Supersymmetry Reduction of N-Extended Supergravities in Four Dimensions

CERN-TH/2001-300 hep-th/0110277 October, 2001

Supersymmetry reduction of N-extended supergravities in four dimensions.

Laura Andrianopoli1, Riccardo D’Auria1 and Sergio Ferrara1,2,3,4

1 Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino and Istituto Nazionale di Fisica Nucleare (INFN) - Sezione di Torino, Italy 2 CERN Theoretical Division, CH 1211 Geneva 23, Switzerland 3 Istituto Nazionale di Fisica Nucleare (INFN) - Laboratori Nazionali di Frascati 4 Department of Physics & Astronomy, University of California, Los Angeles, U.S.A.

Abstract: We consider the possible consistent truncation of N-extended supergravities to lower N 0 theories. The truncation, unlike the case of N-extended rigid theories, is non trivial and only in some cases it is sufficient just to delete the extra N N 0 gravitino − multiplets. We explore different cases (starting with N =8downtoN 0 2) where the reduction implies restrictions on the matter sector. We perform a detailed≥ analysis of the interesting case N =2 N = 1. This analysis finds applications in different contexts of superstring and M-theory−→ dynamics. 1 Introduction

It is well known that for globally supersymmetric theories, with particle content of spin 1 0, 2 , 1 any theory with N supersymmetries can be regarded as a particular case of a theory with a number N 0 supersymmetry–extended multiplets into N 0-multiplets. Of course N-extended supersymmetry is more restrictive than N 0 supergravity theories. Indeed, let us consider a standard N-extended supergravity theory with N 1 gravitini and a given number of matter multiplets with spin 0, 2 , 1: then the N 0-extended supergravity obtained by reduction from the mother theory will no longer be standard 3 because a certain number N N 0 of spin multiplets appear in the decomposition. − 2 Therefore to obtain a standard N 0-extended supergravity one must truncate out at least 3 the N N 0 spin 2 multiplets and all the non-linear couplings that they generate in the supergravity− action. The most known example is N = 2 supergravity in presence of hypermatter [2], [3], [4]. The non-linear couplings of the hypermultiplets generate what is called a “quaternionic geometry” [2]. If we regard the N = 2 hypermultiplets as a pair of N = 1 Wess-Zumino multiplets, what we obtain is incompatible with N = 1 supergravity where the non-linear couplings must describe a K¨ahler-Hodge manifold geometry [5]. Therefore, in order to consistently reduce a N = 2 supergravity to a N = 1 theory, the former theory must have the property that a certain submanifold of the original quaternionic manifold be a K¨ahler-Hodge manifold. Note that in rigid supersymmetry hypermultiplet couplings are described by HyperK¨ahler geometry which is instead compatible with N = 1 supersymmetry. As an illustrative example let us consider maximal N = 8 supergravity in D = 4 [6] trun- cated to lower N 0 supergravities. In this situation the consistent truncation consists in 3 deleting only spin 2 multiplets for sufficiently high N 0 (N 0 =6, 5, 4), but for N 0 4, where the matter sectors begin to appear, the consistent truncation also requires to delete≤ some matter multiplets. We will illustrate how this process of reduction can be understood in group-theoretical and geometrical terms, by requiring that certain geometrical condi- tions dictated by supergravity define some submanifold of the original scalar manifold E7(7)/SU(8) of N = 8 supergravity. Returning to the N =2 N = 1 case, we can show that this generally demands −→ SK a reduction of both special K¨ahler manifold ( (nV )) [3],[7], [8], [4] and quaternionic manifold ( Q(n )), where n and n are theM number of vector multiplets and hypermul- M H V H tiplets respectively. By equipping these manifolds with complex coordinates zi SK (i =1, ,n ) and real coordinates qu Q (u =1, , 4n ) the Riemann tensors∈M are V H b given respectively··· by: ∈M ··· b R = g g + g g C C gmn (1.1) ik` i` k k` i − `n ikm i uv αA βB = Ωx (σ )ABCαβ +IRαβABbb (1.2) R pqUu Ubvbbb −bb2bbpq bxb bb bbb bbpqb where the SU(2) triplet and singlet parts are the SU(2) and Sp(2nH ) curvatures respec- tively. Here A, B =1, 2; x =1, 2, 3 are indices of the fundamental and adjoint represen-

1 tation of SU(2) and α, β... =1, n are indices in the fundamental representation of ··· H Sp(2nH). The K¨ahler metric g = ∂ ∂ ,with = log[i(XΛF F XΛ)], is given in terms of i` i `K K − Λ − Λ Λ e SK (X ,FΛ) which are the holomorphic symplectic sections of e(they are related to the bb b b e Me Λ Λ K2 Λ covariantlye holomrphic symplectic sections (L ,MΛ)by(L ,MΛ)=e (X ,FΛ)). The tensor eC is threefold symmetric and covarianty holomorphic, i.e. C = eKW (with ikm e e ikm e ikm Wikm holomorphic). e e e Q αA On ,bb denotes the vielbein 1-form. Furthermore, we have: bb bb M Ub b b bbb 1 Ωx dωx + xyzωy ωz = iC (σ ) αA βB, (1.3) ≡ 2 ∧ − αβ x ABU ∧U IR α d∆α +∆α ∆γ β ≡ β γ ∧ β =  Aα B + Aγ Bδ CαρΩ . (1.4) − AB U ∧Uβ U ∧U AB ρβγδ where Ωρβγδ is completely symmetric in its four indices. The N =2 N = 1 reduction imposes a number of conditions on the above defined structures, which→ have to be satisfied in order to have a consistent reduction. In par- ticular, we find that the two scalar manifolds SK and Q have to be reduced to the SK KH M Q M submanifolds R(nC ) and (nh) ,wherenC nV nV0 , nh nH M ⊂M M ⊂M ≤ − KH ≤ are the complex dimensions of the two K¨ahler–Hodge manifolds R and and nV0 is the number of N = 1 vector multiplets. M M

We first discuss the two extreme cases nV0 = nV (nC =0)andnV0 =0(nC = nV ). In the first case no N = 1 chiral multiplet coming from N = 2 vector multiplet is retained and all N = 1 vector multiplets may remain. In the second case, all the N = 1 vector multiplets are truncated out, and no restrictions appear on the special-K¨ahler manifold: SK R = . InM the generalM case, let us decompose the coordinates on SK: M zi (zi,zα) (1.5) → and those on Q: b M qu (ws, ws,nt, nt) (1.6) → i s where z (i =1, ,nC )andw (s =1, ,nh) are the holomorphic coordinates in R KH ··· α ··· t M and respectively, and z (α =1, ,nV0 = nV nC )andn (t =1, ,nH nh) are theM holomorphic coordinates in their··· orthogonal complements.− Splitting··· furthermore− ˙ ˙ the N = 2 vector indices Λ (Λ, Λ), where Λ = 1, ,nV0 and Λ=0, 1, ,nC , we find → ···KH ··· the following constraints to be satisfied on R from supersymmetry reduction. e M ×M On R we get, for consistent reduction of the special geometry sector in the ungauged case:M

Cijα =0 (1.7) MR Λ | Λ Λ L R =0,fi R iL R =0 (1.8) |M |M ≡∇ |M Λ˙ Λ˙ fα R αL R =0 (1.9) |M ≡∇ |M The parent (non holomorphic) vector kinetic matrix ,satisfieson : NΛΣ MR

ΛΣ˙ R =0, (1.10) N |M ee 2 1 Furthermore we obtain that ΛΣ R fΛΣ is holomorphic, while Λ˙ Σ˙ has no restric- N |M ≡ 2 N tions and gives the period matrix on R, which is indeed a Special-K¨ahler manifold. For the hypermultiplet sector, theM reduction is more subtle because we have to reduce the holonomy from SU(2) Sp(2nH )toU(1) SU(nh) which corresponds to decompose × × ˙ the SU(2) indices A, B, ... (1, 2) and the Sp(2nH ) indices α, β... (I,I). The following constraints are found on the→ geometrical structure of the manifold→ KH Q M ⊂M

ΩIJKL˙ KH = 0 (1.11) |M 2I 1I˙ KH =( )∗ KH = 0 (1.12) U |M U |M In particular, the second equation implies that the complex scalars of the chiral multi- plets coming from the reduced quaternionic manifold are at most half of the quaternionic dimension of the original N = 2 manifold [9]. The present investigation concerning the N =2 N = 1 reduction is further analyzed in the most general case when isometries of the scalar→ manifolds are gauged. In particular we find that the number of reduced N = 1 vector multiplets and of N = 1 chiral multiplets obtained by truncation of the N = 2 vector multiplets (which are in the adjoint representation of some gauge group G(2)) depend on the gauge group G(1) under which the reduced hypermultiplets are charged. Indeed, if Adj(G(2)) Adj(G(1))+ R(G(1)), then the chiral multiplets coming from N = 2 vector multiplets are→ in R(G(1)). The reduction of the gauge group further implies constraints on the special geometry and quaternionic Killing vectors and prepotentials [10],[4]. For the K¨ahlerian Killing vectors ki and prepotential P 0 we find: Λ Λ

b i α e e kΛ˙ =0,kΛ = 0 (1.13) i i 0 kΛ =ig ∂PΛ = 0 (1.14) 0 6 PΛ˙ =0. (1.15) Furthermore, for the quaternionic Killing vectors ku and SU(2)-valued prepotentials P x, Λ Λ we find: e e s t kΛ˙ =0,kΛ = 0 (1.16) s ss 3 kΛ =ig ∂sPΛ = 0 (1.17) 3 6 PΛ˙ = 0 (1.18) i PΛ =0, (i = 1, 2) (1.19)

The N = 1 D-term and superpotential are respectively given by 1:

Λ 1ΛΣ 0 3 D = 2(Imf)− (P (z, z)+P (w, w)) (1.20) − Σ Σ + KR KH i Λ˙ 1 2 L = e 2 W (z, w)= L P iP (1.21) 2 Λ˙ − Λ˙   KH where R, H are the K¨ahler potentials on R and respectively. ThisK reductionK may find applications andM is in factM related to many interesting aspects of string theory or M theory compactified on a Calabi-Yau threefold. Indeed M-theory 1Particular cases of these formulae have been obtained in [11] - [19].

3 on a Calabi-Yau threefold originates a N = 2 theory in five dimensions [20]. Trivial reduction on S1 would give a N = 2 theory in D = 4. However, if we reduce on the 1 orbifold S /Z2 [21] - [25], then we obtain a N = 1 theory with a particular truncation of the D =5,N = 2 supergravity states. Other applications are related to brane-dynamics where the theory on the brane has lower supersymmetry than the theory on the bulk [26], [27]. A different mechanism is obtained by considering Type IIB theory on a Calaby-Yau threefold in presence of H-fluxes [11] - [19] where also N =1(orN = 0) supersymmetric vacua can be studied. A related issue is the partial supersymmetry breaking of N =2downtoN =1 through a superHiggs mechanism [28], [29]. If one integrates out the massive gravitino, then the theory should become a N = 1 theory. In this case to “integrate out” is in principle different from truncating unless very special situations occur. However in the minimal model studied in reference [28], the resulting N = 1 Lagrangian is a particular case of the general case studied here. The paper is organized as follows: In section 2 we study the decomposition of the N = 8 supergravity multiplet into N 0 < 8 supermultiplets and infer the reduced theories from group-theoretical arguments. In section 3 we extend the analysis to three, five and six dimensional maximal supergrav- ities reduced to eight supercharges. In section 4 we give the interpretation of the reduction procedure in a geometrical setting which will be useful to apply our results to the specific problem of the N =2 N =1 reduction. −→ In section 5 we discuss the constraints coming from supersymmetry when the reduction procedure is applied to ungauged theories. Insection6,whichistheheartofthepaper,wegivetheanalysisoftheN =2 N =1 reduction in detail, also in presence of gauging, both in the vector multiplet and−→ hyper- multiplet sectors. At the beginning of the section we discuss the constraints coming from the gravitino truncation, while in section (6.1) and (6.2) we study the reduction of the N = 2 vector multiplet sector. Subsection (6.3) is devoted to the truncation of the hy- permultiplets sector, while subsection (6.4) discusses further consequences of the gauging. In subsection (6.5) the computation of the reduction of the scalar potential is given, and finally in subsection (6.6) we give examples of supergravity models which realize this con- sistent truncation. The Appendices include some technical details related to the reduction. In particular, in Appendices A and B we show the consistency of the N =8 N = N 0 truncation in the superspace Bianchi identities formalism and we apply it to→ the N =2 N =1 reduction of gauged supergravity. In Appendix C we prove a formula valid for the→ N =2 vector multiplets which is useful for the truncation. Appendix D refers to the reduction of the special-K¨ahler manifolds with special coordinates; Appendix E contains the reduc- tion of an important relation valid on quaternionic geometry in presence of isometries and Appendix F shows the consistency of the reduction of the N = 2 scalar potential to N = 1 and exploits some magic properties of the supersymmetry Ward identities. Finally, Appendix G contains the explicit form of the N =2andN = 1 lagrangians which are left invariant under the supersymmetry transformation laws given in the text.

4 2 N =8 N 0 reduction without gauging −→ Reduction of N = 8 supergravity to 2 N 0 6 offers interesting examples of consistent truncations of standard supergravity [30],[31].≤ ≤ We restrict our analysis to theories whose σ-models are given by symmetric spaces G/H. This includes all the theories with N 0 3 and a subset of the N = 2 theories. The analysis turns out to be particularly simple≥ in all these cases. Let us first consider N 0 =5, 6 where the reduction only involves the graviton multiplet 3 and N N 0 spin 2 multiplets. In the N− =6casetheN = 8 R-symmetry group SU(8) decomposes as:

SU (8) SU (6) U (1) SU (2) (2.1) → × × where SU(2) is the group commuting with the N 0 =6 R-symmetry U(6). Correspond- 3 ingly the N = 8 graviton multiplet decomposes into N 0 =6spin2and 2 multiplets, as follows: 3 1 (2), 8( ), 28(1), 56( ), 70(0)  2 2  −→

3 1 3 1 (2), 6( ), (15 + 1)(1), (20 + 6)( ), (15 + 15)(0) 2 ( ), 6(1), 15( ), 20(0) (2.2)  2 2  ⊕  2 2  3 The hypersurface corresponding to freeze 40 scalars of the spin 2 multiplets is precisely ? the N 0 =6σ-model described by the symmetric space SO (12)/U(6). Therefore by just 3 deleting the two spin 2 multiplets one obtains standard N = 6 supergravity. Let us now consider N 0 = 5. In this case the decomposition of the N = 8 graviton multiplet into N 0 = 5 multiplets, corresponding to the R-symmetry decomposition

SU (8) SU (5) U (1) SU (3) (2.3) → × × is: 3 1 3 1 (2), 8( ), 28(1), 56( ), 70(0) (2), 5( ), 10(1), (10 + 1)( ), (5 + 5)(0)  2 2  −→  2 2  ⊕ 3 1 3 ( ), (5 + 1)(1), (10 + 5)( ), (10 + 10)(0) (2.4) ⊕  2 2  3 If we delete the three spin 2 multiplets we obtain standard N = 5 supergravity, or, 3 geometrically, freezing the 60 scalars inside the spin 2 multiplets corresponds to single out the manifold SU(5, 1)/U(5) E /SU(8). ⊂ 7(7) When N 0 4 a new phenomenon appears since in this case also matter multiplets start ≤ to appear in the decomposition of N = 8 supergravity into N 0-extended supergravities. 3 Therefore in this case deleting the spin 2 multiplets is only a necessary, but not sufficient condition to obtain a consistent N 0-extended supergravity theory. Let us first start with N 0 = 4 (this actually corresponds to compactify a Type II theory in ten dimensions on T T /Z ). The decomposition of the N = 8 graviton multiplet 2 ⊗ 4 2 into N 0 = 4 multiplets, corresponding to

SU (8) SU (4) SU (4) U (1) (2.5) → × × 5 is: 3 1 3 1 (2), 8( ), 28(1), 56( ), 70(0) (2), 4( ), 6(1), 4( ), 2(0)  2 2  −→  2 2  3 1 1 4 ( ), 4(1), (6 + 1)( ), (4 + 4)(0) 6 (1), 4( ), 6(0) (2.6) ⊕  2 2  ⊕  2  3 If we now delete the 4 spin 2 multiplets this is equivalent to freeze 32 scalars. When this occurs the E7(7)/SU(8) manifold reduces to the submanifold (SU(1, 1)/U(1)) SO(6, 6)/SU(4) SU(4), corresponding to the product space of the N = 4 supergrav-× × ity σ-model and the σ-model of 6 vector multiplets. In this case a standard N 0 =4 supergravity coupled to 6 vector multiplets corresponds to a consistent truncation since E SU(1, 1) SO(6, 6). 7(7) ⊃ × The last case we would like to consider is N 0 = 2 where there are two kinds of matter multiplets, namely the vector multiplets and the hypermultiplets. In the standard N =2 theory the corresponding σ-model generally is not a coset, but we limit ourselves to examine this case, namely = G/H. The consistent truncation will now receive severe constraints on the matterM content since the submanifold of the N =8σ-model must factorize as: SK(n ) Q(n ) E /SU(8) (2.7) M V ×M H ⊂ 7(7) wherewehavedenotedwith SK(n )and Q(n ) the special-K¨ahler and quaternionic M V M H manifolds of real dimensions 2nV and 4nH respectively. The decomposition of the N = 8 graviton multiplet gives now: 1 (2), 8(3/2), 28(1), 56( ), 70(0) [(2), 2(3/2), (1)]  2  −→ 1 1 1 6 (3/2), 2(1), ( ) 15 (1), 2( ), 2(0) 20 ( ), 2(0) (2.8) ⊕  2  ⊕  2  ⊕  2  3 We immediately see that deleting the spin 2 multiplets all the scalars survive. So the question is now, how many scalars we must delete so that the scalar submanifold enjoys the above property of reducing to SK(n ) Q(n ). M V ×M H Two immediate solutions are obtained [32]. For nH =0,nV = 15 we find:

SK (n = 15) = SO∗(12)/U(6) E /SU(8) (2.9) M V ⊂ 7(7) which is indeed a special-K¨ahler manifold (coinciding with the σ-model of N =6super- gravity). The other solution is nV =0,nH = 10 for which

Q(n = 10) = E /SU(6) SU(2) E /SU(8) (2.10) M H 6(2) × ⊂ 7(7) which is indeed a quaternionic space. It corresponds to the σ-model obtained by com- pactification of Type IIB on T6/Z3 where only the untwisted states were retained. By c-map of (2.10) we obtain another solution with nV =9andnH = 1 corresponding to Type IIA on T6/Z3 [33]: SU(3, 3) SU(2, 1) E /SU(8) (2.11) SU(3) SU(3) U(1) × SU(2) U(1) ⊂ 7(7) × × ×

6 If we look for other maximal subgroups G G E we find [34] (see Table 3): 1 × 2 ⊂ 7(7) G G = Sp(6,R) G ; SU(1, 1) F ; SU(1, 1) SO(6, 6) ; SU(4, 4) (2.12) 1 × 2 × 2(2) × 4(4) ×

The first two correspond to (nV ,nH )=(6, 2) and to its c-map image (1, 7), namely: Sp(6,R) G 2(2) E /SU(8) (2.13) U(3) × SO(4) ⊂ 7(7) SU(1, 1) F 4(4) E /SU(8) (2.14) U(1) × Usp(6) Usp(2) ⊂ 7(7) × From the last two cases we can obtain a N = 2 truncation of N =4andN =3 supergravities with six and four hypermultiplets respectively: SO(6, 6) SO(6, 4) (2.15) SO(6) SO(6) −→ SO(6) SO(4) × × SU(4, 3) SU(4, 2) (2.16) SU(4) SU(3) U(1) −→ SU(4) SU(2) U(1) × × × × with (nV ,nH )=(1, 6) and (nV ,nH )=(0, 4) respectively. The first, together with its c-map (nV ,nH )=(5, 2) SU(1, 1) SO(2, 4) SO(4, 2) (2.17) U(1) × SO(2) SO(4) × SO(4) SO(2) × × 2 corresponds to Type IIB (Type IIA) on T6/Z4 . The last is a truncation of the (nV ,nH )= (0, 10) case and its c-map is

SU(3, 1) SU(2, 1) (2.18) SU(3) U(1) × SU(2) U(1) × × with (nV ,nH )=(3, 1) ( This is a truncation of the (nV ,nH )=(9, 1) case.). By the decomposition

SO(6, 6) SO(4, 6 p) SO(2,p) (2.19) −→ − × we can obtain the additional cases:

(nV =2,nH =5) p = 1 (2.20)

(nV =4,nH =3) p = 3 (2.21)

(nV =3,nH =4) p = 2 (2.22)

(nV =6,nH =1) p = 5 (2.23)

(nV =7,nH =0) p = 6 (2.24)

Their c-map do not give new models. We note that the case p = 6 is a truncation of the (nV ,nH )=(15, 0) case and that the case p = 5 is not a subcase of the (nV ,nH )=(9, 1)

2 SO(4,2) SU(2,2) Note that SO(4) SO(2) SU(2) SU(2) U(1) . × ∼ × × 7 Usp(2,2) case because the corresponding quaternionic manifold is in this case Usp(2) Usp(2) which is × not the same of the (nV ,nH )=(9, 1) case. In conclusion we have found eleven “maximal” cases: the cases (nV ,nH )=(15, 0), (6, 1) which have no c-map counterpart, the case (nV ,nH )=(3, 4) which is self conjugate under c-map and four pairs conjugate under c- map, namely:

c map (nV =6,nH =2) − (nV =1,nH = 7) (2.25) c←→map (nV =5,nH =2) − (nV =1,nH = 6) (2.26) ←→c map (nV =0,nH = 10) − (nV =9,nH = 1) (2.27) c ←→map (n =4,n =3) − (n =2,n = 5) (2.28) V H ←→ V H

Many of these cases can be retrieved from Type II string theories compactified on ZZN orbifolds which preserve one left and one right supersymmetry [33],[35].

3 D =3, D =5and D =6reduction of maximal supergravity to theories with eight supercharges.

The same analysis can be carried out in N = 2 theories (eight supercharges) in D =3, D = 5 (for the cases where the scalars span a symmetric space) and in D =6. In D =5,N = 8 supergravity has a non-linear σ-model E6(6)/USp(8)[30]. We consider only the N =8 N =2case. The 42 scalars,→ decomposed with respect to the N = 2 theory, consist of 14 scalars belonging to vector multiplets and 4 7 = 28 scalars belonging to quaternionic multiplets, giving (n =14,n =0)and(n ×=0,n = 7) models which correspond to SU∗(6) V H V H USp(6) ⊂ E6(6) and F4(4) E6(6) [32]. For each model in D = 4 there is a parent in D =5 USp(8) USp(6) USp(2) ⊂ USp(8) (the above correspond× to the n n = 0 cases). V · H If we now look to spaces with isometry groups G1 G2 E6(6),whereG1, G2 corre- spond to real special geometry and quaternionic geometry× respectively,⊂ we find (see Table 3): G G = SL(3,C) SU(2, 1) (3.1) 1 × 2 × which give rise to SL(3,C) SU(2, 1) E 6(6) (n =8,n = 1) (3.2) SU(3) × SU(2) U(1) ⊂ USp(8) V H × and [34] G G = SL(3, IR ) G (3.3) 1 × 2 × 2(2) giving SL(3, IR ) G E 2(2) 6(6) (n =5,n =2). (3.4) SO(3) × SO(4) ⊂ USp(8) V H If we go through the N = 4 theory we also get the series of six cases SO(1,p) SO(4, 5 p) SO(1, 1) − (n = p +1,n =5 p), 0 p 5. × SO(p) × SO(4) SO(5 p) V H − ≤ ≤ × − (3.5)

8 So we see that there are ten D = 5 cases with similar types of quaternionic manifold as in D = 4 (with the only exception of the nH =10case.).

In D =6theN = 8 ((2, 2) theory) σ-model is SO(5, 5)/SO(5) SO(5). If we decompose the (2, 2) theory with respect to the (1, 0) theory we get 5 tensor× multiplets and 5 hypermultiplets corresponding to SO(1, 5) SO(5, 5) SO(4, 5) SO(5, 5) , . (3.6) SO(5) ⊂ SO(5) SO(5) SO(4) SO(5) ⊂ SO(5) SO(5) × × × These are the nT nH = 0 cases. Again we can· now look at subgroups G G SO(5, 5) where G = SO(1,n )and 1 × 2 ⊂ 1 T G2 is the isometry group of a quaternionic manifold. We find a series analogous to the D = 5 case (3.5), with G = SO(1,p) ,G= SO(4, 5 p)(n = p, n =5 p) (3.7) 1 2 − T V − corresponding to the manifolds SO(1,p) SO(4, 5 p) − (n = p, n =5 p), 0 p 5 (3.8) SO(p) × SO(4) SO(5 p) T H − ≤ ≤ × − which contains also the above mentioned n n = 0 cases (3.6). T · H The reduction of N =8 N = 2 supergravity studied in D =6, 5 and 4 finds a further simplification if we look→ for theories with eight supercharges in D = 3, where the R-symmetry is SU(2) SU(2) . 1 × 2 In fact, if we compactify Type II on a Calabi-Yau threefold times S1,downtoD =3, then Type IIA and IIB become the same theory with 1 2. The N =4σ-model is a ⇔ product of two quaternionic geometries, where nH1 = h1,1 +1,nH2 = h2,1 + 1, the extra quaternion coming from the graviton and graviphoton degrees of freedom. SK More generically, suppose we have a theory which at D =4hasaσ-model (nV ) Q M × (nH ), then its dimensional reduction to D = 3 will give rise to a N =4SU(2)1 M Q1 Q2 Q1 × SU(2)2 theory with σ-model (nH1 = nV +1) (nH2 = nH ), where is the M SK ×M M dual quaternionic manifold of (nV ). Using the previous recipe, if weM look to the D =4,N = 2 theories of section 2 obtained E8(8) from N =8,wecanpredictN = 4 theories at D = 3 which are embedded in the SO(16) σ-model of D =3,N = 16 maximal supergravity. From (nV =15,nH =0)and(nV =0,nH = 10) we respectively obtain:

E7( 5) E8(8) (n =16,n =0), − (3.9) H1 H2 SO(12) SU(2) ⊂ SO(16) × SU(2, 1) E E (n =1,n = 10) , 6(2) 8(8) (3.10) H1 H2 SU(2) U(1) × SU(6) SU(2) ⊂ SO(16) × × G F E (n =2,n =7), 2(2) 4(4) 8(8) . (3.11) H1 H2 SO(4) × USp(6) USp(2) ⊂ SO(16) × Also, by using the embedding [36] E SO(8, 8) we have the further possibility: 8(8) ⊃ E SO(8, 8) SO(4,k) SO(4, 8 k) 8(8) × − (3.12) SO(16) ⊃ SO(8) SO(8) ⊃ SO(4) SO(k) SO(4) SO(8 k) × × × × − 9 with n = k, n =8 k (k =0isasubcaseofthe(n =16,n =0)case,since H1 H2 − H1 H2 SO(4, 8) SU(2) E7( 5).). They are all dimensional reductions of the cases previously studied at×D =4.⊂ − For the decomposition of the isometry group of maximal D = 3 supergravity to maximal subgroups, see Table 3.

Table 1: Decomposition of “duality” groups of maximal D =3, 4, 5 supergravities with respect to maximal subgroups relevant for supergravity reduction.[36],[34] ——————————————— Some Maximal subgroups of E8(8): SO(16) SO(8, 8) E7( 5) SU(2) E − ×SU(2, 1) 6(2) × G2(2) F4(4) ———————————————× Some Maximal subgroups of E7(7): SU(8) SU(4, 4) E SO(2) 6(2) × SO∗(12) SU(2) SO(6, 6) ×SU(1, 1) SU(3, 3) × SU(2, 1) SU(1, 1)× F × 4(4) Sp(6, IR ) G2(2) ———————————————× Some Maximal subgroups of E6(6): USp(8) F4(4) SU∗(6) SU(2) SO(5, 5) ×SO(1, 1) × SL(3, IR ) G2(2) SL(3,C) ×SU(2, 1) ———————————————×

4 Geometrical interpretation

It is interesting to analyze the results of the previous section in geometrical terms, that is to explore the consistency of the reduction of the N =8σ-model E7(7)/SU(8) to the appropriate submanifolds for different values of N 0. A consistent truncation of a manifold of dimension n to a submanifold of dimension n k can be obtained by considering a set of k 1-forms φi, i =1, ,k, which vanish on the− submanifold and such that they are in involution, that is: ··· dφi = θi φj (4.1) j ∧ i where θj are suitable 1-forms on the manifold.

10 To apply this result, known as Frobenius theorem, to our problem we consider the coset representative U of E7(7)/SU(8) in the 56 fundamental representation of E7(7) and the corresponding left invariant 1-form 3: Ω P Γ U 1dU = (4.2) − P Ω ≡   satisfying the Cartan-Maurer equation

dΓ+Γ Γ=0. (4.3) ∧ Here the 28 28 subblocks Ω and P embed the SU(8) connection and the vielbein of E /SU(8).× Introducing indices A, B =1, , 8 we have explicitly: 7(7) ··· Ω 2ω[A δB]; P P (4.4) ≡ [C D] ≡ ABCD A where ω B is the SU(8) connection and PABCD is the vielbein of E7(7)/SU(8), antisym- metric in its four indices and satisfying the reality condition: 1 P =  P PQRS. (4.5) ABCD 24 ABCDP QRS From the Cartan-Maurer equations one easily finds the two structure equations: 1 RA dωA + ωA ωC = P ALMN P (4.6) B ≡ B C ∧ B −3 ∧ BLMN P ABCD dP ABCD 4ω[A P BCD]L =0. (4.7) ∇ ≡ − L A Equation (4.6) gives the SU(8) Lie algebra valued curvature R B in terms of the vielbein of the symmetric coset E7(7)/SU(8) and equation (4.7) expresses the fact that the same manifold is torsionless. Note that, since the coset is symmetric, the Lie algebra connection A ω B is simply related via a structure constant to the Riemannian spin connection. Let us now consider how the vielbein PABCD decomposes under the holonomy reduction SU(8) SU(N 0) U(1) SU(8 N 0). We call a, b, c, =1, ,N0 the indices of −→ × × − ··· ··· SU(N 0)andi, j, k =1, , 8 N 0 the indices of SU(8 N 0). Then the holonomy reduction gives the··· following··· fragments:− −

P P P P P P (4.8) ABCD −→ abcd ⊕ abci ⊕ abij ⊕ aijk ⊕ ijkl where actually some of the fragments can be zero if the number of antisymmetric indices of SU(N 0)orSU(8 N 0) exceeds N 0 or 8 N 0, respectively. Now we observe that Pabcd satisfies equation (4.7)− which gives for this− particular component:

dP abcd 4ω[a P bcd]` 4ω[a P bcd]i =0 (4.9) − ` − i

We see that, in order that equation (4.9) describe a torsionless submanifold with SU(N 0) a × U(1) holonomy, we must set ω i =0andsince 1 Ra dωa + ωa ωc + ωa ωj = P aLMN P (4.10) i ≡ i c ∧ i j ∧ i −3 ∧ iLMN 3We use notations as in ref.[37].

11 we must also impose that, on the submanifold whose vielbeins are P abcd, the curvature a 1 aLMN with mixed indices is zero, namely R i = 3 P PiLMN = 0. Using the decomposition (4.8), equation (4.10) can be rewritten as− follows ∧ 1 dωa = ωa ωc ωa ωj P abcd P i − c ∧ i − j ∧ i − 3 ∧ ibcd 1 1 1 P abcj P P abjk P P ajkl P (4.11) − 3 ∧ ibcj − 3 ∧ ibjk − 3 ∧ ijkl On the basis of the Frobenius theorem, each term on the r.h.s. of (4.11) must be in a involution with ω i; this is satisfied for the terms bilinear in the ω-connections, but not for those involving the vielbein. In order to obtain involution, we must also set to zero some of the vielbein 1-forms and verify that also these are actually in involution. Let us see how we can achieve this result in the various cases. When N 0 =6,Pijkl = Pibjk 0 because we have 4-fold or threefold antisymmetrization of the SU(2) indices. Therefore≡ it is sufficient to set

P P ibcd = 0 (4.12) ibcd ≡ on the submanifold in order to obtain involution, since in this case equation (4.11) reduces to dωa = ωa ωc ωa ωj Ra = 0 (4.13) i − c ∧ i − j ∧ i −→ i We still have to verify that also the vanishing 1-forms Pibcd areininvolutionwiththem- a selves and with ω i. Indeed, from equation (4.7), we find:

abci [a bc]di [a bc]ji i abcd i abcj dP =3ω dP +3ω jP + ω dP + ω jP (4.14)

abcj a and we see that every term in the r.h.s. contains either P or ω i so that we get involution. We note that condition (4.12) is equivalent to impose that the SU(6) U(1) SU(2) representation (20, 0, 2) must be absent in the reduction of the scalar vielbein,× × and this 3 implies that all the 40 scalars of the N 0 =6spin 2 multiplets must be frozen according abcj a to our analysis in the previous section. In conclusion, setting P =0andω i =0,we define a consistent truncation of the N = 8 theory down to a N 0 = 6 theory since the above conditions define a submanifold of holonomy SU(6) U(1) SU(2) whose curvature is easily seen to be given by × × 1 Ra = P almn P . (4.15) b −3 ∧ blmn

The corresponding manifold has dimension 30 and of course coincides with SO∗(12)/U(6). The cases N 0 =5andN 0 = 4 can be treated in exactly the same way. For N 0 =5, equation (4.11) does not contain Pijkl and in order to get involution we have to set

Pabci = Pabij = 0 (4.16) which corresponds to delete, in the holonomy reduction (2.3), the representations (10, 1, 3) and (10, 1, 3) for the vielbein (because of the reality condition(4.5)). According to the − 3 discussion of the previous section, this is equivalent to freeze the 60 scalars of the spin 2

12 a multiplets. Using again equation (4.7), we can immediately verify that Pabci,Pabij and ω i are indeed in involution so that the reduction to the submanifold SU(1, 5)/U(5) is indeed consistent. For N 0 = 4, equation (4.11) contains all the terms bilinear in the vielbeins. However it is sufficient to set P = P ( P abci) = 0 (4.17) abci aijk ≈ to achieve the vanishing of the r.h.s. of (4.11) on the submanifold. This corresponds to delete, in the holonomy reduction (2.6), the representations (4, 1, 4)and(4, 1, 4)in the decomposition of the scalar vielbein, that is to freeze the 32 scalars appearing− in the 3 N 0 =4spin 2 multiplets. Again the structure equation (4.7) can be used to show that a Pabci,Paijk and ω i are in involution so that we get a consistent reduction to the N 0 =4 submanifold SU(1, 1)/U(1) SO(6, 6)/[SU(4) SU(4)]. × × a Finally, in the N 0 = 2 case, in order to have involution for ω i =0,wemusthave 1 1 Ra = P abjk P P ajkl P = 0 (4.18) i −3 ∧ ibjk − 3 ∧ ijkl on the manifold. abjk ajkl If we take P (and its complex conjugate Pijk`)orP vanishing on the submanifold this corresponds to delete the complex representation (1, 1, 15) or the real representation (2, 0, 20) of the holonomy group SU(2) U(1) SU(6).− We may check immediately that × × a in both cases the vanishing vielbein are indeed in involution with ω i andwiththemselves. Indeed:

abci [a bc]di [a bc]ji i abcd i abcj dP =3ω dP +3ω jP + ω dP + ω jP (4.19) ajk` a bjk` a ijk` [j k`]ab [j k`]ia dP = ω bP + ω iP +3ω bP +3ω iP (4.20) and we see that in both cases the involution condition is satisfied. Therefore we have found a consistent reduction to the submanifolds SO∗(12)/U(6) and E6(2)/SU(6) SU(2) which are special-K¨ahler and quaternionic manifolds respectively of maximal holonomy.× The other cases treated group theoretically in the previous section can be handled in an analogous way, provided we reduce the holonomy of the resulting submanifold in a suitable way. We just give an example. Consider the manifold given in equation (2.11), corresponding to (nV ,nH )=(9, 1). We decompose the representation 6 of SU(6) into the representation (3, 1)+(1, 3)ofSU(3) SU(3). Correspondingly, the index i in the 6 of SU(6) is decomposed: ×

i α, α,˙ (α, α˙ =1, 2, 3) (4.21) → where α andα ˙ run on the fundamental rep of the two SU(3) groups. Then we have:

Ra Ra ,Ra (4.22) i → α α˙ and we find:

(1,3,1) (1,3,3) (1,1,3) (2,1,1) (2,3,3) ˙ (2,3,3) a abβγ abβγ˙ abβγ˙ R = P P + P P + P P ˙ α ∧ αbβγ ∧ αbβγ˙ ∧ αbβγ˙ (2,3,3) (2,3,3) (2,1,1) (1,1,3) (1,3,3) (1,3,1) aβγδ˙ aβγ˙ δ˙ aβ˙γ˙ δ˙ + P P ˙ + P P ˙ + P P ˙ ˙ (4.23) ∧ αβγδ ∧ αβγ˙ δ ∧ αβγ˙ δ 13 where we have set on the top of each vielbein the rep of SU(2) SU(3) SU(3) to which it belongs. We see that deleting the vielbein in the reps (1, 3, 1),× (2, 3, 3×)and(1, 1, 3)(and a their complex conjugates) we get R α = 0 so that involution is satisfied. An analogous a computation can be done, with the same conclusions, for R α˙ . Note that the vielbein which survive, Paαβγ and Pabβγ˙ , in the representations (2, 1, 1)and(1, 3, 3) respectively, do in fact describe the vielbein system of the given manifold. The involution of the deleted vielbein is also easily proved. Indeed:

˙ dP abβγ =2ω[a P b]cβγ +2ω[a P b]αβγ +2ω[a P b]˙αβγ 2ω[β P γ]δab 2ω[β P γ]δab (4.24) c ∧ α ∧ α˙ ∧ − δ ∧ − δ˙ ∧ and we see that each term contains at least a 1-form which is zero on the submanifold. It is a simple exercise to verify that one can actually further reduce the holonomy to all the holonomy subgroups of the various cases treated in Section 2 and find con- sistent reduction to the corresponding special-K¨ahler and quaternionic symmetric coset submanifolds.

