1. IIB on K3 £ T 2 /Z 2 orientifold + flux and D3/D7: a supergravity view-point Dr. Mario Trigiante (Politecnico di Torino)

2. Plan of the Talk General overview: Compactification with Fluxes and Gauged Supergravities. Type IIB on K3 x T 2 / Z 2 orientifold + fluxes and D3/D7 branes. N = 2 Gauged SUGRA N = 2, 1, 0 vacua, super-BEH mechanism and no-scale structure. Conclusions +

3. Superstring Theory in D=10M-Theory in D=11 Low-energy Supergravity in D=4 Compactified on R 1,3 £ M 6 Compactified on R 1,3 £ M 7 D=4 SUGRA: plethora of scalar fields moduli from geometry of M) From D=10,11: add fluxes In D=4: gauging Realistic models from String/M-theory ) V( ) 0, (predictive, spontaneous SUSY, cosmological constant…)

4. Type II flux-compactifications (+branes): very tentative (and rather incomplete) list of references Type II on: Hep-th/ CY 3 (orientifold) Michelson ; Gukov, Vafa, Witten Taylor, Vafa; Curio, Klemm, Kors, Lust DallAgata; Louis, Micu Kachru, Kallosh, Linde, Trivedi; Frey Giryavets,Kachru,Tripathy,Trivedi Grana,Grimm,Jockers, Louis; DAuria, Ferrara, M.T.;. Grimm, Louis ; Lust, Reffert, Stieberger; Smet, Van den Bergh 9610151; 9906070; 9912152; 0012213; 0107264; 0202168; 0301240; 0308156; 0312104; 0312232;; 0401161; 0403067 ; 0406092; 0407233; K3 x T 2 /Z 2 Orientifold Tripathy, Trivedi; Koyama, Tachikawa, Watari Andrianopoli, DAuria, Ferrara,Lledo Angelantonj,DAuria, Ferrara, M.T. DAuria, Ferrara, M.T. 0301139; 0311191; 0302174 0312019; 0403204; T 6 /Z 2 Orientifold Frey, Polchinski Kachru, Schulz, Trivedi DAuria, Ferrara, Vaula DAuria, Ferrara, Lledo,Vaula DAuria, Ferrara, Gargiulo,M.T.,Vaula Berg, Haak, Kors 0201029; 0201028; 0206241; 0211027; 0303049; 0305183; T p-3 x T 9-p /Z 2 orientifold Angelantonj, Ferrara, M.T. 0306185; 0310136; IIB on T 6 from N=8 de Wit, M.T., Samtleben 0311224;

5. IIB on K3 x T 2 /Z 2 - orientifold with D3/D7: Type IIB bosonic sector: g MN,, B (2) NS-NS R-R C (0),C (2),C (4) (B (2),C (2) )´ (B (2) ) 2 2 SL(2,R) u global symmetry: u = C (0) - i e - 2 Compactification to D=4 and branes: x 0 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 M 1,3 K3T2T2 xx £ £ £ £ - - - - - - £ £ £ £ £ £ £ £ - - n 3 D3 n 7 D7 Low-en. brane dynamics: SYM (Coulomb ph.) on w.v. A r, y r = y r,8 + i y r,9 (r=1,…, n 3 ) A k, x k = x k,8 + i x k,9 (k=1,…,n 7 )

6. T 2 : {x p } (p=8,9) Basis of H 2 (K3,R): { I }, I = {m, a} m=1,2,3 a=1,…,19 Complex struct. moduli ( 2 ) Kaehler moduli (J 2 ) (except Vol(K3)) ( e m a ) $ L(e) 2 Complex struct.: Volume: Moduli from geometry of internal manifold K3 manifold (CY 2 ): { x 4, x 5, x 6, x 7 } !

7. world-sheet parity I 2 (T 2 ): x p !- x p Orientifold proj. wrt I (-) F L N=2 SUGRA in D=4 (ungauged) ) Define complex scalar s = C (4) K3 – i Vol(K3) E Scalars in non-lin. -model G A 0 A 1 S A 2 t A 3 u A k x k A r A,r y r n v = 3 + n 7 +n 3 A,1 C m, A,a e m a, C a 20 M scal = M SK [L(0,n 3,n 7 )] x MQMQ [] 2 (2,2) = 4 of SL(2) u x SL(2) t = SO(4) (,p) = 0,…,3] Surviving bulk fields

8. Geometry of M SK : Hodge-Kaehler manifold, locally described by choice of coordinates {z i } (i=1,…,n v ) and by a 2 (n v +1) -dim. section (z) of a holomorphic symplectic bundle on M SK which fixes couplings between {z i } and the vector field-strengths: n v Global symmetries: G = Isom( M scal ) Non-linear action on scalars Linear action F G g¢g¢ F G Sp(2 (n v +1),R) E/M duality promotes G to global sym. of f.eqs. E B. ids. g =2 G AB CD fixes E/M action of G on vector of f. strengths

