Hypermultiplet gaugings and supersymmetric solutions from 11D and massive IIA supergravity on H$^{(p,q)}$ spaces
Adolfo Guarino
hep-th













Université Libre de Bruxelles (ULB) and International Solvay Institutes,

Service  de Physique Théorique et Mathématique, 

Campus de la Plaine, CP 231, B-1050, Brussels, Belgium.








Supersymmetric AdS_4, AdS_2×Σ_2 and asymptotically AdS_4 black hole solutions are studied in the context of non-minimal 𝒩=2 supergravity models involving three vector multiplets () and Abelian gaugings of the universal hypermultiplet moduli space. Such models correspond to  subsectors of the SO(p,q) and ISO(p,q) gauged maximal supergravities that arise from the reduction of 11D and massive IIA supergravity on 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  and massive IIA holography are mentioned in passing.





04.65.+e, 11.25.Mj 
 e-mail: aguarino@ulb.ac.be




Hypermultiplet gaugings and supersymmetric solutions
 from 11D and massive IIA supergravity on H^(p,q) spaces

    Adolfo Guarino
    December 30, 2023
==============================================================================================================









§ MOTIVATION



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 <cit.>. 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 <cit.>. The space-time metric takes the form


    d s^2 = - e^2U  d t^2 + e^-2U  d r^2 + r^2   dΩ_Σ_2 ,


with


    e^2 U =  ( r/L_AdS_4 -  L_AdS_4/2   r)^2  ,


and dΩ_Σ_2=d θ^2 + sinh^2(θ)   d ϕ^2 being the Riemann  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_AdS_2^2 = 14  L_AdS_4^2. 

This black hole is a solution of the equations of motion that follow from the cosmological Einstein–Maxwell 


    ℒ =  ( 12  R - V ) *  1 - 12 ℋ∧ * ℋ .


Endowing the Lagrangian (<ref>) with 𝒩=2 local supersymmetry requires the cosmological constant to be negative and also a mass term for the gravitini fields in the theory <cit.>. Furthermore, supersymmetry fixes the cosmological constant to V=-3    L_AdS_4^-2=-3    |μ|^2 in terms of the mass μ of a (single) complex gravitino, and renders the black hole magnetically charged <cit.> (see also <cit.>), 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 <cit.>. This is an RG flow across dimensions[See <cit.> for a generalisation to asymptotically AdS black branes in various dimensions.] 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 ,

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 <cit.>, the low-energy limit of M-theory, and ten-dimensional massive IIA supergravity <cit.>. Specific examples have been studied in the context of ABJM theory <cit.> and GJV/SYM-CS duality <cit.> (and its generalisation of <cit.>) involving reductions of M-theory and massive IIA strings on various compact spaces <cit.>. More concretely, when placing the SCFTs on S^1 ×Σ_2, a counting of supersymmetric ground states using the topologically twisted index of <cit.> at large N was shown to exactly reproduce the Bekenstein—Hawking entropy associated with the AdS_4 black hole in (<ref>)-(<ref>).

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 <cit.> and massive IIA strings <cit.>. 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 <cit.> down to a four-dimensional SO(8) gauged maximal supergravity <cit.>, and further truncated to its U(1)^4 invariant subsector <cit.>. 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 ten-dimensional massive IIA supergravity when reduced on a compact six-sphere H^(7,0)≡S^6 <cit.> down to a four-dimensional ISO(7) gauged maximal supergravity <cit.> of dyonic type <cit.>, and further truncated to its U(1)^2 invariant subsector <cit.>. 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 (<ref>)-(<ref>) have been constructed in such M-theory <cit.> and massive IIA <cit.> 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 <cit.>. 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 <cit.>. We provide a systematic characterisation and a unified description of the U(1)^2 invariant sectors associated with such  and massive IIA reductions[Einstein-scalar systems associated with ℤ_2×SO(3) and SU(3) invariant sectors of such H^(p,q) reductions were presented in <cit.>.]. The resulting models describe non-minimal 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 ten-dimensional backgrounds,  are of interest for  and massive IIA holography.




