead to the actual conditions for N = 2 vacua. Since they must hold on any spinor satisfying the projection (5.4), we infer that r P a ik kj + Xkj = 0 ; kj + Xkj = 0 : Recalling the de nition of the shift matrices (2.15), (2.16) and displaying the SO(5) gamma matrices, these equations can be rewritten as kj + Xp( p)kj = 0 ; p 2 3 ^am( m)ik + f^amn( mn)ik kj + Xp( p)kj = 0 : Working out the contractions of the USp(4) indices, (5.19) is equivalent to 2 The rst can be combined with the identity P mnXn = 0 (following from the de nition of Xn and the fact that P m[nP pqP rs] trivially vanishes in ve dimensions) to give (5.5), while the second is already the same as (5.6). Separating the di erent USp(4) representations, it is straightforward to see that (5.20) is equivalent to (5.7), (5.8). This concludes our proof. The derivation above assumed that Xe does not vanish. When Xe = 0 the solution may preserve eight Poincare supercharges, which is the situation considered in section 3. However, it may still be possible to have N = 2 AdS5 vacua with vanishing Xe . This still requires the existence of a unit vector X, however now unrelated to the Xe de ned in (5.1), projecting out half of the spinor degrees of freedom as in (5.4). For this to be compatible with the gravitino equation we also need that Xij and Pij commute, which is equivalent to demanding P mnXn = 0. The rest of the analysis of the supersymmetry equations is unchanged, hence conditions (5.5){(5.8) still hold and the AdS radius is given by ` 1 = p2P mnPmn. (5.17) (5.18) (5.19) (5.20) (5.21) Review of conditions for minimally supersymmetric ows After having identi ed models admitting both N = 4 and N = 2 AdS5 vacua, we will be interested in describing supersymmetric domain walls connecting them. Away from the xed points, the domain wall should preserve just four Poincare supercharges, namely the minimal amount of supersymmetry on R1;3. The necessary and su cient conditions for such domain walls in half-maximal supergravity were given in [28].11 Here we summarize them. The conditions use the same vector X and the same superpotential W de ned in section 5.1, however now the scalars are non-constant and depend on the radial coordinate. In addition to solving the ow equations HJEP06(218) one has to impose the following constraints along the ow: (5.22) (5.23) (5.24) (5.25) (5.26) (5.27) (5.28) (5.29) (5.30) W 1P+mn = 0 ; ^amXm = 0 ; f^+mna 4 W 2 P+pqf^+pqa P+mn = 0 ; P+mn = P mn 1 2 2 1 mnpqrPpqXr where we have introduced and f^+mna = ( pm both living in the four-dimensional space orthogonal to X and being anti-self-dual.12 The superpotential (5.10) can also be written as W = 2pP+mnP+mn. One can then use (5.26) to show that @ P+ is proportional to P+, and is therefore analogous to (5.28). We are interested in constructing domain wall solutions interpolating between an N = 4 and an N = 2 AdS5 vacuum. Thus one of the restrictive N = 4 conditions (3.1){(3.4).13 The other xed point instead has to satisfy the N = 2 conditions (5.5){(5.8). One can see that the latter are in fact equivalent to the constraints (5.25){(5.28), together with the condition that the superpotential is extremized.We xed points has to satisfy the now proceed to discuss two explicit examples which display all these features. 11The analysis of [28] was restricted to an embedding tensor satisfying M = 0. Recall that we are also assuming this condition here as it is necessary for a fully supersymmetric AdS5 vacuum. Also note that in [28] two superpotentials W were constructed, depending on the preserved supersymmetry; without loss of generality here we choose W = W+. 12The \+" subscript comes from the original de nitions in [28]. Although expressed in a slightly di erent form, these constraints are equivalent to eqs. (3.29), (3.31), (3.32) in [28]. 13Notice that the vector Xem has to vanish there, so that the four Poincare supersymmetries preserved along the ow can be enhanced to eight (plus the conformal supersymmetries). A model with one N = 2 vacuum An example of a supersymmetric domain wall solution connecting a maximally supersymmetric AdS5 vacuum to an N = 2 AdS5 vacuum is the well-known Freedman-GubserPilch-Warner (FGPW) ow [2]. This was originally constructed in the SO(6) maximal supergravity, where the UV vacuum is the standard SO(6) invariant critical point, while the IR N = 2 vacuum is the one rst found in [36]. As discussed in [2] this domain wall solution can also be described in half-maximal gauged supergravity by a model with two N = 4 vector multiplets and a gauging determined by the truncation of SO(6) maximal supergravity. Here we extend the FGPW model allowing for a more general gauging. We could also allow for an arbitrary number of vector multiplets as done in section 3 when studying ows between two N = 4 vacua (see [15] for such an extension of the FGPW model), however all essential features of the ow are already captured by a model with two multiplets, so we restrict to that. We choose the embedding tensor as p ; 2 67 = 2 g 1 ; (5.31) where g and are parameters. The vectors A1; A2; A3 gauge SU(2), A0 gauges U(1), while A4; A5; A6; A7 are eaten up by tensor elds. The FGPW model obtained by truncating SO(6) maximal supergravity has = 2, so that 67 = 45. In this case the fully superymmetric vacuum has a complex modulus, parameterizing the space SU(1; 1)=U(1). Since the conditions of section 3.1 are satis ed, we have a fully supersymmetric solution at the origin of the scalar manifold for any value of . In order to obtain an N = 2 vacuum at some other point of the scalar manifold, we break the SO(3) rotations in the 1; 2; 3 directions by mixing the 1; 2 and 6; 7 directions on SO(5;2) the scalar manifold. We thus parameterize the SO(5) SO(2) coset representative as14 HJEP06(218) (5.32) (5.33) V = e 2 t16 2 t27 = BB ^12 = ^45 = ^17 = ^67 = p2g g p ; 2 p2g ^26 = p2g 1 cosh2 ; 0 cosh B B B B B B B B sinh cosh 0 0 0 0 0 sinh 1 sinh cosh ; f^367 = g sinh2 ; The dressed embedding tensor (2.8) then reads: 14We could introduce two di erent scalars but the N = 2 vacuum conditions would set them equal. where by 6,7 we are denoting the values taken by the a index. For the unit vector de ning the USp(4) ! SU(2) projection of the supersymmetries we nd Xm = sign( ) 3m. The metric on the space spanned by the scalars ; in this case is while the scalar potential is V = g2 cosh2 4 2 sinh2 2 (cosh(2 ) 3) : + 1 4 The N = 2 vacuum conditions (5.5), (5.8) are satis ed automatically. Eq. (5.6) gives In addition to the N = 4 AdS5 solution we obtain the N = 2 AdS5 solution cosh2 = Note that the latter only exists for j j > 1 since only then we have a real scalar . For j j ! 1 the N = 2 AdS5 vacuum merges with the N = 4 vacuum at the origin. In the N = 2 vacuum most of the gauge symmetries are broken. From (5.11) we see that the non-trivial scalar covariant derivatives around this vacuum are: D 63 = d 63 D 73 = d 73 g g r j j j j 3 3 1 2 A ; 1 1 A ; (5.34) (5.35) (5.36) (5.37) (5.38) (5.39) 1 p 2 1A0 A3 , corresponding D( 62 71) = d( 62 71) 3 2 gp 2 + j j 2 p 2 1A0 + A (5.40) The vector elds on the right hand side get a mass through the Stuckelberg mechanism. The vectors in the rst two lines are just two of the gauge vectors of the gauged SU(2). The N = 2 vacuum is invariant under the combination to the U(1) R-symmetry. N = 2 vacua. The superpotential reads g 3 We can now move on to study the supersymmetric ow connecting the N = 4 and the W = + ; (5.41) where we are assuming g > 0. It is easy to check that with the parameterization (5.32) of the coset representative, the constraints (5.25){(5.28) are satis ed. This means that it is g 6 consistent to assume that the only owing scalars are ; . One can also check that the scalar potential and the superpotential are related as V = in agreement with (2.20). The ow equations (3.29), (3.30) for the scalar elds read From now on we assume without loss of generality that > 0 so that we can remove the absolute values. Let us call the operators dual to the two scalars O and O . Expanding around the N = 4 vacuum one nds that the dimensionless masses of the two scalars are Using the standard AdS/CFT relation m2`2 = 4) this implies that the dimensions of the dual