9. Special coordinate basis sc (z): z i = X i /X 0 ; F 0 = - F; F i = F / z i sc (z) does not reproduce right couplings, i.e. right duality action of G of f. strengths ! Sp – rotation to correct (z) in new Sp-basis: s X = 0 ) F Correct duality action of G: Non-pert. pert.SL(2) u pert.SL(2) t Non-pert. SL(2) s A r A k A If (n 3 =0, n 7 =n) or (n 3 =n, n 7 =0), M SK [L(0,n 3,n 7 )] ! Symmetric:

10. Switching on fluxes: hs internal q-cycle F (q) i 0 Fluxes surviving the orientifold projection: (dB (2), dC (2) )´ (F p I Æ dx p ) F (3) 0 ) Local symmetries in D=4 N=2 SUGRA : C (4) kinetic term in D=10 F (5) Æ * F (5) (F (5) = dC (4) + F Æ F ) ( C I – f I A ) 2 Stueckelberg-coupling in D=4 Local translational invariance: C I ! C I + f I 4–dim. abelian gauge-group: G = { X } $ A ; A ! A + Integer ; fixed by tadpole cancellation condition.

11. In Isom( M Q )=SO(4,20) 22 translational global symmetries {Z I }: Gauge group generators X are 4 combinations of Z I defined by the fluxes: C I ! C I + X = f I Z I = f m Z m+ h a Z a Gauging: promote G ½ G to local symmetry of action Vector fields in co-Adj (G) ! gauge vectors Fermion/gravitino SUSY shifts Fermion/gravitino mass terms V( ) 0 (bilinear in f. shifts) ! r = + A X (minimal couplings) SUSY of action)

12. Action of X on hyper-scalars q u described by Killing vecs. k u expressed in terms of momentum maps P x (x=1,2,3: SU(2) holonomy index): 2 k u R x uv =r v P x k m =f m ; k a =h a P x / e L(e) -1 x m f m + L(e) -1 x a h a ] gaugino > 0 + hyperino > 0gaugino > 0 + gravitino < 0 Scalar potential: Vacua: bosonic b.g. ´ 0, V( 0 ) = 0 SUSY preserving vacua, 9 killing spin. (Fermi) = 0

13. SUSY vacua A,1 / X P x x AB B =0 A,a / (f m L -1 a m + h b L -1 a b ) X A = 0 / X P x x AB B =0 / g i j D j X P x x AB B =0 A,a ) e a m f m = e m a h a = 0; h a X =0 Equations for Killing spinor A K3 c.s. moduli fixing P x / e f x T 2 c.s. t fixing axion/dilaton u fixing ; ) condition on fluxes

14. N=2 vacua: / X f x x AB B = 0 8 A ) f x ´ 0 Flux has no positive norm vecs. in 3,19 h a X =0 has solution ) h a at most 2 indep. vecs. h 2 a=1 =g 2, h 3 a=2 =g 3 : h a X =0 ) e m a h a =0 ) e x a=1,2 ´ 0 t, u fixed s, x k, y r moduli C a=1,2 Goldstone eaten by A 2,3 ) a=1,2 hypers V( 0 )´ 0 (independent of moduli), effective theory is no-scale X 2 = X 3 = 0, t = u t 2 = -1+x k x k /2

15. N=1, 0 vacua: f 0 m=1 =g 0, f 1 m=2 =g 1 h 2 a=1 =g 2, h 3 a=2 =g 3 e a m f m = e m a h a =0 ) e x a=1,2 ´ 0; e x=1,2 a ´ 0 C m=1,2, C a=1,2 Goldstone b. ) a=1,2 hypers 2 Killing spin. : = 0, = 0 f 3 =0: flux at most 2 norm > 0 vecs.in 3,19 (primitivity of G (3) ) =x = 0 ) x k = 0, i.e. D7 branes fixed at origin of T 2 K3 c.s.fix ) Mass to A 0,1,2,3 h a X =0 ) X 2 = X 3 = 0 t = u = - i,

16. Moduli: s, y r ; C m=3 +i e C a +i e m=3 a, (a 1,2) M scal = x Superpotential (classical): W( 0 ) / e [X ( P 1 +i P 2 )] |0 / g 0 -g 1 (moduli indep.) g 0 = g 1 (N=1) g 0 g 1 (N=0) V 0 (moduli) ´ 0 (no-scale) More general N=1 vacua: g 2 SL(2) t £ SL(2) u : t = u = -i ! t 0, u 0 f, h m t = u = -i ) f=g.f, h=g.h m t = t 0, u = u 0

17. Conclusions Discussed instance of correspondence between flux compactification and gauged supergravity. Starting framework for studying more general situations pert. and non-pert.effects [Becker, Becker et al.; Kachru, Kallosh et al.] gauging compact isometries ! hybrid inflation [Koyama et al.] extended N=2 theory with tensor fields (some C I undualized) [DAuria et.al]

18. Vector kinetic terms described by complex matrix N (z, z) N constructed from (z): Section (z) in the new basis: