Hypermultiplet gaugings and supersymmetric solutions from 11D and massive IIA supergravity on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {H}^{(p,q)}$$\end{document}H(p,q) spaces

Supersymmetric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {AdS}_{4}$$\end{document}AdS4, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {AdS}_{2} \times \Sigma _{2}$$\end{document}AdS2×Σ2 and asymptotically \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {AdS}_{4}$$\end{document}AdS4 black hole solutions are studied in the context of non-minimal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}=2$$\end{document}N=2 supergravity models involving three vector multiplets (STU-model) and Abelian gaugings of the universal hypermultiplet moduli space. Such models correspond to consistent subsectors of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {SO}(p,q)$$\end{document}SO(p,q) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {ISO}(p,q)$$\end{document}ISO(p,q) gauged maximal supergravities that arise from the reduction of 11D and massive IIA supergravity on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {H}^{(p,q)}$$\end{document}H(p,q) spaces down to four dimensions. A unified description of all the models is provided in terms of a square-root prepotential and the gauging of a duality-hidden symmetry pair of the universal hypermultiplet. Some aspects of M-theory and massive IIA holography are mentioned in passing.

Asymptotically anti-de Sitter (AdS 4 ) black holes in minimal N = 2 gauged supergravity have recently been connected to a universal renormalisation group (RG) flow for a large class of three-dimensional N = 2 superconformal field theories (SCFTs) using holography [1].The relevant (universal) AdS 4 black hole is static, extremal (thus T = 0 ) and of Reissner-Nordström (R-N) type with zero mass and a hyperbolic horizon Σ 2 = H 2 [2].The space-time metric takes the form ds 2 = −e 2U dt 2 + e −2U dr 2 + r 2 dΩ Σ2 , with and dΩ Σ2 = dθ 2 + sinh 2 (θ) dφ 2 being the Riemann surface element on Σ 2 = H 2 .The metric asymptotes an AdS 4 geometry with radius L AdS 4 when r → ∞ , and conforms to AdS 2 × H 2 in the near-horizon region r → r h = L AdS 4 / √ 2 after a shift of the radial coordinate r → r + r h and the identification L 2 AdS2 = 1 4 L 2 AdS4 .This black hole is a solution of the equations of motion that follow from the cosmological Einstein-Maxwell Lagrangian Endowing the Lagrangian (3) with N = 2 local supersymmetry requires the cosmological constant to be negative and also a mass term for the gravitini fields in the theory [3].Furthermore, supersymmetry fixes the cosmological constant to V = −3 L −2 AdS4 = −3 |µ| 2 in terms of the mass µ of a (single) complex gravitino, and renders the black hole magnetically charged [2] (see also [4]), i.e.H = p sinh(θ) dθ ∧ dφ , with the constant flux p being also set by supersymmetry.
The AdS 4 black hole described above is gauge/gravity dual to a universal RG flow in field theory [1].This is an RG flow across dimensions 1 from a three-dimensional N = 2 SCFT (dual to the asymptotic AdS 4 geometry) placed on H 2 and with a topological twist along the exact superconformal R-symmetry, to a one-dimensional superconformal quantum mechanics (dual to the AdS 2 factor of the black hole near-horizon geometry).Such a universal RG flow admits various holographic embeddings in eleven-dimensional (11D) supergravity [6,7], the lowenergy limit of M-theory, and ten-dimensional massive IIA supergravity [8].Specific examples have been studied in the context of ABJM theory [9] and GJV/SYM-CS duality [10] (and its generalisation of [11]) involving reductions of M-theory and massive IIA strings on various compact spaces [1].More concretely, when placing the SCFTs on S 1 × Σ 2 , a counting of supersymmetric ground states using the topologically twisted index of [12] at large N was shown to exactly reproduce the Bekenstein-Hawking entropy associated with the AdS 4 black hole in (1)- (2).
Extensions to non-minimal N = 2 supergravity coupled to matter multiplets, namely, n v vector multiplets and n h hypermultiplets, have also been investigated in the context of M-theory [13][14][15] and massive IIA strings [16,17].In the M-theory case, the relevant gauged supergravity is the so-called STU-model with (n v , n h ) = (3, 0) and Fayet-Iliopoulos gaugings with equal gauging parameters g Λ = g .This model arises from 11D supergravity when reduced on a compact seven-sphere H (8,0) ≡ S 7 [18] down to a four-dimensional SO(8) gauged maximal supergravity [19], and further truncated to its U(1) 4 invariant subsector [20,21].The gauging parameter g is then identified with the inverse radius of S 7 .In the massive IIA case, the relevant gauged supergravity has (n v , n h ) = (3, 1) and involves Abelian gaugings of the universal hypermultiplet moduli space with gauging parameters g and m .This model arises from tendimensional massive IIA supergravity when reduced on a compact six-sphere H (7,0) ≡ S 6 [22] down to a fourdimensional ISO (7) gauged maximal supergravity [23,24] of dyonic type [25], and further truncated to its U(1) 2 invariant subsector [26].The gauging parameter g is again identified with the inverse radius of S 6 whereas m corresponds to the Romans mass parameter.Supersymmetric AdS 4 black holes generalising the one in (1)-( 2) have been constructed in such M-theory [27] and massive IIA [26,28] non-minimal N = 2 supergravity models.
In this note we build upon the above results and study non-minimal N = 2 supergravity models that arise from 11D and massive IIA supergravity when reduced on H (p,q) = SO(p, q)/SO(p − 1, q) homogeneous spaces with various (p, q) signatures.In the case of 11D supergravity, the zero-mass sector recovers an electrically-gauged maximal supergravity in four dimensions with SO(p, q) gauge group and p + q = 8 [29].In the case of ten-dimensional massive IIA supergravity, the reduction on H (p,q) spaces down to four dimensions yields a dyonically-gauged maximal supergravity with ISO(p, q) = CSO(p, q, 1) gauge group and p + q = 7 [30][31][32].We provide a systematic characterisation and a unified description of the U(1) 2 invariant sectors associated with such M-theory and massive IIA reductions 2 .The resulting models describe nonminimal N = 2 supergravity with (n v , n h ) = (3, 1) and involve Abelian gaugings of the universal hypermultiplet moduli space.Supersymmetric AdS 4 , AdS 2 × Σ 2 and universal AdS 4 black hole solutions are systematically studied which, upon uplifting to eleven-or tendimensional backgrounds, are of interest for M-theory and massive IIA holography.