operators are15 O = 2 + 2 ; O = 2 : 0 = 0 = 3 2 3 2 j j cosh2 + Along the RG ow there is operator mixing and in the IR SCFT we have two new eigenstates of the dilatation operator. The corresponding operator dimensions are O1 = 3 + 25 72 2 + ; O2 = 1 + For any > 1 we have that O1 > 4 and thus this is always an irrelevant operator. For O2 one nds 2 4 < O2 O2 4; 1 6; 2:5 < 2:5 ; < 1 : The ratio of central charges (in the planar limit) of the dual SCFTs is cIR = cUV VN =2 VN =4 3=2 = 27 (2 + )3 : Since the N = 2 vacuum only exists for of central charges [2, 14] is realized only for exactly the one where one marginal coupling in the dual SCFT. 15One could in principle choose the other root of the quadratic equation for O , i.e. O = 2 2 . This however violates the unitarity bound, > 1, for 1 < < 2. Moreover for = 2 we know from the FGPW model that O = 3 which is obeyed for the choice in (5.46). > 1, we nd that the well-known 27=32 ratio = 2. As already noticed, this value is also nds a modulus for the N = 4 vacuum, corresponding to a dashed red/blue lines are the values for the scalars at the IR N = 2 AdS5 vacuum at r ! The UV N = 2 AdS5 vacuum is at r ! 1 with = 2. Right : a contour plot of the superpotential as a function of the scalars p6 log = 0 there. We have xed g = 1 and (horizontal = p6 log axis) and (vertical axis) together with a parametric plot of the numerical solution for the scalars from the left panel. Let us compare the ratio in (5.49) with the ratio of central charges from equation (2.22) in [19] where we x z = 1 for the UV theory (this corresponds to the Maldacena-Nun~ez N = 2 solution) and the goal is to map the parameter z from [19] to the parameter in (5.49). From [19] we nd cIR = cUV 9z2 1 + (1 + 3z2)3=2 16z2 (5.50) : z 2 One can now nd a map between z2 and . The explicit expression is not very illuminating but one nds that z2 = 0 is mapped to = 2 and z2 = 1 is mapped to = 1. Moreover the map is monotonic, i.e. if we restrict ourselves to 0 1 we have to restrict to be in the range 2 1. This suggest that our model with two vector multiplets may describe holographic RG ows between the N = 2 Maldacena-Nun~ez vacuum and some of the N = 1 vacua with jzj < 1 studied in [19]. The ow equations (5.43) for this model can be integrated numerically. This is illustrated in gure 2. It is clear from this gure that there is a smooth domain wall solution which interpolates between the N = 4 and N = 2 AdS5 vacua. To understand better what drives the ow we can expand the BPS ow equations near the N = 4 AdS5 vacuum in the UV. The linearized expansion of the BPS equations depends on the value of . For > 2 we nd c e (2 2= )r=` ; 1 + c e 2r=` ; (5.51) while for 1 < 3(2 (5.52) Using (5.46) we conclude that the RG ow is driven by a source term for the operator O O proportional to the constant c . The constant c is related to the vev of the operator which is dynamically generated along the RG ow. The expression in (5.51) has the expected form for scalar elds with masses as in (5.45). The result in (5.52) however is di erent since for 1 < < 2 one should keep quadratic (and higher order) terms in in the linearized expansion of the di erential equation for the scalar in (5.43). The case = 2 should be treated separately and the linearized expansion of the BPS equations near the N = 4 AdS5 then reads c e r=` ; 3` 1 + 4 c2 re 2r=` + c e 2r=` : This is the behavior of an RG ow triggered by sources for operators of dimensions 3 and 2. This behavior was also observed in section 5 of [2]. The regular numerical solution displayed in gure 2 xes a particular relation between the constants c and c which depends on the value of . Now we turn our attention to reproducing the ratio (5.49) between the central charges from a eld theory argument. This can be viewed as a generalization of the results in [14] which is reproduced by selecting = 2 above. To this end suppose that we have a deformation of the N = 2 SCFT dual to the N = 4 AdS5 vacuum in the UV which is such that the resulting RG ow ends in an N = 1 SCFT with a superconformal R-symmetry given by the following linear combination of the Cartan generators of the UV SU(2) U(1) R-symmetry RNIR=1 = 1 + 3 RN =2 + 4 3 2 I3 : Using this superconformal R-symmetry and the anomaly relations in (4.2) one readily nds the following relation between the UV and IR central charges (5.53) (5.54) (5.55) (5.56) (5.57) For = 1=2 the result above reproduces the anomaly calculation in [14]. When the UV theory has aUV = cUV, such as in SCFTs with a weakly coupled gravity dual, one nds that the relations in (5.55) reduce to aIR = cIR = ( + 1)( 2)2 aUV : This suggests that to reproduce the supergravity result found in (5.49) above we have to make the identi cation16 16Unitarity and the a-theorem imply that 2 > > 0 which is mapped to the range 1 > > 1 in the supergravity analysis. aIR = (1 + cIR = ( 2 3)aUV ( + 1)cUV ; 1)aUV + ( + 1)(4 3 2)cUV : = This indeed turns out to be the case since as we show in appendix B the combination of gauge elds which are massless at the IR N = 2 vacuum in (5.39) corresponds to the generator which after the identi cation in (5.57) reduces to (5.54). As an additional consistency check one can show that the charge of the scalar under the supergravity gauge eld corresponding to the UV superconformal R-symmetry generator, i.e. the one in (5.54) with = 0, is 43 (1 + 1). This should correspond to the superconformal R-charge of the operator O in the dual SCFT. It is generally expected that operators dual to supergravity scalar elds belong to chiral multiplets and thus the conformal dimension of O should be determined by its R-charge via the relation (5.58) (5.59) 3 2 4 3 (1 + 1) = 2 + 2 : It is reassuring to nd that this result nicely agrees with the one obtained in (5.46) from an explicit evaluation of the mass of the scalar . A model with two N = 2 vacua We now consider a more involved model displaying an N = 4 vacuum and two distinct = 2 vacua. Since this is similar to the previous example we studied we keep the presentation short. We take four vector multiplets and choose the embedding tensor as p ; 2 67 = p 2 g 1 ; 1 89 = p 2 g 2 : 1 (5.60) For simplicity, we assume g > 0, 1 > 0, 2 > 0. We parameterize an SO(5) SO(4) coset element in terms of the scalar elds , as SO(5;4) V = e 2 cos (t16+t27) 2 sin (t18+t29) 0 B B B = BBBB sh cos 0 B ch 0 0 0 0 0 0 0 0 ch cos2 + sin2 sh cos 0 0 0 0 0 0 sh2 2 sin 2 sh cos 0 0 0 0 0 0 ch cos2 + sin2 sh2 2 sin 2 sh sin 0 0 0 0 0 0 sh2 2 sin 2 ch sin2 + cos2 sh sin 0 0 0 0 0 0 sh2 2 sin 2 ch sin2 + cos2 1 C C C C C C C C A ch 0 0 0 0 0 0 sh cos 0 0 0 sh sin 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 3 q 21 + 1 2 ; VN(1=) 2 = g 2 6 1 2=3(2 + 1)2 ; (5.61) Again we have Xm = 3m. In addition to the usual N = 4 vacuum at the origin with cosmological constant V = 3 g2, we obtain two N = 2 vacua by solving the supersymme2 try conditions in a way similar to the example in section 5.3. The rst N = 2 vacuum is 3 = 1 ; = 0 ; e 2 = 1 + 2 1 + 2 while the second is 3 = 2 ; 2 1 3 2 = 1 + 2 2 + 2 q 22 + 2 2 ; VN(2=) 2 = Note that for the two vacua to be distinct we need 1 6= 2. In addition it is simple to nd the ratio of central charges of the dual SCFTs. If we assume that 2 > 1 > 1 we nd the N = 4 vacuum is in the UV, the vacuum in (5.61) is intermediate and the vacuum in (5.62) is in the deep IR, 6 2 2=3(2 + 2)2 : (5.62) cI(R1) = VN =4 27 1 (2 + 1)3 VN(2=) 2 ! 3=2 VN =4 = 27 2 (2 + 2)3 : (5.63) The metric on the space spanned by the three scalars ; ; is The expression for the scalar potential is not particularly illuminating, however it can easily be recovered using (2.20) and the superpotential W = 3 6 Let us now discuss possible supersymmetric ows connecting the three supersymmetric vacua in this model. The constraints (5.25){(5.28) are satis ed, so a ow involving ; ; will not require switching on other scalars. The superpotential above generates the following ow equations for the scalar elds: (5.64) (5.65) (5.66) 0 = 0 = 0 = 3 2 2 cosh2 + 2 3 1 1 cos2 + 2 1 sin2 sinh2 3 ; 2 2 1 1 1 cos2 + 2 1 sin2 1 1 2 sin(2 ) : 1 sinh(2 ) ; There are ows from the N = 4 vacuum to either one of the N = 2 vacua with = 0 (5.61) or =2 (5.61). These ows have a constant value for the scalar and can be constructed numerically in a way very similar to the one described at the end of section 5.3. On the other hand, in order to ow from the vacuum in (5.61) to the one in (5.62) the scalar has to ow. This seems to imply that the numerical integration of the BPS ow equations is nely tuned and it is more challenging to construct these ows numerically. This is most likely related to the fact that both vacua in (5.61) and (5.62) are saddle points of the potential V . 6 Discussion In this paper we studied the general structure of supersymmetric AdS vacua in half-maximal ve-dimensional gauged supergravity as well as possible supersymmetric domain-wall solutions that connect them. Our results have a direct application to holography where they translate into constraints on the possible conformal vacua and RG ows of four-dimensional N = 2 SCFTs with a gravity dual. The approach we took in this work is \bottom-up", i.e. we eschewed any reference to a particular embedding of the gauged supergravity into string or M-theory and studied the general structure of the ve-dimensional theory. On one hand this allowed us to obtain very general results that should be applicable to all four-dimensional N = 2 SCFTs with a holographic dual, but on the other hand leaves the question open to what are concrete realizations in ten or eleven dimensions. For instance the domain wall connecting two supersymmetric AdS5 vacua with sixteen supercharges studied in section 3.5 should imply ow connecting two N = 2 SCFTs with a gravity dual. We provided further evidence for this claim with the anomaly calculation in section 4, however we are not aware of an explicit example of such an RG ow either in a \top-down" model arising from string or M-theory or in eld theory. A potential realization of this N = 2 RG ow might be provided by the theories of class S, i.e. N = 2 SCFTs arising from M5-branes compacti ed on a punctured Riemann surface, discussed in [37]. The vev deformation of the UV N = 2 SCFTs which reduces the SU(2)R R-symmetry and the SU(2)F avor symmetry to the diagonal subgroup (preserved all along the ow) may be provided by an appropriate \Higgsing of a puncture" on the Riemann surface. It was furthermore shown in [37] how to describe this class of strongly interacting N = 2 SCFTs holographically in Mtheory. What is missing to connect this set-up to our results is a well-de ned prescription to assign a given ve-dimensional gauged supergravity theory to any of the AdS5 elevendimensional solutions in [37]. It will be interesting to understand how to make such a link. We should also stress that the results presented in section 4 for the conformal anomalies of the UV and IR N = 2 SCFTs are valid beyond the supergravity approximation. It may be useful to emphasize that the IR central charges aIR and cIR in section 4 are those of the full IR SCFT. As a consequence of the partial spontaneous breaking of the UV global symmetry, the IR theory will contain a free sector made of Goldstone bosons in addition to the interacting sector.17 In class S theories it is known how to separate the contributions of the Goldstone bosons from the rest, see e.g. [38]. We were also able to describe general constraints for the existence of AdS5 vacua and domain-walls with eight supercharges in a gauged supergravity theory with at least one AdS5 vacuum with 16 supercharges. These results should be useful to understand RG ows between N = 2 and N = 1 SCFTs in four dimensions. The model with two vector multiplets discussed in section 5.3 is a particularly simple example of our general results which nevertheless is rich enough to capture interesting physics. For = 2 this model provides a holographic realization of the universal eld theory RG ow discussed in [14]. A well-known \top-down" example of this RG ow is provided by the N = 1 mass deformation of N = 4 SYM [1, 2], as well as its Zk orbifold [15, 16, 39]. It is widely expected that this universal RG ow should connect also the N = 2 and N = 1 Maldacena-Nun~ez SCFTs arising from M5-branes wrapping a smooth Riemann surface [17]. These theories have holographic dual AdS5 vacua but there is no known domain wall solution connecting 17We thank Prarit Agarwal for useful discussions on this. them. The supergravity solution with = 2 described in section 5.3 should serve as a ve-dimensional e ective description of this holographic RG ow. It will certainly be very interesting to embed this ve-dimensional model into a consistent truncation of elevendimensional supergravity. We are not aware of an explicit embedding of the model with 6= 2 in section 5.3 into string or M-theory. However it is natural to conjecture that it may be describing holographic RG ows between the N = 2 Maldacena-Nun~ez SCFT and one of the N = 1 SCFTs with 0 < jzj < 1 studied in [18, 19]. By the same token we can speculate that the model with one N = 4 and two N = 2 vacua described in section 5.4 may describe holographic RG ows connecting the N = 2 Maldacena-Nun~ez vacuum with two of the N = 1 theories with jzj < 1 in [18, 19]. To establish these conjectures rigorously one has to show how to construct a consistent truncation for the eleven-dimensional supergravity solutions of [18, 19] to ve-dimensional gauged supergravity. Partial progress in this direction was presented in [40], however the solution to the full problem is still out of reach. Finally we would like to point out that various interesting conjectures about the structure of RG ows in quantum eld theory were presented in [41, 42]. Supersymmetric CFTs with holographic duals and the RG ows connecting them provide a natural playing ground to explore these conjectures and we hope that some of our results may be useful in this context. Acknowledgments We would like to thank Prarit Agarwal, Chris Beem, Sergio Benvenuti, Fridrik Gautason and Parinya Karndumri for useful discussions. We are particularly grateful to Nick Halmagyi for participating at the initial stages of the development of this project and for many important discussions. The work of NB is supported in part by an Odysseus grant G0F9516N from the FWO, by the KU Lueven C1 grant ZKD1118 C16/16/005, and by the Belgian Federal Science Policy O ce through the Inter-University Attraction Pole P7/37. The work of H.T. was supported by the EPSRC Programme Grant EP/K034456/1. A Uniqueness of half-maximal AdS solutions in various dimensions Half-maximal gauged supergravity theories in di erent dimensions share a very similar structure. Their matter content and their couplings are completely xed by the number of vector multiplets and the embedding tensor specifying the gauge group. Therefore a natural question is the possible existence of a no-go result for multiple N = 4 vacua within half-maximal supergravity in dimension other than ve, similar to the one obtained in section 3.2. Indeed, in this appendix we show that, again under the assumption that the only compact subgroup of the gauge group is the R-symmetry of the vacuum, an analogous proof holds in dimensions four, six and seven. In more general situations it is natural to expect that there may be two distinct N = 4 vacua in four, six and seven dimensions. This should be viewed as a generalization of the ve-dimensional results presented in section 3.3. It should then be possible to exhibit holographic RG ows connecting these distinct AdS vacua analogous to the ones studied in section 3.5. Indeed, examples of such ows in sixHJEP06(218) so that H have the same properties as Hnc in ve dimensions, see (3.6), but with the novelty that H+ and H are electrically and magnetically gauged, respectively. In the AdS4 vacuum we nd again the breaking H ! SU(2) : In the holographically dual 3d N = 4 SCFT, SU(2)+ SU(2) is the R-symmetry group. Hc is again compact and semi-simple and is gauged under vector multiplet gauge bosons. It corresponds to the group of avor symmetries in the dual SCFT. The embedding tensor has components f MNP (while M have to vanish in the N = 4 vacuum). If we de ne fmnp = VM mVN nVP p( f MNP f+MNP ) ; where is the SL(2) complex scalar in the gravity multiplet, then the N = 4 supersymmetry conditions read and seven-dimensional half-maximal gauged supergravity have been studied in [43, 44]. It will be interesting to study this further and understand these holographic RG ows from the point of the dual SCFT. A.1 Four dimensions In four dimensions, fully supersymmetric AdS vacua in half-maximal supergravity have been discussed in [6]. There it was shown that the gauge group of N = 4 AdS vacua is G = H+ H Hc SO(6; n) ; (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) VM mVN nVP af MNP = 0 ; 6 1 "mnpqrsfqrs = f123 = f456 = i fmnp : 1 3 2 p i : Using the quadratic constraints and the symmetries of the scalar manifold one can take The cosmological constant is V = 3 fmnpfmnp = j j2. 2 Let us x one N = 4 AdS4 vacuum to be at the origin, and let us assume that Hc is trivial. Then we can argue analogously to the ve-dimensional case that because of (A.4), the following identities hold (up to SO(6) rotations) in the second vacuum VM VM VM 1 = 2 = 3 = M M M N N 1 ; VM N N 2 ; VM N N 3 ; VM 4 = ~ M N N 4 ; 5 = ~ M N N 5 ; 6 = ~ M N N 6 ; where and ~ describe the embedding of SU(2) into H , respectively, which correspond to Goldstone directions in VM mix since they are electrically and magnetically gauged. m, cf. (3.12). Note that the two SU(2) gauge groups cannot where H SO(3; m) and H0 m) for some m n. As in lower dimensions, this gauge group is spontaneously broken in a supersymmetric vacuum to its maximal compact subgroup, which turns out to be where SO(3) is gauged by three of the four graviphotons and corresponds to the Rsymmetry group of the dual CFT, while Hc SO(n m) corresponds to avor symmetries. The supersymmetry constraints on the embedding tensor re ect the discussion of the gauge group and are given by by brie y reviewing [10]. The gauge group is In six dimensions, half-maximally supersymmetric AdS vacua are the only allowed supersymmetric AdS vacua and have been constructed and studied in [10, 45{47]. Let us start for m = 1; 2; 3. The gauge coupling g and the mass m~ of the two-form in the gravity multiplet together also determine the cosmological constant via Again, if we x one half-maximal AdS6 vacuum to sit at the origin and we assume that Hc is empty, we can argue from the third equation in (A.10) that the vielbein (VM 0; VM m; VM a) of any other N = 4 AdS6 vacuum must be related by the following embedding G = H SO(1; n H0 SO(4; n) ; SO(3) Hc ; VM VM VM VM m VN nVP 0f MNP = 0 ; VN 0VP af MNP = 0 ; VN nVP af MNP = 0 ; VN nVP pf MNP = g "mnp ; V = 20 m~ 2 g 3m~ 3=2 : VM VM VM VM 0 = 1 = 2 = 3 = M M M M N N 0 ; a = ~bfb0a : where has the same form as in (3.12) and therefore describes the embedding of SO(3) into H0. Similarly, is a transformation in SO(1; n m) whose non-vanishing components are 0a and a0 given by Again, the transformations and are precisely the Goldstone modes of the model, and thus the N = 4 vacuum is unique. When Hc contains an SO(3) subgroup, multiple supersymmetric AdS6 solutions preserving all sixteen supercharges can be found. A supersymmetric ow between two such solutions was constructed in [43]. (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) Supersymmetric AdS vacua of half-maximal supergravity in seven dimensions have been discussed in [7]. Analogous to lower dimensions, the gauge group is of the form G = H Hc SO(3; n) ; (A.14) where H is spontaneously broken in the AdS vacuum to its maximal compact subgroup SO(3), which is gauged by graviphotons and corresponds to the R-symmetry of the dual CFT. The compact group Hc SO(n) corresponds to avor symmetries in the CFT. This result is found by inspecting the supersymmetry conditions imposed on the embedding tensor components fMNP ; M . These read: with given by (3.12), which corresponds to shifts by a Goldstone boson, establishing uniqueness of the supersymmetric AdS7 vacuum. Also in this case, when Hc contains an SO(3) subgroup, one can have multiple AdS7 solutions preserving sixteen supercharges, as well as supersymmetric ows connecting them, see [44] for an example. where the gauge coupling constant g determines the cosmological constant. Again, if Hc is trivial the only transformations that leave these