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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. +. - PowerPoint PPT Presentation
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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:
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