An SO(3) × SO(3) invariant solution of D=11 supergravity
Hadi Godazgar, Mahdi Godazgar, Olaf Krueger, Hermann Nicolai, Krzysztof Pilch
aa r X i v : . [ h e p - t h ] J a n DAMTP-2014-55AEI-2014-056
An SO(3) × SO(3) invariant solution of D = 11 supergravity Hadi Godazgar ⋆ , Mahdi Godazgar † , Olaf Kr¨uger ‡ ,Hermann Nicolai § and Krzysztof Pilch ¶ ⋆ † DAMTP, Centre for Mathematical Sciences,University of Cambridge,Wilberforce Road, Cambridge,CB3 0WA, UK ‡§ Max-Planck-Institut f¨ur Gravitationsphysik,Albert-Einstein-Institut,Am M¨uhlenberg 1, D-14476 Potsdam, Germany ¶ Department of Physics and Astronomy,University of Southern California,Los Angeles, CA 90089, USA ⋆ [email protected], † [email protected], ‡ [email protected], § [email protected], ¶ [email protected] 19, 2014 Abstract
We construct a new SO(3) × SO(3) invariant non-supersymmetric solution of the bosonic fieldequations of D = 11 supergravity from the corresponding stationary point of maximal gauged N = 8 supergravity by making use of the non-linear uplift formulae for the metric and the3-form potential. The latter are crucial as this solution appears to be inaccessible to traditionaltechniques of solving Einstein’s field equations, and is arguably the most complicated closed formsolution of this type ever found. The solution is also a promising candidate for a stable non-supersymmetric solution of M-theory uplifted from gauged supergravity. The technique that wepresent here may be applied more generally to uplift other solutions of gauged supergravity. ontents S . . . . . . . . . . . . . . . . . . . . . 62.2 Invariant tensors for the SO(3) × SO(3) solution . . . . . . . . . . . . . . . . . . . . . 92.3 The solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 × SO(3) invariants 11 × SO(3) solution of gauged supergravity 165 The SO(3) × SO(3) solution of D = 11 supergravity 18 α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 × SO(3) invariants 39E Ambient coordinate embedding 42 Introduction
Kaluza-Klein theory plays an important role as an organising framework in supergravity relatinghigher and lower-dimensional theories to one another as well as providing a tool by which to derivenew theories by dimensional reduction. Nevertheless, one is confronted with some challenging issues,such as the question of whether a lower-dimensional theory can be obtained from a reduction of ahigher-dimensional one, and if so, whether the reduction is consistent. That is, whether all solutionsof the lower-dimensional theory can be mapped onto a subset of the higher-dimensional solutions.How this is done in practice, i.e. how one uplifts solutions to higher dimensions, is yet another levelof complication. Indeed, examples of such results are rare and are mainly confined to truncationswith relatively simple scalar sectors.Eleven-dimensional supergravity compactified on a seven-sphere is one example in which progresshas been made; the four-dimensional theory associated with this reduction being maximal SO(8)gauged supergravity. Recently, an uplift ansatz has been derived for the seven-dimensional com-ponents of the 3-form potential in terms of the (pseudo)scalars of the gauged theory [1, 2]. Thiscomplements the uplift ansatz for the seven-dimensional components of the metric given in Ref. [3].Together, these ans¨atze give a new method for constructing solutions of D = 11 supergravity, andit is the purpose of the present paper to explicitly demonstrate the utility of this new method.Indeed, without the new uplift formula for the internal flux it is basically impossible to constructthe solution to be presented in this paper, or to derive any other solutions of this type that aremore complicated than those already in the literature (see for example Refs. [4–6, 3, 7]). This isbecause in all previous examples of solutions corresponding to critical points, the symmetry of thesolution reduces the equations of motion to a set of ODEs. In particular, if one obtains the metricvia the metric lift ansatz, the equations for the components of the flux field strength are algebraicand usually easy to solve. The analysis becomes even simpler if one has supersymmetry, where theODEs are first order, as is the case for the G [3] and SU(3) × U(1) [7] solutions.The ans¨atze can be applied to obtain a very general class of solutions of D = 11 supergravity.In particular, they facilitate the uplifting of all stationary points to Freund-Rubin compactifications[8] with flux, viz. E M A ( x, y ) = ∆ − / ( y ) ◦ e µα ( x ) 00 e ma ( y ) , F MNP Q = F µνρσ = i f FR ◦ η µνρσ F mnpq = F mnpq ( y ) , , otherwise Ψ M = 0 , (1.1)with the corresponding metric G MN dX M dX N = ∆ − ◦ η µν dx µ dx ν + g mn dy m dy n , (1.2)where ( x µ , y m ) are coordinates on the four and compact seven-dimensional spacetimes, respectively; ◦ e µα ( x ) (with corresponding metric ◦ η µν ) is the vierbein of the maximally symmetric four-dimensionalspacetime with corresponding alternating tensor ◦ η µνρσ ; e ma ( y ) (with corresponding metric g mn ) is2he siebenbein of the compact space and f FR is a constant. In what follows we consider the siebenbeinto be that of a deformed round seven-sphere, i.e. e ma ( y ) = ◦ e mb ( y ) S ba ( y ) , (1.3)where ◦ e ma (with corresponding metric ◦ g mn ) corresponds to the siebenbein on a round seven-sphereof inverse radius m and the deformation parameter S has determinant ∆,∆( y ) = det S ab ( y ) . (1.4)The uplift ans¨atze are derived within the context of the SU(8) invariant reformulation of the D = 11 theory [9], whereby eleven-dimensional fields are decomposed in a 4 + 7 split, such thatone can loosely talk of them as having external/internal indices. Note that SU(8) is the localenhanced symmetry obtained in the toroidal reduction of D = 11 supergravity to four dimensions,with associated global group E [10]. Importantly, however, no truncation is assumed and thereformulation remains on-shell equivalent to D = 11 supergravity [11]. The SU(8) structures in thereformulation are obtained by an analysis of the D = 11 supersymmetry transformations in such a4 + 7 split, and by the enlargement of the original SO(7) tangent space symmetry to a full chiralSU(8) symmetry; the R-symmetry of N = 8 supergravity.The uplift ans¨atze for the internal metric and flux are derived by comparing the supersymmetrytransformations of particular components of the eleven-dimensional fields, namely those with a single“four-dimensional” index: the graviphoton B µm and A µmn , which contain the internal metric and3-form potential components, and the supersymmetry transformation of the associated vectors infour dimensions, which are given in terms of the (pseudo)scalar expectation values.In this paper, we demonstrate the utility of the uplift ans¨atze by applying them to the onlyknown stable non-supersymmetric solution of the gauged theory [12, 13]: the SO(3) × SO(3) invariantstationary point [14]. This yields a new solution of D = 11 supergravity: see equations (2.20, 2.22)and (7.29, 7.30) for the solution in stereographic and ambient coordinates, respectively. This solution,to our knowledge, is the most non-trivial closed form solution of this type ever found (inspectionof the explicit formulae in section 5 of this paper will probably immediately convince readers ofthe correctness of this claim). Indeed, the remarkable efficiency of the uplift formulae is clearlydemonstrated by the fact that it is significantly simpler to write down the solution than to verifythat it does indeed satisfy the D = 11 equations of motion.Note that there are many known stable non-supersymmetric compactifications of D = 11 su-pergravity of the form AdS × M (see e.g. Ref. [15]) or indeed AdS × M [16–18], or even purelyeleven-dimensional solutions, such as for example, the eleven-dimensional Schwarzschild-Tangherlinisolution [19]. However, the solution we construct here is the first such solution, as far as we areaware, with non-trivial internal flux and uplifted from maximal gauged supergravity. While we can-not comment on the eleven-dimensional stability of the solution, the fact that the compactification isstable [12] in the sense of Breitenlohner-Freedman (BF) [20] is promising. Eleven-dimensional stabil-ity would be established by demonstrating that the fluctuations associated with higher Kaluza-Kleinstates also remain above the BF bound. 3he SO(3) × SO(3) invariant stationary point is a distinguished solution of the gauged theory.Not only is it the only known stable non-supersymmetric solution, but it also has the most negativevalue of the cosmological constant of all known stable points and several unstable points [13] and is,therefore, likely [21] to be an attractive IR fixed point for many flows in the world-volume theoryon M2-branes [22]. One example of an RG flow in which this solution is the IR fixed point isthat considered in Ref. [12], where the UV fixed point is given by the maximally symmetric SO(8)invariant stationary point [8, 23]. The study of such RG flows is important in so-called top-downholographic applications to condensed matter systems (see e.g. Refs. [24, 25]).The SO(3) × SO(3) invariant solution is an example of a compactification of the form (1.1).Therefore, the uplift ans¨atze for the metric and internal flux given in Refs. [3, 1] suffice. In this case,the eleven-dimensional field equations R MN = g MN F P QRS − F MP QR F N P QR , (1.5) E − ∂ M ( EF MNP Q ) = √ iη NP QR ...R S ...S F R ...R F S ...S , (1.6)reduce to [3] R µν = (cid:16) f FR ∆ + F mnpq F mnpq (cid:17) δ νµ , (1.7) R mn = − F mpqr F npqr + (cid:16) F pqrs F pqrs − f FR ∆ (cid:17) δ nm , (1.8) ◦ D q (cid:0) ∆ − F mnpq (cid:1) = √ f FR ◦ η mnpqrst F qrst , (1.9)where R µν and R mn denote components of the eleven-dimensional Ricci tensor R M N , ◦ D m denotesa background covariant derivative and ◦ η m ...m is the permutation tensor with respect to the metric ◦ g mn . All seven-dimensional indices in the equations above are raised with g mn , except for ◦ η m ...m ,whose indices are raised with ◦ g mn . We parametrise AdS and the seven-sphere such that ◦ R µν = 3 m ◦ g µν , ◦ R mn = − m ◦ g mn . (1.10)There are three constants in (1.7)-(1.10), namely, m , m and f FR . It is convenient to choose m as the overall scale of the solution, since it is simply related to the coupling constant, g , of the D = 4 theory [26], m = g √ . (1.11)The remaining two constants are determined by the value of the scalar potential, P cr = − P ∗ g , atthe stationary point, or, equivalently, the cosmological constant of the solution in four dimensions, m = 2 P ∗ m . (1.12)The value of the f FR parameter can be obtained from the uplift formulae in [26, 27] or the upliftansatz for the internal components of the 6-form dual [28, 29]. In particular, it has been conjectured We use the conventions of [9]. f FR = P ∗ √ m . (1.13)However, a general proof of (1.13) beyond explicit examples remains an open problem. It isstraightforward to verify that for vanishing scalar fields one recovers the maximally supersymmetric AdS × S Freund-Rubin solution [8] given by (1.10) with m = 2 m , f FR = ± √ m (1.14)and no internal flux.The outline of the paper is as follows: in section 2.1 we provide the necessary background in orderto be able to present the solution without dealing with the technical details. Then, in section 2.2, weintroduce the objects in terms of which we find the solution, which is presented in section 2.3. Forthe reader who is simply interested in the solution, and not the technical details of its derivation,section 2 is sufficient.In section 3, we state identities satisfied by the SO(3) × SO(3) tensors – an outline of the derivationof the identities is given in appendix B. The metric ansatz gives ∆ − g mn and some of the identitieslisted in section 3.2 are used to invert this to find the metric, g mn , in section 5. Furthermore, theidentities are also used to find and simplify the expression for the 3-form potential, A mnp , from theflux ansatz in section 5. The majority of the identities are, however, used, in section 6, to verifythat the field equations are satisfied.We present the SO(3) × SO(3) invariant stationary point of D = 4 maximal supergravity [14] insection 4. In particular, we recapitulate the scalar profile of the SO(3) × SO(3) invariant stationarypoint, which is uplifted by means of the ans¨atze, in section 5, to give the internal components of themetric and 3-form potential of the eleven-dimensional solution.In section 6, we verify that the solution found in section 5 satisfies the D = 11 supergravityfield equations. Given the general arguments that guarantee that the ans¨atze obtained from theuplift formulae solve the equations, this is not strictly necessary. However, we do this in order todemonstrate the full complexity of the solution as well as to give the reader further confidence thatthe uplift formulae do indeed provide bona fide solutions of the D = 11 equations.Finally, in section 7 we re-express the eleven-dimensional solution in terms of ambient and lo-cal coordinates, which are better adapted to the isometry of the solution than the stereographiccoordinates on S used in section 5.In order to set conventions, we review some basic material, largely contained in Ref. [9], inappendix A. For comparison, we list the identities satisfied by the SO(8) and SO(7) tensors forthe G and SU(4) − solutions in appendix C. In appendix D, we demonstrate explicitly that thesolution can indeed be expressed solely in terms of a single set of (anti-)selfdual SO(8) tensors, asargued in section 2. In the final appendix, E, we give an explicit representation of seven-dimensionalΓ-matrices and an embedding of R ⊕ R in R , which is used in section 7.5 Overview S The (pseudo)scalars of the maximal gauged supergravity in four dimensions parametrise the non-compact coset E / SU(8). In the unitary gauge, the group elements of the coset are given by thescalar 56-bein [30] V ( x ) = exp φ IJKL ( x ) φ IJKL ( x ) 0 ! = u IJ KL ( x ) v IJKL ( x ) v IJKL ( x ) u IJ KL ( x ) ! ∈ E , (2.1)where φ IJKL ≡ φ ∗ IJKL is a complex, selfdual tensor field: φ IJKL = 124 ε IJKLMNP Q φ MNP Q . (2.2)The uplift formulae for the internal metric and 3-form potential [3, 1] are then written in terms ofthe 56-bein, V ( x ), and the Killing vectors, K IJm , and 2-forms, K IJmn , on S as follows: (cid:0) ∆ − g mn (cid:1) ( x, y ) = 18 K m IJ ( y ) K n KL ( y ) h (cid:0) u MN IJ + v MNIJ (cid:1) (cid:0) u MN KL + v MNKL (cid:1) i ( x ) , (2.3)and (cid:0) ∆ − g pq A mnp (cid:1) ( x, y ) = − √ iK IJmn ( y ) K q KL ( y ) h (cid:0) u MN IJ − v MNIJ (cid:1) (cid:0) u MN KL + v MNKL (cid:1) i ( x ) . (2.4)In writing these and similar formulae we will adopt and apply the following convention consistentlythroughout this paper: The raising or lowering of indices on any geometric object on S , is always done by means of theround S metric ˚ g mn and its inverse. By contrast, to raise or lower indices on the physical fieldsof D = 11 supergravity (as they appear for instance in (1.5) and (1.6)), we always employ the fullmetric g mn and its inverse. This means, in particular, that on the right hand side of the above equations we have K mIJ ≡ ◦ g mn K IJn and so on.The full metric g mn ( x, y ) is then obtained by inverting and peeling of the determinant factorusing ∆ − = det(∆ − g mn ◦ g np ) . (2.5)For the 3-form field, A mnp ( x, y ), one must then insert the result for the densitised metric, ∆ g qr , onthe right hand side of (2.4).Formulae (2.3) and (2.4) are off-shell in the sense that they give the internal metric, g mn , and the3-form potential, A mnp , for any configuration of the scalar fields of the maximal gauged supergravityembedded in eleven-dimensional supergravity. In particular, note that the full antisymmetry of A mnp in (2.4) is not manifest, but can be established by means of the E properties of the 56-bein V ,and is thus independent of whether the equations of motion are satisfied or not [1]. For conventions and properties of the Killing spinors and tensors, see appendix A. φ IJKL ( x ), the geometricquantities g mn ( x, y ) and A mnp ( x, y ). To gain a better perspective on this problem, let us firstdiscuss the construction in a more general context before we specialise to SO(3) × SO(3) symmetricconfigurations below. For the most general configuration that has no symmetries at all the scalarfield configuration would of course involve the full set of 35 scalars and 35 pseudoscalars. However,we are here interested in specific configurations preserving some symmetry, for which we can restrictattention to φ IJKL ( x ) = X r λ ( r ) ( x )Φ ( r ) IJKL + i X s µ ( s ) ( x )Ψ ( s ) IJKL (2.6)where (cid:8) Φ ( r ) IJKL (cid:9) and (cid:8) Ψ ( s ) IJKL (cid:9) form a basis of invariant real selfdual and real anti-selfdual 4-forms(when we are dealing with real tensors the position of the indices
I, J, ... does not matter). If oneis looking for stationary points preserving a given symmetry, the scalar manifold is accordinglyparametrised by coordinates (cid:8) λ ( r ) , µ ( s ) (cid:9) . Simple examples of invariant 4-forms (for which the labels r and s are not needed) areΦ IJKL = C + IJKL , Ψ IJKL = 0 for SO(7) + symmetry;Φ IJKL = 0 , Ψ IJKL = C − IJKL for SO(7) − symmetry;Φ IJKL = C + IJKL , Ψ IJKL = C − IJKL for G symmetry . For the SO(3) × SO(3) solution we are about to construct, there are two invariant selfdual and twoinvariant anti-selfdual 4-forms, which are given in (2.16) below. In order to rewrite the solutionin terms of geometric objects adapted to the (deformed) S geometry, we define a set of invarianttensors via ξ ( r ) m = 116 Φ ( r ) IJKL K IJmn K n KL , ξ ( r ) mn = −
116 Φ ( r ) IJKL K IJm K KLn , ξ ( r ) = ˚ g mn ξ ( r ) mn (2.7)for the scalars, and S ( s ) mnp = 116 Ψ ( s ) IJKL K IJmn K KLp (2.8)for the pseudoscalars. By virtue of their definition and the (anti-)selfduality properties of the invari-ant 4-forms, these tensors satisfy the relations ◦ D m ξ = 2 m ξ m , ◦ D m ξ ( r ) n = 6 m ξ ( r ) mn − m ξ ( r ) ˚ g mn , ◦ D m ξ ( r ) np = 13 m (cid:16) ◦ g np ξ ( r ) m − ◦ g m ( n ξ ( r ) p ) (cid:17) , ◦ D m S ( s ) npq = 16 m ˚ η mnpqrst S ( s ) rst (2.9) Of course, at the stationary point, we can group all scalars and pseudoscalars into single SO(8) invariant objectswith associated SO(7) tensors, defined in an analogous manner to those defined in (2.7) and (2.8). In this case, one isguaranteed that the solution may be written solely in terms of these reduced set of SO(7) tensors. However, the resultwill not, in general, take a ‘nice’ form (see appendix D for a demonstration of this for the SO(3) × SO(3) invariantsolution). r and s . Furthermore, we have the inversion formulaeΦ ( r ) IJKL = 16 ξ ( r ) K [ IJm K m KL ] − ξ ( r ) mn K [ IJm K KL ] n + 112 ξ ( r ) m K [ IJmn K n KL ] , Ψ ( s ) IJKL = 12 S ( s ) mnp K mn [ IJ K p KL ] , (2.10)which are, again, valid separately for all r and s .Now, for any specific set of invariant 4-forms we will need further identities. First, such identitiesare needed to perform the exponentiation required for the calculation of u IJ KL , v
IJKL and theircomplex conjugates in (2.1). Second, we need these identities to solve the uplift formulae for g mn and A mnp and to bring the resulting expressions into a manageable form.The simplest examples, again, are provided by the SO(7) ± and G solutions for which theinvariant 4-forms C ± IJKL obey C ± IJMN C ± MNKL = 12 δ IJKL ± C ± IJKL , (2.11)i.e. their contractions either reproduce the same 4-forms or give the identity. The general case ismore complicated because any product of 4-forms may produce new invariant tensors that are not4-forms. The simplest example here is the G solution that depends on both C + IJKL and C − IJKL , aswell as the product C + IJMN C − MNKL , which defines a new invariant tensor (which is not a 4-form);this object then completes the list of G invariant tensors. A more complicated example is the tensor F IJ defined in (2.18) and further invariant objects for the SO(3) × SO(3) solution. Consequently wewill need to evaluate products such asΦ ( r ) IJMN Φ ( r ′ ) MNKL , Φ ( r ) IJMN Ψ ( s ) MNKL , Φ ( r ) IJMN Ψ ( s ) MNP Q Φ ( r ′ ) P QKL , etc. (2.12)and either reduce them to previously defined expressions or add them as new objects to the list ofinvariant tensors. The procedure stops when all products or contractions reproduce objects alreadycontained in the list; exploiting all such identities should enable us to compute u IJ KL and v IJKL ina closed form.Furthermore, as we will explain below in much detail for the SO(3) × SO(3) case, the identitiessatisfied by the above invariants entail a corresponding hierarchy of identities for the geometrictensors introduced in (2.7) and (2.8). The main use of these identities will be in carrying out theinversion required to derive the metric and 3-form from the uplift formulae (2.3) and (2.4) andin bringing the resulting expressions into a manageable form. This last step is necessary for theverification of the D = 11 field equations which would otherwise be unmanageably complicated.Before proceeding let us comment on another point. In Kaluza-Klein theory one is usuallyinterested in calculating the mass spectrum of a given compactification, and the massless states inparticular. This requires a linearised expansion of the metric (2.3) and the 3-form potential (2.4) inthe scalar fluctuations around a given vacuum. For the maximally symmetric S compactificationwe thus have [23, 31] g mn ( x, y ) = ◦ g mn ( y ) + X A IJKL ( x ) Y IJKLmn ( y ) + . . . ,A mnp ( x, y ) = X B IJKL ( x ) Y IJKLmnp ( y ) + . . . , (2.13)8here A IJKL and B IJKL are the 35 scalar and 35 pseudoscalar fields of N = 8 supergravity ( φ IJKL = A IJKL + iB IJKL ), and, where the ellipses denote massive modes. The corresponding eigenmodeshave been known for a long time [23, 31] Y IJKLmn ( y ) = K [ IJm K KL ] n − ◦ g mn K p [ IJ K KL ] p , Y IJKLmnp ( y ) = K [ IJ [ mn K KL ] p ] . (2.14)The formulae (2.3) and (2.4) are thus the consistent non-linear extensions of the above formulae (it is straightforward to check that the linearised formulae follow directly from (2.3) and (2.4) byexpanding the latter to first order in the scalar and pseudoscalar fields). One can therefore askwhether it is possible to directly ‘exponentiate’ the formulae (2.13). The above discussion showsthat this is indeed possible for restricted configurations if one has enough tensor identities at hand. × SO(3) solution
The SO(3) × SO(3) subgroup of SO(8), which is the symmetry of the stable stationary point inmaximal gauged supergravity, is defined by the following branchings of the three fundamental rep-resentations: v −→ ( , ) + ( , ) + 2 × ( , ) , s,c −→ × ( , ) . (2.15)In the conventions that we are using, the eight gravitini, ψ I , and the Killing spinors, η I , on S ,transform under v . We choose the two SO(3) groups to act on the subspaces defined by I = 1 , , I = 6 , ,
8, respectively. Then the four invariant noncompact generators of E are given bythe tensors Y + IJKL = 4! (cid:0) δ IJKL + δ IJKL (cid:1) , Y − IJKL = 4! (cid:0) δ IJKL + δ IJKL (cid:1) ,Z − IJKL = 4! (cid:0) δ IJKL − δ IJKL (cid:1) , Z + IJKL = 4! (cid:0) δ IJKL − δ IJKL (cid:1) , (2.16)where Y + IJKL and Z + IJKL are selfdual, while Y − IJKL and Z − IJKL are anti-selfdual. In section 5, weshow that the simplest and most symmetric form of the solution is obtained in terms of the followinginvariants defined by these tensors: ξ m = 116 Y + IJKL K IJmn K n KL , ξ mn = − Y + IJKL K IJm K KLn , ξ = ˚ g mn ξ mn ,ζ m = 116 Z + IJKL K IJmn K n KL , ζ mn = − Z + IJKL K IJm K KLn , ζ = ˚ g mn ζ mn ,S mnp = 116 Y − IJKL K IJ [ mn K KLp ] , T mnp = 116 Z − IJKL K IJ [ mn K KLp ] , (2.17)as well as two additional tensors F m = F IJ K IJm , F mn = F IJ K IJmn , F IJ = δ IJ , (2.18) Cf. definitions (2.7) and (2.8). F m = 118 ˚ η mnpqrst S npq T rst , ◦ D m F n = − m F mn . (2.19)Note that, as emphasised before, the objects defined in (2.17) and (2.18) belong to S . Hence, theirindices are raised and lowered with ˚ g mn and its inverse, for instance ξ mn ≡ ˚ g mp ˚ g nq ξ pq . We are now in a position to state the main result of this paper, which is an explicit uplift ofthe solution at the SO(3) × SO(3) stationary point of the scalar potential written in terms of thegeometric quantities introduced above. The solution below is presented in its simplest and the mostsymmetric form. We refer the reader to section 5 for a more general form of the solution which, inparticular, includes an additional parameter, α , corresponding to an accidental U(1) symmetry ofthe potential. The solution below is for α = − π/ g mn = ∆ (cid:2) ( X + Z )˚ g mn − X ξ mn + Z ζ mn ) + 2 f m f n (cid:3) , (2.20)where f m = 6 F m − √ ξ m + √ ζ m . (2.21)The 3-form flux is A mnp = ∆ √ h(cid:16) √ Z ( X − Z ) − X − Z (cid:17) S mnp + (cid:16) √ X ( X − Z ) + 5 X + Z (cid:17) T mnp + 127˚ η mnpqrst ( Z ξ q − X ζ q ) (cid:0) Z S rst + X T rst (cid:1) − (cid:0) X ξ [ m + Z ζ [ m (cid:1) S np ] q ξ q i , (2.22)where the warp factor, ∆, is given by∆ = 36 X + 10 X Z + Z , (2.23)with X = 2 ξ − √ , Z = 2 ζ − √ . (2.24)The solution is now complete modulo two constants, which as discussed in the introduction, aredetermined by the value of the potential, P ∗ , at the stationary point using (1.12) and (1.13). Forthe SO(3) × SO(3) point, P ∗ = 14. Hence, m = 283 m , f FR = 7 √ m . (2.25)In particular, the fact that the value of f FR given above, as determined by equation (1.13), is consistentwith a solution of the equations of motion is further evidence for the validity of this conjecturedrelation (1.13) between f FR and the potential. The remaining constant, m , sets the overall scale ofthe solution.One should note that the metric and the 3-form potential in (2.20) and (2.22) are obtained by anapplication of the identities derived in section 3 to simplify the “raw” expressions that follow fromthe uplift formulae. We refer the reader to section 5 for details of the derivation and to section 7 foranother form of the solution in which the geometry of the internal space is perhaps more transparent.10 Identities for SO(3) × SO(3) invariants
In this section, we present in a systematic way a set of identities for the geometric objects ξ , ζ , ξ m , ζ m , ξ mn , ζ mn , S mnp , T mnp , F m , F mn , (3.1)defined in (2.17) and (2.18). These identities are crucial for the discussion in subsequent sections,in particular, we need them to derive, in section 5, the simplified form of the solution in (2.20)and (2.22) and to verify that the equations of motion are satisfied in section 6. They are also ofinterest on their own as the starting point for identifying the SO(3) × SO(3) geometry underlying oursolution. Such an identification would allow one to construct a large class of new solutions in whichthe underlying internal manifold is not necessarily the round seven-sphere in much the same way asis done when extending the SU(4) − solution [6] to arbitrary Sasaki-Einstein manifolds [32, 6, 33].The identities we are looking for fall into two broad categories: (i) generic identities, which areproved using only the (anti-)selfduality property of the underlying SO(8) tensors and propertiesof the Killing vectors/spinors; (ii) identities specific to the objects (3.1). These are proved byexploiting the concrete SO(3) × SO(3) invariant form of the SO(8) tensors Y ± IJKL , Z ± IJKL and F IJ defined in (2.16) and (2.18). The identities in this section follow from the particular dependence of the SO(7) tensors (3.1) definedin (2.17) and (2.18) on the Killing vectors/spinors. They do not require specific knowledge of howthe underlying SO(8) tensors are defined. We refer the reader to Refs. [5, 34, 35, 2] for proofs andfurther details.Equations (2.17) and (2.18) can be inverted using the completeness property of the Γ-matrices.This yields, cf. (2.10), Y + IJKL = 16 ξK [ IJm K m KL ] − ξ mn K [ IJm K KL ] n + 112 ξ m K [ IJmn K n KL ] , Y − IJKL = 12 S mnp K [ IJmn K KL ] p ,Z + IJKL = 16 ζK [ IJm K m KL ] − ζ mn K [ IJm K KL ] n + 112 ζ m K [ IJmn K n KL ] , Z − IJKL = 12 T mnp K [ IJmn K KL ] p ,F IJ = 18 F m K IJm + 116 F mn K IJmn . (3.2)Similarly, the background covariant derivative of the SO(7) tensors can be computed using theKilling spinor equation (A.8)˚ D m ξ = 2 m ξ m , ˚ D m ξ n = 6 m ξ mn − m ξ ˚ g mn , ˚ D p ξ mn = 13 m (cid:0) ˚ g mn ξ p − ˚ g p ( m ξ n ) (cid:1) , ˚ D m ζ = 2 m ζ m , ˚ D m ζ n = 6 m ζ mn − m ζ ˚ g mn , ˚ D p ζ mn = 13 m (cid:0) ˚ g mn ζ p − ˚ g p ( m ζ n ) (cid:1) , ˚ D m S npq = 16 m ˚ η mnpqrst S rst , ˚ D m T npq = 16 m ˚ η mnpqrst T rst , ˚ D n F m = m F mn , ˚ D p F mn = 2 m ˚ g p [ m F n ] . (3.3) All identities in section 2.1 fall into this category.