conditions invariant are M = 0 ; VM VM VN nVP af MNP = 0 ; VN nVP pf MNP = g "mnp ; VM VM VM 1 = 2 = 3 = M M M The generator of the IR U(1)R symmetry In this appendix we show that the generator of the U(1) R-symmetry at the IR xed point of the holographic ow discussed in section 5.3 is given in eld theory units by RNIR=1 = We can extract the information we need from the action of the supergravity gauge covariant derivative on the spinor parameter i. The general form of the gauge covariant derivative was given in eq. (2.7). When acting on the spinor parameter, this reads: D = r 1 4 ( A^mf^mnp np + A^af^anp np + A0 ^np np) ; where r is the covariant derivative in the ungauged supergravity theory and we are suppressing the USp(4) indices on the spinor as well as on the SO(5) gamma matrices. (A.15) (A.16) (B.1) (B.2) Before coming to the IR vacuum, let us consider the vacuum at the origin of the scalar manifold, preserving sixteen supercharges. Recalling the form (5.31) of the embedding tensor, we have at that point: D = r 1 4 gAm"mnp np 2 2 g p A0 45 where in this equation the indices m; n; p run over 1; 2; 3 only. The embedding of the SU(2) U(1) R-symmetry of the N = 4 vacuum in USp(4) is such that we have the following identi cation: 45 = RN =2 ; 1 4 "mnp np = Im ; m = 1; 2; 3 : Therefore the covariant derivative can be written as D = r gAmIm g 2 2 p A0RN =2 with = 3 1 , we nd and 12 is the generator of the IR R-symmetry, which should be understood as the linear N =2 and I3 we are after. In addition, when in the main text we discussed the gauge symmetries being broken, we found that the combination Now let us consider the supersymmetric ow discussed in section 5.3. Since we have found there that Xij = ( 3)ij, the supersymmetries being preserved along the ow are + = 1+ 3 . This also implies 45 + = 2 12 +. Acting with the projector 1+ 3 on (B.2) 2 to select these supersymmetries and using the expression for the dressed components of the embedding tensor given in (5.33), we arrive at D + = r + g 2 + p 1 2 A3 cosh2 0 A ( 2 sinh2 ) 12 + : At the UV vacuum = 0 and this yields D + = r + g 2 2 p A0RN =2) + : Of the two symmetries generated by RN =2 and I3, one linear combination is preserved along the ow, while another one is spontaneously broken, with the associated gauge eld becoming massive. The symmetry that is preserved is manifest by evaluating the covariant derivative at the IR vacuum. Recalling that the latter is characterized by cosh2 D + = r + AIR 12 + ; AIR = (2 + ) g 6 1 2 1A0 A 3 ; Abroken = g(p2 1A0 + A3) (B.3) (B.5) (B.7) = +32 , (B.8) (B.9) (B.10) is massive (this is determined up to an overall normalization that will not matter). Inverting the relation between AIR; Abroken and A0; A3 we obtain Plugging this in (B.7), we nd that the generator multiplying AIR is (B.1), which is what we wanted to show. As an additional consistency check of our results, let us retrieve the ratio of central charges by studying the topological term in supergravity. After ignoring all other vector elds, the relevant Chern-Simons term of half-maximal supergravity is A 0 = A 3 = 2 3g 1 3g Abroken + AIR Abroken AIR 2 + 2 + LCS IR LCS 32p2 g3(2 + )3 AIR ^ F IR ^ F IR : (B.11) (B.12) (B.13) (B.14) If we also discard the vector becoming massive in the IR vacuum, the remaining Chern Simons term is The coe cient of this term in the supergravity Lagrangian is proportional to the cubic R-symmetry anomaly of the IR superconformal R-symmetry which gives the leading contribution to the aIR = cIR conformal anomaly. The analogous Chern-Simons term in the UV can be obtained by setting ! 1 in (B.13) to nd UV LCS 3227pg32 AUV ^ F UV ^ F UV : Taking the ratio of the two coe cients in (B.13) and (B.14) above we obtain the same result as the central charge ratio in (5.49) computed by comparing the IR and UV values of the cosmological constant. Open Access. 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Nikolay Bobev, Davide Cassani, Hagen Triendl. Holographic RG flows for four-dimensional \( \mathcal{N}=2 \) SCFTs, Journal of High Energy Physics, 2018, 86, DOI: 10.1007/JHEP06(2018)086