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IIB on K3 £ T 2 /Z 2 orientifold + flux and D3/D7: a supergravity view-point

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

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IIB on K3 £ T 2 /Z 2 orientifold + flux and D3/D7: a supergravity view-point

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  1. IIB on K3£ T2/Z2 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 T2/ Z2 orientifold + fluxes and D3/D7 branes. + N = 2 Gauged SUGRA • N = 2, 1, 0 vacua, super-BEH mechanism and no-scale structure. • Conclusions

  3. Low-energy • D=4 SUGRA: plethora of scalar fields  (moduli from geometry of M) From D=10,11: add fluxes • Realistic models from String/M-theory )V()  0, In D=4: gauging (predictive, spontaneous SUSY, cosmological constant…) Superstring Theory in D=10 M-Theory in D=11 Compactified on R1,3£ M7 Compactified on R1,3£ M6 Supergravity in D=4

  4. Type II flux-compactifications (+branes):very tentative (and rather incomplete) list of references

  5. M1,3 x K3 x T2 x0 x1 x2 x3x4 x5 x6 x7x8 x9 R-R Arm, yr = yr,8+i yr,9 (r=1,…, n3) NS-NS u = C(0)- i e - f2 Akm, xk = xk,8+i xk,9 (k=1,…,n7) IIB on K3 x T2/Z2 - orientifold with D3/D7: • Type IIB bosonic sector: gMN, , B(2) C(0),C(2),C(4) (B(2),C(2))´ (B(2)) 2 2 • SL(2,R)u global symmetry: • Compactification to D=4 and branes: Low-en. brane dynamics: SYM (Coulomb ph.) on w.v. n3 D3 £ £££ - - - - - - n7D7 £ £££££££- -

  6. K3 manifold (CY2): {x4, x5, x6, x7} ! Basis of H2(K3,R): {wI},I = {m, a} m=1,2,3 a=1,…,19 Complex struct. moduli (W2) Kaehler moduli (J2) (except Vol(K3)) ( ema) $ L(e) 2 Complex struct.: Volume: Moduli from geometry of internal manifold • T2 : {xp} (p=8,9)

  7.  I2 (-)FL • Orientifold proj. wrt Surviving bulk fields [L = (a,p) = 0,…,3] L 2 (2,2) = 4 of SL(2)u x SL(2)t = SO(4) zA,1 Cm, Gmn yAm A0m zA,a ema , Ca A1m lA,1 S A2m lA,2 t A3m lA,3 u Akm lA,k xk Arm lA,r yr [ ] MQ  = world-sheet parity ) I2 (T2): xp ! - xp ß N=2 SUGRA in D=4 (ungauged) Define complex scalars = C(4)K3– i Vol(K3)E nv = 3 + n7 +n3 20 Scalars in non-lin. s-model Mscal = MSK [L(0,n3,n7)] x

  8. Linear action g¢ A B g = 2 G C D s E/M duality promotes G to global sym. of f.eqs. E B. ids. Fmn Fmn Gmn Gmn Global symmetries: Non-linear action onscalars G = Isom(Mscal) Sp(2(nv+1),R) Geometry ofMSK : Hodge-Kaehler manifold, locally described by choice of coordinates {zi} (i=1,…,nv) and by a 2 (nv+1) -dim. section W(z) of a holomorphic symplectic bundle on MSK which fixes couplings between {zi} and the vector field-strengths: (L,S=0,..., nv ) W fixes E/M action of G on vector of f. strengths

  9. ALm Akm Arm SL(2)s Non-pert. Non-pert. SL(2)t pert. SL(2)u pert. Non-pert. Special coordinate basis Wsc(z): zi = Xi /X0 ; F0= - F; Fi = ¶F / ¶ zi Wsc (z) does not reproduce right couplings, i.e. right duality action of G of f. strengths ! Sp – rotation to correct W(z) Correct duality action of G: W in new Sp-basis: ¶s XL= 0 ) F If (n3=0, n7=n) or (n3=n, n7=0),MSK [L(0,n3,n7)] !Symmetric:

  10. Integer ; fixed by tadpole cancellation condition. Switching on fluxes:hsinternal q-cycle F(q)i 0 • Fluxes surviving the orientifold projection: (dB(2), dC(2) )´ (Fa I pwIÆ dxp) • F(3)  0 )Local symmetries in D=4 N=2 SUGRA : C(4) kinetic term in D=10 Stueckelberg-coupling in D=4 F(5)Æ*F(5) (F(5) = dC(4) +eab FaÆ Fb) ( CI– fLI ALm)2 Local translational invariance: CI ! CI + fLIxL ; ALm! ALm+¶mxL 4–dim. abelian gauge-group: G= { XL} $ ALm

  11. In Isom(MQ)=SO(4,20) 22 translational global symmetries {ZI}: CI ! CI + x I Gauge group generators XL are 4 combinations of ZI defined by the fluxes: XL= fLI ZI = fLm Zm+ hLa Za Gauging: promote G ½G to local symmetry of action ß • Vector fields in co-Adj (G) ! gauge vectors • ¶m!rm = ¶m + ALm XL (minimal couplings) Fermion/gravitino SUSY shifts Fermion/gravitino mass terms V(f) ¹ 0 (bilinear in f. shifts) • SUSY of action )

  12. Action of XL on hyper-scalars qu described by Killing vecs. kuLexpressed in terms of momentum maps PLx (x=1,2,3: SU(2) holonomy index):2 kuL Rxuv=rvPLx kmL=fmL; kaL=haL PLx / ej [L(e)-1 xm fmL+ L(e)-1 xa haL] Scalar potential: gaugino > 0 + hyperino > 0 gaugino > 0 + gravitino < 0 Vacua:bosonic b.g. <F (x)>´F0, ¶F V(F0) = 0 SUSY preserving vacua , 9 killing spin.e : de(Fermi)F0= 0

  13. delA,i/ gi jDj XLPLxsx ABeB =0 SUSY vacua Equations for Killing spinor eA dezA,a/ (fLm L-1 am+ hLb L-1 ab) XLeA = 0 dezA,1/ XLPLxsx ABeB =0 deyAm/ XLPLxsx ABeB =0 dezA,a = 0 ) eam fLm = ema hLa = 0; hLa XL=0 • K3 c.s. moduli fixing • PLx / ejfLx • T2 c.s. t fixing • axion/dilaton u fixing deyAm= 0 ; delA,i= 0 )condition on fluxes

  14. t = u • t2= -1+xk xk/2 X2 = X3 = 0 , Ca=1,2 Goldstone eaten by A2,3m N=2 vacua: fLx ´ 0 ß deyAm/ XL fLxsx ABeB = 0 8eA ) Flux has no positive norm vecs. in G3,19 hLa XL=0 has solution ) hLa at most 2 indep. vecs. h2a=1=g2, h3a=2=g3 : t, u fixed s, xk, yr moduli hLa XL=0 ) a=1,2 hypers ema hLa =0 ) exa=1,2´ 0 ) V(F0)´ 0 (independent of moduli) ,effective theory is no-scale

  15. f 3L=0: flux at most 2 norm > 0 vecs.in G3,19 (primitivity of G(3)) , Cm=1,2, Ca=1,2 Goldstone b. Massto Am0,1,2,3 t = u = - i hLa XL=0 ) X2 = X3 = 0 N=1, 0 vacua: e2 Killing spin. : deyAm = 0 , delA,i = 0 f0m=1=g0, f1m=2=g1 h2a=1=g2, h3a=2=g3 delA,i=x= 0 ) xk = 0, i.e. D7 branes fixed at origin of T2 ) a=1,2 hypers eam fLm = ema hLa =0 ) exa=1,2´ 0; ex=1,2a ´ 0 ) K3 c.s.fix

  16. Moduli: s, yr ; Cm=3+i ej, Ca +i em=3a, (a¹ 1,2) Mscal = x Superpotential (classical): g0 = g1 (N=1) W(F0) / e-j [XL (P1L+i P2L)]|0/g0-g1(moduli indep.) g0 ¹ g1 (N=0) V0(moduli) ´ 0 (no-scale) More general N=1 vacua:g 2 SL(2)t£ SL(2)u : t = u = -i ! t0, u0 f , h m t = u = -i f’=g.f , h’=g.h m t = t0, u = u0 )

  17. Conclusions • Discussed instance of correspondence between flux • compactificationand 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 CI undualized) • [D’Auria et.al]

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

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