§ 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 <cit.>


    [                                                             ℒ                                                             = ( R/2 - V ) *  1 - K_ij̅  Dz^i∧ * Dz̅^j̅ - h_uv  Dq^u∧ * Dq^v;                                                                                                                           +                       1/2 ℐ_ΛΣ ℋ^Λ∧ * ℋ^Σ +  1/2 ℛ_ΛΣ ℋ^Λ∧ℋ^Σ;                                                                                                                           +                1/2Θ^Λα ℬ_α∧ d 𝒜̃_Λ + 1/8 Θ^Λα Θ_Λ^β ℬ_α∧ℬ_β . ]


In this note we focus on models with three vector multiplets and the universal hypermultiplet <cit.>, namely, (n_v, n_h)=(3,1).

The three complex scalars z^i 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 


    ds_SK^2 = K_ij̅  dz^i dz̅^j̅ = 1/4∑_idz^i  dz̅^i̅/(Im z^i)^2 ,


where K_i j̅ = ∂_z^i∂_z̅^j̅ K and K=-log(i ⟨ X, X̅⟩) is the real Kähler potential. The latter is expressed in terms of an Sp(8) symplectic product ⟨ X, X̅⟩=X^MΩ_MNX̅^N=X_ΛX̅^Λ-X^ΛX̅_Λ of holomorphic sections X^M(z^i)=(X^Λ,F_Λ) that satisfy F_Λ = ∂ℱ/∂ X^Λ (Λ=0,1, 2, 3) for a homogeneous prepotential of degree-two ℱ(X^Λ). Our choice of sections


    X^M =  (-z^1 z^2 z^3,-z^1,-z^2,-z^3,1, z^2 z^3, z^3 z^1, z^1 z^2)  ,

 
is compatible with a square-root prepotential


    ℱ = -2 √(X^0   X^1   X^2   X^3) ,


and restricts the range of the z^i scalars to the Kähler cone


    i ⟨ X, X̅⟩ = 8  Im z^1 Im z^2 Im z^3 > 0  .


The condition in (<ref>) leaves two different domains: either the three Im z^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 (<ref>) are given by ℛ_ΛΣ=Re(𝒩_ΛΣ) and ℐ_ΛΣ=Im(𝒩_ΛΣ) in terms of a complex matrix


    𝒩_ΛΣ = F̅_ΛΣ+ 2 i Im(F_ΛΓ)X^Γ  Im(F_ΣΔ)X^Δ/Im(F_ΩΦ)X^ΩX^Φ ,


with F_ΛΣ=∂_Λ∂_Σℱ. They can be used to define a symmetric, real and negative-definite scalar matrix


    ℳ(z^i) = ( 
    [ ℐ + ℛℐ^-1ℛ     -ℛℐ^-1;    - ℐ^-1ℛ       ℐ^-1 ])  ,


that will appear later on in Sec. <ref>. The vector field strengths are given by


    ℋ^Λ = d 𝒜^Λ - 12 Θ^Λα ℬ_α ,


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 (<ref>) provided that the couplings Θ^Λα≠ 0, namely, if magnetic charges (see eq.(<ref>) below) are present in the theory <cit.>. 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  mass parameter.


The quaternionic Kähler (QK) manifold spanned by the four real scalars q^u=( ϕ ,σ , ζ , ζ̃ ) in the universal hypermultiplet is ℳ_QK=SU(2,1)/SU(2)×U(1). The metric on this QK space reads


    [                                                             ds_QK^2                                                                   = d ϕ   d ϕ + 1/4 e^4 ϕ[ d σ + 12 ζ⃗ ℂ  d ζ⃗]  [ d σ + 12 ζ⃗ ℂ  d ζ⃗];                                                                                                                                       +                                           1/4  e^2ϕ    d ζ⃗   dζ⃗ , ]


with ℂ=(  [   0   1; - 1   0 ]) and ζ⃗=(ζ,ζ̃). In this note we are exclusively interested in gauging Abelian isometries of ℳ_QK as dictated by an embedding tensor Θ_M^α <cit.>. They have associated Killing vectors k^u_α and yield covariant derivatives in (<ref>) of the form


    Dz^i = d z^i and  
    D q^u = d q^u - 𝒜^M Θ_M^α   k_α^u .



Lastly it turns also convenient to introduce symplectic Killing vectors 𝒦_M≡Θ_M^α  k_α and moment maps 𝒫_M^x≡Θ_M^α  P_α^x in order to maintain symplectic covariance <cit.>. The scalar potential in (<ref>) can then be expressed as <cit.>


    [                                                       V                                                       =                           4  𝒱^M 𝒱̅^N 𝒦_M^u  h_uv 𝒦_N^v;                                                                                                               + 𝒫^x_M 𝒫^x_N( K^ij̅  D_i𝒱^M  D_j̅𝒱̅^N  - 3  𝒱^M 𝒱̅^N)  , ]


in terms of rescaled sections 𝒱^M≡ e^K/2  X^M and their Kähler derivatives D_i𝒱^M=∂_z^i𝒱^M + 1/2 (∂_z^i K)𝒱^M. 





§ ABELIAN HYPERMULTIPLET GAUGINGS FROM REDUCTIONS ON H^(P,Q)



The metric (<ref>) on the special QK manifold associated with the universal hypermultiplet lies in the image of a  <cit.> with a trivial special Kähler base. There are three isometries k_α={ k_σ  ,  k_σ  ,   k_𝕌} of (<ref>) 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


    [                                 k_σ                                   =                              -∂_σ ,;                                 k_σ                                   =     σ ∂_ϕ - (σ^2 - e^-4 ϕ - U)  ∂_σ;                                     - [  σζ⃗ - ℂ  (∂_ζ⃗ U)   ]^T ∂_ζ⃗ , ]


with


    U = 1/16   |ζ⃗|^4   +  1/2  e^-2 ϕ  |ζ⃗|^2   .


The remaining isometry k_𝕌 corresponds to a model-dependent duality symmetry and, for the M-theory and massive IIA models studied in this note, it is given by


    [               k_𝕌                 = ζ̃ ∂_ζ - ζ ∂_ζ̃ . ]


Note that, while k_σ + k_σ and k_𝕌 are compact Abelian isometries of (<ref>), the combination k_σ - k_σ turns to be non-compact. 

The triplet of moment maps P_α^x associated to the Killing vectors in (<ref>) and (<ref>) can be obtained from the general construction of <cit.>. The Killing vectors in (<ref>) have associated moment maps of the form


    P^x_σ =
    (
    [           0;           0; -1/2  e^2 ϕ ])  ,


and


    P^x_σ =
    (
    [                       - e^-ϕ ζ̃ + e^ϕ (   - σ ζ  +1/4   |ζ⃗|^2  ζ̃ );                       e^-ϕ ζ   + e^ϕ (   - σ ζ̃ - 1/4   |ζ⃗|^2   ζ ); -1/2  e^-2 ϕ -1/2  e^2 ϕ σ ^2 -1/32  e^2 ϕ   |ζ⃗|^4   + 3/4   |ζ⃗|^2 ])  ,


whereas the moment maps for the isometry in (<ref>) take the simpler form


    P^x_𝕌 =
    (
    [                 e^ϕ ζ̃;                - e^ϕ ζ; 1 - 1/4  e^2 ϕ  |ζ⃗|^2 ])  .









In order to fully determine the 𝒩=2 supergravity model, one must still specify the gauge connection entering the covariant derivatives in (<ref>). This is done by an embedding tensor Θ_M^α of the form


    Θ_M^α = (
    [ Θ_Λ^α;      ; Θ^Λ α ])
    = (
    [ g_0   0;   0 g_1;   0 g_2;   0 g_3;    ; m_0   0;   0 m_1;   0 m_2;   0 m_3 ])  ,


where the various electric g_Λ and magnetic m_Λ charges are displayed in Table <ref>. The covariant derivatives in (<ref>) reduce to


    [                                            D z^i                                                =                                          d z^i ,;                                            D q^u                                                = d q^u - (g_0 𝒜^0 + m_0 Ã_0)   k_1^u - 𝒜  k_2^u , ]


with 𝒜=∑_i g_i 𝒜^i + ∑_i m_i 𝒜̃_i. However, the specific isometries k_1 and k_2 to be gauged in (<ref>) depend on the M-theory or massive IIA origin of the models:


    [                M-theory : k_1 =k_σ + (-1)^p q   k_σ                         ,                   k_2=k_𝕌                         ,;             Massive IIA :                 k_1 = k_σ                         ,                  k_2= k_𝕌                         .;                           ]


The (-1)^p q sign in k_1 for the M-theory models depends on the signature of the H^(p,q) space employed in the . Moreover, as a consequence of (<ref>), 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.






§ 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.




 §.§ AdS_4 solutions


An AdS_4 vacuum solution with radius L_AdS_4 is describing a space-time geometry of the form


    d s^2 = - r^2/L^2_AdS_4  d t^2 + L^2_AdS_4/r^2  d r^2 + r^2  dΩ_Σ_2 ,


where dΩ_Σ_2=d θ^2 + ( sin√(κ) θ/√(κ))^2   d ϕ^2 is the surface element of Σ_2={𝕊^2   (κ=+1)   ,  ℍ^2   (κ=-1) }. Being a maximally symmetric solution, only scalars are allowed to acquire a non-trivial and constant vacuum expectation value that extremises the potential in (<ref>). In addition, preserving 𝒩=2 supersymmetry requires the vanishing of all fermionic supersymmetry variations. This translates into the conditions <cit.>: 


    [                               X^M 𝒦_M                                     =                                     0                                     ,; [  ∂_z^i X^M + (∂_z^i K) X^M ]  𝒫^x_M                                     =                                     0                                     ,;                              S_𝒜ℬ ϵ^ℬ                                     =                           1/2 μ ϵ^∗_𝒜                                     , ]


where S_𝒜ℬ=1/2 e^K/2 X^M𝒫^x_M (σ^x)_𝒜ℬ is the gravitino mass matrix,  (σ^x)_𝒜ℬ are the Pauli matrices and |μ|=L^-1_AdS_4.




  §.§.§ M-theory


In the SO(p,q) models with k_1=k_σ + (-1)^pq   k_σ the first condition in (<ref>) yields 


    σ = 0  and (e^-2 ϕ + 1/4  |ζ⃗|^2)^2 = (-1)^pq .


The last equation in (<ref>) excludes models with p   q ∈odd, namely, the SO(7,1) and SO(5,3) models. For those models with p   q ∈even, the first condition in (<ref>) additionally gives  

    ( ∑_i g_i  z^i + g_0 ∏_i z^i)  ζ⃗ = 0  .


Extremising the scalar potential in (<ref>) subject to the conditions in (<ref>) imposed by supersymmetry yields two AdS_4 solutions that preserve a different residual () gauge symmetry. 


The first solution fixes the scalars z^i in the vector multiplets as


    z^i= ±   i  (g_0  g_1  g_2  g_3)^1/2g_0  g_i ,


and involves a trivial configuration of the scalars in the universal hypermultiplet


    e^ϕ= 1   and   |ζ⃗|^2=0  .


This solution possesses a realisation in two models:


    SO(8)     :      z^1=z^2=z^3=  i   , 
    
    SO(4,4)     :      z^1=z^2=-z^3=  -i   .


In both cases


    L^2_AdS_4 = 1/2   g^2 ,


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 <cit.>, and the dual SCFT is identified with ABJM theory <cit.> at low (k=1,2) Chern–Simons levels k and -k.

The second solution fixes the scalars z^i at the values


    z^i= ±   i  √(3) (g_0  g_1  g_2  g_3)^1/2g_0  g_i ,


and involves a non-trivial configuration of the universal hypermultiplet independent of the gauging parameters


    e^ϕ= √(3)√(2) and    |ζ⃗|^2=4/3 .


This solution has a realisation in the same models as before:


    SO(8)     :      z^1=z^2=z^3=  i  √(3) , 
    
    SO(4,4)     :      z^1=z^2=-z^3=  -i  √(3)  .


In both cases


    L^2_AdS_4 = 2/3 √(3)  g^2 ,


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) models. The associated U(1) gauge symmetry is thus preserved and to be identified with the  of the dual field theory. In the compact SO(8) case, this solution was studied in <cit.> and uplifted to a background of 11D supergravity in <cit.>. Its dual SCFT was identified in <cit.> 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 (<ref>)-(<ref>) and (<ref>)-(<ref>) is compatible with the Kähler cone condition (<ref>) for a physically acceptable solution in 𝒩=2 supergravity in four dimensions.






  §.§.§ Massive IIA


In the ISO(p,q) models the first condition in (<ref>) yields


    ( ∑_i g_i  z^i)   ζ⃗ = 0
     and ∏_i z^i = m_0/g_0 .


The extremisation of the scalar potential in (<ref>) subject to the supersymmetry conditions in (<ref>) yields this time a unique AdS_4 solution. It has


    [                              (Rez^i)^3                                      =      -18 g_1   g_2   g_3g_i^3 m_0g_0 ,;                              (Imz^i)^3                                      = ±3 √(3)8 g_1   g_2   g_3g_i^3 m_0g_0 , ]


for the z^i scalars in the vector multiplets, together with |ζ⃗|^2=0 and a non-trivial dilaton


    e^6 ϕ= 8   g_1   g_2   g_3g_0   m_0^2 ,


in the universal hypermultiplet. Note that (<ref>) requires g_0   g_1   g_2   g_3 > 0 which is satisfied only by the ISO(7) and ISO(4,3) models (see Table <ref>). The corresponding solutions are given by


    ISO(7)     :      z^1=z^2=z^3= (mg)^1/3  e^i 2π/3 , 
    
    ISO(4,3)     :      z^1=z^2=-z^3= (mg)^1/3  e^-i 2π/3 ,


together with


    e^ϕ= √(2) (g/m)^1/3 .


Both configurations (<ref>)-(<ref>) satisfy the Kähler cone condition (<ref>) and yield 


    L^2_AdS_4 = 1√(3)  g^-7/3  m^1/3 .


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 <cit.>[Note that z^i_here = e^i π/3  z_<cit.>^i as a consequence of the sign choice m_here=-m_<cit.>.] and uplifted to a background of massive IIA supergravity in <cit.>. The dual three-dimensional SCFT was also identified in <cit.> as a super Chern–Simons-matter theory with simple gauge group SU(N) and level k given by the Romans mass .





 §.§ AdS_2×Σ_2 solutions




Let us focus on AdS_2×Σ_2 vacuum solutions with radii L_AdS_2 and L_Σ_2. The corresponding space-time geometry is specified by a metric of the form


    d s^2 = - r^2/L^2_AdS_2  d t^2 + L^2_AdS_2/r^2  d r^2 + L_Σ_2^2  dΩ_Σ_2 .


The metric in (<ref>) also describes the horizon of a static and extremal black hole, like the one in (<ref>)-(<ref>), 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 (<ref>) that is compatible with the space-time symmetries takes the form <cit.> (see <cit.> for a tensor gauge-equivalent choice):


    [                                    𝒜^Λ                                      =  𝒜_t^Λ(r)   d t - p^Λ cos√(κ) θ/κ  d ϕ                                      ,;                                   𝒜̃_Λ                                      = 𝒜̃_t_Λ(r)   d t - e_Λ cos√(κ) θ/κ  d ϕ                                      ,;                                    ℬ_α                                      =       b_α(r)  sin√(κ) θ/√(κ)  d θ∧ d ϕ                                      . ]


As a consequence of (<ref>) and (<ref>), this sector of the theory is encoded in a vector of charges 𝒬 of the form 𝒬^M = ( 𝔭^Λ, 𝔢_Λ)^T with


    𝔭^Λ=p^Λ - 1/2 Θ^Λ α  b_α and 𝔢_Λ=e_Λ + 1/2 Θ_Λ^α  b_α ,


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 (<ref>). 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 <cit.> (see also <cit.>)


    𝒬    =   κ  L_Σ_2^2 Ω ℳ 𝒬 ^x 𝒫^x - 4  Im(𝒵̅ 𝒱)  , 
    
    L_Σ_2^2L_AdS_2    =  -2  𝒵  e^-i β , 
    
    ⟨𝒦^u , 𝒱⟩    =  0  ,


defined in terms of a central charge 𝒵(z^i)=⟨𝒬 , 𝒱⟩, the scalar matrix ℳ(z^i) in (<ref>) and 𝒬^x = ⟨𝒫^x , 𝒬⟩. The phase β is associated with the complex function


    W=e^U (𝒵 + i  κ  L_Σ_2^2  ℒ)= |W|   e^i β ,


which depends on the central charge 𝒵(z^i) and a superpotential ℒ(z^i,q^u)=⟨𝒬^x𝒫^x , 𝒱⟩. 

