N = 4 gauged supergravity and a IIB orientifold with fluxes

We analyse the properties of a spontaneously broken D = 4, N = 4 supergravity without cosmological constant, obtained by gauging translational isometries of its classical scalar manifold. This theory offers a suitable low energy description of the super-Higgs phases of certain type-IIB orientifold compactifications with 3-form fluxes turned on. We study its N = 3, 2, 1, 0 phases and their classical moduli spaces and we show that this theory is an example of no-scale extended supergravity.


Introduction
Spontaneously broken supergravities have been widely investigated over the last 25 years, as the supersymmetric analogue of the Higgs phase of spontaneously broken gauge theories [1]- [7].
We recall that, when N supersymmetries are spontaneously broken to N < N supersymmetries, then N − N gravitini acquire masses by absorption of N − N Goldstone fermions. The theory in the broken phase will then have N manifest supersymmetries with N − N gravitini belonging to massive multiplets of the residual N supersymmetries. However, unlike gauge theories, the super-Higgs phases of local supersymmetries require more care because these theories necessarily include gravity.
Therefore, by broken and unbroken supersymmetry, we mean the residual global supersymmetry algebra in a given gravitational background solution of the full coupled Einstein equations.
A particularly appealing class of spontaneously broken theories are those which allow a Minkowski background, because in this case the particle spectrum is classified in terms of Poincaré supersymmetry, and the vacuum energy (cosmological constant) vanishes in this background.

71.3
In all these models, the massive vector bosons, partners of the massive gravitini, are again associated to spontaneously broken translational isometries (R 27 ⊂ E 7 (7) in the case of N = 8 supergravity) of the scalar manifold of the unbroken theory.
Many variants of the Scherk-Schwarz breaking and their stringy realization have been studied in the literature [32]- [34].
Spontaneously broken supergravities, by using dual versions of standard extended supergravities, where again translational isometries of the scalar manifold of the ungauged theory are gauged, were studied in [35] as a N > 2 generalization [36] of the original model which allowed the N = 2 −→ 1 hierarchical breaking of supersymmetry.
In the string and M -theory context, no-scale supergravity models were recently obtained as a low energy description of orientifold compactification with brane fluxes turned on [37]- [52]. The natural question arises as to which low energy supergravity corresponds to their description and how the Higgs and super-Higgs phases are incorporated in the low energy supergravity theory.
It was shown in a recent investigation [53], extending previous analysis [15,35,36], that the main guide to study new forms of N -extended gauged supergravities is to look for inequivalent maximal lower triangular subgroups of the full duality algebra (the classical symmetries of a four-dimensional N -extended supergravity) inside the symplectic algebra of electric-magnetic duality transformations [54].
Indeed, different maximal subgroups of the full global (duality) symmetry of a given supergravity theory allow us in principle to find all possible inequivalent gaugings [10,55,56].
In the case of a type-IIB superstring compactified on a T 6 /Z 2 orientifold [42,43] the relevant embedding of the supergravity fields corresponds to the subgroup SO(6, 6) × SL(2, R), which acts linearly on the gauge potentials (six each coming from the N −S and the R−R 2-forms B µi , C µi i = 1 . . . 6). It is obvious that this group is GL(6, R)×SL(2, R), where GL(6, R) comes from the moduli space of T 6 while SL(2, R) comes from the type-IIB SL(2, R) symmetry in ten dimensions. This means that the twelve vectors are not in the fundamental 12 of SO(6, 6) but rather a (6 + , 2) of GL(6, R) × SL(2, R), where the '+' refers to the O(1, 1, ) weight of GL(6, R) = O(1, 1) × SL (6, R). Their magnetic duals are instead in the (6 − , 2) representation. Note that, instead, in the heterotic string the 12 vectors g µi , B µi i = 1 . . . 6 are in the (6 + + , 6 + − ) and their magnetic dual in the (6 − + , 6 − − ) representation, where the lower plus or minus refers to the R of GL(6, R) and the upper plus or minus refers to the R of SL(2, R).
The gauged supergravity, corresponding to this symplectic embedding, was constructed in [35] but the super-Higgs phases were not studied.
In the present paper we study these phases, derive the mass spectrum in terms of the four complex gravitino masses and analyse the moduli space of these phases and their relative unbroken symmetries.
Connection with supergravity compactification on the T 6 /Z 2 orbifold with brane fluxes is discussed.

71.4
The major input is that the 15 axion fields B ΛΣ = −B ΣΛ , Λ, Σ = 1 . . . 6 related to the 15 translational isometries of the moduli space SO(6, 6)/SO(6)×SO(6) are dual to a compactified R − R 4-form scalars (B ΛΣ = 1 4! ΛΣ∆ΓΠΩ C ∆ΓΠΩ ). Moreover the charge coupling of N = 4 dual supergravity of [35]: identifies the supergravity coupling f ΛΣ∆α with the 3-form fluxes coming from the term [42,43] dC of the covariant 5-form field strength, where H α is taken along the internal directions and integrated over a non-trivial 3-cycle †. This paper is organized as follows. in section 2 we describe the geometry underlying the N = 4 supergravity in the dual basis chosen by the type-IIB orientifold compactification.
In section 3 we describe the ungauged and gauge theory in this basis: the main ingredient is to rewrite the supergravity transformation laws in an unconventional way in terms of the reduced manifold GL(6)/SO (6) and the 15 axion fields B ΛΣ . This allows us to compute the fermion shifts in terms of which the potential can be computed.
In section 4 we analyse the potential. We show that it is semidefinite positive and find the extremum which stabilizes the dilaton and the GL(6)/SO(6) scalar fields, except for three fields related to the radii of In section 5 we compute the mass spectrum of the gravitini and the vector fields. It is shown that the four complex gravitino masses precisely correspond to the (3, 0) + 3(2, 1) decomposition of the real 3-form flux matrix F ΛΣ∆ = L α f ΛΣ∆α (L α coset representative of SL(2, R)/SO(2)).
In section 6 the reduction of the massive and massless sectors of the different super-Higgs phases are described. In particular it is shown that, by a given choice of the complex structure, the N = 3 supergravity corresponds to taking as nonzero only the (3, 0) part of the holomorphic components of F ΛΣ∆ .
In section 7 we give the conclusions, while in the appendix we give the explicit representation of the SU(4) gamma matrices used in the text.

Geometry of the N = 4 scalar manifold
We start from the coset representative of SO(6, 6)/SO(6) × SO (6) written in the following form [57,58]: Note that the representation 12 of so(6, 6) decomposes as 12 → 6 +1 + 6 −1 ,, thus containing six electric and six magnetic fields, and the bifundamental of so(6, 6) + sl(2, R) decomposes as (12, 2) = (6 +1 , 2) electric + (6 −1 , 2) magnetic . In particular, we see that sl(2, R) is totally electric. The 12 vectors gauge an Abelian subgroup of the 15 + translations. The left invariant 1-form L −1 dL ≡ Γ turns out to be Now we extract the connections ω d and ω, where ω d is the connection of the diagonal SO(6) d subgroup and ω is its orthogonal part. We get so that the total connection Ω = ω d + ω is By definition the vielbein P is defined as so that we get In the following we will write Ω and P as follows: where (2.12) Note that For the SU(1, 1)/U (1) factor of the N = 4 σ model we use the following parametrizations: (2.14) Introducing the 2-vectors Introducing the left-invariant sl(2, R) Lie algebra valued 1-form: one can easily determine the coset connection 1-form q and the vielbein 1-form p: (2.20) Note that we have the following relations:

The gauging (turning on fluxes)
In the ungauged case the supersymmetry transformation laws of the bosonic and fermionic fields can be computed from the closure of Bianchi identities in superspace and turn out to be 21, a m being the scalar fields parametrizing the coset GL(6)/SO (6).
The position of the SU(4) index A on the spinors is related to its chirality as follows: The previous transformations leave invariant the ungauged Lagrangian that will be given elsewhere together with the solution of the superspace Bianchi identities. Our interest is, however, in the gauged theory where the gauging is performed on the Abelian subgroup T 15 of translations.
It is well known that, when the theory is gauged, the transformation laws of the fermion fields acquire extra terms called fermionic shifts which are related to the gauging terms in the Lagrangian and enter in the computation of the scalar potential [5,7,59].

71.7
Let us compute these extra shifts for the gravitino and spin 1 2 fermions in the supersymmetry transformation laws. Since we want to gauge the translations, according to the general rules, we have to perform the substitution where k ijΛα are the Killing vectors corresponding to the 15 translations, with ij a couple of antisymmetric world indices and Λα denoting the adjoint indices of GL(6) × SL(2, R). Since the 'coordinates' B ij are related to the axion B ΛΣ by we get where f ΛΣΓα are numerical constants. Therefore, the gauged connection affects only ω and not ω and we have Therefore, if we take the Bianchi identities of the gravitino where ω 1 is the composite connection of the SU(4) ∼ SO(6) R symmetry acting on the gravitino SU(4) index, and then, since As for the supersymmetry transformations (3.1) we do not report here the procedure used to determine the fermion shifts in the gauged Bianchi identities which, as mentioned before, will be given elsewhere. It is sufficient to say that, according to a well known procedure, the cancellation of the extra term appearing in (3.16) requires an extra term in the superspace parametrization of the gravitino curvature. This in turn implies a modification of the space-time supersymmetry transformation law of the gravitino, obtained by adding the following extra term to δψ Aµ : where we have defined f IJKα = f ΛΣΓα E I Λ E J Σ E K Γ and the symmetric matrix S AB is (one-half) the gravitino mass matrix entering the Lagrangian.
Recalling the selfduality relation Γ IJK = i 3! IJKLMN Γ LM N and introducing the quantities the gravitino gauge shift can be rewritten as Analogous computations in the Bianchi identities of the left handed gaugino and dilatino fields give the following extra shifts: These results agree, apart from normalizations, with [35].

The scalar potential
The Ward identity of supersymmetry [5,7].
allows us to compute the scalar potential from knowledge of the fermionic shifts S AB , N AB , Z IB A computed before, equations (3.17), (3.21) and (3.22) †. We obtain where we have made explicit the dependence on the GL(6)/SO(6) scalar fields and N ΛΣ is defined by Another useful form of the potential, which allows the discussion of the extrema in a simple way, is to rewrite equation (4.2) as follows: where we have used equations (3.18) and (3.19). From (4.4) we see that the potential has an absolute minimum with vanishing cosmological constant when F −IJK = 0.
In order to have a theory with vanishing cosmological constant the two SL(2, R) components of f αΛΣΓ cannot be independent. A general solution of F −IJK = 0 is given by setting where α is a complex constant. In real form we have (4.6) The solution of (4.5) is 71.9 In the particular case α = 1, (4.5) reduces to which is the constraint imposed in [35]. In this case the minimum of the scalar potential is given by or, in terms of the L α fields, If we take a configuration of the GL(6)/SO (6) fields where all the fields a m = 0 except the three fields ϕ 1 , ϕ 2 , ϕ 3 , parametrizing O(1, 1) 3 , then the matrix E I Λ has the form given in section 6 (equation (6.21) and in this case The minimum condition can be also retrieved in the present case by observing that in the case α = 1 the potential takes the simple form Using the explicit form of f IJK as given in the next section (equations (5.1) and (5.2)), the potential becomes Therefore V = 0 implies which is satisfied by equation (4.9) where we have taken into account equation (2.14). Note that equations (4.14) and (4.9), giving the extremum of the potential, fixes the dilaton field. In contrast, the extremum of the potential with respect to ϕ 1 , ϕ 2 , ϕ 3 , does not fix these fields, since the corresponding extremum gives the condition V ≡ 0. This shows the no-scale structure of the model. The a m fields are instead stabilized at a m = 0. All the corresponding modes get masses (for N = 1, 0).
When α = 1, a simple solution of equation (4.5) is to take f −ΛΣ∆  [42]. The general solution contains, besides α, four complex parameters, since f ΛΣ∆ 1 has at most eight non-vanishing components.
In string theory, f 1 and f 2 satisfy some quantization conditions which restrict the value of α [42,43].
It is interesting to see what is the mechanism of cancellation of the negative contribution of the gravitino shift to the potential which makes it positive semidefinite. For this purpose it is useful to decompose the gaugino shift (3.22) in the 24-dimensional representation of SU (4) into its irreducible parts 20 + 4. Setting (4. 19) In this way the irreducible parts of the fermion shifts are all proportional to F ±IJK Γ IJK , namely When one traces the indices AB in (4.1) one sees that the contributions from the gravitino shifts and from the 4 of the gaugino shifts are both proportional to |F −IJK | 2 and since, on general grounds they have opposite sign, they must cancel against each other. Vice versa, the square of the gaugino shift in the 20 representation and the square of the dilatino shift are both proportional to |F +IJK | 2 , that is, to the scalar potential. Indeed It then follows that the χ A(4) are the four Goldstone fermions of spontaneously broken supergravity. These degrees of freedom are eaten by the four massive gravitini in the super-Higgs mechanism. This cancellation reflects the no-scale structure of the orientifold model as discussed in [42,43]. It is the same kind of cancellation of F and D terms against the negative (gravitino square mass) gravitational contribution to the vacuum energy that occurs in Calabi-Yau compactification with brane fluxes turned on [38,41,48].

Mass spectrum of the gravitini and vector fields
Let us now compute the masses of the gravitini. As we have seen in the previous section the extremum of the scalar potential is given by It follows that the gravitino mass matrix S AB at the extremum takes the values (5.6) From (5.5) we may derive an expression for the gravitino masses at the minimum of the scalar potential very easily, going to the reference frame where S AB is diagonal. Indeed it is apparent that this corresponds to choosing the particular frame corresponding to the diagonal Γ IJK . As is shown in the appendix, the diagonal Γ IJK correspond to Γ 123 , Γ 156 , Γ 246 Γ 345 and their dual. It follows that the four eigenvalues µ i + iµ i , i = 1, . . . , 4 of S AB are Here we have set f −IJK 1 ≡ f −IJK . Furthermore L 2 , computed at the extremum (see equation (4.7)), is a function of α. The gravitino mass squared m 2 i is given by m 2 i = µ 2 i + µ 2 i . The above results (5.7) and (5.3) take a more elegant form by observing that they use a complex basis: where the tensor f IJK1 ≡ f IJK takes the following components: (5.14) Therefore, the 20 entries of f ΛΣ∆ 1 are reduced to eight.

71.12
In this holomorphic basis the gravitino mass eigenvalues assume the rather simple form: (Note that the role of µ 1 + iµ 1 , µ 2 + iµ 2 , µ 3 + iµ 3 , µ 4 + iµ 4 can be interchanged by changing the definition of the complex structure, (5.8), that is permuting the roles of E x,y,z and E x,y,z .) Let us now compute the masses of the 12 vectors. We set here for simplicity α = 1. Taking into account that the mass term in the vector equations can be read from the kinetic term of the vectors and of the axions in the Lagrangian, namely −i)) the vector equation of motion gives a square mass matrix proportional to Q Λα,LΣβ : (5.20) which is symmetric in the exchange Λα ←→ Σβ. The eigenvalues of Q Λα,Σβ can be easily computed and we find that they are twice degenerate. In terms of the four quantities ) the six different values turn out to be proportional to Note that for N = 3, 2 six and two vectors are, respectively, massless, according to the massless sectors of these theories described in section 6.

Reduction to lower supersymmetry N = 4−→ 3, 2, 1, 0
Since the supergravities with 1 ≤ N < 4 are described by σ models possessing a complex structure, it is convenient to rewrite the scalar field content of the N = 4 theory in complex coordinates as already done for the computation of the gravitino masses.

71.13
We recall that we have 36 scalar fields parametrizing SO(6, 6)/SO(6) × SO(6) that have been split into 21 fields g IJ = g JI parametrizing the coset GL (6) We may take advantage of the complex structure of this manifold, by rotating the real frame {e I }, I = 1 . . . 6 to the complex frame defined in (5.8). In this frame we have the following decomposition for the scalar fields in terms of complex components: In the presence of the translational gauging, the differentials of the axionic fields become covariant and they are obtained by the substitution: Since in the N = 4 −→ 3 truncation the only surviving massless moduli fields are B i +ig i , then the 3 + 3 axions {B ij , B ı  } must become massive, while δB i must be zero. We see from equation (6.4) that we must put to zero the components Looking at equations (5.15) we see that these relations are exactly the same which set µ 1 + iµ 1 = µ 2 + iµ 2 = µ 3 + iµ 3 = 0 and µ 4 + iµ 4 = 0, confirming that the chosen complex structure corresponds to the N = 3 theory. Note that the corresponding g i fields partners of B i in the chosen complex structure parametrize the coset O(1, 1) × SL(3, C)/SU (3). Actually the freezing of the holomorphic g ij gives the following relations among the components in the real basis of g IJ : g 14 = g 25 = g 36 = 0 (6.7) g 11 − g 44 = 0, g 22 − g 55 = 0, g 33 − g 66 = 0 (6.8) g 12 − g 45 = 0, g 13 − g 46 = 0, g 23 − g 56 = 0 (6.9) g 15 + g 24 = 0, g 16 + g 34 = 0, g 26 + g 35 = 0. The massless g i and B i are instead given by the following combinations: g xx = 1 2 (g 11 + g 44 ), g yy = 1 2 (g 22 + g 55 ), g zz = 1 2 (g 33 + g 66 ) (6.14) If we now consider the truncation N = 4 −→ 1 the relevant coset manifold is (SU(1, 1)/U (1)) 3 which contains three complex moduli. To obtain the corresponding complex structure, it is sufficient to freeze g i , B i with i = j. In particular the SU(1, 1) 3 can be decomposed into O(1, 1) 3 ⊗ s T 3 where the three O(1, 1) and the three translations T 3 are parametrized by g xx , g yy , g zz and B xx , B yy , B zz , respectively.
These axions are massless because of equation (5.14). (Note that the further truncation N = 1 −→ 0 does not alter the coset manifold SU(1, 1) 3 since we have no loss of massless fields in this process.) In this case we may easily compute the moduli dependence of the gravitino masses. Indeed, O(1, 1) 3 , using equations (6.8) and (6.14), will have as coset representative the matrix where we have set g 11 = e 2ϕ 1 , g 22 = e 2ϕ 2 , g 33 = e 2ϕ 3 , the exponentials representing the radii of the manifold T 2 (14) × T 2 (25) × T 2 (36) . We see that in the gravitino mass formula (3.17) the vielbein E I Λ reduces to the diagonal components of the matrix (6.21). A straightforward computation then gives We see that the square of the gravitino masses goes as . Note that this is different from what happens in the Kaluza-Klein compactification, where the gravitino mass squared goes as 1 R 2 Im SIm τ , where 1 Im S = g 2 string and the complex structure τ = i is a constant, so that µ 2 We note that in the present formulation, where we have used a contravariant B ΛΣ as basic charged fields, the gravitino mass depends on the T 6 volume. However, if we made use of the dual 4-form C ΛΣΓ∆ , as it comes from type IIB string theory, then the charge coupling would be given in terms of * f α ΛΣΓ and the gravitino mass matrix would be trilinear in E Λ I instead of E I Λ . Therefore all our results can be translated into the new one by replacing

Conclusions
In this paper we have shown that a non-standard form of N = 4 supergravity, where the full SO(6, n) symmetry is not manifest, nor even realized linearly on the vector field strengths [53], is the suitable description for a certain class of IIB compactifications in presence of 3-form fluxes.
Since the super-Higgs phases of N -extended supergravities solely depend on their gauging, the use of a dual formulation [35] is crucial, where the linear symmetry acting on the vector fields (n = 6) is GL(6, R) × SL(2, R) rather than SO (6,6), thus allowing the gauging of a subalgebra T 12 inside the T 15 (see equation (1.1)), the latter being a nilpotent Abelian subalgebra of SO (6,6). For a choice of complex structures on T 6 = T 2 ×T 2 ×T 2 the four complex gravitino masses are proportional to the (3, 0) and three (2, 1) fluxes of 3-forms. N = 3 supergravity corresponds to setting to zero the three (2, 1)-form fluxes, N = 2 and 1 supergravities correspond to the vanishing of two or one (2, 1)-form.
The scalar potential is non-negative and given by the square of the supersymmetry variation of the component 20 of the 24 SU(4) (reducible) representation of the six gaugini (6 × 4 = 20 + 4) of the six matter vector multiplets of the SO(6, 6) symmetric supergravity. Indeed the positive contribution of the component 4 of the gaugino just cancel in the calculation of the potential the negative contribution of the spin 3 2 gravitini. The classical moduli space of the N = 3, 2, 1 (or 0) are, respectively, the following three complex manifolds: with six, two, or zero massless vectors, respectively. Note in particular that the N = 2 −→ 1 phases correspond to a spontaneously broken theory with one vector and two hypermultiplets, which is the simplest generalization [22] of the model in [20,21].

71.16
It is curious to observe that the moduli space of the N = 0 phase is identical to the moduli space of the N = 0 phase of N = 8 spontaneously broken supergravity via Scherk-Schwarz dimensional reduction [28]- [31,53]. The moduli spaces (7.1) and (7.2) of the Scherk-Schwarz N = 8 dimensional reduced case occur as N = 2 broken phases (depending on the relations among the masses of the gravitini).
The main difference is that, in Scherk-Schwarz breaking, the gravitini are 1 2 -BP S saturated, while here they belong to long massive multiplets [42,53]. This is related to the fact that the 'flat group' which is gauged is Abelian in the N = 4 (orientifold) theory and non-Abelian in the Scherk-Schwarz dimensional reduced N = 8 theory.
We have considered here the effect of supergravity for the IIB orientifold only for the part responsible for the super-Higgs phases. If one adds n D3 branes, that will correspond to adding n matter vector multiplets [35] which, however, will not modify the supersymmetry breaking condition. Then, the σ model of the N = 3 effective theory will be SU(3, 3 + n)/SU (3) × SU(3 + n) × U (1) [60] and will also contain, as moduli, the 'positions' of the n D3 branes [42].

Acknowledgments
SF would like to thank the Dipartimento di Fisica, Politecnico di Torino for its kind hospitality during the completion of this work.
RD'A would like to thank the Theoretical Division of CERN for its kind hospitality during the completion of this work.
This work was supported in part by the European Community's Human Potential Program under contract HPRN-CT-2000-00131 Quantum Space-Time, in which RD'A and SV are associated to Torino University. The work of SF has also been supported by the DOE grant DE-FG03-91ER40662, Task C.