UCLA/96/TEP/3 POLFIS-TH 02/96 NSF-ITP-96-11 CERN-TH/95-350 hep-th/xx9603004 General Matter Coupled N=2 Sup ergravity 1 2 y 2;4 L. Andrianop oli , M. Bertolini , A. Ceresole 2 35 6 R. D'Auria ,S.Ferrara and P.Fre 1 Dipartimento di Fisica, Universita di Genova, via Do decaneso 33, I-16146 Genova, Italy 2 Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy 3 CERN, CH 1211 Geneva 23, Switzerland 4 Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA 5 DepartmentofPhysics, University of California, Los Angeles, CA 90024, USA 6 Dipartimento di Fisica Teorica, UniversitadiTorino, via P. Giuria, 1 I-10125 Torino, Italy Abstract The general form of N = 2 sup ergravity coupled to an arbitrary number of vector multiplets and hyp ermultiplets, with a generic gauging of the scalar manifold isometries is given. This ex- tends the results already available in the literature in that we use a co ordinate indep endent and manifestly symplectic covariant formalism which allows to cover theories dicult to formulate within sup erspace or tensor calculus approach. We provide the complete lagrangian and sup er- symmetry variations with all fermionic terms, and the form of the scalar p otential for arbitrary quaternionic manifolds and sp ecial geometry, not necessarily in sp ecial co ordinates. Our results can b e used to explore prop erties of theories admitting N = 2 sup ergravityaslow energy limit. y Fellowby Ansaldo Ricerche srl, C.so Perrone 24, I-16152 Genova Supp orted in part by DOE grant DE-FGO3-91ER40662 Task C., EEC Science Program SC1*CT92- 0789, NSF grant no. PHY94-07194, and INFN. 1 Intro duction Impressive results over the last year on non p erturbative prop erties of N = 2 sup ersym- metric Yang-Mills theories[1, 2 ] and their extension to string theory[3, 4, 5, 6 ] through the notion of string-string duality[7 ], have used the deep underlying mathematical structure of these theories and its relation to algebraic geometry. In the case of N =2vector multiplets, describing the e ectiveinteractions in the Ab elian (Coulomb) phase of a sp ontaneously broken gauge theory, Seib erg and Witten [1] have shown that p ositivity of the metric on the underlying mo duli space identi es the geometrical data of the e ective N = 2 rigid theory with the p erio ds of a particular torus. In the coupling to gravityitwas conjectured by some of the present authors [3, 4] and later con rmed by heterotic-Typ e I I duality[8, 9, 10, 11 ], that the very same argument based on p ositivity of the vector multiplet kinetic metric identi es the corresp onding geometrical data of the e ective N = 2 sup ergravity with the p erio ds of Calabi-Yau threefolds. On the other hand, when matter is added, the underlying geometrical structure is much richer, since N = 2 matter hyp ermultiplets are asso ciated with quaternionic geometry[12, 13 , 14 ], and charged hyp ermultiplets are naturally asso ciated with the gaug- ing of triholomorphic isometries of these quaternionic manifolds [15]. It is the aim of this pap er to complete the general form of the N = 2 sup ergravity lagrangian coupled to an arbitrary numberofvector multiplets and hyp ermultiplets in presence of a general gauging of the isometries of b oth the vector multiplets and hyp er- multiplets scalar manifolds. Actually this extends results already obtained years ago by some of us [15], that in turn extended previous work by Bagger and Witten on ungauged general quaternionic manifolds coupled to N = 2 sup ergravity[12 ], by de Wit, Lauwers and Van Pro eyen on gauged sp ecial geometry and gauged quaternionic manifolds obtained by quaternionic quotient in the tensor calculus framework [16], and by Castellani, D'Auria and Ferrara on covariant formulation of sp ecial geometry for matter coupled sup ergravity [17 ]. This pap er rstly provides in a geometrical setting the full lagrangian with all the fermionic terms and the sup ersymmetry variations. Secondly, it uses a co ordinate in- dep endent and manifestly symplectic covariant formalism which in particular do es not require the use of a prep otential function F (X ). Whether a prep otential F (X ) exists or not dep ends on the choice of a symplectic gauge[4 ]. Moreover, some physically interesting cases are precisely instances where F (X ) do es not exist[4]. Of particular relevance is the fact that we exhibit a scalar p otential for arbitrary quaternionic geometries and for sp ecial geometry not necessarily in sp ecial co ordiantes. This allows us to go b eyond what is obtainable with the tensor calculus (or sup erspace) approach. Among many applications, our results allow the study of general conditions for sp ontaneous sup ersymmetry breaking in a manner analogous to what was done for N =1 matter coupled sup ergravity [18 ]. Many examples of sup ersymmetry breaking studied in the past are then repro duced in a uni ed framework. Recently the p ower of using simple geometrical formulae for the scalar p otential was exploited while studying the breaking of half sup ersymmetries in a particular simple mo del, using a symplectic basis where F (X ) is not de ned[19]. The metho d has p oten- tial applications in string theory to study non p erturbative phenomena such as conifold 1 transitions [20], p-forms condensation [21] and Fayet-Iliop oulos terms [19, 22 ]. Although the sup ersymmetric Lagrangian and the transformation rules lo ok quite involved, all the couplings, the mass matrices and the vacuum energy are completely xed and organized in terms of few geometrical data, such as the choice of a gauge group G, and of a sp ecial Kahler SK(n ) and of a Quaternionic manifold Q(n ) describing V H the self-interactions of the n vector and n hyp ermultiplets resp ectively, whose direct V H pro duct yields the full scalar manifold of the theory MSK(n ) Q(n ) : (1.1) V H If G is non-ab elian, it must b e a subgroup of the isometry group of the scalar manifold SK(n ) with a blo ck diagonal immersion in the symplectic group Sp(2n +2;IR)of V V electric-magnetic duality rotations. An expanded version of this pap er, with particular attention to the geometrical prop- erties of the scalar manifolds, the rigidly sup ersymmetric version and further related issues is given in [23]. 2 Resume and Glossary of Sp ecial and Quaternionic Geometry Here we collect some useful formulae for sp ecial and quaternionic geometry, following i closely the conventions of [24]. The n complex scalar elds z of N =2vector multiplets V are co ordinates of a sp ecial Kahler manifold, that is a Kahler-Ho dge manifold SK(n ) V with the additional constraint on the curvature ? pp ? ? ? ? ? ? ? ? ? R = g g + g g C C g ; (2.1) ij kl ij kl il kj ik p j l p ? ? where g = @ @ K is the Kahler metric, K is the Kahler p otential and C is a com- ij i j ik p pletely symmetric covariantly holomorphic tensor. We remind that the Levi-Civita con- nection one form and the Riemann tensor are given by ? i i k i il i i ? ? = dz ; = g @ g ;R = @ : (2.2) ? j kl k j kj kj jk l jl AKahler-Ho dge manifold has the prop erty that there is a U (1) bundle L whose rst Chern class coincides with the Kahler class. This means that lo cally the U (1) connection Q can b e written as i ? i i ? Q = (@ Kdz @ Kdz ) : (2.3) i i 2 i The covariant derivative of a generic eld , that under a Kahler transformation K ! 1 i i K + f + f transforms as ! exp[ (pf + p f )] is given by 2 p j j j k j D = @ + + @ K ; i i i ik 2 p j j j ? ? ? D @ = @ K : (2.4) + i i i 2 ? j p = p). Note that has weight(p; p). Since (In the following we always have ? ? C is covariantly holomorphic and has weight p = 2, it satis es D C =(@ ik p q ik p q ? @ K)C =0. q ik p 2 A more intrinsic and useful de nition of a sp ecial Kahler manifold can b e given by constructing a at 2n + 2-dimensional symplectic bundle over the Kahler -Ho dge manifold V whose generic sections (with weight p =1) V =(L ;M ) = 0;:::;n ; (2.5) V are covariantly holomorphic 1 ? ? ? D V =(@ @ K)V =0 (2.6) i i i 2 and satisfy the further condition i<V;V>=i(L M M L )=1 ; (2.7) 0 11 where <; > denotes a symplectic inner pro duct with metric chosen to b e . 11 0 De ning U = D V =(f ;h ), and intro ducing a symmetric three-tensor C by i i i ij k i ? kk ? D U = iC g U ; (2.8) i j ij k k one can show that the symplectic connection D V = U i i ? kk ? D U = iC g U i j ij k k ? ? D U = g V i j ij D V = 0 (2.9) i is at, provided the constraint (2.1) is veri ed. Furthermore, the Kahler p otential can b e computed as a symplectic invariant from eq. (2.7). Indeed, intro ducing also the holomorphic sections K=2 K=2 = e V = e (L ;M )= (X ;F ) ? @ = 0 (2.10) i eq.