II. N = 2 SUPERGRAVITY MODELS
The bosonic sector of N = 2 supergravity coupled to n v vector multiplets and n h hypermultiplets is described by a Lagrangian of the form [33] In this note we focus on models with three vector multiplets and the universal hypermultiplet [34], namely, (n v , n h ) = (3, 1) .The three complex scalars z i = −χ i + i e −ϕi in the vector multiplets (i = 1, 2, 3) serve as coordinates in the special Kähler (SK) manifold M SK = [SU(1, 1)/U(1)] 3 .The metric on this manifold is given by where ) is the real Kähler potential.The latter is expressed in terms of an Sp(8) symplectic product X, for a homogeneous prepotential of degree-two F (X Λ ) .Our choice of sections is compatible with a square-root prepotential and restricts the range of the z i scalars to the Kähler cone The condition in (8) leaves two different domains: either the three Imz i are positive, or two of them are negative and the third one is positive.
The kinetic terms and the generalised theta angles for the vector fields in (4) are given by R ΛΣ = Re(N ΛΣ ) and I ΛΣ = Im(N ΛΣ ) in terms of a complex matrix with F ΛΣ = ∂ Λ ∂ Σ F .They can be used to define a symmetric, real and negative-definite scalar matrix that will appear later on in Sec.IV B. The vector field strengths are given by and incorporate a set of tensor fields B α where α is a collective index running over the isometries of the scalar manifold that are gauged.Importantly, the tensor fields enter the Lagrangian in (4) provided that the couplings Θ Λα = 0 , namely, if magnetic charges (see eq.( 13) below) are present in the theory [33].Tensor fields will play a role in the case of massive IIA reductions on H (p,q) spaces as magnetic charges are induced by the Romans mass parameter.
The quaternionic Kähler (QK) manifold spanned by the four real scalars q u = ( φ , σ , ζ , ζ ) in the universal hypermultiplet is M QK = SU(2, 1)/SU(2) × U(1) .The metric on this QK space reads with In this note we are exclusively interested in gauging Abelian isometries of M QK as dictated by an embedding tensor Θ M α [33].They have associated Killing vectors k u α and yield covariant derivatives in (4) of the form ) Lastly it turns also convenient to introduce symplectic Killing vectors K M ≡ Θ M α k α and moment maps P x M ≡ Θ M α P x α in order to maintain symplectic covariance [35].The scalar potential in (4) can then be expressed as [33,36] in terms of rescaled sections V M ≡ e K/2 X M and their Kähler derivatives

III. ABELIAN HYPERMULTIPLET GAUGINGS FROM REDUCTIONS ON H (p,q)
The metric (12) on the special QK manifold associated with the universal hypermultiplet lies in the image of a c-map [37][38][39] with a trivial special Kähler base.There are three isometries k α = {k σ , k σ , k U } of ( 12) that play a role in our reductions of 11D and massive IIA supergravity on H (p,q) spaces.The isometry k σ corresponds to a model-independent (axion shift) duality symmetry.On the contrary, the isometry k σ corresponds to a hidden symmetry.Together, they form a duality-hidden symmetry pair (conjugate roots of the global symmetry algebra) given by with The remaining isometry k U corresponds to a modeldependent duality symmetry and, for the M-theory and massive IIA models studied in this note, it is given by gauging g0 g1 g2 g3 m0 mi k1 k2 Embedding tensor (21) and gauged isometries.For the sake of definiteness, and without loss of generality, we are taking p ≥ q as well as g > 0 and m > 0 .
Note that, while k σ + k σ and k U are compact Abelian isometries of ( 12), the combination k σ − k σ turns to be non-compact.
The triplet of moment maps P x α associated to the Killing vectors in ( 15) and ( 17) can be obtained from the general construction of [40].The Killing vectors in (15) have associated moment maps of the form and whereas the moment maps for the isometry in (17) take the simpler form In order to fully determine the N = 2 supergravity model, one must still specify the gauge connection entering the covariant derivatives in (13).This is done by an embedding tensor Θ M α of the form where the various electric g Λ and magnetic m Λ charges are displayed in Table I.The covariant derivatives in (13) reduce to with A = i g i A i + i m i Ãi .However, the specific isometries k 1 and k 2 to be gauged in (22) depend on the M-theory or massive IIA origin of the models: M-theory : The (−1) pq sign in k 1 for the M-theory models depends on the signature of the H (p,q) space employed in the reduction.Moreover, as a consequence of ( 23), only duality symmetries k σ and k U appear upon reductions of massive IIA supergravity whereas also the hidden symmetry k σ does it in the reductions of M-theory.Note also that, in the massive IIA case, the resulting gaugings are of dyonic type: they involve both electric and magnetic charges.

IV. SUPERSYMMETRIC SOLUTIONS
The M-theory and massive IIA non-minimal N = 2 supergravity models presented in the previous section possess various types of supersymmetric solutions.

A. AdS4 solutions
An AdS 4 vacuum solution with radius L AdS4 is describing a space-time geometry of the form where . Being a maximally symmetric solution, only scalars are allowed to acquire a non-trivial and constant vacuum expectation value that extremises the potential in (14).In addition, preserving N = 2 supersymmetry requires the vanishing of all fermionic supersymmetry variations.This translates into the conditions [41]: where S AB = 1 2 e K/2 X M P x M (σ x ) AB is the gravitino mass matrix, (σ x ) AB are the Pauli matrices and |µ| = L −1 AdS4 .

M-theory
In the SO(p, q) theories with k 1 = k σ + (−1) pq k σ the first condition in (25) yields σ = 0 and The last equation in ( 26) excludes theories with p q ∈ odd, namely, the SO(7,1) and SO(5,3) theories.For those theories with p q ∈ even , the first condition in (25) additionally gives Extremising the scalar potential in (14) subject to the conditions in (25) imposed by supersymmetry yields two AdS 4 solutions that preserve a different residual (unbroken) gauge symmetry.The first solution fixes the scalars z i in the vector multiplets as and involves a trivial configuration of the scalars in the universal hypermultiplet This solution possesses a realisation in two theories: SO(8) : SO (4,4) : In both cases and the two vectors g 0 A 0 and A remain massless (k 1 = k 2 = 0), thus preserving the full U(1) 1 × U(1) 2 gauge symmetry of the models.In the compact SO(8) case, this solution uplifts to the maximally supersymmetric Freund-Rubin vacuum of 11D supergravity [42], and the dual SCFT is identified with ABJM theory [9] at low ( k = 1, 2 ) Chern-Simons levels k and −k .
The second solution fixes the scalars z i at the values and involves a non-trivial configuration of the universal hypermultiplet independent of the gauging parameters This solution has a realisation in the same theories as before: SO(8) : SO (4,4) : In both cases and only the linear combination of vectors − 3 g 0 A 0 + A remains massless (k 1 = k 2 ) in the SO(8) and SO(4, 4) theories.The associated U(1) gauge symmetry is thus preserved and to be identified with the R-symmetry of the dual field theory.In the compact SO(8) case, this solution was studied in [43,44] and uplifted to a background of 11D supergravity in [45,46].Its dual SCFT was identified in [47] as the infrared fixed point of an RG flow from ABJM theory triggered by an SU(3) invariant mass term in the superpotential.The fixing of the scalars z i in the vector multiplets to the values in ( 30)-( 31) and ( 35)-( 36) is compatible with the Kähler cone condition (8) for a physically acceptable solution in four dimensions.

Massive IIA
In the ISO(p, q) theories the first condition in (25) The extremisation of the scalar potential in (14) subject to the supersymmetry conditions in (25) yields this time a unique AdS 4 solution.It has for the z i scalars in the vector multiplets, together with | ζ| 2 = 0 and a non-trivial dilaton in the universal hypermultiplet.Note that (40) requires g 0 g 1 g 2 g 3 > 0 which is satisfied only by the ISO (7) and ISO(4,3) theories (see Table I).The corresponding solutions are given by ISO (7) : ISO (4,3) : together with Both configurations ( 41)-( 42) satisfy the Kähler cone condition ( 8) and yield The vector A remains massless (k 2 = 0) and so the U(1) 2 gauge symmetry is preserved.This symmetry is holographically identified with the R-symmetry of the dual field theory.In the ISO(7) case, this solution was presented in [10,24] 3 and uplifted to a background of massive IIA supergravity in [10,22].The dual threedimensional SCFT was also identified in [10] as a super Chern-Simons-matter theory with simple gauge group SU(N ) and level k given by the Romans mass parameter.

B. AdS2 × Σ2 solutions
Let us focus on AdS 2 × Σ 2 vacuum solutions with radii L AdS2 and L Σ2 .The corresponding space-time geometry is specified by a metric of the form The metric in (45) also describes the horizon of a static and extremal black hole, like the one in ( 1)-( 2), so these solutions are sometimes referred to as black hole horizon solutions.
An ansatz for the fields in the vector-tensor sector of the Lagrangian (4) that is compatible with the spacetime symmetries takes the form [26] (see [35] for a tensor gauge-equivalent choice): 3 Note that z i here = e i π 3 z i [10] as a consequence of the sign choice m here = −m [10] .
As a consequence of ( 11) and (46), this sector of the theory is encoded in a vector of charges Q of the form which generically depends on vector charges (p Λ , e Λ ) as well as on the θ-ϕ components b α of the tensor fields in (46).The latter play a role only in the massive IIA models where the gaugings are of dyonic type.
In the presence of hypermultiplets, quarter-BPS black hole horizon solutions with constant scalars require the set of algebraic equations [35] (see also [48]) defined in terms of a central charge Z(z i ) = Q, V , the scalar matrix M(z i ) in (10) and Q x = P x , Q .The phase β is associated with the complex function which depends on the central charge Z(z i ) and a superpotential L(z i , q u ) = Q x P x , V .With the aim of constructing later on asymptotically AdS 4 black holes of the universal type in (1)-( 2), we are concentrating on the configurations of the z i scalars found in the previous section to be compatible with AdS 4 solutions preserving N = 2 supersymmetry.

M-theory
As for the case of AdS 4 solutions, the third condition in (48) automatically discards AdS 2 × Σ 2 solutions in the SO(7, 1) and SO(5, 2) theories.Nevertheless, black hole horizon solutions with non-zero magnetic charges p Λ exist in all the remaining SO(p, q) theories.
In the case of a trivial configuration of the universal hypermultiplet, various solutions are found for the SO(8) , SO(4, 4) and SO(6, 2) a,b theories.Of special interest will be those of the SO(8) and SO(4, 4) theories with the scalars z i being fixed at the values in (30) and (31), respectively.These solutions are supported by charges of the form SO(8) : SO (4,4) : and both have a hyperbolic horizon Σ 2 = H 2 (κ = −1), a phase β = 0 in (49) and radii given by Q ISO( 7) ISO(6, 1) ISO(5, 2) ISO(4, 3)  For the SO(6,2) a theory there are solutions with the scalars z i being also fixed as in (30) and (31).They are respectively supported by charges of the form SO(6,2) a : and both have a spherical horizon Σ 2 = S 2 (κ = 1), a phase β = 0 in (49) and radii given by Analogue solutions can also be found in the SO(6,2) b theory.
In the case of requiring a non-trivial configuration of the universal hypermultiplet, only the SO(8) and SO (4,4) theories turn out to accommodate black hole horizon solutions.These have the scalars z i fixed as in (35) and (36) and the universal hypermultiplet set as in (34).The solutions are supported by charges of the form SO (8) : SO (4,4) : and both have a hyperbolic horizon Σ 2 = H 2 (κ = −1), a phase β = 0 in (49) and radii given by Lastly, additional AdS 2 × Σ 2 solutions equivalent to the ones presented above are obtained upon replacing Q → −Q and β → β + π .

Massive IIA
Quarter-BPS AdS 2 × Σ 2 solutions exist in all the ISO(p, q) theories for different values of the vector of charges Q (see Table II).The solutions have non-zero magnetic p Λ and electric e Λ charges in (47), and require | ζ| 2 = 0 and a non-trivial value of the dilaton e φ in the universal hypermultiplet.
For the ISO (7) and ISO(4, 3) theories the scalars z i are fixed at the values in (41) and (42), respectively, whereas the dilaton takes the value in (43).In both cases the solution has a hyperbolic horizon Σ 2 = H 2 (κ = −1) and radii given by For the ISO(5, 2) and ISO(6, 1) theories the scalars z i are also fixed at the values in ( 41) and (42), respectively.However the dilaton in the universal hypermultiplet is fixed at the value The solutions have a spherical horizon Σ 2 = S 2 (κ = +1) and radii given by Once again, a set of equivalent solutions is obtained upon replacing Q → −Q and β → β + π .

