Isometries, gaugings and N $$ \mathcal{N} $$ = 2 supergravity decoupling

CPHT-RR040.072016 arXiv:1611.00964v2 [hep-th] 29 Nov 2016 Isometries, gaugings and N = 2 supergravity decoupling Ignatios Antoniadis,1,2 Jean-Pierre Derendinger,2 P. Marios Petropoulos3,1 and Konstantinos Siampos2 1 Laboratoire de Physique Théorique et Hautes Energies, Sorbonne Universités, CNRS UMR 7589, UPMC Paris 6, 4 place Jussieu, 75005 Paris, France 3 2 Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland Centre de Physique Théorique, Ecole Polytechnique, CNRS UMR 7644, Université Paris-Saclay, 91128 Palaiseau Cedex, France A BSTRACT We study off-shell rigid limits for the kinetic and scalar-potential terms of a single N = 2 hypermultiplet. In the kinetic term, these rigid limits establish relations between fourdimensional quaternion-Kähler and hyper-Kähler target spaces with symmetry. The scalar potential is obtained by gauging the graviphoton along an isometry of the quaternion-Kähler space. The rigid limits unveil two distinct cases. A rigid N = 2 theory on Minkowski or on AdS4 spacetime, depending on whether the isometry is translational or rotational respectively. We apply these results to the quaternion-Kähler space with Heisenberg ⋉ U (1) isometry, which describes the universal hypermultiplet at type-II string one-loop. Contents Introduction 1 1 Hyper-Kähler and quaternionic manifolds with a symmetry 3 2 The kinetic term and the rigid limits 6 3 The scalar potential 11 3.1 Potential and spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The decoupling limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Spaces with Heisenberg isometry 15 4.1 The kinetic term and its three distinct rigid limits . . . . . . . . . . . . . . . . . 15 4.2 The scalar potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Conclusion 25 A Pseudo-Fubini–Study metric 26 B Four-dimensional Kähler spaces with an isometry 27 B.1 Four-dimensional Kähler spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 27 B.2 Hyper-Kähler spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 B.3 Scalar-flat spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 B.4 Einstein spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Introduction The gravity decoupling is a subject of interest in supergravity and string theory. It is a precious source of information about the spectrum in models where supersymmetry is rigidly realized, controls the curvature of the background metric, and can shed light on the supersymmetry breaking mechanism. In the framework of local N = 2 supersymmetry, the hypermultiplet scalar dynam- ics is captured in σ-models with four-dimensional quaternion-Kähler target spaces [1], and interaction potentials obtained by gauging some symmetries. Similarly, global N = 2 supersymmetry requires hyper-Kähler spaces [2]. These are Ricci-flat Kähler spaces and for one hypermultiplet Riemann-self-dual. When supersymmetry is locally realized, the scalar curvature of the quaternion-Kähler −2 space is directly proportional to the gravitational constant k2 = 8πMPlanck . The decou- pling limit consists in taking this coupling constant to zero i.e. MPlanck → ∞, which de1 forms the quaternion-Kähler geometry into a hyper-Kähler one. Such a limiting process must smoothly interpolate between the two geometries, and its description requires care. It implies to simultaneously “zooming-in” in order to recover non-trivial hyper-Kähler geometries, as performed in Ref. [3] for quaternion-Kähler spaces with Heisenberg isometry (this symmetry was discussed in Ref. [4]). Bridging hyper-Kähler and quaternion-Kähler four-dimensional spaces has been discussed in some specific cases, including the quaternionic quotient method [5, 6], or the eightdimensional hyper-Kähler cone technique [7, 8]. Later, a general correspondence between quaternionic manifolds with an isometry and hyper-Kähler manifolds with a rotational symmetry, endowed with a hyperholomorphic connection, was found in [9] within a mathematical framework. This correspondence was further pursued in [10, 11], and developed from a more physical perspective in [12]. Finally progress has been made in the rigid limit of special quaternionic Kähler manifolds [13], constructed through the local c-map, reducing to hyper-Kähler spaces constructed by the rigid c-map [14, 15]. It it obvious that several quaternion-Kähler spaces can have the same rigid limit. For instance, SO(1, 4)/SO(4) and SU (1, 2)/SU (2) × U (1) lead to flat hyper-Kähler spaces. It is also true that one quaternion- Kähler space can have several rigid limits. A point that we discuss in this work. More recently, a systematic pattern for connecting general quaternion-Kähler and hyperKähler spaces with symmetries was introduced in [16]. The aim of the present article is to recast this method in more general terms, including in particular the scalar potential and its behaviour along the decoupling, its critical points and their supersymmetry properties, as well as the value of the cosmological constant. We will also discuss alternative decoupling limits, setting the control of the symmetry at the hyper-Kähler level. We will first discuss hyper-Kähler and quaternion-Kähler spaces with symmetries, emphasizing a simple relationship between pairs of such spaces, which translates into the coupling or decoupling of gravity. This holds for the kinetic term of hypermultiplet scalars. The behaviour of the potential will be analyzed next, when this potential is obtained by gauging (for simplicity) the graviphoton along an isometry of the quaternion-Kähler space. Two separate regimes will be studied: the case where the decoupling of gravity leaves a rigid N = 2 theory on Minkowski background, and the alternative where the spacetime is AdS4 . For all these cases, we systematically study the mass spectrum. We start with a short review on hyper-Kähler and quaternionic manifolds with a symmetry, Sec. 1. In Sec. 2, we investigate gravity decoupling limits of quaternionic manifolds with a symmetry, leading to hyper-Kähler spaces. The analysis of the scalar potential by gauging the graviphoton is performed in Sec. 3. Finally, we study extensively in Sec. 4 the quaternion-Kähler space with Heisenberg ⋉ U (1) isometry and its decoupling limits. Two appendices follow, including the discussion of the pseudo-Fubini–Study metric, which describes the universal hypermultiplet at string tree-level, Sec. A, and an alternative exhibition 2 of generic, Ricci-flat, scalar-flat or Einstein four-dimensional Kähler spaces with a holomorphic isometry, Sec. B. 1 Hyper-Kähler and quaternionic manifolds with a symmetry A hyper-Kähler space in four dimensions is a Kähler manifold with self-dual Riemann tensor: Ruvxy − 1 ε wz Rwzxy = 0 . 2 uv (1.1) The indices u, v, . . . run from 1 to 4, and we have introduced ε uvxy = p det g ǫuvxy with ǫ0123 = 1. This space is Ricci-flat and endowed with 3 covariantly constant anti-self-dual 2-forms JK . These form a triplet of SU (2) complex structures normalized to satisfy ( JK )u x ( JL ) xv = −δKL δuv − ε KLM ( JM )uv , 3 ∑ K =1 y (1.2) vy ( JK )uv ( JK ) x y = gux gvy − δu δxv + ε ux . In the following, we will assume the existence of isometries. As a consequence of Bianchi identity, a Killing vector ξ satisfies ∇ x ∇v ξ u = Ruvxy ξ y . (1.3) Its (anti)-self-dual covariant derivatives1 k± uv 1 = 2   1 wz ∇u ξ v ± ε uv ∇w ξ z , 2 (1.4) obey remarkable identities, 1 ± uv gxy k2 , k2± := k± , uv k 4 ±  1 1 u v z ∇ x ξ z ∇y ξ − gxy ∇u ξ v ∇ ξ , = 2 4 ± guv k± ux k vy = ∓ guv k± ux k vy (1.5) ∓ guv gxy k± ux k vy = 0 , valid irrespective of the nature of the space. The self-duality condition (1.1) can be recast using (1.3) as ∇ x k− uv = 0 , 1 The (anti)-self-dual components of a 2-form Auv are defined as A± uv = 3 (1.6) 1 2   Auv ± 12 ε uvwz Awz . leading to ∂ x k2− = 0 =⇒ k2− = c , (1.7) where c is a non-negative constant. Consequently, in hyper-Kähler spaces, a Killing vector is translational if k− uv = 0, and rotational otherwise [17–19]. In order to clarify the meaning of translational versus rotational isometry, we evaluate the Lie derivative on the complex structures with respect to the Killing vector ξ: Lξ JK