With the aim of constructing later on asymptotically AdS_4 black holes of the universal type in (<ref>)-(<ref>), 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 (<ref>) automatically discards AdS_2×Σ_2 solutions in the SO(7,1) and SO(5,2) models. Nevertheless, black hole horizon solutions with non-zero magnetic charges p^Λ exist in all the remaining SO(p,q) models. 


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 models. Of special interest will be those of the SO(8) and SO(4,4) models with the scalars z^i being fixed at the values in (<ref>) and (<ref>), respectively. These solutions are supported by charges of the form


    SO(8)     :        p^0=-p^1=-p^2=-p^3=14 g , 
    SO(4,4)     :      p^0=p^1=p^2=-p^3 = - 14 g ,


and both have a hyperbolic horizon Σ_2=ℍ^2 (κ = - 1), a phase β=0 in (<ref>) and radii given by


    L^2_AdS_2 = 18  g^-2 ,   L^2_ℍ^2 = 14  g^-2 .


For the SO(6,2)_a model there are solutions with the scalars z^i being also fixed as in (<ref>) and (<ref>). They are respectively supported by charges of the form 


    SO(6,2)_a     :      p^0=1g      ,       p^1=p^2=p^3=0  ,
    SO(6,2)_a     :      p^0=p^1=p^2=0       ,       p^3=1g ,


and both have a spherical horizon Σ_2=𝕊^2 (κ = 1), a phase β=0 in (<ref>) and radii given by


    L^2_AdS_2 = 12  g^-2 ,  
    L^2_𝕊^2 = 12  g^-2 .


Analogue solutions can also be found in the SO(6,2)_b model.

In the case of requiring a non-trivial configuration of the universal hypermultiplet, only the SO(8) and SO(4,4) models turn out to accommodate black hole horizon solutions. These have the scalars z^i fixed as in (<ref>) and (<ref>) and the universal hypermultiplet set as in (<ref>). The solutions are supported by charges of the form


    SO(8)     :      p^0 = -3p^1 = -3p^2 = -3p^3 = 12 g , 
    SO(4,4)     :      p^0 = 3p^1 = 3p^2 = -3p^3 = - 12 g  ,


and both have a hyperbolic horizon Σ_2=ℍ^2 (κ=-1), a phase β=0 in (<ref>) and radii given by


    L^2_AdS_2 = 16 √(3)  g^-2 ,     L^2_ℍ^2 = 13 √(3)  g^-2 .


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) models for different values of the vector of charges 𝒬 (see ). The solutions have non-zero magnetic 𝔭^Λ and electric 𝔢_Λ charges in (<ref>), and require  |ζ⃗|^2=0 and a non-trivial value of the dilaton e^ϕ in the universal . 

For the ISO(7) and ISO(4,3) models the scalars z^i are fixed at the values in (<ref>) and (<ref>), respectively, whereas the dilaton takes the value in (<ref>). In both cases the solution has a hyperbolic horizon Σ_2=ℍ^2 (κ=-1) and radii given by


    L^2_AdS_2 = 1/4 √(3)  g^-7/3  m^1/3 ,   
    L^2_ℍ^2 = 1/2 √(3)   g^-7/3  m^1/3 .


For the ISO(5,2) and ISO(6,1) models the scalars z^i are also fixed at the values in (<ref>) and (<ref>), respectively. However the dilaton in the universal hypermultiplet is fixed at the value


    e^ϕ = √(2)/√(3) (g/m)^1/3 .


The solutions have a spherical horizon Σ_2=𝕊^2 (κ=+1) and radii given by


    L^2_AdS_2 = 3 √(3)/4  g^-7/3  m^1/3 ,  
     L^2_𝕊^2 =3 √(3)/14   g^-7/3  m^1/3 .



Once again, a set of equivalent solutions is obtained upon replacing 𝒬→ -𝒬 and β→β + π.





