Domain wall seeds in CSO-gauged supergravity

Journal of High Energy Physics, Jun 2017

Gravitational domain wall solutions in gauged supergravity are often constructed within truncations that do not include vectors. As a consequence the gauge group is only a global symmetry of this truncation. The consistency of the truncation requires the restriction to solutions with vanishing Noether charge under this global symmetry, since otherwise vector fields are sourced. We show that this has interesting consequences for the orbit structure of the solutions under the global symmetries. We investigate this for CSO(p, q, r)-gaugings in various dimensions with scalar fields truncated to the \( \mathrm{S}\mathrm{L}\left(n,\mathbb{R}\right)/\mathrm{SO}(n) \) subcoset. We prove that the seed solution — which generates all other solutions using only global transformations — has a diagonal coset matrix. This means that there exists a transformation at the boundary of the geometry that diagonalises the coset matrix and that this same transformation also diagonalises the whole flow as a consequence of the vanishing charge.

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Domain wall seeds in CSO-gauged supergravity

Accepted: June Domain wall seeds in CSO-gauged supergravity Juan Diaz Dorronsoro 0 1 2 Harold Erbin 0 1 Thomas Van Riet 0 1 2 0 F-75231 , Paris , France 1 Celestijnenlaan 200D B-3001 Leuven , Belgium 2 Instituut voor Theoretische Fysica, K.U. Leuven Gravitational domain wall solutions in gauged supergravity are often constructed within truncations that do not include vectors. As a consequence the gauge group is only a global symmetry of this truncation. The consistency of the truncation requires the restriction to solutions with vanishing Noether charge under this global symmetry, since otherwise vector elds are sourced. We show that this has interesting consequences for the orbit structure of the solutions under the global symmetries. We investigate this for CSO(p; q; r)-gaugings in various dimensions with scalar elds truncated to the SL(n; R)=SO(n) subcoset. We prove that the seed solution | which generates all other solutions using only global transformations | has a diagonal coset matrix. This means that there exists a transformation at the boundary of the geometry that diagonalises the coset matrix and that this same transformation also diagonalises the whole ow as a consequence of the vanishing charge. Global Symmetries; Supergravity Models; AdS-CFT Correspondence 1 Introduction 2 3 4 5 6 CSO-gauged SUGRA Supersymmetric domain wall ows Noether charges Normal forms 5.1 5.2 Normal form for SO(p; q) gaugings Normal form for CSO(p; q; r) gaugings Discussion A Noether currents B Normal forms and vanishing Noether charges C Jordan normal forms and SO(p; q) any solution to the equations can be mapped to another solution under the action of G. This leads to the concept of orbits of solutions. Studying the orbits of solutions has been a very active eld of research over the past decades in the context of black hole solutions to ungauged supergravity, see [3, 4] for some original references. Less well studied is the orbit structure for domain wall solutions in gauged supergravity. This structure is expected to be somewhat more complicated since the symmetry group in gauged supergravity is not as big as in ungauged supergravity. The diminished symmetry is a consequence of the gauging which makes a subgroup of the { 1 { isometry local and destroys the rest1 of the original global symmetry by the introduction of a scalar potential (see for instance [5]). Often one is interested in domain walls with maximally symmetric wall spaces and with all elds with non-trivial spin (apart from the metric) set to zero. The resulting equations of motion enjoy then the gauge group as a global symmetry. For this to be consistent the scalar elds should not source any vector elds. This implies that the charge of the scalars has to vanish. This is a necessary and su cient condition for the truncation to be valid and has to be checked separately when a solution of the scalar-metric system is found, since this truncation is not guaranteed to be consistent. We are therefore interested in understanding the orbit structure of solutions with zero charge under global gauge rotations, which at rst sight seems harder than the orbit structure under local rotations. In this paper we initiate a rst systematic study of this in the context of a popular class of truncations of gauged supergravity theories in D dimensions with 2 < D < 10: the CSO(p; q; r)-gaugings of theories with as scalar coset SL(n; R)= SO(n) where n = p + q + r. A review of these theories, their solutions, and their link with (string theory) compacti cations can be found in [6]. We prove here that all solutions in these theories can be found by transforming, under the global symmetry group, the solutions with diagonal coset matrices. In other words, we show that there exists a rotation matrix inside the global symmetry group that diagonalises the coset matrix at the boundary and that this same matrix diagonalises the whole ow. This is non-trivial since a priori the transformation that diagonalises the matrix could depend on the position along the ow. It is also not evident that one can diagonalise the coset matrix using non-compact gauge groups. In this paper we prove both claims. Since these statements can be confusing, let us consider a reasoning that can be found (indirectly stated) in the literature on susy domain walls for SO(n)-gaugings, see for instance [7]. The argument is that prior to truncating the vectors, one can use the local rotation symmetry to diagonalise the coset matrix everywhere. Hence it is su cient to look at solutions with diagonal matrices. These solutions do not source vectors since their SO(n)-charge vanishes. The susy solution then obeys the typical ow equation where i are the scalars, Gij the metric on eld space, W a function of the scalars called the real superpotential and z is the coordinate transverse to the wall. For diagonal coset matrices only n 1 scalars are switched on and this equation is easily solved [7]. But imagine now that we started solving (1.1) for a general coset matrix with all o -diagonal elements switched on and that the solutions have zero charge, as required for consistency. Then the resulting solution is expected to be diagonalisable using a local SO(n)-transformation. Such a transformation will switch on vectors AI that are pure gauge, such that the eld strengths still vanish. Hence one would expect the diagonal solution to solve the ow equation with @z replaced by a covariant derivative that includes the vectors. But since the charge vanishes 1Not necessarily all the rest of the global symmetry is destroyed. What is left is the normaliser of by the following action V ( ) ; where Gij denotes the metric on the scalar manifold X and V is the scalar potential. All maximal and half-maximal gauged supergravities allow a further consistent truncation of the scalars such that the scalar manifold X is X = SL(n; R) SO(n) or X = GL(n; R) SO(n) : For example in N = 8; D = 4 maximal gauged SUGRA the scalar manifold is E7(7)= SU(8), describing 70 real scalars. Those 70 can be split into 35 scalars and 35 pseudo-scalars. The 35 scalars span SL(8; R)= SO(8). Maximal supergravities in D = 7 or higher do not need a truncation in order to be of the kind (2.2) (or be products of such manifolds). We are particularly interested in CSO(p; q; r) gaugings as they form a \canonical" set of gaugings in maximal supergravity [6]. Well-known examples are the SO(n) gaugings originating from sphere compacti cations of type II supergravity in D = 10 or of D = 11 supergravity. For instance SO(8) comes from 11D sugra on S7, SO(6) from IIB on S5, SO(5) from 11D sugra on S4, SO(3) from IIA on S2 or 11D on S3 (as a group space, not as a coset). The SO(p; q) gauge groups can be found from generalised hyperbolic compacti cations [8]. Consider the coset representative L of SL(n; R)= SO(n) in its fundamental representation. It is multiplied by the left with the SL(n; R) isometry transformations and from the right with the SO(n) isotropy group: L ! Lh 2 SL(n; R) ; h 2 SO(n) : It turns out useful to work with the symmetric coset element, M unit determinant, is positive de nite, and invariant under the isotropy group. Under the = LLT , which has SL(n; R) isometry group it transforms as a bilinear form (2.1) (2.2) (2.3) (2.4) The metric on SL(n; R)= SO(n), which has SL(n; R), as an isometry group is given by The coset can be extended to GL(n; R)= SO(n) with an extra scalar ' as follows 1 2 1] : HJEP06(217)9 The scalar potential for CSO-gaugings can be written as where a is a number that depends on the dimension and the matrix is diagonal where p + q + r = n. The combinations of n and D for which a vanishes are also theories for which the scalar ' is absent. The global symmetry group, in presence of the scalar potential, is the group that leaves the bilinear form invariant. When r > 0 this is larger than the CSO(p; q; r)-group, as we will explain below. 3 Supersymmetric domain wall ows The metric ansatz for Minkowski-sliced domain wall solutions is ds2 = f 2(z)dz2 + g2(z)ds2D 1 ; where ds2D 1 is the metric on Minkowski space. The warpfactors g; f and the scalars i are only functions of z. The warpfactor f (z) is pure gauge and can be chosen freely. From the action (2.1) one can derive the following e ective action for domain-wall solutions: Z Se = dz " 2 1 _2 + 1 4 Tr(M_ 1M_ ) + (D 1)(D 2) g_ 2 g g2(D 1)ea Tr( M M) + ea [Tr( M)]2 : g2(D 1) 2 Varying (3.2) gives the general second order equations of motion for M: 1 d f gD 1 dz f 1gD 1 M 1 _ M = ea' 4( M) 2 2Tr( M) M Tr( M)2 + Tr M 4 n { 4 { # 2 n (2.5) (2.6) (2.7) (2.8) (2.9) (3.1) (3.2) 2 : (3.3) Supersymmetric solutions solve rst-order ow equations derived from a speci c real superpotential W : (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) is equivalent to (3.2). and is given by:2 where The matrix form of the rst-order equations has not appeared earlier in the literature to our knowledge. The proof proceeds in the usual way by squaring the action. In the gauge f = gD 1, we nd that the following squared e ective action, The diagonal solution was constructed in [9] (see also [7, 10]) in the gauge f = g3 D 1 2(D 2) W : V = 2 (D 4(D 1) 2) W 2 : a gD 1Tr( M) : The potential can be written in terms of W as follows: For the class of theories discussed here we have W = ea'=2 Tr[ M]. Both V and W are manifestly invariant under the CSO gauge group. The ow equation for the scalars (3.4) can be written as an equivalent rst-order equation for the matrix M as follows: HJEP06(217)9 ds2 = h(3 D)=(2D 4)dz2 + h1=(2D 4)ds2D 1 ; e' = h a=4 ; M = h1=ndiag(1=h1; : : : ; 1=hn) ; hi = 2 iz + `i2 ; h = h1 : : : hn ; with i the i'th diagonal entry of (hence 1 or 0) and `i an arbitrary integration constant. In the context of maximal supergravity, these ows were shown to preserve half of the supersymmetries [7, 9, 10]. In the next sections, we will argue that all susy Minkowski-sliced domain wall solutions are related to this diagonal solution through a global transformation. The same also holds for all non-susy solutions with vanishing Noether charges: the seed will again be diagonal. 2In the case n = 3 the solution with all axions turned on was found in [11], but we do not require it here since we only care about seed solutions. { 5 { The e ective action (3.2) for nding domain walls corresponds to a normal Hamiltonian system in classical mechanics with the spatial coordinate z playing the role of time. Since the e ective action has the gauge group G = CSO(p; q; r) as a global symmetry, there must be a set of conserved charges equal to the dimension of G, which remain constant along the ow in z. These conserved charges can always be written in terms of momenta and generalised coordinates and hence provide a set of rst-order equations. We now show that for supersymmetric solutions in our models these rst-order equations contain no new information since they are implied by (3.7). To gain some intuition we rst consider the case without potential. The equations of motion for the scalars in the gauge f = gD 1 decouple from the metric and are derived from the geodesic action The geodesic equations for the scalars in M are summarized as (4.1) (4.2) (4.3) (4.4) (4.5) Sg = Z dz Gij _i _j : d dz (M 1M_ ) = 0 : M 1M_ = Q ; M(z) = M(0)eQz : { 6 { This can be integrated once with Q any traceless matrix. This equation is further integrated as So remarkably all geodesics on SL(n; R)= SO(n) can be found in the language of the symmetric coset matrix in a trivial manner.3 From (4.3) one nds that Q is in the Lie algebra of SL(n; R) since it is traceless and corresponds to the matrix of Noether charges. Once we deform the geodesic motion by the potential V , only those Noether charges corresponding to the symmetry preserved by V should survive. Hence the Noether charge matrix corresponding to the CSO gauged supergravity theories should be related to the projection of the general Q 2 Lie(SL(n)) to Lie(CSO(p; q; r)). An explicit computation of the Noether charge carried out in appendix A yields Q = M_ M 1 M 1 _ M : Substituting the rst-order equation (3.7) into (4.5) one nds that supersymmetric solutions have vanishing charge. Hence the supersymmetric solutions source no vectors, consistent with our truncation. Moreover, one can explicitly check that Q (which is also conserved) is an element of the CSO algebra. We already explained in the Introduction that we expect solutions with zero Noether charges to behave in a special way under global transformations of the gauge group. Namely 3This is in sharp contrast with the algorithms that are developed to integrate the equations at the level of the scalars [12{14] where the solutions can be found algorithmically but the computations and expressions are complicated. a solution whose coset matrix can be diagonalised at some value for z using a gauge transformation will be diagonal for all values of z. We have included a computationally explicit proof of this in appendix B for SO(p; q)-gaugings. For gaugings with contracted gauge groups the story complicates somewhat in the sense that a bigger transformation is needed to diagonalise a solution. But again the vanishing of the Noether charge will turn out su cient to argue that all solutions are global transformations of the solution with diagonal coset matrix. The reasoning uses the equations of motion (3.3), which can be schematically written as M = F (M) : (4.6) Since the matrix