Scholarly article on topic 'String instantons, fluxes and moduli stabilization'

String instantons, fluxes and moduli stabilization Academic research paper on "Physical sciences"

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Nuclear Physics B
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{"String instantons" / S-duality / "Flux compactifications"}

Abstract of research paper on Physical sciences, author of scientific article — P.G. Cámara, E. Dudas, T. Maillard, G. Pradisi

Abstract We analyze a class of dual pairs of heterotic and type I models based on freely-acting Z 2 × Z 2 orbifolds in four dimensions. Using the adiabatic argument, it is possible to calculate non-perturbative contributions to the gauge coupling threshold corrections on the type I side by exploiting perturbative calculations on the heterotic side, without the drawbacks due to twisted moduli. The instanton effects can then be combined with closed-string fluxes to stabilize most of the moduli fields of the internal manifold, and also the dilaton, in a racetrack realization of the type I model.

Academic research paper on topic "String instantons, fluxes and moduli stabilization"

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Nuclear Physics B 795 (2008) 453-489

String instantons, fluxes and moduli stabilization

P.G. Cámara a-*, E. Dudasab, T. Maillarda, G. Pradisic

a Centre de Physique Théorique, 1 Ecole Polytechnique, F-91128 Palaiseau, France b LPT, 2 Bat. 210, Université de Paris-Sud, F-91405 Orsay, France c Dipartimento di Fisica, Université di Roma "Tor Vergata" and INFN — Sez. Roma II, Via della Ricerca Scientifica 1, 00133 Roma, Italy

Received 30 October 2007; accepted 27 November 2007

Available online 4 December 2007


We analyze a class of dual pairs of heterotic and type I models based on freely-acting Z2 x Z2 orbifolds in four dimensions. Using the adiabatic argument, it is possible to calculate non-perturbative contributions to the gauge coupling threshold corrections on the type I side by exploiting perturbative calculations on the heterotic side, without the drawbacks due to twisted moduli. The instanton effects can then be combined with closed-string fluxes to stabilize most of the moduli fields of the internal manifold, and also the dilaton, in a racetrack realization of the type I model. © 2007 Elsevier B.V. All rights reserved.

PACS: 11.25.Wx; 11.25.Sq; 11.25.Mj

Keywords: String instantons; S-duality; Flux compactifications

1. Introduction and conclusions

In recent years new ways to compute non-perturbative effects in string theory were developed, based on Euclidean ^-branes (E^-branes) wrapping various cycles of the internal manifold of string compactifications [1-7]. Some of the instanton effects have an interpretation in terms of gauge theory instantons, whereas others are stringy instanton effects whose gauge theory counter-

* Corresponding author.

E-mail address: (P.G. Cámara).

1 Unité mixte du CNRS, UMR 7644.

2 Unité mixte du CNRS, UMR 8627.

0550-3213/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2007.11.026

part is still under investigation. (For recent reviews on instanton effects in field and string theory, see, e.g., [8].) Whereas the former effects are responsible for the generation of non-perturbative superpotentials via gauge theory strong IR dynamics [9] and of moduli potentials satisfying various gauge invariance constraints [10], the latter could be responsible for generating Majorana neutrino masses or the ^-term in MSSM [4,5], as well as for inducing other interesting effects at low energy [7].

The purpose of the present paper is to present a class of examples based on freely-acting Z2 x Z2 orbifold models, that adds two new ingredients to the discussion, trying to go deeper into the non-perturbative effects analysis. The first new ingredient is the heterotic-type I duality [11], which exchanges perturbative and non-perturbative regimes. As is well known [12], it is possible to construct freely-acting dual pairs with N = 1 supersymmetry in four dimensions which preserve the S-duality structure. As we show explicitly here, the dual pairs can have a rich non-perturbative dynamics exhibiting both types of effects mentioned above. The heterotic-type I duality allows, for example, to obtain the exact E1 instantonic summations on the type I side for the non-perturbative corrections to the gauge couplings using the computation of perturbative threshold corrections on the heterotic side.3 Second, non-perturbative effects also play a potentially important role in addressing the moduli field stabilization issue. Closed string fluxes were invoked in recent years in the framework of type IIB and type IIA string compactifications, following the initial proposal of [14] to try to stabilize all moduli fields, including the dilaton. The combination of closed string fluxes and freely-acting orbifold actions has the obvious advantage of avoiding to deal with twisted-sector moduli fields, absent in our construction. We show that, besides the Ramond-Ramond (RR) three-form fluxes, also metric fluxes can be turned on in our freely-acting type I models, requiring new quantization conditions and the twisting of the coho-mology of the internal manifold. The low-energy effective description is equivalent to the original one, with the addition of a non-trivial superpotential. Moreover, our string constructions allow naturally racetrack models with dilaton stabilization [15]. We show how they can be combined with closed string fluxes and stringy instanton effects in order to stabilize most of the moduli fields of the internal manifold.

The plan of the paper is as follows. In Section 2 we discuss the geometric framework of the freely acting Z2 x Z2 orbifolds. In Section 3 we display the explicit type I descendants obtained by quotienting the orbifold with the geometric world-sheet parity operator. Besides some variations of the simplest class with orthogonal gauge groups, we also construct the corresponding heterotic duals in Section 4. In Section 5, we report the calculation of the threshold corrections to the gauge couplings both for the heterotic and for the type I models. The details of the calculations are reported in Appendices A-D. In particular, we verify that the moduli dependence of the non-perturbative corrections on the type I side is in agreement with the conjectured form [16]. In Section 6 we analyze the instanton contributions in the type I framework, that are combined with closed string fluxes in Section 7 in order to attain the stabilization of most of the moduli of the compactification manifold. In particular, in Section 7 we describe an example in which the dilaton can be also stabilized, due to a natural racetrack realization of the type I model in combination with closed metric and RR three-form fluxes.