 §.§ AdS_4 black hole solutions


Extremal R–N black holes interpolating between the (charged version <cit.> 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 (<ref>)-(<ref>) 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) models. The first one involves a trivial configuration of the universal hypermultiplet and interpolates between an AdS_4 geometry with radius (<ref>) in the ultraviolet (UV at r →∞) and an AdS_2×ℍ^2 geometry with radii (<ref>) in the infrared (IR at r → r_h). The case of the SO(8) model arising from H^(8,0)=S^7 has been extensively studied in the literature, see e.g. <cit.>, also from a holographic perspective <cit.>. The second version involves a non-trivial configuration of the universal hypermultiplet (<ref>) and interpolates between an AdS_4 geometry with radius (<ref>) in the UV and an AdS_2×ℍ^2 geometry with radii (<ref>) in the IR[See <cit.> for a similar model based on the Q^111 truncation of M-theory <cit.>.]. For the SO(8) model, it would be interesting to perform a holographic counting of AdS_4 black hole microstates in the field theory context of <cit.>.

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) models. It involves a non-trivial value of the dilaton field in the universal hypermultiplet (<ref>) and interpolates between an AdS_4 geometry with radius (<ref>) in the UV and an AdS_2×ℍ^2 geometry with radii (<ref>) 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 <cit.> and connected to a universal RG flow across dimensions in <cit.> (see <cit.>) using holography.




§ 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 models based on non-compact H^(p,q) reductions. Solutions of such models often lie in a different domain of the Kähler cone than their counterparts based on 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 branes wrapping a Riemann surface <cit.>. 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 duality-covariant formulation of 11D <cit.> and massive IIA supergravity <cit.> in terms of an exceptional field theory.  this framework, general uplifting formulae have been systematically derived for the consistent truncation of  and massive IIA on spheres[See <cit.> for a recent derivation of the complete D=11 embedding of SO(8) gauged maximal supergravity using the method introduced in <cit.> for the D=10 embedding of the ISO(7) theory.] and hyperboloids <cit.> 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 to backgrounds of 11D and massive IIA supergravity involving H^(p,q) spaces. 


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  version of the SO(8) theory <cit.>. In this case, and unlike for the original STU-model with only vector multiplets <cit.>, the presence of the universal hypermultiplet makes the  in (<ref>) 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 <cit.>. It is therefore interesting to understand the structure of AdS_4 black hole solutions in this setup <cit.> for which a higher-dimensional description remains  <cit.>. We hope to come back to these and related issues in the near future.





Acknowledgements: This work is partially supported by the ERC Advanced Grant “High-Spin-Grav" and by F.R.S.-FNRS through the conventions PDRT.1025.14 and IISN-4.4503.15.







§ APPENDIX: EMBEDDING IN MAXIMAL SUPERGRAVITY





In this appendix we provide the embedding of the non-minimal 𝒩=2 supergravity models analysed in this note into 𝒩=8 (maximal) supergravity <cit.>. In its ungauged version, maximal supergravity possesses an E_7(7) global symmetry group. This group plays a central role in systematically constructing the gauged versions of the theory using the embedding tensor formalism <cit.>: in a gauged maximal supergravity, a specific subgroup of E_7(7) is promoted from global to local in what is known as the gauging procedure. In this note we focus on SO(p,q) and ISO(p,q) subgroups of SL(8) ⊂E_7(7). These gauged supergravities appear upon reduction of 11D and  supergravity on H^(p,q) spaces. 




Let us start by introducing a fundamental SL(8) index A=1,...8. In the SL(8) basis, the E_7(7) generators t_α=1,...,133 have a decomposition 133→63 + 70. These are the 63 generators t_A^B of SL(8), with vanishing trace t_A^A=0, together with 70 generators t_ABCD=t_[ABCD]. The fundamental representation of E_7(7) decomposes as 56→28 + 28', which translates into a splitting of the E_7(7) fundamental index of the form _𝕄→ _[AB]⊕ ^[AB]. The entries of the 56 × 56 matrices [t_α]_𝕄^ℕ are given by


    [                         [t_A^B]_[CD]^[EF]                                         = 4  ( δ_[C^B δ_D]A^EF + 1/8 δ_A^B δ_CD^EF)                                         ,;                         [t_A^B]^[EF]_[CD]                                         =                       - [t_A^B]_[CD]^[EF]                                         , ]


for the SL(8) generators t_A^B. The generators t_ABCD completing SL(8) to E_7(7) take the form