F (M) is diagonal for diagonal M, if both M and its rst derivative are diagonal at some point, the whole ow will remain diagonal. We will show that if we require the Noether charge to vanish, we can diagonalize M and M_ at a reference value (e.g. z = 0) using a transformation inside the global symmetry group of the action. This ensures that all solutions with vanishing Noether charge (and hence all the solutions consistent with the truncation) can be obtained through a global symmetry transformation from the diagonal seed solutions. The intuition behind this is simple. Consider for instance a spherically symmetric system in classical mechanics for which the angular momenta vanish. Since angular momentum is linked to rotation, there exists a frame in which the system has no rotation and all angular variables vanish at all times. Let us now extend this logic to our context. If it is possible to diagonalise M(0) using SO(p; q) then the general coset element can be written as L = P D, with D diagonal and P in SO(p; q). The degrees of freedom are explicitly separated into `radii' sitting in D and `angular variables' residing in P. From the point of view of the scalar potential the variables P are cyclic. If the variables in P are then zero in a certain frame (basis) at z = 0 they will remain zero throughout the ow. Hence the transformation that brings one to the frame of vanishing angles at the boundary brings us to vanishing angles along the ow. 5 Normal forms We now set out to prove the main claim made in the introduction that all solutions of the CSO-gaugings with zero charge can be found performing a global transformation on solutions with diagonal coset matrices. We rst discuss SO(p; q) gaugings and then we treat the contracted CSO algebras. 5.1 Normal form for SO(p; q) gaugings We start by arguing that any coset matrix M evaluated at the reference value z = 0 can be diagonalised using an SO(p; q) transformation. Using the result form the previous section this then implies that the solution will be diagonal globally. It is well-know that M(0) can be diagonalised using an SO(n)-transformation since it is a symmetric bilinear form. We want to generalise this to SO(p; q)-transformations with p + q = n. The statement is not true for general symmetric bilinear forms, but relies on { 7 { M(0) being positive de nite. Our proof uses a detour via the normal form of the coset element of SL(n; R)= SO(p; q) constructed in [ 15, 16 ]. To make this connection, note that any symmetric matrix M can be written as with the invariant bilinear form under SO(p; q) de ned earlier and A some generalised symmetric matrix, which means Such a matrix A can be regarded as being in the Lie algebra of SL(n; R)= SO(p; q). It was shown in [ 15, 16 ] that one can bring A into its Jordan normal form J through a similarity The proof of this proceeds by showing that one can always nd a basis for in which the normal form of a generalised symmetric matrix is itself generalised symmetric: J This means that the similarity transformation is inside SO(p; q). The detailed proof of this T = J . statement can be found in [ 15, 16 ]. In appendix C we show that, due to M being positive de nite, the normal form J has to be strictly diagonal. After this we get M = A = J 1 = J T ; and we therefore see that we can indeed diagonalize M with an SO(p; q) transformation. 5.2 Normal form for CSO(p; q; r) gaugings For the CSO(p; q; r) gaugings the global symmetry group is larger than the gauge group. The CSO group can be obtained by exponentiating the Lie algebra elements gij given by [6] whereas the global symmetries of the action are larger and contain all the transformations that leave invariant This second group is larger, since even though one has for any choice of constants ij , not all the matrices form exp ij gij . Indeed, (5.6) only implies that which leave invariant are of the (gij )kl = [ki j]l ; = T : exp ij giTj exp ij gij = = Ar (p+q) 2 SO(p; q), A is an arbitrary r by (p + q) matrix and B is an arbitrary r by { 8 { (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) Here, M1 has size (p + q) (p + q) and M3 has size r previous section, we know that there is a matrix is diagonal. Let us transform M with the matrix r. From the arguments of the 2 SO(p; q) such that M1(0) T = D1 where for the moment we keep B arbitrary. Then we get M(0) T = D1 The matrix S is symmetric, and moreover it is positive de nite.4 From the properties of S we know that there exists an SO(r) transformation O which brings it into a diagonal form: We will argue now that we can diagonalize any susy ow through a global transfor mation. To this aim, let us split M(0) into blocks as follows: M(0) 1 = ( ) ( ) ( ) S 1 ! { 9 { M(0) = M1(0) M2(0)! M2T (0) M3(0) with di > 0. If we then let we see that This shows that using the transformation we just described, we get block of the inverse