11e stress once more that both (3.2) and (3.3) do not depend on the particular forms of the SO(8)tensors Y ± IJKL , Z ± IJKL and F IJ . The starting point for proving the identities satisfied by the SO(7) tensors and listed in tables 1-7 arevarious contraction identities for the SO(8) tensors Y ± IJKL , Z ± IJKL and F IJ . The latter follow directlyfrom the definitions of these tensors in Eqs. (2.16) and (2.18), and can be split into several groupsdepending on the number of factors and the number of contractions. Each group then gives rise todifferent types of SO(7) identities. The identities given in this section are sufficient for determiningthe internal components of the metric and 3-form potential from the uplift ans¨atze and proving thatthe metric and 3-form potential thus obtained solve the field equations. A. Double contraction identities between two of the Y ± IJKL and Z ± IJKL tensors: Y + IJMN Y + MNKL = Z − IJMN Z − MNKL , Y − IJMN Y − MNKL = Z + IJMN Z + MNKL , (3.4) Y + IJMN Z + MNKL = Z − IJMN Y − MNKL , Y − IJMN Z − MNKL = Z + IJMN Y + MNKL , (3.5)and Y + IJMN Y − MNKL = Z − IJMN Z + MNKL , Y − IJMN Y + MNKL = Z + IJMN Z − MNKL ,Z + IJMN Y − MNKL = Y − IJMN Z + MNKL , Y + IJMN Z − MNKL = Z − IJMN Y + MNKL . (3.6)Note that each set of (anti-)selfdual tensors, Y ± IJKL and Z ± IJKL , respectively, do not in themselveslead to simple quadratic identities, but are instead related to each other via quadratic relations. Thisis pertinent to the discussion in section 2.1 and appendix D, where it is argued that one can alwaysmake do with a single set of (anti-)selfdual tensors at the price of working to higher order. Here wesee that there are no self-contained set of quadratic identities for a single set of (anti-)selfdual tensors.Therefore, the result is that one must work with expressions that are higher-order in tensors—asillustrated explicitly in appendix D. This is to be contrasted with the previously known uplifts wherethe situation is simpler, see table 8. In the case of the G invariant quantities, there are quadraticrelations between the single set of (anti-)selfdual tensors. While in the slightly more complicatedSU(4) − example, the single set of (anti-)selfdual 4-form tensors close on a 2-form tensor, rather thananother set of 4-form tensors. More generally, for stationary points with even less symmetry thelesson seems to be that one must include enough (anti-)selfdual tensors in order to have quadraticrelations between the tensors. Otherwise, the metric and 3-form potential will not be expressible atmost quadratically in the SO(7) tensors. B. Double contraction identities with triple factors: Y + IJMN Y + MNP Q Y + P QKL = 4 Y + IJKL , Z + IJMN Z + MNP Q Z + P QKL = 4 Z + IJKL , (3.7)12s well as Y − IJMN Y − MNP Q Y − P QKL = 4 Y − IJKL , (3.8) Y + IJMN Y − MNP Q Y + P QKL = 0 , Y − IJMN Y + MNP Q Y − P QKL = 0 , (3.9) Y − IJMN Y + MNP Q Y + P QKL + Y + IJMN Y + MNP Q Y − P QKL = 4 Y − IJKL , (3.10) Y + IJMN Y − MNP Q Y − P QKL + Y − IJMN Y − MNP Q Y + P QKL = 4 Y + IJKL , (3.11)and analogous identities obtained by replacing Y by Z in the above identities. C. Identities involving the F IJ tensor: Y + IKLM Z + JKLM = Z − IKLM Y − JKLM = 12 F IJ , (3.12) Y ± IJKL F KL = Z ± IJKL F KL = 0 , (3.13)8 Y ± [ IJK | M | F M L ] = ± Z ± IJKL , Z ± [ IJK | M | F M L ] = ∓ Y ± IJKL . (3.14)Given the identities for the SO(8) tensors, it is clear from the inversion formulae (3.2) that theseidentities imply identities satisfied by the SO(7) tensors in (3.1). We list these identities in tables1–4. Note that we do not use the cubic identities (3.11) in deriving the SO(7) tensor identities—theywill be used in section 4 to exponentiate the 56-bein in the unitary gauge.While it is correct that the SO(7) tensor identities in tables 1–4 are a consequence of substitutingthe inversion formulae into the SO(8) tensor identities (3.4)–(3.7) and (3.12), (3.14), it is ratherlaborious to obtain these identities by the said method—at least without the aid of a computerprogram. In appendix B, we sketch a simpler proof for these identities. Furthermore, in the appendixwe explain how the identities listed in tables 5–7 are derived from the identities in tables 1–4. Despitethe fact that the derivation of these identities is quite an involved task, we have tried to present theidentities as systematically as possible. In particular, the order in which the identities are presentedis such as to indicate the fact that identities listed prior to a given identity may have been used toderive or simplify that identity. This means that, for instance, we have included an identity thatmay be obtained by contracting another identity, allowing the reader to check the consistency ofthe two. In any case, here we limit the explanation of the derivations to the comments in the tablecaptions, sketching a derivation of the identities in appendix B.13 able 1 (i) ξ mn ξ mn = 32 + ξ , ξ p ξ p = 9 − ξ , ζ mn ζ mn = 32 + ζ , ζ p ζ p = 9 − ζ (ii) S mnp S mnp = 6 , T mnp T mnp = 6(iii) ξ mn ξ n = 0 , ζ mn ζ n = 0(iv) ξ mp ξ np = (cid:18) − ξ (cid:19) ˚ g mn + ξ ξ mn − ξ m ξ n , ζ mp ζ np = (cid:18) − ζ (cid:19) ˚ g mn + ζ ζ mn − ζ m ζ n (v) S mpq S npq = (cid:18) − ζ (cid:19) ˚ g mn − ζ m ζ n + 2 ζ ζ mn , T mpq T npq = (cid:18) − ξ (cid:19) ˚ g mn − ξ m ξ n + 2 ξ ξ mn (vi) ˚ η mnqrstu T qrs T tup = 8 ξ [ m ξ n ] p − ξξ [ m ˚ g n ] p , ˚ η mnqrstu S qrs S tup = 8 ζ [ m ζ n ] p − ζζ [ m ˚ g n ] p (vii) S mnr S pqr = 2 ζ [ m [ p ζ n ] q ] + (cid:18) − ζ (cid:19) δ mnpq − ζ [ m ζ [ p δ n ] q ] (viii) T mnr T pqr = 2 ξ [ m [ p ξ n ] q ] + (cid:18) − ξ (cid:19) δ mnpq − ξ [ m ξ [ p δ n ] q ] Identities derived from (3.4) and (3.7).
Table 2 (i) ξ m ζ m = − ξζ, ξ mn ζ mn = 16 ξζ, S mnp T mnp = 0(ii) ˚ η mnpqrst S npq T rst = 18 F m (iii) ξ mn ζ n = ξ ζ m − ζ ξ m + 32 F m , ζ mn ξ n = − ξ ζ m + ζ ξ m − F m (iv) ξ mp ζ np = − ξζ ˚ g mn − ξ m ζ n + 16 ( ζξ mn + ξζ mn ) + 14 F mn (v) S mpq T npq = − ξζ ˚ g mn −
118 ( ξ m ζ n + ζ m ξ n ) + 13 ( ζξ mn + ξζ mn ) − F mn (vi) ˚ η npqrstu S mqr T stu = − ξ m [ n ζ p ] − ζ m [ n ξ p ] + 23 ζ ˚ g m [ n ξ p ] + 23 ξ ˚ g m [ n ζ p ] − g m [ n F p ] (vii) ˚ η npqrstu T mqr S stu = − ξ m [ n ζ p ] − ζ m [ n ξ p ] + 23 ζ ˚ g m [ n ξ p ] + 23 ξ ˚ g m [ n ζ p ] + 6˚ g m [ n F p ] (viii) S mnr T pqr + T mnr S pqr = − ξζδ mnpq − ξ [ m ζ [ p δ n ] q ] − ζ [ m ξ [ p δ n ] q ] + 4 ξ [ m [ p ζ n ] q ] Identities derived from (3.5) and (3.12).14 able 3 (i) S mnp ξ p + T mnp ζ p = 0 , S mnp ζ p = T mnp ξ p = 0(ii) S qmn ζ pq = S q [ mn ζ p ] q = ζ S mnp − η mnpqrst ζ q S rst (iii) T qmn ξ pq = T q [ mn ξ p ] q = ξ T mnp − η mnpqrst ξ q T rst (iv) 4 ζ qr T rmn − η qmnstuv ζ s T tuv = 8 S sq [ m ξ n ] s − ξS qmn (v) 4 ξ qr S rmn − η qmnstuv ξ s S tuv = 8 T sq [ m ζ n ] s − ζT qmn Identities derived from (3.6).
Table 4 (i) S mnp F np = 0 , S mnp F p = 112˚ η mnpqrst S pqr F st (ii) T mnp F np = 0 , T mnp F p = 112˚ η mnpqrst T pqr F st (iii) S q [ mn F p ] q = 23 T mnp + 118˚ η mnpqrst S qrs F t (iv) T q [ mn F p ] q = − S mnp + 118˚ η mnpqrst T qrs F t Identities derived from (3.13) and (3.14).