    [ [t_ABCD]_[EF][GH]                 =   2/4! ϵ_ABCDEFGH                 ,; [t_ABCD]^[EF][GH]                 =    2  δ_ABCD^EFGH                 . ]


The electric SO(p,q) and dyonic ISO(p,q) gaugings of maximal supergravity belong to SL(8)⊂E_7(7) and are specified by an embedding tensor <cit.> of the form 


    Θ_AB^C_D = 2  δ_[A^C η_B]D , Θ^AB^C_D = 2  δ^[A_D η̃^B]C .


The matrices η_AB and η̃^AB  associated with the different gaugings are collected in Table <ref>.




The scalar sector of maximal supergravity consists of 70 fields spanning a coset space E_7(7)/SU(8). However, in this note we concentrate on non-minimal 𝒩=2 supergravity models associated with the U(1)^2 invariant sector of the theory. The scalars in this sector span an [SL(2)/SO(2)]^3  × SU(2,1)/(SU(2)×U(1)) coset space associated with the following E_7(7) generators. The [SL(2)/SO(2)]^3 factor is associated with


    [                                                          H_φ_1                                                              = t_4^4 + t_5^5 + t_6^6 +  t_7^7 - t_1^1 - t_8^8 - t_2^2 - t_3^3                                                              ,;                                                          H_φ_2                                                              = t_2^2 + t_3^3 + t_6^6 +  t_7^7 - t_1^1 - t_8^8 - t_4^4 - t_5^5                                                              ,;                                                          H_φ_3                                                              = t_2^2 + t_3^3 + t_4^4 +  t_5^5 - t_1^1 - t_8^8 - t_6^6 - t_7^7                                                              ,;                                                          g_χ_1                                                              =       t_1238
     , 
    g_χ_2   =     t_1458
     , 
    g_χ_3   =     t_1678                                                              . ]


The SU(2,1)/(SU(2)×U(1)) factor is associated with


    [                                       H_ϕ                                         = 12  (t_8^8-t_1^1) 
    
     , 
    g_σ    =    t_8^1                                         ,;                                       g_ζ                                         =       t_8357   - t_8346 - t_8256 - t_8247                                         ,;                                      g_ζ̃                                         =         t_8246 - t_8257 - t_8347 - t_8356                                         . ]


Using the above generators, the coset representative 𝒱=𝒱_SK×𝒱_QK is obtained upon the exponentiations


    [                                                    𝒱_SK                                                       =               ∏_i e^-12  χ_i  g_χ_i    e^1/4 φ_i  H_φ_i                                                       ,;                                                    𝒱_QK                                                       = e^σ   g_σ    e^-6  (ζ  g_ζ + ζ̃  g_ζ̃)     e^-2  ϕ  H_ϕ                                                       . ]


Starting from the representative 𝒱∈E_7(7)/SU(8), the scalar matrix ℳ_𝕄ℕ entering the Lagrangian of 𝒩=8 supergravity <cit.> is obtained as ℳ=𝒱 𝒱^t. Plugging this matrix ℳ into the kinetic terms of the maximal theory, e^-1ℒ_kin=1/96Tr [ D_μℳ  D^μℳ^-1], we obtain the kinetic terms for the 𝒩=2 models in (<ref>) and (<ref>) upon identifying z^i=-χ_i+ i   e^-φ_i. Note that this parameterisation of the U(1)^2 invariant scalar coset in maximal supergravity is not compatible with configurations for which Imz^i <0. The scalar potential and covariant derivatives of the 𝒩=2 models can be obtained from the ones of the maximal theory upon identifying the electric vectors 𝒜^AB=𝒜^[AB] in the maximal theory as {𝒜^0,𝒜^1,𝒜^2,𝒜^3 }≡{𝒜^18 , 𝒜^23 , 𝒜^45 , 𝒜^67} (and similarly for the magnetic vectors 𝒜̃_AB), and then truncating away the non-singlet fields. We have verified that the 𝒩=8 results precisely match the 𝒩=2 results obtained from (<ref>) and (<ref>).






