Flat deformations of type IIB S-folds

Type IIB S-folds of the form $\textrm{AdS}_{4} \times \textrm{S}^1 \times \textrm{S}^5$ have been shown to contain axion-like deformations parameterising flat directions in the 4D scalar potential and corresponding to marginal deformations of the dual S-fold CFT's. In this note we present a group-theoretical characterisation of such flat deformations and provide a 5D interpretation thereof in terms of $\mathfrak{so}(6)$-valued duality twists inducing a class of Cremmer--Scherk--Schwarz flat gaugings in a reduction from 5D to 4D. In this manner we establish the existence of two flat deformations for the $\mathcal{N}=4$ and $\textrm{SO}(4)$ symmetric S-fold causing a symmetry breaking down to its $\textrm{U}(1)^2$ Cartan subgroup. The result is a new two-parameter family of non-supersymmetric S-folds which are perturbatively stable at the lower-dimensional supergravity level, thus providing the first examples of such type IIB backgrounds.


Motivation and summary of results
S-fold backgrounds of type IIB supergravity of the form AdS 4 × S 1 × S 5 [1] have recently received much attention due to their holographic interpretation as new strongly coupled threedimensional conformal field theories (CFT's) on a localised interface of super-Yang-Mills (SYM) [2].On the gravity side, S-folds have been obtained in two complementary manners: i) as AdS 4 vacua of an effective four-dimensional dyonically-gauged supergravity with gauge group [SO(1, 1) × SO (6)] R 12 and electromagnetic deformation parameter c [1,3,4].ii) as limiting Janus solutions of an effective five-dimensional gauged supergravity with gauge group SO (6) [5,6,7].Both approaches rely on the consistency of the reduction of type IIB supergravity on S 5 [8] as well as on S 1 × S 5 [1], respectively, the latter incorporating an additional non-trivial SL(2) IIB hyperbolic monodromy [9] in the reduction ansatz along S 1 that depends on the electromagnetic parameter c and is responsible for the SO(1, 1) factor in the gauge group.
After the N = 4 and SO(4) symmetric S-fold originally obtained in [10] using the 4D approach, various multi-parametric families of S-folds were obtained in [4] preserving different amounts of supersymmetry (N = 0, 1, 2) and of residual symmetry.These families generically depend on a set of axion-like parameters, denoted χ's in [4], which correspond to flat directions of the scalar potential of the four-dimensional [SO(1, 1) × SO (6)] R 12 gauged supergravity, and thus to exactly marginal deformations in the holographic S-fold CFT duals.The family of N = 0 S-folds depends on three such parameters χ 1,2,3 which specify the matrix that controls the residual symmetry group of the corresponding S-fold solution.This group ranges from U(1) 3 at generic values of χ 1,2,3 to SO(6) ∼ SU(4) at the special case χ ij = 0 .The family of N = 1 S-folds depends on the axions in (1.1) which are this time subject to the condition χ 1 + χ 2 + χ 3 = 0 .The residual symmetry group of the S-fold solution ranges from U(1)2 at generic values of the axions to SU(3) at the special case χ ij = 0 .Lastly, the family of N = 2 S-folds depends on the axions in (1.1) subject now to the identifications χ ≡ χ 1 = −χ 3 and χ 2 = 0 .The symmetry group of the S-fold solution ranges from U(1) 2 at a generic value of the axion χ to SU(2) × U(1) at the special case χ ij = 0 . 1 One quickly realises that turning on the axions χ ij induces a sequential symmetry breaking from the largest possible symmetry at χ ij = 0 down to its Cartan subgroup.The aim of this note is to disclose the universal character of the axion-like parameters χ ij ∈ so (6) in the families of type IIB S-folds, and to understand why they give rise to flat directions in the scalar potential and, therefore, to potential marginal deformations in the dual S-fold CFT's.To this end we will take advantage of the E 7 (7) /SU (8) coset structure of the scalar manifold in the N = 8 supergravity multiplet.Being an homogeneous space, any two scalar field configurations are connected by an E 7 (7) transformation.Using the formally E 7 (7) covariant formulation of the maximal 4D gauged supergravities provided by the embedding tensor formalism [13], it is always possible to map an S-fold solution with non-vanishing axions χ ij = 0 in the gauged supergravity with gauge group G = [SO(1, 1) × SO (6)] R 12  to a new S-fold solution this time with vanishing axions χ ij = 0 .However, E 7(7) -covariance requires having to act on the embedding tensor Θ itself which fully specifies the gauging and the couplings in the supergravity Lagrangian.Then, bringing the axions to χ ij = 0 implies having to change the original gauged supergravity to a new one based on a new gauge group G specified by a new embedding tensor Θ .This method of mapping solutions belonging to different theories has already been exploited in the past to chart the landscape of AdS 4 vacua in half-maximal [14] as well as maximal [15,16,17] supergravities in four dimensions.
As we will show in this note, and upon application of the above procedure, the new theory will be a superposition of the original [SO(1, 1) × SO( 6)] R 12 dyonically-gauged supergravity and a class of Cremmer-Scherk-Schwarz (CSS) flat gaugings [18].More concretely, three out of the four mass parameters in a CSS gauging are directly mapped into the axion-like parameters χ ij ∈ so(6) in (1.1), i.e. m 1,2,3 = χ 1,2,3 .On the contrary, being associated to a non-compact so(1, 1) duality twist [1], the electromagnetic parameter c is not part of the CSS gauging.The fourth mass parameter in a CSS gauging is thus absent, i.e. m 4 = 0 , and we arrive at the general structure We will show the connection between the axion-like parameters χ ij ∈ so(6) and the existence of flat directions in the scalar potential of the original [SO(1, 1) × SO(6)] R 12 gauged supergravity.We will do it by proving that the contribution of (δΘ) CSS in (1. 2) to the scalar potential induced by Θ identically vanishes when performing a group-theoretical truncation of the scalar field content to the G χ 0 -invariant sector, where G χ 0 is a Cartan subgroup of G 0 ⊂ SO(6) chosen to commute with χ ij and G 0 is the largest symmetry group of the axion-vanishing solution within a family of S-folds.
As an illustration of the method, we establish the existence of two flat deformations χ 1,2 of the N = 4 and SO(4) symmetric S-fold in [1] which lie outside the SO(4)-invariant sector of the theory.These two flat deformations are responsible for the breaking of the SO(4) symmetry of the N = 4 S-fold down to its U(1) × U(1) Cartan subgroup.Setting one of the deformation parameters to zero, e.g.χ 2 = 0 , a known class of N = 2 S-folds with U(1) < l a t e x i t s h a 1 _ b a s e 6 4 = " E + I B P 2 b I d 3 3 N T 0 9 z P s 5 < l a t e x i t < l a t e x i t s h a 1 _ b a s e 6 4 = " q A j Q Figure 1: Two-dimensional parameter space (χ 1 , χ 2 ) of S-fold solutions induced by axion-like flat deformations of the N = 4 and SO(4) symmetric S-fold (green dot).The blue and red lines correspond to special choices of parameters χ 1,2 and produce supersymmetry (blue, χ = 0 ) and residual symmetry (red, χ 1 = χ 2 ) enhancements.These two special lines define the boundary of the parameter space.
symmetry is recovered 2 .(Anti-)identifying the two parameters yields a one-parameter family of non-supersymmetric, yet perturbatively stable, S-folds with SU(2) × U(1) symmetry. 3 Lastly, a generic choice of the two parameters χ 1,2 produces non-supersymmetric S-folds with U(1) 2 symmetry which are also perturbatively stable at the lower-dimensional supergravity level.The invariance of the entire setup under the reflection χ i → −χ i and the exchange χ ↔ χ 2 reduces the parameter space of the S-fold solutions to the octant depicted in Figure 1.
On the other hand, the geometric interpretation of the axion-like parameter χ in the family of N = 2 S-folds was presented in [20].By looking at the ten-dimensional uplift of the entire family of S-folds, the parameter was shown to be compact, i.e. χ ∈ [0, 2π/T ) , and to induce a non-trivial χ-dependent fibration of S 5 over the T -periodic S 1 .This interpretation holds for the parameters in (1.1) determining the other families of N = 0, 1 S-folds: they are also compact and induce a non-trivial monodromy on the internal S 5 when moving around S .The specific monodromy element h(χ 1,2,3 ) controls the patterns of symmetry breaking as classified by the mapping torus T h (S 5 ) [21].It would then be interesting to uplift the flat deformations χ 1,2 of the N = 4 and SO(4) symmetric S-fold to ten dimensions using E 7(7)covariant Exceptional Field Theory techniques along the lines of [1,3,4].It is a reasonable expectation that the axions χ 1,2 can be interpreted in terms of non-trivial monodromies on S when moving around S 1 , and that the patterns of symmetry breaking they induce are still controlled by the mapping torus.This would render the axions compact, i.e. χ 1,2 ∈ [0, 2π/T ) , and so the parameter space they span (shadow region in Figure 1). 2 It corresponds to setting (χ, ϕ) = ( χ 1 √ 2 , 1) in the general two-parameter family of N = 2 and U(1) 2 symmetric S-folds of [12]. 3The existence of non-supersymmetric AdS4 critical points in the [SO(1, 1) × SO(6)] R 12 gauged maximal supergravity has already been established in [15], and further investigated in [3,4,12] by looking at the Z 3  2invariant sector of the theory.However all such non-supersymmetric AdS4 extrema of the scalar potential exhibit BF instabilities [19].
The note is structured as follows.In section 2 we introduce the [SO(1, 1) × SO(6)] R 12 dyonically-gauged maximal 4D supergravity and provide a characterisation of its consistent flat deformations as induced by a set of axions χ ij = 0 .To illustrate the general method, we work out explicitly the set of flat deformations of the N = 4 S-fold with SO(4) symmetry and find new classes of non-supersymmetric S-fold solutions.In section 3 we provide a 5D interpretation of the flat deformations in terms of duality twists inducing CSS gaugings in a reduction from 5D to 4D.We also present a group-theoretical unification of the axion-like deformations χ ij and the electromagnetic parameter c , emphasising their similarities and main differences.Two appendices accompany the note.Appendix A collects our conventions on the e 7(7) algebra.Appendix B contains a proof of the result (2.26) used in the main text.
2 The maximal [SO(1, 1) × SO(6)] R 12 gauged supergravity 2.1 Maximal 4D supergravities: gaugings and scalar potential The bosonic sector of the maximal N = 8 supergravity multiplet in four dimensions consists of the spin-2 metric field g µν , a set of 28 spin-1 fields A µ Λ with Λ = 1, . . .28 (we will also introduce their magnetic duals Ãµ Λ ), and 70 spin-0 fields serving as coordinates in a coset space M scal = E 7(7) /SU (8).Electric and magnetic spin-1 fields can be arranged into with M = 1, . . ., 56 being an index in the fundamental 56 representation of E 7 (7) : the global duality symmetry underlying the four-dimensional ungauged maximal supergravity.This symmetry acts non-linearly on the scalar fields (which are not charged under the U(1) 28  abelian gauge group spanned by the spin-1 fields) and, by virtue of supersymmetry, induces an Sp(56) linear transformation on the spin-1 sector in (2.1) that elevates the global E 7 (7) symmetry of the scalar sector to a symmetry of the field equations and Bianchi identities.In this manner, the index M is also identified with the 56 of the Sp(56) electromagnetic group of the theory. 4e will be interested in the so-called gauged maximal supergravities [13].These are theories in which a non-abelian subgroup G of the duality group E 7 (7) is promoted from global to local (gauge).In addition to the minimal couplings of the spin-0 fields to the spin-1 fields, consistency of the gauging procedure requires the introduction of a non-trivial scalar potential as well as scalar-dependent fermion mass terms in the Lagrangian.Moreover, for dyonic gaugings involving magnetic vector fields Ãµ Λ , a topological term is also required together with a set of auxiliary two-form tensor fields which do not carry an independent dynamics (they are dual to spin-0 fields).All the modifications introduced by the gauging of G ⊂ E 7 (7) are encoded in an object called the embedding tensor Θ M α where the index α = 1, . . ., 133 denotes the adjoint representation of E 7 (7) .Consequently, the embedding tensor transforms as 56 × 133 = 912 + . . .where the ellipsis stands for additional irreps that are projected out by the so-called linear or representation constraints of the theory [13].In addition, consistency of the gauge structure requires an additional set of quadratic constraints of the form [13] Ω where Ω MN is the skew-symmetric Sp(56) invariant matrix.