matrix M(0) 1. Indeed, We now argue why this transformation will diagonalise the coset matrix at all values for z if the Noether charge vanishes. This result is not implied from the results in the previous section since we only require that the Noether charge under CSO rotations are zero, whereas the -transformation above is typically inside a larger symmetry group. Hence we have to prove that the -transformation will diagonalise the solution for all values of z. 4The way to see this last point is by remarking that S is the combination that appears on the lower-right whenever S is not singular. The matrix S is also known as the Schur complement of M1(0), and the matrix M(0) is positive de nite if and only if M1(0) and its Schur complement are both positive de nite. After performing a transformation as the one we described, we can write M = D1 0 ! The Noether charge is therefore Q = M = M_ 2T (0) s sM_ 1(0)D1 D1M_ 1(0) s sM_ 2(0)! D1 0 ! where s is a diagonal matrix with p times +1 and q times know that M_ 2(0) must vanish, and therefore we have 1. If this charge vanishes, we (5.17) (5.18) (5.19) (5.20) (5.21) (5.22) (5.23) Suppose therefore that M is a solution of the second order equations of motion such that Let us expand such a solution around the reference value z = 0 as Q = M_ M 1 M z + O(z2) : 0 ; After applying the global transformations we described above, we have shown that both the coset matrix and its rst derivative can be made diagonal at the reference value z = 0. Due to the structure of the second order equations of motion, the matrix will preserve its diagonal form along the whole ow. This concludes the proof that any consistent solution (therefore with vanishing Noether charge) can be brought into a diagonal form through a global symmetry transformation. In this frame, the equations of motion for the upper-left diagonal part of the matrix M are precisely the same equations of motion of an SO(p; q) gauging, and we can use the throughout the ow. This implies that M_ 1(0) = D_ 1 is of course diagonal. same arguments of the previous section to show that this part of M will remain diagonal As for the lower-right component of M, we remark that since M_ 3(0) is symmetric, we can diagonalize it through an SO(r) transformation Or such that OrM_ 3(0)OrT = D_ 3. Transforming now M as we get around z = 0 We have investigated the space of domain wall ows in CSO(p; q; r)-gauged supergravities with scalar elds truncated to the coset SL(n; R)= SO(n) (or GL(n; R)= SO(n)), where n = p + q + r. We have emphasized how the gauge group is a global and not a local symmetry once the vectors are truncated and that this requires the CSO(p; q; r)-charge to be zero. We then showed that all the zero-charge solutions of the theory can be found by letting the global symmetry act on solutions with diagonal coset matrices. As the vanishing of the Noether charge is a necessary consistency condition for any solution of the truncated theory, we have proved that all the consistent solutions of CSO gaugings (within our truncation) can be obtained through a global rotation of a diagonal solution. All supersymmetric domain walls with diagonal coset matrices were constructed in [6, 7, 9, 10]. For supersymmetric ows the results we found were implicitly known for SO(n) gaugings. For all other CSO(p; q; r)-gaugings our ndings were to our knowledge not explained in the literature.5 We emphasize that this result is more surprising for the gaugings with contracted gauge groups (r > 0). In that case the global transformation that diagonalises the ows are outside of the gauge group, but still within the global symmetry of the action. In the extreme case r = n, there is no gauging and hence no scalar potential so that the ows describe geodesic curves on SL(n; R)= SO(n). Those curves were known to be diagonalisable using SL(n; R) (see for instance [ 16 ]). The reason the ow can be diagonalised when 0 < r < n is a mixture between the di erent reasonings used for the extremes r = 0 and r = n. This work is a rst step towards classifying (supersymmetric) domain wall ows in gauged supergravity. To achieve that goal, one should go beyond the SL(n; R)= SO(n)truncation.6 We then expect that a very non-trivial orbit structure should arise. Acknowledgments We thank Nikolay Bobev, Fridrik Gautason for useful discussions and especially Adolfo Guarino and Mario Trigiante for discussions and essential feedback on a earlier draft. The work of JDD and TVR is supported by the FWO odysseus grant G.0.E52.14N and by the C16/16/005 grant of the KULeuven. We furthermore acknowledge support from the European Science Foundation Holograv Network and the COST Action MP1210 `The String Theory Universe'. A Noether currents tions M ! M T , with Consider the e ective action (3.2). It is invariant under global CSO(p; q; r) transformaan element of the CSO(p; q; r) group such that T = . 