3 See [13] for earlier work on instanton effects and heterotic-type I duality.

2. The freely-acting orbifold and its moduli

From the point of view of the target space, we take a T6 torus (y' = y' + 1) with vielbein vectors e' = e' M dyM and metric given by

ds2 = £ e',xe'». (2.1)

This has to be SL(6, Z) invariant. Therefore, performing a general rotation of the lattice vectors one may write a basis as follows4

e6 = R6 dy6, (2.2)

e5 = R5 (dy5 + a56 dy6), (2.3)

e4 = R4(dy4 + a45 dy5 + a46 dy6), (2.4)

e3 = R3 (dy3 + a34 dy4 + a35 dy5 + a36 dy6), (2.5)

e2 = R2 (dy2 + a23 dy3 + a24 dy4 + a25 dy5 + a26 dy6), (2.6)

e1 = R1 (dy1 + a12 dy2 + a13 dy3 + a14 dy4 + a15 dy5 + a16dy6). (2.7)

Modding by the orbifold action will break the SL(6, Z) symmetry to a smaller subgroup. We define the generators (g, f, h} of the Z2 x Z2 freely-acting orbifold as,

(y1 ,y2,y3,y4,y5,y6) - (y1 + 1/2,y2, -y3, -y4, -y5 + 1/2, -y6), (2.8)

(y1 ,y2,y3,y4,y5,y6) - (-y1 + 1/2, -y2,y3 + 1/2,y4, -y5, -y6), (2.9)

(y1 ,y2,y3,y4,y5,y6) - (-y1, -y2, -y3 + 1/2, -y4,y5 + 1/2,y6). (2.10)

Notice that these orbifold operations have no fixed points due to the shifts, hence they act freely (see, e.g., [17]). Moreover, for objects localized in the internal space, as will be the case for the E1' instantons to be discussed in Section 6, orbifold operations will generate inevitably instanton images. This has non-trivial consequences on the instanton spectra, as we shall see later on.

In order for the lattice vectors (2.2)-(2.7) to transform covariantly with respect to the orbifold action, it is required that

a45 = a46 = a35 = a36 = a23 = a24 = a25 = a26 = a^ = a14 = a15 = a^ = 0. (2.11) A basis of holomorphic vectors can thus be introduced in the form

Z1 = e1 + ie2 = Ri (dy1 + iU1 dy2), (2.12)

Z2 = e3 + ie4 = R3 (dy3 + iU2 dy4), (2.13)

Z3 = e5 + ie6 = R5(dy5 + iU3 dy6), (2.14) where we have defined

246 R 1 R 3 R 5

U1 = - ia12, U2 = R3 - ia34, U3 = R5 - ia56. (2.15)

4 We use the notation y', i = 1,..., 6, to denote the internal compact dimensions and x', i = 0,..., 3, for the non-

compact space-time dimensions.

Hence, the moduli space of the untwisted sector matches precisely the one of an ordinary Z2 x Z2, given by the three complex structure moduli, Ui, together with the three Kahler moduli, Ti, which result from the expansion of the complexified Kahler 2-form in a cohomology basis of even 2-forms,

Jc = e-* J + iC2 = T1 dy1 A dy2 + T2 dy3 A dy4 + T3 dy5 A dy6. (2.16)

Making use of (2.12)-(2.14), the real parts of the Kahler moduli can be seen to be

Re T1 = e-* R1R2, Re T2 = e-*R3R4, Re T3 = e-*R5R6. (2.17)

The effective theory contains also, as usual, the universal axion-dilaton modulus

S = e-*y[Ri + ic, (2.18)

where c is the universal axion. On the other hand, since there are no fixed points in the orbifold action, we expect the twisted sector to be trivial. We shall see in next section, from the exchange of massless modes in the vacuum amplitudes, that this is indeed the case. The internal space of the orbifold is therefore completely smooth and can be interpreted as a Calabi-Yau space with Hodge numbers (hn,h2i) = (3, 3). The corresponding type IIB string theory on this orbifold space has the standard left-right worldsheet involution QP as a symmetry, which we use, following [18,19], in order to construct type I freely-acting orbifolds.

3. Type I models: Vacuum energy and spectra

3.1. Type I with orthogonal gauge groups

We briefly summarize here some of the results of [18]. Following the original notation, the Z2 x Z2 orbifold generators of Eqs. (2.8)-(2.10) can be written as

g = (P1,-1, -P3), f = (-P1 ,P2,-1), h = (-1, -P2,P3), (3.1)

where Pi represents the momentum shift along the real direction y2'-1 of the ith torus. We consider the type I models obtained by gauging the type IIB string with QP, the standard worldsheet orientifold involution. The spectrum can be read from the one-loop amplitudes [20]. In particular, the torus partition function is5

. d t 1

t3In14 4

iToo + Tog + Toh + Tof

+ I Too + Tog - Toh - Tof |2(-1)m1 Ar + I Too - Tog + Toh - Tof |2(-1)m3 A3

& 2 U2

5 There is an overall normalization that is explicitly written in Appendix A. For other conventions concerning orien-

tifolds, see, e.g., the reviews [21].

+ ¡Too - Tog - Toh + Tof |2(-1)m2Ä2

I ,2 Anl

+ |Tgo + Tgg + Tgh + Tgf 1

& 2 &4

+ 1 Tgo + Tgg - Tgh - Tgf |2(-1)mi

"1 + 2

2 „«3 + 2

+ 1 Tho + Thg + Thh + Thf 1

& 2 &4

+ ¡Tho - Thg + Thh - Thf |2(-1)m3

& 2 &3

& 2 &3

+ | Tfo + Tfg + Tfh + Tff |2^"2 +2

& 2 &4

+ | Tfo - Tfg - Tfh + Tff |2(-1)m2 ^32+1

4n2 2n

& 2 &3

while the Klein bottle, annulus and Möbius strip amplitudes read in the direct (loop) channel respectively as

f id2 1 (Too + Tog + Toh + Tof ){P1P2P3 + (-1)m1 P1W2W3

+ W1(-1)m2 P2W3 + W1 W2(-1)m3 P3},

f dt 1 f 2

J t^M 7w(Too + Tog + Toh + Tof )P1 P2P3

+ gN(Too + Tog - Toh - Tof )(-1)m1 P1—2

+ ¿N(Too - Tog + Toh - T0f)(- 1)m3 P3 —T

+ fN (Too - Tog - Toh + Tof)(-1)m2 P2&- }'

f dt IN \ „

J tn 1 (Too + Tog + Toh + Tof )P1 P2P3

+ (Too + Tog - Toh - Tof)(-1)m1 ^^ + (Too - Tog + Toh - Tof)(-1)m3 P3

& 22 4n2

+ (Too - Tog - Toh + Tof)(-1)m2 P2-4

4n 2 &2 (3.5)

Some comments on the notation are to be made. In the torus amplitude, F is the fundamental domain and the Ai are the lattice sums for the three compact tori, whereas the shorthand notation (-1)miAni+1/2 indicates a sum with the insertion of (-1)mi along the momentum in y2'-1, with the corresponding winding number shifted by 1 /2. Pi and Wi in (3.3)-(3.5) are respectively the momentum and winding sums for the three two-dimensional tori. More concretely, using for the geometric moduli the conventions of the previous section, one has6

Pi = exp

nt . . 2

|m - iUml2

(-1)miPi =J2 (-1)m exp

(Re Ti)(Re Ut)

(Re Ti)(Re Ui)

\m' - iUim|2

Moreover, in (3.5) hatted modular functions define a correct basis under the P transformation extracting a suitable overall phase [20]. Indeed, the moduli of the double-covering tori are t = (it/2 + 1/2) for the Mobius-strip amplitude, t = 2it for the Klein-bottle amplitude and t = it/2 for the annulus amplitude. In Appendix B we give the definition of the characters used in Eqs. (3.3)-(3.5) in terms of [SO(2)]4 characters.

It is worth to analyze the effects of the freely-acting operation on the geometry of the models. In general, Z2 x Z2 orientifolds contain O9-planes and three sets of O5' -planes defined as the fixed tori of the operations QP o g, QP o f, QP o h, each wrapping one of the three internal tori T' . In our freely-acting orbifold case, the overall O5' -plane charges are zero and the O5' -planes couple only to massive (odd-windings) states. A geometric picture of this fact can be obtained T-dualizing the two directions the O5' planes wrap, so that they become O3' -planes. In this way, the freely acting operation replaces the O3i,- planes by (O3i,+-O3i,-) pairs, separated by half the lattice spacing in the coordinate affected by the free action. Since there are no global background charges from O5' -planes, the model contains only background D9 branes. Finally, the Chan-Paton D9 charges are defined as,

In = no + ng + nh + nf, gN = no + ng - nh - nf,

hN = no - ng + nh - nf, fN = no - ng - nh + nf, (3.8)

with IN = 32 fixed by the tadpole cancellation condition. The massless spectrum has N = 1 supersymmetry. The gauge group is SO(no) ® SO(ng) ® SO(nh) ® SO(n f ), with chiral multiplets in the bifundamental representations

(no, ng, 1, 1) + (no, 1, nf, 1) + (no, 1, 1, nh) + (1, ng, nf, 1)

+ (1, ng, 1, nh) + (1,1, nf, nh). (3.9)

The existence of four different Chan-Paton charges can be traced to the various consistent actions of the orbifold group on the Chan-Paton space or, alternatively, to the number of independent sectors of the chiral conformal field theory. It can be useful for the reader to make a connection with the alternative notation of [23]. The original Chan-Paton charges can be grouped into a 32 x 32 matrix X. In this Chan-Paton matrix space, the three orbifold operations g, f and h act via matrices Yg,Yf,yh which, correspondingly to (3.8), are given by

Yg = (Ino,Ing, -IHf, -Inh),

6 In what follows we set the string tension a' = 1/2.

Yf — (Ino, Ing ,Inf, Ink X

Yh — (Ino, -Ins, -Inf,Ink), (3.10)

where Ino denote the identity matrix in the no x no block diagonal Chan-Paton matrix, and the same for the other multiplicities n. For ng — nh — nf — 0 one recovers a pure SO(32) SYM with no extra multiplets, a theory where gaugino condensation is expected to arise. Finally, let us notice that even if perturbatively no,ng,nf ,nh can be arbitrary positive integers subject only to the tadpole condition no + ng + nf + nh — 32, non-perturbative consistency asks all of them to be even integers.

3.2. Type I racetrack model

In a variation of the previous SO(32) model, we may add a discrete deformation along one of the unshifted directions, similar to a Wilson line A2 — (e2nta) along y2, with a — (0p, 1/232-p) and breaking SO(32) ^ SO(p) ® SO(32 - p). The annulus and Mobius amplitudes, (3.4) and (3.5), get correspondingly modified to the following expressions:

A — / i^H ^ + q2^Pm1 + 2pqpm\ +1 ]pmi P2P3(Too + Tog + Toh + f

n g \ [ (p2 + q2) pm """ 2pqpm+aJ pmip 2p 3(o -r o ■

+ (p2 + q2) [(-1)m2 P2(Too - Tog - Toh + T0f)

+ (-1)m3 P3(Too - Tog + Toh - Tof)}^

+ (-1)m1 [(p2 + q2)Pm[ + 2pqPm[+ 1 ]Pmi (Too + Tog - Toh - f [, (3.11)

M —--— PxP2P3(Too + Tog + Toh + Tof)

+ (-1)m1 Pl(Too + Tog - Toh - Tof)—2

m1 (T T T T )

i I \ (oo + (.og (■oh (.of ) 2

)m2P 4

+ (-1)m2 P2(Too - Tog - Toh + Tof )-2

+ (-1)m3P3(Too - Tog + Toh - Tof)4n2 (3.12)

As mentioned, — p + q — 32,

Pm'j Pmj = P1, and (3.13)

Pm1 + i Pm1 ^ I] exp

jm'- ¡^im + 1/2|2

(Re 7! )(Re tfO


Hence, the resulting SO(p) ® SO(32 - p) gauge group is accompanied by a pure N — 1 SYM theory on both factors, leading to a racetrack scenario with two gaugino condensates. Indeed, in the four-dimensional effective supergravity Lagrangian, the tree-level gauge kinetic functions on

the two stacks of D9 branes are equal,

fsO(p) = fsO(q) = S, (3.15)

where S is the universal dilaton-axion chiral multiplet. Gaugino condensation on both stacks then generates the non-perturbative superpotential

Wnp = Ap)e-aPS + A(q)e-aqS, (3.16)

where A(p} = (p - 2) exp(2nik/(p - 2)) and A() = (q - 2) exp(2nil/(q - 2)), with k = 1,...,p — 2 and l = 1,...,q - 2, provide the requested different phases of the SYM vacua [24]. Moreover, ap = 2/(p - 2) (aq = 2/(q - 2)) is related to the one-loop beta function of the SO(p) (SO(q)) SYM gauge factor. In addition to the massless states, the model contains massive states, in particular a massive vector multiplet in the (p, q) bifundamental representation, with a lowest mass of the order of the compactification scale Mc ~ 1/R. Since the four-dimensional effective theory is valid anyway below Mc, these states are heavy and their effects on the low-energy physics can be encoded in threshold effects which we shall compute later on.

An interesting question is the geometrical interpretation of the present model.7 The natural interpretation is in terms of a Wilson line breaking of the SO(p) ® SO(32 - p) model. The absence of scalars describing positions of the branes corresponding to each SO factor indicates that the corresponding branes are fractional and, as such, cannot move outside the fixed points. However, by giving vev's to the scalars in bifundamentals, one converts fractional branes into regular branes. The resulting gauge group is SO(2P) ® SO(16 - P), where the first factor comes from the branes sitting at the fixed point, while the second factor describe brane pairs in the bulk having scalars in the symmetric representation corresponding to their positions. Moving the bulk branes to another fixed-point, one gets, as usual, an enhancement of the gauge group to SO(2P)® SO(32 - 2P).

3.3. Type I with unitary groups

It is interesting to analyze the non-perturbative dynamics of the gauge theory on the D9 branes in the case of an orbifold action on the Chan-Paton space that produces unitary gauge groups. This can be done in a very simple way by choosing a different Chan-Paton assignment compared to (3.8). Consider the same cylinder amplitude (3.4) equipped with the following parametrization of the Chan-Paton charges:

In = n + n + m + m, gN = n + n - m - m,

fy = i(n -n + m - m), Hn = i(n -n - m + m). (3.17)

The Möbius amplitude has to be changed for consistency into

f dt IN[ ^

m = -j tv~8~\(r°0 + T°g +Xoh + T°f)P1 P2P3

+ (Too + Tog - T°h - T°f)(-1)m1 P1 —

7 E.D. is grateful to C. Angelantonj and M. Bianchi for illuminating discussions on this and the other string models presented in the present paper.

- (Too - Tog + Toh - T0f)(-1)mPs-^

- (Too - Tog - Toh + Tof)(-1)m2P2%(3.18)

where the changes of sign in the D9-O52 and D9-O53 propagation, needed to enforce the unitary projection, are interpreted as discrete Wilson lines on the D9 branes in the last two torii [20]. The massless open string amplitudes,

Ao + Mo = (nn + mm)Too +

n(n - 1) + n(n - 1) + m(m - 1) + m(m - 1)

2 2 2 2 + (nm +n m)t0f + (nm + nm)toh, (3.19)

exhibit the spectrum of an N = 1 supersymmetric U(n) ® U(m) theory, with n + m = 16 due to the (D9/O9) RR tadpole cancellation condition. Matter fields fall into massless chiral multiplets in the representations

n(n — 1) n(n — 1) \ / m(m — 1) m(m — 1) ( ) + —--, 1 + 1, —(-- + —--

2 2 ) V 2

+ (n, m) + (ñ, m) + (n, m) + (n, m). (3.20)

Notice that the choice m = 0 with a gauge group U(16), in contrast to the SO(32) case, is not pure SYM, since it contains massless chiral multiplets in the (120 + 120) representation.

The gauge theory on D9 branes is not really supersymmetric QCD with flavors in the fundamental and antifundamental representation, whose non-perturbative dynamics is known with great accuracy [9]. One way to get a more interesting example is the following. Moving p D9 branes out of the total 16 to a different orientifold fixed point not affected by the shift, one gets a gauge group U(n) ® U(m) ® U(p), with n + m + p = 16. Strings stretched between the p D9 branes and the remaining n + m are massive, and therefore they disappear from the effective low-energy gauge theory, whereas the U(n) ® U(m) gauge sector has the massless spectrum displayed in (3.20). Choosing n = 3 and m = 1, a gauge group SU(3) ® U(1)2 results, together with a factor U(12) decoupled from it. Using the fact that the antisymmetric representation of SU(3) coincides with the antifundamental 3, one ends up with an SQCD theory with gauge group SU(3) and Nf = 3 flavors of quarks-antiquarks. This is the regime Nc = Nf = N described in [25], where the composite mesons M = QQ and baryons (antibaryons) B = Q \ ■■■ Qn (B = QQ i ■■■ QQ n) have a quantum-deformed moduli space such that

detM - BB = A2N, (3.21)

where A2N = exp(-8^2/g2) is the dynamical scale of the SU(3) gauge theory. As a consequence, the deformation in (3.21) originates only from the one-instanton contribution.

4. Heterotic dual models

4.1. Heterotic SO(32) model

Due to the freely-acting nature of the type I orbifold, according to the adiabatic argument [12] the S-duality between the type I and the SO(32) heterotic string is expected to be preserved.

In this section we explicitly construct the heterotic S-dual of the SO(32) type I model.8 The natural guess is to use the same freely-acting orbifold generators with a trivial action on the internal gauge degrees of freedom, consistently with the fact that in its type I dual the action on the Chan-Paton factors is trivial as well. There is however one subtlety, already encountered in similar situations and explained in other examples in [12]. Modular invariance forces us to change the geometric freely-orbifold actions (2.8)-(2.10) into a non-geometric one. Let us consider for simplicity one circle of radius R and one of the geometric shift in (2.8)-(2.10)

X ^ X + nR. (4.1)

Our claim is that its S-dual on the heterotic side is the non-geometric action9

nR na' nR na'

XL ^ ^ + -2- + ^ XR ^ XR + - - ^ (4.2)

In order to prove this claim, we use the fermionic formulation of the sixteen-dimensional het-erotic gauge lattice, with 16 complex fermions. Guided by the type I dual model, we take a trivial orbifold action on the 16 gauge fermions. The adiabatic argument of [12] allows identification of the orbifold action only in the large radius limit, where the shift (4.2) is indistinguishable from (4.1). In the twisted sector of the theory, the masses of the lattice states (m, n) are shifted according to

(m,n) ^ (m + s\,n + ), (4.3)

where ^s^) = (1 /2, 0) for (4.1) and (sx,sj) = (1 /2,1 /2) for (4.2). The Virasoro generators of the left and right CFT's are

Lo = N + 2 x|- -1 - ^J + 2 x ( ^ + 1

12 24/ \24 12

Lo = N +10 x(-^ + 2 x 214, (4.4)

where N (N) contains the oscillator contributions whereas the other terms are the zero-point energy in the NS sector from the spacetime and the gauge coordinates. Level-matching in the twisted sector is then

L0 - L0 = N - N + ^ = —(m + sx)(n + s[) (mod 1). (4.5)

This is possible only for (sx,s^) = (1 /2, 1 /2) which therefore fixes (4.2) to be the correct choice. The S-dual of the type I freely-acting SO(32) is then defined by the modular invariant torus amplitude

. d2T 1

n2n2 4

(Too + Tog + Toh + Tof)AiA2A3

+ (Too + Tog - Toh - Tof )(-1)m1+n1 Ax

Û 2 U2

8 We are grateful to M. Bianchi and E. Kiritsis for helpful discussions and comments on this point.

9 As shown recently [26], such asymmetric shifts in type I models are consistent only if they act in an even number of coordinates.

.m2 + 2 ,«2 + 2 + (Tfo + Tfg + Tfh + r/f)^2

& 2 &4

(tgo + tgg Tgh Tgf)( 1)

m\+n\ j^m1 + 2 ,n1 + 2

- (Tho - Thg + Thh - Tf )(-1)m3+n3 ^3+2 ,n3+2

- (Tfo - Tfg - Tfh+Tff )(-i)m2+n2 2 ,n2+2

4n2 2n

& 2 &3

X (032 + S32). (4.6)

Indeed, the massless spectrum matches perfectly with its type I counterpart. Compared to its type I S-dual cousin, the heterotic model has the same spectrum for the Kaluza-Klein modes, whereas it has a different spectrum for the winding modes. This is precisely what is expected from S-duality [11], which maps KK states into KK states, whereas it maps perturbative winding states into non-perturbative states in the S-dual theory.

4.2. Dual heterotic models with orthogonal gauge groups

In the fermionic formulation, the dual of the type I SO(no) ® SO(ng) ® SO(nh) ® SO(nf), n0 + ng + nf + nh = 32 can be constructed by splitting the 16 complex fermions of the gauge lattice into n0/2 + ng/2 + nf/2 + nh/2 groups. We then embed the orbifold action into the gauge lattice as shown in Table 1.

Level matching in this case can be readily worked out with the result, in the g, f and h twisted sectors respectively

Lo - Lo = N - N- 4 +

5 no + ng ,

g - (m1 + S1)(«1 + ) (mod 1),

5 no + nf ,

Lo - Lo = N - N - - + o f - (m2 + siKni + 4) (mod 1),

4 16 2

5 no + nh ,

Lo - Lo = N - N - - + o h - (m3 + s3)(n3 + s3) (mod 1). 4 16

The various possibilities are then as follows

• no + ng = 8 (mod 8) ^ s1 = s'1 = 1 /2,

• no + ng = 4 (mod 8) ^ s1 = 1 /2, s'1 = 0,

Table 1

Orbifold actions in the gauge degrees of freedom in the fermionic formulation

Orb. actions SO(no) SO(ng) SO(nf) SO(nh)

g + + - -

f + - + -

h + - - +

and similarly for the other pairs no + nf, no + nh. It is interesting to notice the restrictions on the rank of the gauge group. While the restriction on the even SO(2n) gauge factors was expected from the beginning, the above conditions are actually stronger.

Let us take a closer look to the particular case of the gauge group SO(p) ® SO(q) with p + q — 32, in order to better understand this point. The corresponding setting is n o — p, ng — q and nf — nh — 0. Level matching in the f and h twisted sectors reads

Lo - Lo — N - N -- + JL — -(m + si)(n + s2) (mod 1), (4.8)

which leads to the following options:

• p — 8 (mod 8) ■

• p — 4 (mod 8) ■

si — S2 — 1 /2, si — 1/2, S2 — 0.

Surprisingly, we do not find solutions for p — 2 (mod 2). We can only speculate that, perhaps, a more subtle orbifold actions on the gauge lattice and/or the introduction of discrete Wilson lines could help in finding the p — 2 models, which the dual type I models suggest that have to exist.

For the first case, p — 8,16,24, it is convenient, in the fermionic formulation of the gauge degrees of freedom, to define the following characters

Xo — OpOq + CpCq, Xv — VpVq + SpSq, Xs — OpCq + CpOq, Xc — VpSq + SpVq.

The complete partition function of the heterotic model is then

T|n2n2 4

(Too + Tog + Toh + Tof

+ (Too + Tog - Toh - Tof)(-1)m1+n1 A1

ft 2 U2

, s ,m1 + 2 ,n1 + A

+ (Tgo + Tgg + Tgh + Tgf)A1 2 2

& 2 U4

+ (Tgo + Tgg - Tgh - Tgf)(-1)"

"1+2 4n2 2-

(Xo + Xv)

+ [(Too - Tog + Toh - Tof)(-1)'

+ (Too - Tog - Toh + T0f )(-1)m2+n2 A2]

(Xo - Xv)

r, , .M3 +1 ,«3 +1

+ [(The + Thg + Thh + Thf)A3 2 2

/ \ Am2 + 2,n2 + 2 1

+ (Tfo + Tfg + Tfh + Tff)A2 2 2 J

ft 2 ft4

(Xs + Xc)

- (-l)q/8[(Tho - Thg + Thh - Thf)(-\)m3+n3Arm3+1 ,n3+1

+ (Tfo - Tfg - Tfh + Tff)(-Dm2+n2Am22+2,n2+ 2]

(Xs - Xc)


As for the SO(32) model, the whole KK spectrum precisely match the corresponding one on the type I S-dual side, whereas the massive winding states and the massive twisted spectra are, as expected, quite different. On the other hand, for the second case p = 4, 12, 20, the correct characters are

Xo = OpOq + CpCq, Xv = VpVq + SpSq, Xs = VpCq + SpOq, Xc = OpSq + CpVq. The complete partition function is now d2T 1

T^n2^2 4

(Too + Tog + Toh + Tof)A1 A2A3

+ (Too + Tog - Toh - Tof)(-1)m1+n1 A1

, s .mj + 2,«1 + 2

+ (Tgo + Tgg + Tgh + Tgf)A1 2 2

+ (Tgo + Tgg - Tgh - Tgf)(-1)m1+n1 A + [(Too - Tog + Toh - Tof)(-1)m3 A3 + (Too - Tog - Toh + Tof)(-l)m2 A2]

ft 2 U2

ft 2 ft4

n1 + 2 4n2 2-

(Xo + Xv)

[, s .m-3,n-3 + 2

+ [(Tho + Thg + Thh + Thf)A3

, / , , , \ .m2,n2 + 2-+ (Tfo + Tfg + Tfh + Tff)A2

ft2 ft2

(Xo - Xv)

(Xs + Xc)

- (-r)(p+4)/8[(Tho - Thg + Thh - Thf )(-1)m3a223,«3+2 + (Tfo - Tfg - Tfh + Tff)(-1)m2a222,«2+1 ]

(Xs - Xc)



It should be noticed that while the KK spectra are actually the same for the two cases p = 4 and p = 8 (mod 8), they are very different in the massive winding sector, in perfect agreement with the modular invariance constraints (4.7).

We expect that appropriate orbifold action in the sixteen-dimensional gauge lattice will also produce the S-dual of the type I racetrack and of the unitary gauge group cases, discussed in the

previous sections. The required action, however, cannot correspond to a standard Wilson line in the adjoint of the gauge group, but rather to a non-diagonal action in the Cartan basis, like the ones considered in [27].

5. Threshold corrections to the gauge couplings

In this section we perform the one-loop calculation of the threshold corrections to the gauge couplings of some of the models described in the previous sections. The effective field theory quantities can be then easily extracted from the one-loop computation. The threshold correction A2 is generically written as

1-loop 8,

8. with

4n 2 2"

+ ¿2,a, (5.1)

^2,a = J — Ba(T) (5.2)

for the heterotic string, and

A2,a = J — Ba(t) (5.3) o

for the type I string. In these expressions, Ba flows in the infrared to

ba = -3Ta(G) + J2 Ta(r), (5.4)

the one-loop beta function for the gauge group factor Ga, with r running over the gauge group representations with Dynkin index Ta(r). From the one-loop expression of the gauge coupling it is possible to extract [30] the holomorphic gauge couplings fa(Mi), where Mi denote here collectively the moduli chiral (super)fields, using the relation [31]

4n2 ba Ml ca Ta(G) 2, 2N

= Re fa + b^ log ML + JLk + JiG ln ga"2(M2)

gl(v2) ja 4 M2 4

-J2 —¡T lndet Zr(M)2, (5.5)

where K is the Kahler potential, Zr is the wave-function normalization matrix for the matter fields and ca = Y,r Ta(r) - Ta(G). With this definition, the holomorphic non-perturbative scale Aa of an asymptotically-free gauge theory (ba < 0) is given by

Aa = MPe l baI. (5.6)

5.1. Type I SO (no) ® SO(ng) ® SO(n f) ® SO(nh) model

For the computation of threshold corrections to the gauge couplings in the freely-acting type I model with orthogonal gauge groups, we make use of the background field method [28-30].

Therefore, we introduce a magnetic field along two of the spatial non-compact directions, say F23 = BQ.ln the weak field limit, the one-loop vacuum energy can be expanded in powers of B, providing

A(B) = Ao + ■

A2 + ■■

For supersymmetric vacua A0 = 0, and the quadratic term accounts exactly for the threshold corrections in Eq. (5.1).

In the presence of F23, the oscillator modes along the non-compact complex plane x2 + ix3 get shifted by an amount e such that

ne = arctan(nqLB) + arctan(nqRB) ~ n(qL + qR)B + O(B3),

where qL and qR are the eigenvalues of the gauge group generator Q, acting on the Chan-Paton states localized at the two endpoints of the open strings. In the vacuum energy, the contribution of the non-compact bosons and fermions gets replaced by

^(0|T) -a(re\r)


for a = 2, 3, 4

in the annulus and Mobius amplitudes. In addition, the momentum operator along the non-compact dimensions becomes,

-(po)2 + (pi )2 + (2n + l)e + 2e^23,


where ^23 is the spin operator in the (23) direction, while n is an integer that labels the Landau levels. The supertrace operator becomes now

STr ^^^^^+qR)B ( d2p

bos ferm



where (qL + qR)B/2n is the density of the Landau levels and the integral is performed only over the momenta in the non-compact directions x0 and x1.

The details of the computation can be found in Appendix C.l. Collecting the results obtained there, and assuming Q to be in a U(1) inside SO(no), SO(ng), SO(nf) or SO(nh), the moduli dependent threshold corrections for the respective gauge couplings can be written as follows,

A2,o = -^(Q2)

(2 - gtf) n Re U1 + log

(Re U1)(Re T1 )fz2

-4 (2ÏU1)

+ (2 - fN)\n Re U2 + log

+ (2 - hN)[n Re U3 + log

(Re U2)(Re Tzf

(Re U3)(Re T?)^2

-4 -4 (2iU2) n3 -2-

-4 (2iU3) n3 -2n

A2,g = -4Tr (Q2)

(2 - gNM nReU1 + log

+ (2 + fN)\n Re U2 + log

(Re U2)(Re Tz)z

(Re U1)(Re T1 )f

n3 -2-

(2iU 1 )



+ (2 + kN){n Re U3 + log

(Re Us)(Re

-4 (2iUs) n3


a2j = -iTr(e2)

(2 + gN)ln Re Ui + log

(Re Ui)(Re Ti)^2


+ (2 - /N^n Re U2 + log + (2 + kN)(n ReU3 + log

(Re U2)(Re Tz)^2

(Re Us)(Re T3)^

-4 -4 (2iU2) n3 -2-

-4 (2iU3) n3 -2-


A2k = -4T< Q2)

(2 + gN) n Re Ui + log

(Re Ui)(Re Ti)/x2

-4 (2iUi)

+ (2 + /N)[n Re U2 + log

+ (2 - kN)[n Re U3 + log

(Re U2)(Re T2

(Re Us)(Re T3K

-4 -4 (2iU2) n3 -2-

-4 (2iU3) n3 -2-


The ¡3-function coefficients can also be extracted in the form

bo = -[3(no - 2) - (n/ + ng + nk)], bg = -[3(ng - 2) - (n/ + no + nk)], b/ = -[3(n/ - 2) - (no + ng + nk)], bk = -[3(nk - 2) - (n/ + ng + no)], and, using the definition (5.5), the holomorphic one-loop gauge kinetic functions are then


/o = * + 2

(2 - gN) log---(2iUi) + (2 - /n) log 3 (2iU2)

+ (2 - kN) log

gnVil2n3 -4


enUs |2n3


/g = *

(2 - gN) log U9 3 (2iUi) + (2 + /n) log ^ 3 (2iU2)



+ (2 + kN) log

enU-3 |2n3


// = * + 2

(2 + gN) log -n-/^ (2iu0 + (2 - /n) log-Jks (2lU2)

+ (2 + kN) log

enU'3 |2n3


/k = * +

(2 + gN) log 3 (2iUi) + (2 + /n) log 3 (2iU2)

+ (2 - kN) log

gnVil2n3 -4


enU-3 l2n3



It is very important to stress the linear dependence of the above threshold corrections on the (n Re Ui) factors. Indeed, the presence of such terms in a loop contribution may seem surprising.

However, expanding the factor #4n-3, it can be realized that this term exactly cancels the contributions coming from the factor q1/24 contained in the n-function. Thus, the total dependence on the moduli of the threshold corrections turns out to be exclusively of logarithmic form. This phenomenon can be physically understood making the observation that, beyond the Kaluza-Klein scale, N = 4 supersymmetry is effectively recovered. Therefore, in the large volume limit only logarithmic corrections in the moduli should be present. The price one has to pay is that modular invariance in the target space is lost, as evident from the above expressions. The breaking of modular invariance in the target space by the shift Z2 x Z2 orbifold is very different from what happens in the ordinary Z2 x Z2 case where, beyond the Kaluza-Klein scale, the effective supersymmetry for each sector is still N = 2. The threshold corrections in that case turn out to be proportional to (ReU)log \n(iU)\4. Therefore, they preserve modular invariance, but have a non-logarithmic dependence on the moduli, due to the term q1/24 inside the n-function.

5.2. Type I racetrack model

The details of the calculation can be found again in Appendix C.2. Using the background field method, the moduli dependent part of the gauge coupling threshold corrections is given by

Ai,P —^TrQ2 )

(2 - p)J2(nReUj + log j

(Re Uj)(Re Tj

-4 (2iUj)

+ q log

(2iU i )

(4iU i )

+ n Re Ui )


together with a similar expression for the SO(q) factor, with the obvious replacements. The corresponding j-function coefficients of the SO(p) and SO(q) gauge group factors are

bp = -3(p - 2), bq = -3(q - 2), and the one-loop holomorphic gauge functions read

2- p fp =CI p

fp = S + log-TUT^T(2iUi) - q

2 "6 enUi/2n3 '

2 - q -4

fq = S I-r^ log (2iUi) -


log ~nU-J2^(2U) - log e^3 (4iUi)

enui n

log (2lUi) - log (4lUi)

2 ^ fe enUi/2n3 ~~ "" 2 i=i 1

The non-perturbative superpotential can be written, in analogy with (3.i6),

Wnp = Ap(Ui)e~apS + Aq(Ui)e~aqS,


q - 2 '

Il e-nUi/2 Î4 (2iUi) .i=i n

I1 e-nUi/2 n (2iUi) L i=i n

enUi/2 (2iUi)n- (4iUi )

enUi/2 (2iUi)^ (TiUi )

n 3 -4

. (5.2i)

a„ =

5.3. Heterotic SO(32) model

For the heterotic string, several procedures are available in literature to extract the threshold corrections [32-34]. The general expression for the threshold corrections to the gauge couplings, valid in the DR renormalization scheme, is given by

, d2T i 1 A2,a =1 — — -2 E 9T 1 a,p=0,1/2


where Qa is the charge operator of the gauge group Ga, and C^} is the internal six-dimensional partition function, which, for the particular case of the SO(32) model, can be read from (4.6). As noticed in [33], only the N = 2 sectors of the theory contribute to the moduli dependent part of this expression.