uv =  ∇ v



( JK )uw ξ w − ∇u ( JK )v w ξ w = ∂v ( JK )uw ξ w − ∂u ( JK )v w ξ w , ∇ ξ ( J )w − ( J ) w ∇ ξ = [∇ξ, J ] , w v u w K v K u K uv (1.8) where the bracket stands for the ordinary commutator of matrices (not to be confused with the Lie bracket). The latter expression trivializes for a translational isometry [19] Lξ JK = [k+ , JK− ] = 0 , (1.9) because [ A− , B+ ] = 0 for any pair of 2-forms A and B. Therefore a translational Killing vector ξ is triholomorphic, leaving the three complex structures invariant. In addition, Eqs. (1.8) and (1.9) ensure the existence of a triplet of Killing potentials (moment maps) K I defined as 1 ( J I )uw ξ w = − ∂u K I . 2 (1.10) For a rotational isometry, we can always find a basis of complex structures such that Lξ J1 = J2 , Lξ J2 = − J1 , Lξ J3 = 0 . (1.11) Consequently a rotational Killing vector ξ is simply holomorphic since only one complex structure remains invariant. In addition, (1.8) and (1.11) ensure the existence of a Killing potential K defined as 1 1 ( J3 )uw ξ w = − ∂u K =⇒ ξ u = ( J3 )uv ∂v K , 2 2 (1.12) and the hyper-Kähler metric gHK uv satisfies the relation gHK uv =
1 w z δu δv + ( J⊥ )uw ( J⊥ )vz ∇w ∇z K , 2 (1.13) where J⊥ is any complex structure orthogonal to J3 . Diagonalizing the latter selects Kähler coordinates with gHK ab = 0 , = ∇ a ∇b K = ∂a ∂b K = K ab . gHK ab̄ Thus the Killing potential of J3 is the Kähler potential for J⊥ [27]. (1.14) Adapting a coordinate τ along the orbits of the Killing field as ξ = ∂τ , the hyper-Kähler 4 metric reads: 1 (dτ + ω )2 + V dℓ2 , V dℓ2 = γij dX i dX j , X i = ( X, Y, Z ) . ds2HK = (1.15) When ξ is a translational Killing vector, we can use the Gibbons–Hawking frame [17] γij = δij , ∇V = ∇ × ω . (1.16) The complex structures read in this case:   J1 = −i a ∧ b − a ∧ b , with 1 a= √ 2 √   J3 = −i a ∧ a + b ∧ b , (1.17) 1 √ V (dx + i dy) . b= √ 2 (1.18) J2 = a ∧ b + a ∧ b ,  i V dz + √ (dτ + ω ) , V Alternatively, for a rotational Killing vector, the Gibbons–Hawking frame (1.16) is traded for the Boyer–Finley one [18, 19]:2
dℓ2 = dZ2 + eΨ dX 2 + dY 2 , 1 ΨZ , 2 1 1 ω = ΨY dX − ΨX dY , 2  2 Ψ ΨXX + ΨYY + e = 0. V= (1.19) ZZ The complex structures are given by (1.17) with 1 a= √ 2 √  i V dz + √ (dτ + ω ) , V ei τ √ Ψ V e /2 (dx + i dy) . b= √ 2 (1.20) Notice the explicit τ-dependence, necessary for (1.11) to hold. Finally, the Killing potential K for ξ = ∂τ is obtained using (1.12), (1.17) and (1.20): K = 2Z . (1.21) We now turn to quaternion-Kähler spaces. These are Einstein and conformally self-dual:3 Wuvxy − 1 ε wz Wwzxy = 0 . 2 uv 2 (1.22) Indices on scalar functions denote ordinary partial derivatives. Four-dimensional quaternion-Kähler spaces are in general not Kähler, but there are exceptions. These include SU (3)/SU (2) × U (1) and SU (1, 2)/SU (2) × U (1) (see App. A). The Kähler structure introduces a canonical orientation and self-duality is not equivalent with anti-self-duality. In particular, a four-dimensional Kähler metric which is Weyl anti-self-dual has vanishing scalar curvature [22]. 3 5 We will here normalize the scalar curvature to R = −12. Assuming again the existence of an isometry and using Eq. (1.3), the Einstein-space condition Ruv = −3guv and (1.22), we find: ∇ x k− vu = −
1 gux gvy − guy gvx − ε uvxy ξ y , 2 (1.23) instead of (1.6). Contrary to what happens for hyper-Kähler spaces, the distinction between translational and rotational Killings is no longer relevant here, and the Gibbons–Hawking or the Boyer–Finley forms (1.15), (1.16) and (1.19) are replaced by the Przanowski–Tod frame [28–31], where4   
 1 2 2 Ψ 2 2 (dτ + ω ) + V dZ + e dX + dY , V   dω = VX dY ∧ dZ + VY dZ ∧ dX + V eΨ dX ∧ dY , Z   ΨXX + ΨYY + eΨ = 0 , 2V = 2 − Z ΨZ . ds2QK 1 = 2 Z (1.24) ZZ A straightforward computation shows that the coordinate Z is related to the anti-self-dual covariant derivative of the Killing field ξ = ∂τ : 1 − uv = k2− = k− . uv k Z2 (1.25) 2 The kinetic term and the rigid limits As discussed in Ref. [16], using any solution of the Toda equation, one can build both a quaternion-Kähler space with symmetry, expressed à la Przanowski–Tod, and a hyperKähler space in the Boyer–Finley frame. This sets a simple one-to-one relationship among these spaces. Although this relationship sounds formal, it supports a deeper geometric interpretation: the hyper-Kähler member of the pair appears actually as a zooming-in of the quaternionic member around the fixed point of the isometry that supports the fiber in the Przanowski–Tod frame. This isometry survives in the hyper-Kähler space as a simply holomorphic symmetry. From a physical viewpoint, as we will see soon, the limiting procedure at hand goes along with the gravity decoupling limit. In order to elaborate on the geometric picture of the above correspondence, we recall that

integrability condition for ω, given by the linearized Toda equation ∂2X + ∂2Y V + ∂2Z VeΨ = 0, is actually a consequence of the last equations in (1.24). 4 The 6 the kinetic term reads:5 K = 1 u w G µν gQK uw ∂µ q ∂ν q . 2k2 (2.1) Here, Gµν is the spacetime metric and xµ the associated coordinates, whereas gQK uv is the quaternionic target-space, coordinated with qu . Notice that qu , gQK uv and Gµν are dimensionless, whereas ∂µ have dimension one and K dimension four. The gravity decoupling limit of the quaternion-Kähler space, reached at k → 0, should be taken in a zoom-in manner in order to avoid the trivialization of the geometry into flat space. In that aim we introduce the following redefinitions: b −δ, Z = αZ b, V = δV b, X = αX τ = α δ τb , b, Y =αY b, Ψ = αΨ in the Przanowski–Tod metric (1.24), leading to    1 b 2 2 2 2 αΨ b b b b dX + dY , τ + ω ) + V dZ + e (db b V   b ∧ dZ b, b + Vb dZ b ∧ dX b + V eαΨb dX b ∧ dY dω = VXb dY Y b Z   b Ψb , b = 1 Ψ b + 1 2 − αZ V Z 2 Z 2δ ds2QK and b, ω = αδω α2 δ = b − δ )2 (α Z (2.2)  − uv k2− = k− = uv k From Przanowski–Tod to Boyer–Finley α2 δ 2 . b − δ )2 (α Z (2.3) (2.4) The kinetic term (2.1) with the insertion of (2.3), remains finite in the double-scaling limit α = 1, k → 0, δ → ∞, k2 δ = 1 , µ̃2 (2.5) where µ̃ is an arbitrary finite mass scale. In this limit, (2.1) reads: K = µ̃2 µν HK G guw ∂µ qu ∂ν qw , 2 (2.6) where gHK uv is the hyper-Kähler space in Boyer–Finley frame (1.15) and (1.19), corresponding to the solution of the Toda equation used in the quaternionic metric gQK uw of (2.1). Furthermore, using (2.4) we find that k2− remains non-vanishing in the double-scaling limit (2.5). 