D-brane Instantons on the T^6/Z_3 orientifold
arXiv:0704.0784v2  [hep-th]  31 May 2007
ROM2F/2007/09
D-brane Instantons on the T 6/Z3 orientifold
Massimo Bianchi, Francesco Fucito and Jose F. Morales
Dipartimento di Fisica, Universitá di Roma “Tor Vergata”
I.N.F.N. Sezione di Roma II
Via della Ricerca Scientifica, 00133 Roma, Italy
Abstract
We give a detailed microscopic derivation of gauge and stringy instan-
ton generated superpotentials for gauge theories living on D3-branes at Z3-
orientifold singularities. Gauge instantons are generated by D(-1)-branes
and lead to Affleck, Dine and Seiberg (ADS) like superpotentials in the ef-
fective N = 1 gauge theories with three generations of bifundamental and
anti/symmetric matter. Stringy instanton effects are generated by Euclidean
ED3-branes wrapping four-cycles on T 6/Z3. They give rise to Majorana
masses, Yukawa couplings or non-renormalizable superpotentials depending
on the gauge theory. Finally we determine the conditions under which ADS
like superpotentials are generated in N = 1 gauge theories with adjoints,
fundamentals, symmetric and antisymmetric chiral matter.
http://arxiv.org/abs/0704.0784v2
Contents
1 Introduction 1
2 The Gauge Theory 4
3 D(-1) Instantons 6
3.1 D3-D(-1) in flat space . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 D(-1)-D3 at the C3/Z3-orientifold . . . . . . . . . . . . . . . 7
4 ADS-like superpotential 9
4.1 D3-D(-1) one-loop vacuum amplitudes . . . . . . . . . . . . . 9
4.2 Sp(6)× U(2) superpotential . . . . . . . . . . . . . . . . . . . 14
4.3 U(4) superpotential . . . . . . . . . . . . . . . . . . . . . . . 15
5 ED3-instantons 17
5.1 D3-ED3 one-loop vacuum amplitudes . . . . . . . . . . . . . 19
5.2 The superpotential . . . . . . . . . . . . . . . . . . . . . . . . 21
6 ADS superpotentials: a general analysis 22
7 Conclusions 24
1 Introduction
Our understanding of non perturbative effects in four dimensional supersym-
metric gauge theories (SYM) has dramatically improved in recent years. This
is due mainly to the observation that integrals over the moduli space of gauge
connections localize around a finite number of points[1]. These techniques
have been applied to the study of multi-instanton corrections to N = 1, 2, 4
supersymmetric gauge theories in R4 [2, 3, 4, 5, 6, 7, 8, 9, 10] (see [11, 12] for
reviews of multi-instanton techniques before localization and complete lists
of references). In the D-brane language language, the dynamics of the gauge
theory around the instanton background is described by an effective theory
governing the interactions of the lowest energy excitations of open strings
ending on a bound state of Dp-D(p+4) branes. For the case of N = 2, 4
SYM the multi-instanton action has been derived via string techniques in
[13, 14].
In [15], D-brane techniques have been applied to the computation of the
Affleck, Dine and Seiberg (ADS) superpotential [16, 17] for N = 1 SQCD
with gauge group SU(Nc) and Nf = Nc − 1 massless flavours and Sp(2Nc)
with 2Nf = 2Nc flavours. The N = 1 gauge theory is realized on the four-
dimensional intersection of Nc coloured and Nf flavour D6 branes. Chiral
matter comes from strings connecting the flavor and color D6 branes. Instan-
tons in the U(Nc) gauge theory are realized in terms of ED2 branes parallel to
the stack of Nc D6-branes. By careful integrating the supermoduli (massless
strings with at least one end on the ED2) the precise form of the ADS su-
perpotential was reproduced in the low energy, field theory limit α′ → 0. In
the recent literature ED2-brane instantons in intersecting D6-brane models
have received particular attention in connection with the possibility of gen-
erating a Majorana mass for right handed neutrinos and their superpartners
[18, 19, 20, 21, 22]. The field theory interpretation of this new instanton effect
is far from clear and it is the subject of active investigation. In this paper
we present a detailed derivation of these new non perturbative superpoten-
tials in N = 1 Z3-orientifold models. Investigations of stringy instantons on
N = 1 Z2 × Z2 orientifold singularities appeared recently in [23].
We study SYM gauge theories living on D3 branes located at a Z3-
orientifold singularity. There are two choices for the orientifold projection
[24, 25, 26, 27, 28] realized by two types of O3-planes1. They lead to anomaly
free2 chiral N = 1 gauge theories with gauge groups SO(N − 4)× U(N) or
Sp(N +4)×U(N) and three generations of chiral matter in the bifundamen-
tal and anti/symmetric representation of U(N). The archetype of this class
can be realized as a stack of 3N + 4 D3-branes and one O3−-plane sitting
on top of an R6/Z3 singularity. This system can be thought of as a T-dual
local description (near the origin) of the T 6/Z3 type I string vacuum found
in [40]. The lowest choices of N lead to U(4) or U(5) gauge theories with
three generations of chiral matter in the 6 and 10 + 5∗ that are clearly of
phenomenological interest in unification scenarios [45, 46, 47, 48] 3.
In [49] the U(4) case was studied and the form of the ADS-like super-
1We will only consider O3±-planes, not the more exotic Õ3
-planes [29, 30, 31].
2Factorizable U(1) anomalies are cancelled by a generalization of the Green-Schwarz
mechanism [32, 33, 34, 35, 36, 37, 38, 37] that may require the introduction of generalized
Chern-Simons couplings [39].
3Only the U(4) case can be realized in the compact Z3 orientifold. In general the Chan-
Paton group is SO(8 − 2n) × U(12 − 2n) ×Hn where Hn = U(n)
3, SO(2n), U(n), U(1)n
depending on the choice of Wilson lines [28, 41, 42, 43, 44].
potential was determined combining holomorphicity, U(1) anomaly, dimen-
sional analysis and flavour symmetry. Stringy instanton effects were also
considered. Very much as for worldsheet instantons in heterotic strings
[50, 51, 52, 53, 54], these genuinely stringy instantons give rise to super-
potentials that do not vanish at large VEV’s of the open string (charged)
‘moduli’.
Here we derive the non-perturbative superpotentials from a direct in-
tegration over the D-instanton super-moduli space. Gauge instantons are
described in terms of open strings ending on D(-1) branes while stringy in-
stantons are given by open strings ending on euclidean ED3 branes wrapping
a four cycle inside the Calabi Yau . The open strings connecting the stack
of D3 branes to D(-1) and ED3 branes have four and eight mixed Neumann-
Dirichelet directions respectively. This ensures that the bound state is su-
persymmetric. The superpotential receives contribution from disk, one-loop
annulus and Möbius amplitudes ending on the D(-1) or ED3 branes. We find
that ADS superpotentials are generated only for two gauge theory choices
U(4) and Sp(6)×U(2) inside the Z3-orientifold class. Stringy instantons leads
to Majorana masses in the U(4) case, Yukawa couplings in the U(6)×SO(2)
gauge theory and non-renormalizable couplings for U(2N + 4) × SO(2N)
gauge theories with N > 3.
The plan of the paper is as follows.
In section 2 we review the gauge theories coming from a stack of D3
branes at a C3/Z3 orientifold singularity. In Section 3 we consider non-
perturbative effects generated by D(-1) gauge instantons, corresponding to
ADS-like superpotential in the low energy limit. A detailed analysis of one-
loop vacuum amplitudes and the integrals over the supermoduli is presented
for SYM theories with gauge groups Sp(6) × U(2) and U(4). In section 5,
we consider stringy instanton effects generated by ED3-branes. Once again
a detailed analysis of the the one-loop string amplitudes and the integrals
over the supermoduli is presented. In section 6 we present a “complete” list
of N = 1 SYM theories with matter in the adjoint, fundamental, symmetric
and antisymmetric representation of the gauge groups (U, SO, Sp) which
exhibit a non perturbatively generated ADS superpotential.
We conclude with some comments and directions for future investigation
in Section 7.
2 The Gauge Theory
The low energy dynamics of the open strings living on a stack of N D3-branes
in flat space is described by a N = 4 U(N) SYM gauge theory. In the N = 1
language the fields are grouped into a vector multiplet V = (Aµ, λα, λ̄α̇) and
3 chiral multiplets ΦI = (φI , ψIα), I = 1, 2, 3 , all in the adjoint of the gauge
group.
We consider the D3-brane system at a R6/Z3 singularity. At the singular-
ity the N D3-branes group into stacks of Nn fractional branes with n = 0, 1, 2
labelling the conjugacy classes of Z3. The gauge group U(N) decomposes
n U(Nn). More precisely, denoting by γθ,N the projective embedding of
the orbifold group element θ ∈ Z3 in the Chan-Paton group and imposing
= 1 and γ†
= γ−1
one can write
N0×N̄0
, ωh 1
N1×N̄1
, ω̄h 1
N2×N̄2
) (2.1)
with N =
nNn. The resulting gauge theory can be found by projecting
the N = 4 U(N) gauge theory under the Z3 orbifold group action:
V → γ
V γ−1
ΦI → ω γ
ΦI γ−1
ω = e2πi/3 (2.2)
Keeping only invariant components under (2.2) one finds the N = 1 quiver
gauge theory
V : N0N̄0 +N1N̄1 +N2N̄2
ΦI : 3×
N0N̄1 +N1N̄2 +N2N̄0
(2.3)
with gauge group
n U(Nn) and three generations of bifundamentals. More
precisely V and ΦI are N ×N block matrices (N =
Nn) with non trivial
Nn × N̄m blocks given by (2.3). Under Z3 a block Nn × N̄m transform as
ωn−m. These non-trivial transformation properties are compensated by the
space-time eigenvalues of the corresponding field (ω0 for V and ω for ΦI )
making the corresponding component invariant under Z3.
Next we consider the effect of introducing an O3±-plane. Woldsheet
parity Ω flips open string orientations and act on Chan-Paton indices as
Nn ↔ N̄−n where subscripts are always understood mod 3. This prescription
leads to
Ω : N0 ↔ N̄0 N1 ↔ N̄2 (2.4)
The choices of O3±-planes correspond to keep states with eigenvalues Ω = ±1
and lead to symplectic or orthogonal gauge groups4.
We start by considering the O3− case. Keeping Ω = − components from
(2.3) one finds
V : 1
N0(N0 − 1) +N1N̄1
ΦI : 3×
N0N̄1 +
N1(N1 − 1)
(2.5)
This follows from (2.3) after identifying the mirror images N̄0 = N0, N̄2 = N1,
and antisymmetrizing the resulting block matrix. (2.5) describes the field
content of a N = 1 SYM with gauge group SO(N0)×U(N1) and three chiral
multiplets in the
( , ¯) + (•, )
For general N0, N1 the U(N1) gauge theory is anomalous. The anomaly
is a signal of the presence of a twisted RR tadpole [34, 35]. Focusing on
a local description near the orientifold singularity one can relax the global
tadpole cancellation condition [55, 56]. These models can be thought as
local descriptions of a more complicated Calabi Yau near a Z3 sigularity.
Cancellation of the twisted RR tadpole can be written as [40]
= −4 ⇒ N0 = N1 − 4 (2.6)
and ensures the cancellation of the irreducible four-dimensional anomaly
I(F ) ∼ [−N0 + (N1 − 4)] trF
3 = 0 (2.7)
Finally the running of the gauge coupling constants is governed by the β
functions with one-loop coefficients5
β0 = 3 ℓ(
N0(N0 − 1))− 3N1 ℓ(N0)
(N0 −N1 − 2) = −9 (IR free)
β1 = 3 ℓ(N1N̄1)− 3N0 ℓ(N̄1)− 3 ℓ(
N1(N1 − 1))
(−N0 +N1 + 2) = +9 (UV free) (2.8)
with βn refering to the n
th-gauge group. The last equalities arise after im-
posing the anomaly cancellation (2.6). As expected, β0 + β1 = 0 since the
ten-dimensional dilaton does not run.
4In the compact case, realized in terms of D9-branes and O9-plane on T 6/Z3, the
orthogonal choice is dictated by global tadpole cancellation. Turning on a quantized NS-
NS antisymmetric tensor [28, 41, 29] leads to symplectic groups.
5Here trRT
aT b = ℓ(R), i.e. ℓ(N) = 1
, ℓ(NN̄) = N and ℓ(1
N(N± 1)) = 1
(N ± 2).
The case Ω = + works in a similar way. The resulting N = 1
quiver has gauge group Sp(N0) × U(N1) and three chiral multiplets in the
[( , ¯) + (•, )]. The U(N1) is anomaly free for N0 = N1 + 4 and the one-
loop β function coefficients are given by β0 = +9 (UV free) and β1 = −9 (IR
free).