5 Consistency constraints from supersymmetry

In the previous sections we have analyzed the effects of truncating out some of the su- percharges in the supergravity theories. In particular, in section 3 we have considered the effects of the reduction of the holonomy group for the various supermultiplets at the linearized level, while in section four we have studied the consequences of such a reduction on the scalar sectors. We still have to analyze if the consistency found at the level of σ-model in the geo- metrical analysis can be extended to the full supersymmetric level. For this purpose, we analyze the supersymmetry transformation laws of N =8super- gravity, when the R-symmetry gets reduced from SU(8) to SU(N 0) U(1). They are, neglecting three fermions terms: ×

δV a = iψAγa + h.c. (5.1) µ − µ A ν ρ B δψAµ = µA + TAB− νργµ γ  (5.2) ∇ | α µ D µν δχABC = PABCD,α∂µφ γ  + T[−AB µν γ C] (5.3) | ΛΣ ΛΣ A B ABC δAµ = fAB ψµ  + χ γµC + h.c. (5.4) α ABCD,α  δφ = P χABCD + h.c. (5.5)

(the SU(8) indices A, run from 1 to 8). We use the same notation as in reference [37]: ··· we call U the coset representative of E7(7)/SU(8) parametrized as follows:

1 f +ih f +ih U = (5.6) √2 f ih f ih  − −  ΛΣ where fAB and hΛΣAB (Λ, Σ, =1, , 8) are labelled by couples of antisymmetric indices ΛΣ andAB with Λ, Σ=1··· ,···8andA, B =1 , 8. Therefore they describe 28 28 sub-blocks of the 56 56 symplectic··· matrix (coinciding··· with the fundamental 56 × × representation of E7(7)). Note that U transforms on the left as the 56 representation of

14 E and on the right as the 28 28 of SU(8) . 7(7) ⊕ In terms of f and h, the 2-form TAB is given by:

i 1 1 T = (f − ) F ΛΣ = h F ΛΣ f ΛΣ (5.7) AB −2 ABΛΣ 2 ΛΣAB − AB GΛΣ   where is the magnetic counterpart of the field-strength F ΛΣ. The spinor fields ψ GΛΣ Aµ and χABC are the N = 8 left-handed gravitinos and dilatinos respectively. Finally, the covariant derivative acting on the spinors is defined as follows:

 =  + ω B (5.8) ∇ A D A A B B where ωA is the SU(8) connection and µ denotes the Lorentz covariant derivative. Let us first analyze the gravitino decomposition.D We want to reduce the theory to an N 0 8 one. Therefore, to reduce the R-symmetry SU(8) SU(N 0) U(1), we decompose ≤ → × the holonomy indices A, (a, i)witha =1, ,N0 and i =1, , 8 N 0.Wethen ··· ⇒ ··· ··· − have to truncate out (to set to zero) the 8 N 0 gravitinos ψ and the corresponding − iµ supersymmetry parameters i.Weget:

b ν ρ b δψaµ = µa + ωa b + Tab− νργµ γ  (5.9) D | a ν ρ a δψiµ = ωi a + Tia− νργµ γ  0 (5.10) | ≡ The second equation, consistency condition for the truncation, implies

a ωi =0,Tia− = 0 (5.11)

The first condition in (5.11) confirms the restriction of the scalar σ-models found in the previous section from the geometrical analysis, while the second one kills the vector su- perpartners of the erased gravitinos at the full interaction level. Then what is left, equation (5.9), is the correct transformation law for the survived 1 1 i − ΛΣ i − ΛΣ gravitini, provided Tab = 2 (f )abΛΣF (and Tij = 2 (f )ijΛΣF for N 0 =6) describe the correct expression− for the (dressed) graviphotons− in the reduced theory, 1 i − Λ Tab = 2 (f )abΛF ,withΛ running on the appropriate representation of the U du- ality group− of the reduced theory4. e e To this aim, let use first recall that, in all N-extended theories, the electric and magnetic field-strengths transform in a representation of the U duality group whose dimension is the same as the fundamental representation of the embedding symplectic group Sp(2nv) [38] (nv is the total number of vectors). Let us consider separately the cases N 0 =5, 6, where all the vectors are graviphotons, from the N 0 4 cases, where matter vectors are present. ≤ In the former cases, note that E7(7) (the isometry group of N = 8 theory) contains, as maximal subgroups: SO∗(12) SU(2) and SU(5, 1) SU(3). The duality groups for the × × N 0 =6, 5areSO∗(12) and SU(5, 1) respectively. The rep 56,inwhichtheN = 8 vectors field strengths and their duals lie, decomposes respectively as follows (see also Table 2):

E SO∗(12) SU(2) 56 (32, 1)+(12, 2) (5.12) 7(7) → × → E SU(5, 1) SU(3) 56 (20, 1)+(6, 3)+(6, 3) (5.13) 7(7) → × →

15 Table 2: Duality reduction in D =4 G = E G G G G G 7(7) 1 × 2 → 1 × 2 N = 6 (# vect. = 16) SO∗(12) SU(2) 56 (32, 1) + (12, 2) N = 5 (# vect. = 10) SU(5, 1) × SU(3) 56 (10, −→1) + (10, 1) + (6, 3) + (6, 3) N = 4 (# vect. = 12) SO(6, 6)) ×SU(1, 1) → 56 (12, 2) + (32, 1) × −→ N = 2 (# vect. = nV +1) n =0 E U(1) 56 1+10 +27+27 V 6(2) × → nV =15 SO∗(12) SU(2) 56 (32, 1) + (12, 2) n =9 SU(3, 3) ×SU(2, 1) 56 (20→ , 1) + (6, 3) + (6, 3) V × → nV =6 Sp(6, IR ) G2(2) 56 (1, 14) + (6, 7) n =2 SU(1, 1) × F 56 → (4, 1) + (2, 26) V × 4(4) →

We note that in each case only a subset of the 56 field-strengths is transformed only with respect to the (reduced theory) duality group, while it is a singlet of the SU(8 N 0)com- muting group, and this immediately identifies the electric and magnetic field− strengths which remain in the gravitational multiplet after truncation. (Indeed this exactly repro- duces the counting at the linearized level, since we expect to have, in the gravitational multiplet of the N 0 = 6 (respectively N 0 = 5) theory, 16 (respectively 10) electric field strengths parametrized by Tab,Tij (respectively by Tab). Therefore, in performing the truncation, we also have to decompose the representations of the N = 8 U duality group with respect to its maximal subgroups as in (5.12), (5.13), and to keep only the irrepses, in the decomposition, which are singlets under the commuting group, as shown in Table 2. Note that this prescription automatically guarantees the consistency of the truncation, since the objects to be truncated out (in particular the (12, 2) (respectively (6, 3)+(6, 3)) field strengths given by Tai and their magnetic duals), being in a non trivial representation of the commuting group SU(8 N 0), can never mix with those which have been kept, which are instead singlets. − Let us now consider the matter coupled theories, and in particular N 0 = 4 (the N 0 =2 case is similar). Here the argument is reversed with respect to the higher N 0 theories, but with analogous conclusions. Indeed, the U duality group for the N 0 = 4 theory is SU(1, 1) SO(6,n), and, for n = 6, it is indeed a maximal subgroup of the N =8U duality group,× (no commuting subgroup). Note that the U-duality group is now factor- ized into the S-duality group SU(1, 1), which mixes electric with magnetic field strengths, and the electric T-duality group SO(6, 6). We have, for the decomposition of the 56 of E SU(1, 1) SO(6, 6): 7(7) → × 56 (2, 12)+(1, 32) (5.14) →

In this case it is the (2, 12) field strengths (given by the six graviphotons Tab and the six matter vectors Tij, together with their magnetic counterpart) which have to be retained, since they have the appropriate transformation property under the U duality group, while the extra 32 field-strengths (given by Tai and its magnetic dual), which are spinors under SO(6, 6), have to be truncated out and do indeed belong to the extra gravitini multiplets. 4With abuse of language, we call U duality group the continuous group whose restriction to the integers is the U duality group of the theory.

16 A similar argument as given previously still works for the consistency; indeed the field- strengths in the (1, 32), spinors under SO(6, 6), can be set to zero consistently since they cannot mix with the other field-strengths which are not in the spinor representation of SO∗(12). As far as the transformation laws for the vectors, scalars and spin one half fields are concerned, one sees that the decomposition confirms the results of the analysis at the linearized level given in section 3, as summarized in Table 3.

Table 3: Decomposition of N =8intoN = N 0 supergravity multiplet N 0 multiplet max spin multiplicity 8 (gµν ,ψAµTAB µν ,χABC ,PABCD) 2 1 | 6 (gµν ,ψaµTab µν ,Tij µν,χabc,χaij ,Pabcd,Pabij) 2 1 | | 3 (ψiµTai µν ,χabi,Pabci) 2 | 2 5 (gµν ,ψaµTab µν ,χabc,χijk,Pabcd,Paijk) 2 1 | 3 (ψiµTai µν ,Tij µν ,χabi,χaij ,Pabci,Pabij ) 3 | | 2 4 (gµν ,ψaµTab µν ,χabc,Pabcd,Pijk`) 2 1 | 3 (ψiµTai µν ,χabi,χijk,Pabci,Paijk) 4 | 2 (Tij µν ,χaij ,Pabij) 1 6 | 3 (gµν ,ψaµTab µν ,χabc) 2 1 | 3 (ψiµTai µν ,χabi,Pabci,Pijk`) 5 | 2 (Tij µν ,χaij ,Pabij ,Paijk) 1 10 | 2 (gµν ,ψaµTab µν ) 2 1 | 3 (ψiµTai µν ,χabi) 6 | 2 (Tij µν ,χaij ,Pabij) 1 15 | 1 (χijk,Paijk) 2 10 1 (gµν ,ψaµ) 2 1 3 (ψiµTai µν ) 7 | 2 (Tij µν ,χaij ) 1 21 | 1 (χijk,Paijk) 2 35

For the case N =8 N = 2, we see from Table 2 that the vectors belonging to the six 3 → spin 2 multiplets and to those vector multiplets which are truncated out are tied together 3 by an irrep. of G1 G2. This means that to delete only the spin 2 multiplets would be inconsistent. × The same analysis applies to theories in higher dimensions and, for the D =5case, the duality reduction, for some interseting cases, is given in Table 4.

Table 4: Duality reduction in D =5 G = E G G G G G 6(6) 1 × 2 → 1 × 2 N = 2 (# vect. = nV +1) n =0 F 27 1+26 V 4(4) → n =14 SU∗(6) SU(2) 27 (15, 1) + (6, 2) V × → nV =8 SL(3,C) SU(2, 1) 27 (3, 30, 1) + (1, 3, 30)+(30, 1, 3) n =5 SL(3, IR× ) G → 27 (6, 1) + (3, 7) V × 2(2) →

17 6 N =2 N =1reduction −→ This section is devoted to a thorough analysis of the consistent truncation of N =2 supergravity down to N = 1 in four dimensions. The N =2 N = 1 reduction of the supersymmetry transformation laws presents different features−→ in the vector multiplet and in the hypermultiplet sectors. The vector multiplet case is simpler since the special geometry is already a K¨ahler-Hodge geometry while for hypermultiplets we are confronted with the more difficult task of reducing a quaternionic manifold to a K¨ahler-Hodge one. Note that, differently from what done in the preceeding sections, where we discussed only ungauged theories, the present reduction is given at the level of the complete N =2 gauged theory. In the first two subsections we begin to analyze the reduction in the vector multiplet sector, where much of the special geometry relations are needed. In subsection 6.3 we analyze the reduction in the hypermultiplet sector. In both cases the geometrical approach discussed in section 3 will be essential for the discussion. The other subsections are devoted to a careful analysis of the implications of the gauging, to the reduction of the scalar potential and to the discussion of some explicit examples. The reduction is obtained by truncating the spin 3/2 multiplet containing the second gravitino ψµ2 and the graviphoton. Here and in the following we use the notations both for N =2andN = 1 supergravity as given in reference [39], the only differences being that we use here hatted world indices i, ı =1, ,nV and tilded gauge indices Λ=0, 1, ,nV for quantities in the N =2 vector multiplets··· (since we want to reserve the notation··· Λ and i, i for the indices of the breducedb N = 1 theory) and that the holomorphice matrix appearing in the kinetic term of the vectors in the N = 1 theory will be renamed as follows:

(zi) f (zi). (6.1) N ΛΣ ≡ ΛΣ Let us write down the supersymmetry transformation laws of the N = 2 theory, up to 3-fermions terms [4]:

Supergravity transformation rules of the (left–handed) Fermi fields:

ν B δψ =  + i gS η +  T − γ  (6.2) Aµ ∇µ A AB µν AB µν iA i µ A i µν AB  iAB δλ =ic z γ  + G− γ   + gW  (6.3) ∇µ µν B B δζ =iBβ qu γµA C + gNA  (6.4) bα Uu b∇µ bAB αβ α Ab where: B µ A = µA + ωµ A B + µA (6.5) ∇ D | Q and the SU(2) and U(1) 1-form “gauged” connections are respectively given by: c b b

ω B = ω B + g AΛ P x (σx) B , (6.6) A A (Λ) Λ A = + g AΛ Pe0 , (6.7) b Q Q (Λ) e Λ e i b = ∂ dzei ∂ dzı (6.8) ie e ı Q −2  K − K  b b b b 18 ω B, are the SU(2) and U(1) connections of the ungauged theory. Moreover we have: A Q zi = ∂ zi + g AΛki (6.9) ∇µ µ (Λ) µ Λ u u Λ u µqb = ∂µqb + g Ae kb (6.10) ∇ (eΛ) µ eΛ e e e Supergravity transformation rules of the Bose fields:

δVa = i ψ γa A i ψA γa  (6.11) µ − Aµ − µ A Λ Λ AB Λ A B δAµ =2L ψAµB +2L ψµ  AB e +if ΛeλiAγ B  +ifeΛ λı γ  AB (6.12) i µ AB ı A µ B δzi = λiAe b e b (6.13) b A b ı bı A δzb = λA (6.14) u u α A αβ AB δqb = b ζ  + C  ζ B . (6.15) UαA β   Here Tµν− appearing in the supersymmetry transformation law of the N = 2 left-handed gravitini is the “dressed” graviphoton defined as:

Σ Λ T − 2iIm L F −. (6.16) µν ≡ NΛΣ µν e e while ee i i Γ Λ G − = g f Im F − (6.17) µν −  NΓΛ µν are the “dressed” field strengths ofb the vectorsbb e inside thee vector multiplets. Moreover the iAB A b ee fermionic shifts SAB, W and Nα are given in terms of the prepotentials and Killing vectors of the quaternionic geometry as follows: b 1 1 S =iP LΛ i P xσx LΛ (6.18) AB 2 AB Λ ≡ 2 Λ AB W iAB =iP AB gif Λe+ ABki LΛ e (6.19) Λ e  eΛ Λ e bN A =2A kbubLe b (6.20) α Ueαu Λ b e α α ue Λ NA = 2 Au k L (6.21) − U e Λ e We recall that the Killing vectors ki and ku aree related to the prepotentials by: Λ Λ b ki =iegi∂ P 0e (6.22) Λ  Λ bu 1bb x vu x k = Ω | P ; λ = 1 (6.23) Λe 6λ2 b e ∇v Λ −

x where Ωuv is the SU(2)-valuede 2-form defined in Sectione (6.3) below, and that the prepo- tential P 0 satisfies: Λ P 0LΛ = P 0LΛ = 0 (6.24) e Λ Λ e e e 19e Since we are going to compare the N = 2 reduced theory with the standard N =1 supergravity, we also quote the supersymmetry transformation laws of the latter theory [5],[40]. We have, up to 3-fermions terms: N =1transformation laws

δψ µ = µ + Qµ +iL(z, z)γµε• (6.25) • D • • i i Λ i µ i δχ =i∂µz + g(Λ)Aµ kΛ γ ε + N ε (6.26) b • • Λ ( )Λ µν Λ  δλ = µν− γ ε +iD ε (6.27) • • •a F δVµ = iψ γµε• +h.c. (6.28) − • Λ 1 Λ δAµ =iλ γµε• +h.c. (6.29) 2 • δzi = χiε (6.30) • where is defined in a way analogous to the N = 2 definition (6.7) and: Q b 1 (z,z) L(z, z)=W (z) e 2 KV (6.31) i i N =2g  L (6.32) Λ ∇ 1 D = 2(Imf )− P (z, z) (6.33) − ΛΣ Σ and W (z), (1)(z, z),PΣ(z, z),fΛΣ(z) are the superpotential, K¨ahler potential, Killing pre- potential andK vector kinetic matrix respectively [40], [5], [41]. Note that for the gravitino and gaugino fields we have denoted by a lower (upper) dot left-handed (right-handed) chirality. For the spinors of the chiral multiplets χ, instead, left-handed (right-handed) chirality is encoded via an upper holomorphic (antiholomorphic) world index (χi,χı). The supersymmetric lagrangians which are left invariant by these transformation laws are given in Appendix G. To perform the truncation we set A=1 and 2 successively, putting ψ2µ = 2 =0,and we get from equation (6.2):

1 ν 1 δψ1µ = µ1 µ1 ωµ 1 1 +igS11ηµν γ  (6.34) D − Q − | b b 1 ab where denotes the Lorentz covariant derivative (on the spinors, µ = ∂µ 4 ωµ γab), while, forD consistency: D −

1 ν 1 δψ2µ 0= ωµ 2 1 + i gS21ηµν Tµν− γ  (6.35) ≡ − | −   For a consistent truncation in theb ungauged case we must set to zero the graviphoton:

Λ T − = TΛF − =0, (6.36) e where e T 2iIm LΛ (6.37) Σ ≡ NΛΣ 2 is the projector on the graviphoton [42], and the componente ω1 of the SU(2) connection 1-form: e ee 2 ω1 = 0 (6.38)

20 In the gauged case we have the further constraints: i i S = P x(σx) LΛ = P 3LΛ =0, (6.39) 21 2 Λ 12 2 Λ ω 2 = ω 2 + g AΛeP x(σx) 2 e g AΛP x(σx) 2 =0. (6.40) 1 1 e (Λ) Λ e1 ≡ (Λ) Λ 1 e e while no further restrictionb comes frome (6.7)e since thee form ofe the gauged U(1) connection should not change in the reduced theory. Comparing (6.25) with (6.34), we learn that we must identify:

ψ1µ ψ µ (6.41) ≡ • 1  (6.42) ≡ • Furthermore, gS = i g P x(σx) LΛ must be identified with the superpotential of the 11 2 (Λ) Λ 11 N = 1 theory, that is to the covariantly holomorphic section L of the N =1K¨ahler-Hodge e manifold. Therefore we havee e [11] - [19]: i i L(q, z,z)= g P x(σx) LΛ = g P 1 iP 2 LΛ (6.43) 2 (Λ) Λ 11 2 (Λ) Λ − Λ   e e We will show in the following (Sectione e (6.4)) that, aftere consistente e reduction of the special- K¨ahler manifold SK and of the quaternionic σ-model Q, L will in fact be a covariantly holomorphic functionM of the K¨ahler coordinates ws of theM reduced manifold KH Q M ⊂M and of some subset zi of the scalars zi of the N = 2 special-K¨ahler manifold SK. ∈MR M The condition on the graviphoton T − = 0 will be analyzed in subsection (6.1), while 2 b the condition ω1 = 0 will be discussed in section (6.3) and the constraints appearing in the gauged theory will be analyzed in section (6.4). Here and in the following we will denote by SK and Q the special-K¨ahler and quaternionic manifolds of the N = 2 theory whileM the special-K¨M ahler and K¨ahler-Hodge SK Q KH manifolds obtained by reduction of and will be denoted by R and respectively. M M M M

6.1 Reduction of the N =2vector multiplet sector

Let us now consider the gaugino transformation laws. When 2 =0weget:

δλi1 =i ziγµ1 + W i11 (6.44) ∇µ 1 i2 i µν i21 δλb = G− bγ  + gWb  (6.45) − µν 1 1 where, using (6.19) b b b

W i21 =iP 3 gif Λ ki LΛ (6.46) Λ  − Λ Wbi11 =iP 11gbbifeΛ = bP 2e iP 1 gif Λ (6.47) Λe  e Λ − Λ    b bb e bb e From eqs. (6.44) and (6.45) we immediatelye b see thate thee spinorsb λi1 transform into the scalars zi (and should therefore give rise to N = 1 chiral multiplets) while the spinors λi2 b b b 21 i transform into the matter vectors field strengths G−µν (and should then be identified with the gauginos of the N = 1 vector multiplets). b However, before entering the details of the identification, we have to discuss the im- plications of putting to zero the graviphoton T −, equation (6.36). We observe that this condition gives a constraint on the scalar and vector content of the N = 1 reduced theory, that is on the number of chiral and vector multiplets which are retained after truncation.

Now, since the graviphoton projector TΛ (6.37) is a scalar field dependent quantity, the request that equation (6.36) is verified all over the manifold can be trivially realized i 0 either by setting to zero all the scalars z ande the graviphoton Aµ, which implies on the Λ 0 Λ symplectic section L (L =1;L =0b, Λ=1, nV ), or, alternatively, by truncating Λ ⇒ ··· out all the vectors Ae ,leavinganN = 1 theory with only chiral matter content. There is however a more interesting and non trivial way to satisfy equation (6.36), by e imposing a suitable constraint on the set of vectors and of scalar sections which can be retained in the reduction. Indeed, if we decompose the index Λ labelling the vectors into ˙ ˙ two disjoint sets Λ (Λ, Λ), Λ=1, ,nV0 = nV nC ; Λ=0, 1, ,nC , we may satisfy the relation (6.36) as⇒ an “orthogonality··· relation”− between thee subset··· Λ running on the retained vectors ande the subset Λ˙ running on the retained scalar sections. That is we set:

Λ˙ Fµν = 0; (6.48) Im LΣ = T = 0 (6.49) NΛΣ Λ e Λ˙ We note that if we delete the electric fielde strengths F − we must also delete their magnetic counterpart Σ˙ Σ − = ˙ ˙ F − + ˙ F − = 0 (6.50) GΛ˙ N ΛΣ N ΛΣ so that we must also impose ˙ =0. (6.51) NΛΣ Then, the constraint (6.49) reduces to Im LΣ = 0 (6.52) NΛΣ which implies LΣ = 0 (6.53) since the vector-kinetic matrix Im ΛΣ has to be invertible. Note that conditions (6.52) and (6.53)N imply a reduction of the N = 2 scalar manifold SK SK R, since it says that some coordinate dependent sections on have to be Mzero in→M the reduced theory. M Let us decompose the world indices i of the N = 2 special-K¨ahler σ-model as follows: i (i, α), with i =1, ,nC , α =1, ,nV0 = nV nC ,wherenC and nV0 are respectively ⇒ ··· ···b − the number of chiral and vector multiplets of the reduced N = 1 theory while nV is the bnumber of N = 2 vector multiplets. Then from eq (6.57) it follows that the metric on R is pulled back to the following form [4], [42]: M ˙ Λ˙ Σ g = 2f Im ˙ ˙ f (6.54) i − i NΛΣ  To examine further the implications of the reduction of the special-K¨ahler manifold to the submanifold , it is convenient to write the special geometry objects using flat MR 22 indices. We then define a set of K¨ahlerian vielbeins P I = P I dzi on SK together with i M their complex conjugates. Performing the reduction, they decompose as: P I (P I,PA), b b b ⇒ where I and A are flat indices in the submanifold R and onb its orthogonal complement M i b respectively. By an appropriate choice of coordinates, we call z the coordinates on R, α I A M z the coordinates on the orthogonal complement. Then we may set Pα =0,Pi =0,so that the metric g = P IP I has only components g ,g , while g =0. i i  i αβ iα i2 i2 α2 Then, if we decomposeb b the gauginos λ (λ ,λ ), the above truncation implies, by supersymmetry, λbbi2 = 0b and,b for consistency,⇒ b i2 i µν i21 δλ = G− γ  + gW  =0. (6.55) − µν 1 1 i Setting Gµν− =0gives:

i i Λ Σ i Λ Σ G− = g L Im F − = g L Im F − = 0 (6.56) µν − ∇ NΛΣ µν − ∇ NΛΣ µν b e e implying b ee LΛ = f Λ =0. (6.57) ∇  Moreover, W i21 = 0 implies: 3 i PΛ˙ =0,kΛ˙ =0. (6.58) Note that the integrability condition of equation (6.57) is:

Λ kk Λ kk Λ αα Λ i jL =iC g L =iCijkg L +iCijαg aL =0. (6.59) ∇ ∇ ijk ∇k ∇k ∇ bb where Cijk is the 3-index symmetricb b tensor appearing in the equations defining the special geometry (see e.g. ref. [42],[4]). Since the first term on the r.h.s. of equation (6.59) is zero on R (equation (6.57)), equation (6.59) is satisfied by imposing: M

Cijα = 0 (6.60) so that only the N = 2 special-K¨ahler manifolds satisfying the constraint (6.60) are suitable for reduction. Note that, since Cijα is defined as a symplectic scalar product [3],[7],[8],[4] in terms of the Λ symplectic section U =(L ,MΛ): e C =< U, U>, (6.61) e ijα ∇i∇j ∇α it follows that Λ˙ C =0 U =0 L =0, M ˙ =0. (6.62) ijα ⇒∇α ⇒∇α ∇α Λ The same constraint (6.60) can also be retrieved by looking at the integrability condi- tions of the N = 2 special geometry as given in [37]. The relevant ones for our discussion are the following:

P I = dP I +i P I + ωI P J = 0 (6.63) ∇ Q∧ J ∧ RJb (dωb+ ω ω)Jb = Pb P Jb iKδJ CJ CL (6.64) I ≡ ∧ I Ib∧ − I − L ∧ I b b b b b b b b 23b b b b where is the K¨ahler connection 1-form, K = d is the K¨ahler 2-form, ωI is the Q Q J J SU(nV )-Lie algebra valued connection and the 1-form C can be written in termsb of the L b 3-world indices symmetric tensor Cijk, whose propertiesb are given in ref. [37], via: b CJbbb= P JiP jC dzk. (6.65) L L ijk Let us restrict the previous equationsb to thebb submanifoldb b . From the vanishing of the b b bbb R torsion, eq (6.63), we find: M

P I = dP I +i P I + ωI P J + ωI P A = 0 (6.66) ∇ Q∧ J ∧ A ∧ P A = dP A +i P A + ωA P J + ωA P B =0. (6.67) ∇ Q∧ J ∧ B ∧ With the same procedure illustrated in the general discussion of section 4 and in the example of section (6.1), we easily find that the vanishing of the torsion on R implies ωI = 0, from which it follows, taking into account the Frobenius theoremM and the A|MR definition of RJ : I b b J J J I J B R R = PA P C C A C C A =0. (6.68) A|M ∧ − I ∧ − B ∧ Now, expanding the vielbein and the C-tensor along the differentials of the coordinates, we easily find

J Ji α j β k ` R R = P P CijkC α` + CiβkC α` dz dz =0, (6.69) A|M − A ∧   where we have set to zero the terms in the external directions dzα,andtheC-terms containing both holomorphic and antiholomorphic indices, which are zero already because of the N = 2 special geometry properties [3],[4]. Again, we see that equation (6.69) is satisfied by imposing the same condition (6.60) on the special-K¨ahler manifold.

6.2 N =2vector multiplets N =1matter multiplets −→ Let us now discuss the precise identification of the N = 1 matter multiplets obtained by reduction of the N = 2 vector multiplets. From the above analysis we have found that the indices labelling N = 1 chiral and vector multiplets are not related anymore, as it was instead the case in the N = 2 theory. As far as equation (6.44) is concerned, we immediately see that, after reduction of the index i and comparison with the corresponding N = 1 formula (6.26), we can make the following identification: b λi1 = χi (6.70) i11 i 1 2 i Λ˙ gW = N =ig ˙ P iP g f (6.71) (Λ) Λ˙ − Λ˙    i1 that is we may interpret the λ as nC N = 1 chiral spinors belonging to N = 1 left-handed i i chiral multiplets (χ ,z ), i =1,...,nC . It can be easily verified that the consistency condition λα1 =0 δλα1 = 0 (6.72) ⇒ 24 gives α kΛ = 0 (6.73) Λ˙ using fα =0. Let us now discuss the N = 1 vector multiplets coming from the truncation. The transformation law for the nV + 1 vectors of the N = 2 theory (6.12) becomes, after truncation:

δAΛ = if Λλi2γ 1 if Λλα2γ 1 + h.c. = if Λλα2γ 1 + h.c. (6.74) µ − i µ − α µ − α µ δAΛ˙ = if Λ˙ λi2γ 1 if Λ˙ λα2γ 1 + h.c. = 0 (6.75) µ − i µ − α µ where in (6.74) we have used (6.57). Eq. (6.75) is consistently zero if we put λi2 =0, Λ˙ Λ˙ 5 since fα R = αL R =0 . |M ∇ |M We note that while the gauge index ΛoftheN = 2 gaugino runs over nV +1 values (because of the presence of the graviphoton) the indices Λ and α take only nV0 nV values. In particular, the index of thee graviphoton A0 belongs to the orthogonal subset≤ ˙ Λ=0, 1, ,nC , so that the graviphoton is automatically projected out. To match··· the corresponding N = 1 formula (6.29) we have to set:

Λ Λ α2 λ 2fα λ . (6.78) • ≡−

Now, we observe that we may trade the gaugino world index i =1, ,nV with a vector index Λ already at the level of the N = 2 theory, by defining ··· b e λΛA 2f ΛλiA. (6.79) ≡− i e e b Here the gauge index ΛoftheN = 2 gauginosb runs over nV + 1 values (because of the presence of the graviphoton) while the index i takes only nV values. The extra gaugino, e say λ0, is actually spurious, since λΛA satisfies: b T λeΛA = 2T f ΛλiA = 0 (6.80) Λ − Λ i where e e b e e b T 2iIm LΣ, (6.81) Λ ≡ NΛΣ due to the special geometry relation e e ee Im LΛf Σ =0. (6.82) NΛΣ i Note that T is the projector on the graviphotone e field-strength, according to equation Λ ee b (6.16) [42]. 5This followse by looking at the expression of the N =2K¨ahler metric [4]

g = 2Im f Λf Σ (6.76) i − NΛΣ i  by requiring that its mixed component giα is zero. Indeed,e aftere reduction we get bb ee b b Λ˙ Σ˙ 0=g = 2Im ˙ ˙ f f (6.77) iα − NΛΣ i α Σ˙ implying fα =0

25 Using special geometry, one can see that the transformation law for the N = 2 gaugini (6.3) can be rewritten in terms of the λΛA, up to 3-fermions terms, as:

ΛA Λ Σ µν ABe ΛΣ 0 AB AB δλ = P F − γ   2iU P  + P  (6.83) Σ µν B − Σ Σ B   e e e ee where P Λ is the projectore on the matter-vector fielde strengthse and U ΛΣ atensorof Σ special geometry.They are defined below in equations (6.87),(6.86). The derivation of e ee formula (6.83)e is given in Appendix C. The above formulae allow us to perform the reduction of the gaugino λΛ2 straightfor- wardly. Indeed, setting A =2andΛ = Λ, we have e Λ Λ2e Λ i2 Λ α2 λ λ = 2f λ = 2fα λ (6.84) • ≡ − i − Λ b since in the reduced theory fi =0,andthen:b

Λ Λ Σ µν ΛΣ 0 3 δλ = P Fµν− γ  2iU PΣ + PΣ  (6.85) • Σ • − •   Let us now apply the following relations of special geometry [42] to the present reduction:

ΛΣ Λ ij Σ 1 1 ΛΣ Λ Σ U f g f = (Im )− L L (6.86) ≡ i j −2 N − b h iee e PeΛe e2UbΛΓeIm = δΛ iT LΛ. e (6.87) Σ ≡−b b NΣΓ Σ − Σ e e e ee Λ˙ Λ After decomposing the indicese and using fαee R =e fi eR = giα R = 0 we have [4]: |M |M |M

ΛΣ Λ αβ Σ 1 1 ΛΣ U f g f = (Im )− ; (6.88) ≡ α β −2 N h i˙ ˙ ˙ Λ˙ Σ˙ Λ˙ ij Σ˙ 1 1 ΛΣ Λ Σ˙ U f g f = (Im )− L L ; (6.89) ≡ i j −2 N − h i Λ 1 ΛΓ Λ P =(Im − ) Im = δ ; (6.90) Σ N NΣΓ Σ ˙ Λ˙ Λ˙ Λ P ˙ ==δ ˙ iT ˙ L . e (6.91) Σ Σ − Σ e Eq. (6.85) can then be rewritten as:

Λ Λ µν 1 ΛΣ 0 3 δλ = Fµν− γ +i Im − PΣ + PΣ  . (6.92) • N •      0 We observe that the prepotential PΣ, which gives the special-K¨ahler manifold contri- bution to the D-term, can be given an explicit form in terms of N = 2 objects. Indeed, let us recall that P 0 has the following general form, as shown in equation (C.9) of Appendix Σ C: 0 Γ i ∆ e P = 2iIm f k L (6.93) Σ − NΣΓ i ∆ which gives, after reduction: e b e e ee b e ˙ P 0 = 2iIm f Γkα L∆. (6.94) Σ − NΣΓ α ∆˙

26 On the other hand, using the following N = 2 special geometry property:

f Γki =iP 0 LΓ f Γ LΣ (6.95) i ∆ ∆ − ∆Σ e b e e e (f Γ are the structure constantb ofe the N e= 2 gaugeee group G(2)) by contracting with L∆ ∆Σ and reducing it to the submanifold , we also find [4]: e MR e ee ˙ P 0 =2iIm f Γ L∆LΣ˙ . (6.96) Σ NΣΓ ∆˙ Σ˙ In conclusion we get the final form of the gaugino transformation law for the N =1 theory as: Λ Λ µν Λ δλ = µν− γ  +iD  , (Λ = 1, ,n) (6.97) • N=1 F • • ···   where, in order to retrieve the transformation law (6.27) we have set

Λ 1 ΛΣ 0 3 D (Im − ) P + P . (6.98) ≡ N Σ Σ   In order to show that equation (6.97) is the correct N = 1 transformation law of the gauginos we have still to prove that ΛΣ is an antiholomorphic function of the scalar i N fields z , since the corresponding object of the N = 2 special geometry ΛΣ is not. For this purpose we observe that in N = 2 special geometry the following identityN holds (at 6 least when a N = 2 prepotential function exists )[42]: ee

= F 2iT T (LΓImF L∆) (6.99) NΛΣ ΛΣ − Λ Σ Γ∆ where the matrix F is holomorphic. e e ΛΣ ee ee e e ee If we now reduce the indices ΛΣ we find: ee Γ˙ ∆˙ =eFe 2iT T (L ImF ˙ ˙ L ) F (6.100) NΛΣ ΛΣ − Λ Σ Γ∆ ≡ ΛΣ since TΛ = 0 is precisely the constraint (6.49). Therefore ΛΣ is antiholomorphic and the D-term (6.98) becomes: N

Λ Λ α21 1 i ΛΣ 0 3 D 2if W = 2(Imf − (z )) P + P . (6.101) ≡ α − Σ Σ   wherewehavedefined 1 F (zi)= f (zi) (6.102) ΛΣ 2 ΛΣ in order to match the normalization of the holomorphic kinetic matrix of the N =1 theory appearing in equation (6.33).

We observe that for choices of symplectic sections such that the function FΛΣ does not exist, the relation (6.100) does not hold, but still has to be antiholomorphic on . NΛΣ MR Un explicit example will be given in section (6.6). ee As a final observation, we note that the above reduction on the indices of the N =2 Killing vectors gives rise to ki (ki ,kα,ki ,kα). The Killing vectors ki gauge the Λ Λ Λ Λ˙ Λ˙ Λ ⇒ α isometries of the submanifold R. On the other hand, k are zero on the submanifold, b Λ since they correspond to isometriesM orthogonal to ; ki are also zero because we have e MR Λ˙ 6In Appendix D we will discuss the reduction with special coordinates.

27 projected out the corresponding vectors. Finally, kα are in general different form zero, Λ˙ 0 and enter in the definition of PΣ, equation (6.94). These conclusions can be formally retrieved by analyzing the reduction of the special geometry identity [4]

2ig ki k = f ΓP 0. (6.103) i [Λ Σ] ΛΣ Γ b b e One can easily verify that if we setbbΛ=Λ;e e Σ =ee Σ thene one retrieves the analogous of relation (6.103) on R provided we set M e e α 0 kΛ =0; PΛ˙ =0. (6.104) For Λ=Λ;Σ=Σ˙ equation (6.103) is identically satisfied provided we add to the previous condition the further constraint e e i kΛ˙ =0. (6.105) Finally, when Λ=Λ;˙ Σ=Σ,˙ equation (6.103) reduces to the relation:

e e 2ig kα kβ = f ΓP 0 (6.106) αβ [Λ˙ Σ]˙ Λ˙ Σ˙ Γ which has to be satisfied all over the manifold . MR 6.3 Reduction of the hypermultiplet sector Let us analyze the sector of the hypermultiplets when the reduction is implemented. The scalars of the hypermultiplets belong to a quaternionic manifold Q. A quaternionic manifold Q has a holonomy group of the following type [43], [44],M [10]: M Hol( Q)=SU(2) (quaternionic) M ⊗H Sp(2n , IR) (6.107) H⊆ H Introducing flat indices A, B, C =1, 2 α, β, γ =1, .., 2n that run, respectively, in { }{ H} the fundamental representations of SU(2) and Sp(2nH , IR ) ( nH is the number of hyper- multiplets) we introduce the vielbein 1-form [4] Aα = Aα(q)dqu (6.108) U Uu such that h = Aα BβC  (6.109) uv Uu Uv αβ AB where Cαβ = Cβα and AB = BA are, respectively, the flat Sp(2nH )andSp(2) SU(2) invariant− metrics. The vielbein− Aα is covariantly closed with respect to the∼ x U SU(2)-connection ω (x =1, 2, 3) and to the Sp(2nH, IR)-Lie Algebra valued connection ∆αβ =∆βα: i Aα d Aα + ωx (σ )A Bα ∇U ≡ U 2 x B ∧U +∆α Aβ = 0 (6.110) β ∧U x AB AC x B x B where (σ ) =  (σ )C and (σ )A are the standard Pauli matrices. Furthermore Aα satisfies the reality condition: U Aα Bβ ( )∗ =  C (6.111) UAα ≡ U AB αβ U 28 The supersymmetry transformation laws of the fields in the hypermultiplets are given in equation (6.4) and (6.15), that we rewrite here using tangent-space indices for the quaternionic variation:

αAδqu = ζαA + CαβABζ  (6.112) Uu β B δζ =iBβ qu γµA C + gNA  (6.113) α Uu ∇µ AB αβ α A δζα =iAα qu γµ + gNαA (6.114) Uu ∇µ A A Let us see what happens to equations (6.112),(6.113),(6.114), when the truncation is implemented. First of all let us note that the scalars in N = 1 supergravity must lie in chiral multiplets, andhaveingeneralaK¨ahler-Hodge structure. It is therefore required that the holonomy of the quaternionic manifold be reduced:

Hol Q SU(2) Sp(2n ) Hol KH U(1) SU(n). (6.115) M ⊂ × H → M ⊂ ×     Therefore the SU(2) index A =1, 2andtheSp(2nH ) index have to be decomposed accordingly. We set α (I,I˙) U(1) SU(n ) Sp(2n ). Since the vielbein Aα → ∈ × H ⊂ H U satisfy the reality condition (6.111), we have, in U(nH ) indices :

1I 2I˙ ( )∗ = C ˙ U1I ≡ U II U 2I 1I˙ ( )∗ = C ˙ (6.116) U2I ≡ U − IIU

0 CIJ˙ where we have used the decomposition of the symplectic metric Cαβ = with CJI˙ 0 ! C ˙ = C ˙ = δ ˙. IJ − JI IJ From equation (6.116) one finds that it is sufficient to refer to the 2nH complex vielbein 1I , 2I since the ones with dotted indices are related to them by complex conjugation. LetU usU first examine the torsion-free equation obeyed by the quaternionic vielbein written in the decomposed indices: i i d 1I + ω3 1I + (ω1 iω2) 2I U 2 ∧U 2 − ∧U +∆I 1J +∆I 1J˙ = 0 (6.117) J ∧U J˙ ∧U i i d 2I (ω1 +iω2) 1I ω3 2I U − 2 ∧U − 2 ∧U +∆I 2J +∆I 2J˙ = 0 (6.118) J ∧U J˙ ∧U For the N = 1 reduced K¨ahler–Hodge scalar manifold, the holonomy has to be U(1) × SU(nH ), with a non trivial U(1)-bundle, whose field-strength has to be identified with the K¨ahler form. Since in the N = 2 quaternionic parent theory there is a similar non trivial SU(2)-bundle, whose field-strength has to be identified with the Hyper-K¨ahler form, we assume that the U(1) part of the holonomy should be valued in the U(1) subgroup of the SU(2) valued connection of N = 2 quaternionic holonomy group. From equations (6.117), (6.118) we see that, setting

1 2 I ω = ω =∆J˙ = 0 (6.119)

29 we get two K¨ahler-Hodge manifolds whose respective vielbeins obey the torsionless equa- tions for each submanifold. Let us now check the involution property dictated by the Frobenius theorem. As we know from section 3, this amounts to demand that the curvatures of the connections set to zero, equation (6.119), must satisfy the constraints of being also zero on the submanifold. That is we must have: 1 2 I Ω =Ω =IRJ˙ = 0 (6.120) where the SU(2) curvature Ωx is given by 7 1 Ωx dωx + xyzωy ωz =iλC (σ ) αA βB (6.122) ≡ 2 ∧ αβ x ABU ∧U α while the Sp(2nH ) curvature IR β is given by: IR α d∆α +∆α ∆γ (6.123) β ≡ β γ ∧ β = λ Aα B + Aγ Bδ CαρΩ , ABU ∧Uβ U ∧U AB ρβγδ where Ωαβγδ is a completely symmetric 4-index tensor [2]. From equation (6.122) we see that the constraint (6.120) for involution is satisfied iff

1I 2I = 0 (6.124) U ∧U that is if, say, the subset 2I = 1I˙ ∗ of the quaternionic vielbein is set to zero on a U U submanifold KH Q.   When conditionM (6.124)⊂M is imposed, our submanifold has dimension at most half the dimension of the quaternionic manifold (in the following we always refer to the maximal case, where I =1, ,nH )andtheSU(2) connection is reduced to a U(1) connection, whose curvature on··· KH is: M 3 1I 1I 1I Ω KH =iλ 1I =iλ (6.125) |M U ∧U U ∧ U so that the SU(2)-bundle of the quaternionic manifold is reduced to a U(1)-Hodge bundle for the n dimensional complex submanifold spanned by the n complex vielbein 1I . H H U The truncation corresponds therefore to select a nH -complex dimensional submanifold KH Q 1I spanned by the vielbein and to ask that, on the submanifold, the 2nH extraM degrees⊂M of freedom are frozen, thatU is:

2I KH =( 1I˙)∗ KH =0. (6.126) U |M U |M s KH t Calling w (s =1, nH )asetofnH holomorphic coordinates on and n (t = ··· M KH 2nH +1, , 4nH )asetof2nH real coordinates for the space orthogonal to ,wesee that equation··· (6.126), which can be rewritten as: M

2I 2I s 2I s 2I t KH = s dw + s dw + t dn KH =0, (6.127) U |M U U U |M 7 x x x  Note that Ω = λKuv with Kuv given in terms of the three complex structures by:

x x u v K = Kuvdq dq x x ∧w Kuv = huw(J )v . (6.121) The scale λ is fixed by supersymmetry of the Lagrangian and in our conventions is λ = 1. − 30 implies: 2I 2I s KH = s KH = 0 (6.128) U |M U |M since: t dn KH =0. (6.129) |M On the other hand, we also have:

1I t KH =0. (6.130) U |M since the vielbein 1I is tangent to the submanifold. Let us note that the conditions (6.120) on the curvaturesU Ω1, Ω2 imposed on the submanifold do not imply that all their components are also zero there, and indeed from (6.128), (6.130) and the definition (6.122) it follows: 1 1 2 2 Ωss KH =Ωtt KH =Ωss KH =Ωtt KH = 0 (6.131) |M 0 |M |M 0 |M 1 2 while the mixed components Ωst KH ,Ωst KH (together with their complex conjugates 1 2 |M |M Ωst KH ,Ωst KH ) are different from zero. We also observe that, when the truncation is performed,|M also|M the mixed components of the metric are zero:

1I hst KH = s 1I t KH =0. (6.132) |M U U | |M   From (6.128), (6.130) and (6.122) it also follows that the only components different from zero of the 2-form Ω3 are Ω3 and Ω3 . ss tt0 Let us now analyze in detail whether the involution of the constraint ∆I =0is J˙ satisfied:

I 1I 2K 2I 1K IR = λC ˙ + J˙ JK U ∧U −U ∧U II˙ 1K  2L 1K 2L˙ 1K˙ 2L˙ +2C Ω ˙ ˙ +2 Ω ˙ ˙ ˙ + Ω ˙ ˙ ˙ ˙ = 0(6.133) U ∧U IJKL U ∧U IJKL U ∧U IJKL h i After imposing (6.126), the first line in equation (6.133) is automatically zero, and equa- tion (6.133) is reduced to the constraint:

II˙ 1K 2L˙ 4C Ω ˙ ˙ ˙ = 0 (6.134) U ∧U IJKL Furthermore, let us note that when the constraint (6.133) is imposed, the Sp(2nH ) holon- omy gets reduced to U(1) SU(n ). The U(n ) curvature becomes: × H H IJ˙ 1I 2K˙ IK˙ JL˙ 1M 2N˙ IR = λ + C C Ω ˙ ˙ (6.135) U ∧U U ∧U NKML 4s+3 4s+4 Choosing cordinates such that q = q =0,s =0,...,nH 1 we may introduce complex coordinates ws = q1+4s+iq2+4s with K¨ahler 2-form K =Ω3−which is automatically 3 closed. The U(1) connection of the Hodge bundle is given by ωs as can be ascertained from the reduced form of equation (6.117) expressing the vanishing of the torsion on the K¨ahler-Hodge submanifold KH: M i 1I d 1I + ω3 1I +∆I 1J = 0 (6.136) ∇U ≡ U 2 ∧U J ∧U In conclusion, what we have found is that the conditions for the truncation of a quaternionic manifold (spanning the scalar sector of nH N = 2 hypermultiplets) to a K¨ahler–Hodge one (spanning the scalar sector of nh N = 1 chiral multiplets) are the following:

31 2I = ω1 = ω2 =∆I =08 •U J˙ The quaternionic manifold cannot be generic; in particular, the completely sym- • metric tensor Ωαβγδ Sp(2nH ), appearing in the Sp(2nH ) curvature, must have the following constraint∈ on its components:

ΩI˙JK˙ L˙ =0. (6.137)

The resulting submanifold, denoted by KH, has at most n complex dimensions [9] M H and is of K¨ahler–Hodge type, with K¨ahlerian vielbein P I , P I (for its normalization, see equation (6.150) below): 1 1I dqu P I dws (6.138) Uu −→ √2 s 1 2I˙dqu P I dws (6.139) Uu −→ √2 s

s (where w ,s =1, nH are complex coordinates on the reduced K¨ahler manifold) and U(1) U(n ) curvature··· given by: × H i I u 1I v Ω3δI +IRI . (6.140) RJ ≡Rv Uu U1J −→ 2 J J Once the dimension of the manifold has been truncated, the only constraint on the quaternionic manifold is given by equation (6.137). Let us therefore discuss how general it is, and which quaternionic manifolds satisfy it. First of all we note that the family of symmetric spaces Sp(2m +2)/Sp(2) Sp(2m)has × a vanishing Ω-tensor, Ωαβγδ = 0 [10], and hence a fortiori satisfy our requirement. Furthermore we can now show that the special quaternionic manifolds obtained by c-map [45] from special-K¨ahler manifolds do indeed satisfy the condition:

ΩI˙JK˙ L˙ = 0 (6.141) Indeed the tensor (6.141) appears in equation (6.133) multiplied by the product of the vielbein 1K 2L˙ . The same sub-block of the Sp(2n) curvature is denoted in [45] by r A . Now,U it is∧U easy to recognise that the set of n + 1 complex vielbein (v, ea) of [45] have 0 B to be identified with our vielbein 1K . However, no wedge product of type v ea nor of type ea eb appear in r A , whichU means that the corresponding coefficient Ω∧ =0. 0 B I˙JK˙ L˙ Therefore,∧ all the special quaternionic manifolds (including non symmetric quaternionic spaces) can be reduced to K¨ahler-Hodge manifolds in a way consistent with our procedure. There are however other symmetric spaces which do not correspond to c-map of special- K¨ahler manifolds, yet they satisfy our constraints. Indeed consider the following reduction from quaternionic to K¨ahler-Hodge manifolds:

SO(4,n) SO(2,n ) SO(2,n ) 1 2 (6.142) SO(4) SO(n) −→ SO(2) SO(n ) × SO(2) SO(n ) × × 1 × 2 8One can easily ascertain that this kind of truncation is actually consistent on the basis of the Frobenius theorem. Indeed, all the objects which have been set to zero, that is 2I ,ω1,ω2, Ω1, Ω2, ∆I ,RI are in J˙ J˙ involution, on the basis of the structure equations of the manifold andU of the relevant Bianchi Identities.

32 where (n1 + n2 = n). We see that they satisfy our constraints. Indeed, the K¨ahler- Hodge manifold on the right of the correspondence in equation (6.142) is apparently a submanifold of the corresponding quaternionic with half dimension. Therefore the conditions for the validity of the Frobenius theorem have to be satisfied, in particular equation (6.137). Indeed, for symmetric spaces we can compute explicitly the Ω-tensor by comparing the general formula of the Riemann tensor for symmetric spaces:

uv αA βB 1 αA βB h γC δD ts u v = f | hfγC δD [t s] , (6.143) R U U −2 | U U

αA βB (where we have denoted by f | γC the structure constants of the isometry group of the symmetric manifold K = G/H, the index h running on the Lie algebra of H, the couple of indices Aα labelling the coset generators) with its general form in the case of quaternionic manifolds: i uv αA βB = Ωx (σ )ABCαβ +IRαβAB. (6.144) R tsUu Uv − 2 ts x ts One easily obtains

λ i AC BD h Ωαβγδ = (CαγCβδ + CαδCβγ)   fC α β D hf A γ δ B (6.145) − 2 − 4 { | } | { | } where the curly brackets mean symmetrization of the corresponding indices. Using equation (6.145), we have explicitly verified the validity of (6.137) in the case of omega-tensor appearing in (6.142). These quaternionic reductions explicitly appear in some effective lagrangians coming from superstring theory models [46].

We still have to analyze the effects of the reduction on the hyperini and on the super- symmetry transformation laws. They become, after putting 2 =0:

1I δqu = ζI 1 (6.146) Uu 2I u IJ˙ δq = C ζ ˙ (6.147) Uu − J 1 2J˙ u µ 1 1 I ∗ δζ =i C ˙ q γ  + gN  = δζ (6.148) I Uu IJ ∇µ I 1 2I u µ 1 1  J˙∗ δζ˙ =i C ˙ q γ  + gN  = δζ (6.149) J Uu JI ∇µ J˙ 1   Choosing the normalization in such a way to match the normalization of the kinetic terms of the N = 1 theory, we set: 2J˙ 1 C ˙ = P (6.150) U IJ √2 Is 1 N 1 = P N s (6.151) I √2 Is ζs √2P sIζ = √2gssP I ζ , (6.152) ≡ I s I 2J˙ u 1 s u CIJ˙ µq KH = PIs µw (6.153) U ∇ |M √2 ∇ which implies: 2J˙ u 1 s u CIJ˙δq KH = P Isδw (6.154) U |M √2

33 s where ζ denote chiral left-handed spinors with holomorphic world indices, PIs are the vielbein of the K¨ahler-Hodge manifold KH and ws its holomorphic coordinates. We observe that due to the definition (6.150)M the 2-form Ω3 defined in equation (6.125) is one half the K¨ahler 2-form on KH. In that way we obtain the standardM formulae for the N = 1 supersymmetry transformation laws of the chiral multiplets (ζs,ws), that is:

s s µ s δζ =iµw γ • + N  (6.155) ∇ • δws = ζs (6.156) • where s sJ 1 sJ 1J˙ t Λ N √2 g P N =2√2 g P C ˙ k L . (6.157) ≡ (Λ) J (Λ) JJ Ut Λ Note that the shift term N s is indeed different from zero, but dependse only on the isome- tries of the projected out part ofe the quaternionice manifold 9. Thee explicit N = 1 form of the gauging contribution will be given in the next section. From equation (6.147), however, we see that the condition 2I = 0 implies that the subset U of nH hyperinos ζI˙ have to be truncated out. Consistency of the truncation in equation (6.149) implies

1 1I u Λ 1I s Λ N 2g C ˙ k L =2g C ˙ k L = 0 (6.158) J˙ ≡ (Λ) IJ Uu Λ (Λ) IJ Us Λ e e g ks LΛ = 0. The restrictionse on the theorye imposede by thise constraint will be discussed (Λ) Λ in the subsectione 6.4. e e 6.4 Further consequences of the gauging The truncation N =2 N = 1 implies, as we have seen in the previous subsections, a number of consequences→ that we are now going to discuss, and in particular: The D-term of the N = 1-reduced gaugino λΛ = 2f Λλi2 is: • − i Λ i21 1ΛΣ 3 s 0 i D = W = 2g (Imf)− P (w )+P (z ) (6.159) − (Λ) Σ Σ   The N = 1-reduced superpotential, that is the gravitino mass, is: • i Λ˙ 1 2 L(z, w)= g ˙ L P iP (6.160) 2 (Λ) Λ˙ − Λ˙   The fermion shifts of the N = 1 chiral spinors χi = λi1 coming from the N =2 • gaugini are: N i =2gi L (6.161) ∇ The fermion shifts of the N = 1 chiral spinors ζs coming from N = 2 hypermultiplets • are: ˙ s t Λ 1I˙ s N = 4g ˙ k L . (6.162) − (Λ) Λ˙ Ut U2I˙ 9 1I˙ 1I˙ s 1I˙ t 1I˙ Indeed, the request = s dw + h.c. + t dn = 0 implies s = 0 but does not impose U |M U U |M U any restriction on the components orthogonal to the retained submanifold.

34 In order for the shifts given in eqs. (6.160), (6.161), (6.162) to define the correct trans- formation laws of the N = 1 theory, we still have to show that the superpotential L is covariantly holomorphic with respect to the ws coordinates:

L = 0 (6.163) ∇s and that the N s shift for the chiral multiplets coming from the quaternionic sector can be written with the standard expression for an N = 1 chiral multiplets shift, that is as:

N s =2ggss L (6.164) ∇s These features do indeed follow, as a consequence of the reduction SU(2) U(1) in the holonomy group. Indeed: → i L = LΛ P x (σx) 2 =ikt LΛ˙ Ωx (σx) 2 (6.165) ∇s 2 ∇s Λ 1 Λ˙ st 1 e Now, recalling that: e

x x 2 α 2β I 2J˙ I 2J˙ t s Ω (σ ) =2 C =4 C ˙ =4 C ˙dn dw (6.166) 1 U1 ∧U αβ U1 ∧U IJ U1tUs IJ ∧ x x 2 x x 2 we immediatly get: Ωst (σ )1 = 0 while Ωst (σ )1 =0,sothat sL = 0 follows. Let us now compute N s explicitly:6 ∇

s sJ 1 ss J 1J˙ t Λ N = √2 P gN =4g C ˙g k L J (Λ) JJ U2sUt Λ i Λ˙ e =4g C gss Ωx (σx)1 kt L (Λ) JJ˙ 2 ts e 2 Λ˙ e ˙ ˙ = i g gss P x(σx)1 LΛ = i g gss P 1 +iP 2 LΛ − e(Λ) ∇s Λ˙ 2 − (Λ) ∇s Λ˙ Λ˙ ss   =2g g sL, (6.167) (Λ)e ∇ e that is it has the right expressione for an N = 1 chiral shift.

Let us now discuss the implications of the gauging constraints (6.39), (6.40) and (6.158) on the N = 1 theory obtained by reduction, that is the consistency of the truncation of the second gravitino multiplet δψµ2 = 0 and of the spinors ζI˙ in the hypermultiplets sector for the gauged theory:

ω 2 =0 = g AΛ P 1 iP 2 = 0 (6.168) 1 ⇒ (Λ) Λ − Λ Λ 3  S12 =0 = g LeP = 0 (6.169) b ⇒ (Λ)e Λ e e se Λ δζ ˙ =0 = g k L = 0 (6.170) I ⇒ (Λ)e Λ e Since the vectors of the N = 2 theory which aree not gauged do not enter in the previous e e equations we may limit ourselves to consider the case where the index Λ runs over the adjoint representation of the N = 2 gauge group. If we call G(2) the gauge group of the N = 2 theory and G(1) G(2) the gauge group of the corresponding N =e 1 theory, then we have that the adjoint⊆ representation of G(2) decomposes as

Adj(G(2)) Adj(G(1))+R(G(1)), (6.171) ⇒ 35 where R(G(1)) denotes some representation of (G(1)) (the representation R(G(1))isof course absent for G(1) = G(2)). The gauged vectors of the N = 1 theory are restricted to the subset AΛ generating Adj(G(1)) (that is the index Λ is decomposed as Λ (Λ, Λ),˙ with Λ Adj{ (G}(2) and Λ˙ R(G(1))). → This decomposition∈ of the∈ indices is of course the same ase the one used in analyzinge the consequences of the constraint (6.36) in section (6.2). In particular, the graviphoton index Λ = 0 always belongs to the set Λ˙ since the graviphoton A0 is projected out. The quaternionic Killing vectors of the N = 2 theory then decompose as e ku ks ,ks ,kt ,ks ,ks ,kt . (6.172) Λ ⇒{ Λ Λ Λ Λ˙ Λ˙ Λ˙ } Obviously, we must have that ks = 0 since the Killing vectors of the reduced submanifold eΛ˙ have to span the adjoint representation of G(1). Viceversa, the Killing vectors with world index in the orthogonal complement, kt ,mustobeykt = 0, while kt are in general Λ Λ Λ˙ different from zero. Indeed, the isometries generated by kt would not leave invariant the Λ hypersurface describing the submanifolde KH Q. These properties will be in fact M ⊂M confirmed in Appendix E, by a careful analysis of the reductione of the quaternionic Ward identities. Coming back to the implications of the constraints (6.168), (6.169), (6.170), they can be rewritten, using the results of section (6.2), as follows:

g AΛ P 1 iP 2 = 0 (6.173) (Λ) Λ − Λ  Λ˙ 3 g(Λ)˙ L PΛ˙ = 0 (6.174) ˙ s Λ g(Λ)˙ kΛ˙ L = 0 (6.175) Since we have found that ks = 0, equation (6.175) is identically satisfied. Λ˙ Eq.s (6.173) and (6.174) are satisfied by requiring:

1 2 3 PΛ = PΛ =0; PΛ˙ = 0 (6.176) Then the superpotential of the theory is given by [11] - [19]: i L = LΛ˙ (z, z) P 1(w, w) iP 2(w, w) . (6.177) 2 Λ˙ − Λ˙   We are left with an N = 1 theory coupled to nV0 vector multiplets (Λ = 1, ,nV0 )and ˙ ··· nC +nH chiral multiplets (Λ=0, 1, ,nC ) with superpotential (6.177). All the isometries of the scalar manifolds are in principle··· gauged since the D-term of the reduced N =1 0 3 theory depends on PΛ(z, z)+PΛ(w, w). In the particular case where the gauge group G(1) of the N = 1 reduced theory is the same as the gauge group G(2) of the N = 2 parent theory, the index Λ˙ takes only the value zero and all the scalars are truncated out (LΛ =0,L0˙ = 1). The vectors AΛ are retained in the truncation while A0 is projected out. In this case the superpotential reduces to: i L = L0 P 1 iP 2 . (6.178) 2 0 − 0   0 Moreover, from equation (6.96) we have that in this case the prepotential PΛ =0,andthe 3 D-term depends only on PΛ(w, w). We then have an N = 1 theory coupled to nV vector

36 multiplets and nH chiral multiplets, with gauged isometries and superpotential (6.178). 1 2 Note that when P0 iP0 is constant, (6.178) gives a constant F-term. This case can only be obtained in absence− of hypermultiplets. Indeed, from the general quaternionic formula [47] 1 n P x = Ωx ukv (6.179) H Λ −2 uv∇ Λ we see that if nH = 0 a Fayet-Iliopoulos term, as well as a constant F-term, is excluded 6 x e e [47], since a constant P0 is not compatible with the covariance of the r.h.s. under SU(2) and the gauge group. Even when the theory is ungauged (ku =0)aconstantP x is still Λ 0 excluded for n = 0, since in this case equation (6.179) reduces to n P x = 0, implying H H Λ P x =0. 6 Λ e x 10 If nH = 0, then a constant PΛ is possible (N = 2 Fayet–Iliopoulos term) [48]e ,provided thee gauge group is abelian (otherwise it breaks the gauge group) and provided it satisfies the identity xyz y z  PΛPΣ = 0 (6.180) which follows from the general quaternionic Ward identity [4], [10]: 1 1 1 Ωx ku kv + xyzP yP z f ΓP x = 0 (6.181) λ uv Λ Σ 2 Λ Σ − 2 ΛΣ Γ in absence of hypermultiplets. When we reduce the theory to N =1,aconstantvalueofP i ξi =0(i=1, 2) and P 3 Λ˙ ≡ Λ˙ 6 Λ ≡ ξ3 = 0 are both compatible with all the constraints (6.173)-(6.175); in particular LΛξ3 =0 Λ 6 Λ and AΛξi = 0 implying the presence of a N = 1 Fayet-Iliopoulos term corresponding to Λ e ξ3 , and a constant F-term corresponding to ξi . e Λ e Λ˙ e 6.5 N =2 N =1scalar potential → Let us now compute explicitely the reduction of the scalar potential of the N = 2 theory down to N =1.TheN = 2 scalar potential is given by:

N=2 i  u v Λ Σ 1 1 ΛΣ Λ Σ x x x x Λ Σ = g k k +4huvk k L L + (Im − ) L L P P 3P P L L i Λ Σ Λ Σ Λ Σ Λ Σ V   −2 N −  − b b e e ee e e (6.182)e e while the Nbb=e 1e scalar potentiale e can be written in terms of the covariantlye e holomorphice e superpotential L as:

N=1 `` 2 1 Λ Σ =4 `L `Lg 3 L + ImfΛΣD D , (6.183) V ∇ ∇ − | | 16  where the holomorphic index ` runs over all the scalars of the theory. Before performing the reduction it is instructive to work out in detail the supersymmetry Ward identity involving the scalar potential [49], [50]:

δA N=2 = 12SACS + g W iACW  +2N AN α. (6.184) BV − CB i BC α B 10 0 An N = 2 Fayet–Iliopoulos term coming from PΛ is excludedb b by the Ward identity (6.95). bb

37 Instead of taking the trace of (6.184) on the SU(2) indices A, B, thus recovering the potential (6.182), one can alternatively write down the stronger relations:

δ1 N=2 = N=2 = 12S1C S + g W i1C W  +2N 1N α (6.185) 1V V − C1 i 1C α 1 b b δ2 N=2 = N=2 = 12S2C S + gbbW i2C W  +2N 2N α. (6.186) 2 V V − 2C i C2 α 2 and furthermore: b b bb δ2 N=2 =0= 12S2C S + g W iC2W  +2N 2N α. (6.187) 1 V − 1C i C1 α 1 b 3 Λ When we pass to the truncated theory, the matrixbb SAB becomes diagonal (S12 PΛL = 0) and its eigenvalues are the masses of the 2 gravitini: ∼

L 0 SAB = . (6.188) 0 L ! where: e i L = LΛ˙ (P 1 iP 2) (6.189) 2 Λ˙ − Λ˙ i L = LΛ˙ ( P 1 iP 2) (6.190) 2 − Λ˙ − Λ˙ so that: e

1 ˙ L 2 = S S11 = LΛ˙ LΣ P xP x +i P 1P 2 P 2P 1 (6.191) | | 11 4 Λ˙ Σ˙ Λ˙ Σ˙ − Λ˙ Σ˙ 1 ˙ h  i L 2 = S S22 = LΛ˙ LΣ P xP x i P 1P 2 P 2P 1 . (6.192) | | 22 4 Λ˙ Σ˙ − Λ˙ Σ˙ − Λ˙ Σ˙ h  i The difference betweene the 2 gravitino mass eigenvalues can be written in terms of the fermionic shifts as: i L 2 L 2 = P 1P 2 P 2P 1 = S1C S S2C S | | −| | 2 Λ˙ Σ˙ − Λ˙ Σ˙ C1 − C2 1  e = g W i1C W  +2N 1N α g W i2C W  2N 2N α . (6.193) 12 i 1C α 1 − i 2C − α 2   Let us now perform the reduction. Using, e.g., equation (6.185) and recalling that S =0andN 1 = 0 (see equation (6.158)), we find : 12 I˙

N=2 N=1 11 i11  i12  1 I → = 12S S + g W W + W W +2N N (6.194) 11 i 11 12 I 1 V −   b b b b Using equations (6.47), (6.46), the firstbb two terms of equation (6.194) give:

˙ ˙ 12S11S = 3P i P i LΛ˙ LΣ +3i P 2P 1 P 1P 2 LΛ˙ LΣ = 12LL (6.195) − 11 − Λ˙ Σ˙ Λ˙ Σ˙ − Λ˙ Σ˙ −   g W i11W  = P 1 +iP 2 P 1 iP 2 U Λ˙ Σ˙ =4gkl L L (6.196) i 11 Λ˙ Λ˙ Σ˙ − Σ˙ ∇k ∇l    b b bb 38 For the term g W i21W  we obtain: i 21 b b 1 1 g W i21W  b=b 2Im f Λf ΣW i21W  = Im DΛDΣ = Imf DΛDΣ (6.197) i 21 − NΛΣ i  21 −2 NΛΣ 4 ΛΣ b b e e b b wherebb we have reduced theee b indicesb according to the results of subsection (6.3) and used equations (6.101), (6.100), (6.102). To compute the last term in equation (6.194) we use equation (6.151) and (6.164) and we find 2N 1N I = g N sN s =4gss L L (6.198) I 1 ss ∇s ∇s Collecting all the terms we find that the reduction of the N = 2 scalar potential gives:

N=2 N=1 i ss 1 Λ Σ → =4 3LL + g iL L + g sL sL + ImfΛΣD D (6.199) V − ∇ ∇ ∇ ∇ 16  which coincides with the scalar potential (6.183) of the N = 1 theory, where we have decomposed the indices according to the fact that the σ-model is a product manifold . We note that our computation of the reduction of the scalar potential has been per- formed by first reducing the N = 2 fermionic shifts to N = 1 and then computing the potential. Of course, we could also have performed the computation by directly com- puting the reduction of each term of the N = 2 potential. In the latter case, to obtain the desired results requires some non trivial computations. In particular, there are some subtleties related to the observation that the N = 2 potential does not contain “interfer- ence” contributions of the form P 0P x or P x P y , while such terms are instead present in Λ Σ [Λ Σ] the N = 1 potential, given the form (6.159) of the D-term and (6.160) of the superpo- tential. To solve the puzzle and recovere e thee precisee correspondence between the N =2 and N = 1 theories, one has to use several times the reduced forms of the Ward identities of quaternionic and special-K¨ahler geometries, the definition of the quaternionic Killing vectors [39], [47], [51] and the expression that the special geometry prepotential gets in the reduction, equation (6.96). The explicit computation is given in Appendix F.

6.6 Examples of truncation to N =1gauged supergravity As an application of the formalism developed in this section, we can now consider reduc- tion on N =8toN = 1 or in general of N = 2 theories down to N =1. ThesimplestcaseistoconsiderN = 2 special-K¨ahler manifolds which are also N =1 Hodge-K¨ahler, or submanifolds of half the dimension of quaternionic manifolds which are “dual” (under c-map) to special-K¨ahler. We first consider “dual quaternionic manifolds” which are symmetric spaces; they were all given in Table 4 of ref. [33]. This immediately gives the N =2 N = 1 reduction of → GQ theories with only hypermultiplets as follows: It is interesting to note that if Q = , M HQ SU(1,1) G SK = U(1) ×H then HQ = SU(2) Gc,whereGc is the compact form of G!. FromM the previous× table we can immediately× obtain N = 1 truncations of N =8super- (N=1) gravity with (nV ,nH ) replaced by (nV ,nC = nV + nQ). In all these models (unless nQ =0)theK¨ahler-Hodge manifold will be of the form SU(1, 1) SK(n ) SK(n 1) . (6.200) V × Q − × U(1)

39 Table 5: N =2 N =1 N =2(nV =0, Q (dimQ = n) N →=1(nV =0, KH (dimC = n) U(2,nM+1) SU(1,1) M U(1,n) U(2) U(n+1) U(1) U(1) U(n) SO(4,n+1)× SU(1,1) SU(1,×1) ×SO(2,n 1) − SO(4) SO(n+1) (n 2) U(1) U(1) SO(2) SO(n 1) × ≥ × × × − G2(2) SU(1,1) SU(1,1) SO(4) U(1) × U(1) F4(4) SU(1,1) Sp(6,IR) USp(6) USp(2) U(1) U(3) × × E6(2) SU(1,1) SU(3,3) SU(6) SU(2) U(1) SU(3) SU(3) U(1) × × × × E7( 8) SU(1,1) SO∗(12) − SO(12) SU(2) U(1) U(6) × × E8( 24) SU(1,1) E7( 26) − − E7 SU(2) U(1) E6 SO(2) × × ×

As a simple example, motivated by string construction [46], for the application of the results of the previous sections, we consider a N =4,D = 4 matter coupled supergravity with gauge group SO(n)(n even). The σ-model of the scalars in presence of gauging is given by: n(n 1) − SU(1, 1) SO(6, 2 ) n(n 1) , (6.201) U(1) × SO(6) SO( − ) × 2 and the content of the scalar sector can be encoded in the vielbein 1-form PABI where the antisymmetric couple AB labels the irrep. 6 SU(4) and I labels the fundamental n(n 1) ∈ − representation of SO( 2 ). This N = 4 theory is reduced to N = 2 through the action of a ZZ2 group and to N =1 by the action of ZZ2 ZZ20 . The generators of ZZ2 ZZ20 in the R-symmetry group SU(4) are given by: × × 10 0 0 1000 01 0 0 0 eiπ 00 α =   ; β =   (6.202) 00eiπ 0 0010      00 0 eiπ   000eiπ          so that two gravitinos are singlets with respect to ZZ2 and one gravitino is invariant with respect to ZZ2 ZZ20 . To obtain× charged matter in the N =4 N = 2 reduction, we implement the action → of ZZ2 on the gauge group. Let us make the following decomposition

SO(n) ZZ2 SO(n ) SO(n ) (6.203) −→ A × B so that, under the action of ZZ2:

n n A ⇒ A n αn (6.204) B ⇒ B and then

Adj(SO(n )) ZZ2 Adj(SO(n )) A −→ A Adj(SO(n )) ZZ2 Adj(SO(n )) B −→ B (n ,n ) ZZ2 α(n ,n ). (6.205) A B −→ A B

40 Correspondingly, for the group SU(4) we have:

4 ZZ2 α4 −→ 2 ZZ2 2 (6.206) 1 −→ 1

The scalars transforming non trivially under ZZ2 are projected out and we are left with the coset manifold:

n (n 1) n (n 1) A A− B B − SU(1, 1) SO(2, 2 + 2 ) SO(4,nAnB) n (n 1) n (n 1) (6.207) U(1) × SO(2) SO( A A− + B B − ) × SO(4) SO(nAnB) × 2 2 × where the first two factors define an N = 2 special-K¨ahler manifold and the last factor is a quaternionic manifold. In order to obtain an N = 1 supergravity theory, the gauge groups SO(nA)and SO(nB) are further decomposed as follows: SO(n ) SO(n ) SO(n ) (6.208) A → 1 × 2 SO(n ) SO(n ) SO(n ) (6.209) B → 3 × 4 andwedefinetheactionofZZ20 as: n n ,nβn 1 ⇒ 1 2 ⇒ 2 n n ,nβn . (6.210) 3 ⇒ 3 4 ⇒ 4

This induces an action of ZZ ZZ0 on the decomposition of the gauge group: 2 × 2 Adj(SO(n)) ZZ2 Adj(SO(n )) + Adj(SO(n )) + (n ,n ) −→ A B A B α ZZ2 ZZ × 20 Adj(SO(n )) + Adj(SO(n )) + Adj(SO(n )) + Adj(SO(n )) + −→ 1 1 2 1 3 1 4 1 +(n1,n2, 1, 1)β +(1, 1,n3,n4)β +(n1, 1,n3, 1)α +

+(n1, 1, 1,n4)αβ +(1,n2,n3, 1)αβ +(1,n2, 1,n4)α (6.211) In equation (6.211) we have labelled each representation with indices 1,α,β,αβ whose meaning is that the corresponding representation is invariant or transforms under α, β or αβ respectively. That is the representations Adj(SO(n )) are invariant under ZZ ZZ0 , I 2 × 2 while the remaining bifundamental representations (nI ,nJ ) transform as follows:

(n1,n3); (n2,n4) transform under α

(n1,n2); (n3,n4) transform under β

(n2,n3); (n1,n4) transform under αβ (6.212)

With the same notation, let us now consider the ZZ ZZ0 action on the 6 of SU(4): 2 × 2 6 ZZ2 4 +2 −→ α 1 ZZ2 ZZ × 20 (2 +2 )+2 (6.213) −→ α αβ β n(n 1) − Joining the information coming from the the decomposition of SU(4) and SO( 2 ) we see that the scalars which remain invariant under the action of ZZ ZZ0 are given 2 × 2 41 by the vielbein in the following representations: P2α(n1,n3);P2α(n2,n4); P2β(n1,n2);P2β (n3,n4);

P2αβ(n1,n4);P2αβ (n2,n3). This means that the special-K¨ahler manifold reduces to:

n (n 1) n (n 1) A A− B B− SU(1, 1) SO(2, 2 + 2 ) SU(1, 1) SO(2,n1n2 + n3n4) n (n 1) n (n 1) . U(1) × SO(2) SO( A A− + B B − ) → U(1) × SO(2) SO(n1n2 + n3n4) 2 2 × × (6.214) while the quaternionic manifold splits as follows: SO(4,n n ) SO(2,n n + n n ) SO(2,n n + n n ) A B 1 3 2 4 1 4 2 3 . SO(4) SO(nAnB) → SO(2) SO(n1n3 + n2n4) × SO(2) SO(n1n4 + n2n3) × × × (6.215) Let us now comment this result. From the analysis in section (6,2) we have learnt that when we reduce a gauged N =2 theory to N =1(withG(2) G(1)) the surviving scalars from the vector multiplets sector are those which are in the representation→ R(G(1)) according to equation (6.171), while all the scalars in the adjoint representation of G(1) are truncated out. Precisely this happens in our case. Indeed, from equation (6.214) the irreps (n1,n2)and(n3,n4)belongtothe left over representations in equation (6.211). Furthermore, all the other bifundamental rep. belong to the scalars coming from the quaternionic sector, according to equation (6.215). Note that the total dimensional of the product manifold of equation (6.215) is exactly half the dimension of the parent quaternionic manifold, according to the general result found in section (6.1). It is interesting to observe that the same kind of result appears in the decomposition N =4 N = 2 described by equation (6.207). In fact, the reduced product manifold in equation→ (6.207) has a σ-model whose scalars belong again to the representation R =(nA,nB) left over in the reduction of the adjoint representation of the N = 4 gauge group. Other examples can be obtained [35] from heterotic strings compactified on ZZN orb- ifolds with reduced non abelian gauge group E6. We finally observe that the N = 2 special-K¨ahler manifold in the l.h.s. of (6.214) can Λ Σ Λ Σ be parametrized with the symplectic section (L ,MΛ)=ηΛΣSL (with L L ηΛΣ =0 and η =(1, 1, 1, , 1)) where a prepotential F does not exist [52]. In this case the ΛΣ − ··· − e e e e N = 2 vector kinetic matrix has the form: e ee ee ee LΛ ΛΣ =(S S) ΦΛΦΣ + ΦΛΦΣ + SηΛΣ ;ΦΛ . (6.216) N − ≡ Λ   L eLΛ q When we performee the truncatione to eN = 1,e thee sectionsee LΛ becomee zero,e and the N =1 e vector kinetic matrix takes the form: = Sη , (6.217) NΛΣ ΛΣ that is it becomes antiholomorphic in the complex scalar S parametrizing the manifold SU(1,1) U(1) .

Acknowledgements

We acknowledge interesting discussions with I. Antoniadis, A. Ceresole, G. Dall’Agata, M.A. Lled´o and R. Varadarajan. One of the authors (S.F.) warmly thanks the Department

42 of Physics of the Politecnico di Torino, where part of this work has been done. The work of L. A. and R. D. has been supported in part by the European Commission RTN network HPRN-CT-2000-00131 (Politecnico di Torino). The work of S. F. has been supported in part by the European Commission RTN network HPRN-CT-2000-00131, (Laboratori Nazionali di Frascati, INFN) and by the D.O.E. grant DE-FG03-91ER40662, Task C.

Appendix A: Supersymmetry reduction from super- space Bianchi identities

We want now to show that the constraints found at the level of supersymmetry transfor- mation laws are actually sufficient to guarantee the closure of the supersymmetry algebra of the reduced theory. We prove this statement by considering the reduction of the superspace Bianchi identities of the N = 8 theory (which, as is well known, is equivalent to the “on-shell” closure of the supersymmetry algebra). The N = 8 Bianchi identities are [53],[37] (we omit the wedge product symbols among the products of forms):

Rpq V +iψ γpρA iρ γpψA = 0 (A.1) ∧ q A − A 1 ρ + Rpqγ ψ R Bψ = 0 (A.2) ∇ A 4 pq A − A B ΛΣ ΛΣ A B ΛΣAB F 2fAB ρ ψ 2f ρAψB + ∇1 − − 1 f ΛΣCDP ψAψB f ΛΣP ABCDψ ψ = 0 (A.3) −2 ABCD − 2 CD A B 1 ( χ ) 3R Lχ + Rpqγ χ = 0 (A.4) ∇ ∇ ABC − [A BC]L 4 pq ABC P = 0 (A.5) ∇ ABCD in terms of the supercovariant field-strengths: i T p V p ψ γpψA = 0 (A.6) ≡D − 2 A pq pq p rq R dω ω rω (A.7) ≡ − A ΛΣ AB F ΛΣ dAΛΣ + f ΛΣψ ψB + f | ψ ψ (A.8) ≡ AB A B ρ ψ + ω Bψ (A.9) A ≡D A A B χ χ +3ω Lχ (A.10) ∇ ABC ≡D ABC [A BC]L R B dω B + ω C ω B. (A.11) A ≡ A A C Note that all the fields are actually superfield 1-forms whose restriction at θ = dθ =0 gives the ordinary space–time fields. To show how the Bianchi identities of the N = 8 theory reduce to the Bianchi identities of the N = N 0 theory, we just work out the example of the N =8 N = 6 reduction. The other cases can be analyzed in analogous way. → First of all we see that, by decomposing the R-symmetry indices as in section 2 and setting ψi =0(i =7, 8), the supercovariant field-strengths get reduced as follows: the

43 superspace bosonic vielbein V p and the spin connection ωpq (p, q denote space–time flat indices) remain untouched by the reduction, and the same happens of course for the Lorentz curvature Rpq and supertorsion T p. As far as the gravitinos are concerned, we find: b ρa ψa + ωa ψb (A.12) ≡D a 0=ρi = ωi ψa (A.13) a which implies ωi = 0, consistently with what we found in section 4. As a consequenc, the gravitinos Bianchi identities reduce to: 1 ρ + Rpqγ ρ + R bψ = 0 (A.14) ∇ a 4 pq a a b which is the correct Bianchi identity for an N = 6 gravitino, while consistency of the truncation implies: ρ = R aψ =0 R a = 0 (A.15) ∇ i i a → i again in agreement with the σ-model results. Let us analyze the spin one-half sector. It gives χ = χ +3ω dχ + ω iχ (A.16) ∇ abc D abc [a bc]d [a bc]i χ = χ 2ω dχ +2ω jχ + ω dχ + ω jχ (A.17) ∇ abi D abi − [a b]di [a b]ij i dab i jab χ = χ ω dχ +2ω dχ + ω dj χ . (A.18) ∇ aij D aij − a dij [i j]ad i jab i Since ωa = 0, we see that the last equation is satisfied only setting χiab = 0, (as already known from section 2, since they belong to the gravitino multiplets truncated out). What is left is the spin one-half sector of the N = 6 theory. It is now straightforward to see that the Bianchi identities for χabc and χaij reduce, after imposing again the constraint i ωa = 0, to the corresponding N = 6 Bianchi identities, while the Bianchi identity for χabi is, consistently, identically satisfied. The analysis of the scalar sector PABCD and its Bianchi identity is identical to what has been already discussed in section 4, and does not deserve further analysis. Finally, the Bianchi identity for the vector field strengths, with ψi = ρi = 0, reduces to: 1 ΛΣ cd a 1 ΛΣ ij a 1 ΛΣ ci a F ΛΣ = 2f ΛΣρaψb f | P ψ ψb f | P ψ ψb f | P ψ ψb. (A.19) ∇ − ab − 2 abcd − 2 abij − 2 abci Here, the scalar vielbein Pabci = 0 according to the discussion of section 2 and 4. Further- more, the reduction of the couple of indices ΛΣ goes according to what we have discussed in section 5. Since the duality group acts now on the electric and magnetic field-strengths in the representation 32 of SO∗(12), we simply substitute the couple ΛΣ with an index r r r running from 1 to 16. Note that the corresponding quantities fab,fij are 16 16 sub- blocks of the 32 32 matrix U, which has exactly the same form of equation× (5.6), but valued in Sp(32,×IR), which gives the embedded coset representative.

Appendix B: Consistency of the Bianchi identities for N =2 N =1gauged theory in D =4. → In the same spirit of the analysis of section (5.1), it is easy to show that the closure of Bianchi identities of the N = 2 theory implies the consistent closure of the reduced N =1

44 theory. The definition of the supercurvatures and superspace Bianchi identities for the N =2 theory have been given in ref [4] (Appendix A). We have to reduce these objects to their N = 1 expressions, and to show that they coincide with the definitions of the supercurvatures and superspace Bianchi identities for the N = 1 theory. We quote in the following their standard expression. Curvatures of N =1gauged theory

T a V a iψ•γaψ 0 (B.1) • ab ≡Dab − a cb ≡ R = dω ω cω (B.2) − i ρ = ψ = ψ + Qψ (B.3) • ∇ • D • 2 • i R χi = χi = χi + Γi bχj Qχi (B.4) ∇ D j − 2   1 F Λ = dAc Λ + CΛ Ab ΣAΓ b (B.5) 2 ΣΓ i λΛ = λΛ + QλΛ + CΛ AΣλΓ (B.6) ∇ D 2 ΣΓ zi = dzi + g bki AΛ (B.7) ∇ (Λ) Λ where the gauged connections are defined as:

Γi =Γi + g ki AΛ (B.8) j j (Λ)∇j Λ Λ bQ = Q + g(Λ)PΛA . (B.9) The ungauged connection Q is givenb by

Q = Q zi + Q zı. (B.10) i∇ ı∇ Bianchi identities of N =1gauged theory

ab a a R Vb iψ•γ ρ +iψ γ ρ• = 0 (B.11) − • • Rab = 0 (B.12) D 2 1 ab i ψ + γabR ψ ψ = 0 (B.13) ∇ • 4 • − 2K • 1 i 2χi + γ Rabχi + Ri bχj + χi = 0 (B.14) ∇ 4 ab j 2K Λ F = 0b b (B.15) ∇ 1 i 2λΛ + γ RabλΛ λΛ CΛ AΣλΓ = 0 (B.16) ∇ 4 ab − 2K − ΣΓ 2 i i Λ z g k F = 0b (B.17) ∇ − (Λ) Λ In the ungauged case, it is straightforward to see that the conditions found in the text from the analysis of the reduction of the quaternionic sector and of supersymmetry trans- formation laws are indeed necessary and sufficient, after setting ψ2 = ρ2 = 0, for reducing the N = 2 supercurvatures and Bianchi identities to the corresponding N = 1 expres- sions.

45 We only observe that in the covariant differential of ζα and its Bianchi identity, after decomposition of the index α =(I,I˙), we get, as integrability condition: 1 i 2ζ + Rabγ ζ + Kζ +IR J ζ =0, (since IR J˙ = 0). (B.18) ∇ I 4 ab I 2 I I J I This equation can be converted in world indices on KH using equation (6.150). Using further the reduction of equation (6.144) one then recoversM the correct N =1result,in terms of the Riemann curvature of the KH manifold, with M (N=1) = (N=2) +Ω3. (B.19) K K Note that Ω3 is one half of the K¨ahler form of the K¨ahler-Hodge manifold KH. As far as the gauged theory is concerned, we observe that the ungaugedM conditions

α α 1 2 1 2 J˙ J˙ Γ i = R i = ω = ω =Ω =Ω =∆I = RI = 0 (B.20) become the corresponding ones for the gauged quantities

α α 1 2 1 2 J˙ J˙ Γ i = R i = ω = ω = Ω = Ω = ∆I = RI =0. (B.21)

Recalling the definitionb ofb the hattedb b quantities,b b we findb thatb the following objects must be zero:

g AΛ kα = g F Λ kα = 0 (B.22) (Λ) Dj Λ (Λ) Dj Λ g AΛ(P 1 e iP 2)=g F Λe(P 1 iP 2) = 0 (B.23) (Λ) e Λ − Λe (Λ)e bΛ e− Λ Λ ev u AI e g(Λ)A ∂uk | v AJ˙ = 0 (B.24) e ΛU e U | e e e e e The previous conditionse can bee analyzed in the light of the results obtained in section 6, and it is straightforward to see that they are actually satisfied. Thus the reduced theory has the correct integrability conditions.

Appendix C: A useful formula for the N =2gaugino transformation law

In this Appendix we show how to retrieve equation (6.83) from (6.3). To avoid a too heavy notation, we write in this Appendix the world indices and gauge indices without hat and tilde, since we are not going to perform any reduction. We are interested in trading the world index i of the gauginos λiA with a gauge index Λ, through the definition:

λΛA 2f ΛλiA (C.1) ≡− i

However, the gauge index of the N = 2 theory runs over nV + 1 values (because of the presence of the graviphoton) while the index i takes only nV values. The extra gaugino, say λ0, is actually spurious, since, as discussed in section (6.2), λΛA satisfies:

ΛA TΛλ =0 (C.2)

46 where TΛ is the projector on the graviphoton field-strength, according to equation (6.16) Λ [42]. In order to show that the nV gauginos λ do appropriately transform into the nV matter-vector field strengths, let us now calculate the susy transformation law of the new fermions λΛA, which, up to 3-fermions terms, is:

ΛA Λ iA Λ i Σ Γ µν AB iAB δλ = 2f δλ = 2f g f Im F − γ ε + W  (C.3) − i − i −  NΓΣ µν B h i Now we use the following relations of special geometry [42]:

g = 2f ΛIm f Σ (C.4) ij − i NΛΣ j δi = gijg = 2gijf ΛIm f Σ ` `j − ` NΛΣ j

ΛΣ Λ ij Σ 1 1 ΛΣ Λ Σ U f g f = (Im )− L L . (C.5) ≡ i j −2 N − h i 1 Im LΛLΣ = (C.6) NΛΣ i −2 Eq. (C.3) can then be rewritten as:

ΛA ΛΣ Γ AB Λ ` ∆ AB ΛΣ x x AB δλ = 2U Im F − ε 2f k L ε 2iU P (σ )  (C.7) NΣΓ µν − ` ∆ − Σ B h i Now, using the definition of the special geometry Killing vectors

i i 0 kΛ =ig ∂PΛ (C.8) we have: 2f Λk` L∆ =2if Λg``∂ P 0 L∆ = 2if Λg``f ∆P 0 = 2iU ΛΣP 0 (C.9) ` ∆ ` ` ∆ − ` ` ∆ − Σ where we have used the special geometry formulae [4]:

P 0LΛ = P 0LΛ =0,fΛ LΛ. (C.10) Λ Λ i ≡∇i Therefore equation (C.7) becomes:

ΛA ΛΣ Γ µν AB 0 AB x x AB δλ = 2U Im F − γ ε +i P ε + P (σ )  (C.11) − NΣΓ µν − Σ Σ B h  i Let us now set

P Λ 2U ΛΣIm = δΛ +2Im LΛLΣ = δΛ iT LΛ (C.12) Γ ≡− NΣΓ Γ NΓΣ Γ − Γ Λ Λ Λ t Λ P Σ = δΣ +iT ΣL =(P ) Σ (C.13) where TΛ is defined by equation (6.81) and satisfies [42] :

T LΛ = i; T f Λ = 0 (C.14) Λ − Λ i Then we have: Λ Σ Λ Λ P ΣP Γ = P Γ; TΛPΓ = 0 (C.15) Λ Therefore P Γ is the projector orthogonal to the graviphoton, that is it projects the nv +1 vector field-strengths onto the nv field-strengths of the vector multiplets.

47 We can then rewrite equation (C.11) as:

ΛA Λ Σ µν AB ΛΓ 0 AB x x AB δλ = P F − γ ε +iU P ε + P (σ )  (C.16) Σ µν − Γ Γ B   which is the equation given in the text. Note that

ΛA Λ ΣA Λ Λ Σ λ = P Σλ ; fi = P Σfi . (C.17)

It is useful to write down the explicit decomposition of the field strength F Λ into the graviphoton and matter vectors part, that is:

Λ Λ Σ Λ Σ Fµν− =iL TΣFµν− + P ΣFµν− . (C.18) Equation (C.16) becomes:

Λ Λ Γ µν AB 1ΣΓ 0 AB x x AB δλ = P Σ Fµν− γ ε +i(Im )− PΓ ε + PΓ (σ ) B (C.19) • N − h  i where we see, as expected, that the gauginos λΛA do transform only into the matter-vector Λ Σ field strengths P ΣFµν− . Hence, equation (C.19) intrinsically defines only nV independent Λ Σ gauginos transforming into the N = 2 field strengths (P ΣFµν− ).

Appendix D: Reduction of special geometry in special coordinates

If we choose special coordinates for special geometry [3],[8],[42],[52], then the indices Λ and i are identified by the fact that e b XΛ ti = , Λ=˙ i, Λ=α (D.1) X0 e   b 1 and a prepotential F (X) exists such that f(t)= (X0)2 F (X), with:

0 i ∂f X F0 =2f t fi , fi = (D.2) − ∂ti ! X0F = ∂ f. b (D.3) i i b b b

Furthermore, b b i i e−K =i 2f 2f +(t t )(fi + f i) . (D.4)  − −  The constraints that define the submanifold b become:b MR b b

Wijα = ∂i∂j∂αf =0 (D.5) ∂F XΛ = = ∂ XΛ = ∂ XΛ˙ = ∂ f = ∂ f =0 (D.6) ∂XΛ i α i Λ α Λ˙ where we used the fact that α =0. K |MR 48 In this basis = ∂ ∂ f and the K¨ahler potential on is: N ΛΣ Λ Σ MR

R i i e ee−K =i 2f 2f +(t t )(fi + f ) . (D.7) − − i h i Note that FΛ = 0 implies ∂αf = 0 which in turn implies ∂α∂if =0,Wαij = MR MR MR | | i i α | ∂α∂i∂jf = 0. Therefore the most general form for f is (t (t ,z )): |MR ⇒ i α α1 αn b f(t ,z )=f(t)+ z z fα1 αn (t). (D.8) ··· n 2 ··· X≥ For the manifold SU(1, 1)/U(1) SO(2,n)/[SO(2) SO(n)] used in section (6.6), 0 1 n 1 × n n × 1 n n with coordinates (t ,t , ,t 0 ,z , ,z − 0 ), the reduced manifold (z , ,z − 0 )=0is ··· ··· ··· parametrized with coordinates (t0,t1, ,tn0 ) and the holomorphic prepotential is [48],[42]: ···

n0 n n0 f(ti,zα)=it0 η titj − δ zαzβ , [η =(1, 1, , 1)] (D.9)  ij − αβ  ij − ··· − Xi=1 Xi=1   in accordance to equation (D.8).

Appendix E: Reduction of the quaternionic Ward iden- tity

We derive here the conditions on the quaternionic prepotentials and Killing vectors, dis- cussed in Section (6.5), from the reduction of the quaternionic Ward identity (6.181), which is essential for the validity of the N = 2 supersymmetric Ward identity involving the scalar potential, that is the relation [4]: 1 1 1 Ωx ku kv + xyzP yP z f ΓP x =0 (E.1) λ uv Λ Σ 2 Λ Σ − 2 ΛΣ Γ After projection, and using the just found results P i =0;P 3 = 0, it decomposes in a set Λ Λ˙ of equations

Λ=Λ;Σ=Σ • e e 1 1 1 Ωi kukv + ijP j P 3 f ΓP i =0 (E.2) λ uv Λ Σ 2 Λ Σ − 2 ΛΣ Γ 1 1 1 e Ω3 kukv + ijP i P j f ΓP 3 =0 (E.3) λ uv Λ Σ 2 Λ Σ − 2 ΛΣ eΓ e Since P j = 0, and since f Γ˙ = 0 (because G(1) G(2)), thene equation (E.2) gives Λ ΛΣ ⊂ t kΛ =0, (E.4) as indeed was expected from geometrical considerations. Then, equation (E.3) becomes 1 1 Ω3 ks ks f ΓP 3 =0. (E.5) λ ss Λ Σ − 2 ΛΣ Γ 49 Setting λ = 1 and recalling that Ωss is half of the K¨ahler form of the reduced submanifold,− we recognize that equation− (E.5) expresses the Poissonian realization 3 KH of the Lie algebra of the prepotentials PΛ on the K¨ahler-Hodge submanifold , namely: M P 3,P3 = f ΓP 3. (E.6) { Λ Σ}P ΛΣ Γ Λ=Λ;Σ=Σ˙ • e e 1 1 1 Ωi ks kt ijP 3P j f Γ˙ P i =0 (E.7) λ st Λ Σ˙ − 2 Λ Σ˙ − 2 ΛΣ˙ Γ˙ 1 1 1 Ω3 kukv + ijP i P j f ΓP 3 =0 (E.8) λ uv Λ Σ˙ 2 Λ Σ˙ − 2 ΛΣ˙ Γ e Eq. (E.7) gives a relation which has to be valid everywheree on the submanifold. Since Ω3 =0,P 3 = P i = 0, and considering (E.4), then equation (E.8) is identically st Λ˙ Λ satisfied.

Λ=Λ;˙ Σ=Σ˙ • e e 1 1 1 Ωi ks kt ijP 3P j f Γ˙ P i =0 (E.9) λ st Λ˙ Σ˙ − 2 Λ˙ Σ˙ − 2 Λ˙ Σ˙ Γ˙ 1 1 1 3 t t0 ij i j Γ 3 Ω k ˙ k ˙ +  P ˙ P f ˙ ˙ P = 0 (E.10) λ tt0 Λ Σ 2 Λ Σ˙ − 2 ΛΣ Γ Eq. (E.9) is identically satisfied for f Γ˙ , while equation (E.10) is a relation to be Λ˙ Σ˙ satisfied all over the submanifold.

Appendix F: Computation of the N =2 N =1scalar → potential

We want to solve here a puzzle raised in the text about the scalar potential. In the N =2 theory, the scalar potential has the form [4]:

N=2 11 i11  i21  1 I = 12S S11 + gi W W 11 + W W 21 +2NI N1 V −   i  u vb Λb Σ bΛΣ bx x x x Λ Σ = g k k +4huvk k L L + U P P 3P P L L . (F.1) i Λ Σ bb Λ Σ Λ Σ Λ Σ   − b b e e ee e e which manifestly does notbb containe e contributionse e antisymmetrice e in thee quaternionice prepo- tentials or Killing vectors, nor it has interference terms P 0P x between quaternionic and Λ Σ special-K¨ahler isometries. On the other hand, the N = 1 scalar potential: e e

N=1 `` 2 1 Λ Σ =4 `L `Lg 3 L + ImfΛΣD D , (F.2) V ∇ ∇ − | | 16  does instead contain both kinds of interference contributions, given the form of the su- perpotential L = i LΛ˙ (P 1 iP 2) and of the D-term DΛ = 2(Imf) 1ΛΣ(P 0 + P 3)which 2 Λ˙ Λ˙ − Σ Σ appear quadratically in (F.2).− −

50 The interference contributions in (F.2) have therefore to cancel each other. As we are going to show, this does indeed happen, in a way which involves non trivially the properties obeyed by the special-K¨ahler and quaternionic Killing vectors. Let us analyze and reduce separately the various contributions to the N = 2 potential, using all the constraint relations found in Section 6.