5Aside some comments in [11] about the case n = 3 where the claim can be found that \the change of SL(n; R)-frame" diagonalises the solution. 6Another extension could be to look at the !-deformations of the standard gaugings [17] within the SL(n; R)= SO(n)-truncation. Using the results of [ 18 ] it seems that the ndings of this paper still go through such that again the diagonal solutions are the seed solutions. Normal forms and vanishing Noether charges Here we prove explicitly for SO(p; q) gaugings that the matrix that diagonalises M at z = 0 diagonalises M throughout the ow if the Noether charge Q vanishes. Inserting into (4.5) and requiring Q = 0 yields 2! = D !D 1 + D 1 !D; 2 so(p; q): If we let D = diag(d1; : : : ; dn), the previous equations becomes in components (A.1) ij gij . (A.2) (A.3) (A.4) (A.5) (A.6) (B.1) (B.2) (B.3) (B.4) In nitesimally we can write Here, gij label the generators of the algebra and ij are transformation parameters, which are taken to be in nitesimal. In order to nd the conserved charges we allow and follow the standard Noether procedure. To simplify the notation, we write h ij = ij (z) The two last terms of the action remain invariant even when h = h(z), so that we only consider the kinetic term and nd: Integrating by parts, throwing away a total derivative and by the standard argument of letting ij be constant, we get a set of conserved charges The generators of the CSO group are [6] A short computation then reveals that Qij = Tr M_ M 1gij ; _ Qij = 0 : (gij )kl = [ki j]l : Q = M_ M 1 M We recall that for ! 2 so(p; q), all the diagonal elements of ! vanish. If all the eigenvalues of M are di erent (so that di 6= dj for all i; j), then (B.3) further implies that !ij = 0, so that ! = 0 and What happens when some of the eigenvalues of M are equal? Suppose for instance that we are dealing with the extreme case in which all the eigenvalues are the same, so that also di = dj for all i; j. Then we get M = D T = D( T so that M is diagonal for any , and there is no need for diagonalizing it in the rst place. We could then trivially keep constant in this case too. The argument for the constancy of when only some of the eigenvalues of M are HJEP06(217)9 equal, follows from realizing that the only components !ij which can di er from zero are those for which di = dj . However, in that case, the transformation will leave D invariant anyway: it will correspond to a trivial reshu ing of the equal di's in D. Indeed, if without loss of generality we arrange the elements of D in k groups D = (d1; : : : ; d1; d2; : : : ; d2; : : : ; dk; : : : ; dk) ; (B.5) (B.6) (B.7) (B.8) (B.9) (B.10) (B.11) (B.12) (B.13) ! will be block diagonal: and we get with c being constant and ::: CC C ; C A : : : 0 C 0 !k (z) = c !(z) ; 0 0 This z-dependent part of the transformation acts however trivially on D: We see therefore that even when there are repeated eigenvalues, M can be diagonalised with a constant SO(p; q) transformation throughout the ow. Jordan normal forms and SO(p; q) Let us construct the basis for the bilinear form that makes the standard Jordan normal form J generalised symmetric. This standard form is given by 0 J = BBBBB 0::: J2 ::: CC C C A 0 i 1 then from the generalised symmetric condition we see that must be of the form where i has the same size as the Jordan block to which it corresponds and is equal to 1, if its size is 1, or with Ji = BBB ::: B :: i = BBB : B0 : : 1 0 : : : 0 otherwise. Using all the previous, we can write [ 16 ] We will now argue that J is strictly diagonal, and not just block diagonal. Since M = LL T is positive de nite, we must have that J = 1 M( 1)T , is also positive de nite. Clearly, the matrix J is block diagonal, and in order to be positive de nite, each of its blocks Ji i must be positive de nite. Suppose that there is at least one non-diagonal block Ji in the Jordan normal form for A. The product Ji i is then Ji i = BBB 0 : : : : : : : : : ::: CCC : i 0 CC 0 C C A De ning the vector7 v = (1; 0; : : : ; 0; 1= i), we have vT Ji iv = ( 1 2 if Ji has size 2 if Ji has size r 2 ; r with r > 2 ; 7We recall that i 6= 0 since det(J ) =det[ 1M( 1)T ] 6= 0. from which we deduce that Ji i is not positive de nite. We have hence shown by contradiction that J must be diagonal. To unify this discussion with the one of the previous subsection, we will write D = J . We know then that we can always diagonalize M using an SO(p; q) transformation: M(z) = D(z) takes the standard diagonal form. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. U.K. (2012). [1] D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge, Kaluza-Klein Theories, Commun. Math. 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Juan Diaz Dorronsoro, Harold Erbin, Thomas Van Riet. Domain wall seeds in CSO-gauged supergravity, Journal of High Energy Physics, 2017, 97, DOI: 10.1007/JHEP06(2017)097