5 Out of the full supergravity action, we consider the following part S= Z p − g d4 x (K − V + LEH ) , LEH = 1 k2  R −Λ 2  , where K , V and LEH correspond to the kinetic, potential and Einstein–Hilbert terms respectively. 7 Thus, the original quaternionic isometry is mapped onto a simply holomorphic Killing vector ∂τ . Other isometries of the quaternionic metric also survive in the hyper-Kähler limit, if they commute with ∂τ . Additional isometries may also exist. Several remarks are in order here regarding the implementation of the double-scaling limit (2.5). This limit consists of zooming-in around Z → −∞ in the quaternion-Kähler space. The latter is described, in the Przanowski–Tod representation, by a solution Ψ( X, Y, Z ) of the Toda equation, and Z → −∞ is the fixed locus of the isometry generated by ∂τ , as the norm of this Killing vanishes in that limit. The double-scaling limit (2.5) is consistent prob Z b αZ b ( X, b Y, b ) = 1 Ψ(α X, b αY, b − δ) introduced in (2.2) makes sense when α → 1 and vided Ψ α δ → ∞. Being a solution of a partial differential equation, Ψ( X, Y, Z ) is actually a function of X + X0 , Y + Y0 , Z + Z0 , where X0 , Y0 , Z0 are arbitrary constants. This freedom makes it pos- sible to tune Z0 so as to absorb δ before the limit is taken. In this way, the limit does not affect Ψ( X, Y, Z ) and the very same function can thus be used on the two sides of the limit. This is why the double-scaling limit under consideration is equivalent to the one-to-one correspondence among quaternionic and hyper-Kähler spaces with symmetry set in the beginning of the present section, and based on the use of a given solution of the Toda equation. From this perspective, the relationship at hand can either be interpreted as a decoupling of gravity when starting from a quaternion-Kähler space, or as a coupling to gravity, when starting from a hyper-Kähler σ-model endowed with a simply holomorphic Killing vector sustaining a rigid N = 2 model. It should finally be stressed that the double-scaling limit under investigation can be taken for any isometry of the quaternion-Kähler space. This provides as many decoupling limits of gravity as symmetries in the σ-model, not all inequivalent though. We will come back to that in the examples of Sec. 4 and App. A, when discussing in particular the fate of the symmetries along the decoupling. From Przanowski–Tod to Gibbons–Hawking Starting with a quaternion-Kähler space with symmetry in the Przanowski–Tod representation, we can reach in some cases another hyperKähler space. This zoom-in limit is not necessarily taken in the same area of the manifold as the previous limit.6 Instead of the double-scaling limit (2.5), we perform the following triple-scaling limit on the kinetic term (2.1) with redefinitions (2.2): k → 0, α → 0, δ → ∞, k2 δ 1 = 2. 2 µ̃ α (2.7) The kinetic term is still given by (2.6), where now gHK uv is a hyper-Kähler space in the Gibbons– Hawking frame (1.15), (1.16). Hence, the original quaternionic isometry generated by ∂τ becomes triholomorphic in the hyper-Kähler limit, where indeed k2− and k− uv vanish (see (2.4)). 6 See the discussion at the end of Section 4.1. 8 Again, additional isometries may exist, as in the previous double-scaling limit. The rigid limit under consideration is again a zooming-in around Z → −∞, where the norm of ∂τ vanishes, further restricted to X = Y = 0. The limiting hyper-Kähler space exists b αZ b = 1 ∂ b Ψ(α X, b αY, b ) makes sense when α → 0. This requirement sets restricas long as V 2α Z tions to the Przanowski–Tod geometries that allow for the triple-scaling limit, contrary to the previous double-scaling limit, which always exists. Despite that, non-trivial examples can be successfully worked out. In conclusion, although the original Przanowski–Tod isometry is generically simply holomorphic in the limiting hyper-Kähler space, the option for a triholomorphic limit exists in certain instances. Alternative limit to Gibbons–Hawking Before closing this section, we would like to mention an alternative possibility for reaching a hyper-Kähler Gibbons–Hawking space from a Przanowski–Tod quaternionic geometry. This limit is peculiar, because when it exists, it always leads to the same space, namely the unique hyper-Kähler invariant under Heisenberg ⋉ U (1) symmetry [16, 33]. Assume that a line ( X0 , Y0 , Z0 ) exists in a quaternion-Kähler space of the Przanowski–Tod type (1.24), such that    V ( X0 , Y0 , Z0 ) = 0 ⇔ Z∂Z Ψ|0 = 2 ,      ω X |0 = ω Y |0 = ω Z |0 = 0 ,   Ψ( X0 , Y0 , Z0 ) = ψ0 , ∂Z ( Z∂Z Ψ) |0 = −2ψZ ,     ∂ ( Z∂ Ψ) | = −2ψ eψ0/2 , ∂ ( Z∂ Ψ) | = −2ψ eψ0/2 , 0 0 Z Y Z X Y X (2.8) with ψ0 , ψX , ψY and ψZ finite, potentially vanishing constants. In the neighborhood of this b Z: b Y, b line, we define the coordinates X, X = X0 + (kµ̃ ) /3 e− 2 ψ0/2 b, X Y = Y0 + (kµ̃ ) /3 e− 2 ψ0/2 b, Y 2 b, Z = Z0 + (kµ̃ ) /3 Z (2.9) and we expand V and ω at linear order, since we are ultimately interested in the scaling limit k → 0. We find that   2 b + ψZ Z b + ψY Y b = (kµ̃ )2/3 V b, V ≈ (kµ̃ ) /3 ψX X   (2.10) 4 b dZ b = (kµ̃ )4/3 ω b + ψY Z b dX b + ψZ X b dY b. ω ≈ (kµ̃ ) /3 ψX Y Introducing finally τ = (kµ̃ ) /3 τb , 4 (2.11) we can proceed with the decoupling limit k → 0 in the kinetic term (2.1) and we find the 9 rigid limit (2.6) with the hyper-Kähler metric 1 = 2 Z0 ds2HK    1 2 2 2 2 b b dX b + dY + dZ b b) + V τ+ω , (db b V b and ω b are linear in the hated coordinates with where both V bV b ×ω b=∇ b = {ψX , ψY , ψZ } . ∇ (2.12) (2.13) b Z b Y, b } for Thanks to the relation (2.13), it is always possible to trade the coordinates {τb, X, {t, x, y, z}: −2/3 t = Z0 r0/3 1 −2/3 r0/3 −2/3 r0/3 −4/3 r0 x = Z0 y = Z0 z = Z0 1 1   b , b + sin ϕ0 Y b + sin ϑ0 cos ϕ0 X cos ϑ0 Z    b b b − sin ϑ0 Z + cos ϑ0 cos ϕ0 X + sin ϕ0 Y ,   b , b + cos ϕ0 Y − sin ϕ0 X  −1/3 where f = (τb + f ) ,     1 b sin2 ϕ0 cos ϑ0 X b 2 + r0 Y b + cos ϕ0 sin ϑ0 Z b b2 − Y r0 cos ϑ0 sin 2ϕ0 X 4 and (r0 , ϑ0 , ϕ0 ) are constants defined as ψX = r0 sin ϑ0 cos ϕ0 , ψY = r0 sin ϑ0 sin ϕ0 , ψZ = r0 cos ϑ0 . The metric thus reads: ds2HK =
1 (dz + x dy)2 + t dt2 + dx2 + dy2 , t (2.14) which is the unique hyper-Kähler space invariant under Heisenberg ⋉ U (1) symmetry [16, 33], generated by (X , Y , Z , M ) obeying [X , Y ] = Z , [M , X ] = Y , [M , Y ] = − X , (2.15) and realized as X = ∂ x − y∂z , Y = ∂y , Z = ∂z , M = y ∂ x − x ∂y +
1 2 x − y2 ∂ z . 2 (2.16) The Killing fields X , Y , Z are triholomorphic (translational) whereas M is simply holomorphic (rotational). If conditions (2.8) are met, the gravity decoupling limit under consideration provides the 10 specific hyper-Kähler space (2.14). This occurs for example in the two-parameter family of U (1) × U (1)-symmetric quaternion-Kähler spaces obtained by quaternionic quotient based upon gauging Y and Z inside the Sp(2, 4) of the N = 2 hypermultiplet manifold [16]. This family of quaternion-Kähler spaces contains the sub-family of the Heisenberg ⋉U (1) spaces resulting from a Z gauging [3]. Finally, it is useful to exhibit Kähler coordinates for the hyper-Kähler space (2.14). There are at least two inequivalent Kähler coordinate systems adapted to the isometry at hand. In the first one, the action of M is not holomorphic: Φ = t +iy, T = −t x + i z , K= T+T