3 D(-1) Instantons
There are two sources of supersymmetric instanton corrections in the D3
brane gauge theory: D(-1)-instantons and Euclidean ED3-branes wrapping
four cycles on T 6/Z3. Both are point-like configurations in the space-time
and can be thought of as D(-1)-D3 and ED3-D3 bound states with four and
eight directions with mixed Neumann-Dirichlet boundary conditions.
3.1 D3-D(-1) in flat space
Gauge instantons in SYM can be efficiently described in terms of D(-1)-
branes living on the world-volume of D3-branes [57]. As before, we start
from the N = 4 case: a bound state of N D3 and k D(-1) branes in flat
space. In this formalism instanton moduli are described by the lowest energy
modes of open strings with at least one end on the D(-1)-brane stack. The
gauge theory dynamics around the instanton background can be described
in terms of the U(k) × U(N) 0-dimensional matrix theory living on the D-
instanton world-volume. In particular, the ADHM constraints [58] defining
the moduli space of self-dual YM connections follow from the F- and D-
flatness condition in the matrix theory [57].
The instanton moduli space is given by the D(-1)D3 field content
(aµ, θ
α , χa, D
c, θ̄Aα̇) kk̄
(wα̇, ν
A) kN̄
(w̄α̇, ν̄
A) Nk̄ (3.1)
with µ = 1, . . . , 4, α, α̇ = 1, 2 (vector/spinor indices of SO(4)), a = 1, . . . , 6,
A = 1, . . . , 4 (vector/spinor indices of SO(6)R), c = 1, . . . , 3. The matrices
aµ, χa describe the positions of the instanton in the directions parallel and
perpendicular to the D3-brane respectively, wα is given by the NS open D3-
D(-1) string (instanton sizes and orientations), Dc are auxiliary fields and
θAα , θ̄Aα̇, ν
A are the fermionic superpartners.
The D3-D(-1) action can be written as [59]
Sk,N = trk
SG + SK + SD
(3.2)
SG = −[χa, χb]
2 + iθ̄α̇A[χ
AB, θ̄
cDc (3.3)
SK = −[χa, aµ]
2 + χaw̄
α̇wα̇χa − iθ
αA[χABθ
α ] + 2iχAB ν̄
SD = i
−[aαα̇, θ
αA] + ν̄Awα̇ + w̄α̇ν
θ̄α̇A +D
w̄σcw − iη̄cµν [a
µ, aν ]
with χAB ≡
T aABχa, T
AB = (η
AB, iη̄
AB) given in terms of the t’Hooft symbols
and g20 = 4π(4π
2α′)−2 gs. The action (3.3) follows from the dimensional
reduction of the D5-D9 action in six dimensions down to zero dimension.
As a consequence our subsequent results hold up to some computable non
vanishing numerical constant.
In the presence of a v.e.v. for the six U(N) scalars ϕa in the D3-D3 open
string sector we must add to Sk,N
Sϕ = trk
w̄α̇(ϕaϕa + 2χaϕa)wα̇ + 2iν̄
AϕABν
(3.4)
The multi-instanton partition function is
Zk,N =
e−Sk,N−Sϕ =
VolU(k)
dχ dD da dθ dθ̄dw dν e−Sk,N−Sϕ
In the limit g0 ∼ (α
′)−1 → ∞, gravity decouples from the gauge theory and
the contributions coming from SG are suppressed. The fields θ̄α̇A, D
c become
Lagrange multipliers implementing the super ADHM constraints
θ̄α̇A : ν̄
Awα̇ + w̄α̇ν
A − [aαα̇, θ
αA] = 0
Dc : w̄σcw − iη̄cµν [a
µ, aν ] = 0 (3.5)
3.2 D(-1)-D3 at the C3/Z3-orientifold
Let us now consider in turn the Ω and then the Z3 projection.
The effect of introducing an O3±-plane in the D(-1)-D3 system corre-
sponds to keep open string states with eigenvalue ΩI = ±, Ω being the
worldsheet parity and I a reflection along the Neumann-Dirichlet directions
of the Dp-O3 system [8]. On D(-1) string modes, I acts as a reflection in the
spacetime plane
I : aµ → −aµ θ
α → −θ
α (3.6)
leaving all other moduli invariant. In addition consistency with the D3-O3
projection requires that the D(-1) strings are projected in the opposite way
with respect to the D3-branes[60] . From the gauge theory point of view this
corresponds to the well known fact that SO(N) and Sp(N) gauge instantons
have ADHM constraints invariant under Sp(k) and SO(k) respectively.
We start by considering the O3− case. After the ΩI projection the sur-
viving fields are
(aµ, θ
k(k− 1)
(Dc, χI , χ̄I , θ̄Aα̇)
k(k+ 1)
(wα̇, ν) kN . (3.7)
Since we are dealing with a SO(N) gauge theory the Dc moduli are projected
in the adjoint of Sp(k). This is also the case for all the other moduli even
under I while the odd ones, (aµ, θ
α ), turn out to be antisymmetric.
Let us now consider the Z3 projection. Out of the six χa one can form
three complex fields χI with eigenvalues ω under Z3 and their conjugate
χ̄I . To embed the Z3 projection into SU(4) we decompose the spinor index
A = (0, I), with I = 1, . . . , 3 and the zeroth direction along the surviving
N = 1 supersymmetry. The D3 and D(-1) gauge groups SO(N) and Sp(k)
break into SO(N0)×U(N1) and Sp(k0)×U(k1) respectively with N0 (k0) the
number of fractional D3 (D(-1)) branes invariant under Z3 and N1 (k1) those
transforming with eigenvalue ω. More precisely, the projective embedding
of the Z3 basic orbifold group element θ in the Chan-Paton group can be
written
N0×N0
, ωh 1
N1×N̄1
, ω̄h 1
N̄1×N1
k0×k0
, ωh 1
k1×k̄1
, ω̄h 1
k̄1×k1
) (3.8)
After projecting under Z3 the symmetric/antisymmetric matrices in (3.7)
break into km × k̄n, km × N̄n or Nm × k̄n each transforming with eigenvalue
ωm−n. In addition fields with up(down) index I transform like ω(ω̄). Keeping
only the invariant components one finds
(aµ; θ
k0(k0 − 1) + k1k̄1
k1(k1 − 1) + k0k̄1
(Dc; θ̄0α̇)
k0(k0 + 1) + k1k̄1
(χ̄I ; θ̄Iα̇)
k̄1(k̄1 + 1) + k0k1
k1(k1 + 1) + k0k̄1
(wα̇; ν0) k0N0 + k1N̄1 + k̄1N1
νI k0N̄1 + k̄1N0 + k1N1 (3.9)
Notice that the Z3 eigenvalues of the Chan-Paton indices in the r.h.s. of (3.9)
compensate for those of the moduli in the l.h.s. making the field invariant
under Z3. In addition (odd)even components under I are (anti)symmetrized
ensuring the invariance under ΩI.
The multi-instanton action follows from that of N = N0+2N1 D3 branes
and k = k0 + 2k1 D(-1) instanton in flat space (3.2) with U(N) and U(k)
matrices restricted to the invariant blocks (3.9).
The results for O3+ can be read off from (3.9) by exchanging symmetric
and antisymmetric representations.
4 ADS-like superpotential
4.1 D3-D(-1) one-loop vacuum amplitudes
Non-perturbative superpotentials can be computed from the instanton mod-
uli space integral [11, 13, 18]
SW = e
〈1〉D+〈1〉A+〈1〉M µβnkn
e−Sk,N−Sϕ (4.1)
The integration is over the instanton moduli space, M, 〈1〉D is the disk
amplitude and 〈1〉A,M are the one-loop vacuum amplitudes with at least
one end on the D(-1)-instanton. The factor µβnkn , µ being the energy scale,
comes from the quadratic fluctuations around the instanton background and
as we will see it combines with a similar contribution coming from the moduli
measure to give a dimensionless SW .
The terms in front of the integral in (4.1) combine into
SW = Λ
e−Sk,N−Sϕ (4.2)
Λknβn = e2πiknτn(µ) µβnkn τn(µ) = τn −
(4.3)
the one-loop renormalization group invariant and the running coupling con-
stant respectively and τn refers to the complexified coupling constant of the
nth gauge group. µ0 is a reference scale.
More precisely, the disk amplitude and one-loop amplitudes yields
e〈1〉D = e2πiknτn τn =
e〈1〉A+〈1〉M =
)−βnκn
+ . . . (4.4)
with dots refering to threshold corrections that will not be considered here.
To see (4.4) we should compute the following one-loop amplitudes
〈1〉A = −
Tr[(1 + (−)F )(1 + θ + θ2) qL0−a]
AD(−1)D3 = −A0,D(−1)D3 ln
+ . . .
〈1〉M = −
Tr[ ΩI (1 + (−)F )(1 + θ + θ2) qL0−a]
MD(−1) = −M0,D(−1) ln
+ . . . (4.5)
In the above formula µ enters as a UV regulator in the open string chan-
nel (see [61] for details) and A0,M0 are the massless contributions to the
amplitudes.
We start by considering the O3− projection. It is important to notice that
only the annulus with one end on the D(-1) and one on the D3 contributes
to these amplitudes since D(-1)-D(-1) amplitudes cancel due to the Riemann
identity. One finds
AD(−1),D3 =
trγθ,ktrγθ,N
ϑ[αβ ]
ϑ[αβ+hi ]
(k0 − k1)(N0 −N1) + . . .
MD(−1) =
trγθ2,k
ϑ[αβ ]
ϑ[αβ+hi ]
= −3(k0 − k1) + . . . (4.6)
The sum runs over the even spin structures and cαβ = (−)
2(α+β). The term
comes from the (b, c) and (β, γ) ghosts while the extra five thetas in the
numerator and denominator describe the contributions of the ten fermionic
and bosonic worldsheet degrees of freedom. We adopt the shorthand notation
h ] ≡ ϑ[
h ]/(2 cosπh) to describe the massive contribution of a periodic
boson to the partition function. hi = (
) denote the Z3-twists while
the extra 1
-shifts in the annulus account for the D(-1)-D3 open string twist
along Neuman-Dirichlet directions while 1
twists in the Möbius come from
the I-projection. In addition we used the fact that the contribution of the
unprojected sector is zero after using the Riemann identity while that of the
θ- and θ2-projected sectors are identical explaining the overall factor of 2.
The extra factor of 2 in the annulus comes from the two orientations of the
string. The second line displays the massless contributions. We use the Chan
Paton traces
1,k = k0 + 2k1 trγθ,k = k0 − k1
1,N = N0 + 2N1 trγθ,N = N0 −N1 (4.7)
that follows from (2.1) and the first few terms in the theta expansions
ϑ[0h] = 1 + q
2 2 cos 2πh+ . . . ϑ[
h ] = q
8 2 cosπh+ . . .
η = q
24 + . . . (4.8)
From (4.6) one finds
A0 +M0 =
(k0 − k1)(N0 −N1 − 2) = knβn (4.9)
with βn the one-loop β coefficients given in (2.8). Plugging (4.9) into (4.5)
results into (4.4). The fact that the β function coefficients are reproduced by
the instanton vacuum amplitudes is a nice test of the instanton field content
(3.9).
Now let us determine the dependence of the instanton measure on the
string scaleMs ∼ α
′ −1/2. The scaling of the various instanton moduli follows
from (3.3):
D, g0 ∼M
s χa, ϕa ∼ Ms wα̇, aµ ∼M
νA, θAα ∼M
s θ̄Aα̇ ∼M
s (4.10)
Collecting from (3.9) the number of components of the various moduli enter-
ing in the instanton measure one finds 6
e−Sk,N−Sϕ ∼ M−βnkns
knβn = −2nD − nχ + na + nw +
nθ̄ −
(k0 − k1)(N0 −N1 − 2) (4.11)
Notice that this factor precisely combines with that in (4.2) leading to a di-
mensionless SW as expected. This simple dimensional analysis can be used to
determine the form of the allowed ADS superpotentials in the gauge theory.
A superpotential is generated if and only if the integral over the instanton
moduli space reduce to an integral over x
0 describing the center of the in-
stanton and θα its superpartner. More precisely
SW = Λ
e−Sk,N−Sϕ = c
d4x0d
Λknβn
ϕknβn−3
(4.12)
where c is a numerical constant. Whether c is zero or not depends on the
presence or not of extra fermionic zero modes besides θ. Notice that the power
of ϕ is completely fixed requiring that SW is dimensionless. The precise form
of the superpotential requires the evaluation of the moduli space integral and
will be the subject of the next section. The superpotential follows from (4.12)
after promoting ϕI to the chiral superfield ΦI and x0, θα to the measure of
the superspace
SW = c
d4xd2θ
Λknβn
Φknβn−3
(4.13)
6We recall that fermionic differentials scale as the inverse of the dimension of the
fermion itself. This explains the extra minus sign in (4.11).
A superpotential of type (4.13) is generated whenever [63, 64, 16, 17]
〈λ2 ϕknβn−3〉 6= 0 (4.14)
Each scalar ϕ soaks two fermionic zero-modes and each gaugino λ one zero
mode7. The condition (4.14) translates into
dimMF = 2knβn − 4 (4.15)
with dimMF the fermionic dimension of the instanton super-moduli space.
The number of fermionic zero modes can be read off from (3.9)
dimMF = nθ + nν − nθ̄
= k0(3N1 +N0 − 2) + k1[2N1 + 3(N0 +N1 − 2)]
= k0(4N0 + 10) + k1(8N0 + 14) (4.16)
where we used the fact that θ̄α̇A enter as a Lagrangian multiplier imposing
the fermionic ADHM constraint and therefore subtracts degrees of freedom.
The last line in (4.16) follows from using the anomaly cancellation condition
N1 = N0 + 4. The result (4.16) is consistent with the Atiyah-Singer index
theorem that states
dimMF = 2k0
N0(N0 − 1)) + 3N1ℓ(N0)
ℓ(N1N̄1) + 3N0ℓ(N1) + 3ℓ(
N1(N1 − 1))
= k0(3N1 +N0 − 2) + k1[2N1 + 3(N0 +N1 − 2)] (4.17)
Combining (4.15) and (4.16) one finds
k1 − 7k0 − 1
k0 + 2k1
(4.18)
One can easily see that the only non-negative solution for N0 is
N0 = 0 k0 = 0 k1 = 1
We conclude that in the class of U(N0+4)×SO(N0) SYM theories describing
the low-energy dynamics of D3-branes on the Z3 orientifold only the U(4)
7This can be seen by explicitly solving the equations of motion of the gaugino and the
ϕ-field in the instanton background [59]. In particular the source for the scalar field comes
from the Yukawa coupling LY uk = gYMϕ
†ψλ in the gauge theory action.
theory with three chiral multiplets in the antisymmetric leads to an ADS-like
superpotential generated by gauge instantons.
The counting can be easily repeated for the Sp(N1+4)×U(N1) cases by
exchanging symmetric and antisymmetric representations in (3.9) as required
by the presence of the O3+-plane. The results are
knβn = 9(k0 − k1)
dimMF = k0(4N1 + 6) + k1(8N1 + 18)
3k0 − 9k1 − 1
k0 + 2k1
(4.19)
One can easily see that the only non-negative solution is
N1 = 2 k0 = 1 k1 = 0
We conclude that in this class, only the gauge theory Sp(6)×U(2) with three
chiral multiplets in the ( , ¯) + (•, ) admits an ADS-like superpotential
generated by instantons.
The aim of the rest of this section is to compute SW . The integral (4.12)
will be evaluated in turn for the Sp(6)× U(2) and U(4) case.
4.2 Sp(6)× U(2) superpotential
We first consider the O3+ case, i.e. the Sp(6) × U(2) gauge theory with
three chiral multiplets in the [(6, 2̄) + (1, 3)]. The instanton moduli is given
by (3.9) after flipping symmetric/antisymmetric representations in order to
deal with the symplectic projection. Plugging k0 = 1, k1 = 0, N0 = 6, N1 = 2
into (3.9) one finds the the surviving fields
aµ, w
, θ0α, ν
0u0 , νIu1 (4.20)
with u0 = 1, ..6, and u1 = 1, 2, whose position from lower to upper has been
switched in this section for notational convenience as we will momentarily
see. In particular both θ̄0α̇ andD
c are projected out (the D(-1) “gauge” group
is O(1) ≈ Z2 in this case) and therefore no ADHM constraint survives. The
instanton action reduces then to
S = SK + Sϕ = w
α̇ ϕ̄Iu0u1ϕ
I u1v0wα̇v0 + ν
Iu1ν0u0ϕ̄Iu0u1 (4.21)
Here and below we omit numerical coefficients that can be always reabsorbed
at the end in the definition of the scale. The integrations over wα̇u0 , ν
, νIu1
are gaussian and the final result, up to a non vanishing numerical constant,
can be written as
SW = Λ
d4ad2θ
det6×6 (ϕ̄Iu1,u0)
det6×6 (ϕ̄Iu1,u0ϕ
Iu1,v0)
d4ad2θ
det6×6 (ϕIu1,u0)
(4.22)
where we have exploited the possibility of combining I and u1 in one ‘bi-
index’ Iu1 so as to get a range of six values. For the sake of simplicity we
have dropped the subscript 0 denoting bare scalar fields. In the following
scalar fields entering in formulae involving Λ will be always understood to be
bare. The last step makes use of det(AB) = det(A)det(B).
4.3 U(4) superpotential
We now consider the O3− case, i.e. the U(4) gauge theory with three chiral
multiplets in the 6. Setting k0 = 0, k1 = 1, N0 = 0, N1 = 4 in (3.9) the
surviving fields can be written as
I[uv]
, ϕ̄I[uv](0) , aµ(0) , χ̄I(−2) , χ
(+2) , D
(0) , w
u(+1) , w̄
α̇(−1)
θ0α(0) , θ̄0α̇(0) , θ̄α̇I(−1) ; ν
u(+1) , ν̄
(−1) , ν
(+1) (4.23)
with u = 1, ..4 and the charge q under U(1)k1 is denoted in parentheses.
Plugging into (3.3) (after taking α′ → 0) one finds
S = SB + SF (4.24)
where
ν̄0uwuα̇ + w̄
θ̄α̇0 + ν
Iu wuα̇ θ̄
I + χ̄Iν
Iu + νIuϕ̄Iuvν̄
SB = w̄
α̇ϕ̄Iuwϕ
Iwvwα̇v + ϕ
Iuvwα̇uwvα̇χ̄I + ϕ̄Iuvw̄
uα̇w̄vα̇χ
I + w̄uα̇wuα̇χ̄Iχ
+Dc w̄σcw (4.25)
As before we omit numerical coefficients. The integral over Dc leads to a δ
function on the ADHM constraints
d8wd8w̄δ3(w̄σcw) =
dρ ρ9d12U (4.26)
In the r.h.s of (4.26) we have solved the ADHM constraints in favor of w and
U defined by
wuα̇ = ρUuα̇ w̄
uα̇ = ρ Ūuα̇ Ūuα̇Uuβ̇ = δ
(4.27)
The coset representatives Uα̇u parameterizes the SU(4)/SU(2) orientations
of the instanton inside the gauge group. The fermionic integrations lead to
the determinant
∆F = ρ
8 ǫu1u2u3u4ǫv1v2u5u6ǫv3v4v5v6Xu1v1u2v2 Xu3v3u4v4 Yu5v5 Yu6v6 (4.28)
Xu1v1u2v2 = ǫ
I1I2I3χ̄I1ϕ̄I2u1v1ϕ̄I3u2v2
Yuv = U
u Uα̇u (4.29)
The bosonic integrals are more involved. For arbitrary choices of the scalar
VEV’s ϕ̄I and ϕ
I , even along the flat directions of the potential, the inte-
gration over U represents a challenging if not a prohibitive task. Fortunately
choosing ϕIuv = ϕηIuv, ϕ̄Iuv = ϕ̄η
Iuv, the full ϕ-dependence can be factor-
ized. SU(4) gauge and SU(3) ‘flavor’ invariance can then be used to recover
the full answer. After the rescaling
ρ2 → ρ2/(ϕϕ̄) χI → ϕχI χ̄I → ϕ̄χ̄I (4.30)
The integral becomes
SW = Λ
d4x0d
(4.31)
with I the ϕ-independent integral
dρρ9 d12U d3χd3χ̄∆F e
S̃B = −ρ
2(1 + ηIuvYuvχI + η̄IuvȲ
uvχI + χ̄Iχ
I) (4.32)
and ∆F given again by (4.28) but now in terms of
Xu1v1u2v2 = ǫ
I1I2I3χ̄I1 η̄I2u1v1 η̄I3u2v2
Finally one can restore the gauge covariance of (4.31) by noticing that there
is a unique SU(4)c × SU(3)f singlet in the symmetric tensor of six ϕ
det3×3[ǫu1..u4ϕ
Iu1u2ϕJu3u4]
Therefore one can replace ϕ6 in (4.31) by this singlet. The superpotential
follows after replacing ϕI → ΦI
SW = c
d4xd2θ
det3×3[ǫu1..u4Φ
Iu1u2ΦJu3u4]
(4.33)
where c is a computable non-zero numerical coefficient.
5 ED3-instantons
Let us now consider the ED3-D3 system. We restrict ourselves to the compact
case T 6/Z3 and consider the ED3 fractional instanton wrapping a four-cycle
Cn inside T
6/Z3. We start by considering the O3
−-orientifold projection.
The zero modes of the Yang-Mills fields in the instanton background can
be described as before in terms of open strings with at least one end on
the ED3. Open strings connecting ED3 and D3 branes have 8 Neumann-
Dirichlet directions therefore the zero-mode dynamics of the ED3-D3 system
is equivalent to that of the D7-D(-1) bound state. The instanton action can
be found starting from that of the N = (8, 0) sigma model describing the
low energy dynamics of a D1-D9 bound state in type I [65] reduced down to
zero dimensions. In flat space the D(-1)-D7 action reads
S = trk
Sg + SK + SD
(5.1)
Sg = −[χ, χ̄]
2 + Θ̃ȧχΘ̃ȧ +DcDc
SK = −[χ,Xm][χ̄, Xm] + Θ
aχ̄Θa + ν(χ + ϕ)ν
SD = Θ̃
ȧXmΓ
a +DcΓ̂cmn[Xm, Xn] (5.2)
with m = 1, . . . , 8v, a = 1, . . . , 8s, ȧ = 1, . . . , 8c, c = 1, , . . . , 7. We denote
by ϕ = mI(Cn)ϕ
I , the gauge scalar parametrizing the position of the D3-
brane along the direction perpendicular to the 4-cycle Cn. Here Γ
ȧa, Γ̂
are gamma matrices of SO(8) and SO(7) respectively. The introduction
of the auxiliary fields Dc has broken the manifest SO(8) invariance of the
action that will be further broken by the Z3-projection. In (5.2), Xm and
χ, χ̄ describe the position of the D(-1)-instanton in the directions longitudinal
and perpendicular to the D7-brane respectively while Θa, Θ̃ȧ are the fermionic
superpartners grouped according to the their chirality along the Dirichlet-
Dirichlet χ-plane. Unlike the D(-1)-D3 case, in the case of 8 Neumann-
Dirichlet directions Ω acts in the same way on the D(-1) and D7 Chan-Paton
indices. This implies that Dc transform in the adjoint of SO(k) if we take
the D7 gauge symmetry to be SO(N). In addition I acts as
I : Xm → −Xm Θ
a → −Θa (5.3)
Fields with eigenvalues ΩI = − are then in the following representations of
SO(k)× SO(N)
(χ, χ̄, Dc, Θ̃ȧ) 1
k(k− 1)
(Xm,Θ
k(k+ 1)
ν kN (5.4)
Fields even under I transform in the adjoint of SO(k) while odd fields tran-
form in the symmetric representation. For k = 1, N = 32 the D(-1)-D7
system or equivalently the D1-D9 bound state describes the S-dual version
of the fundamental heterotic string on T 2. k > 1 bound states correspond to
multiple windings of the heterotic string [65].
The field Dc implements the one-real D and three complex F flatness
conditions
V = −
DcDc = −g20
m,n=1
[Xm, Xn]
2 = 0 (5.5)
Dc = −1
mn[Xm, Xn] (5.6)
An explicit choice of Γ matrices in D = 7 is given by (a = 1, 2, 3)
Γa8×8 = iσ1 ⊗ η
4×4 Γ
8×8 = iσ3 ⊗ η̄
4×4 Γ
8×8 = iσ2 ⊗ 14×4 (5.7)
As in section 3 Z3 acts both on spacetime and Chan-Paton indices. Chan-
Paton indices decompose as N → N0 + N1 + N̄1 and k → k0 + k1 + k̄1.
Spacetime indices on the other hand decompose as
8v = 4 + 2ω + 2ω̄
8s = 2 + 2ω + 4ω̄
8c = 2 + 2ω̄ + 4ω
7 = 3 + 2ω + 2ω̄ (5.8)
In addition χ, ν transform with eigenvalue ω under Z3. Combining with (5.4)
one finds the Z3-invariant components
χ, χ̄ 1
k1(k1 − 1) + k0k̄1 + h.c.
Dc 3(1
k0(k0 − 1) + k1k̄1) + 2
k1(k1 − 1) + k0k̄1 + h.c.
Θ̃ȧ 2
k0(k0 − 1) + k1k̄1
k̄1(k̄1 − 1) + k0k1)
k1(k1 − 1) + k0k̄1)
k0(k0 + 1) + k1k̄1
k1(k1 + 1) + k0k̄1 + h.c.
k0(k0 + 1) + k1k̄1
k1(k1 + 1) + k0k̄1)
k̄1(k̄1 + 1) + k0k1)
ν k0N̄1 + k1N1 + k̄1N0 (5.9)
5.1 D3-ED3 one-loop vacuum amplitudes
ED3 generated superpotentials can be computed following the same steps as
in section 4.1. The disk amplitude can be written as
e〈1〉D = e2πiknτ̃n τ̃n = i
4πV4(Cn)
g2n α
(C4 + C0 ∧R ∧ R) (5.10)
τ̃n describes the coupling of closed string moduli to the ED3 instanton wrap-
ping the 4-cycle Cn with volume V4(Cn). We remark that closed string states
in the Z3-twisted sectors flow in the ED3-ED3 cylinder amplitude and there-
fore τ̃n is function of both untwisted and twisted closed twisted moduli. This
is not surprising since the volume of the cycle depends also on the volume of
the exceptional cycles that the ED3 wraps.
The annulus and Möbius amplitudes are given by
AED3,D3 =
ϑ[αβ ]
2trγθ,ktrγθ,N
ϑ[αβ−2h1 ]
+ trγ
1,ktrγ1,N
ϑ[αβ ]
k0N1 −
k1(N0 +N1) + . . .
MED3 = −
ϑ[αβ ]
2trγθ2,k
ϑ[αβ−2h1 ]
+ trγ
ϑ[αβ ]
= 3k0 + k1 + . . . (5.11)
The origin of the various contributions is the same that in the D(-1)-D3
system. Now the D3-ED3 open strings have 8 Neumann-Dirichlet directions
explaining the extra 1
twists in the annulus amplitude. On the other side,
the I projection accounts for the 1
-shift in the Möbius amplitude. Notice
that unlike the D(-1)-D3 case, the unprojected amplitude tr1, now gives a
non-trivial contribution.
Collecting the contributions from (5.11) one finds
Λ̃knbn = µknbn e〈1〉D+〈1〉A+〈1〉M = µknbn e2πikn τ̃n(µ) (5.12)
τ̃n(µ) = τ̃n −
(5.13)
A0 +M0 = knbn =
k0(6−N1) +
k1(2−N0 −N1) (5.14)
The interpretation of the bn as the one-loop β function coefficients of the τ̃n
coupling, though tantalizing, is not clear to us. We will now check that knbn
reproduces the right scale dependence of the instanton measure. The scaling
of the various instanton moduli follows from (5.2):
D, g0 ∼M
s χ, χ̄, ϕ ∼Ms Xm ∼ M
ν, Θa ∼M−1/2s Θ̃
ȧ ∼ M3/2s (5.15)
Collecting from (5.9) the number of degrees of freedom entering in the
instanton supermoduli measure one finds
e−Sk,N ∼ M−knbns
knbn = −2nD − nχ + nX +
nΘ̃ −
k0(6−N1) +
k1(2−N0 −N1) (5.16)
As in the previous case we write the instanton generated superpotential as
the moduli space integral
SW = Λ̃
e−Sk,N−Sϕ =
d4x0d
2θ Λ̃knbn ϕ−knbn+3 (5.17)
After promoting ϕ→ Φ and x0, θα to the measure of the superspace one finds
the ED3 generated superpotential
d4xd2θ Λ̃knbn Φ−knbn+3 (5.18)
The main difference with respect to the D(-1) instantons is that now ϕ enters
into Sϕ (5.2) only through the coupling to the ν-fermions. This implies that
in order to get a non zero result from the fermionic integral in (5.17) only the
ν’s and the two fermionic zero modes θα ∈ Θ
a should survive the orientifold
projections. From (5.9) one can easily see that this implies k0 = 1, k1 = 0.
The same counting shows that no solutions are allowed in the Sp(N) case.
5.2 The superpotential
Here we evaluate the instanton moduli space integral for the SO(N0)×U(N1)
case. From our analysis above the relevant cases are k0 = 1, k1 = 0.
The surviving fields in (5.9) are
θα ∈ Θ
0 ∈ Xm νu (5.19)
with u = 1, ...N1. The instanton action reduces to
S = νuϕ
uvνv (5.20)
The superpotential is then given by the integral
SW = Λ̃
d4x d2θ dN1ν e−νϕν (5.21)
After integration over ν and lifting ϕ→ Φ to the superfield one finds
SW = c Λ̃
d4x d2θ ǫu1....uN1Φ
u1u2Φu3u4...ΦuN1−1uN1 (5.22)
where c is a non vanishing numerical constant. Notice that the result
is non-trivial only when N1 is even. The superpotentials (5.22) are non-
renormalizable for N1 > 6 and grow for large vacuum expectation values
where the low energy approximation breaks down. The only exceptions are
Majorana masses U(4) + 3
Yukawa couplings SO(2)× U(6) + 3 ( , ¯) + 3 (•, ) (5.23)
Notice that both instanton generated Yukawa couplings involve only the mat-
ter in the antisymmetric representation.
6 ADS superpotentials: a general analysis
Here we consider a general N = 1 gauge theory with gauge group U(N) and
nAdj, nf/n̄f , nS/n̄S, nA/n̄A number of chiral multiplets in the adjoint, fun-
damental, symmetric and anti-symmetric representations (and their complex
conjugates) respectively.
The cubic chiral anomaly, one-loop β function and number of fermionic
zero modes in the instanton background of the gauge theory can be written
Ianom = nf− + nS−(N + 4) + nA−(N − 4) = 0 (6.1)
β1−loop = 3N −NnAdj −
nf+ −
nS+(N + 2)−
nA+(N − 2)
dimMF = k [2N + 2NnAdj + nf+ + nS+(N + 2) + nA+(N − 2)]
nf± = nf ± n̄f nS± = nS ± n̄S nA± = nA ± n̄A (6.2)
The condition for an Affleck, Dine and Seiberg like superpotential [16, 17] to
be generated was determined in section 4.1 to be
dimMF = 2kβ − 4 (6.3)
Combining (6.1) and (6.3) one finds
β1−loop = 2N +
(6.4)
nf− = −nS−(N + 4)− nA−(N − 4)
nf+ = 2N −
− 2NnAdj − nS+(N + 2)− nA+(N − 2)
Remarkably the β function in a theory admitting an instanton generated su-
perpotential depends only on the rank of the gauge group. A simple inspec-
tion shows that a superpotential is generated only for k = 1 and nAdj = 0.
The complete list follows from a scan of any choice of nS±,nA± such that
n+ ≥ |n−| and n+ ≥ 0. One finds
U(N) +Nf ( + ¯ ) Nf ≤ N − 1
U(N) + + (N − 4)¯ +Nf ( + ¯ ) Nf ≤ 2
U(4) + 2 +Nf ( + ¯ ) Nf ≤ 1
U(4) + 3
U(5) + 2 + 2¯ (6.5)
The inequalities are saturated for gauge theories satisfying (6.3) and (6.4),
while the lower cases are found by decoupling quark-antiquark pairs via mass
deformations.
The generalization to SO(N)/Sp(N) gauge groups is straightforward. In
these cases there is no restriction coming from anomalies since representations
are real. The β function and the number of fermionic zero modes in the
instanton background are given by
β1−loop =
(N ± 2)− 1
nS(N + 2)−
nA(N − 2)
dimMF = k [N ± 2 + nf + nS(N + 2) + nA(N − 2)]
with upper sign for Sp(N) and lower sign for SO(N) gauge groups. Imposing
(6.3) one finds
β1−loop = N ± 2 +
(6.6)
nf = N ± 2−
− nS(N + 2)− nA(N − 2)
The list of solutions is even shorter
SO(N) +Nf Nf ≤ N − 3 k = 2
Sp(N) +Nf Nf ≤ N k = 1
Sp(N) + + 2 k = 1
(6.7)
Notice that k = 1, respectively k = 2, are the basic instantons in Sp(N),
respectively SO(N), since the instanton symmetry groups are in these cases
SO(k), respectively Sp(k).
7 Conclusions
In the present paper, we have given a detailed microscopic derivation of non-
perturbative superpotentials for chiral N = 1 D3-brane gauge theories living
at Z3-orientifold singularities. We considered both unoriented projections
leading to SO(N1− 4)×U(N1) and Sp(N1+4)×U(N1) gauge theories with
three generations of chiral matter in the representations ( , ¯) + (•, ) and
( , ¯) + (•, ) respectively.
The U(4) case was studied in details in [49] and describes the local physics
of type I theory near the origin of T 6/Z3 with SO(8)× U(12) gauge group
broken by Wilson lines. In the present T-dual setting, there are two sources
of non-perturbative effects: D(-1) and ED3 instantons. The former realize
the standard gauge instantons and lead to Affleck, Dine and Seiberg like
superpotentials. The latter lead to Majorana masses or non-renormalizable
superpotentials and were ignored till very recently [18, 19, 20, 21, 15, 49, 22,
Our explicit instanton computations confirm the form of ADS and stringy
superpotentials proposed in [49] on the basis of holomorphicity, dimensional
analysis U(1) anomaly and flavour symmetry. We show that ADS super-
potentials are generated only for the U(4) and Sp(6)× U(2) gauge theories
in the Z3-orientifold list. The precise form of the superpotential is derived
from an integration over the instanton super-moduli space. Like in [15], the
β function running of gauge couplings are reproduced from vacuum ampli-
tudes given in terms of annulus and Möbius amplitudes ending on the in-
stantons. The same analysis is performed for “stringy instantons” generated
by Euclidean ED3-branes (dual to ED1-strings in type I theory) wrapping
holomorphic four-cycles on T 6/Z3. A detailed microscopic analysis of the
multi-instanton super-moduli space encompasses massless open string states
with a least one end on the ED3-instanton. We show the generation of Ma-
jorana mass terms for the open string chiral multiplets in the U(4) case,
Yukwa couplings for the SO(2)×U(6) gauge theory and non-renormalizable
superpotentials for SO(N0)×U(N0 +4) gauge theories. The field theory in-
terpretation of the β function coefficients generated by the one-loop vacuum
amplitudes for open strings ending on the ED3-instantons is one of the most
interesting open question left by our instanton super-moduli space analysis.
As previously observed, the invariance under anomalous U(1)’s results from
a detailed balance between the charges of the open strings involved and the
axionic shift of a closed string R-R modulus from the twisted sector.
Our present analysis has some analogies with the recent ones [18, 19, 20,
21, 15, 22, 23] that have focussed on ED2-branes at D6-brane intersection.
As stressed in [49], one immediate advantage of the viewpoint advocated
here is the consistency of the local description. Indeed, imposing twisted
tadpole cancellation [34, 35] the models presented here and all closely related
settings of D-branes at singularities (not necessarily of the Zn kind) give
rise to anomaly free theories, while this is not necessarily the case for the
‘local’ models with intersecting D-branes. We can envisage the possibility of
extending our analysis to other Zn singularities [55, 56] or even to Gepner
models [66, 67, 68] where many if not all ingredients, such as the brane actions
from gauge kinetic functions including one-loop threshold effects [69, 70, 71,
72], are available.
In the present paper we have not addressed phenomenological implica-
tions of the stringy instanton effects we have analyzed in detail. We hope to
be able to investigate these issues in this or similar contexts with D-branes
at singularities, where the rigidity of the cycles is well understood and allows
for the correct number of fermionic zero-modes. Clearly additional (closed
string) fluxes neeeded for moduli stabilization [73, 74] may change some of
our present conclusions.
Acknowledgments
It is a pleasure to thank P. Anastasopoulos, R. Argurio, C. Bachas, M.
Bertolini, M. Billo, G. Ferretti, M.L. Frau, A. Kumar, E. Kiritsis, I. Kle-
banov, S. Kovacs, A. Lerda, L. Martucci, I. Pesandro, R. Russo and M.
Wijnholt for valuable discussions. Special thanks go to G. Pradisi for collab-
oration on the computation of the string amplitudes and useful exchanges.
During completion of this work, M.B. was visiting the Galileo Galilei In-
stitute in Arcetri (FI) and thanks INFN for hospitality and support. M.B.
is very grateful to the organizers and participants to the workshop “String
and M theory approaches to particle physics and cosmology” for creating
a very stimulating atmosphere. This work was supported in part by the
CNRS PICS no. 2530 and 3059, INTAS grant 03-516346, MIUR-COFIN
2003-023852, NATO PST.CLG.978785, the RTN grants MRTNCT- 2004-
503369, EU MRTN-CT-2004-512194, MRTN-CT-2004-005104 and by a Eu-
ropean Union Excellence Grant, MEXT-CT-2003-509661.
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