Table 5 (i) F m ξ m = ζ, F m ζ m = − ξ, F m ξ mn = 16 ζ n + ξ F n , F m ζ mn = − ξ n + ζ F n (ii) F mn ξ n = − ζ m − ξF m , F mp ξ pn = ζ g mn − F m ξ n − ζ mn + ξ F mn (iii) F mn ζ n = ξ m − ζF m , F mp ζ pn = − ξ g mn − F m ζ n + ξ mn + ζ F mn (iv) F m F m = 1 , F mn F n = 0 , F mp F pn = F m F n − δ mn F -tensor identities derived by contractions of the equations in (iii) and (iv)in table 2 with ξ m , ζ m , F m , ξ mq , ζ mq and F mq .15 able 6 (i) ξ sm S nps + ζ sm T nps = ξ s [ m S np ] s + ζ s [ m T np ] s (ii) ξ s [ m S np ] s = 19 ( ζT mnp + 2 ξS mnp ) − η mnpqrst (2 ζ q T rst + ξ q S rst )(iii) ζ s [ m T np ] s = 19 ( ξS mnp + 2 ζT mnp ) − η mnpqrst (2 ξ q S rst + ζ q T rst )Identities derived from the equations in (iv) and (v) in table 3. Table 7 (i) ˚ η mnpqrst ξ p ζ q S rst = 6 ζS mnp ξ p + 54 S mnp F p , ˚ η mnpqrst ζ p ξ q T rst = 6 ξT mnp ζ p − T mnp F p (ii) ˚ η mnpqrst F p ζ q S rst = 6 S mnp ξ p + 6 ζS mnp F p , ˚ η mnpqrst F p ξ q T rst = − T mnp ζ p + 6 ξT mnp F p (iii) ˚ η mnpqrst F p ξ q S rst = 6 ξS mnp F p , ˚ η mnpqrst F p ζ q T rst = 6 ζT mnp F p Identities derived by contractions of the equations in (ii)–(iii) in table 3 with ξ p , ζ p and F p ; and contractions of (iii) and (iv) in table 4 with ξ m and ζ m , respectively. × SO(3) solution of gauged supergravity
In the unitary gauge defined in equation (2.1), the u and v matrices are of the form u IJ KL = ∞ X n =0 n )! [( φφ ∗ ) n ] IJKL , v
IJKL = ∞ X n =0 n + 1)! [ φ ∗ ( φφ ∗ ) n ] IJKL . (4.1)For an SO(3) × SO(3) invariant configuration, the most general parametrisation of the scalar andpseudoscalar expectation value φ IJKL is given by the SO(3) × SO(3) invariant quantities defined inequation (2.16) φ IJKL = λ (cid:2) cos α (cid:0) Y + IJKL + iY − IJKL (cid:1) − sin α (cid:0) Z + IJKL − iZ − IJKL (cid:1)(cid:3) , (4.2)where the parameter α may be freely chosen without loss of generality. This is because, while therelevant SO(3) × SO(3) invariant truncation of the theory contains two complex scalars, the potentialcorresponding to this truncation is invariant under an extra U(1) symmetry that lies outside thegauge group, namely SO(8) [21]. The α parameter corresponds to this U(1) freedom that leaves thepotential invariant. In what follows we will choose to keep the value of α general. Interestingly, froman eleven-dimensional perspective we find that α corresponds to a coordinate transformation of theeleven-dimensional solution along the seven compactified directions (see section 5.3).16n exponentiating the scalar expectation value φ IJKL to find the u and v matrices, it is useful todefine Π = 18 (cid:0) Y + + iY − (cid:1) (cid:0) Y + − iY − (cid:1) = 18 (cid:0) Z + − iZ − (cid:1) (cid:0) Z + + iZ − (cid:1) , (4.4)which, using the cubic identities (3.7) and (3.11), satisfies the following propertiesΠ = Π , Π ∗ IJKL = Π
KLIJ . (4.5)Therefore, Π is a hermitean projector, and (cid:0) Y + − iY − (cid:1) Π = Y + − iY − , (cid:0) Z + + iZ − (cid:1) Π = Z + + iZ − . (4.6)In particular, using identities (3.7), we find that φφ ∗ = 2 λ Π , φ ∗ Π = φ ∗ . (4.7)Hence, the u and v matrices may be written as follows u IJ KL = δ KLIJ + ( c − IJKL , (4.8) v IJKL = s √ (cid:2) cos α ( Y + − iY − ) − sin α ( Z + + iZ − ) (cid:3) IJKL , (4.9)where c = cosh( √ λ ) , s = sinh( √ λ ) . The scalar potential for the scalar λ reads P = − g s − s − , (4.10)and, indeed, does not depend on α .The SO(3) × SO(3) invariant stationary point is given by d P ds = 0 , (4.11)and corresponds to [14] c = √ , s = 2 . (4.12)This stationary point is the only known stable non-supersymmetric stationary point of D = 4maximal supergravity [12, 13]. In fact, there clearly exists another stationary point correspondingto s → − s , that is s = −
2. From the perspective of the D = 11 solution this corresponds to A mnp → − A mnp under which the equations of motion (1.7)-(1.9) are invariant. We will take s = 2henceforth, while keeping this in mind. In what follows, we make use of the short-hand notation
A B = (
A B ) IJKL = A IJMN B MNKL . (4.3) The SO(3) × SO(3) solution of D = 11 supergravity Given the scalar profile of the SO(3) × SO(3) invariant solution of the gauged theory described in theprevious section, the eleven-dimensional SO(3) × SO(3) solution is simply constructed by applyingthe uplift formulae (2.3) and (2.4) for the internal metric and 3-form potential [3, 1]. In this sectionwe present the details of the calculation leading to the solution in its simplified form.
We apply the uplift formula (2.3) to evaluate the metric from the data at the SO(3) × SO(3) invariantstationary point. The Sp(56) property of the u and v matrices [36] u MN IJ u MN KL − v MNIJ v MNKL = δ KLIJ , (5.1) u MN IJ v MNKL − v MNIJ u MN KL = 0 , (5.2)can be used to rewrite the scalar part of the metric ansatz (2.3) as follows (cid:0) u MN IJ + v MNIJ (cid:1) (cid:0) u MN KL + v MNKL (cid:1) = − δ KLIJ + 2Re (cid:0) u MN IJ u MN KL + v MNIJ u MN KL (cid:1) . (5.3)Substituting in the expressions for u and v , equations (4.8) and (4.9), we find thatRe (cid:0) u MN IJ u MN KL + v MNIJ u MN KL (cid:1) = δ KLIJ + s Re(Π
IJKL ) + sc √ (cid:0) cos α Y + − sin α Z + (cid:1) IJKL . (5.4)Contracting the expression above with K m IJ K n KL and using the completeness relation (B.1) torewrite the expression in terms of SO(7) tensors gives∆ − g mn ( x, y ) =˚ g mn + s (cid:20) g m [ n ξ q ] ξ q + 2 ξ mp ξ np + S mpq S npq + 19˚ g m [ n ζ q ] ζ q + 2 ζ mp ζ np + T mpq T npq (cid:21) − √ sc (cos αξ mn − sin αζ mn ) . (5.5)Using the SO(7) identities in table 1, the above expression reduces to∆ − g mn = (cid:20) c − s
18 ( ξ + ζ ) (cid:21) ˚ g mn − s
18 ( ζ m ζ n + ξ m ξ n ) + s X ξ mn + Z ζ mn ) , (5.6)where X ( α ) = ξs − √ c cos α, Z ( α ) = ζs + 3 √ c sin α. (5.7)The first four lines of equations in tables 1 and 2 and the identities in table 5 can be used toinvert the densitised metric (still for arbitrary α )∆ g mn = 1 X + 2 c X Z + Z + Y h X + Z )˚ g mn − s ( X ξ mn + Z ζ mn ) + s f m f n i , (5.8)where X ( α ) = √ α ξs − c , Z ( α ) = −√ α ζs − c , Y ( α ) = s (cos α − sin α )( ξ − ζ ) , (5.9)18nd f m ( α ) = √ c cos α ζ m + √ c sin α ξ m + 3 sF m . (5.10)We can calculate the warp factor, ∆, using (2.5), by evaluating the variations∆ g mn δ (∆ − g mn ) , (5.11)with respect to α and λ . After simplifying (5.11) using identities in tables 1, 2 and 5, one canintegrate back to obtain ∆, with the overall normalisation fixed by requiring that ∆ = 1 for λ = 0.This gives ∆ = 36 X + 2 c X Z + Z + Y . (5.12)This completes the derivation of the uplifted metric tensor, g mn , for arbitrary values of λ and α . As before, we simplify the scalar part of the flux ansatz (2.4) using the Sp(56) property of the u and v matrices (cid:0) u ijIJ − v ijIJ (cid:1) (cid:0) u ij KL + v ijKL (cid:1) = δ IJKL + 2 i Im (cid:0) u ijIJ u ij KL − v ijIJ u ijKL (cid:1) . (5.13)For the u and v matrices corresponding to the SO(3) × SO(3) invariant sectorIm (cid:0) u ij IJ u ij KL − v ijIJ u ijKL (cid:1) = s Im(Π
IJKL ) + sc √ Y − IJKL . (5.14)Contracting the above expression with K IJmn K q KL and making use of the completeness relation (B.1),the flux ansatz (2.4) gives∆ − g pq A mnp ( x, y ) = s √ (cid:18) S sq [ m ξ n ] s − ξS qmn − ξ qr S rmn + 19˚ η qmnstuv ξ s S tuv − T sq [ m ζ n ] s + 43 ζT qmn + 4 ζ qr T rmn − η qmnstuv ζ s T tuv (cid:19) + 16 sc (cos α S qmn + sin α T qmn ) . (5.15)Upon use of the identities in tables 3 and 6, the expression above simplifies significantly:∆ − g pq A mnp = s √ (cid:18) sS sq [ m ξ n ] s − sT sq [ m ζ n ] s − X S qmn + 13 Z T qmn (cid:19) . (5.16)Multiplying the above equation by the metric and substituting the expression (5.8) for ∆ g pq , andmaking full and repeated use of the SO(7) identities in section 3.2, the resulting expression reduces19o A mnp = ∆ √ (cid:20) − (cid:16) s c ) X + s c √ ζ (sin α ξ + cos α ζ ) (cid:17) S mnp + (cid:16) s c ) Z − s c √ ξ (sin α ξ + cos α ζ ) (cid:17) T mnp + s η mnpqrst ( Z ξ q − X ζ q ) (cid:0) Z S rst + X T rst (cid:1) − s (cid:0) X ξ [ m + Z ζ [ m (cid:1) S np ] q ξ q (cid:21) . (5.17)with ∆ given in (5.12).Note that while it is clear that the metric obtained from the ansatz (2.3) is manifestly symmetricin its indices, this is not the case for the 3-form potential (2.4). However, as is shown in Ref. [1], theantisymmetry property of the 3-form potential is guaranteed to hold even off-shell for any values ofthe scalar fields as is the case for the 3-form potential in (5.17).This concludes the uplift of the SO(3) × SO(3) stationary point to D = 11 supergravity. It isindeed remarkable that such a complicated solution as this one can be so simply derived in thematter of a few calculational steps. α As remarked earlier, from the point of view of gauged supergravity we are free to choose α withoutloss of generality, because of an accidental U(1) symmetry of the potential that is outside thegauge group. This is a novel feature of the SO(3) × SO(3) invariant truncation and is absent forother truncations for which the higher dimensional uplift is known. There ought to be a way ofunderstanding this redundancy in the choice of α from an eleven-dimensional perspective. Giventhat in the four-dimensional theory the U(1) transformation does not lead to a different stationarypoint, it must be the case that for any choice of α the uplifted solutions are equivalent, viz. they arerelated by coordinate transformations as we demonstrate here. Specifically, we find that a shift inthe parameter α corresponds to a diffeomorphism in the seven compactified dimensions, in the sensethat δ α (∆ g mn ( α )) = L V (∆ g mn ( α )) , δ α ( A mnp ( α )) = L V ( A mnp ( α )) , (5.18)with the generating vector field V V = − m F m ˚ D m . (5.19)This allows us to pick any particular value of α : checking the equations of motion for that particularvalue then implies that the equations are also satisfied for other values of α . Henceforth, we choose Note that, while V is a Killing vector on the background internal space, corresponding to the round S , it is nolonger a Killing vector in the deformed space given by the metric g mn . In deforming the round seven-sphere to obtainthe SO(3) × SO(3) invariant solution, the number of Killing vector fields reduces from 28 to 6; these are given by K , K , K , K , K and K .
20o fix the value of α , α = − π , (5.20)so that the metric (5.8) is symmetric under the interchange of tensors defined with respect to Y ± IJKL and Z ± IJKL . In this case, sin( α ) = − √ , cos( α ) = 1 √ , Y = 0 , (5.21) X = X ≡ X = ξs − c, Z = Z ≡ Z = ζs − c, (5.22)and the metric determinant is: ∆ = 36 / (cid:0) X + 2 c X Z + Z (cid:1) − / . (5.23)In summary, at the stationary point values given by equation (4.12), we find the internal metricand 3-form potential given in equations (2.20) and (2.22). It is only at the stationary point values,given in equation (4.12), that these expressions solve the equations of motion (1.7)–(1.9). Note alsothat with the choice of α given in this section, the metric is indeed symmetric under the interchangeof tensors defined using invariants Y ± IJKL and Z ± IJKL , while the 3-form is antisymmetric. Giventhe symmetric form of the solution for the choice of α = − π/
4, this is the solution that we workwith in order to verify that the field equations are satisfied.
