Singular limits in STU supergravity

We analyse the STU sectors of the four-dimensional maximal gauged supergravities with gauge groups ${\rm SO(8)}$, ${\rm SO(6)}\ltimes\mathbb{R}^{12}$ and $[{\rm SO(6)}\times{\rm SO(2)}]\ltimes\mathbb{R}^{12}$, and construct new domain-wall black-hole solutions in $D=4$. The consistent Kaluza-Klein embedding of these theories is obtained using the techniques of Exceptional Field Theory combined with the 4$d$ tensor hierarchies, and their respective uplifts into $D=11$ and type IIB supergravities are connected through singular limits that relate the different gaugings.

Such consistent truncations are in general difficult to construct, but when they exist they constitute a fundamental tool for obtaining solutions in ten and eleven dimensions, as the consistency of the truncation guarantees that every solution of the lower-dimensional gauged supergravity can be embedded into a configuration that solves the equations of motion of the parent theory.This approach has been particularly fruitful in holographic set-ups, where the gauged supergravity techniques have made possible the construction of hundreds of different AdS solutions -see [11][12][13][14][15] for recent surveys -that are dual to different conformal field theories (CFTs) and can be used, among other things, as a playground to check Swampland conjectures, such as the AdS conjecture [16] or the CFT distance conjecture [17].Furthermore, the relative simplicity of the lower-dimensional theories has allowed the analysis of other holographically relevant phenomena, such as black holes [18,19] describing finite temperature states in the CFT, or domain walls realising CFT interfaces [20,21] and RG-flows between different CFTs [22,23].Additionally, the existence of these consistent truncations also facilitates even some intrinsically higher-dimensional computations such as the spectrum of masses of the KK modes [24].
Recently, reformulations of string theory based on its duality symmetries [25,26] have played a pivotal rôle in our understanding of consistent truncations [27,28].In fact, at present there are theorems [29] that guarantee the existence of these reductions for wide classes of theories and backgrounds in terms of suitable factorisations of duality-covariant fields.However, explicit KK Ansätze for the standard higher-dimensional metric and fluxes keeping the entire dependence on the lower-dimensional fields and their derivatives are often not known due to the intricate nature of the dictionary relating the original supergravities to the duality-covariant counterparts.A convenient way to describe the embedding of these non-trivial profiles makes use of the tensor hierarchy, which is a supplement to the p-forms in the ungauged supergravity that are introduced so as to guarantee that the gauged theory is formally covariant under the original global symmetry group [30][31][32][33].The redundancies so introduced can be eliminated at the level of the field strengths, and an explicit KK Ansatz in terms of the original fields and their derivatives can thus be obtained [34][35][36].These extra forms allow us to trade some complicated dualisations with respect to the internal metric for much simpler dualisations with respect to the four-dimensional metric.
It has been observed [15,37] that there exists a non-trivial network of gauged supergravitites in four dimensions that are connected by singular limits of their moduli.When the original gauged supergravity admits an uplift to string theory, one can follow this limit also in higher dimensions [38,39], and thereby connect one consistent truncation to another.This technique can also be employed to construct new solutions from known ones [40].In this paper, we employ these relations to describe the truncation of type IIB supergravity on AdS 4 × S 5 × S 1 .There are in fact several inequivalent truncations on this geometry, and in this work we focus on the ones in which the resulting gaugings are SO(6) ⋉ R 12 [41] and [SO(6) × SO(2)] ⋉ R 12 [42,43].The latter can be understood as a modification of the former where an extra vector is dyonically coupled to the matter fields.Even though the consistency of these truncations is well known [44], complete KK Ansätze are unavailable.For simplicity, we focus here on the truncation of these four-dimensional theories down to the STU sector so that only six scalars and four vectors can have a non-trivial profile.The STU model obtained from the SO(6) ⋉ R 12 theory can be obtained as a singular limit of the STU model corresponding to the SO(8) gauging of [19], and we show that the S 5 × S 1 background in type IIB can be obtained as a singular limit of eleven-dimensional supergravity on the S 7 followed by a circle reduction and T-duality.The explicit Ansatz embedding both theories in type IIB is presented in equations (4.32)- (4.37).
The rest of the paper is structured as follows.In the next section we introduce the 4d models which we are going to analyse, and discuss some of their properties and their reformulation in terms of the tensor hierarchy.Section 3 briefly reviews E 7(7) -Exceptional Field Theory (ExFT) [45] and the description of consistent truncations down to maximal gauged supergravities in the language of generalised Scherk-Schwarz reductions [28], which will be subsequently used to describe the uplift of the STU models into M-theory on S 7 and type IIB supergravity on S 5 × S 1 , and the consistency of these embedding is explicitly checked in a simpler sub-truncation.New solutions in these gaugings are then constructed in section 5 and we conclude by discussing some possible future directions and include further technical details in two appendices.

