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

IIB on K3£ T2/Z2 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 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

+

Superstring Theory in D=10 M-Theory in D=11

Low-energy

Supergravity in D=4

Compactified onR1,3£ M6

Compactified onR1,3£ M7

•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…)

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

Type II on: Hep-th/

CY3

(orientifold)

•Michelson ; Gukov, Vafa, Witten •Taylor, Vafa; Curio, Klemm, Kors, Lust• Dall’Agata; Louis, Micu• Kachru, Kallosh, Linde, Trivedi; Frey• Giryavets,Kachru,Tripathy,Trivedi• Grana,Grimm,Jockers, Louis;

• D’Auria, 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 T2/Z2

Orientifold

Tripathy, Trivedi; Koyama, Tachikawa, Watari

Andrianopoli, D’Auria, Ferrara,Lledo’

Angelantonj,D’Auria, Ferrara, M.T.

D’Auria, Ferrara, M.T.

0301139; 0311191;

0302174

0312019;

0403204;

T6 /Z2

Orientifold

Frey, Polchinski

Kachru, Schulz, Trivedi

D’Auria, Ferrara, Vaula’

D’Auria, Ferrara, Lledo’,Vaula’

D’Auria, Ferrara, Gargiulo,M.T.,Vaula’

Berg, Haak, Kors

0201029;

0201028;

0206241;

0211027;

0303049;

0305183;

Tp-3 x T9-p/Z2

orientifold

Angelantonj, Ferrara, M.T.

Angelantonj, Ferrara, M.T.

0306185;

0310136;

IIB on T6 from N=8 de Wit, M.T., Samtleben 0311224;

IIB on K3 x T2/Z2 - orientifold with D3/D7:

•Type IIB bosonic sector: gMN, , 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:

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9

M1,3 K3 T2x x

£ £ £ £ - - - - - -

£ £ £ £ £ £ £ £ - -

n3 D3

n7 D7

Low-en. brane dynamics: SYM (Coulomb ph.) on w.v.

Ar, yr = yr,8+i yr,9

(r=1,…, n3)

Ak, xk = xk,8+i xk,9

(k=1,…,n7)

• T2 : {xp} (p=8,9)

Basis of H2(K3,R): {I}, I = {m, a}m=1,2,3

a=1,…,19

Complex struct. moduli (2)Kaehler moduli (J2)

(except Vol(K3))

( ema) $ L(e) 2

Complex struct.:

Volume:

Moduli from geometry of internal manifold

• K3 manifold (CY2): {x4, x5, x6, x7} !

world-sheet parity

I2 (T2): xp ! - xp

• Orientifold proj. wrt I (-)FL

N=2 SUGRA in D=4(ungauged)

)

Define complex scalar s = C(4)

K3 – i Vol(K3)E

Scalars in non-lin. -model

G

A0

A1

S

A2

t

A3

u

Ak

xk

Ar

A,r

yr

nv = 3 + n7 +n3

A,1

Cm,

A,a

ema , Ca

20

Mscal = MSK [L(0,n3,n7)] x MQ[ ]

2 (2,2) = 4 of SL(2)u x SL(2)t = SO(4)

(,p) = 0,…,3]

Surviving bulk fields

Geometry of MSK : Hodge-Kaehler manifold, locally described by

choice of coordinates {zi} (i=1,…,nv) and by a 2 (nv+1) -dim. section (z) of a

holomorphic symplectic bundle on MSK which fixes couplings between {zi}

and the vector field-strengths:

nv

Global symmetries:

G = Isom(Mscal)

Non-linear action on scalars

Linear actionF

G

g¢F

G

Sp(2(nv+1),R) E/M duality promotes

G to global sym. of f.eqs. E B. ids.

g = 2 GA B

C D

fixes E/M action of G on vector of f. strengths

Special coordinate basis sc(z): zi = Xi /X0 ; F0= - ; Fi = / zi

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 )

Correct duality action of G:

Non-pert.pert.SL(2)u

pert.SL(2)t

Non-pert.Non-pert.SL(2)s

Ar Ak

A

If (n3=0, n7=n) or (n3=n, n7=0), MSK [L(0,n3,n7)] ! Symmetric:

Switching on fluxes: hsinternal q-cycle F(q)i 0

• Fluxes surviving the orientifold projection: (dB(2), dC(2) )´ (Fp I Æ dxp)

•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) +FÆ F)

( CI– fI A

)2

Stueckelberg-coupling in D=4

Local translational invariance: CI ! CI + fI

4–dim. abelian gauge-group: G = { X} $ A

; A! A

+

Integer ; fixed by tadpole cancellation condition.

In Isom(MQ)=SO(4,20) 22 translational global symmetries {ZI}:

Gauge group generators X are 4 combinations of ZI defined by the fluxes:

CI ! CI +

X= fI ZI = fm Zm+ ha Za

Gauging: promote ½G to local symmetry of action

• Vector fields in co-Adj () ! gauge vectors

Fermion/gravitino SUSY shiftsFermion/gravitino mass terms

V() 0 (bilinear in f. shifts)

• ! r = + A X(minimal couplings)

• SUSY of action )

Action of X on hyper-scalars qu described by Killing vecs. ku

expressed in terms of momentum maps Px (x=1,2,3:

SU(2) holonomy index): 2 kuRx

uv=rvPx

km=fm

; ka=ha

Px / eL(e)-1 x

m fm+ L(e)-1 x

a ha]

gaugino > 0 + hyperino > 0 gaugino > 0 + gravitino < 0

Scalar potential:

Vacua: bosonic b.g. < (x)>´ 0, V(0) = 0

SUSY preservingvacua , 9 killing

spin.(Fermi)= 0

SUSY vacua

A,1/ X Px x ABB =0

A,a/ (fm L-1 am+ h

b L-1 ab) XA = 0

/ X Px x ABB =0

/ gi j Dj X Px x ABB =0

A,a ) eam fm = em

a ha = 0; h

a X=0

Equations for Killing spinor A

• K3 c.s. moduli fixing• P

x / efx

• T2 c.s. t fixing• axion/dilaton u fixing

; )condition on fluxes

N=2 vacua:

/ X fx x ABB = 0 8 A )

fx ´ 0

Flux has no positivenorm vecs. in 3,19h

a X=0 has solution ) ha at most

2 indep. vecs. h2a=1=g2, h3

a=2=g3 :

ha X=0 )

ema h

a =0 ) exa=1,2´ 0

t, u fixed s, xk, yr moduli

Ca=1,2 Goldstone eaten by A2,3

) a=1,2 hypers

V(0)´ 0 (independent of moduli) , effective theory is no-scale

X2 = X3 = 0 , • t = u• t2= -1+xk xk/2

N=1, 0 vacua:

f0m=1=g0, f1

m=2=g1

h2a=1=g2, h3

a=2=g3

eam fm = em

a ha =0 ) ex

a=1,2´ 0; ex=1,2a ´ 0

Cm=1,2, Ca=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 ) xk = 0, i.e. D7 branes fixed at origin of T2

K3 c.s.fix

) Mass to A0,1,2,3

ha X=0 ) X2 = X3 = 0 t = u = - it = u = - i,

Moduli: s, yr ; Cm=3+i eCa +i em=3a, (a 1,2)

Mscal = x

Superpotential (classical):

W(0) / e [X (P1+i P2

)]|0 / g0-g1 (moduli indep.)

g0 = g1 (N=1)

g0 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 mt = u = -i

) f’=g.f , h’=g.h m t = t0, u = u0

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 CI undualized)

[D’Auria et.al]

Vector kinetic terms described by complex matrixN(z, z)

Nconstructed from (z):

Section (z) in the new basis: