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
Compactified on
R1,3£ M7
Compactified on
R1,3£ M6
Supergravity in D=4
•D=4 SUGRA: plethora of scalar fields  (moduli from geometry of M)
• Realistic models from String/M-theory
) V()  0 ,
From D=10,11:
add fluxes
(predictive, spontaneous SUSY, cosmological constant…)
In D=4:
gauging
Type II flux-compactifications (+branes):
very tentative (and rather incomplete) list of references
Type II on:
CY3
(orientifold)
K3 x T2/Z2
Orientifold
T6 /Z2
Orientifold
Tp-3 x T9-p/Z2
orientifold
IIB on T6 from N=8
Hep-th/
•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;
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;
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;
Angelantonj, Ferrara, M.T.
Angelantonj, Ferrara, M.T.
0306185;
de Wit, M.T., Samtleben
0311224;
0310136;
IIB on K3 x T2/Z2 - orientifold with D3/D7:
•Type IIB bosonic sector:
NS-NS
R-R
gMN, , B(2)
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:
M1,3
x
K3 x
Low-en. brane dynamics:
SYM (Coulomb ph.) on w.v.
T2
x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
n3 D3
n7 D7
£ £ £ £
-
-
-
-
-
-
£ £ £ £ £ £ £ £
-
-
Arm, yr = yr,8+i yr,9
(r=1,…, n3)
Akm, xk = xk,8+i xk,9
(k=1,…,n7)
Moduli from geometry of internal manifold
• K3 manifold (CY2): {x4, x5, x6, x7} !
Basis of H2(K3,R):
{wI},
Complex struct. moduli (W2)
Kaehler moduli (J2)
I = {m, a}
m=1,2,3
a=1,…,19
( ema) $ L(e) 2
(except Vol(K3))
• T2 :
Complex struct.:
{xp} (p=8,9)
Volume:
• Orientifold proj. wrt
W I2 (-)FL
Surviving bulk fields
W = world-sheet parity
I2 (T2): xp !
- xp
)
[L = (,p) = 0,…,3]

L 2 (2,2) = 4 of SL(2)u x SL(2)t = SO(4)
N=2 SUGRA in D=4
(ungauged)
Gmn
yAm
A0m
Define complex scalar s = C(4)K3 – i Vol(K3)E
nv = 3 + n7 +n3
20
zA,1
A1m A2m A3m Akm Arm
lA,1 lA,2 lA,3 lA,k lA,r
S
t
u
xk
yr
Mscal = MSK [L(0,n3,n7)]
Cm,
x
MQ[
zA,a
ema , Ca
Scalars
in non-lin.
s-model
]
Global symmetries:
Non-linear action on scalars
G = Isom(Mscal)
Linear action
g=
A
C
B
D
2G
s
Fmn
Gmn
Fmn
g¢
Gmn
Sp(2(nv+1),R)
E/M duality promotes
G to global sym. of
f.eqs. E B. ids.
Geometry of MSK :
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:
W fixes E/M action of G on vector of f. strengths
(L, S=0,..., nv )
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)
ALm
Correct duality action of G:
W in new Sp-basis: s XL= 0
Akm
SL(2)s
Non-pert. Non-pert.
SL(2)t
pert.
SL(2)u
pert.
)
Arm
Non-pert.
F
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) )´ (F
Integer ;
I
p
wI Æ dxp)
fixed by tadpole cancellation
condition.
•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)
I
(m CI– fL ALm)2
(F(5) = dC(4) +eb FÆ Fb)
I
Local translational invariance: CI ! CI + fL xL
4–dim. abelian gauge-group:
G = { XL} $ ALm
; ALm ! ALm +m xL
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
½ G to local symmetry of action

• Vector fields in co-Adj (G) ! gauge vectors
• m ! rm = m + ALm XL (minimal couplings)
• SUSY of action )
Fermion/gravitino SUSY shifts
Fermion/gravitino mass terms
V()  0 (bilinear in f. shifts)
Action of XL on hyper-scalars qu described by Killing vecs.
kuL expressed in terms of momentum maps PLx (x=1,2,3:
SU(2) holonomy index): 2 kuL Rxuv=rvPLx
PLx
kmL=fmL; kaL=haL
/ ej [L(e)-1 xm fmL+ L(e)-1 xa haL]
Scalar potential:
gaugino > 0 + hyperino > 0
Vacua: bosonic b.g.
SUSY preserving
vacua
gaugino > 0 + gravitino < 0
<F (x)>´ F0, F V(F0) = 0
, 9 killing spin.e : de(Fermi)F0 = 0
SUSY vacua
Equations for Killing spinor eA
de zA,a/ (fLm L-1 am+ hLb L-1 ab) XLeA = 0
de zA,1/ XL PLx sx ABeB =0
de yAm / XL PLx sx ABeB =0
de lA,i / gi j Dj XL PLx sx ABeB =0
de zA,a = 0 ) eam fLm = ema hLa = 0;
• K3 c.s. moduli fixing
• PL x / ej fLx
hLa XL=0
• T2 c.s. t fixing
• axion/dilaton u fixing
de yAm = 0 ; de lA,i = 0 ) condition on fluxes
N=2 vacua:
fL x ´ 0
de yAm / XL fLx sx ABeB = 0
8 eA
)
hLa XL=0 has solution ) hLa at most
2 indep. vecs. h2a=1=g2, h3a=2=g3 :
hL
a
em
XL=0
a hL
a =0
)
)
X2 =
X3 =
exa=1,2´
0 ,
0

Flux has no positive
norm vecs. in G3,19
•t=u
• t2= -1+xk xk/2
t, u fixed
s, xk, yr moduli
Ca=1,2 Goldstone
)
eaten by A2,3 m
a=1,2
hypers
V(F0)´ 0 (independent of moduli) , effective theory is no-scale
N=1, 0 vacua:
e2 Killing spin. :
f0m=1=g0, f1m=2=g1
h2a=1=g2, h3a=2=g3
de yAm =
0 , de
lA,i =
0
de lA,i=x = 0 ) xk = 0,
hLa XL=0 ) X2 = X3 = 0
,
Cm=1,2, Ca=1,2 Goldstone b. )
eam fLm = ema hLa =0
)
f 3L=0: flux at
most 2 norm > 0
vecs.in G3,19
(primitivity of G(3))
i.e. D7 branes fixed
at origin of T2
t=u=-i
Mass to Am0,1,2,3
exa=1,2´ 0; ex=1,2a ´ 0
a=1,2
hypers
)
K3 c.s.fix
Moduli: s, yr ; Cm=3+i ej, Ca +i em=3a, (a 1,2)
Mscal =
x
Superpotential (classical):
g0 = g1 (N=1)
W(F0) / e-j [XL (P1L+i P2L)]|0 / g0-g1 (moduli indep.)
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
m
t = 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 matrix
NLS constructed from W(z):
Section W(z) in the new basis:
NLS(z, z)