11 2 12S S11 12 L − ⇒− | | ˙ ˙ = 3P i P i LΛLΣ˙ 6iP 1 P 2 LΛ˙ LΣ (F.3) − Λ˙ Σ˙ − [Λ˙ Σ]˙

g W i11W  U ΛΣ 4gi L L i 11 ⇒ ∇i ∇ ˙ b b ee = gi LΛ˙ LΣP i P i 2iP 1 P 2 U Λ˙ Σ˙ bb ∇i ∇ Λ˙ Σ˙ − [Λ˙ Σ]˙ ˙ ˙ = gi LΛ˙ LΣP i P i +2iP 1 P 2 LΛ˙ LΣ (F.4) ∇i ∇ Λ˙ Σ˙ [Λ˙ Σ]˙

i21  1ΛΣ 0 0 3 3 ΛΣ 0 3 g W W (Imf)− P P + P P +2U P P (F.5) i 21 ⇒ Λ Σ Λ Σ Λ Σ   where we have used equationb b (C.9) of Appendix C, the identity of special geometry (6.103) bb 0 and the definition of the prepotential PΓ , equation (6.96).

2N 1N I 4gss L L I 1 ⇒ ∇s ∇s = gss P i P i +2igss P i P i (F.6) ∇s Λ˙ ∇s Σ˙ ∇s [Λ˙ ∇s Σ]˙ The last term in equation (F.6) is transformed using the definition of quaternionic Killing vectors: 2kv Ωx = P x (F.7) Λ uv ∇u Λ the realization of the SU(2) algebra on the curvatures Ωx:

hstΩx Ωy = λ2δxyh + λxyzΩz (F.8) us tw − uw uw and the normalization chosen for the metric on KH: M 1 h = g . (F.9) ss 2 ss After some calculations we get:

˙ 2igss P i P i =4iΩ3 kt ht LΛ˙ LΣ ∇s [Λ˙ ∇s Σ]˙ tt [Λ˙ Σ]˙ ˙ 1 2 i Γ 3 Λ˙ Σ = 4iP[Λ˙ PΣ]˙ f Λ˙ Σ˙ PΓ L L  − 2  ˙ 1 2 Λ˙ Σ 3 0 1ΛΣ =4iP P L L 2P P (Imf)− (F.10) [Λ˙ Σ]˙ − Λ Σ where we have applied the quaternionic Ward identity (6.181) discussed in Appendix D 0 to the present case and the definition of the prepotential PΛ, equation (6.96). Collecting together all the terms in eqs. (F.3), (F.4), (F.5), (F.10), we find that the antisymmetric 0 3 parts in Λ, Σ and the two terms in PΛPΣ cancel against each other identically.

51 Appendix G: The N =2and N =1Lagrangians in D =4

For reference of the reader we give here the Lagrangian of the N = 2 theory and of the N = 1 theory as given in reference [39]11. The N = 1 lagrangian is, up to four-fermions terms:

1 N=1 1 Λ Σµν +Λ +Σµν i µ  (detV)− = +i f − − f + g z z L −2R ΛΣFµν F − ΛΣFµν F i∇µ ∇ µνλσ   • + ψµγσDνψ λ ψ µγσDνψλ• + √ g • − • −   1 Λ Λ • µ Σ µ Σ + f ΛΣλ γ µλ fΛΣλ γ µλ• 8 ∇ • − • ∇ 1  i µ   µ i µ ν i µ j • µ ν j i i gi χ γ µχ + χ γ µχ gi ψ ν γ γ χ z + ψνγ γ χ µz − 2 ∇ ∇ − • ∇ ∇  Σ  Σ   +Λ µ ν Λ • µ ν iImfΛΣ µν λ γ ψ• + µν− λ γ ψ − F • F •   i Λ i µν Σ +Λ ı µν Σ ∂ifΛΣ µν− χ γ λ ∂ıf ΛΣ µν χ γ λ• − 8 F • − F   • µν µν +2Lψµγ ψν• +2Lψµ γ ψν • • j 1 Λ Λ i µ i  µ • µ µ +igi N χ γ ψµ• + N χ γ ψ µ + PΛ λ γ ψ µ λ γ ψµ•| • 2 • − •     i j ı  Λ Σ Λ Σ + ijχ χ + ıχ χ + ΛΣλ λ + ΛΣλ •λ • M M M • • M Λ i Λ ı + Λiλ χ + Λıλ •χ (z, z,q) M • M −V i where the kinetic matrix fΛΣ is a holomorphic function of z , and the mass matrices , , are given by: Mij MΛΣ MΛi = L (G.1) Mij ∇i∇j i = N i∂ (G.2) MΛΣ 8 iN ΛΣ 1 1 M = i Im ∂ DΣ k g (G.3) Λi − 4 NΛΣ i − 2 Λ i Λ 1 Λ i µνρσ Λ Λ wherewehaveset µν± = 2 ( µν 2  ρσ), µν being the field-strengths of the vectors Λ F F ± F F Aµ . Note that, since the scalar manifold is a K¨ahler-Hodge manifold, all the fields and the bosonic sections have a definite U(1) weight p under U(1). We have

p(V a)=p(AΛ)=p(zi)=p(g )=p( )=p(DΛ)=p(P )=p( )=0 µ i NΛΣ Λ V 1 p(ψ )=p(χı)=p(λΛ)=p(ε )= • • • 2 i Λ 1 p(ψ•)=p(χ )=p(λ •)=p(ε•)= −2 p(L)=p( )=p( )=1 Mij MΛΣ p(L)=p( )=p( )= 1(G.4) Mı MΛΣ − 11Some misprints of ref. [39] have been corrected

52 Accordingly, when a covariant derivative acts on a field Φ of weight p it is also U(1) covariant (besides possibly Lorentz, gauge and scalar manifold coordinate symmetries) according to the following definitions:

Φ=(∂ + 1 p∂ )Φ ; Φ=(∂ 1 p∂ )Φ (G.5) ∇i i 2 iK ∇i∗ i∗ − 2 i∗ K where (z, z)istheK¨ahler potential. On theK other hand, the N = 2 Lagrangian, up to four-fermions terms, is:

µνλσ 1 N=2 1 µ i  u µ v  A A (detV )− = R + g z z + h q q + ψ γ ρ ψ γ ρ L −2 i∇ ∇µ uv∇µ ∇ √ g µ σ Aνλ − Aµ σ νλ −   i g λiAγµ λ + λ γµ λiA i ζαγµ ζ + ζ γµ ζα − 2 i ∇µ A A ∇µ − ∇µ α α ∇µ  Λ Σµν +Λ  +Σµν  µ iA  +2i − − + g z ψ λ N ΛΣFµν F −NΛΣFµν F − i∇µ A  µ iA  n 2 Aα quψ ζ + g zλ γµν ψ +2 αA quζ γµν ψ +h.c. − Uu ∇µ A α i∇µ Aν Uu ∇µ α Aν Λ Σ Aµ Bν Σ ı ν µ AB o + − Im [4L ψ ψ  4if λ γ ψ  {Fµν NΛΣ AB − ı A B 1 + f ΣλiAγµν λjB LΣζ γµν ζ Cαβ ]+h.c. 2∇i j AB − α β } iAB  µ A α µ ++igiW λAγµψB +2iNα ζ γµψA + αβζ ζ + α ζ λiB + λiAλjB +h.c.] (z, z,q). (G.6) M α β M iB α Mij AB −V Furthermore LΛ(z, z) are the covariantly holomorphic sections of the special geometry, Λ Λ Λ fi iL and the kinetic matrix ΛΣ is constructed in terms of L and its magnetic dual≡∇ according to reference [4]. TheN normalization of the kinetic term for the quaternions depends on the scale λ of the quaternionic manifold for which we have chosen the value 1 αβ α λ = 1. Finally, the mass matrices of the spin 2 fermions , AB ij, iA (and their hermitian− conjugates) and the scalar potential are givenM by: M M V αβ = αA βB ε [ukv] LΛ (G.7) M −Uu Uv AB ∇ Λ α = 4 α ku f Λ (G.8) M iB − UBu Λ i Λ l? 1 Λ =  g ? f k + iP f (G.9) MAB ik AB l [i k] Λ 2 ΛAB ∇i k

(z, z,q)=g2 g ki k +4h ku kv LΛLΣ + gif Λf Σ x x 3LΛLΣ x x . (G.10) V i Λ Σ uv Λ Σ i  PΛPΣ − PΛPΣ h  i The U(1)-K¨ahler weight of the Fermi fields is 1 P (ψ )=P (λı )=P (ζ )= . (G.11) A A α 2 References

[1] J. Strathdee, “Extended Poincar´e Supersymmetry”, Int.J.Mod.Phys.A2 (1987) 273 [2] J. Bagger and E. Witten, “Matter Couplings in N =2Supergravity”, Nucl.Phys. B222(1983)1

53 [3] B. de Wit, P.G. Lauwers and A.Van Proeyen, “Lagrangians of N =2Supergravity - Matter Systems”, Nucl.Phys. B255 (1985) 569

[4] L.Andrianopoli, M.Bertolini, A.Ceresole, R.D’Auria, S. Ferrara, P.Fr´e, “General Matter Coupled N=2 Supergravity”, Nucl.Phys. B476 (1996) 397, hep-th/9603004; L.Andrianopoli, M.Bertolini, A.Ceresole, R.D’Auria, S. Ferrara, P.Fr´e, T.Magri, “N =2Supergravity and N =2SuperYang-Mills Theory on General Scalar Man- ifolds: Symplectic Covariance, Gaugings and the Momentum Map”, J.Geom.Phys. 23 (1997) 111; R. D’Auria, S. Ferrara, P. Fr´e, “Special and Quaternionic Isometries: General Couplings in N =2Supergravity and the Scalar Potential”, Nucl. Phys. B359 (1991) 705

[5] J. Bagger and E. Witten, Quantization of Newton’s Constant in Certain Supergravity Theories”, Phys.Lett. B115 (1982) 202; “The Gauge Invariant Supersymmetric Non Linear Sigma Model”, Phys.Lett. B118 (1982) 103

[6] E.Cremmer,B. Julia “The N =8Supergravity Theory.I. The Lagrangian”, Phys.Lett. 80B (1978) 48

[7] A.Strominger, “Special Geometry”, Comm. Math. Phys. 133 (1990) 163

[8] L. Castellani, R. D’Auria and S. Ferrara, “Special K¨ahler Geometry: an Intrinsic Formulation from N =2Space–Time supersymmetry”, Phys. Lett. 241B (1990) 57; “Special Geometry without Special Coordinates”, Class. Quantum Grav. 7 (1990) 1767.

[9] D.V.Alekseevsky, S.Marchiafava, “Hermitian and K¨ahler Submanifold of a quater- nionic K¨ahler Manifold”, Vienna, Preprint ESI 827 (2000).

[10] K.Galiski, “A Generalization of the Momentum Mapping Construction for Quater- nionic K¨ahler Manifolds”, Commun.Math.Phys. 108 (1987) 117

[11] J.Polchinski, A. Strominger, “New Vacua for Type II String Theory”, Phys.Lett. B388 (1996) 736; hep-th/9510227

[12] J. Michelson, “Compactifications of Type IIB Strings to Four Dimensions with Non- trivial Classical Potential”, Nucl.Phys. B495 (1997) 127-148, hep-th/9610151

[13] T.R. Taylor, C. Vafa, “RR Flux on Calabi-Yau and Partial Supersymmetry Breaking”, Phys.Lett. B474 (2000) 130-137; hep-th/9912152

[14] G.Curio, A. Klemm, D. Luest, S. Theisen, “On the Vacuum Structure of Type II String Compactifications on Calabi-Yau Spaces with H-Fluxes”, Nucl.Phys. B609 (2001) 3-45; hep-th/0012213

[15] S.B. Giddings, S. Kachru, J. Polchinski, “Hierarchies from Fluxes in String Com- pactifications”, hep-th/0105097

[16] Klaus Behrndt, Carl Herrmann, Jan Louis, Steven Thomas, “Domain walls in five dimensional supergravity with non-trivial hypermultiplets”, JHEP 0101 (2001) 011

54 [17] Klaus Behrndt, Gabriel Lopes Cardoso, Dieter Lust, “Curved BPS domain wall so- lutions in four-dimensional N=2 supergravity”, Nucl.Phys. B607 (2001) 391-405

[18] G. Dall’Agata, “Type IIB supergravity compactified on a Calabi-Yau manifold with H-fluxes”, hep-th/0107264

[19] M.Gra˜na, J.Polchinski, “Supersymmetric Three-Form Flux Perturbations on AdS5”, Phys.Rev. D63 (2001) 026001; hep-th/0009211

[20] A.C. Cadavid, A. Ceresole, R. D’Auria, S. Ferrara, “11-Dimensional Supergrav- ity Compactified on Calabi-Yau Threefolds”, Phys.Lett. B357 (1995) 76-80; hep- th/9506144

[21] P. Horava, E. Witten, “Eleven-Dimensional Supergravity on a Manifold with Bound- ary”, Nucl.Phys. B475 (1996) 94; hep-th/9603142; P. Horava, E. Witten, “Heterotic and Type I String Dynamics from Eleven Dimensions, Nucl.Phys. B460 (1996) 506; hep-th/9510209

[22] E. Witten, “Strong Coupling Expansion Of Calabi-Yau Compactification”, Nucl.Phys. B471 (1996) 135

[23] A. Lukas, B.A. Ovrut, K. S. Stelle, D. Waldram, “Heterotic M-theory in Five Di- mensions”, Nucl.Phys. B552 (1999) 246

[24] J. Bagger, F. Feruglio, F. Zwirner, “Generalized symmetry breaking on orbifolds”, hep-th/0107128

[25] I. Antoniadis, M. Quir´os, “On the M-theory Description of Gaugino Condensation”, Phys.Lett. B416 (1998) 327-333, hep-th/9707208

[26] J. Bagger, F. Feruglio, F. Zwirner, “Brane-induced supersymmetry breaking”,hep- th/0108010

[27] E. Bergshoeff, R. Kallosh, A. Van Proeyen, “Supersymmetry in Singular Spaces”, JHEP 0010 (2000) 033; “Supersymmetry of RS bulk and brane”, Fortsch.Phys. 49 (2001) 625

[28] S. Ferrara, L. Girardello, M. Porrati, “Spontaneous Breaking of N =2to N =1 in Rigid and Local Supersymmetric Theories”, Phys.Lett. B376 (1996) 275-281; hep- th/9512180

[29] P. Fre’, L. Girardello, I. Pesando, M. Trigiante, “Spontaneous N =2 N =1 Local Supersymmetry Breaking with Surviving Local Gauge Group”, Nucl.Phys.→ B493 (1997) 231

[30] E. Cremmer, in Supergravity ’81, eds. S.Ferrara and J.G. Taylor, p.313.

[31] B. Julia, in Superspace and Supergravity, ed. S.W. Hawking and M. Rocek (Cam- bridge, 1981),p.331.

55 [32] M. G¨unaydin, P. Sierra, P.K. Townsend, “ Exceptional Supergravity Theories and the Magic Square”, Phys. Lett. B133 (1983) 72; “ The Geometry of N =2Maxwell- Einstein Supergravity and Jordan Algebras”, Nucl. Phys. B242 (1984) 244; “ Gauging the D = 5 Maxwell-Einstein Supergravity Theories: More on Jordan Algebras”,Nucl. Phys. B253 (1985) 573; E. Cremmer, A. Van Proeyen, “Classification Of Kahler Manifolds In N=2 Vector Multiplet Supergravity Couplings”, Class. Quant. Grav. 2 (1985) 445.

[33] S. Cecotti, S. Ferrara, L. Girardello, “Geometry of Type II Superstrings and the Moduli of Superconformal Field Theories”, Int.Jou. of Mod.Phys.A,10 (1989) 2475

[34] R. Slansky, Group Theory for Unified Model Building”, Phys.Rep.79 (1981) 1

[35] S. Ferrara, M. Porrati, “ The Manifolds of Scalar Background Fields in ZN Orbifolds”, Phys.Lett. B216 (1989) 289

[36] R. Gilmore, “Lie Groups, Lie Algebras and some of their Applications”, Jhon Wiley and Sons, New York, (1974).

[37] L. Andrianopoli, R. D’Auria, S. Ferrara, “U-Duality and Central Charges in Various Dimensions Revisited”, Int.J.Mod.Phys. A13 (1998) 431, hep-th/9612105

[38] M.K. Gaillard, B. Zumino, “Duality Rotations for Interacting Fields”, Nucl.Phys. B193 (1981) 221

[39] R. D’Auria and S. Ferrara, “On fermion masses, gradient flows and potential in supersymmetric theories”, JHEP 05 (2001) 034; hep-th/0103153

[40] E. Cremmer, S. Ferrara, L. Girardello, A. Van Proeyen, “Yang-Mills Theories with local supersymmetry”, Nucl.Phys. B212 (1983) 231

[41] J.A. Bagger, “Coupling the Gauge Invariant Supersymmetric Non Linear Sigma Model to Supergravity”, Nucl.Phys. B211 (1983) 302

[42] A. Ceresole, R. D’Auria and S. Ferrara, “The Symplectic Structure of N=2 Su- pergravity and its Central Extension”, Nucl.Phys.Proc.Suppl. 46 (1996) 67-74; hep- th/9509160

[43] D.V. Alekseevsky, “Riemannian Spaces with Exceptional Holonomy Groups”, Funct.Anal.Applic. 2 (1968) 97

[44] J.A. Wolf, “Complex Homogeneous Compact Manifolds and Quaternionic Symmetric Spaces”, J.Math.Mech. 14 (1965) 1033

[45] S.Ferrara, S.Sabharwal, “Quaternionic Manifolds For Type Ii Superstring Vacua Of Calabi-Yau Spaces”, Nucl.Phys. B332 (1990) 317

[46] S. Ferrara, L. Girardello, C. Kounnas, M. Porrati, “Effective Lagrangians for Four Dimensional Superstrings”, Phys.Lett. B192 (1987) 368

56 [47] A. Ceresole, G. Dall’Agata, R. Kallosh, A. Van Proeyen, Hypermultiplets, Do- main Walls and Supersymmetric Attractors”, Phys.Rev. D64 (2001) 104006, hep- th/0104056

[48] E. Cremmer, C. Kounnas, A. Van Proeyen, J.P. Derendinger, S. Ferrara, B. de Wit, L. Girardello, “Vector Multiplets Coupled To N=2 Supergravity: Superhiggs Effect, Flat Potentials And Geometric Structure”, Nucl.Phys. B250 (1985) 385

[49] S.Ferrara and L. Maiani, “An Introduction to supersymmetry breaking in extended Supergravity, Proceedings of SILARG V, Bariloche, Argentina,(1985), O.Bressan, M.Castagnino and V.Hamity editors, World Scientific (1985)

[50] S.Cecotti, L. Girardello and M.Porrati, “Constraints On Partial Superhiggs”, Nucl.Phys B268 (1986) 295

[51] D.V. Alekseevsky, V. Cort´es, C. Devchand, A. Van Proeyen, “Flows on quaternionic- K¨ahler and very special real manifolds”, hep-th/0109094

[52] A. Ceresole, R. D’Auria, S. Ferrara, A. Van Proeyen, “Duality Transformations in Su- persymmetric Yang-Mills Theories coupled to Supergravity”, Nucl.Phys. B444 (1995) 92

[53] L. Castellani, R. D’Auria, P. Fr´e, “Supergravity and Superstrings, A Geometric Per- spective”, World Scientific, 1991

57