Gauged STU Supergravity
We consider D = 4 theories that arise as a truncation of N = 8 gauged supergravity [46,47] by requiring invariance under the maximal torus of the relevant gauge group -see appendix A for the expression of the generators of these Cartan subalgebras in terms of the generators of E 7 (7) .The bosonic field content of these theories consists of the metric, four vectors and six (pseudo)scalars corresponding to the scalar manifold parametrised by u i = χ i − ie −ϕ i , with i = 1, 2, 3.The bosonic sector of the Lagrangians of these STU supergravities can then be written as The scalar kinetic terms read and the vector kinetic terms are given by with a = 1, 2, 3, 4.These non-minimal couplings can be extracted from the symmetric coset representative of the maximal theory in (A.9) via the block decomposition in (A.12) after the identifications in (A.11).The result can be given as the period matrix 1 This period matrix recovers the kinetic and Chern-Simons terms in [48] under χ there i = −χ here i and the following redefinition of the vector fields: (2.5) In section 4.1, this same relabelling applies to the coordinates µ a , φ a , and similarly to Wa and Za.
In the STU truncation, these embedding tensors induce Fayet-Iliopoulos gaugings, whose potentials read The potential (2.12a)only admits one critical point.It sits at the scalar origin and corresponds to the SO(8) maximally supersymmetric solution.In turn, the potentials (2.12b) do not possess any extremum in this sector.
Even though the V (6c) potential is blind to the value of the magnetic coupling m, the fermion couplings in this theory do depend on it.A similar situation has been previously encountered [51] in the STU truncation of the dyonically-gauged SO(8) supergravity.In fact, the truncated theory is not supersymmetric for non-vanishing m.On the other hand, the electric cases are gauged N = 2 supegravities coupled to three vector multiplets.To see this, observe that theories with N = 2 supersymmetries can necessarily be recovered in the canonical perspective of [52,53] in terms of special Kähler and quaternionic structures.For the scalar manifold (2.1), the special holomorphic section can be taken as in terms of the special holomorphic coordinates in (2.1).This section describes the geometry of the scalar manifold and encodes its Kähler potential as with C M N the symplectic form on Sp(8, R).Purely Fayet-Iliopoulos gaugings have a potential [52] with V M = e −K/2 Ω M (z) a section of the special U(1)-bundle, g i ī = ∂ i ∂īK the hermitean metric associated to the Kähler potential, and ϑ M the embedding tensor describing how the U(1) gauge group sits into the SU(2) R-symmetry group.The potential obtained from the truncation of the SO (8) gauging is given in this language by whereas the embedding tensor corresponding to V (6) is given by ϑ (6)  M = g − 1, 0, 0, 0, 0, 0, 1, 1 . (2.17) For the electric theories, the N = 2 supersymmetry variations and fermionic mass-like terms in the Lagrangian associated to these embedding tensors agree with the truncation of the N = 8 fermion shifts associated to (2.11).However, for the dyonic gauging in (2.11), the fermion shifts carry dependences on m which can not be recovered in the N = 2 language.
The potentials in (2.12b) can be obtained from V (8) by means of a singular scaling.For that, the scalars must be redefined as the gauge coupling as g → e −k/2 g, and the gauge fields must also be scaled as whilst the metric remains invariant.The singular limit k → ∞ on (2.2) after these redefinitions maps the Lagrangian L (8) into L (6) , also including the fermion couplings.