In this section, we verify that the SO(3) × SO(3) invariant solution does indeed satisfy the fieldequations of D = 11 supergravity, equations (1.7)–(1.9). It is a surprising fact that the verificationforms by far the most involved part of the work and requires the use of many of the identities listed insection 3.2. In comparison, finding the solution using the non-linear ans¨atze is fairly straightforward.This is a testimony to the power of the uplift ans¨atze, which are non-linear. From the perspectiveof the SU(8) invariant reformulation, it is clear that the ans¨atze should lead to internal metric and3-form potential components that satisfy the D = 11 supergravity equations of motion. This isbecause they have been derived by the use of supersymmetry transformations which are first orderequations, rather than second order as in the case of the field equations. Moreover, the highly non-linear problem of relating the scalars of the D = 4 maximal gauged supergravity to the componentsof the internal metric and 3-form has been linearised by packaging the components of the D = 11fields in the generalised vielbeine. The relation between the scalars of the D = 4 theory and thegeneralised vielbeine is a linear one. Both of the simplifications alluded to above mean that while thederivation of the solution is relatively simple, its verification in the context of the original formulationof D = 11 supergravity [11] becomes non-trivial. Note that under this interchange we also have F m → − F m , F mn → − F mn . We refer the reader to the first equation in table 3 for the antisymmetry of the last term in equation (2.22).
21n order to verify the Einstein and Maxwell equations (1.7)–(1.9), we make use of the computeralgebraic manipulation program
FORM [37] to simplify the expressions for the Ricci tensor and the4-form field strength.
We begin by computing the components of the eleven-dimensional Ricci tensors R µν and R mn thatappear in the equations of motion, (1.7) and (1.8), and whose indices are raised with the full metric, g MN . Denoting g µν ( x, y ) = ∆( y )˚ g µν ( x ) , g µν ( x, y ) = ∆ − ( y )˚ g µν ( x ) , (6.1)the Christoffel symbols with mixed index components areΓ ρmn = Γ pmν = 0 , (6.2)Γ pµν = − g pq ∂ q g µν = 12 (cid:16) ∆ − ˚ D q ∆ (cid:17) g pq g µν , (6.3)Γ ρµn = 12 g ρσ ∂ n g µσ = − (cid:16) ∆ − ˚ D n ∆ (cid:17) δ ρµ . (6.4)Moreover, for convenience, we defineˆΓ pmn = Γ pmn − ˚Γ pmn = 12 g pq (cid:16) ˚ D m g nq + ˚ D n g mq − ˚ D q g mn (cid:17) . (6.5)The relevant components of the eleven-dimensional Riemann tensor are R µνρσ = − ∂ ρ Γ µσν + ∂ σ Γ µρν − Γ µρM Γ Mσν + Γ µσN Γ Nρν = ˚ R µνρσ − Γ µρm Γ mσν + Γ µσn Γ nρν = ˚ R µνρσ + 12 (∆ − ˚ D p ∆)(∆ − ˚ D q ∆) g pq δ µ [ ρ g σ ] ν , (6.6) R µmνn = − ∂ ν Γ µnm + ∂ n Γ µνm − Γ µνp Γ pnm + Γ µnρ Γ ρνm = −
12 ˚ D n (∆ − ˚ D m ∆) δ µν + 12 ˆΓ pmn (∆ − ˚ D p ∆) δ µν + 14 (∆ − ˚ D n ∆)(∆ − ˚ D m ∆) δ µν , (6.7) R mµnν = g mp g µρ R ρpνn , (6.8) R mnpq = ˚ R mnpq − ˚ D p ˆΓ mqn + ˚ D q ˆΓ mpn − ˆΓ mpr ˆΓ rqn + ˆΓ mqr ˆΓ rpn , (6.9)where ˚ R µνρσ and ˚ R mnpq denote the Riemann tensors of the background AdS and round seven-sphere, respectively. The associated Ricci tensors in our conventions are given in (1.10).It is now straightforward to obtain the expressions for the relevant components of the Ricci22ensor, R µν = R ρµρν + R pµpν = 3∆ m g µν + g µν g mn (cid:18) (∆ − ˚ D m ∆)(∆ − ˚ D n ∆) −
12 ˚ D n (∆ − ˚ D m ∆) + 12 ˆΓ pmn (∆ − ˚ D p ∆) (cid:19) , (6.10) R mn = R pmpn + R ρmρn = − m ˚ g mn − ˚ D p ˆΓ pnm + ˚ D n ˆΓ ppm − ˆΓ ppr ˆΓ rnm + ˆΓ pnr ˆΓ rpm + (∆ − ˚ D m ∆)(∆ − ˚ D n ∆) − D n (∆ − ˚ D m ∆) + 2ˆΓ pmn (∆ − ˚ D p ∆) . (6.11)In fact, it is more convenient for us to directly calculate ∆ − R µν = ∆ − R µρ g ρν and ∆ − R mn = R mp (∆ − g pn ). Using the expression for the internal metric given in equation (2.20) and the expres-sion for the determinant (2.23) as well as equations (3.3) and the SO(7) identities in section 3.2,∆ − R µν = 3 m δ νµ + m ∆ (cid:18) X − X Z − X Z − X Z + 91 Z + 24 √ X + Z )(19 X − X Z + 19 Z ) + 1260(5 X + 2 X Z + 5 Z ) (cid:19) δ νµ , (6.12)∆ − R mn = m ∆ (cid:0) A ( X , Z ) δ nm + A ( X , Z ) ξ mn + A ( Z , X ) ζ mn + A ( X , Z ) F mn + A ( X , Z ) ξ m ξ n + A ( Z , X ) ζ m ζ n + A ( X , Z ) F m F n + A ( X , Z ) ξ m ζ n + A ( Z , X ) ζ m ξ n + A ( X , Z ) ξ m F n − A ( Z , X ) ζ m F n + A ( X , Z ) F m ξ n − A ( Z , X ) F m ζ n (cid:1) . (6.13)Recall that in our conventions, the index n on the left hand side is raised with the inverse metric g mn , while on the right hand side we use the inverse metric on the round S , ◦ g mn . The coefficientfunctions in the above equation are as follows: A ( X , Z ) = 2 √
53 ( X + Z ) (cid:0) X − X Z − X Z − X Z + 17 Z (cid:1) + 403 (cid:0) X − X Z − X Z − X Z + 13 Z (cid:1) + 40 √ X + Z ) (cid:0) X − X Z + 17 Z (cid:1) − X + 2 X Z + 5 Z ) ,A ( X , Z ) = − √ X − Z ) − (cid:0) X − X Z − X Z − Z (cid:1) − √ X + Z ) (cid:0) X − X Z − X Z − Z (cid:1) ,A ( X , Z ) = 10080( X − Z ) + 96 √ X − Z ) (cid:0) X + 10 X Z + 7 Z (cid:1) + 200( X − Z )( X + Z ) ,A ( X , Z ) = 672 √ X + 13 Z ) + 32(45 X − X Z + 79 Z )+ 8 √ (cid:0) X + 43 X Z −
X Z + 17 Z (cid:1) , ( X , Z ) = − X + 10 X Z + Z ) − √ X + Z )( X − X Z + Z ) ,A ( X , Z ) = − √ X + 5 Z ) − X − X Z − Z ) − √ (cid:0) X − X Z + 75 X Z + 25 Z (cid:1) ,A ( X , Z ) = 336 √ X + 10 X Z + Z ) + 16 (cid:0) X − X Z −
X Z − Z (cid:1) ,A ( X , Z ) = − X + 13 Z ) − √ X − X Z + 47 Z ) − (cid:0) X + 116 X Z − X Z + 66 Z (cid:1) . Note that, like the metric, both R µν and R mn are symmetric under the interchange of tensorsdefined using Y ± IJKL and Z ± IJKL , definitions (2.17). In this section, we calculate the 4-form field strength F mnpq = 4! ˚ D [ m A npq ] (6.14)of the 3-form potential given in equation (2.22). Using the equations for the derivatives of the SO(7)tensors (3.3) F mnpq = 2 √ m ∆ (cid:20) B ( X , Z ) ξ [ m S npq ] − B ( Z , X ) ζ [ m T npq ] + B ( X , Z ) ζ [ m S npq ] − B ( Z , X ) ξ [ m T npq ] + B ( X , Z )˚ η mnpqrst S rst − B ( Z , X )˚ η mnpqrst T rst + ˚ η mnpqrst S rsu ξ u (cid:0) B ( X , Z ) ξ t + B ( Z , X ) ζ t + B ( X , Z ) F t (cid:1) (cid:21) , (6.15)where we have simplified some expressions using the SO(7) identities in section 3.2 and B ( X , Z ) = 3( X + 10 X Z + 49 Z ) − √ X − Z )( X + 11 Z ) ,B ( X , Z ) = 3(5 X + 2 X Z + 5 Z ) + √ X − X Z − X Z − Z ) ,B ( X , Z ) = −
18 ( X + 2 Z )( X − X Z + 5 Z ) + √
548 ( X + Z )( X + 10 X Z + Z ) ,B ( X , Z ) = − √
52 ( X − Z ) ,B ( X , Z ) = 15( X − Z ) . Raising the indices on F mnpq using the inverse metric g mn poses the greatest challenge from acomputational point of view. Therefore, we choose to calculate it using the following method∆ − F mnpq = 4!∆ (∆ − g r [ m )(∆ − g n | s | ) (cid:20) ∆ − g p | t | ˚ D r (cid:16) ∆ − A q ] st (cid:17) − ˚ D r (cid:16) ∆ − g p | t | (cid:17) ∆ − A q ] st (cid:21) . (6.16) See footnote 8. α = − π/
4, and simplifying the resulting expression using equations(3.3) and the SO(7) identities in section 3.2 gives∆ − F mnpq = √ m ∆ (cid:20) C ( X , Z ) ξ [ m S npq ] − C ( Z , X ) ζ [ m T npq ] + C ( X , Z ) ζ [ m S npq ] − C ( Z , X ) ξ [ m T npq ] + C ( X , Z )˚ η mnpqrst S rst − C ( Z , X )˚ η mnpqrst T rst + ˚ η mnpqrst S rsu ξ u ( C ( X , Z ) ξ t + C ( Z , X ) ζ t + C ( X , Z ) F t ) (cid:21) , (6.17)where C ( X , Z ) = 2243 Z −
89 (3 + √ Z ) (cid:0) X + 10 X Z + Z (cid:1) ,C ( X , Z ) = − X Z + 89 (15 + √ X ) (cid:0) X + 10 X Z + Z (cid:1) ,C ( X , Z ) = − X + 5 Z ) − √ (cid:0) X + 16 X Z + 17 Z (cid:1) − X (cid:0) X + 10 X Z + Z (cid:1) ,C ( X , Z ) = − X + 2 √ (cid:0) X + 10 X Z + Z (cid:1) ,C ( X , Z ) = 0 . The field strength of A , F mnpq , and F mnpq also share the antisymmetry property of A mnp underthe interchange of tensors defined from Y ± IJKL and Z ± . This allows us to derive an expression for∆ − F mpqr F npqr = − − R mn + 4 m ∆ (cid:18) X + 35 X Z + 178 X Z + 35 X Z + 14 Z + 3 √ X + Z )(19 X − X Z + 19 Z ) + 634 (29 X − X Z + 29 Z ) (cid:19) δ nm , (6.18)where we have used the expressions for F mnpq and F mnpq , equations (6.15) and (6.17), respectively,as well as equation (6.13) and the SO(7) identities in section 3.2.Finally, contracting the indices in the equation above and using the expression for R mn inequation (6.13) as well the SO(7) identities gives∆ − F mnpq F mnpq = 16 m ∆ (cid:0) X + 35 X Z + 178 X Z + 35 X Z + 14 Z + 3 √ X + Z )(19 X − X Z + 19 Z ) − X − Z )(3
Z − X ) (cid:1) . (6.19) Using equations (6.12), (6.13), (6.18), (6.19) and (2.23), it is now straightforward to show that theEinstein equations (1.7) and (1.8) are satisfied for the values of m and f FR given in (2.25). Finally,using the equations for the derivatives of the SO(7) tensors (3.3) to differentiate (6.17) as well asthe SO(7) identities in section 3.2, we find that the Maxwell equation (1.9) is also satisfied.25 Solution in ambient coordinates
The solution presented in the previous sections is given in terms of quantities defined on the roundseven-sphere. In particular, the metric, g mn , in (2.20) is written as a deformation of the metric, ◦ g mn , on the round seven-sphere. Furthermore, the tensors (2.17) are defined in terms of the Killingspinors on S . While this is necessary for obtaining the solution via the uplift ans¨atze (2.3) and(2.4) for the metric and flux, it is perhaps not the most natural form in which to express the solutiongiven its isometry. In this section, we present the solution in a form in which the action of theSO(3) × SO(3) is more manifest.
To find the relation between the coordinates on the round seven-sphere and coordinates that we willuse in this section, we introduce coordinates x A on R , where A = 1 , . . . ,
8. Then the seven-sphereis defined by m x · x = 1 , (7.1)where in this section we use the notation x · x ≡ x A x A . It is straightforward to see that the aboverelation is solved by m x m = 2 y m | y | , m x = 1 − | y | | y | , (7.2)which define stereographic coordinates y m on the round seven-sphere of inverse radius m (with | y | ≡ y m y m ). The relations in the previous section can be viewed as being written in precisely sucha coordinate system. Hence, in the previous sections the line element on the round S is given by ds = ◦ g mn dy m dy n . (7.3)In fact, the induced metric on the seven-sphere can easily be calculated by substituting equations(7.2) into the flat line element on R , whereupon we find that ◦ g mn = 4 m (1 + | y | ) δ mn . (7.4)A convenient choice for the siebenbein is ◦ e a = − m (1 + | y | ) dy a . (7.5)Instead of viewing the action of SO(3) × SO(3) in stereographic coordinates, we can now view itsaction as an action of SO(3) × SO(3) ≃ SO(4) on two four-dimensional subspaces of R in ambientcoordinates x A . More precisely, we can view R as the direct sum R ⊕ R and decompose x = ( u, v ),where u, v ∈ R such that SO(4) acts separately on u and v . No confusion should arise between these and the u and v matrices that parametrise the scalars in the gaugedtheory, described in section 4. × SO(3) invariant tensors in the previous section, written in terms of Killing spinorson the round S , can be expressed in ambient coordinates as follows. In terms of Killing spinors,the 1-form duals of Killing vectors on S are [3] K IJ = K IJa ◦ e a . (7.6)However, since the Killing vectors, K IJa , generate SO(8) in , they are related by triality to gener-ators of SO(8) in the vector representation. Or, equivalently, in terms of their 1-form duals K IJ = − m IJAB K AB , K AB = − m Γ ABIJ K IJ , (7.7)where K AB = 2 x [ A dx B ] . (7.8)Furthermore, K IJ (2) ≡ K IJab ◦ e a ∧ ◦ e b = 12 Γ IJAB d K AB . (7.9)Now we can use these relations to determine the SO(3) × SO(3) invariant tensors in ambient coordi-nates. We start with the scalar invariant ξ defined in (2.17), and substitute for K IJm using relation(7.7) ξ = m Y + IJKL Γ IJKLAB x A x B = 3 m Γ AB x A x B . (7.10)Note that since the exterior derivative in K AB , definition (7.8), is with respect to stereographiccoordinates, we also use relations (7.2) in deriving the above result. Similarly, ζ = 3 m Γ AB x A x B . (7.11)Naively, there are three scalar invariants that can be formed from u and v . However, note that fromequation (7.1) u · u + v · v = 1 . (7.12)Therefore, we only have two scalar invariants u · u − v · v, u · v and without loss of generality we can pick an embedding of the R in R where ξ = − u · u − v · v ) , ζ = − u · v. (7.13)For an explicit embedding where the above relations hold see appendix E. Note that any otherembedding will correspond to a rotation between u and v , which in the present representation, seeappendix E, is given by Γ AB , viz . Γ : u v, v
7→ − u. (7.14)This freedom is represented by the parameter α in section 5, which is related to the rotation anglebetween u and v. In the four-dimensional theory, this corresponds to a redundancy in the description27f the SO(3) × SO(3) invariant stationary point and not an invariance. As was shown in section 5.3,this is reflected in the fact that the uplift of all these points correspond to the same solution up tocoordinate transformations.Given the expressions for ξ and ζ in ambient coordinates, it is now straightforward to find thetensors ξ a and ζ a in ambient coordinates by differentiating expressions (7.13) and using equationsin (3.3): m ξ a ◦ e a = − u · du − v · dv ) , m ζ a ◦ e a = − v · du + u · dv ) . (7.15)The remaining invariant 1-form F a , (2.18), is found using equations (7.7), (7.8) and the third equationin (E.6), m F a ◦ e a = v · du − u · dv. (7.16)We may again differentiate the tensors ξ a and ζ a to obtain expressions for the symmetric tensors ξ ab and ζ ab , respectively, in ambient coordinates. However, we will instead find these expressions byother means, which will be applicable also to the derivation of the tensors S abc and T abc . Using equation (7.7), we rewrite ξ ab ◦ e a ◦ e b = − m Y + IJKL Γ IJAB Γ KLCD K AB K CD . (7.17)Note that the indices on Y + fully antisymmetrise the indices on the Γ-matrices. Hence we can makeuse of the following identity [30]:Γ [ IJAB Γ KL ] CD = 12 (cid:18) Γ IJ [ AB Γ KLCD ] − ǫ ABCDEF GH Γ IJEF Γ KLGH (cid:19) + 23 δ [ C | [ B Γ IJKLA ] | D ] , (7.18)which is a consequence of SO(8) triality and is a decomposition of the object on the left hand sideinto its anti-selfdual (first term) and selfdual part (second term). Moreover, noting that in theexpression for ξ ab the combination of Γ-matrices contracts with a selfdual tensor, Y + IJKL , we obtain ξ ab ◦ e a ◦ e b = m AB K AC K CB . (7.19)Finally using (7.8) and the first equation in (E.6), we find that m ξ ab ◦ e a ◦ e b = ( v · v ) dv · dv − ( u · u ) du · du, (7.20)where we have also used u · du + v · dv = 0 , (7.21)which follows from (7.12). Similarly, we also find m ζ ab ◦ e a ◦ e b = − h u · v ( du · du + dv · dv ) + du · dv i . (7.22)We determine S abc and T abc in an analogous way. For example, S (3) ≡ S abc ◦ e a ◦ e b ◦ e c = − m Y − IJKL Γ IJAB Γ KLCD K AB (2) ∧ K CD . (7.23)28ence, we can again use identity (7.18), but in this case the anti-selfdual part of the decompositiongiven in equation (7.18) survives and we obtain S (3) = − m AB Γ CD x [ A dx B ∧ dx C ∧ dx D ] , (7.24)which can be evaluated using the Γ-matrices and the embedding given in appendix E. All in all, weobtain m S (3) = − h ǫ ( u, dv, dv, dv ) + ǫ ( v, du, du, du ) + 3 ǫ ( u, du, du, dv ) + 3 ǫ ( v, du, dv, dv ) i , (7.25) m T (3) = − h ǫ ( u, du, du, du ) − ǫ ( v, dv, dv, dv ) i , (7.26)where we have introduced the convenient notation ǫ ( u, du, du, dv ) ≡ ǫ ijkl u i du j ∧ du k ∧ dv l . (7.27)It is clear that there are two more invariant 3-forms, ǫ ( u, du, dv, dv ) , ǫ ( v, du, du, dv ) , (7.28)that do not appear in the expression for S (3) or T (3) . However, these invariant 3-forms as well as the3-forms in S (3) and T (3) do appear in the expression for the internal 3-form potential given below. In terms of the ambient coordinates introduced above, the solution (2.20)-(2.22) reads: ds = ∆ m h + c (6 c − s ( ζ + ξ )) ( du · du + dv · dv ) + s ( sξ − c ) ( du · du − dv · dv )+ 2 s ( sζ − c ) du · dv + 16 s f i , (7.29)and A (3) = √ m h + s (cid:0) c − c s (2 ζ + 2 ξ + 3) − cs ( ζ − ξ + 3) + ζs (cid:1) ǫ ( u, du, du, du ) − s (cid:0) c + c s ( − ζ − ξ + 3) + cs ( ζ − ξ −
3) + ζs (cid:1) ǫ ( v, dv, dv, dv ) − s ( c + s ) (cid:0) c − cs ( ζ + ξ + 3) + ξs (cid:1) ǫ ( u, du, du, dv ) − s ( c − s ) (cid:0) c − cs ( ζ + ξ − − ξs (cid:1) ǫ ( v, du, dv, dv ) − s ( c − s ) (cid:0) c − cs ( ζ + ξ + 3) + ξs (cid:1) ǫ ( v, du, du, du )+ s ( c + s ) (cid:0) − c + cs ( ζ + ξ −
3) + ξs (cid:1) ǫ ( u, dv, dv, dv ) − s ( c + s )( ζs − c ) ǫ ( u, du, dv, dv )+ 3 s ( c − s )(3 c − ζs ) ǫ ( v, du, du, dv ) i , (7.30) K.P. would like to thank N. Bobev, A. Kundu and N. Warner for a collaboration which independently led to themetric in the ambient form presented here [38]. ds = g ab ◦ e a ◦ e b , A (3) = 16 A abc ◦ e a ∧ ◦ e b ∧ ◦ e c , (7.31) m f ≡ m f a ◦ e a = 3 c ( u · du − v · du − u · dv − v · dv ) + 3 s ( v · du − u · dv ) (7.32)and with c and s set to their stationary values (4.12). We conclude this section with a construction of local coordinates on S using the Euler angles of theSO(3) × SO(3) isometry group and the two scalar invariants, ξ and ζ . To this end let us consider S as a subspace of 2 × Z = z − iz z + iz − z + iz z + iz ! , z j = u j + iv j , (7.33)satisfying 12 Tr ZZ † = u · u + v · v = 1 (7.34)and det Z = −
13 ( ξ + iζ ) . (7.35)Then the SO(4) action on C is the same as the action of SU(2) × SU(2) on such matrices givenby Z −→ R Z R † , (7.36)under which both (7.34) and (7.35) remain invariant.We use the Euler angles for the two SU(2)s defined by R j ( θ j , φ j , ψ j ) = e i ( φ j + ψ j ) cos θ j − e i ( φ j − ψ j ) sin θ j e − i ( φ j − ψ j ) sin θ j e − i ( φ j + ψ j ) cos θ j ! , j = 1 , . (7.37)By an SU(2) × SU(2) transformation, one can bring Z to a diagonal form, Z d ( ρ, ϕ ) = √ e i ( ϕ + π ) cos ρ
00 sin ρ ! , ≤ ρ ≤ π , ≤ ϕ ≤ π , (7.38)where ( ρ, ϕ ) parametrise a disk of radius π/
2. Using (7.35), we find ξ = 3 sin ρ cos ϕ , ζ = 3 sin ρ sin ϕ (7.39)so that have | ξ | , | ζ | ≤
3, which is consistent with identities (i) in table 1 [5].At a generic point, we have Z = R ( θ , φ , ψ ) Z d ( ρ, ϕ ) R ( θ , φ , ψ ) † . (7.40)Clearly, Z is invariant under ψ i → ψ i + χ , which shows that a typical orbit is isomorphic with thecoset SU (2) × SU (2) U (1) , (7.41)30here U (1) is the diagonal subgroup. The local coordinate system on S is now comprised of theangles ρ and φ that parametrise a disk and the Euler angles θ , φ , θ , φ and ψ = ψ − ψ on thecoset. The range of these angles are0 ≤ ρ ≤ π , ≤ θ , ψ ≤ π , ≤ ϕ , φ , φ , θ ≤ π . (7.42)Let us also introduce the left invariant forms on SU (2) × SU (2) , σ j ) = sin ψ j dθ j − cos ψ j sin θ j dφ j ,σ j ) = − cos ψ j dθ j − sin ψ j sin θ j dφ j ,σ j ) = dψ j + cos θ j dφ j , (7.43)satisfying dσ j ) = σ j ) ∧ σ j ) , etc., and define σ ± ( j ) = σ j ) ± iσ j ) . (7.44)These forms are then pulled-back onto the coset by setting ψ = − ψ = ψ/
2, such that σ , σ , σ , σ , σ ≡ σ − σ , (7.45)yield a local frame, σ a , a = 1 , . . . ,
5, along the orbits of the SO(4) isometry.The round metric on S in these coordinates reads d ˚ s = 14 m h dρ + sin ρ dϕ + (cid:0) σ +(1) σ − (1) + σ +(2) σ − (2) (cid:1) − sin ρ (cid:0) σ +(1) σ − (2) + σ +(2) σ − (1) (cid:1) + (cid:0) cos ρ dϕ − σ (cid:1) i . (7.46)The geometric objects (3.1) are the scalars given by (7.39), the vectors: m (cos ϕ ξ a + sin ϕ ζ a ) ◦ e a = 32 cos ρ dρ ,m (cos ϕ ζ a − sin ϕ ξ a ) ◦ e a = 32 sin ρ dϕ , (7.47)the symmetric tensors: m (cos ϕ ξ ab + sin ϕ ζ ab ) ◦ e a ◦ e b = −
14 sin ρ (cid:0) cos ρ dϕ − σ (cid:1) σ + 14 sin ρ (cid:0) σ +(1) σ − (1) + σ +(2) σ − (2) (cid:1) + 116 (cos(2 ρ ) − (cid:0) σ +(1) σ − (2) + σ +(2) σ − (1) (cid:1) ,m (cos ϕ ζ ab − sin ϕ ξ ab ) ◦ e a ◦ e b = 14 dρ (cos ρ dϕ − σ ) + i ρ (cid:0) σ +(1) σ − (2) − σ +(2) σ − (1) (cid:1) , (7.48) Note that at the center of the disk ξ = ζ = 0 and we simply reproduce the explicit construction of T , in [39]. m (cid:0) cos ϕ S abc − sin ϕ T abc (cid:1) ◦ e a ∧ ◦ e b ∧ ◦ e c = 316 × n − i dρ ∧ h(cid:0) σ +(1) ∧ σ − (1) + σ +(2) ∧ σ − (1) (cid:1) − sin ρ (cid:0) σ +(1) ∧ σ − (2) + σ +(2) ∧ σ − (1) (cid:1)i + cos ρ (cos ρ dϕ − σ ) ∧ (cid:0) σ +(1) ∧ σ − (2) − σ +(2) ∧ σ − (1) (cid:1)o m (cid:0) cos ϕ T abc + sin ϕ S abc (cid:1) ◦ e a ∧ ◦ e b ∧ ◦ e c = 3 i × n (cos ρ dϕ − σ ) ∧ h(cid:0) σ +(1) ∧ σ − (2) + σ +(2) ∧ σ − (1) (cid:1) − sin ρ (cid:0) σ +(1) ∧ σ − (1) + σ +(2) ∧ σ − (1) (cid:1)i + sin ρ σ ∧ h(cid:0) σ +(1) ∧ σ − (1) + σ +(2) ∧ σ − (1) (cid:1) − sin ρ (cid:0) σ +(1) ∧ σ − (2) + σ +(2) ∧ σ − (1) (cid:1)io . (7.49)We also have that m F a ◦ e a = − (cid:0) dϕ − cos ρ σ ) ,m F ab ◦ e a ∧ ◦ e b = −