C. AdS4 black hole solutions
Extremal R-N black holes interpolating between the (charged version [49] of) AdS 4 and AdS 2 × Σ 2 solutions previously found can be constructed.These black holes have constant scalars and can be viewed as non-minimal M-theory and massive IIA incarnations of the universal black hole in (1)-( 2) with a hyperbolic horizon.
In the context of M-theory reduced on H (p,q) spaces, two versions of such a black hole occur in each of the SO(8) and SO (4,4) theories.The first one involves a trivial configuration of the universal hypermultiplet and interpolates between an AdS 4 geometry with radius (32) in the ultraviolet (UV at r → ∞ ) and an AdS 2 × H 2 geometry with radii (52) in the infrared (IR at r → r h ).The case of the SO(8) theory arising from H (8,0) = S 7 has been extensively studied in the literature, see e.g.[2,27,50], also from a holographic perspective [1,[13][14][15].The second version involves a non-trivial configuration of the universal hypermultiplet (34) and interpolates between an AdS 4 geometry with radius (37) in the UV and an AdS 2 × H 2 geometry with radii (58) in the IR 4 .For the SO(8) theory, it would be interesting to perform a holographic counting of AdS 4 black hole microstates in the field theory context of [47].
In the context of massive IIA reduced on H (p,q) spaces, there exists a universal black hole both in the ISO (7) and ISO(4, 3) theories.It involves a non-trivial value of the dilaton field in the universal hypermultiplet (43) and interpolates between an AdS 4 geometry with radius (44) in the UV and an AdS 2 × H 2 geometry with radii (59) in the IR.In the case of the ISO(7) theory arising from H (7,0) = S 6 , such a black hole has recently been constructed in [26,28] and connected to a universal RG flow across dimensions in [1] (see [16,17]) using holography.

V. SUMMARY AND FINAL REMARKS
In this note we have investigated supersymmetric AdS 4 , AdS 2 × Σ 2 and universal AdS 4 black hole solutions in non-minimal N = 2 supergravity models with three vector multiplets (STU-model) and Abelian gaugings of the universal hypermultiplet.We have performed a systematic characterisation of N = 2 models that arise from 11D and massive IIA supergravity when reduced on H (p,q) spaces down to four dimension.More concretely, the models correspond to the U(1) 2 invariant sector of the SO(p, q) (M-theory) and ISO(p, q) (massive IIA) gauged maximal supergravities resulting upon reduction of the higher-dimensional theories.In M-theory models, the gaugings involve a duality-hidden symmetry pair of the universal hypermultiplet.In contrast, only duality symmetries of the universal hypermultiplet are gauged in massive IIA models.Supersymmetric solutions turn to populate different domains of the Kähler cone both in the M-theory and massive IIA cases.
Future lines to explore are immediately envisaged.The first one is the higher-dimensional description of the solutions based on non-compact H (p,q) reductions.Such solutions often lie in a different domain of the Kähler cone than their counterparts in sphere reductions.In addition, for the case of sphere reductions, it would also be interesting to have a higher-dimensional picture of the AdS 4 black holes involving a non-trivial universal hypermultiplet as the gravitational backreaction of bound states of M-/D-branes wrapping a Riemann surface [52,53].To this end, connecting the four-dimensional fields and charges to higher-dimensional backgrounds of 11D and massive IIA supergravity is required.A suitable framework to obtain such uplifts is provided by the dualitycovariant formulation of 11D [54] and massive IIA supergravity [55] in terms of an exceptional field theory.Using this framework, general uplifting formulae have been systematically derived for the consistent truncation of M-theory and massive IIA on spheres5 and hyperboloids [32,57] down to a gauged maximal supergravity in four dimensions.It would then be interesting to uplift the N = 2 supergravity models constructed in this note, and therefore any of their solutions, to backgrounds of 11D and massive IIA supergravity on H (p,q) .Lastly the N = 2 formulation of M-theory models presented in this note can be straightforwardly used to describe SO(p, q) models with dyonic gaugings, the prototypical example being the ω-deformed version of the SO(8) theory [58].In this theory, and unlike for the original STU-model with only vector multiplets [59], the presence of the universal hypermultiplet makes the Lagrangian in (4) sensitive to the electric-magnetic deformation parameter ω and affects the structure of supersymmetric solutions.For instance, two inequivalent N = 2 AdS 4 solutions preserving an SU(3) × U(1) symmetry in the full theory appear at generic values of ω [60].It is therefore interesting to understand the structure of AdS 4 black hole solutions in this setup [61] for which a higher-dimensional description remains elusive [62].We hope to come back to these and related issues in the near future.

TABLE II .
Vector charges supporting supersymmetric AdS2 × Σ2 solutions in the massive IIA models.