Tensor and duality hierarchies
The equations of motion of gauged D = 4, N = 8 supergravity can be written in a formally covariant E 7(7) -covariant formulation [30,31,54] if one introduces a set of redundant fields in the so-called tensor hierarchy.For generic gaugings of D = 4, N = 8 supergravity, one requires [30,31] one-forms in the 56 representation of E 7 (7) , two-forms in the 133 and three-forms in the 912, together with a set of four-forms that will not play a rôle in the following.These redundancies can be eliminated at the level of the field strengths through a chain of dualities that relate them to the original fields and their derivatives [32,33], so that combinations of the equations of motion for the original fields are recovered from the Bianchi identities for the forms in the tensor hierarchy.In the following, we show how these reformulations apply to the STU supergravities of interest.
The equations of motion stemming from (2.2) for the one-forms and scalars are which can be interpreted as Bianchi identities for two-and three-form field strengths, respectively.For gaugings inside SL(8, R), as the ones considered in this work, the p-forms in the E 7 (7) tensor hierarchy are conveniently decomposed into one-forms A AB , ÃAB , twoforms B A B , B ABCD , and three-forms C AB , CAB , C A BCD , CA BCD in the 28 ⊕ 28 ′ , 63 ⊕ 70 and 36 ⊕ 36 ′ ⊕ 420 ⊕ 420 ′ of SL(8, R).Their associated field strengths are given by and similarly for the four-forms H AB (4) , H(4)AB , H (4)A BCD , HA (4) BCD .Here the covariant derivatives are given by the SL(8, R) decomposition of with (t α ) (R) the E 7 (7) generators in the appropriate representation.For instance, (2.23) These field strengths obey the Bianchi identities which, together with the E 7( 7) duality relations [32,33] recover the equations of motion for the vectors and scalars.Further details on the E 7( 7) generators (t α ) M N , structure constants X M N P and scalar representative M M N can be found in appendix A.
Demanding invariance under the H algebra in (A. and extend the index as a = (p, 4).If we restrict our attention to the class of gaugings in (2.10), we can further truncate consistently the field content to 4+4 one-forms : A a , Ãa , 3+6 two-forms : B p , B ab , with Cpq = Cqp .The field strengths for these gauge potentials can be obtained by implementing these truncations at the level of the SL(8, R) field strengths in (2.21).The resulting expressions read and similarly for H (4)ab and H(4)ab .In (2.29) there are no sums unless explicitly indicated, and the embedding tensor components θ a and ξ a are related to θ AB and ξ AB in (2.11) following the same pattern as the three-forms in (A.14).We have also introduced the totally symmetric tensor From their definition in terms of potentials, it is easy to check that the field strengths satisfy the Bianchi identities (2.31) Implementing the restrictions in (A.11)-(A.15)on (2.25), we can express the dual field strengths in terms of the original STU fields and their derivatives.The magnetic two-form field strengths are then given by with W in (2.8) and (2.33) Similarly, the three-form field strengths are (2.34) for the ones in the 63 of SL(8, R), and for the ones in the 70.The expressions for the four-form field strengths in terms of the scalars depend on the choice of embedding tensor.Employing (2.25) for the class of gaugings in (2.10), we obtain and (2.37)The forms in (2.36) can be used to reproduce the potentials in (2.12) as As noted before, inserting the duality relations (2.32) into the Bianchi identities (2.31), we obtain the scalar-Maxwell equations in (2.20) stemming from the Lagrangian (2.2).Similarly, the Bianchi identities for the three-form field strengths are identically verified if the equations of motion for the scalars are satisfied.The derivatives of the scalars that appear in equations of motion for the dilatons are retrieved from ) and, finally, the non-minimal couplings to the vector fields follow from in the equation for the dilations, and in the one for the axions.
Turning now our attention to the singular limit relating the different gaugings, under the scaling (2.18) the three-form field strengths in (2.34) stay invariant, the magnetic two-form field strengths have opposite scaling to the electric forms in (2.19), and the electric four-form field strengths reduce to with H(4)a the four-forms given by (2.36) with x = 0 and x = 0.The scaling of the other four-forms, which will not be relevant in the following, can be computed in the same way from (2.36) and (2.37).
In the sequel, we will embed the previous D = 4 gauged supergravities into type IIB and D = 11 supergravity using a duality-covariant reformulation of the latter higherdimensional theories known as Exceptional Field Theory, and encoding part of the dependence on the 4d fields in terms of a subset of the forms in the tensor hierarchy.

