CERN-TH/2003–136 SPIN-03/19 ITP-03/31 hep-th/0306185

New D =4gauged supergravities from N =4orientifolds with fluxes

1, 1,2, 3,[ Carlo Angelantonj †, Sergio Ferrara ‡ and Mario Trigiante

1 CERN, Theory Division, CH 1211 Geneva 23, Switzerland 2 INFN, Laboratori Nazionali di Frascati, Italy 3 Spinoza Institute, Leuvenlaan 4 NL-3508, Utrecht, The Netherlands ? [ e-mail: [email protected] , ∗ [email protected], [email protected]

Abstract

We consider classes of T6–orientifolds, where the orientifold projection contains

an inversion I9 p on 9 p coordinates, transverse to a Dp–brane. In absence of − − fluxes, the massless sector of these models corresponds to diverse forms of N =4 supergravity, with six bulk vector multiplets coupled to N = 4 Yang–Mills theory on the branes. They all differ in the choice of the duality symmetry corresponding to different embeddings of SU(1, 1) SO(6, 6+n)inSp(24+2n, R), the latter being × the full group of duality rotations. Hence, these Lagrangians are not related by local field redefinitions. When fluxes are turned on one can construct new gaugings of N = 4 supergravity, where the twelve bulk vectors gauge some nilpotent algebra which, in turn, depends on the choice of fluxes. 1. Introduction

New string or M–theory models are obtained turning on n–form fluxes, which allow, in general, the lifting of vacua, supersymmetry breaking and moduli stabilisation [1]–[24]. Examples of such new solutions are IIB and IIA orientifolds [25, 26, 27, 28, 29], where the orientifold projection (in absence of fluxes) preserves N =4orN = 2 supersymmetries. Recently, the T /Z orientifold with N = 4 supersymmetry [8, 9] and K T /Z 6 2 3 × 2 2 orientifold [24] with N = 2 supersymmetry have been the subject of an extensive study. In these cases, turning on NS–NS and R–R three–form fluxes allows to obtain new string vacua with vanishing vacuum energy, reduced supersymmetry and moduli stabilisation [7, 8, 9, 21, 24]. These features can all be understood in terms of an effective gauged su- pergravity, where certain axion symmetries are gauged [31, 30, 32]. These are generalised no–scale models [33, 34]. In the present investigation, we consider more general four–dimensional orientifolds with fluxes (both in type IIB and IIA) where the orientifold projection involves an inver-

sion I9 p on 9 p coordinates, transverse to the Dp–brane world–volume, thus generalising − − the T6/Z2 orientifold (with p = 3) constructed by Frey–Polchinski [8] and Kachru–Shulz– Trivedi [9] (see also [10] for a derivation of the complete low–energy supergravity from T–dialysed Type I theory in ten dimensions). Interestingly, their low–energy descriptions are all given in terms of N = 4 supergravity with six vector supermultiplets from the closed–string sector, coupled to an N = 4 Yang–Mills theory living on the Dp–brane world–volume. However, despite the uniqueness of N = 4 supersymmetry, the low–energy actions crucially differ in the choice of the manifest “duality symmetries” of the Lagrangian, since different sets of fields survive the orientifold projection, and therefore different symme- tries are manifestly preserved. Leaving the brane degrees of freedom aside, these duality symmetries are specified by their action on the (twelve) bulk vectors. Actually, N =4su- pergravity demands that such symmetries be contained in SU(1, 1) SO(6, 6) [35, 36] and × act on the vector field strengths and their duals as symplectic Sp(24, R) transformations [37]. On the other hand, the symmetries of the Lagrangian correspond to block–lower– triangular symplectic matrices, whose block–diagonal components have a definite action on the vector potentials [38, 39, 40]. For instance, in the orientifold models containing an

I9 p inversion, the block–diagonal symmetries always include GL(9 p, R) GL(p 3, R), − − × − as maximal symmetry of the GL(6, R) associated to the moduli space of the six–torus met- rics. The lower–triangular block contains the axion symmetries of the R–R scalars and of the NS–NS ones originating from the B–field, whenever present1.

1 For example, the latter is not present in the p = 3 case, i.e. the T6/Z2 orientifold.

1 In the sequel, we describe all nilpotent algebras Np [41], corresponding to axion sym-

metries of the R–R and NS–NS scalars for all orientifold models. All Np’s are nilpotent subalgebras of so(6, 6), are generically non–abelian and contain central charges. There are four of them in type IIB (p =3, 5, 7, 9) with dimensions 15, 23, 23, 15 respectively, while there are only three of them in type IIA (p =4, 6, 8) of dimensions 20, 24, 20, respectively. A common feature of these algebras is that they always contain fifteen R–R axionic symmetries, while the extra symmetries correspond to NS–NS B–field axions in the bi–fundamental of GL(9 p, R) GL(p 3, R). − × − A further R–R axion symmetry originates from the SU(1, 1), which acts as electric– magnetic duality on the gauge fields living on the brane world–volume. The corresponding axion field can be identified with the Cp 3 R–R field, as dictated by the coupling −

Cp 3 F F, (1) − ZΣp+1 ∧ ∧ where F is the two–form field strength of gauge fields living on the branes. Turning on fluxes in the orientifold models (three- and five–form fluxes in type IIB, two- and four–form fluxes in IIA) corresponds to a “gauging” in the corresponding su- pergravity Lagrangian, whose couplings are dictated by the particular choice of fluxes.

Non–abelian gaugings may also occur corresponding to subalgebras of Np,orquotient

algebras Np/Z,whereZ are some of the central generators of Np. As an illustrative example, let us consider the p =7typeIIBorientifolddefinedinsec- i ab ijkl tion 2, where the non–vanishing NS–NS and R–R fluxes are Haij ,Faij ,G =   Gabjkl (a, b =5, 6andi, j =1,...,4), and let us look at terms involving the axions coming from the B and four–form fields, Bia and Cijab = Cijab. Inspection of the three–form kinetic term reveals a non–abelian gauge coupling proportional to

ab iµ jν √ gHaij Hµνb g g g , (2) − as well as axion gauge couplings proportional to

ab iµ j` √ gHaij Hµb` g g g , (3) − together with similar expressions for the F –three form. Such terms come also from the reduction of type IIB four–form field. In addition, when a five–form flux Gi is turned on an axion gauge coupling emerges of the type

k ` ∂µCij + ijk` G Gµ . (4)

` `i where Gµ = g giµ are the Kaluza–Klein vectors. We report here only a preliminary analysis of the deformation of the N = 4 supergravity due to these new gaugings.

2 In the present paper we do not address either the question of unbroken supersymme- tries or the question of moduli stabilisation, which would require the knowledge of the scalar potential and a study of the fermionic sector. However, we can anticipate that cer- tain moduli are indeed stabilised in all these models, since a Higgs effect is taking place as suggested by the presence of charged axion couplings.

The paper is organised as follows: in section 2, we review the four-dimensional T6/Z2 orientifold models, their spectra and their allowed fluxes. In section 3, the N =4 supergravity interpretation is given for the ungauged case (absence of fluxes) and the

duality symmetries exposed. The Np algebras are exhibited as well as their action on the vector fields. In section 4, we give a preliminary description of gauged supergravity, for the particular case of type IIB orientifolds with some three–form fluxes turned on. In section 5 some conclusions are drawn. Finally, in appendix some useful formulae needed to compute the quadratic part of the vector field strengths in the Lagrangian, are given.

2. N =4orientifolds: spectra and fluxes

In this section we review the construction of orientifold models preserving N =4 supersymmetries in D = 4 [29]. This is the simplest setting for orientifold constructions, and consists of modding out type II superstrings by the world–sheet parity Ω [25]. Follow- ing [28, 29], the orientifold projection can be given a suggestive geometrical interpretation in terms of non–dynamical defects, the orientifold O–planes, that reflect the left–handed and right–handed modes of the closed string. Actually, one can combine world–sheet parity with other (geometrical) operations. In general, this can affect the nature of the orientifold planes, that, in the simplest instance of a bare Ω have negative tension and R–R charge, and are (9 + 1)–dimensional (O9 planes) since they have to respect the full Lorentz symmetry preserved by Ω. In the present paper, we are interested in the class

of models generated by the ΩI9 p generator, where I9 p denotes the inversion on 9 p − − − coordinates. Of course, ΩI9 p must be a symmetry of the parent theory, and this is the − case of type IIB for p odd, and of type IIA for p even. Actually, ΩI9 p reflects the action − of T-duality in orientifold models. Indeed, T-duality itself can be thought of as a chiral parity transformation X X ,X X , (5) L → L R →− R and conjugates Ω so to get 1 T9 pΩT9−p = ΩI9 p . (6) − − − As a result, the full ten–dimensional Lorentz symmetry is now broken to the subgroup

SO(1,p) SO(9 p), and the closed–string sector involves O9 p planes sitting at the fixed × − − points of the orbifold T9 p/I9 p. The associated open-string sector will then correspond − −

3 to open strings with Dirichlet boundary conditions along T9 p, i.e. open strings ending on − D(9 p) branes. As usual, tadpole conditions will fix the rank of the Chan–Paton gauge − group, i.e. the total number of D-branes. In the present paper, however, we shall not be concerned with open–string degrees of freedom and we shall concentrate our analysis solely on the closed-string degrees of freedom. Before we turn to the description of specific models, a general comment is in order.

An important requirement in the construction is that the orientifold group be Z2, i.e. its generator ΩI9 p must square to the identity. Although Ω has always 1 eigenvalues, and − 2 ± thus Ω = 1, this is not the case for I9 p. For example, for p =7I2 would correspond to − a π rotation on a two–plane and, although its action on the bosonic degrees of freedom is real and assigns to them a plus or minus sign according to the number of indices along the two–plane, its eigenvalue on spinors is eiπΣ,whereΣ = 1 are the two helicities. Thus, ± 2 it does not square to the identity, but rather to ( 1)F ,withF the (total) space–time − fermion number. Therefore, in this case the orientifold projection needs be modified by the inclusion of ( 1)FL ,withF the left-handed space–time fermion number [47]. We are − L thus dealing with the four-dimensional orientifolds

9 p − FL [ 2 ] (Tp 3 T9 p) /ΩI9 p ( 1) , (7) − × − − −   9 p where − denotes the integer part of (9 p)/2. Here we have decomposed the six-torus 2 − as  

T6 = Tp 3 T9 p , (8) − × − since I9 p only acts on the coordinates of T9 p, while leaves invariant those along Tp 3. − − − As we shall see, this is a natural decomposition since, in the orientifold, we are left with the perturbative symmetry GL(p 3) GL(9 p) of the compactification torus. To − × − fix the notation, in this paper we shall label coordinates on the T6 withapairofindices (i, a), where i =1,...,p 3 counts the coordinates not affected by the space parity (those − coordinates that would be longitudinal to the branes), while a =1,...,9 p runs over − the coordinates of T9 p (orthogonal to the branes). As usual, Greek indices µ,ν,... will − label coordinates on the four–dimensional Minkowski space–time. At this point, it is better to consider the cases p odd or p even separately. In the first 9 p − FL [ 2 ] case, ΩI9 p ( 1) is a symmetry in type IIB, while in the latter case it is properly − − defined within type IIA.

2.1. IIB orientifolds In type IIB superstring we have to consider four cases, corresponding to the allowed choices p =9, 7, 5, 3. The massless ten-dimensional fields have a well defined parity with

4 respect to Ω:

even : GMN ,φ,CMN , (9) (+) odd : BMN ,C,CMNPQ , (10)

where GMN is the metric tensor, φ the dilaton, BMN the Kalb–Ramond two-form, and 2 Cp+1 are the R–R (p + 1)-forms . Henceforth, it is straightforward to select the four- dimensional excitations that survive the orientifold projection. In fact, after splitting the

ten-dimensional index M in the triple (µ, i, a) labelling M1,3 Tp 3 T9 p,itisevidentthat × − × − the fields with an odd (even) number of a–type indices are odd (even) under the action

FL of I9 p. On the other hand, when present, ( 1) assigns a plus sign to the NS-NS states − − (which originate from the decomposition of the product of two bosonic representations of SO(8)) and a minus sign to the R–R states (which originate from the decomposition of the product of two spinorial representations of SO(8)). At the end, aside from the four–dimensional metric tensor, one is left with the massless (bosonic) degrees of freedom listed in table 1.

Table 1: Massless degrees of freedom for the IIB orientifolds

p scalars vectors i 9 gij, φ, Cµν , Cij Gµ, Ciµ i 7 gij, gab, φ, Bia, C, Cia, Cijkl, Cijab Gµ, Baµ, Caµ, Cijkµ i 5 gij, gab, φ, Bia, Cµν , Cij, Cab, Ciabc Gµ, Baµ, Ciµ, Cabcµ

3 gab, φ, C, Cabcd Baµ, Caµ

However, in orientifold models it happens often that fields which are odd under the projection can be consistently assigned with a (quantised) background value for the fields themselves, or for their field strengths. For example, in the p = 7 case the NS–NS fields

Bij and the R–R fields Cij are both odd with respect to the orientifold projection and,

thus, their quantum excitations are projected out. However, acting on them with a ∂a derivative changes their parity, and thus (quantised) fluxes along the internal directions,

Haij and Faij , can be incorporated in the model. Repeating a similar analysis for the other cases yields the allowed fluxes listed in table 2.

2.2. IIA orientifolds

2 (+) Actually, the four-form C4 is constrained to have a self–dual field strength, a peculiarity of type IIB

5 Table 2: Allowed fluxes for the IIB orientifolds. F , H and G fluxes are associated to the

B, C2 and C4 fields

p fluxes 9 none

7 Hija, Fija, Gijkab

5 Habc, Fiab, Hija, Gijabc

3 Habc, Fabc

Type IIA superstring selects p even, and thus leaves us with the three cases p =8, 6, 4. Although a bare Ω is not a symmetry in type IIA, we can nevertheless assign a well defined parity to the massless ten-dimensional degrees of freedom:

even : GMN ,φ,CM , (11)

odd : BMN ,CMNP . (12)

As before, GMN is the metric tensor, φ the dilaton, BMN the Kalb–Ramond two-form,

while in this case the R–R potentials Cp+1 carry an odd number of indices. The additional

FL action of I9 p and, eventually, of ( 1) thus yields the massless degrees of freedom listed − − in table 3. Also in this case one can allow for (quantised) fluxes along the compactification torus, as summarised in table 4.

Table 3: Massless degrees of freedom for the IIA orientifolds

p scalars vectors i 8 gij, g99, φ, Bi9, Ci, C9µν , Cij9 Gµ, Cµ, Ci9µ, B9µ i 6 gij, gab, φ, Bia, Ca, Ciµν , Cijk, Ciab Gµ, Baµ, Cijµ, Cabµ 4 4 g44, gab, φ, B4a, C4, Caµν , Cabc Gµ, Baµ, Cµ, C4aµ

6 Table 4: Allowed fluxes for the IIA orientifolds. F , H and G fluxes are associated to the

B, C1 and C3 fields

p fluxes

8 Hij9, Gijk9

6 Haij, Habc, Fia, Gijab

4 Habc, Fab, G4abc

3. N =4supergravity interpretation of T6 orientifolds: manifest duality transformations and Peccei–Quinn symmetries.

The four–dimensional low–energy supergravities of N = 4 orientifolds (in the absence of fluxes) can be consistently constructed as truncations of the unique four–dimensional N = 8 supergravity which describes the low–energy limit of dimensionally reduced type

II superstrings. Its duality symmetry group E7(7) acts non linearly on the 70 scalar fields, and linearly, as a Sp(56, R) symplectic transformation, on the 28 electric field strengths and their magnetic dual. In this framework an intrinsic group–theoretical characterisation of the ten–dimensional origin of the four–dimensional fields is indeed achieved. In the so– called solvable Lie algebra representation of the scalar sector [41, 42], the scalar manifold

Mscal =exp(Solv(e7(7))) (13) is expressed as the group manifold generated by the solvable Lie algebra Solv(e7(7)) defined through the Iwasawa decomposition of the e7(7) algebra:

e7(7) = su(8) + Solv(e7(7)) . (14)

In this framework, there is a natural one–to–one correspondence between the scalar fields and the generators of Solv(e7(7)). The latter consists of the 7 generators Hp of the e7(7) σn Cartan subalgebra, parametrised by the T6 radii Rn = e together with the dilaton φ, and of the shift generators corresponding to the 63 positive roots α of e7(7),whichare in one–to–one correspondence with the axionic scalars that parametrise them. This cor- respondence between Cartan generators and positive roots on one side and scalar fields on the other, can be pinpointed by decomposing Solv(e7(7)) with respect to some rele- vant groups. For instance, the duality group of maximal supergravity in D dimensions is E11 D(11 D) and therefore, in the solvable Lie algebra formalism, the scalar fields in − − the D–dimensional theory are parameters of Solv(e11 D(11 D)). Since e11 D(11 D) e7(7), − − − − ⊂

7 decomposing Solv(e7(7)) with respect to Solv(e11 D(11 D)) it is possible to characterise the − − higher–dimensional origin of the four–dimensional scalars. Moreover, in four dimensions

the group SL(2, R) SO(6, 6)T E , SO(6, 6)T being the isometry group of the T × ⊂ 7(7) 6 moduli–space, acts transitively on the scalars originating from ten–dimensional NS–NS

fields of type II theories. These scalars therefore parametrise Solv(sl(2, R)+so(6, 6)T ).

Henceforth, decomposing Solv(e7(7)) with respect to Solv(sl(2, R)+so(6, 6)T )onecan achieve an intrinsic characterisation of the NS–NS or R–R ten–dimensional origin of the four–dimensional scalar fields, the R–R scalars (and the corresponding solvable genera-

tors) transforming in the spinorial representation of SO(6, 6)T . Finally, depending on whether we interpret the four–dimensional maximal supergravity as tied to type II super-

gravities on T6 or D =11supergravityonT7, the metric moduli are acted on transitively

by GL(6, R)g or GL(7, R)g subgroups of E7(7), respectively. Therefore, in the two cases

the metric moduli parametrise Solv(gl(6, R)g)orSolv(gl(7, R)g) and thus, decompos-

ing Solv(e7(7)) with respect to these two solvable subalgebras, depending on the higher– dimensional interpretation of the four–dimensional theory, we may split the axions into metric moduli of the internal torus and into scalars deriving from dimensional reductions of ten- or eleven–dimensional tensor fields. The latter will parametrise nilpotent genera- tors transforming in the corresponding tensor representations with respect to the adjoint

action of GL(6, R)g or GL(7, R)g. As a result of the above decompositions, we are able

to characterise unambiguously each parameter of Solv(e7(7)) as a dimensionally reduced field. Let us consider the dimensional reduction of type II supergravities. As far as the axionic scalars are concerned the correspondence with roots can be summarised in terms 73 of an orthonormal basis p of R : { }

Cn n ...n a + n + ...n , (15) 1 2 k ↔ 1 k n1...nkm1...m6 k Cn1n2...nkµν a + m1 + ...m6 k , ( − =0), (16) ↔ − 6 Bnm n + m , (17) ↔ Bµν √2  , (18) ↔ 7 Gnm n m , (n = m) , (19) ↔ − 6 where 6 1 1 a = n +  . (20) − 2 √2 7 Xn=1

In our notation, the so(6, 6)T roots have the form n m ,where1 n

8 in tensor representations of GL(6, R)g, and this, in turn, defines the GL(6, R)g represen-

tation of the corresponding scalar. For instance, the Cn1...nk parametrises the generator T n1...nk = E whose transformation property under GL(6, R) is a+n1 +...nk g

n1...nk 1 n1 nk m1...mk g GL(6, R)g : g T g− = g m g m T . (21) ∈ · · 1 ··· k

The roots corresponding to R–R fields are spinorial with respect to SO(6, 6)T and, de- pending on whether the number of their indices is even or odd, they belong to the root

system of two e7(7) algebras which are mapped into each other by the SO(6, 6)T outer automorphism (T–duality) [43, 44]. These two systems naturally correspond to the re- duction of IIB and IIA superstrings, that are indeed related by T-dualities. Hence, the

T6 metric moduli in the type IIA or B descriptions, are acted upon transitively by two inequivalent GL(6, R)g subgroups of E7(7): in the former case GL(6, R)g is contained in

SL(8, R) E , while in the latter case GL(6, R)g is contained in the maximal sub- ⊂ 7(7) group SL(3, R) SL(6, R)g of E . As far as the R–R scalars are concerned, the two × 7(7) representations differ in the SO(6, 6)T chirality of the 32 spinorial positive roots odd + 1 1 IIA : 32− = (  ...  )+  , { 2 ± 1 ± 6 √2 7} z even}| + { IIB : 32+ = 1 (  ...  )+ 1  . (22) { 2 ± 1 ± 6 √2 7} z }| { Similarly, vector potentials, and their corresponding duals, are in one–to–one correspon-

dence with weights W of the 56 of E7(7) in the two representations discussed above:

Cn ...n µ w + n + ...n , 1 k ↔ 1 k 1 Bmν n  , ↔ − √2 7 n 1 G n  , µ ↔− − √2 7 where 6 1 w = n . (23) − 2 Xn=1 The dual potentials correspond to the opposite weights W . − The above axion–root (Φ α) and vector–weight (Aµ W ) correspondences can be ↔ ↔ retrieved also from inspection of the scalar and vector kinetic terms in the dimensionally reduced type IIA or type IIB Lagrangians [43, 45, 46] on a straight torus, which have the form:

~ µ~ dilatonic scalars: ∂µh ∂ h, − · 1 2 α h µ axionic scalars: e− · (∂µΦ ∂ Φ) , − 2 · 1 2 W h µν vector fields: e− · Fµν F , − 4 9 where 6 ~ 1 1 h = σn (n +  ) φa, (24) √2 7 − 2 Xn=1

and, as usual, Fµν = ∂µAν ∂ν Aµ. − α h A generic axion Φ and its dilatonic partner e · can be thought of as the real and imaginary parts of a complex field z spanning an SL(2, R)/SO(2) submanifold, where the SL(2, R) group is defined by the root α. In the models describing type II strings on

Tp 3 T9 p orientifolds, the real part of the complex scalar z spanning the SL(2, R)/SO(2) − × − factor in the scalar manifold is Ci1...ip 3 ,wherei1,...ik label the directions of Tp 3,as − − dictated by the coupling in eq. (1). From eqs. (15) and (24) one can then verify that p 7 α h − φ Im(z)=e · =Volp 3 e 4 ,whereVolp 3 denotes the volume of Tp 3. The scalar Im(z) − − − defines the effective four–dimensional coupling constant of the super Yang–Mills theory on Dp–branes through the relation:

1 p 7 −4 φ 2 = Vp 3 e . (25) gYM −

The embedding of the N = 4 orientifold models Tp 3 T9 p (in absence of fluxes) − × − inside the N = 8 theory (in its type IIA or IIB versions) is defined by specifying the embedding of the N = 4 duality group SL(2, R) SO(6, 6) inside the N =8E one. × 7(7) As far as the scalar sector is concerned, this embedding is fixed by the following group requirement:

SO(6, 6) GL(6, R)g = O(1, 1) SL(p 3, R) SL(9 p, R) . (26) ∩ × − × − Condition (26) fixes the ten–dimensional interpretation of the fields in the ungauged N = 4 models (except for the cases p =3andp = 9) which, for a given p, is indeed consistent with the bosonic spectrum resulting from the orientifold reductions listed in the previous section. In the p =3andp = 9 cases, the two embeddings are characterised by a different interpretation of the scalar fields, consistent with the T6/Z2 orientifold reduction in the presence of D3orD9 branes. We shall denote these two models by T T and 0 × 6 T T , respectively. In these cases, equation (26) in the solvable Lie algebra language 6 × 0 amounts to requiring that metric moduli are related either to the Tp 3 metric gij or to − the T9 p metric gab. The scalar field parameterising the Cartan generator of the external − SL(2, R) factor is given in eq. (25), while the metric modulus corresponding to the O(1, 1) in eq. (26) is (modulo an overall power)

9 p 11 p O(1, 1) (Vp 3) − (V9 p) − . (27) ↔ − − R The axions not related to the T6 metric moduli consist of Ci1...ip 3 in the external SL(2, )/SO(2) − factor, (p 3) (9 p) moduli Bia in the bifundamental of SL(p 3, R) SL(9 p, R)and15 − − − × − 10 R–R moduli which we shall generically denote by CI and which span the maximal abelian I ideal T of Solv(so(6, 6)). The scalars Bia and CI parametrise a 15 + (p 3) (9 p)di- { } − − mensional subalgebra Np of Solv(so(6, 6)) consisting of nilpotent generators only. In figure 1theso(6, 6) Dynkin diagrams for the various models and the corresponding intersections with gl(6, R)g, represented by sl(p 3, R)+sl(9 p, R) subdiagrams, are illustrated. As − − far as the scalar fields are concerned, the Tp 3 T9 p models within the same type IIA or − × − IIB framework are mapped into each other by so(6, 6)T Weyl transformations, which can

be interpreted as T–dualities on an even number of directions of T6. We con now turn to the detailed analysis of each (IIB or IIA) orientifold model.

IIB Superstring

β β β β β 5 1 2 3 4

β 6 × × × × T0 T60 or T62 T T T4 T4 T2

IIA Superstring

× × × T15 T T33 T T5 T1

Figure 1: SO(6, 6) Dynkin diagrams for the Tp 3 T9 p models. The shaded subdiagrams − × − define the groups SL(p 3, R) SL(9 p, R) acting transitively on the metric moduli. − × − The empty circles define simple roots corresponding to the metric moduli gij,gab,the

grey circle denotes a simple root corresponding to a Kalb–Ramond field Bia and the black circle corresponds to a R–R axion.

3.1. T T IIB orientifold with D7 branes. The ungauged version. 4 × 2 Solvable algebra of global symmetries. The following model (with p = 7) describes the bulk sector of IIB superstring compactified on a (T T )/Z orientifold with D7 branes 4 × 2 2 wrappedontheT4. To this end, we describe the embedding of the scalar sector of the corresponding

N = 4 model within the N = 8 by expressing the so(6, 6) Dynkin diagram βn in 4 { } terms of the simple roots of e7(7)

β =   , 1 1 − 2 4 In our conventions β1 is the end root of the long leg and β5,β6 the symmetric roots

11 β =   , 2 2 − 3 β =   , 3 3 − 4 β4 = 4 + 5 ,

β5 = 5 + 6 , − 6 1 1 β = ( n)+  = a. 6 − 2 √2 7 Xn=1

According to eq. (15), the root β6 corresponds to the ten–dimensional R–R scalar C0, and thus identifies the type IIB duality group SL(2, R)IIB. The Dynkin diagram of the external SL(2, R) factor in the isometry group consists, instead, of the single root

β = a + 1 + 2 + 3 + 4 . (28)

It is useful to classify the positive roots according to their grading with respect to three relevant O(1, 1) groups generated by the Cartan operators Hβ,Hλ4 ,Hλ6 and parametrised by the moduli β h, h ,h: · 4 6 β h O(1, 1) e · = V , 0 → 4 h 1 1 O(1, 1) e 4 =(V ) 4 (V ) 2 , 1 → 4 2 h6 φ O(1, 1) e = e− , (29) 2 → n n n where we have denoted by λ the so(6, 6) simple weights, λ βm = δ . O(1, 1) is · m 0 generated by the Cartan generator of the external SL(2, R)andO(1, 1)1,O(1, 1)2 are in GL(4, R) GL(2, R), the former corresponding to the metric modulus given in eq. × (27). In table 5 we list the axionic fields of the model together with the corresponding generator of Solv(sl(2, R)) + Solv(so(6, 6)), for each of which the O(1, 1)3 grading and the SL(4, R) SL(2, R) representations are specified. The indices i, j and a, b label as × usual the directions of the torus which are longitudinal (T4) and transverse (T2)tothe D–branes.

The fields Bia and Cia transform in the representation (4, 4)ofSL(4, R) SO(2, 2) × where SO(2, 2) = SL(2, R) SL(2, R) , and therefore will be collectively denoted by Φλ, × IIB i where λ =(α, a)=1, 2, 3, 4labelsthe4 of SO(2, 2), with a choice of basis corresponding to the invariant metric ηλσ = diag(+1, +1, 1, 1). Its expression in terms of the fields − − Bia and Cia is

λ 1 Φ = Ci Bi ,Bi + Ci ,Bi + Ci , Bi + Ci . (30) i √2 { 2 − 1 2 1 1 2 − 2 1} We shall use the same notation for the corresponding generators, T i T 1ia,T2ia . { λ}≡{ } i ij From the assigned gradings one can conclude that the generators T0, Tλ and T close a 23–dimensional nilpotent solvable subalgebra N7 of Solv(so(6, 6)). The non–trivial

12 Table 5: Axionic fields for the T T IIB orientifold, generators of Solv(so(6, 6)), O(1, 1)3 4 × 2 gradings, and SL(4, R) SL(2, R)representations. × GL(4) GL(2)–rep. generator root field dim. × — T i j,a b (ib) gij,gab 7 (0,0,0) { − − } { } (1, 1)(0,0,1) T0 a C0 1 1ia (4, 2)(0,1,0) T i + a Bia 8 2ia (4, 2)(0,1,1) T a + i + a Cia 8 ij (6, 1) T a +(i + a)+(j + b) Cij ab Cij ab 6 (0,2,1) ≡ (1, 1) T β = a +  +  +  +  Cijkl c 1 (2,0,0) 1 2 3 4 ≡

commutation relations are determined by the grading and the index structure of the generators, and read

T ,Ti = M λ0 T i , 0 λ λ λ0  i j  ij T ,T = ηλλ T . (31) λ λ0 0   λ where Mλ 0 is a nilpotent generator acting on the 4 of SO(2, 2) which, for our choice of basis, can be cast in the form 0 10 1 − −  1010 λ0 1 Mλ = 2 . (32)  0101    10 10  − −  Infinitesimal transformations. Let us consider now the infinitesimal transformations of the scalar fields generated by T0, Tλi and Tij. For simplicity we shall restrict our analysis λ ij to those points in the moduli space where the only non-vanishing scalars are Φi , C and C. The corresponding coset representative thus takes the simple form

ij λ i L =exp Cij T exp Φi Tλ exp (CT0) , (33)

and its associated left–invariant one–form is

1 1 1 i λ 1 ij L− dL =(L− ∂0L) dC +(L− ∂λL) dΦi +(L− ∂ L) dCij

λ λ0 λ0 i 1 ij λ ij = T0 dC + dΦ (δ CMλ ) T + T dΦ Φλj + T dCij . (34) i λ − λ0 2 i

In general, the action of an element TΛ on the coset representative can be expressed as:

1 α 1 L− TΛL = kΛL− ∂αL, (35)

13 where the kΛ are the corresponding Killing vectors. In the case at hand, from eq. (31), we can derive

1 λ λ i 1 λ λ ij L− T L = T + Φ M 0 T + Φ Φ 0 M T , 0 0 i λ λ0 2 i j λλ0

1 i λ0 λ0 i ij L− TλL = δλ CMλ Tλ + T Φjλ ,  −  0 1 ij ij L− T L = T , (36) and, thus, read the non–vanishing components of the Killing vectors

k = ∂ + Φλ M λ0 ∂i , 0 0 i λ λ0 i i 1 ij kλ = ∂λ + 2 Φjλ ∂ , kij = ∂ij , (37) where ∂ ij ∂ i ∂ ∂0 = ,∂= and ∂λ = λ . (38) ∂C ∂Cij ∂Φi 0 λ i ij Therefore, under the infinitesimal diffeomorphism ξ k0 + ξi kλ + ξijk the fields transform as follows:

δC = ξ0 ,

λ λ 0 λ0 λ δΦi = ξi + ξ Φ Mλ0 , 1 λ δCij = ξij + 2 ξ[i Φj]λ . (39)

Scalar kinetic terms. Since all the quantities of our gauging are covariant with respect to SO(2, 2) GL(4, R) it is useful to define the (full) coset representative in the following × way ij λ i E L =exp Cij T exp Φi Tλ exp (cT) , (40)

where E is the coset representative of the submanifold SO(2, 2) GL(4, R) E O(1, 1) . (41) ∈ 0 × SO(2) SO(2) × SO(4) × The scalar kinetic terms are computed by evaluating the components of the vielbein 1 P = L− dL G/H : | 1 ˆˆıˆ λˆ ˆˆı ˆ L− dL G/H = Pˆıˆ T + Pˆı Tˆ + P T + PE , (42) | λ where the restriction to G/H amounts to select the non–compact isometries of the scalar manifold, PE is the algebra–valued vielbein of the submanifold (41). Finally, the hatted generators denote the non–compact component of the corresponding solvable generator. The kinetic Lagrangian for the scalar fields is then

1 µ 1 λˆ λµˆ 1 µ 2 Lscal = 2 Pµ P + 2 Pˆıµ Pˆı + 4 Pˆıµˆ Pˆıˆ +Tr(PE) , (43) Xˆıλˆ Xˆıˆ

14 where

Pµ = ∂µc, λˆ λ i λˆ Pˆıµ =(∂µΦi ) E ˆı Eλ , 1 λ λ i j Pıµ = ∂µCij + (∂µΦ Φjλ ∂µΦ Φiλ) E ı E  . (44) ˆˆ 4 i − j ˆ ˆ   i Vector fields. The twelve vector potentials are Baµ,Caµ,Gµ,Cijkµ. As before, we λ λ λ shall collectively denote by A the pair Baµ,Caµ ,andbyF = dA the corresponding µ { } field strengths. To avoid confusion, we shall then adopt the following notation for the i i i ijkl ˜ ˜ ˜ remaining field strengths: F = dG and F =  dCjkl.Moreover,Fλ, Fi and Fi will denote the “dual” field strengths, obtained by varying the Lagrangian with respect to the λ i electric ones, not to be confused with the four–dimensional Hodge duals ∗F , ∗F and i ∗F . Following [37], we can then collect the field strengths and their duals in a symplectic vector λ i i ˜ ˜ ˜ F , F ,F, Fλ, Fi, Fi . (45) { } In table 6, we list the field strengths and their duals as they appear in the symplectic section, together with their O(1, 1)3 gradings and the corresponding weights of the 56 of

E7(7).

Table 6: Field strengths, O(1, 1)3 gradings, and corresponding weights.

Sp–section O(1, 1)3–grading weight 1 1 F a ( 1, 0, ) a  1 − − 2 − √2 7 1 F2a ( 1, 0, 2 ) w + a i − 1 1 F ( 1, 1, ) i  − − − 2 − − √2 7 i 1 F (1, 1, 2 ) w + j + k + l 1a − −1 1 F˜ (1, 0, ) a +  2 − √2 7 2a 1 F˜ (1, 0, ) w a − 2 − − ˜ (1, 1, 1 )  + 1  Fi 2 i √2 7 1 F˜i ( 1, 1, ) w j k l − 2 − − − −

Under a generic nilpotent transformation

λ i ij ξT + ξ0 T0 + ξi Tλ + ξijT , (46) the field strengths transform as

λ λ i λ0 λ δF = ξ F + ξ F Mλ , − i 0 0 15 δF i =0, δF i = ξ F i ,

λ0 λ0 λ0 i δF˜λ = ξηλλ F ξ Mλ F˜λ ηλλ ξ F , 0 − 0 0 − 0 i ˜ ˜ λ ˜ j δFi = ξ Fi + ξ Fλ 2 ξij F , − i − ˜ λ0 λ j δFi = ξ F ηλλ +2ξij F . (47) − i 0 We then deduce that the electric subalgebra is R ge = o(1, 1)(0,0) + so(2, 2)(0,0) + gl(4, )(0,0) +(1, 1)(2,0) +(4, 4)(0,1) +(1, 6)(0,2) , (48) where o(1, 1) is the generator of O(1, 1) , and the grading refers to O(1, 1) O(1, 1) . (0,0) 0 0 × 1 The group O(1, 1)2 is now included inside SO(2, 2) and, in what follows, we shall not consider its grading any longer. Furthermore, we identify T as the generator in (1, 1)(2,0), i ij Tλ and T are associated to (4, 4)(0,1) and (6, 1)(0,2), respectively. The interested reader may find in appendix the explicit symplectic realisation of the generators of N7,aswell as the computation of the vector kinetic matrix.

3.2. T T IIB orientifold with D5 branes. The ungauged version 2 × 4 Solvable algebra of global symmetries. In this second model the relevant axions are d Bia,Cab,Ciabc C ,Cµν c and Cij = ij c0, and can be associated to the following ≡ i ≡ choice of simple roots

β =   , 1 1 − 2 β2 = 2 + 3 , β =  +  , 3 − 3 4 β =  +  , 4 − 4 5 β =  +  , 5 − 5 6 β6 = a + 3 + 4 , for the subalgebra so(6, 6) e . The Dynkin diagram of the external SL(2, R) consists, ⊂ 7(7) instead, of the single root

β = a + 1 + 2 , (49) whose corresponding axion is Cij, according to eq. (15). The triple grading, this time, refers to the O(1, 1)3 group generated by the three

Cartan Hβ,H 2 ,H 6 and parametrised by the moduli β h, h ,h: λ λ · 2 6 β h φ O(1, 1)0 e · = V2 e− 2 , → 1 1 φ h2 O(1, 1)1 e =(V2) 2 (V4) 4 e 2 , → 1 φ h6 O(1, 1) e =(V ) 2 e− 2 , (50) 2 → 4 16 where, as usual, O(1, 1)0 is in the external SL(2, R), while O(1, 1)1 and O(1, 1)2 are contained in GL(2, R) GL(4, R). × In table 7 we list the axionic fields of this model, together with the corresponding generator of Solv(so(6, 6)), for each of which the O(1, 1)3 grading is specified, as well as their SL(2, R) SL(4, R)representations ×

Table 7: Axionic fields for the T T IIB orientifold, generators of Solv(so(6, 6)), O(1, 1)3 2 × 4 gradings, and SL(2, R) SL(4, R)representations. × GL(2) GL(6)–rep. generator root field dim. × — T(0,0,0) i j,a b (ib) gij,gab 7 ab { − − } { } (1, 6)(0,0,1) T α7 + a + b Cab 6 ia (2, 4)(0,1,0) T i + a Bia 8 i d (2, 4)(0,1,1) Td α7 + i + a + b + c Ci 8

(1, 1)(0,2,1) T α7 + i + j + a + b + c + d Cµν = c 1

(1, 1)(2,0,0) T 0 β Cij = c0 1

ia i ab Also in this case, the generators T , T , Ta and T close a 23–dimensional solvable subalgebra of SO(6, 6)

ia a i ab N5 = cT + Bia T + Ci Ta + Cab T , (51) whose algebraic structure is encoded in the non–vanishing commutators

ia bc abcd i T ,T =  Td , (52)  ia j ij a T ,Td =  δd T. (53)   The corresponding coset representative reads

cT B T ia Ca T i C T ab L = e e ia e i a e ab , (54) while its left–invariant one–form is

1 ab i d ia ij a abcd i L− dL = Tdc+ T dCab + Td dCi +(T +  Cj T +  Td Cbc) dBia . (55) The transformation properties of the axionic scalars can be deduced from

1 L− TL = T, 1 i i ij L− TaL = Ta +  Bja T, 1 ia ia ij a abcd i L− T L = T +  Cj T +  Td Cbc , 1 ab ab abcd i L− T L = T +  Bid Tc , (56)

17 which identify the Killing vectors

k = ∂, i i ij ka = ∂a +  Bja∂, kia = ∂ia , ab ab abcd i k = ∂ +  Bid∂c , (57) where ∂ i ∂ ia ∂ ab ∂ ∂ = ,∂a = a ,∂= ,∂= . (58) ∂c ∂Ci ∂Bia ∂Cab ia a i ab Hence, under the infinitesimal diffeomorphism ξT + ξia T + ξi Ta + ξab T , one has

ij a δc =  ξi Bja + ξ, a abcd a δCi =  ξbc Bid + ξi ,

δBia = ξia ,

δCab = ξab . (59)

1 cd For later convenience we shall define the generator Tab = 4 abcd T , and the correspond- 1 cd − ing parameter ξab = abcd ξ , in terms of which the relation (52) reads − 4 ic c i Tab,T = δ[a Tb] . (60)   i a Vector fields. The vector fields of this model are Gµ,Ciµ Baµ,Cµ, and we name the corresponding field strengths and their duals by

i a ˜ ˜i ˜ a ˜ Fµν ,Fiµν , Haµν ,Fµν , Fiµν , Fµν , Hµν , Faµν . (61)

In the table 8 we list the field strengths and their duals as they appear in the symplectic 3 section, together with their O(1, 1) gradings, and the corresponding E7(7) weights.

ab The transformation laws under a generic nilpotent transformation ξ0 T 0 +ξT+ξ Tab + ia a i ξia T + ξi Ta can be deduced from the grading and weight structures. One finds

δF i =0, j δFi = ξ0 ij F , i δHa = ξia F , a ab a i δF = ξ Hb ξ F , − i ˜ ˜j a ˜ ˜ a δFi = ξ0 ij F + ξ Fa ξia H + ξFi , i − ˜i ij a ij a i δF =  ξja F  ξ Ha + ξ F − j ˜ a a ab ˜ a ij δH = ξ0 F + ξ Fb + ξi  Fj , ˜ ij δFa = ξ0 Ha ξai  Fj . (62) − 18 Table 8: Field strengths, O(1, 1)3 gradings, and corresponding weights.

Sp–section O(1, 1)3–grading weight i 1 1 F ( 1, 1, ) i  µν − − − 2 − − √2 7 1 Fiµν (1, 1, 2 ) w + i − − 1 1 Haµν ( 1, 0, ) a  − − 2 − √2 7 a 1 F ( 1, 0, ) w + b + c + d µν − 2 ˜ (1, 1, 1 )  + 1  Fiµν 2 i √2 7 i 1 F˜ ( 1, 1, ) w + j + a + b + c + d µν − 2 ˜ a 1 1 H (1, 0, ) a +  µν 2 − √2 7 1 F˜aµν (1, 0, ) w + i + j + a − 2

The explicit symplectic representation of the N5 generators together with the computation of the vector kinetic matrix N may be found in appendix.

3.3. T T and T T IIB orientifolds with D3 and D9 branes. The ungauged version. 0 × 6 6 × 0 The T T model in the presence of D3–branes, with and without fluxes was con- 0 × 6 structed in [32, 30, 31]. The structure of the T T model, on the other hand, is somewhat 6 × 0 trivial, since there is no room for fluxes to be turned on. For completeness, here we shall confine ourselves to the description of their embeddings within the N = 8 theory, and to the identification of the solvable algebras N3 and N9, together with their action on scalar and vector fields. Solvable algebra of global symmetries: the T T model. The embedding of the 0 × 6 sl(2, R)+so(6, 6) algebra inside e7(7) is defined by the following identification of the simple roots:

β =  +  , 1 − 1 2 β =  +  , 2 − 2 3 β =  +  , 3 − 3 4 β =  +  , 4 − 4 5 β =  +  , 5 − 5 6 β6 = a + 1 + 2 + 3 + 4 , (63) for the so(6, 6) component, and β = a, (64)

19 for the sl(2, R) one. The correspondence axion–root is quite simple and is summarised in table 9.

Table 9: Axionic fields for the T T IIB orientifold, generators of Solv(so(6, 6)), O(1, 1)2 0 × 6 gradings and GL(6, R)representations.

GL(6)–rep. generator root field dim.

— T(0,0) a b (a>b) gab 15 { − } ab { abcdef} 15 , Tab a + c + d + e + f C  Ccdef 15 (0 1) ≡ 1(2,0) T β C0 = c 1

In this case, the grading is with respect to the pair of O(1, 1) groups generated by

Hβ,Hλ6 and corresponding to the following moduli:

β h φ O(1, 1) e · = e− , 0 → λ6 h O(1, 1) e · = V . (65) 1 → 6

The nilpotent algebra N3, generated by Tab, acts as Peccei–Quinn translations on the R–R scalars Cab δCab = ξab . (66)

The vector fields are Caµ and Baµ, and the symplectic section of the corresponding ˜a ˜ a field strengths Faµν and Haµν and their magnetic duals Fµν , Hµν is listed in table 10.

Table 10: Field strengths, O(1, 1)2 gradings, and corresponding weights.

Sp–section O(1, 1)2–grading weight 1 Faµν (1, 2 ) w + a − 1 1 Haµν ( 1, ) a  − − 2 − √2 7 ˜a 1 F ( 1, ) w a µν − 2 − − ˜ a 1 1 H (1, ) a +  µν 2 − √2 7

ab The duality action of an infinitesimal transformation ξ Tab + ξT is then

δFa = ξ Ha ,

20 δHa =0, ˜a ab δF = ξ Hb , ˜ a ab ˜a δH = ξ Fb ξ F . (67) − − Solvable algebra of global symmetries: the T T model. The embedding of the sl(2, R)+ 6 × 0 so(6, 6) algebra inside e7(7) is defined by the following identification of the simple roots

β =   , 1 1 − 2 β =   , 2 2 − 3 β =   3 3 − 4 β =   , 4 4 − 5 β =   , 5 5 − 6 β6 = a + 5 + 6 , (68)

for the so(6, 6) component, and 6

β = a + n , (69) Xn=1 for the sl(2, R) one. The correspondence axion–root is quite simple, and is summarised in table 11.

Table 11: Axionic fields for the T T IIB orientifold, generators of Solv(so(6, 6)), O(1, 1)2 6× 0 gradings, and GL(6, R)representations.

GL(6)–rep. generator root field dim.

— T(0,0) i j (i

1(2,0) T β Cµν = c 1

In this case, the grading is with respect to a pair of O(1, 1) groups generated by

Hβ,Hλ6 and corresponding to the following moduli:

β h φ O(1, 1) e · = V e 2 , 0 → 6 λ6 h 1 3 φ O(1, 1) e · =(V ) 2 e− 4 . (70) 1 → 6 ij The nilpotent algebra N9, generated by T , acts as Peccei–Quinn translations on the R–R

scalars Cij,

δCij = ξij . (71)

21 i The vector fields are Ciµ and Gµ, and the symplectic sections of the corresponding i ˜i ˜ field strengths Fiµν and Fµν and their magnetic duals Fµν , Fiµν are listed in table 12.

Table 12: Field strengths, O(1, 1)2 gradings, and corresponding weights.

Sp–section O(1, 1)2–grading weight 1 Fiµν ( 1, 2 ) w + i i − 1 1 F ( 1, ) i  µν − − 2 − − √2 7 i 1 F˜ (1, ) w i µν − 2 − − ˜ (1, 1 )  + 1  Fiµν 2 i √2 7

ij The duality action of an infinitesimal transformation ξij T + ξT is then

j δFi = ξij F , δF i =0, δF˜i = ξ F i , ˜ ˜j δFi = ξij F + ξFi . (72)

As a result, the electric group contains the whole SO(6, 6), as for the heterotic string

on T6. In other words, there are no Peccei–Quinn isometries in SO(6, 6) which could be gauged. This feature is consistent with the fact that this model does not allow fluxes, and usually fluxes translate into local Peccei–Quinn invariances in the low–energy supergravity description.

3.4. T T IIA orientifold with D4–branes. 1 × 5 Solvable algebra of global symmetries. The embedding of the sl(2, R)+so(6, 6) algebra inside e7(7) is defined by the following identifications of simple roots:

β1 = 1 + 2 , β =  +  , 2 − 2 3 β =  +  , 3 − 3 4 β =  +  , 4 − 4 5 β =  +  , 5 − 5 6 β6 = a + 2 + 3 + 4 , (73)

22 for the so(6, 6) factor, and

β = a + 1 , (74) for the sl(2, R) one. The correspondence axion–root is quite simple, and is summarised in table 13.

Table 13: Axionic fields for the T T IIA orientifold, generators of Solv(so(6, 6)), 1 × 5 O(1, 1)3 gradings, and GL(5, R)representations.

GL(5)–rep. generator root field dim.

— T(0,0,0) a b (a>b) gab 10 { − } { } ab 10(0,0,1) Tab a + c + d + e Ccde C 10 a ≡ 5(0,1,0) T 1 + a B1a Ba 5 ≡ e 5 , , Te a +  + a + b + c + d Cµνa C 5 (0 1 1) 1 ≡ 1(2,0,0) T β C1 = c 1

3 In this case the grading is with respect to the O(1, 1) group generated by Hβ,Hλ1 ,Hλ6 and parametrised by the moduli β h, h ,h: · 1 6 β h 3 φ O(1, 1)0 e · = V1 e− 4 , → 1 φ h1 O(1, 1)1 e = V1 (V5) 5 e 2 , → 3 φ h6 O(1, 1) e =(V ) 2 e− 4 . (75) 2 → 5

a The generators T , Ta and Tab close a twenty–dimensional nilpotent subalgebra N4 of Solv(so(6, 6)): a a ab N4 = Ba T + C Ta + C Tab , (76) whose algebraic structure is encoded in the non–vanishing commutator

c c [Tab,T ]=T[aδb] . (77)

The corresponding coset representative reads

a a ab L = eC Ta eBa T eC Tab ecT E , (78) where the E factor parametrises the submanifold:

SL(5, R) O(1, 1) O(1, 1) O(1, 1) . (79) 0 × 1 × 2 × SO(5)

23 a a ab A generic element ξa T + ξ Ta + ξ Tab of N4 then induces the following transformations on the axionic scalars

a a ab δC = ξ + ξ Bb ,

δBa = ξa , δCab = ξab . (80)

1 Vector fields. The vector fields of this model are Cµ,Gµ,C1aµ,Baµ, and we name the 1 corresponding field strengths Fµν , Fµν , F1aµν , Haµν . The symplectic section of the field strengths and their duals is

1 ˜ ˜ ˜1a ˜ a Fµν , F ,Faµν , Haµν , Fµν , F µν , F , H , (81) { µν 1 1 µν µν } 3 and in table 14 we give their O(1, 1) gradings and the corresponding E7(7) weights.

Table 14: Field strengths, O(1, 1)3 gradings, and corresponding weights.

vector O(1, 1)3–grading weight 1 Fµν (1, 1, ) w − − 2 F 1 ( 1, 1, 1 )  1  µν − − − 2 − 1 − √2 7 1 F1aµν (1, 0, 2 ) w + 1 + a − 1 1 Haµν ( 1, 0, ) a  − − 2 − √2 7 1 F˜µν ( 1, 1, ) w − 2 − ˜ (1, 1, 1 )  + 1  F1µν 2 1 √2 7 1a 1 F˜ ( 1, 0, ) w  a µν − 2 − − 1 − ˜ a 1 1 H (1, 0, ) a +  µν 2 − √2 7

a a ab The action of infinitesimal duality transformation ξa T + ξ Ta + ξ Tab + ξT on the symplectic section is

δF = ξ F 1 , δF 1 =0,

δF1a = ξa F + ξ Ha , 1 δHa = ξa F , ˜ ˜1a a δF = ξa F + ξ Ha , − ˜ ˜ a a ˜ δF = ξa H ξ F a ξ F, 1 − − 1 − 24 ˜1a a 1 ab δF = ξ F ξ Hb , − − ˜ a a ab ˜1a δH = ξ F + ξ F b ξF . (82) 1 −

The explicit symplectic realisation of the N4 generators together with the computation of the vector kinetic matrix can be found in appendix.

3.5. T T IIA orientifold with D6–branes. 3 × 3 Solvable algebra of global symmetries. The embedding of the sl(2, R)+so(6, 6) algebra inside e7(7) is defined by the following identification of the simple roots β =   , 1 1 − 2 β =   , 2 2 − 3 β3 = 3 + 4 , β =  +  , 4 − 4 5 β =  +  , 5 − 5 6 β6 = a + 4 (83) for the so(6, 6) factor, and

β = a + 1 + 2 + 3 , (84) for the sl(2, R) one. The correspondence axion–root is quite simple and is summarised in table 15.

Table 15: Axionic fields for the T T IIA orientifold, generators of Solv(so(6, 6)), 3 × 3 O(1, 1)3 gradings, and GL(3, R) GL(3, R)representations. ×

GL(3) GL(3)–rep. generator root field dim. × — T(0,0,0) i j,a b (ib) gij,gab 6 { − − } { ab } (1, 3)(0,0,1) Tab a + c C 3 ia (3, 3)(0,1,0) T i + a Bia 9 i a (3, 3)(0,1,1) Ta a + i + b + c Cibc Ci 9 ij ≡ (3, 1) T i + j + a + b + c Ckµν Cij 3 (0,2,1) ≡ (1, 1) T β Cijk c 1 (2,0,0) ≡

The triple grading refers to three O(1, 1) groups generated by Hβ,Hλ3 ,Hλ6 and parametrised by the moduli β h, h ,h: · 3 6 β h φ O(1, 1) e · = V e− 4 , 0 → 3 25 h 1 1 φ 3 3 3 2 O(1, 1)1 e =(V3) (V30) e , → 1 3 h6 φ O(1, 1) e =(V 0) 3 e− 4 . (85) 2 → 3

ia i ij The generators T , Tab, Ta and T form now a 24–dimensional solvable subalgebra

N6 of Solv(so(6, 6)):

ia ab a i ij N6 = Bia T + C Tab + Ci Ta + Cij T , (86) whose algebraic structure is encoded in the non–vanishing commutators

ic i c Tab,T = T[aδb] ,  j ij a Tia,Tb = T δb . (87)   A possible choice for the coset representative is then

C T ij Ca T i B T ia Cab T cT L = e ij e i a e ia e ab e E , (88) with E parameterising the submanifold GL(3, R) GL(3, R) O(1, 1) . (89) 0 × SO(3) × SO(3)

ij a i ia ab Under an infinitesimal transformation ξij T + ξi Ta + ξia T + ξ Tab of N6 the variation of the axionic scalars is

a a ab δCi = ξi + ξ Bib , a δCij = ξij + ξa[i Cj] ,

δBia = ξia , δCab = ξab . (90)

i i ijk a abc Vector fields. The vector fields of this model are Gµ,Cµ =  Cjkµ, Baµ, Cµ =  Cbcµ, i i a and we name the corresponding field strengths Fµν , Fµν , Haµν , Fµν . The symplectic section of the field strengths and their duals is

i i a ˜ ˜ ˜ a ˜ F ,F , Haµν ,F , Fiµν , Fiµν , H , Faµν , (91) { µν µν µν µν } 3 and in table 16 we give their O(1, 1) gradings and the corresponding E7(7) weights.

ij a i ia ab The action of an infinitesimal duality transformation ξij T +ξi Ta+ξia T +ξ Tab+ξT on the symplectic section is

δF i =0, δF i = ξ F i ,

26 Table 16: Field strengths, O(1, 1)3 gradings, and corresponding weights.

Sp–section O(1, 1)3–grading weight i 1 1 F ( 1, 1, ) i  µν − − − 2 − − √2 7 i 1 Fµν (1, 1, 2 ) w + j + k − − 1 1 Haµν ( 1, 0, ) a  − − 2 − √2 7 a 1 F ( 1, 0, ) w + b + c µν − 2 ˜ (1, 1, 1 )  + 1  Fiµν 2 i √2 7 i 1 F˜ ( 1, 1, ) w j k µν − 2 − − − ˜ a 1 1 H (1, 0, ) a +  µν 2 − √2 7 ˜ 1 Faµν (1, 0, ) w b c − 2 − − −

i δHa = ξia F , a a i δF = ξab Hb + ξi F , ˜ ˜ a a ˜ j ˜ δFi = ξia H ξ Fa 2 ξij F ξ Fi , − − i − − ˜ a a j δFi = ξia F + ξi Ha +2ξij F , ˜ a ab ˜ a i a δH = ξ Fa + ξi F + ξF , ˜ i δFa = ξia F + ξ Ha . (92)

The explicit symplectic realisation of the N6 generators together with the computation of the vector kinetic matrix can be found in appendix.

3.6. T T IIA orientifold with D8–branes 5 × 1 Solvable algebra of global symmetries. The embedding of the sl(2, R)+so(6, 6) algebra inside e7(7) is defined by the following identification of the simple roots

β =   , 1 1 − 2 β =   , 2 2 − 3 β =   , 3 3 − 4 β =   , 4 4 − 5 β5 = 5 + 6 ,

β6 = a + 5 , (93)

27 for the so(6, 6) factor, and

β = a + 1 + 2 + 3 + 4 + 5 , (94) for the sl(2, R) one. The correspondence axion–root is quite simple and is summarised in table 17.

Table 17: Axionic fields for the T T IIA orientifold, generators of Solv(so(6, 6)), 5 × 1 O(1, 1)3 gradings, and GL(5, R)representations.

GL(5)–rep. generator root field dim.

— T(0,0,0) i j,a b (i

The triple grading refers to three O(1, 1) groups generated by Hβ,Hλ5 ,Hλ6 ,allcom- muting with SL(5, R), and parametrised by the moduli β h, h ,h: · 5 6 β h φ O(1, 1)0 e · = V5 e 4 , → 1 φ h5 O(1, 1)1 e =(V5) 5 V1 e 2 , → 1 3 h6 φ O(1, 1) e =(V ) 5 e− 4 . (95) 2 → 5 i i ij The generators T 0 , T and T form now a twenty–dimensional solvable subalgebra N8 of Solv(so(6, 6)): i i ij N8 = Bi6 T 0 + Ci T + Cij T , (96) whose algebraic structure is encoded in the non–vanishing commutator

i j ij T ,T0 = T . (97)   A possible choice for the the coset representative is then

ij i i L = eCij T eBi6 T 0 eCi T ecT E , (98) with the E parameterising the submanifold: SL(5, R) O(1, 1) O(1, 1) O(1, 1) . (99) 0 × 1 × 2 × SO(5)

28 ij i i Under an infinitesimal transformation ξij T + ξi T + ξi0 T 0 of N8 the variation of the axionic scalars is

δCij = ξ[i Bj]6 + ξij ,

δBi6 = ξi0 ,

δCi = ξi . (100)

i Vector fields. The vector fields of this model are Gµ, Ci6µ, Cµ, Baµ, and we name the i corresponding field strengths Fµν , Fi6µν , Fµν , H6µν . The symplectic section of the field strengths and their duals is

i ˜ ˜i6 ˜ ˜ 6 F ,Fi µν ,Fµν , H µν , Fiµν , F , Fµν , H , (101) { µν 6 6 µν µν } 3 and in table 18 we give their O(1, 1) gradings and the corresponding E7(7) weights.

Table 18: Field strengths, O(1, 1)3 gradings, and corresponding weights.

Sp–section O(1, 1)3–grading weight i 1 1 F ( 1, 1, ) i  µν − − − 2 − − √2 7 1 Fi6µν ( 1, 1, 2 ) w + i + 6 − 1 Fµν ( 1, 1, 2 ) w − − 1 1 H µν ( 1, 1, )   6 − − 2 6 − √2 7 ˜ (1, 1, 1 )  + 1  Fiµν 2 i √2 7 ˜i6 1 Fµν (1, 1, 2 ) w i 6 − −1 − − − F˜µν (1, 1, ) w − 2 − H˜ 6 (1, 1, 1 )  + 1  µν − 2 − 6 √2 7

ij i i The action of an infinitesimal transformation ξij T + ξi T + ξi0 T 0 + ξT on the sym- plectic section is

δF i =0, j δFi = ξi H ξ0 F + ξij F , 6 6 − i i δF = ξi F , − i δH6 = ξi0 F , ˜ ˜ 6 ˜j6 δFi = ξi F ξ0 H + ξij F + ξFi , − i 6 δF˜i6 = ξ F i ,

29 ˜ ˜i6 δF = ξi0 F + ξ H6 , ˜ 6 ˜i6 δH = ξi F + ξF. (102) −

The explicit symplectic realisation of the N8 generators, together with the computation of the vector kinetic matrix can be found in appendix.

4. Fluxes and gauged supergravity: local Peccei–Quinn symmetry as gauged duality transformations.

In the present section we consider the deformation of N = 4 supergravity induced by the presence of fluxes. We shall restrict our analysis, here, only to IIB orientifolds with some (three–form) fluxes turned on, while we shall defer the study of more general fluxes and of the gauge structure of other models elsewhere.

Differently to what happened in the well–studied T6/Z2 orientifolds, non–abelian gauged supergravities (for the bulk sector) now emerge, due to the presence of gauge fields originating from the ten–dimensional metric, and of axionic scalars associated to the NS–NS two–form B.

4.1. The T T IIB orientifold model 4 × 2 λ In this model, the allowed three–form fluxes are H = Haij ,Faij , and are in corre- ij { } spondence with the representation (4, 6) of SO(2, 2) GL(4, R). The grading simply +2 × counts the number of indices along the internal T4 and, more specifically, is associated to the subgroup O(1, 1) GL(4, R). As mentioned in the introduction, inspection of the 1 ⊂ dimensionally reduced three–form kinetic term indicates for the four–dimensional theory i λ a gauge group Gg with connection Ωg = Xi Gµ + XλAµ and the following structure:

λ [Xi,Xj]=HijXλ . (103)

We may identify the gauge generators with isometries as follows:

λ j Xi = H T , − ij λ λ 1 λ0 ij X = 2 Hij T . (104)

Using relations (31) and the property

λ 1 λ 1 H Hi `λ = H Hk`λ Hijkl , (105) k[j ] 2 ij − 4 λ ij where H = Hij Hλ , one can show that the generators defined in (104) fulfil the following algebraic relations λ 1 [Xi,Xj]=H Xλ HTij , (106) ij − 4 30 which coincide with (103) only if H = 0 which amounts to the condition that F(3) T6 ∧ H(3) = 0 (this condition is consistent with a constraint found in [48] on the embeddingR matrix of a new gauge group in the = 8 theory, which seems to yield an = 8 “lifting” N N of the type IIB orientifold models Tp 3 T9 p discussed here). Under this condition the − × − gauge group is indeed contained in the isometry group of the scalar manifold. Moreover it can be verified that under the duality action of the gauge generators defined in (104)

the vector fields transform in the co–adjoint of the gauge group Gg and thus provide a

consistent definition for the gauge connection Ωg . The variation of the gauge potentials under an infinitesimal transformation with parameters ξλ,ξi reads

λ i λ j λ δAµ = ξ Hij Gµ + ∂µξ , i i δGµ = ∂µξ ,

δCijkµ =0, (107)

and is compatible with the following non–abelian field strengths

λ λ λ λ i j F = ∂µA ∂ν A H G G , µν ν − µ − ij µ ν i i i F = ∂µG ∂νG , µν ν − µ i ijkl F =  (∂µCjklν ∂νCjklµ) . (108) µν − λ The Cij and Φi scalars are also charged and, up to rotations, subject to shifts

1 λ 1 k λ δCij = H ξλ ξ H Φj λ, 2 ij − 2 k[i ] λ λ j δΦi = Hij ξ , (109)

and their kinetic terms are modified accordingly by covariantisations

1 λ 1 k λ DµCij = ∂µCij Hij λ A + G H Φj λ, − 2 µ 2 µ k[i ] λ λ λ j DµΦ = ∂µΦ H G . (110) i i − ij µ

Chern–Simons terms. The gauge group consists of Peccei–Quinn transformations that shift the real part of the vector kinetic matrix N (the generalised theta angle). In [49],[50], it was shown that such a local transformation is a symmetry of the Lagrangian provided suitable generalised Chern–Simons terms are introduced. In the case at hand, the new contribution to the Lagrangian is

µνρσ H Aλ Gi ∂ Cj0 + 1 H Hλ Gi Cj0 Gk G` , (111) Lc.s. λij0 µ ν ρ σ 8 λij0 k` µ ν ρ σ ∝   corresponding to the non–vanishing entries

Cλ, ij = Hλij and Ci, λj = Hλij (112) 0 − 0 0 0 31 where, in general, the coefficients CΓ,ΛΣ define the moduli–independent gauge variation of the real part of the kinetic matrix N

Γ δξ Re NΛΣ = ξ CΓ,ΛΣ . (113)

4.2. Type T T IIB orientifold model. 2 × 4 Let us consider the T T model in presence of the fluxes Hija = ij Ha and Fiab. 2 × 4 These fluxes appear as structure constants

a [Xi,Xj]=ij Ha X , a ab [Xi,X ]=Fi Xb , (114)

a g i a a of the gauge algebra Gg Xi,X ,Xa with connection Ω = G Xi + Baµ X + C Xa, ≡{ } µ µ µ all other commutators vanishing. The identification

ab a X0 = F Tab + Ha T , i − i i a ab i X 0 = F T ,X0 = Ha T, (115) i b a − of the gauge generators with the isometries of the solvable algebra N5, reproduces only a contracted version of the algebra (114) in which three of the central charges Xa vanish and we are left with Xa0 = Ha T .Ifwedenoteby X⊥ = Xa / Xa0 these three − { a} { } { } central generators, we see that the subgroup G 0 = X0,X 0,X0 of the isometry group g { i a} which is gauged coincides with the quotient:

G 0 Gg/ X⊥ , (116) g ≡ { } that amounts to imposing the vanishing of the central terms on all fields. On the other hand, transformations generated by the operators in (115) induce isom- etry transformations with parameters:

ξia = ξi Ha , − ξab = ξi F ab , − i a ba ξi = ξb Fi , a ξ = Haξ , (117) − j i where ξi = ij ξ . Using eqs. (62) and (117), one can then verify that the vectors Gµ, a Baµ and Cµ transform in the co–adjoint representation of Gg under the duality action

32 a g generated by Xi,X ,Xa , so that the above definition of the gauge connection Ω is { } µ consistent:

i j i δBaµ = ξ G ij Ha + ∂µξa = ξi G Ha + ∂µξa , µ − µ a i ba i ba a δC = ξ Bbµ F G ξb F + ∂µξ , µ i − µ i i i δGµ = ∂µξ . (118)

Notice that the action of the central charges Xa amounts just to a gauge transformation on a Cµ. These ten vectors can therefore be used to gauge the group Gg, and the non–abelian field strengths read

i j Haµν = ∂µBaν ∂ν Baµ ij Ha G G , − − µ ν a a a ab i ab i F = ∂µC ∂ν C + F G Bbν F G Bbµ , µν ν − µ i µ − i ν i i i F = ∂µG ∂ν G . (119) µν ν − µ Since Gg is not part of the global symmetries of the Lagrangian, we should restrict ourselves to the quotient Gg, i.e. we demand that central charges T, Xa vanish on all physical { } fields. The gauge transformations of the scalar fields

a ab ij δc = Ha ξ + ξa F Bjb  , − i a ba j ab δCi = ξb Fi + ξ Fj Bbi ,

δBia = ξi Ha , − i δCab = ξ Fiab , (120) − are then compatible with the covariant derivatives

a ab ij Dµc = ∂µc + Ha C Baµ F Bjb  , µ − i a a ba j ab DµC = ∂µC Bbµ F G F Bbi , i i − i − µ j DµBia = ∂µBia + Giµ Ha , i DµCab = ∂µCab + Gµ Fiab . (121)

Chern–Simons terms. Also in this case local Peccei–Quinn transformations demand the inclusion in the Lagrangian of the Chern–Simons terms

µνρσ i a a i ij ab Lc.s. =  Ha Gµ Ciν ∂ρCσ Ha Cµ Ciν ∂ρGσ  Fj Baµ Ciν ∂ρBbσ  − − 1 ab i k 1 ij ab k + 8 Ha Fk Gµ Ciν Gρ Bbσ 8  Ha Fj Bbµ Ciν Gρ Gkσ , (122) −  corresponding to the non–vanishing components

j j Ci, a = δi Ha , j j Ca, i = δ Ha , − i Ca, ib = ij F ab , (123) − j of the CΓ,ΛΣ coefficients.

33 5. Conclusions and outlooks

In the present paper, we have investigated the symmetries and the structure of sev-

eral T6 orientifolds which, in absence of fluxes, have N = 4 supersymmetries in four dimensions. we have not addressed here the question of vacua with some residual su- persymmetry, that will be the subject of future investigations. All these models lead to different low–energy supergravity descriptions. When fluxes are turned on, the deformed Lagrangian is described by a gauged N = 4 supergravity and fermionic mass–terms and a scalar potential are developed. The low–energy Lagrangians underlying these orientifolds are different versions of gauged N = 4 supergravity with six bulk vector multiplets and additional Yang–Mills multiplets living on the brane world–volume. The gaugings are based on quotients (with respect to some central charges) of nilpotent subalgebras of so(6, 6). These nilpotent subalgebras are basically generated by the axion symmetries associated to R–R scalars and to NS–NS scalars originating from the two–form B–field. Along similar lines, one can also consider new examples of orientifolds with N =2, 1 four–dimensional supersymmetries, with and/or without fluxes.

Acknowledgements M.T. would like to thank H. Samtleben for useful discussions and the Th. Division of CERN, where part of this work has been done, for their kind hos- pitality. The work of S.F. has been supported in part by European Community’s Hu- man Potential Program under contract HPRN-CT-2000-00131 Quantum Space-Time, in association with INFN Frascati National Laboratories and by D.O.E. grant DE-FG03- 91ER40662, Task C. The work of M.T. is supported by a European Community Marie Curie Fellowship under contract HPRN-CT-2001-01276.

Appendix. Symplectic realisation of the solvable generators

In this appendix, we give the coset representatives of our models in the symplectic basis of vector fields. This is needed in order to compute the kinetic matrix NΛΣ,which is a complex symmetric matrix in the space of vectors in the theory. Its imaginary and real parts describe the terms

Λ Σµν 1 µνρσ Λ Σ Im NΛΣ Fµν F + 2 Re NΛΣ  Fµν Fρσ . (124)

34 Model T T The Sp(24, R) representation of the solvable generators in model 1 in 4 × 2 the basis (45) is:

00000 0  00000 0  011000 0 T =   ,  η 0000 0      00000 11   −   00000 0   M T 00 0 00  000000  000000 T =   ,  000 M 00    −   000000    000000   0 (ti )T 0000 − λ  00 0000  00 0000 T i =   , λ  00(ti η)T 000  − λ   i   00 0tλ 00    ti η 00000  − λ  00 0 000  00 0 000  00 0 000 T ij =   , (125)  00 0 000    ij   00 t 000  −   0 tij 0000   where each block is a 4 4 matrix, 11 denotes the identity matrix, η ηλλ and × ≡ 0

i λ0 i λ0 ij i j i j (t )j = δ δ , (t )kl = δ δ δ δ . (126) λ j λ k l − l k The coset representative is

ij λ i A 0 L =exp(CijT )exp(Φ T )exp(cT ) E = , (127) i λ  CD where E parametrises the manifold

SO(2, 2) GL(4, R) E O(1, 1) , (128) ∈ 0 × SO(2) SO(2) × SO(4) ×

35 and can be written in the following general form: ϕ e− E(`) 00 0 0 0 ϕ  0 e− E 00 0 0 eϕ E E  00 000 =  ϕ  , (129)  000e ηE(`)η 00    0000eϕ E 1 0   −   ϕ 1   0000 0e− E−  with

λ SO(2, 2) E ` σ , ( ) ˆ ∈ SO(2) SO(2) R× i GL(4, ) E  , ˆ ∈ SO(4) eϕH O(1, 1) , (130) ∈ 0 the hatted indices being the rigid ones transforming under the isotropy group. The blocks is L read ϕ λ ϕ λ i e− E(`) σˆ e− Φi E ˆ 0 − ϕ i A =  0 e− E ˆ 0  , ce ϕ Ei eϕ Ei  0 − ˆ ˆ  ϕ ϕ j ϕ j ce− E(`)λσˆ ce− Φλj E ˆı e Φλj E ˆı ϕ δ − ϕ j − ϕ j C =  ce− Φδi E σˆ ce− 2 C˜ij E ˆ e 2 C˜ij E ˆ  , (`) − k − k e ϕ Φ E δ e ϕ 2 C˜ Ej 0  − − δi (`) σˆ − ij kˆ  ϕ σˆ e E(`)λ 00 ϕ λ σˆ ϕ 1 ˆ ϕ 1 ˆ D =  e Φ E e E− i ce− E− i  , i (`)λ − e ϕ E 1 ˆ  00− − i  ˜ 1 λ Cij = Cij + 4 Φi Φλj . (131) 1 In the sequel we shall need also the expression of A− : ϕ σˆ ϕ σˆ λ e E(`) λ e E(`) λΦi 0 1 ϕ 1ˆı A− =  0 e E− j 0  . (132) 0 e ϕ cE 1ˆı e ϕ E 1ˆı  − − − j − − j  In terms of the matrices h, f f = 1 A ,h= 1 (C i D) , (133) √2 √2 − the kinetic matrix is expressed as (see [51] and references therein)

Nλλ0 Nλi Nλi0 1 N = hf − =  Nij N 0  , (134) ∗ ij N 00  ∗∗ ij  and is characterised by the following entries:

2ϕ σˆ σˆ Nλλ = i e E ` λ E ` λ + cηλλ , 0 − ( ) ( ) 0 0 2ϕ σˆ σˆ λ0 Nλi = i e E E Φ + cΦλi − (`)λ (`)λ0 i N 0 = Φλi , λi − 2ϕ 2ϕ 2 1 ˆı 1 2ϕ λ σˆ σˆ λ0 λ Nij = i (e + e− c ) E− i E− jˆı + e Φi E(`)λ E(`)λ0 Φj + cΦi Φλj , −   2ϕ 1 ˆı 1 N 0 =ice− E− i E− jı 2 C˜ij , ij ˆ − 2ϕ 1 ˆı 1 N 00 = i e− E− i E− jı . (135) ij − ˆ

36 Model T T The Sp(24, R) representation of the solvable generators in model 2 in 2 × 4 the basis (61) is:

00000000

 ij 0000000  00000000    00000000 T 0 =   ,  00000 00  ij     00000000    000110000      001100000 00 0 0000 0  00 0 0000 0  00 0 0000 0  cd   00δab 0000 0 Tab =   ,  00 0 0000 0      00 0 0000 0    00 0 0000δcd   ab     00 0 0000 0 0 0000000  0 0000000 δa δi 0000000  b j     0 0000000 T ia =   ,  0 00000δi δa 0   − j b   ij a   000 δb 0000  −   0 0000000    0 δa ij 000000  − b  0 0 0 0000 0  0 0 0 0000 0  0 0 0 0000 0  b i   δa δj 0 0 0000 0 T i =  −  , a  0 0 0 0000δi δb   j a   ij b   00 δa 0000 0    0 δb ij 0 0000 0  a     0 0 0 0000 0

37 0 0000000  0 0000000  0 0000000    0 0000000 T =   . (136)  011000000      110000000    0 0000000      0 0000000 The coset representative has the form:

c T cT B T ia Ca T i C T ab A 0 L = e 0 0 e e ia e i a e ab E = , (137)  CD where this time the matrix E describes the submanifold: GL(2, R) GL(4, R) E = O(1, 1) , (138) 0 × SO(2) × SO(4) and has the following form:

ϕ i e− E2 ˆ 0000000 ϕ 1 ˆ  0 e E2− i 000000 e ϕ E 1 ˆb  00− 4− a 00000  00 0e ϕ E a 0000 E  − 4 ˆb  =  ϕ 1 ˆ  , (139)  00 0 0e E− 00 0  2 i   ϕ i   00 0 00e− E2 ˆ 00  ϕ a   00 0 000e E4 ˆ 0   b   eϕ E 1 ˆb   00 0 00004− a 

R i GL(2, ) E  , 2 ˆ ∈ SO(2) R a GL(4, ) E ˆ , 4 b ∈ SO(4) eϕH O(1, 1) . (140) ∈ 0 The blocks A, C, D of L can be conveniently described in terms of the following matrices a a ab ab (B)ia = Bia,(C)i = Ci ,(C ) = C :

ϕ e− E2 000 e ϕ c E eϕ E 1 00 A =  − 0 2 2−  , e ϕ Bt E 0 e ϕ E 1 0  − 2 − 4−  ϕ t ϕ 1 ϕ  e C E2 0 e C E− e E4   − − − 4 −  ϕ t ϕ t 1 ϕ 1 ϕ e− c0 (c+ BC ) E2 e (c+ BC ) E2− e− c0 (C B C ) E4− e− c0 B E4 ϕ t − ϕ − 1 − ϕ  e−  (c+ BC ) E2 0 e−  (C B C ) E4− e−  B E4  C = − ϕ t ϕ t 1 − ϕ − 1 ϕ , e c C E e C E− e c C E− e c E  − 0 2 2 − 0 4 − 0 4  − ϕ t ϕ t 1 ϕ 1  e c B E2 e B E− e c E− 0   − 0 − 2 − 0 4  ϕ 1 ϕ ϕ ϕ 1 e E2− e− c0 E2 e B E4 e (C B C ) E4− ϕ − − D =  0 e− E2 00 , (141) 00eϕ E eϕ E 1  4 C 4−   ϕ 1   00 0 e E4−  38 1 it is also useful to compute A− :

ϕ 1 e E2− 000 ϕ ϕ 1  e− c0 E2 e− E2 00 A− = − . (142) eϕ E Bt 0 eϕ E 0  4 4  ϕ −1 t t ϕ 1 ϕ 1  e E− (C B + C )0 e E− C e E−   4 − 4 4  We then compute the kinetic matrix N whose independent components are:

j a Nij Ni Ni Nia ij ia i  N N N a  N = ∗ ab a , (143)  N N b   ∗∗   Nab   ∗∗∗  where

1 ˆ 1 ˆ 2 ϕ 2 ϕ 2 2 ϕ a b Nij = i E− i E− j (e + e− c0 )+e Bia E a E a Bjb+ − 2 2 4 ˆ 4 ˆ 2 ϕ ca a 1 aˆ 1 aˆ bd b a e ( Bic C + C ) E− a E− b (C Bjd + C ) 2 Ba i C c0 , − i 4 4 j − ( j) j 2 ϕ k j j a kj  Ni = i e− c0 ik E ˆ E ˆ + cδi Bia C  , − 2 k 2 k − k a 2 ϕ b a bc c 1 cˆ 1 cˆ da a Ni =ie Bib E ˆ E ˆ +( Bib C + C ) E− c E− d C + c0 C , 4 b 4 b − i 4 4 i 2 ϕ bc c 1 cˆ 1 cˆ  Nia = i e ( Bib C + C ) E− c E− a c0 Bia , − − i 4 4 − ij 2 ϕ i j N = i e− E ˆ E ˆ , − 2 k 2 k N ia = ij Ca , − j i ij N a =  Bja , ab 2 ϕ ad 1 dˆ 1 dˆ cb a b N = i e C E4− d E4− c C + E4 ˆb E4 ˆb , − −  a 2 ϕ ad 1 dˆ 1 dˆ a N b = i e C E− d E− b + c0 δ b , − 4 4 2 ϕ 1 dˆ 1 dˆ Nab = i e E− a E− b . (144) − 4 4

Model T T The Sp(24, R)representationoftheN generators is the following: 1 × 5 4 0 0 0000 0 0  0 0 0000 0 0 δa 0 0000 0 0  b   a   0 δb 0000 0 0 T a =   ,  0 0 0000 δa 0   − b   a   0 0 0000 0 δb   −   0 0 0000 0 0      0 0 0000 0 0

39 00 000000  00 000000  00 000000    00 000000 Ta =   ,  00 0δb 0000  a   b   00 δa 0 0000  −   0 δb 0 0 0000  − a   b   δa 0 0 0 0000 00 0 0 0000  00 0 0 0000  00 0 0 0000    00 0 0 0000 Tab =    00 0 0 0000      00 0 0 0000    00 0 δcd 0000  − ab   cd   00δab 0 0000 01100 0 0 0 0  0000 0 0 0 0  000110000    0000 0 0 0 0 T =   . (145)  0000 0 0 0 0      0000 110 0 0  −   0000 0 0 0 0    0000 0 0 110  −  We have chosen the coset representative to have the form given in eq. (78). We may choose for the matrix E the following matrix form: eϕ E 00 0 0000 ϕ  0 e− E 000000 ϕ 1 ˆb  00e E− a 00000  ϕ 1 ˆb   00 0e− E− a 00 0 0 E =   , (146)  00 0 0 e ϕ/E 00 0  −   ϕ   00 0 0 0e /E 00    00 0 0 0 0e ϕ Ea 0   − ˆb   00 0 0 0 0 0eϕ Ea   ˆb  where: R a SL(5, ) E ˆ ,E O(1, 1) O(1, 1) , b ∈ 1 × 2 × SO(5) eHϕ O(1, 1) . (147) ∈ 0 1 The blocks A, C, D of L and A− have the following form: { ϕ } ϕ Ee cEe− 00 ϕ  0 Ee− 00 A = ˆ ˆ , B Eeϕ cB Ee ϕ eϕ E 1 b ce ϕ E 1 b  a a − − a − − a   ϕ ϕ 1 ˆb   0 Ba e− 0 e− E− a  40 a ϕ ϕ ba a 1 dˆ 0 Ba C Ee− 0 e− (Bb C + C ) E− a a ϕ a ϕ ϕ ba a 1 dˆ ϕ ba a 1 dˆ Ba C Ee cBa C Ee e (B C + C ) E a e c (B C + C ) E a C =  − − − − b − − − b −  , 0 e ϕ Ca E 0 e ϕ Cab E 1 dˆ  − − − b   a ϕ − a ϕ ϕ ab 1 dˆ ϕ ab 1 dˆ   C Ee cC Ee− e C E− b e− cC E− b  ϕ ϕ a e− /E 0 Ba e− E dˆ 0 ϕ ϕ − ϕ a ϕ a ce /E e /E c Ba e E Ba e E D =  − − − dˆ − dˆ  , 00e ϕ Ea 0  − dˆ   00ce ϕ Ea eϕ Ea   − − dˆ dˆ  ϕ ϕ e− /E ce− /E 00 − ϕ 1  0 e /E 00 A− = , (148) e ϕ E a B ce ϕ E a B e ϕ E a ce ϕ E a  − dˆ a − dˆ a − dˆ − dˆ  − ϕ a − ϕ a  0 e E Ba 0 e E   − dˆ dˆ  from equations (133) and (134) we compute the matrix N : (1a) a NN1 N N (1a) a  N11 N1 N1  N = ∗ (1a)(1b) (1a) b , (149)  N N   ∗∗   N ab   ∗∗ ∗  whose entries are

2 ϕ a b 1 N = i e− (Ba Bb E a E a + ) , − ˆ ˆ E2 1 N =ice 2 ϕ (B B Ea Eb + ) , 1 − a b aˆ aˆ E2 (1a) 2 ϕ a b N =ie− Bb E aˆ E aˆ , a a ba 2 ϕ a b N = C + Bb C i ce− Bb E a E a , − ˆ ˆ 2 ϕ 2 2 ϕ a b 1 N = i(e− c + e )(Ba Bb E a E a + ) , 11 − ˆ ˆ E2 (1a) ba a 2 ϕ a b N = (BbC + C ) i ce− Bb E a E a , 1 − − ˆ ˆ a 2 ϕ 2 2 ϕ a b N1 =i(e− c + e ) Bb E aˆ E aˆ , (1a)(1b) 2 ϕ a b N = i e− E a E a , − ˆ ˆ (1a) b ab 2 ϕ a b N = C +ie− cE a E a , − ˆ ˆ ab 2 ϕ 2 2 ϕ a b N = i(e− c + e ) E a E a , (150) − ˆ ˆ where the asterisks denote the symmetric entries.

Model T T The Sp(24, R)representationoftheN generators is the following: 3 × 3 6 00 0 0000 0  00 0 0000 0  00 0 0000 0  cd   00δab 0000 0 Tab =   ,  00 0 0000 0      00 0 0000 0    00 0 0000δcd   ab     00 0 0000 0 41 0 000000 0  0 000000 0 δaδi 000000 0  b j     0 000000 0 T ia =   ,  0 00000 δi δa 0   − j b   i a   000δjδb 0000    0 000000 0    a i   0 δb δj 00000 0 00000000  00000000  00000000  b i   δaδj 0 0 0000 0  T i =   , a  0000000δi δb   − j a   i b   00δjδa 0000 0     0 δbδi 0 0000 0   a j     00000000 0 0 000000  0 0 000000  0 0 000000    0 0 000000 T ij =   ,  0 2 δij 000000  kl   ij −   2 δkl 0 000000    0 0 000000      0 0 000000 00000 0 00  1100 00 0 00  00000 0 00    00000 0 00 T =   . (151)  00000 1100    −   00000 0 00    000110 0 00      001100 0 00

42 We have chosen the coset representative to have the form given in eq. (88). We may choose for the matrix E the following matrix form:

ϕ i e− E1ˆ 00 00 000 ϕ i  0 e E1ˆ 000000 e ϕ E 1 ˆb  00− 2− a 00 000  00 0e ϕ Ea 0000 E  − 2 ˆb  =  ϕ 1 ˆ  , (152)  00 0 0e E− 000  1 i   ϕ 1 ˆ   00 0 0 0e− E1− i 00  ϕ a   00 0 0 0 0e E ˆ 0   2 b   eϕ E 1 ˆb   00 0 0 0 0 0 2− a  where R i GL(3, ) E1ˆ , ∈  SO(3) 1 R a GL(3, ) E2 ˆb , ∈  SO(3) 2 eϕH SO(1, 1) . (153) ∈ 0 1 The blocks A, C, D of L and A− have the following form: { } ϕ i e− E1ˆ 00 0 ϕ i ϕ i  ce− E1ˆ e E1ˆ 00 A = ˆ , B e ϕ Ei 0 e ϕ E 1 b 0  ia − 1ˆ − 2− a   a ϕ i ϕ ab 1 cˆ ϕ a   Ci e− E1ˆ 0 e− C E2− b e− E2 cˆ  ϕ j ϕ j ϕ a ba 1 cˆ ϕ 1b 2 ce C˜ E 2 e C˜ E ce (C + B C ) E− a e cB E− − − ij 1kˆ − ij 1kˆ − − i ib 2 − − ib 2 dˆ 2e ϕ C˜ Ej 0 e ϕ (Ca + B Cba) E 1 cˆ e ϕ B E a C =  − ij 1kˆ − i ib 2− a − ia 2 cˆ  , ce ϕ Ca Ei eϕ Ca Ei ce ϕ Cab E 1 cˆ ce ϕ E a  − i 1kˆ i 1kˆ − 2− b − 2 cˆ   ce ϕ B Ei eϕ B Ei ce ϕ E 1 cˆ   − ia 1kˆ ia 1kˆ − 2− a 0  ϕ 1 ˆ ϕ 1 ˆ ϕ a ϕ a ba 1 cˆ e E− ce E− e B E e (C + B C ) E− a 1 i − − 1 i − ia 2 dˆ − i ib 2 0 e ϕ E 1 ˆ 00 D =  − 1− i  , 00eϕ Ea eϕ Cab E 1 cˆ  2 dˆ 2− b   ϕ 1 cˆ   00 0 e E2− a  ϕ 1ˆ e E1− i 000 ϕ 1ˆ ϕ 1ˆ 1  ce− E− i e− E− i 00 A− = − 1 1 , eϕ E a B 0 eϕ E a 0  2aˆ ia 2aˆ  ϕ −1cˆ ba a ϕ 1cˆ ab ϕ 1cˆ  e E− a (B C + C )0 e E− a C e E− a   − 2 ib i − 2 2  ˜ 1 a Cij = Cij + 2 Ci Bja . (154)

The vector kinetic matrix can now be calculated and has the following form:

a Nij Nij0 Ni Nia a  Nij00 Ni0 Nia0  N = ∗ ab a , (155)  N N b   ∗∗   Nab   ∗∗∗  whose entries are

a 2 2 ϕ 2 ϕ 1 ˆ 1 ˆ Nij =2cBa i C i (c e− + e ) E− i E− j + ( j) − 1 1 2 ϕ a ba c bc 1 cˆ 1 cˆ 2 ϕ a b e (Ci + Bib C )(Cj + Bjb C ) E2− a E2− c + e Bia Bjb E2 cˆ E2cˆ ,  43 2 ϕ 1 ˆ 1 ˆ N 0 = 2 C˜ij +ice− E− i E− j , ij − 1 1 a a 2 ϕ b cb ad 1 eˆ 1 eˆ a b Ni = cC +ie (C + Bic C ) C E− b E− d + Bib E c E c , − i i 2 2 2 ˆ 2ˆ 2 ϕ  b cb 1 eˆ 1 eˆ  Nia = cBia +ie (C + Bic C ) E− b E− a , − i 2 2 2 ϕ 1 ˆ 1 ˆ N 00 = i e− E− i E− j ij − 1 1 a a Ni0 = Ci ,

Nia0 = Bia , ab 2 ϕ a b ac bd 1 eˆ 1 eˆ N = i e E c E c + C C E− c E− d , − 2 ˆ 2ˆ 2 2 a 2 ϕ ac 1 eˆ 1 eˆ  N b = c i e C E− c E− b , − 2 2 2 ϕ 1 eˆ 1 eˆ Nab = i e E− a E− b . (156) − 2 2

Model T S The Sp(24, R)representationoftheN generators is the following: 5 × 1 8 0 0000 0 00 i  000δj 00 00 i  δj 0000 0 00  −   0 0000 0 00 T i =   ,  0 0000 0 δi 0   j     0 0000 0 00    0 0000 0 00    0 0000 δi 00  − j  00 0 0000 0  00 δi 00 0 0 0 − j  00 0 0000 0  i  i  δj 0000000 T 0 =   ,  00 0 0000 δi   j   −   00 0 0000 0    00 0 00δi 00  j     00 0 0000 0 00000000 ij  δkl 0000000  00000000    00000000 T ij =   ,  0 0000δij 00  kl     00000000    00000000      00000000

44 00000000  00000000  00000000    00000000 T =   . (157)  0110 00000      110000000    000110000      001100000 We have chosen the coset representative to have the form given in eq. (98). We may choose for the matrix E the following matrix form

ϕ i e− E ˆ 0000000 ϕ 1 ˆ  0 e− E− i 00 0 000 ϕ  00e− E 00000  ϕ   000e− /E 0000 E =   , (158)  0000eϕ E 1 ˆ 000  − i   ϕ i   00000e E ˆ 00    000000eϕ/E 0     0000000eϕ E    where: R i SL(5, ) E ,E O(1, 1) O(1, 1) , ˆ ∈ 1 × 2 × SO(5) eϕH SO(1, 1) . (159) ∈ 0 1 The blocks A, C, D of L and A− have the following form { } i E ˆ 000 j 1 ˆ ϕ  (Bi Cj + Cij ) E ˆ E− i Bi ECi/E  A = e− − , C Ei 0 E 0  − i ˆ   i   Bi E ˆ 001/E  j 1 ˆ c (Bi Cj + Cij ) E ˆ cE− i cBi EcCi/E i − ϕ  cE ˆ 000 C = e− , cB Ei 00c/E  i ˆ   cC Ei 0 cE 0   − i kˆ  1 ˆ j E− i (Bi Cj + Cij ) E ˆ Ci/E Bi E i − D = eϕ  0 E ˆ 00 , 0 B Ei 1/E 0  i ˆ   0 C Ei 0 E   − i ˆ  1ˆ E− i 00 0 j j j j 1 ϕ  Eˆı (Cj Bi Cji) Eˆı Eˆı Bj Eˆı Cj  A− = e − − . C /E 01/E 0  i   B E 00 E   − i  The vector kinetic matrix can now be calculated and has the following form

j Nij Ni Ni Ni0 ij i i  N N N 0  N = ∗ , (160)  NN0   ∗∗   N 00   ∗∗∗  45 whose entries are

2 ϕ 1 ˆ 1 ˆ k n 1 2 Nij = i e E− i E− j +(Bi Ck + Cik)(Bj Cn + Cjn) E ˆ E ˆ + Ci Cj + Bi Bj E , −  ` ` E2  j 2 ϕ k j Ni = c i e (Bi Ck + Cik) E ˆ E ˆ , − ` `   2 ϕ k j 1 Ni = i e (Bi Ck + Cik) E ˆ Bj E ˆ + Ci , −  ` ` E2  2 ϕ k j 2 Ni0 =ie (Bi Ck + Cik) E `ˆ Cj E `ˆ + E Bi , ij 2 ϕ i j  N = i e E ˆ E ˆ , − ` ` i 2 ϕ i j N = i e Bj E ˆ E ˆ , − ` ` i 2 ϕ i j N 0 =ie Cj E `ˆ E `ˆ ,

2 ϕ i j 1 N = i e Bi Bj E ˆ E ˆ + , −  ` ` E2  2 ϕ i j N 0 = c +ie Ci Bj E `ˆ E `ˆ , 2 ϕ i j 2 N 00 = i e Ci Cj E ˆ E ˆ + E . (161) − ` `  

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