2 +
3 1 Φ+Φ , 12 Φ+Φ X = − Φ ∂ T − Φ ∂ T , Y = i ( ∂Φ − ∂Φ ) , Z = i ( ∂ T − ∂ T ) , ! 2 

2 T+T 1 1 1 T+T Z, − M = Y + Φ−Φ X + Φ−Φ 2i 2 4 Φ+Φ Φ+Φ (2.17) while it is holomorphic in the second:  
1 1 2 2 2 Ψ = x + iy , U = xy+z , 2t − x − y + i 4 2 1 4 3 Q = U + U + ΨΨ = t2 , K = Q /2 , 2 3
1 i 1 Ψ∂U − Ψ∂U + i (∂Ψ − ∂Ψ ), X = − Ψ∂U − Ψ∂U + ∂Ψ + ∂Ψ , Y = 2 2 2
Z = i (∂U − ∂U ) , M = −i Ψ∂Ψ − Ψ∂Ψ . (2.18) 3 The scalar potential 3.1 Potential and spectrum The scalar potential of N = 2 supergravity theories is obtained by gauging one or sev- eral symmetries realized as isometries on the quaternion-Kähler geometry. These isometries act on the components of hyper- and vector multiplets. The gauging procedure involves in general the graviphoton and possibly other gauge fields in vector multiplets. Here, we will gauge isometries of the hypermultiplet σ-model using only the graviphoton as gauge field. Despite the obvious limitations of such a choice (e.g. partial breaking into N = 1 is impossible without vector multiplets: the massive N = 1 gravitino multiplet includes two massive spin-one fields), its analysis is rich and instructive, as we will see in the following. When considering extra vector multiplets, the output of the gauging depends on whether the isometry acts or not on the vectors at hand: when it does not, one commonly obtains a run-away behaviour, alternatively the scalar potential is more intricate and no generic conclusion can be drawn a priori. We leave this investigation for the future. 11 In N = 2 supergravity, the choice of the symmetry to be gauged is free. In particular, the concept of translational versus rotational isometries is not pertinent in the quaternionKähler target space. This is in contrast with the global-supersymmetry case, and one of our purposes is to analyze how this distinction emerges in the gravity-decoupling limit. As we will see, it is intimately linked to the background spacetime geometry (Minkowski or AdS) dictated by the scalar potential. The gauging procedure works as follows. The hypermultiplet metric is quaternionKähler and admits three complex structures satisfying the quaternionic algebra (1.2). For each isometry generated by a Killing field ξ, one defines the corresponding Killing prepotentials [20]: PI = − 1 ( J I ) uv ∇ u ξ v . 4k2 (3.1) Once an isometry is selected, its gauging with the graviphoton produces a superpotential W expressed in terms of the Killing prepotentials as: W2 = 3 ∑ PI PI . (3.2) I =1 The corresponding scalar potential Vξ takes the form [21]
Vξ = k2 | X 0 |2 −6W 2 + 4guv ∂u W∂v W . (3.3) In the latter expression, X 0 is the compensator for dilation symmetry, gauge-fixed to 1/k. Using (3.1), (3.2), the quaternion algebra (1.2) and some of the identities (1.5), we find (the gravitational coupling k2 should not be confused with the square of the anti-self-dual covariant derivative of the Killing k2− ) W2 = and its derivative7 ∂x W 2 = 1 2 k , 4k4 − (3.4) 1 − y k ξ . k4 xy (3.5) Hence, we retrieve guv ∂u W∂v W = 1 1 1 − x y guv ∂u W 2 ∂v W 2 = 4 2 guv k− guv ξ u ξ v . ux k vy ξ ξ = 2 4W 4k4 k k− (3.6) Expressions (3.4) and (3.6) enable us to produce the following equivalent expressions for the 7 We
− uv = 4k − ξ y . have used (1.23): ∂ x k2− = ∇ x k− uv k xy 12 scalar potential (3.3): 
  k4 −6W 2 + 4 guv ∂u W∂v W ,   3 1  Vξ = 4 −6 k4 ∑ PI PI + guv ξ u ξ v , k  I =1     − 3 k2 + g ξ u ξ v . uv 2 − (3.7) Given the scalar potential (3.7), it is straightforward to investigate its supersymmetric vacuums. Those obey h∂u W i = 0 ⇔ hξ u i = 0 (or equivalently h guv ξ u ξ v i = 0 – the brackets mean as usual that the quantity under consideration is evaluated at the vacuum). The cosmological constant is related to the – generically non-vanishing – value of the potential at the extremum, 1 3 Λ = hVξ i = − 4 hk2− i. 2 k 2k (3.8) Expanding around the vacuum allows to determine the mass matrix: M uv = 1 k2 ux − uw − h g ∂x ∂v Vξ i = 2 k+uw k+ kvw − k+uw k− vw − 2k vw . 2 k (3.9) Equation (3.8) shows that the spacetime has negative curvature R = 4Λ, whereas Eq. (3.9) describes the spectrum of an N = 2 chiral multiplet in AdS4 [23, 24]. Indeed, using the identities (1.5) the mass matrix can be diagonalized with eigenvalues M2B = m2 − 2µ2 + mµ , M2A = m2 − 2µ2 − mµ , m2 = 1 h k2 i , 2k2 + µ2 = 1 Λ hk2− i = − . 2 2k 3 (3.10) Recapitulating, hk2+ i and hk2− i control respectively the physical mass m of the chiral multiplet and the cosmological constant Λ of the anti-de Sitter spacetime. The described vacuum is stable as it is supersymmetric. Furthermore it satisfies the Breitenlohner–Freedman stability bound [23, 24] M2A,B > 3 Λ 4 =⇒  m± µ 2 > 0. 2 (3.11) In (3.10), M2A,B is the coefficient of the Lagrangian mass term. A field with a shift symmetry or a flat direction of the potential correspond to M2A = 0 or M2B = 0. This does not mean that the field is massless. In AdS4 space, a field propagating on the lightcone has a Lagrangian mass term with coefficient 2Λ/3 = −2µ2 . We may then have a hypermultiplet with two flat directions and two massless scalars, if m = ±µ. 13 3.2 The decoupling limits So far our analysis has been confined in the supergravity framework, i.e. with coupling to gravity. We would like now to investigate the decoupling limit, and in particular the behaviour of the above scalar potential (3.7) in the rigid limits presented in Sec. 2. In these decoupling limits as they emerge in the analysis of the kinetic term, the Killing field supporting the fiber of the quaternion-Kähler space in its Przanowski–Tod representation (1.24), ∂τ , plays a preferred role. In particular, the zooming-in is triggered around distinguished points, where the norm of this vector is singular. However, in general, this specific Killing field needs not be the one that enters the gauging procedure. Consequently, the behaviour of the potential in the decoupling limits associated with the kinetic term is not unique and also depends on the isometry chosen for gauging. In the present general analysis, we do not make any assumption regarding extra isometries in the quaternion-Kähler space. Hence, we limit our discussion to the gauging of the graviphoton along the shift isometry ξ = g ∂τ fibering the Przanowski–Tod metric (1.24), with g a dimensionless gauge coupling. In Sec. (4), when dealing with a specific quaternionic space, we will use the option of gauging isometries other than the one carrying the fiber and being responsible for the decoupling in the kinetic term. Using (1.25) with ξ = g ∂τ , we find for the scalar potential (3.7): g2 Vξ = 4 k  3 1 − 2+ 2 2Z Z V  . (3.12) Our goal is to investigate the behaviour of the latter expression when k → 0. As already discussed extensively in Sec. 2 for the kinetic term, this rigid limit must be taken in a zoomin manner. Performing the redefinitions as stated in (2.2), we obtain: gb2 α2 δ2 Vξ = k4 1 3 + − 2 b b b 2( α Z − δ ) ( α Z − δ )2 δV ! , gb = g . αδ (3.13) Performing the double-scaling limit (2.5) on (3.13), assuming that g → ∞ while keeping the coupling constant gb finite, yields the potential8 From Przanowski–Tod to Boyer–Finley Vflat gb2 µ̃2 = 2 k  1 b − 3Z b V  , (3.14) in the presence of a non-vanishing cosmological constant Λ=− 8 The 3b g2 . 2k2 (3.15) subscript “flat” refers to the Ricci-flat nature of the hypermultiplet target space – not to the spacetime. 14 Before proceeding with the alternative rigid limit, we would like to pause and make contact with the general results of Butter and Kuzenko on N = 2 supersymmetric σ-models in AdS4 [25, 26]. According to these authors, rigid N = 2 supersymmetry in AdS4 spacetime requires the target space of the σ-model be a non-compact hyper-Kähler manifold endowed with a simply holomorphic isometry. Let ξ be the simply holomorphic Killing field, and assume a basis for the complex structures obeying (1.11), so that the preserved one is J3 . The latter provides a globally defined Killing potential K as in (1.12). Following Butter and Kuzenko, the scalar potential reads: 2 2 VBK = µ µ̃   1 uv g ∂u K ∂v K − 3 K , 2 HK Λ = −3µ2 . (3.16) Consider a generic hyper-Kähler metric as in Eqs. (1.15) and (1.19), with a rotational isometry ξ = ∂τ and Killing potential (1.21). Inserting the latter into (3.16), we recover (3.14) with gb µ= √ . 2 k From Przanowski–Tod to Gibbons–Hawking We now turn to the triple-scaling limit (2.7), for which the Killing field becomes translational in the hyper-Kähler space. Performed on the scalar potential (3.13), while keeping gb finite, this limit yields Vflat = gb2 µ̃2 . b k2 V (3.17) The Killing ∂τ is translational in the limit at hand, k2− vanishes and so does Λ. The result (3.17) is in agreement with the potential of a hypermultiplet of global N = 2 supersymmetry in Minkowski spacetime [32]. Along the same lines of thought, the alternative decoupling limit (2.9) on (3.12) yields Vflat = where gb = g ( kµ̃)4/3 gb2 µ̃2 b k2 Z02 V , (3.18) is a finite coupling constant when k → 0. 4 Spaces with Heisenberg isometry 4.1 The kinetic term and its three distinct rigid limits As already mentioned in the introduction, Heisenberg symmetry has a distinguished role. Firstly, only two hyper-Kähler spaces exist with this isometry group [33], and one quaternionKähler [16]. Secondly, the latter captures the type-II string one-loop perturbative corrections to the hypermultiplet scalar manifold [4], and can be derived through the quaternionic quotient method by gauging the Z isometry inside the Sp(2, 4) of the N = 2 hypermulti15 plet manifold [3]. The quaternion-Kähler space under consideration has actually extended Heisenberg ⋉ U (1) isometry and its metric reads: ds2QK = 2
2V1 8ρ2 2 2 2 2 dτ + ηdϕ + dρ + dη + dϕ , ( ) V1 V22 V22 2 V2 = ρ − 2σ , V1 = ρ + 2σ > 0 , (4.1) σ = constant. The Heisenberg ⋉ U (1) algebra is realized as [X , Y ] = Z , [M , Y ] = − X , [M , X ] = Y , X = ∂η − ϕ ∂τ , Y = ∂ϕ , M = ϕ ∂η − η ∂ ϕ + Z = ∂τ ,
1 2 η − ϕ2 ∂τ . 2 (4.2) For vanishing σ, this isometry algebra is actually extended to U (1, 2), and the quaternionic space becomes the non-compact SU (2,1)/SU (2)×U (1) (see App. A). In the frame at hand, the space (4.1) is fibered over Z . It was noticed in [3] that the rigid limit of the kinetic term (2.1) with (4.1) is σ-dependent. For σ > 0 two trivial, flat-space rigid limits occur around ρ2 ∼ 0 and ρ2 − 2σ ∼ 0. For σ < 0 a non-trivial limit appears at ρ2 + 2σ ∼ 0. The latter case corresponds precisely to the hyper-Kähler metric (2.14), invariant under the Heisenberg ⋉ U (1) symmetry (2.15). These results can be summarized as follows: - PARAMETRIC REGION σ>0 1. Around ρ2 ∼ 0 we zoom as: ρ= √ σ kµ̃ t , η= √ σ kµ̃ x , ϕ= √ σ kµ̃ y , τ = σz, k → 0, (4.3) and this leads to flat space ds2HK = dx2 + dy2 + dt2 + t2 dz2 , (4.4) where the Killing supporting the fiber in the original quaternionic space, Z ∝ ∂z is here a simply holomorphic (rotational) Killing vector. 2. Around ρ2 − 2σ ∼ 0 we zoom as: 1 η= √ (kµ̃ )2 x , 8σ 1 τ = (kµ̃ )2 z , k → 0 , 2 ρ2 = 2σ + kµ̃ (1 + kµ̃ t) , 1 ϕ= √ (kµ̃ )2 y , 8σ 16 (4.5) and obtain again flat space ds2HK = dx2 + dy2 + dt2 + dz2 , (4.6) where Z ∝ ∂z is now a triholomorphic (translational) Killing vector. - PARAMETRIC REGION σ<0 Around ρ2 + 2σ ∼ 0 we zoom as:   √ 2 2 ρ2 = −2σ 1 + 2 t(kµ̃ ) /3 , η = −2σ x(kµ̃ ) /3 , √ 2 4 ϕ = −2σ y(kµ̃ ) /3 , τ = −2σ z(kµ̃ ) /3 , k → 0 , (4.7) such that the Kretschmann scalar of (4.1), R xuvw R xuvw = 48 (ρ4 + 4σ2 )(ρ8 + 56ρ4 σ2 + 16σ4 ) , (ρ2 + 2σ)6 (4.8) remains finite when k → 0. This limiting procedure results to the hyper-Kähler space (2.14), ds2HK =
1 (dz + x dy)2 + t dt2 + dx2 + dy2 , t (4.9) which is invariant under the Heisenberg ⋉ U (1) symmetry (2.15), (2.16). Therefore the decoupling limit (4.7) preserves all the isometries of the quaternionic ancestor metric (4.1). Along the process, the Killing supporting the fiber, Z ∝ ∂z becomes a triholomorphic (translational) Killing vector. In order to make contact with the general developments presented in Sec. 2, we should recast the quaternionic Heisenberg ⋉ U (1)-invariant metric (4.1) in the Przanowski–Tod form (1.24). This is achieved by keeping τ unaltered, while trading ρ, η, ϕ for X, Y, Z as follows: X = η, Y = ϕ, Z= V2 (ρ) , 2 (4.10) e Ψ = ρ2 . (4.11) and setting ω = η dϕ , V= V1 (ρ) , 2ρ2 The fiber of the Przanowski–Tod is supported by the Killing field Z = ∂τ , which turns rotational or translational in the hyper-Kähler, depending on the decoupling limit. With these conventions, on the one hand, the rigid limits for σ > 0, ρ2 ∼ 0 and ρ2 − 2σ ∼ 0, correspond to (2.5) (Przanowski–Tod to Boyer–Finley i.e. hyper-Kähler limit with rotational Killing Z ) and (2.7) (Przanowski–Tod to Gibbons–Hawking i.e. hyper-Kähler limit with translational Killing Z ). On the other hand, the rigid limit ρ2 + 2σ ∼ 0 for σ < 0 is the alternative Przanowski–Tod to Gibbons–Hawking limit, (2.9) and (2.11), leading to the 17 unique hyper-Kähler space with Heisenberg ⋉ U (1) symmetry, with a translational Killing vector Z supporting the fiber. Notice finally that in all cases, extra Killing fields survive the decoupling limit, either simply holomorphic or triholomorphic, which may be chosen to further recast the hyperKähler metric in another Boyer–Finley or Gibbons–Hawking frame. Such options can be exploited depending on the form of the original scalar potential, and its structure in the decoupling limit. 4.2 The scalar potential The form of the scalar potential depends on which symmetry is gauged, i.e. which element is chosen inside the Heisenberg ⋉ U (1) isometry group of the quaternionic metric (4.1). We will always use the graviphoton for the gauging, as already advertised, and the general formalism of Sec. 3.1. In our analysis, we will systematically investigate the rigid limits. There are always three distinct cases corresponding to the limits (4.3), (4.5) and (4.7), exhibited for the kinetic term (2.1) on the Heisenberg ⋉ U (1)-invariant quaternionic space (4.1), and leading to the hyperKähler spaces (4.4), (4.6) and (4.9). They are associated with the Przanowski–Tod to Boyer– Finley limit (hyper-Kähler limit with rotational Killing), the Przanowski–Tod to Gibbons– Hawking limit (hyper-Kähler limit with translational Killing), and the Przanowski–Tod to Gibbons–Hawking limit with full Heisenberg ⋉ U (1) symmetry. Coming back to the potential term, two distinct situations arise depending on which isometry is gauged: (i) either the field carrying the Przanowski–Tod fiber Z – the potential is Heisenberg ⋉ U (1) invariant, (ii) or any other Killing – the potential is not Heisenberg ⋉ U (1) invariant. In the first instance, the properties of the potential and its associated spectrum at the supergravity level and in the decoupling limits will follow the general classification presented in Sec. 3. For these limits, in particular, we will find (Sec. 3.2) rigid N = 2 supersymmetry with potential (3.14) and anti-de Sitter vacuum, or with potentials (3.17), (3.18) and Minkowski vacuum. The case (ii) is expected to be slightly different, and is worth presenting case by case: M , Z plus M , Y . In the three rigid-supersymmetry limits, we find only anti-de Sitter vacuums for the first, and anti-de Sitter and Minkowski for the second and the third. The Y gauging exhibits a further peculiarity already at the supergravity level, namely a de Sitter N = 0 vacuum, which stands outside of the general analysis of Sec. 3.1, where supersymmetry was unbroken. Gauging the Z isometry Here, the Killing vector supporting the gauging is ξ = g Z (see (4.2)). The scalar potential is obtained thanks to the general formula (3.7): Vξ = 2g2 ρ2 − 6σ , k4 V1 V22 18 (4.12) which is invariant under the action of Heisenberg ⋉ U (1). The potential at hand has an extremum at the origin ρ = 0, the fixed point of ξ, and two flat directions η, ϕ. In order to analyze this extremum, we move from polar to Cartesian coordinates (ρ, τ ) 7→ (q1 , q2 ) : ρ= q
σ q21 + q22 , τ = σ arctan q2 , q1 (4.13) as ρ = 0 is a coordinate singularity of (4.1). Expanding (4.12) at second order around the extremum: gb2 Vξ ≈ 4 k  3 q2 + q22 − − 1 2 2  , (4.14) where gb = g/σ. From the latter, and normalizing with respect to the kinetic term, Eqs. (2.1) and (4.1), we read off the mass terms and the cosmological constant: Mq21 ,q2 = − gb2 2 = Λ, k2 3 2 Mη,ϕ = 0, Λ=− 3b g2 . 2k2 (4.15) These satisfy the Breitenlohner–Freedman stability bound, and fit the general form (3.10) of the mass spectrum of an N = 2 chiral multiplet in AdS4 spacetime, with A = {q1 , q2 }, B = {η, ϕ} and µ = m = √ gb . 2k Besides the flat directions η, ϕ, the fields q1,2 are massless. Next task in our agenda is to analyze the behaviour of the potential (4.12) in the three distinguished rigid limits (4.3), (4.5) and (4.7). - PARAMETRIC REGION σ>0 1. Applying the rigid limit (4.3) to (4.12), we find the potential Vflat = − gb2 µ̃2 2 t . 2k2 (4.16) To this end, we use the generic expression for the potential in the Przanowski– Tod to Boyer–Finley rigid limit, Eq. (3.14), after rewriting the flat space (4.4) in the Boyer–Finley frame along the rotational isometry Z = ∂τ = σ1 ∂z , for which 1 b = t2 . = 2Z b V (4.17) In the present rigid limit, the vacuum is thus an AdS4 spacetime, and the hyperKähler space describes a global N = 2 hypermultiplet. 2. The rigid limit under consideration is now (4.5). Applied to (4.12), it leads to a global N = 2 hypermultiplet in Minkowski spacetime with an irrelevant constant 19 potential Vflat = − where ge = - PARAMETRIC 2σ gb ( kµ̃ )2 REGION ge2 µ̃2 , 2k2 (4.18) is a finite coupling constant when k → 0. σ<0 The scalar potential (4.12) becomes Vflat = gb ge2 µ̃2 , k2 t (4.19) in the rigid limit (4.7), with ge = − 2(kµ̃)4/3 a finite coupling constant at k → 0. Again, the cosmological constant vanishes and we describe a global N = 2 hypermultiplet in Minkowski spacetime. Gauging the M isometry Consider now the isometry of (4.1) generated by ξ = g M . Using (3.7) we find the following scalar potential: g2 Vξ = 4 k  
V2 − 4σ 2
V1 2 3 2 2 2 η +ϕ , − − 2 η +ϕ + 2 V2 2V1 V22 (4.20) invariant under the action of M and Z . This potential has an extremum at the fixed point of ξ, η = ϕ = 0, and two flat directions ρ, τ. We can directly read off the mass terms and the cosmological constant (again normalization with respect to the kinetic term is required): 2 Mρ,τ = 0, 2 Mη,ϕ =− g2 2 = Λ, 2 k 3 Λ=− 3g2 . 2k2 (4.21) Comparing with the general expression (3.10), we identify the fields A = {η, ϕ} and B = {ρ, τ }, whereas µ = m = √g2 k . We find again the spectrum of an N = 2 chiral multiplet in AdS4 , with two massless fields ρ, τ and two massive fields η, ϕ. Let us now consider the usual rigid limits when the dynamics is captured by the potential (4.20). - PARAMETRIC REGION σ>0 The pattern goes as in the previous gauging. We first consider the rigid limits (4.3) or (4.5). We use again Eq. (3.14) after rewriting the flat space (4.4) or (4.6) in the Boyer– Finley frame along the rotational isometry M , with 1 = x 2 + y2 , b V
b = 1 x 2 + y2 . Z 2 3g2 (4.22) We thus find for both limits AdS4 spacetime with Λ = − 2k2 . The hyper-Kähler space 20 describes a a global N = 2 hypermultiplet with potential Vflat = − - PARAMETRIC REGION
g2 µ̃2 x 2 + y2 . 2 2k (4.23) σ<0 Applying the rigid limit (4.7) to (4.20), we utilize Eq. (3.14) after rewriting (2.14) in the Boyer–Finley frame along the rotational isometry M [16]: 1 1 = ( x 2 + y2 ) 2 + t ( x 2 + y2 ) , b 4t V b = 1 t ( x 2 + y2 ) . Z 2 (4.24) We now find a global N = 2 hypermultiplet in AdS4 spacetime with with potential Vflat g2 µ̃2 =− 2 2k  
1 2 2 2 t( x + y ) − x +y , 2t 2 2 (4.25) 3g2 and cosmological constant Λ = − 2k2 . A comment is worth here on the Killing potential K. This is determined using (1.21) and (4.24): b = t ( x 2 + y2 ) . K = 2Z (4.26) In the Kähler coordinates (2.17), the Killing potential K reads: K= 1 K+W +W, 2 1 W = − Φ3 . 6 (4.27) Hence K appears as a Kähler potential and W is a Kähler transformation. Notice also that in the set of Kähler coordinates (2.17), the complex structure J1 is diagonalized while M is not holomorphic. Gauging Z and M isometries This gauging is performed along ξ = g1 Z + g2 M . The scalar potential is determined with (3.7) and (4.1), and is invariant under the action of M and Z : Vξ  
V2 − 4σ 2
2g12 ρ2 − 6σ g22 V1 2 3 2 2 2 = η +ϕ + 4 − − 2 η +ϕ + k4 V1 V22 k 2 V2 2V1 V22
η 2 + ϕ2 (V2 − 4σ) + 3V1 V2 . −2g1 g2 k4 V1 V22 (4.28) This potential has an extremum at ρ = η = ϕ = 0, the fixed point of ξ. To analyze this extremum, we change coordinates from polar to Cartesian as in (4.13), and we expand (4.28) 21 at second-order around the extremum   3( gb1 − g2 )2 1 1 g2 2 2 2 2 Vξ ≈ 4 − + gb1 (3g2 − gb1 )(q1 + q2 ) + (3b g1 − g2 )(η + ϕ ) , 2 2 2σ k (4.29) where gb1 = g1/σ. The mass terms and the cosmological constant are obtained as usual: Mq21 ,q2 = g1 (3g2 − gb1 ) , k2 2 Mη,ϕ = g2 (3b g1 − g2 ) , k2 Λ=− 3( gb1 − g2 )2 , 2k2 (4.30) and satisfy the Breitenlohner–Freedman stability bound. For gb1 6= g2 , this vacuum generi- cally describes the spectrum of a hypermultiplet in AdS4 and fits the general form (3.10) of the mass spectrum with A = {η, ϕ}, B = {q1 , q2 } and g2 − gb1 µ= √ , 2k gb1 + g2 m= √ . 2k (4.31) We now come to consider the rigid limits of the kinetic term (2.1), (4.1) on the potential (4.28). - PARAMETRIC REGION σ>0 1. Applying the rigid limit (4.3) to (4.28), we obtain: Vξ = − µ̃2 3 ( gb1 − g2 )2 − 2k4 2k2