E 7(7) Exceptional Field Theory
The bosonic field content of ExFT [45] is given by with all fields depending on both "external", x µ , µ = 0, . . ., 3, and extended "internal" coordinates, Y M , M = 1, . . ., 56.This dependence and the fields themselves are restricted by the section constraints [45] ( with Q M ∈ {∂ M , B µν M } and the derivatives acting on any combination of fields or gauge parameters in the theory.Here, (t α ) M N are the algebra generators and indices are raised and lowered with the invariant symplectic form Ω M N as These constraints are needed for the generalised Lie derivative to close into a local E 7 (7) gauge algebra.Variations under the latter are given by [45] δ where is the projector onto the adjoint representation and λ(V ) is the weight associated to the generalised vector V M .To solve the constraints (3.2), E 7 (7) can be reduced down to GL(7, R) (M-theory section) or GL(6, R) × SL(2, R) (Type IIB section).After this reduction, the variation δ Λ encodes the behaviour of the different fields under both "internal" diffeomorphisms and gauge transformations.

M-theory section
For D = 11 supergravity, we use conventions in which our fields are subject to the action Under the structure group relevant for a seven-dimensional internal space, the extended ExFT coordinates decompose as2 with y i = Y i8 , etc.The section constraint (3.2) can be solved by imposing keeping only the B µν i components of B µν M .Similarly, the objects in the adjoint representation of E 7(7) break according to (3.9) To make contact with M-theory, one needs to split the D = 11 structure group GL(11, R) ⊃ GL(4, R) × GL(7, R).Then, the metric and three-form of D = 11 supergravity give rise to the following fields: where all of the fields depend on both x µ and y i .The forms C ijk , C ijρ , C iνρ are related to the components of the eleven dimensional Â(3) through the usual Kaluza-Klein decomposition with flattening and unflattening of indices.The C ijklmn components are dual to the external legs C µνρ through with as required by the infinitesimal transformations These fields are then related to the ExFT fields branched according to (3.7) and (3.9).For the tensor-like degrees of freedom, the dictionary reads e µ a = φ 1/4 e µ a , and C µνi are related to B µν α and B µν M .For the D = 4 scalars, (3.15) for the components of the M M N coset representative of E 7(7) /SU (8).Here and throughout, φ = det φ ij .Conversely, the internal components of the three-and six-form can be given as In the following, we will show that in the context of consistent truncations, one can circumvent the dualisation in (3.11) by working in terms of the p-forms of the four-dimensional tensor hierarchy, which also cleanly account for the information corresponding to the 4d vectors A i µ and C µij , and two-forms C µνi .

Type IIB section
Similarly to the D = 11 case, the extended ExFT coordinates decompose under GL(6, R)× SL(2, R) following now with i = 1, . . ., 6 and a = 1, 2. The section constraint (3.2) can be solved by requiring that all fields and parameters only depend on y i and that the only non-zero component of the constrained two-form is B µν i .In turn, the objects in the adjoint representation adhere to Contact with type IIB supergravity is achieved after splitting the ten-dimensional structure group GL(10, R) into GL(4, R) × GL(6, R) so that the bosonic fields read again with all of them depending both on x µ and y i and taking into account the flattening and unflattening of indices with the Kaluza-Klein vector.Here, we use conventions in which the type IIB pseudoaction is given by

21) with Ĉa
(2) = ( B(2) , Ĉ( 2) ).In addition to the equations of motion obtained from (3.20), the self-duality of F( 5) needs also to be imposed.The relation to the ExFT fields branched according to (3.17) and (3.18) then is Similarly, we have e µ a = φ 1/4 e µ a and A µ i = A µ i for the tensor-like components of the metric, and the other p-form contributions we will phrase in terms of objects in the tensor hierarchy when we turn to consistent truncations in the sequel.
To make contact with the S 5 × S 1 configurations, we need to further decompose under GL(5, R) × GL(1, R) × SL(2, R) via i = (i, 6) with i = 1, . . ., 5.This subgroup is common to both SL(8, R) and GL(6, R) × SL(2, R), and therefore its representations can be given in terms of SL(8, R) pairs.In our conventions, and for the coordinates we further introduce y i = ŷi and y 6 = ỹ.

Generalised Scherk-Schwarz Ansätze
The ExFT fields (3.1) can be parametrised in terms of D = 4 N = 8 supergravity fields via a Scherck-Schwarz Ansatz [28] 3 This Ansatz provides a consistent truncation of the ExFT equations of motion down to D = 4 maximal supergravity provided that the twist matrix and scaling factor define a generalised frame For the gaugings under consideration, the twist matrix can be written as with The S 7 reduction [3] can be described via (3.25) [28] in terms of the scaling function The components of the SL(8, R) matrix U A Ā(y) in (3.28) under the splitting (3.7) for local and global indices are with and |y| 2 = δ ij y i y j .In terms of these coordinates, the round metric obtained through (3.15) by setting the scalars to zero reads The uplift of the dyonic CSO gaugings of [42] can also be described via (3.25) with block diagonal twist matrix (3.27) with (3.28) [44].For the [SO(6) × SO(2)] ⋉ R 12 gauging, the scaling function reads and the components of the SL(8, R) matrix U A Ā(y) under the splitting (3.24) for local and global indices are In the following, it is convenient to introduce coordinates (µ a , φ a ), with a = 1, 2, 3, 4, as4 In terms of these coordinates, the metric with (3.25) and (3.30) becomes [48] ds where and with For this gauging, the covariant derivatives on the angles denote the fibering with the four vectors Dφ a = dφ a − gA a .(4.7) with A a given in terms of SL(8, R) objects in (A.11).The three-form potential can be written in terms of the 4d potentials in (2.28) suitably coupled to the S 7 coordinates, and an internal contribution as dictated by (3.15).The link between the tensor hierarchy fields and the sphere coordinates is in fact fixed by their respective SL(8, R) structure, and gauge invariance demands that the different terms combine into the field strengths in (2.29) when acted upon by the exterior derivative.Notably, only a subset of the forms in (2.28), dubbed "restricted tensor hierarchy" in [33,34], enters the KK Ansatz.For the STU truncation, the eleven-dimensional three-form can be decomposed as simply with the overall scaling fixed by the equations of motion.This result agrees with the truncation of [34] through (A.11)-(A.15).The expression for the internal three-form can also be obtained from (3.15) with (3.25) and (3.30), and reads with components C a,bc = C a, [bc] given by (4.10) and shorthands This result matches (4.19) of [48] upon making the identifications in footnote 1.The eleven-dimensional four-form is then produced by the exterior derivative of (4.8), which by using (2.29) can be given as and thus making use of only the original fields appearing in the N = 8 action and their derivatives through the duality relations (2.32)-(2.36),with the last equation particularised for this gauging to (4.13)

Singular limit
Scaling the fields and couplings as in (2.18) and (2.19), the configuration remains finite up to a trombone scaling if we also transform the internal coordinates as with the coordinates (µ ā, φ ā, φ 4 ) invariant.Doing so, the warping (4.4) factorises into with Accordingly, the metric becomes and the three-form components (4.10) transform as Ca,bc dµ a , ( with Ca,12 Ca,41 Therefore, the four-form (4.12) becomes with the H(4)ā in (2.44) reading

IIA reduction and dualisation to IIB
The metric and four-form in (4.17) and (4.21) formally describe a warped compactification on S 5 × R 2 , with the R 2 factor parameterised by (µ 4 , φ 4 ) and the sphere by (µ ā, φ ā) satisfying which follows from (4.2) after taking the k → ∞ limit on (4.14).Introducing coordinates z 1,2 as we can promote the R 2 factor into a two-torus by imposing z 1,2 ∼ z 1,2 + 2πR 1,2 .The eleven-dimensional configuration, (4.17) and (4.21), can then be interpreted as a type IIA supergravity solution upon reducing on one of the circles, say z 2 .The resulting IIA geometry has an S 1 factor that allows one to perform a T-duality transformation and in this way obtain a solution of type IIB supergravity.
To ease our notation, we introduce the following shorthands such that the eleven-dimensional four-form can be written as Both X (2) and X (4) are exact, and a representative potential for X (2) can be read off from (4.25) to be Following the conventions stated in appendix B, the type IIA configuration resulting from reduction of (4.17) and (4.26) on z 2 is and the type IIB solution obtained by T-dualising along the z 1 direction employing the relations (B.8)-(B.9)reads The type IIB configurations that uplift from D = 4 gauged supergravity with gauging specified by (θ (6c) , ξ (6c) ) in (2.11) can also be obtained from (3.25), by employing the twist in (3.33)- (3.35) and the ExFT dictionary.For the STU sector, this configuration is related to the singular limit of the S 7 configurations in M-theory in the last section.To accomplish this, we use the coordinates (µ ā, φ ā) above related to y i as and constrained by (4.23).A coordinate z can also be introduced such that which reduces to ỹ = z in the vanishing m limit.The metric obtained from (3.22) reads where Ξ 2 and H are given in (4.16) and, for this gauging, the covariant derivatives on the angles denote the fibering with the four vectors with A ā = {A 12 , A 34 , A 56 } in terms of SL(6, R) indices.Note that for these flat indices, SL(6, R) ⊂ SL(8, R).In this sector, the axiodilaton vanishes as well as the two-form potentials Finally, the four-form gauge potential can also be derived from the ExFT dictionary.The purely internal contribution can be obtained from (3.22) to be (4.36)We recognise that, stripped of dz, the contributions in the last three lines in (4.36) precisely match the coefficients Cā, bc in (4.19), and the derivatives of the first two lines are dual to the contributions in (4.19) given by the forms in the tensor hierarchy.Therefore, the associated self-dual five-form field strength is given by with Cā, bc given in (4.19), and the field strengths in the tensor hierarchy given by (2.32), (2.34) and (2.36) with x = 0 and x = 1, which coincide with the ones in (4.22).The uplift formulae (4.32)-(4.37)precisely match the type IIB configuration obtained in (4.29) if one identifies the angles on S 1 .Even though the gauge coupling m does not enter into the Kaluza-Klein Ansätze for the type IIB bosonic fields, it mediates the relation between ten-and four-dimensional spinors, as expected from the different fermion couplings observed in four dimensions.
To perform a thorough check of the Ansatz in (4.32)-(4.37),we will restrict our attention to a simpler truncation that identifies the fields as and leaves A 4 (1) unfixed.For the SO(6) ⋉ R 12 gauging, this theory can be obtained as a circle reduction of the N = 4 SU(2) × U(1) gauged theory in D = 5 [55,56] truncated so that the five-dimensional scalar vanishes and the gauge group is reduced as U(1) ⊂ U(1) × U(1) ⊂ SU(2) × U(1).After the circle reduction, one can identify A 4 (1) with the dual of the KK vector.In the following, we only consider configurations with χ = 0, which per (2.31)  with H (2) = dA (1) .To make contact with the STU models in (2.2), the dilaton coupling needs to be set as a = 1/ √ 3, but we find it convenient to keep it unspecified at the four-dimensional level.
For a = 1/ √ 3, we can embed any solution of the 4d theory in ten dimensions.From (4.32), the metric reads dŝ 2  10 = e −ϕ √ (4.40) and the tensor hierarchy fields in (4.42) allow us to write the five-form as .41) using the duality relations (2.32)-(2.36), that reduce to with Ĝ = Ĝμν v μv ν the Einstein tensor for ĝ10 .We find it convenient to expand our tensors in the one-form basis now switch gears to present new black-hole solutions in these theories, first by considering singular limits of previously known solutions and later by directly solving the equations of motion for a suitable Ansatz.

AdS-BH limits
Setting the axions to zero, the potential (2.12a) for the STU truncation of the SO(8) gauging reads V = −8g 2 (cosh ϕ 1 + cosh ϕ 2 + cosh ϕ 3 ) . (5.1) This theory admits four-charge AdS-black hole solutions [19], with Here, dΩ 2 2,κ denotes the line element for the unit radius metric on the sphere, torus and hyperboloid for κ = 1, 0, −1, respectively given by and we also define These black holes have charges5 (5.6) Domain-wall limits of these solutions can be constructed by considering rescalings of the metric and vectors and shifts of the dilatons so that the equations of motion remain invariant up to a change in the scalar potential resulting from the loss of some of the terms in (5.1) after taking a singular limit [40].Taking the potential (5.1) becomes and the equations of motion are solved by (5.2) with now and (5.10) In this case, it is possible to take the g → 0 limit of the potential (5.8) while keeping non-trivial solutions provided that g sinh( √ σ κ β i ) can be kept constant by sending β i to infinity.Note that this cannot be achieved for the hyperbolic horizon (κ = −1) case above, and therefore only domain walls with spherical and toroidal horizons can be constructed with this method.We note that the potential (5.8) after the g → 0 limit corresponds to the STU sector of a maximal supergravity with SO(6) ⋉ R 12 gauging which is related by duality to the one discussed above.In particular, instead of by the embedding tensor Θ (6) in (2.11), it is described by with only non-trivial components in the 420 representation.Using the same scalings as the ones studied in section 4.1, this gauged supergravity can be proved to uplift consistently into an S 5 × T 2 configuration of D = 11 supergravity.
The potential (2.12b) can be similarly obtained by taking so that (5.1) becomes However, in this case the rescaled solution is (5.2) with 2 and (5.14) so that the g → 0 limit is ill-defined.
which describes a charged black hole with domain-wall asymptotics.At r = 0, this solution has a curvature singularity for all possible values of the charge and mass.When (5.25)