12 sin ρ dρ ∧ σ − i ρ (cid:0) σ +(1) ∧ σ − (1) − σ +(2) ∧ σ − (2) (cid:1) . (7.50)Rotations by the angle ϕ to obtain the actual SO(7) tensors (3.1) result in even larger expressions.As expected, the explicit formulae for the metric (2.20) and the 3-form potential (2.22) in these localcoordinates are quite complicated and we will not write them here. One can easily obtain themusing the expressions for the SO(7) tensors given above. In this paper, we have constructed a new and highly non-trivial solution of D = 11 supergravitycorresponding to an uplifting of the SO(3) × SO(3) invariant stationary point of maximal gaugedsupergravity. While this solution is of interest in holographic applications and we hope that readerswill find good use for it, we have endeavored to present the derivation of the solution in such a manneras to lend itself to a more general explanation of uplifting solutions of this type, i.e. Freund-Rubincompactifications with internal flux. The uplifting of any stationary point of the gauged theoryto eleven dimensions will follow the same steps as those presented for the SO(3) × SO(3) invariantstationary point here, except that, clearly, for stationary points with less symmetry, this will be amore cumbersome process with many different invariant forms to consider.Apart from allowing for a direct derivation of uplift formulae, the rewriting of the eleven-dimensional theory in an SU(8) invariant reformulation [9], highlights features of the four-dimensionaltheory in eleven dimensions and makes it possible to prove [26, 27], for example, the consistency ofthe S reduction [8, 23].In recent work [28, 40], the ideas initiated in Ref. [9] are taken to their full conclusion giving anon-shell equivalent reformulation of the D = 11 theory in which features of the global group E are also made manifest. As well as breaking manifest eleven-dimensional Lorentz invariance and32ovariance, one is also compelled to introduce eleven-dimensional dual fields in order to bring outthe E structure.The reformulation of D = 11 supergravity given in Ref. [28] provides a very direct and efficientway of studying the relation between four-dimensional maximal gauged theories and D = 11 super-gravity via a higher-dimensional understanding [28] of the embedding tensor [41–44]. In particular,it allows for a simple analysis of which four-dimensional theories arise as consistent reductions ofthe eleven-dimensional theory (see e.g. [45]). For example, it is very simple to deduce [29] that thenew deformed SO(8) gauged theories of Ref. [46, 47] cannot be obtained from a consistent reductionof the D = 11 theory.In fact, given the success of the reformulations described above, we argue that, generally, themost appropriate setting in which to address questions to do with reductions and consistency isone in which the higher-dimensional theory is reformulated in such a manner as to fully resemble aduality covariant reformulation of the lower-dimensional theory, including both the global and localduality groups.Of particular relevance here is that in the case of the S reduction to the original maximal SO(8)gauged theory [36], Ref. [28] completes the metric and flux ans¨atze and provides full uplift ans¨atzefor any solution of the gauged theory to eleven dimensions, including dynamical solutions with non-trivial x -dependence [29]. The method can, however, be applied more generally. For example, onecan in principle setup a reformulation along the lines of [9, 28] for type IIB supergravity and therebystudy its S truncation—for a recent conjecture on uplift ans¨atze in this case see Ref. [48].An interesting application of these full uplift ans¨atze [28, 29] would be to construct the fullinterpolating solution for a particular RG flow between two stationary points of the potential, suchas the flow between the maximally symmetric SO(8) and the SO(3) × SO(3) invariant stationarypoints considered in Ref. [12].
Acknowledgements : We are grateful to Nikolay Bobev, Arnab Kundu, Chris Pope, Harvey Realland Nick Warner for discussions. M.G., H.G. and K.P. would like to thank the Max-Planck-Institutf¨ur Gravitationsphysik (AEI) and in particular H.N. for hospitality. H.G. and M.G. are supportedby King’s College, Cambridge. H.G. acknowledges funding from the European Research Councilunder the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grantagreement no. [247252]. K.P. was supported in part by DOE grant DE-SC0011687.33
Conventions
We define a set of euclidean, antisymmetric and purely imaginary 8 × † = Γ). Theseare generators of the euclidean Clifford algebra in seven dimensions, { Γ a , Γ b } = 2 δ ab I × . (A.1)We choose a Majorana representation and set the charge conjugation matrix that defines spinorconjugates or raises and lowers spinor indices to be the unit matrix. An explicit representation forthe Γ-matrices is given in appendix E.The Γ-matrices can be used to define the 8 × a ...a i = Γ [ a . . . Γ a i ] (A.2)for i = 2 , . . .
7. Γ a and Γ ab are antisymmetric matrices and Γ abc is symmetric. These 7 + 21 + 35 = 63matrices together with the unit matrix span the vector space of 8 × a ...a = − iη a ...a , (A.3)Γ a ...a = − iη a ...a b Γ b , (A.4)Γ a ...a = i η a ...a bc Γ bc , (A.5)Γ a ...a = i η a ...a bcd Γ bcd . (A.6)Furthermore, it is useful to note that each product of Γ-matrices can be written in terms of the unitmatrix, Γ a , Γ ab and Γ abc .We choose the eight Killing spinors of the round S to be orthonormal,¯ η I η J = δ IJ , η I ¯ η I = I × , (A.7)where ¯ η I = ( η I ) † .The curved Γ-matrices on the round seven-sphere are given by ˚Γ m = ˚ e ma Γ a . Hence, in ourconventions, the Killing spinors satisfy i ˚ D m η I = m m η I . (A.8)The Killing spinors define a set of Killing vectors, 2-forms and tensors: K IJm = i ¯ η I ˚Γ m η J , K IJmn = ¯ η I ˚Γ mn η J , K IJmnp = i ¯ η I ˚Γ mnp η J , (A.9)respectively, whose equivalents are also defined in flat space. Using equation (A.8), the reader maycheck that K IJmn is proportional to the derivative of K IJm ,˚ D n K IJm = m K IJmn , ˚ D p K IJmn = 2 m ˚ g p [ m K IJn ] . (A.10)Note that curved seven-dimensional indices of the Killing vectors and their derivatives are raisedand lowered with the round seven-sphere metric ˚ g mn .As all Γ-matrices are traceless, we find that¯ η I ˚Γ m ...m i η I = 0 (A.11)for i = 1 , . . . ,
6. 34
Derivation of SO(7) tensor identities
In this appendix, we sketch the derivation of the SO(7) identities, listed in tables 1-7, for theSO(3) × SO(3) invariant tensors (3.1) .In the derivations below, we make heavy use of the completeness relation16 δ KLIJ = 2 K IJm K m KL + K IJmn K mn KL , (B.1)as well as the following useful identities [9]:116 Y + IJKL K IJmn K KLp = − ◦ g p [ m ξ n ] , Y + IJKL K IJmn K pq KL = − δ [ p [ m ξ n ] q ] + 23 ξδ pqmn , (B.2)116 Z + IJKL K IJmn K KLp = − ◦ g p [ m ζ n ] , Z + IJKL K IJmn K pq KL = − δ [ p [ m ζ n ] q ] + 23 ζδ pqmn , (B.3)116 Y − IJKL K IJmn K KLpq = − ◦ η mnpqrst S rst , Z − IJKL K IJmn K KLpq = − ◦ η mnpqrst T rst . (B.4)One can verify these using the inversion formulae (3.2). B.1 Derivation of the identities in table 1
Identities (i) and (ii)
Consider the first equation in (3.4) contracted with K IJt K t KL : K IJt Y + IJMN (2 K MNm K m P Q + K MNmn K mn P Q ) Y + P QKL K t KL = K IJt Z − IJMN K MNmn K mn P Q Z − P QKL K t KL , (B.5)where we have used the completeness relation (B.1) and the fact that K IJm K KLn Z − IJKL = 0 by virtueof the fact that K [ IJm K KL ] n is selfdual, while Z − IJKL is anti-selfdual. Now, substituting for the SO(7)tensors using the definitions (2.17) and equation (B.2) gives2 ξ mn ξ mn + 19 ◦ g t [ m ξ n ] ◦ g t [ m ξ n ] = T mnp T mnp , (B.6)which simplifies to ξ m ξ m + 6 ξ mn ξ mn = 3 T mnp T mnp . (B.7)Repeating the above steps, except now contracting the first equation in (3.4) with K IJtu K tu KL gives − ξ + ξ m ξ m + 30 ξ mn ξ mn = 9 T mnp T mnp (B.8)Finally, by contracting the first cubic identity (3.7) with K IJtu K u KL and simplifying as before, exceptthat the completeness relation (B.1) must be used twice, gives (cid:0)
36 + 2 ξ − ξ m ξ m − ξ mn ξ mn (cid:1) ξ t = 0 . (B.9)There are seemingly two cases to consider: first we consider the case in which the expression inthe brackets vanishes. Together, with equations (B.7) and (B.8), we obtain the equations for ξ m ξ m , ξ mn ξ mn and T mnp T mnp in terms of ξ , as they appear in equations in (i) and (ii) in table 1. The35quations derived from considering the second case, ξ m ≡
0, are already contained in equations (i)and (ii). However, in our case, ξ m ξ is three.Note, however, that (3.7) is not used anymore in deriving the identities in table 1.Interchanging Y and Z in the discussion above, or equivalently by considering the second iden-tities in (3.4) and (3.7) gives analogous expressions for ζ m ζ m , ζ mn ζ mn and S mnp S mnp . Identities (iii) and (vi)
This case is similar to the example above. We contract equations (3.4)with K IJmn K pKL . This gives identity (vi). Identity (iii) is obtained upon letting index p = n andnoting that the wedge product of an odd-form with itself vanishes, e.g. ◦ η mnpqrst S npq S rst = 0 . (B.10) Identities (iv) and (v)
These identities are derived by contracting equations (3.4) with K IJm K KLn and K IJmp K np KL . Identities (i)–(iii) are used to simplify the expressions. Identities (vii) and (viii)
Contract identities (3.4) with K IJmn K KLpq and use identities (i)–(v) tosimplify.
B.2 Derivation of the identities in table 2
Identity (i)
The third identity in the line is proved by contracting the last equality in (3.12) by δ IJ . Using the appropriate inversion formulae in (3.2) and K [ IJ [ ab K KL ] c ] K [ IJde K KL ] f = 32 δ abcdef , (B.11)we immediately find S mnp T mnp = 0 . The first two identities are derived by contracting either equation in (3.5) with K IJm K m KL and K IJmn K mn KL . Identity (ii)
Contract the last equality in (3.12) with K m IJ , whereupon we find Z − IKLM Y − JKLM K m IJ = 12 F m . (B.12)We then make use of the inversion formula for Z − IKLM , (3.2) to find Z − IKLM K m IJ = 14 S npq K [ LMnp K qm J | K ] . (B.13)Substituting this expression and the inversion formula for Y − JKLM in equation (B.12), gives therequired result.
Identity (iii)
These are obtained by contracting identity (3.5) with K IJmn K nKL . dentities (iv) and (v) The symmetric, in indices m and n , part of these are derived by contracting(3.5) with K m IJ K n KL and K mp IJ K npKL . The antisymmetric part is derived by contracting (3.12)with K mn IJ , Z + IKLM Y + JKLM K mn IJ = 12 F mn , Z − IKLM Y − JKLM K mn IJ = 12 F mn . (B.14)The evaluation of the left hand side of the above equations using the inversion formulae 3.2 yieldsthe antisymmetric part of the identities. Identities (vi) and (vii)
These identities are derived by contracting identity (3.5) with K IJm K KLnp .Note that the F terms in this expression arise through the use of identities (iii), which have beenused to simplify the expression. Identity (viii)
This is obtained by contracting identity (3.5) with K IJmn K KLpq . B.3 Derivation of the identities in table 3
Identity (i)
These identities are proved by contracting equation (3.6) with K IJm K KLn . Identity (ii)–(v)
These are obtained by contracting equation (3.6) with K IJmn K KLp . B.4 Derivation of the identities in table 4
Identities (i) and (ii)
The required result is obtained by contracting identities (3.13) (with the − sign choice) with K m IJ and K mn IJ . Identities (iii) and (iv)
Contract identities (3.14) (with the − sign choice) with K mn IJ K p KL . B.5 Derivation of the identities in table 5
It would, at first sight, appear that the identities in table 5 are most easily derived analogouslyto the identities in table 4, sketched above, using identities (3.14) except with the + sign choice.However, in fact they can most simply be derived by contracting identities (iii) and (iv) in table 2with ξ m , ζ m , ξ mn , ζ mn , F m and F mn and using identities (i), (iii) and (iv) from table 1 and identities(i), (iii) and (iv) from table 2 to simplify the resulting expressions. Note that the identities must bederived in the order given in table 5 as earlier identities are used to obtain later ones. B.6 Derivation of the identities in table 6
Identity (i)
We add 4 ξ qr S rmn to both sides of equation (iv) in table 3,4 ζ rq T mnr + 4 ξ rq S mnr − η qmnstuv ζ s T tuv = 8 S s [ qm ξ n ] s − ξS qmn . (B.15)Rearranging the above equation, we conclude that ζ rq T mnr + ξ rq S mnr is fully antisymmetric in { q, m, n } . Hence identity (i). Identities (ii) and (iii)
Fully antisymmetrise the indices in identities (iv) and (v) in table 3. Thisleads to a set of simultaneous equations, which can be solved to obtain the result.37 .7 Derivation of the identities in table 7
Identity (i)
These are derived by contracting identities (ii) and (iii) of table 3 with ξ p and ζ p ,respectively and using identities (iii) of table 2. Identity (ii)
Contract (ii) and (iii) of table 3 with F p and use identities (i) of table 5. Identity (iii)
These are the most non-trivial identities to prove. We consider the first of the iden-tities, and the other follows from analogous arguments, or simply interchange symmetry. However,before embarking on the proof, we note that contracting (v) in table 3 with F q and using identity (iii)of table 6 leads to an equation for the sum of the two equations in (iii) and not on each separately.Therefore, we need another method.Contract identity (iii) in table 4 with ξ p . Hence, using identity (i) of table 3, η mnpqrst F p ξ q S rst = 18 S q [ mn F p ] q ξ p . (B.16)In order to find an expression for S q [ mn F p ] q that is amenable to contraction with ξ p , we consider ξ [ mq ζ nr S p ] qr . (B.17)This expression can be simplified in two ways. First, we can use identity (viii) of table 2 to rewrite ξ [ m [ q ζ n ] r ] and the identities in tables 1 and 2 can be used to simplify expression (B.17). Anotherway of simplifying the expression is to observe that, from (ii) in table 3, ξ [ mq ζ nr S p ] qr = ξ [ m | q ζ qr S | np ] r . Hence, we can also rewrite expression (B.17) using identity (iv) of table 2. We can now equate thetwo different expressions to derive27 S q [ mn F p ] q = 2 ξζS mnp + 2(9 − ζ ) T mnp + 3 ζ [ m S np ] q ξ q − ζS q [ mn ξ p ] q − ξS qmn ζ pq + 12 ζT q [ mn ζ p ] q . (B.18)The required identity can be deduced by substituting the above equation into expression (B.16) andsimplifying using the identities listed in the tables. C Comparison of stationary points
In this appendix, we present table 8, which gives a list of the various tensors used to construct otherstationary point uplifts and the associated identities they satisfy.38 able 8
Symmetry SO(8) tensor identities Associated SO(7) tensor identitiesG C ± IJMN C ± MNKL = 12 δ IJKL ± C ± IJKL ξ a ξ a = (21 + ξ )(3 − ξ )6 ξ ab = (3 + ξ ) δ ab − − ξ ) ξ a ξ b S abe S cde = 12 δ abcd + η abcdefg S efg S [ abc S d ] ef = η abcdgh [ e S f ] gh S e [ ab S cd ] f = η abcdgh ( e S f ) gh SU(4) − Y − IJMN Y − MNKL = 8 δ IJKL − F − [ I [ K F − J ] L ] F − KI F − JK = − δ JI Y − MIJK F − ML = Y − M [ IJK F − ML ] = ( Y − M [ IJK F − ML ] ) − K a K a = 1 , K ab K b = 0 K ac K cb = K a K b − δ ab T abc K c = 0 T acd T bcd = 4( δ ab − K a K b ) List of identities satisfied by G and SU(4) − invariant tensors. We use notation where ( X IJKL ) − refers to the anti-selfdual part of tensor X. The SO(7) tensors ξ, S and T are defined according tothe general definitions (2.7) and (2.8), and 4 K a = F IJ K IJa , K ab = F IJ K IJab . In G , the single set of tensors C ± do not close on themselves at the quadratic level, but onecan form new tensors from the contraction of C ± C ∓ . However, the new SO(7) tensors that can bedefined for these objects are related to ξ and S at the quadratic level, hence there is no simplificationin doing this. D Choice of SO(3) × SO(3) invariants
The metric (5.8) and the 3-form potential (5.17) have been derived using two sets of SO(3) × SO(3)-invariant geometric objects on S , namely, ( ξ, ξ m , ξ mn , S mnp ) and ( ζ, ζ m , ζ mn , T mnp ), thatare associated with two sets of (anti-)selfdual SO(8) tensors Y ± IJKL and Z ± IJKL , respectively. Thischoice of invariants is crucial for being able to carry out the simplification of the metric and the3-form potential in sections 5.1 and 5.2 starting with the uplift formulae (2.3) and (2.4), and alsofor the explicit check of the equations of motion in section 6.However, as we have already discussed in section 2.1, one might as well choose to work with asingle set of the geometric objects associated with the particular noncompact generator of E thatparametrises a given stationary point. In our case that means settingΦ IJKL = cos α Y + IJKL − sin α Z + IJKL , Ψ IJKL = cos α Y − IJKL + sin α Z − IJKL , (D.1)39nd expressing the solution in terms of the corresponding set of SO(7) tensors x mn = cos α ξ mn − sin α ζ mn , x m = cos α ξ m − sin α ζ m , x = cos α ξ − sin α ζ , S mnp = cos α S mnp + sin α T mnp . (D.2)To do this one may introduce the complementary set of rotated tensors, z mn , z m , z and T mnp , suchthat ξ mn = cos α x mn + sin α z mn , ζ mn = − sin α x mn + cos α z mn , etc. . (D.3)After rewriting the solution in terms of the rotated tensors, one can check using identities in sec-tion 3.2 that all terms involving the additional tensors either cancel out or can be rewritten in termsof (D.2).The calculation is long and, as one might expect, results in more complicated and less symmetricformulae for the metric and the 3-form potential. The reason for this is that the geometric objectsthat are being eliminated, z mn , . . . , T mnp , are replaced by more complex expressions in terms ofsums of products of tensors that are kept. To illustrate this point, let us consider the warp factor,∆, given in (5.12). At the stationary point (4.12), X + 2 c X Z + Z + Y = 20 h(cid:16) cos(2 α ) + 15 (cid:17) ξ − α ) ξζ − (cid:16) cos(2 α ) − (cid:17) ζ − √
10 (cos α ξ − sin α ζ ) + 540= 24 x − z − √ x + 540= 40 x − √ x + 16 x m x n S mpq S npq + 396 , (D.4)where, to eliminate z , in the last step we used the fact that x m x n S mpq S npq = 9 − ξ − ζ = 9 − x − z , (D.5)which follows from the identities in tables 1, 2 and 5.One may also note that the α -dependence in the first line in (D.4) is completely removed byrewriting the right hand side in terms of the rotated tensors using (D.3). Furthermore, the ro-tated tensors, x mn , . . . , S mnp and z mn , . . . , T mnp , satisfy the same identities as ξ mn , . . . , S mnp and ζ mn , . . . , T mnp , respectively, in tables 1-7. This means that the calculation is precisely the same forall α and thus we may as well set α = 0. The problem then is simply to rewrite the metric (5.8) andthe 3-form potential (5.17) for α = 0, solely, in terms of ξ mn , ξ m and S mnp . With this in mind, wenow turn to the metric tensor (5.8).It can be shown that one can write all SO(7) tensors appearing in the metric in terms of a smallnumber of fields constructed from ξ , ξ m , ξ mn and S mnp only:(i) scalars ξ , Ξ ≡ ξ m ξ n S mpq S npq , (D.6) Throughout this section we assume that c and s are set to their stationary point values. Otherwise, there areadditional terms proportional to z that must be dealt with separately. We have not analysed that case in detail. ξ m , Ξ m ≡ ξ n S mpq S npq , (D.7)(iii) symmetric tensors ˚ g mn , ξ m ξ n , ξ mn , Ξ m Ξ n , (D.8)and Ξ mn = S mpq S npq , e Ξ mn = ξ mp ξ nq S prs S qrs , Ω mn = ξ p ξ q S mpr S nqr , e Ω mn = ξ pq S mpr S nqr , Λ mn = ξ p S qr ( m η n ) pqrstu S stu , e Λ mn = ξ ( p ξ w )( m η n ) pqrstu S wqr S stu . (D.9)Using the identities, one finds that there are two relations between the symmetric tensors. Oneis simple ξ Ξ mn = 6 e Ω mn , (D.10)while the other involves most of the tensors and is quite complicated. We choose the basis of thesymmetric tensors by eliminating Ξ m Ξ n and e Ω mn from the list.Now, the metric (5.8) (with α = 0 and for general s and c ) is g mn = ∆ (cid:2) g ˚ g mn + g ξ mn + g ξ m ξ n + g Ξ mn + g e Ξ mn + g Ω mn + g Λ mn + g e Λ mn (cid:3) , (D.11)where g = − √ cξs − √ cξs + 16 s (cid:0) ξ + Ξ + 126 (cid:1) + 2 (cid:0) ξ + 27 (cid:1) s + 36 ,g = 2 s (cid:16) √ c + ξs (cid:0) s − (cid:1)(cid:17) , g = − s ,g = − s (cid:16) − √ cξs + (cid:0) ξ + 63 (cid:1) s + 108 (cid:17) , g = − s ,g = − s , g = 5 ξs − √ cs , g = − s . (D.12)This completes the proof that the metric tensor can be expressed entirely in terms of a single setof geometric objects, ξ mn , ξ m and S mnp , together with composite tensors that are built from them.A similar result should also hold for the 3-form potential. Since the solution written in this formis clearly quite complicated, we will not discuss this further. It should be clear at this point thatthe more symmetric basis of invariant tensors used throughout the paper is a much better choice fordoing calculations and that it leads to simpler and more symmetric looking formulae.41 Ambient coordinate embedding
In this appendix, we provide an explicit embedding of the R in R . We use the following represen-tation of seven-dimensional Γ-matrices in terms of Pauli matrices:Γ = 1 ⊗ σ ⊗ σ , Γ = 1 ⊗ σ ⊗ σ , (E.1)Γ = σ ⊗ σ ⊗ , Γ = σ ⊗ ⊗ σ , (E.2)Γ = σ ⊗ ⊗ σ , Γ = σ ⊗ σ ⊗ , (E.3)Γ = − σ ⊗ σ ⊗ σ . (E.4)In terms of seven-dimensional Γ-matrices the SO(8) generators Γ AB are ˆΓ ab = Γ ab , ˆΓ a = − i Γ a . (E.5)In this representation,Γ = − σ ⊗ σ ⊗ , Γ = σ ⊗ σ ⊗ , Γ = − iσ ⊗ ⊗ . (E.6)Therefore, we can easily verify that for the embedding given by m x A = { u , u , v , v , − v , − v , u , u } (E.7)the SO(3) × SO(3) invariant scalars, equations (7.10) and (7.11), are ξ = − u · u − v · v ) ζ = − u · v. (E.8)Furthermore, δx A = Γ AB x B = { v , v , − u , − u , u , u , v , v } . (E.9)so the α rotation rotates the u coordinates into the v coordinates, and vice versa . In the expression below we use ˆΓ for the SO(8) generators in the spinor representation and denote the seven-dimensional gamma matrices by Γ to avoid confusion. However, we do not make such a distinction elsewhere. eferences [1] B. de Wit and H. Nicolai, “Deformations of gauged SO(8) supergravity and supergravity ineleven dimensions,” arXiv:1302.6219 [hep-th] .[2] H. Godazgar, M. Godazgar, and H. Nicolai, “Testing the non-linear flux ansatz for maximalsupergravity,” Phys.Rev.
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