gb12 t2 + g22 ( x2 + y2 ) − 3b g1 g2 (t2 + x2 + y2 ) . (4.32) Two distinct cases should be examined, when gb1 6= g2 or gb1 = g2 : (a) For gb1 6= g2 , we find a global N = 2 hypermultiplet in AdS4 spacetime with Λ=− 3( gb1 − g2 )2 2k2 and potential Vflat = − µ̃2 2k2
g1 g2 (t2 + x2 + y2 ) . gb12 t2 + g22 ( x2 + y2 ) − 3b (4.33) To prove this, we utilize (3.14) after rewriting the hyper-Kähler metric (4.4) in the Boyer–Finley frame along the rotational isometry ξ = gb1 Z + g2 M . We find

b = 1 ( gb1 − g2 ) gb1 t2 − g2 x2 + y2 . Z 2
1 = gb12 t2 + g22 x2 + y2 , b V (b) For gb1 = g2 , we obtain the potential:9 Vflat = 9
µ̃2 gb12 2 t + x 2 + y2 , 2 k (4.34) For gb1 = g2 , the isometry is translational, in agreement with a theorem in [19] on commuting rotational isometries, and the rigid limit yields a global N = 2 hypermultiplet in Minkowski spacetime. 22 which corresponds to a massive N = 2 hypermultiplet in Minkowski spacetime. Indeed this can be easily seen by changing coordinates in (4.4) from polar (t, z) to Cartesian ones (q1 , q2 ), as t = 0 is a coordinate singularity of the hyper-Kähler metric (4.4). 2. We now consider the rigid limit (4.5) to (4.28). We rewrite the hyper-Kähler metric (4.6) in the Boyer–Finley frame along the rotational isometry ξ = ge1 Z + g2 M and use (3.14):
1 = ge12 + g22 x2 + y2 , b V
b = 1 ge12 + g22 ( x2 + y2 ) − 2e Z g1 g2 t . 2 (4.35) We obtain a global N = 2 hypermultiplet in AdS4 spacetime with potential Vflat = − 3g2 and Λ = − 2k22 , where ge1 = - PARAMETRIC REGION σ<0 µ̃2 2k2 2σ gb1 ( kµ̃)2
g1 g2 t + g22 ( x2 + y2 ) , ge12 − 6e (4.36) is a finite coupling constant when k → 0. The last rigid limit is (4.7) applied to (4.28). Now, with (3.14), we express the hyperKähler metric (4.9) in the Boyer–Finley frame along the rotational isometry ξ = ge1 Z + g2 M : g1 g2 ( x2 + y2 ) + g22 ( x2 + y2 )(4t2 + x2 + y2 ) 4e g2 − 4e 1 = 1 , b 4t V
b = 1 g2 t −2e g1 + g2 ( x2 + y2 ) . Z 2 (4.37) 3g2 Again, we find a global N = 2 hypermultiplet in AdS4 spacetime with Λ = − 2k22 and potential Vflat =
µ̃2 2 2 2 2 2 2 2 2 2 2 ( 2t − x − y )( x + y ) , + 4e g g ( 3t − x − y ) − g 4e g 2 1 2 1 4tk2 (4.38) gb where ge1 = − 2(kµ̃1)4/3 is a finite coupling constant when k → 0. Gauging the Y isometry This last gauging is based on the isometry ξ = g Y and leads to the scalar potential Vξ = 2g2 η 2 (V2 − 4σ) − 2V1 (V2 + σ) , k4 V1 V22 (4.39) invariant under the action of Y and Z . The potential (4.39) has an extremum at ρ = η = 0, and a flat direction ϕ. As usual, we change coordinates from polar to Cartesian, see Eqs. (4.13), and we expand (4.39) at second 23 order Vξ ≈ g2 3g2 η 2 − 2 4, 4 σk 2σ k (4.40) where σ > 0, required by positivity of the metric (4.1) around the extremum at hand. Using the latter and normalizing with (4.1), we read off the mass terms and the cosmological constant: Mη2 = − 3g2 , σk2 M2ϕ = Mq21 = Mq22 = 0 , Λ= g2 . σk2 (4.41) As advertised earlier on, this corresponds to an N = 0 hypermultiplet in dS4 spacetime with R = 4Λ. Our next task is the analysis of the usual rigid limits on the potential (4.39). - PARAMETRIC REGION σ>0 1. The rigid limit (4.3) applied to (4.39) leads to a global N = 2 hypermultiplet in g gb2 µ̃2 Minkowski spacetime with constant potential Vflat = k2 , where gb = √σ kµ̃ is a finite coupling constant when k → 0. 2. In the alternative rigid limit (4.5), we find again a global N = 2 hypermultiplet gb2 µ̃2 in Minkowski spacetime (with another constant potential Vflat = − 2k2 , and gb = √ 8σ g ( kµ̃)2 - PARAMETRIC a coupling constant, finite in the decoupling). REGION σ<0 Similarly the rigid limit (4.7) of the potential (4.39) gives a global N = 2 hypermultiplet in Minkowski spacetime with potential Vflat = where gb = √ g −2σ ( kµ̃)4/3 gb2 µ̃2 (t2 + x2 ) , k2 t (4.42) remains finite when k → 0. It is worth mentioning, that this potential could be obtained from the N = 1 expression V = µ̃2 ab K Wa W b , k2 (4.43) expressed in terms of a linear holomorphic superpotential: W = gb T , in the Kähler basis (2.17). This linear superpotential breaks supersymmetry as the kinetic term is non-canonical. Conclusion We can now highlight and summarize our results. The core of the present work is the investigation of various off-shell gravity decoupling limits of the N = 2 scalar hypermultiplet. 24 This concerns at the first place the kinetic term, based on a σ-model which has a quaternionKähler target space. Analyzing the decoupling limit, establishes various relationships between quaternionic and hyper-Kähler spaces with symmetry. Equally important is the behaviour of the scalar potential, produced by gauging symmetries with specific vectors. For the general analysis, we have chosen the graviphoton, along a generic isometry of a quaternion-Kähler space of the Przanowski–Tod type. The rigid limits reveal two separate cases: a rigid N = 2 theory on Minkowski or on AdS4 spacetime, depending on whether the isometry is translational or rotational in the hyper-Kähler limit. These results are in agreement with previous results in the literature for global N = 2 in Minkowski and AdS4 spaces [2] and [25, 26]. In order to illustrate our general results, we analyzed extensively the quaternionic metric with Heisenberg ⋉ U (1) isometry, Eq. (4.1). The global N = 2 limits of this space are found to be trivial (flat space) or the hyper-Kähler space (4.9), which is invariant under the Heisenberg ⋉ U (1) symmetry (2.15), (2.16). We further derived the scalar potential by gauging the graviphoton along all possible isometries of the quaternion-Kähler space, (Y , Z , M ) and studied the vacuum structure of the scalar potential together with its on/off-shell rigid limits. Interesting open questions remain at this stage, which have not been addressed in our work. An important one is to gauge isometries of the hypermultiplet σ-model using the graviphoton and a vector multiplet. The latter combination can give access to vacuum solutions which describe the spectrum of N = 1 hypermultiplets in Minkowski or AdS4 spacetimes. Acknowledgements We would like to thank Jean-Claude Jacot for participation in the early stages of this project. We also acknowledge each-other institutes for hospitality and financial support. This work was partially supported by the Germaine de Staël Franco–Swiss bilateral program 2015 (project no 32753SG). K. Siampos would like to thank the TH-Unit at CERN for hospitality and financial support during various stages of this project. He would like also to thank the ICTP, Trieste and the Physics department of the National and Kapodistrian University of Athens for hospitality. 25 A Pseudo-Fubini–Study metric f2 = We shall utilize the results of Secs. 2 and 3 for the pseudo-Fubini–Study metric on CP SU (1,2)/SU (2)×U (1) ds2QK = 2gab dza dzb , gab = K ab ,