As discussed above, the embedding tensor Θ M α ∈ 912 of E 7 (7) fully determines the minimal couplings codified in the gauge connection as well as the scalar potential of a given maximal gauged supergravity.Covariant derivatives take the form with t α being the generators of E 7 (7) in the appropriate representation.Note that (2.3) generically allows for magnetic vector fields to enter the gauge connection provided (2.2) holds.The scalar potential of the theory is given by in terms of the scalar-dependent matrix M MN = (V V t ) MN constructed from the coset representative V ∈ E 7(7) /SU( 8) and the X-tensor which encodes the scalar charges entering the covariant derivative in (2.3).Recall that an index M is raised/lowered using the Ω MN and Ω MN matrices according to the North-West-South-East convention.These matrices satisfy Ω MN Ω MP = δ P N .It will also prove convenient to introduce a scalar-dependent X-tensor known as the T -tensor in terms of which the scalar potential in (2.4) is expressed as From (2.4) and (2.7) one finds that V (X, M ) = V (T ) .As for the X-tensor in (2.5), the T -tensor can be written as where the operation denotes the action of a group element of E 7(7) -in this case the inverse coset representative V −1 -both on the fundamental and the adjoint indices.In terms of the scalar-dependent Ξ-tensor in (2.8) the scalar potential (2.7) can be rewritten as where we made use of the normalisation condition Tr(t α t t β ) = δ αβ and the definition of the Killing-Cartan matrix K αβ = Tr(t α t β ) .Note that the first contribution in the r.h.s of (2.9) is positive definite (sum of squares) while the second one has no definite sign due to the non-compactness of E 7(7) .
In this note we will consider the dyonic gauging of G = [SO(1, 1) × SO( 6)] R 12 .This amounts to a choice of embedding tensor Θ M α of the form with θ AB = g diag( 0 , I 6 , 0 ) and θAB = c diag( −1 , 0 6 , 1 ) , so that only components belonging to the 36 and the 36 are present in the decomposition (2.12).Note the presence of an electric parameter g and a magnetic one c rendering the gauging of dyonic type whenever gc = 0.The quadratic constraint in (2.2) is automatically satisfied by the embedding tensor choice (2.14) with (2.15).The four-dimensional [SO(1, 1) × SO(6)] R 12 maximal gauged supergravity arises from the consistent reduction of type IIB supergravity on S 5 followed by a further S 1 reduction that incorporates an SL(2) IIB hyperbolic twist [1].This suggests to further decompose the E 7 (7) irreps in (2.10)-(2.12)under the maximal subgroup SL(6) × SL(2) × SO(1, 1) ⊂ SL(8) in order to better analyse the gauging.The result is Under this decomposition the electric component of the embedding tensor corresponds to (21, 1) 1 ⊂ 36 whereas the magnetic one to (1, 3) 3 ⊂ 36 .Similarly, the fundamental SL(8) index decomposes as A = (i , a) with i = 2, . . ., 7 being a fundamental SL(6) index and a = 1, 8 a fundamental SL(2) index.As a consequence and, from the embedding tensor components in (2.14)-(2.15),one gets Then the compact SO( 6) factor of the gauge group is gauged electrically, the non-compact SO(1,1) factor is gauged magnetically and the R 12 translations are gauged dyonically provided gc = 0. Various AdS 4 solutions have been found in this theory and subsequently uplifted to Sfold backgrounds of type IIB supergravity [1,3,4].Being maximally symmetric vacua of the theory, vector and auxiliary tensor fields are set to zero resulting in a Lagrangian of Einstein-scalar type where V is the scalar potential in (2.4) which is retained upon setting to zero vector and tensor fields.For the purpose of this note it will be convenient to use the Iwasawa or solvable parameterisation of the scalar coset representative where t h are the 7 generators of the maximal non-compact torus R 7 ⊂ E 7(7) (Cartan subalgebra) and t ê are the 63 positive roots of E 7(7) (computed with respect to a choice of basis for the maximal torus).We will refer to the associated scalars as dilatons φ h and axions χ ê .With the exception of the original N = 4 and SO(4) symmetric S-fold in [10,1], the rest of S-fold solutions presented in [4] within a Z 3 2 -invariant sector of the theory happened to contain moduli fields (axion-like fields χ ê ) representing flat directions in the scalar potential.Understanding the general features of axion-like flat deformations of S-fold solutions will be the goal of this note.

Axion-like flat deformations
Let us consider a vacuum solution of the Lagrangian (2.21), namely, an extremum of the scalar potential (2.4), and denote the associated coset representative at such a vacuum V 0 and its vacuum energy V 0 .Importantly, we will always assume some continuous residual symmetry group G 0 ⊂ SO( 6) ⊂ G at the vacuum and parameterise an element of its algebra g 0 by an antisymmetric constant matrix χ ij .We will show that, starting from the G 0 -invariant vacuum solution V 0 of the theory, then the coset replacement still describes a vacuum solution of the G = [SO(1, 1) × SO(6)] R 12 maximal gauged supergravity with the same value of the cosmological constant V 0 .The new vacuum solution V χ V 0 no longer belongs to the G 0 -invariant sector of the theory, but nevertheless parameterises an entire family of χ-dependent solutions with χ ij corresponding to flat directions in the scalar potential.We will refer to the χ ij parameters as axion-like flat deformations of the original S-fold solution V 0 .
To prove the statement above we will take advantage of the E with non-zero axions V χ V 0 in the G-gauged maximal supergravity can be mapped into an axion-vanishing solution in a different theory with different gauge group G = G specified by an embedding tensor where the denotes the action of the E 7(7) element V χ in (2.23) on the embedding tensor Θ M α ∈ 912 of the original G-gauged theory.At the linear level, the element V χ is generated by t 1ij8 ∈ (15 , 1) 2 .Then, an inspection of Table 1 shows that, upon acting on the original embedding tensor components in (2.14), the action of V χ only produces a single term δΘ ∈ (35, 1) 3 ⊂ 420 that originates from the electric piece (21, 1) 1 ⊂ 36 in the 912 decomposition (2.18). 5Equivalently, only a linear term appears since the SO(1, 1) grading in the decomposition (2.18) solely allows for charges −3 , −1, 1 or +3 , and not +5 or higher which would be the ones produced beyond the linear level.As a result, δΘ is invariant under the action of V χ .An explicit computation of δΘ in (2.24) yields where χ i j = θ ik χ kj with θ ij = g δ ij in (2.15).This comes from the fact that the only terms in the product 56 ⊗ 133 contributing to the (35, 1) 3 are precisely the (15, 1) 1 ⊗ (15 , 1) 2 , the (6 , 2) 1 ⊗ (6, 2) 2 and the (1, 1) 3 ⊗ (35, 1) 0 .It is also worth noticing that δΘ in (2.25) verifies the quadratic constraints (2.2) so it defines a consistent gauging by itself.
Using Θ M α in (2.24) with Θ M α and (δΘ) M α given in (2.20) and (2.25) respectively, it can be proved that where G χ 0 denotes a Cartan subgroup of G 0 commuting with χ i j and V G χ 0 -inv denotes the scalar-dependent coset representative of the G χ 0 -invariant sector of maximal supergravity (which contains the G 0 -invariant vacuum V 0 ).This implies that there always exist flat directions of the scalar potential at the vacuum V 0 parameterised by the axions χ ij that are not captured by the G 0 -invariant sector of the theory.The first equality in (2.26) is a direct consequence of the formally E 7(7) -covariant formulation of the maximal 4D gauged supergravities provided by the embedding tensor formalism.A detailed proof of the second equality in (2.26) is presented in the Appendix B.
2.4 Deforming the N = 4 and SO(4) symmetric S-fold A direct consequence of (2.26) is the existence of two axion-like flat deformations6 χ α (α = 1, 2) of the original N = 4 S-fold with G 0 = SO(4) symmetry, which control the pattern of symmetry breaking down to its G χ 0 = U(1) 2 Cartan subgroup.These flat deformations lie outside the Z 3  2 -invariant sector of the theory investigated in [4] and, adopting the conventions therein, they specify a matrix χ ij of the form An explicit computation of the full scalar and vector mass spectra at the corresponding AdS 4 vacua yields the following results.The normalised spectrum (masses and multiplicities) of scalar fields is given by in terms of the AdS 4 radius L 2 = −3/V 0 = g −2 c .The normalised spectrum (masses and multiplicities) of vector fields reads and contains two massless vectors at generic values of χ α .Lastly, the computation of the eight normalised gravitino masses yields7 By inspection of (2.28)-(2.30)we identify four different classes of flat deformations of the N = 4 and SO(4) symmetric S-fold (see Figure 1): • At generic values of χ 1,2 one finds non-supersymmetric S-folds with a U(1) 2 symmetry which is interpreted as a flavour symmetry in the dual S-fold CFT's.
Note that χ 1 and χ 2 enter (2.28)-(2.30)symmetrically, as expected, and that the scalar mass spectrum in (2.28) does not display any instability associated with a normalised mass mode violating the Breitenlohner-Freedman (BF) bound m 2 L 2 ≥ −9/4 for perturbative stability in AdS 4 [19].Therefore, all the non-supersymmetric S-folds discussed above turn out to be perturbatively stable at the lower-dimensional supergravity level.
Carrying out a study of their higher-dimensional stability by computing the associated Kaluza-Klein (KK) spectra, as done in [20] (following [23]) for the N = 2 S-folds of [4] and the N = 4 S-fold of [1], would help in establishing the perturbative (in)stability of the non-supersymmetric S-folds. 8The reason to perform such a KK spectroscopy study is twofold.Firstly, to search for (super) symmetry enhancements when considering modes up in the KK tower.Secondly, to give support or pose a new challenge to the AdS Swampland Conjecture [26].

5D origin and CSS gaugings
The results in the previous section followed from the very specific flat deformation introduced by the axions χ ij which, as already emphasised, can be alternatively understood in terms of the δΘ deformation tensor in (2.25).The attentive reader might have recognised in (2.25) a structure similar to a CSS gauging [18].This type of gaugings appear upon compactification of 5D supergravity down to 4D.We will now elaborate more on this point.
Let us start by recalling that there is a formally E 6(6) -covariant formulation of the maximal 5D gauged supergravities provided by the embedding tensor formalism [27].The bosonic sector of the maximal N = 8 supergravity multiplet in five dimensions consists of the metric field g µν , a set of 27 vector fields and 27 two-form tensor fields, and 42 scalar fields serving as coordinates in a coset space M scal = E 6(6) /USp (8).The embedding tensor is subject to a set of linear or representation constraints restricting it to the 351 representation.In addition it must also obey a set of quadratic constraints in order to specify a consistent gauging of the theory.To establish a 4D ↔ 5D connection we will perform a group-theoretical decomposition of the E 7 (7) representations in (2.16), (2.17) and (2.18) under the maximal subgroup E 6(6) × SO(1, 1) ⊂ E 7 (7) .This yields ) ) Importantly, while the 351 +1 captures the electric (21, 1) +1 piece in the 4D embedding tensor Θ induced by g , it does not capture the magnetic (1, 3) +3 and (35, 1) +3 pieces induced by c and χ i j .These two pieces are instead contained in the 78 +3 , as it is clear from the SO(1,1) charge.A direct consequence is that the 4D gauging Θ involving the (1, 3) +3 ⊂ 78 +3 (c-terms) and (35, 1) +3 ⊂ 78 +3 (χ-terms) cannot be directly uplifted to an embedding tensor deformation in 5D.These terms are instead generated dynamically by introducing an explicit dependence on the S 1 coordinate (in the form of a duality twist [28]) in the reduction process from 5D to 4D.
A general duality twist takes the form [18,29] φ(x µ , η) = e M η φ(x µ ) , where η is the coordinate along the S 1 , M ∈ e 6(6) and φ is any field in the 5D theory.This type of duality twists has been studied in the context of 5D ungauged supergravity [30].Within this context, the dependence on the η coordinate factorises out in the reduction process as a consequence of M being chosen in the global duality group E 6(6) of the 5D theory.Moreover, choosing M in the maximal compact subalgebra usp(8) ⊂ e 6(6) makes the scalar potential vanish identically in the reduced 4D theory.Our scenario, however, differs from the one just discussed: the theory to begin with is the 5D SO(6)-gauged supergravity.The global duality group E 6( 6) is broken to a local SO(6) and a global SL(2), and the axionlike flat deformation M = χ i j ∈ g 0 ⊂ so(6) leaves invariant the embedding tensor of the SO(6) gauging.Then, in our case, the flat deformations χ i j are expected to describe trivial twists leaving the putative 5D backgrounds locally invariant.
To conclude, group-theoretical arguments put forward in [21] suggest that axion-like deformations should be related to one-form deformations of N = 4 SYM on R 1,2 × S 1 [31].Such one-forms are often discarded within the context of Janus solutions by a gauge-fixing argument without much regard for large gauge transformations.Working out the explicit 5D oxidation of the AdS 4 vacua with χ ij = 0 would be the next step towards testing these ideas.In addition, it would also be interesting to further investigate the interplay between embedding tensor deformations (351 +1 ) and e 6(6) duality twists (78 +3 ) in S 1 compactifications of other 5D maximal supergravities.Also to understand the physical meaning (if any) of the rest of representations appearing in the group-theoretical decomposition of the 912 in (3.3).We leave these and other related questions for future work.As a result, whenever Ξ 1 or Ξ 2 are so(6)-valued (i.e.Ξ 1,2 + Ξ 1,2 t = 0 ), the bilinear (B.5) vanishes identically.

l a t e x i t > N = 4 & SO( 4 )
< l a t e x i t s h a 1 _ b a s e 6 4 = " j M W 4 d P t s n c Y 2 e S K E I W Q / 8 h p 4 H e Y = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k a I 8 F L x 4 r 2 A 9 o Q 9 l s N + 3 S z S b u T o Q S + i e 8 e F D E q 3 / H m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I J H C o O t + O 4 W N z a 3 t n e J u a W / / 4 P

Table 1 :
(8))/SU(8)coset structure of the scalar manifold of maximal supergravity.As discussed in the introduction, the solution Group-theoretical action of the 133 representation of E 7(7) on the 912 using the SL(6) × SL(2) × SO(1, 1) basis.Only pieces belonging to the 912 are displayed.