Discussion
In this paper, we have shown that singular limits of gauged supergravities can offer insights not only into how to construct new solutions in the resulting gauged supergravity, but also on its consistent uplift to higher dimensions if the resulting gauged supergravity can itself be obtained as a consistent truncation.We have exemplified this idea relating the STU sector of the electrically gauged SO(8) supergravity to the STU sector of the CSO theory with the SO(6) ⋉ R 12 gauging, and we have used the known consistent uplift of the former theory into M-theory on S 7 to construct an embedding of the latter into type IIB supergravity on S 5 × S 1 .It would be interesting to investigate if singular limits such as the ones we studied can be used to relate gauged supergravities with larger field contents.
To describe the uplifts of these gauged supergravities, we have employed techniques that exploit the formal duality-covariance of the embedding tensor formulation of the gauging.In particular, apart from making use of the generalised Scherk-Schwarz factorisation of the higher-dimensional fields expressed in the language of Exceptional Field Theory, we have recast all the contributions of p-form fields in D = 4 in terms of the four-dimensional tensor hierarchy (suitably restricted to the STU sector).This has provided a simple way of circumventing some complicated dualisations in ExFT involving the internal metric.To illustrate the power of this technique, we have explicitly checked that the higherdimensional equations of motion follow from the four-dimensional ones, and constructed new families of charged black hole solutions in 4d.These black holes involve non-trivial scalar profiles, and their asymptotics do not approach an AdS solution, but a domain wall.
A close cousin of the dyonic CSO theory we have considered comes equipped with an [SO(6) × SO(1, 1)] ⋉ R 12 gauge group.This theory has been shown to uplift into type IIB supergravity on S 5 × S 1 with a non-geometric patching of the circle [44].This theory possesses a very rich structure of AdS vacua [57], both supersymmetric and nonsupersymmetric, including continuous families realising a holographic conformal manifold [58][59][60][61][62][63].It will be interesting to extend our results to describe more explicitly the uplift of this gauging, and to construct in this way ten-dimensional solutions arising from nontrivial profiles of the fields in the consistent truncation.These new solutions may play an important rôle in understanding the holography of the T[U(N )] theories and the J-folds of N = 4 SYM that are conjectured to be dual to this solution [64].

) 4 = 6g e ϕ √ 3 vol 4 ,
(4.42)    upon using the truncation (4.38) with χ = 0. From(3.20), the Bianchi identity for the five-form and Einstein equations are the only equations to be checked in ten dimensions.Since the axiodilation and two-forms in (4.34) and (4.35) are zero, the equation of motion for the type IIB five-form amounts to demanding that F (5) be closed.In (4.41), it is straightforward to see that this is in turn an immediate consequence of the four-dimensional Bianchi identities and equations of motion from (4.39).For vanishing axiodilaton and two-forms, Einstein equations in 10d in turn reduce to Ĝ = T = − 1 480 Fμρ 1 ρ2 ρ3 ρ4 Fν ρ1 ρ2 ρ3 ρ4 − 1 10 ĝμν Fρ 1 ρ2 ρ3 ρ4 ρ5 F ρ1 ρ2 ρ3 ρ4 ρ5 v μv ν ,
) with Φ u a collective name for the six scalars (ϕ i , χ i ) and G uv the non-linear sigma-model metric in(2.3).The Killing vectors k[h i ] are defined in (A.10).Similarly, the three-forms in(2.35)arerelated to k[e i ] and k[f i ] in (A.10) and encode the derivatives of the axions in their equations of motion.The derivatives of the scalar potential in (2.20) are accounted for by the four-form field strengths as cannot be dyonically charged so as to guarantee H a (2) ∧ H b (2) = 0.The Lagrangian (2.2) for the SO(6) ⋉ R 12 and [SO(6) × SO(2)] ⋉ R 12 gaugings then becomes e −1 L = R − 1 2 (∂ϕ) 2 − 1 4 e aϕ H 2 − 1 4 e −ϕ/a (H 4 ) 2 + 12g 2 e −aϕ .(4.39)