K = − ln 1 − |z|2 − |w|2 , za = (z, w) , (A.1) This is a Kähler–Einstein–WSD space, with R = −12, which describes the universal hyper- multiplet at string tree-level. The Kähler potential is invariant under the action of ξ = α i ( z ∂z − z ∂z ) + β i ( w ∂w − w ∂w ) , α, β ∈ R , (A.2) and through Eq. (3.7) we find the scalar potential (1 − r22 )(4r12 + 3r22 − 3)α2 + (1 − r12 )(4r22 + 3r12 − 3) β2 + 2αβ(3 − 3r12 − 3r22 + r12 r22 ) , 2k4 (1 − r12 − r22 )2 (A.3) 2 2 2 2 with r1 = |z| and r2 = |w| . This potential has an extremum at (z, w) = (0, 0), fixed point of ξ. Expanding it around this point we find at second order Vξ = 1 Vξ ≈ 4 k  3 − (α − β)2 + α (3β − α) r12 + β (3α − β) r22 2  , (A.4) and we read off the masses and the cosmological constant which satisfy the Breitenlohner– Freedman stability bound Mr21 = α (3β − α) , k2 Mr22 = β (3α − β) , k2 Λ=− 3( α − β ) 2 . 2k2 (A.5) This vacuum generically describes the spectrum of a chiral multiplet in an AdS4 spacetime, similarly to (3.10) Mr21 = m2 − 2µ2 − mµ , with Λ = −3µ2 , where: α−β , µ= √ 2k Mr22 = m2 − 2µ2 + mµ , (A.6) α+β m= √ . 2k (A.7) We can now analyze the rigid limit around z, w ∼ 0. The kinetic term (2.1) corresponding to (A.1), has a unique trivial gravity decoupling limit around z, w ∼ 0, where we zoom as: z = kµ̃ b z, b, w = kµ̃ w 26 k → 0, (A.8) leading to flat space   b , b dw ds2HK = 2 db z db z + dw (A.9) while the potential (A.3) truncates to: Vξ = −
3( α − β ) 2 µ̃2 2 2 b b z | + β 3α − β | w | + α 3β − α | . ( ) ( ) 2k4 k2 (A.10) Two cases need to be considered, for α 6= β or α = β : 1. For α 6= β, we find a global N = 2 hypermultiplet in AdS4 spacetime with potential Vflat =
µ̃2 2 2 b b α 3β − α | z | + β 3α − β | w | . ( ) ( ) k2 and a non-vanishing cosmological constant Λ = − 3( α − β ) 2 2k2 (A.11) . To prove this, we utilize (3.14) after rewriting the hyper-Kähler metric (A.9) in the Boyer–Finley frame along the rotational isometry (A.2)
1 b |2 , = 2 α2 | b z |2 + β2 | w b V
b = (α − β) α |b b |2 . Z z |2 − β | w (A.12) 2. For α = β, we have a global N = 2 hypermultiplet in Minkowski spacetime with potential: Vflat =
2µ̃2 2 b |2 , α |b z |2 + | w 2 k (A.13) which corresponds to a massive N = 2 hypermultiplet in Minkowski spacetime. B Four-dimensional Kähler spaces with an isometry We are interested in providing an alternative exhibition of generic, Ricci-flat, scalar-flat or Einstein four-dimensional Kähler spaces with a holomorphic isometry. Hyper-Kähler (Ricciflat), scalar-flat or Einstein solutions appear as Gibbons–Hawking like metrics. For generic and Ricci-flat four-dimensional Kähler space with a holomorphic isometry, see also Ref. [34]. B.1 Four-dimensional Kähler spaces We begin with Kähler complex coordinates T, Φ, with an isometry acting as a shift of ImT and so the Kähler potential takes the form K = K ( T + T, Φ, Φ). A simple rearrangement 27 yields ds2Kähler = K TT d TdT + K TΦ dTdΦ + KΦT dΦdT + KΦΦ dΦdΦ    
2 1 i 1 2 + = K TT dIm T + K TΦ dΦ − K TΦ dΦ (dK T ) + D dΦdΦ , 2K TT K TT 4 (B.1) where D is the determinant of the Kähler metric D := det K ab = K TT KΦΦ − K TΦ K TΦ . (B.2) Next we define a new coordinate Z as K T := 1 ( Z + c) , 2 (B.3) where c is an integration constant that can be absorbed by a Kähler transformation K 7→ K +
c T+T . 2 We may think of (B.3) defining a Legendre transformation as H ( Z + c, Φ, Φ) = with transformation relations HZ = which lead to10 K TT = 1 , 4HZZ
1 ( Z + c) T + T − K ( T + T, Φ, Φ) , 2
1 T+T , 2 K TΦ = − HZΦ , 2HZZ HΦ = − K Φ , K TΦ = − HZΦ , 2HZZ HΦ = − K Φ , KΦΦ = HZΦ HZΦ − HΦΦ , HZZ (B.4) (B.5) (B.6) and the determinant (B.2) equals D=− HΦΦ . 4HZZ (B.7) Putting altogether we find that the line element (B.1) rewrites as ds2Kähler = V = HZZ , τ = ImT , 10 Using 1 4   
 1 , (dτ + ω )2 + V dZ2 + eΨ dX 2 + dY 2 V HXX + HYY , ω = HZY dX − HZX dY , eΨ = − HZZ (B.8) Φ = X +iY,

the Jacobian matrix for the Legendre transformation T + T, Φ, Φ 7→ Z, Φ, Φ , implied by (B.5). 28 with11   dω = VX dY ∧ dZ + VY dZ ∧ dX + V eΨ dX ∧ dY . Z (B.9) These results apply to an arbitrary four-dimensional Kähler metric with an isometry. B.2 Hyper-Kähler spaces To describe a hyper-Kähler metric we impose Ricci-flatness on (B.8) R ab = − (ln D) ab = 0 , (B.10) D = e F( T,Φ)+ F( T,Φ) . (B.11) with general solution Demanding invariance under the shift isometry of ImT, we find: D = e−α(T + T) | f (Φ)|2 , α = constant , (B.12) where f (Φ) can be eliminated by a holomorphic redefinition of Φ, here we use f (Φ) = 1/4. Employing (B.5), (B.7) and (B.12), we find: HXX + HYY + e−2α HZ HZZ = 0 . (B.13) Translational isometry For α = 0, Eq. (B.13) simplifies to the Laplace equation in Cartesian coordinates ( X, Y, Z ) HXX + HYY + HZZ = 0 , (B.14) and the line element is given by ds2HK 1 = 4  
1 2 2 2 2 , (dτ + ω ) + V dX + dY + dZ V ∇V = ∇ × ω =⇒ (B.15) 2 ∇ V = 0. parameterizing a hyper-Kähler space with a translational isometry, the Gibbons–Hawking metric [17]. 11 Whose compatibility relation reads: VXX + VYY + VeΨ 29
ZZ = 0. Rotational isometry For α 6= 0, we evaluate Ψ through (B.8) and (B.13) Ψ = −2α HZ . (B.16) Next we differentiate (B.13) with respect to Z and we get the Toda equation   ΨXX + ΨYY + eΨ ZZ = 0, (B.17) and the line element is given by   
 1 1 2 2 2 2 Ψ , = (dτ + ω ) + V dZ + e dX + dY 4 V 1 1 1 V = − ΨZ , ω = − ΨY dX + ΨX dY , 2α 2α 2α   ds2HK ΨXX + ΨYY + eΨ ZZ (B.18) = 0. parameterizing a hyper-Kähler space with a rotational isometry, the Boyer–Finley metric [18]. B.3 Scalar-flat spaces To describe a scalar-flat space we impose vanishing scalar curvature on (B.8)12 R = 2gab R ab = −2gab (ln D) ab = 0 , (B.19) whose general solution reads D = e−α( T + T)+ΣZ | f (Φ)|2 , α = constant , (B.20) where Σ Z is a solution of ∇24d Σ Z = 0 and we use f (Φ) = 1/4. Utilizing (B.5), (B.7) and (B.20), we find: HXX + HYY + e−2α HZ +ΣZ HZZ = 0 . (B.21) Using the above we obtain Ψ through (B.8) Ψ = −2α HZ + Σ Z . 12 So it is Weyl anti-self-dual, according to footnote 3. 30 (B.22) Next we differentiate (B.21) with respect to Z and we get the Toda equation   ΨXX + ΨYY + eΨ ZZ = 1 V eΨ ∇24d Σ Z = 0 , 4 (B.23) and the line element is given by  
 1 2 2 Ψ 2 2 , (dτ + ω ) + V dZ + e dX + dY V   dω = VX dY ∧ dZ + VY dZ ∧ dX + V eΨ dX ∧ dY , Z   Ψ ΨXX + ΨYY + e = 0. ds2LeBrun 1 = 4  (B.24) ZZ parameterizing a scalar-flat Kähler space with an isometry, the LeBrun metric [35]. B.4 Einstein spaces To describe an Einstein metric we impose on (B.8) R ab = − (ln D) ab = Λ K ab , (B.25) whose general solution reads D = e−α(T + T)−Λ K | f (Φ)|2 , α = constant , (B.26) where we use f (Φ) = 1/4. Employing (B.5), (B.7) and (B.26), we find: HXX + HYY + e−2α HZ −Λ K HZZ = 0 , (B.27) Using the above we derive Ψ through (B.8) Ψ = −2α HZ − Λ K . (B.28) Using the latter and Eqs. (B.3), (B.5), (B.8), we find V = HZZ = − ΨZ . Λ ( Z + c) + 2α (B.29) Then we differentiate (B.27) with respect to Z   ΨXX + ΨYY + eΨ ZZ 1 = − Λ V eΨ ∇24d K = −2Λ V eΨ , 4 31 (B.30) where we have used: ∇24d K = 2gab ∇ a ∇b K = 2gab K ab = 2 gab gab = 8 , Γ ab c = Γ ab c = 0 . (B.31) Plugging (B.29) into (B.30) we find the Pedersen–Poon equation   ΨXX + ΨYY + eΨ ZZ = 2Λ ΨZ eΨ , Λ ( Z + c) + 2α (B.32) and the line element is given by   
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