Again, the details of the computation are relegated to Appendix C.3. The expression for the gauge threshold corrections of the heterotic SO(32) model is

72 3 V2T

A2 = -¿1 [(-Dmi - ^ + ^ + 2

TT i = 1

- (-1)mi +n ZZmi + 2n + 2E4(E2E - EE6) ,


where E2n are the Eisenstein series (given explicitly in Appendix D), and the three toroidal lattice sums, ZZi = | n | 4Ai, read

E (-1)



2nTi det(A) -

n(Re Ti)

T2(Re Ui)

(1 iUi)A

A = ( n1 + § £1 +h

n2 £2



(-1)mi +niZ= Z i

\mi +ni Zmi + 2 ,ni + 2 _

(-1)mi +ni ZT

~mi + 2 ,ni + 2 ~

Z; 2 2 = Z i

Notice that labels the three N = 2 sectors associated to the ith 2-torus, i = 1, 2, 3. Although the full expression (5.23) is worldsheet modular invariant, each of these N = 2 sectors is not worldsheet modular invariant by itself, contrary to what happens in orbifolds with a trivial action on the winding modes.

In the large volume limit, Re Ti > 1, the winding modes decouple and only Kaluza-Klein modes with small q contribute to the integral. In that case, the threshold correction receives contributions only from A matrices with zero determinant in the sector (h,g) = (1, 0), in such a

way that (5.23) becomes

Re Ti »1"

—п Re Ui — log

(Re Ui)(Re Ti)



matching exactly the threshold corrections for the dual type I SO(32) model.

For arbitrary Ti, however, the winding modes do not decouple from the low energy physics and corrections due to worldsheet instantons appear:

Л2 " ^2|ReT.»1(Ui) + Ainsi(Ui,Ti).


They correspond to E1 instanton contributions in the dual type I SO(32) model, and therefore are absent in (5.17).

For example, consider the q ^ 0 contributions to Ainst of winding modes in the sector (h,g) = (1,0). These result in

insi L 0 ]

"—В—1)" logro —

—2nnTi )

+ c.c.


Since the axionic part of Ti in type I corresponds to components of the RR 2-form, C2, it is natural to expect that these contributions come from E1 instantons wrapping n times the (1, 1)-cycle associated to Ti .Notice that the dependence on Ti perfectly agrees with general arguments in [16] for the mirror type IIA picture.

The corresponding holomorphic gauge kinetic function reads



(2iUi) — 2^2(—!)n log(1 — e—2nnTi)


where the dots denote further contributions from Ainst. Hence, the non-perturbative superpotential generated by gaugino condensation receives an extra dependence in the Kahler moduli,

WnP = A(Ui,Ti)e

-, A = П

15 ' 1 1

e—nUi/2 (2iUi)H

1 — e—4n(n+1/2)Ti 1 _ e—4nnTi



Unfortunately, a complete analytic evaluation of the non-perturbative corrections in (5.23) is subtle, as worldsheet modular invariance mix orbits within different N = 2 sectors and the unfolding techniques of [13,33] cannot be applied straightforwardly to this case.

6. Euclidean brane instantons in the type I freely-acting SO(32) model

The model has two types of BPS brane instantons, denoted as E5 and E1. The E5 branes are interpreted as gauge instantons within the four-dimensional gauge theory on the compactifled D9 branes and map, in the heterotic dual, to non-perturbative euclidean NS5 corrections. The E1;

10 We have neglected an extra term coming from the non-holomorphic regularization of Ë2> which in the dual type I

side would presumably correspond to contact contributions in two-loop open string diagrams.

Table 2

Op-planes and D9/Ep branes present in the type I models. A - denotes a coordinate parallel to the Op-plane/Dp-brane, while a • represents an orthogonal coordinate

Coord. 0123456789

D9/O9 - -- -- -- -- -

051 - - - - - - ••••

052 - - - - ## - - ##

053 - - - - •••• - -

E1i •••• - - ••••

EI2 •••••• - - ••

EI3 •••••••• - -

E5 •••• - - - - - -

type I instantons wrapping the internal torus Ti, instead, are stringy instantons from the gauge theory perspective and are responsible, in the heterotic dual, for the perturbative world-sheet instantons effects, that we have computed in Section 5.ii

The configurations of the various Op planes and (D/E)p branes in the models are pictorially provided in Table 2.

6.1. E5 instantons

A convenient way to describe the E5 instantons is to write the partition functions coming from the cylinder amplitudes (for E5-E5 and E5-D9 strings) and the Mobius amplitudes (for E5-O9 and E5-O5i ). In order to extract the spectrum, it is useful to express the result using the subgroup of SO(iO) involved in a covariant description, namely SO(4) x SO(2)3 in our present case. Considering p coincident E5 instantons, one gets

^ c^r a 1 c^r a "l3

A p2 7dt 4;]¡ppp-[;i

AE5-E5 = --2> caj-—\PiP2P3

i6 j t n2 „ n [

+ 2 / , ap 1 * i * 2 * 3 9

i6 J t n2 a n [ n9

+ (-i)mi Pi + (-i)m2 P2

mi p •-p^ .-j-i/2J _

' 5 ,o2

n5 -22

-[ p-i/2i-[ a i-1 p+i/214n2

n5 -22

- [ P+i/21-[ p 1/2 1-[ a 1 4n2

+ (-i)m3 P3 Lj-i/2J Lj J (6.i)

n 5 -22

p f dt 4n2^ 4p+i/J Hp -W n \ppp-[ap] 7T ——T n y, caj-2--¡~~n pip2p3-9-

i60 t 72 ^ n2 -ml n9

11 Notice that generically there will be also massless modes stretching between both kind of instantons, E5 and E1i. From the gauge theory perspective, these modes are presumably responsible of the E1 instanton corrections to the Veneziano-Yankielowicz superpotential, discussed at the end of Section 5.3.

(—1)m1 Pi


ß 1/2 -

+ (-1)m2 P2


a ]#[ß]#[ a


+ (-1)m3 P3

+V] 4ß]

ß-1/2 J

where ca/e are the usual GSO projection coefficients. In terms of covariant SO(4) x SO(2)3 characters, the massless instanton zero-modes content results


E5—O9 '

p(p + 1) p(p — 1) P(P -(V4O2O2O2 — C4C2C2C2) — --S4S2S2 52. (6.3)

From a four-dimensional perspective, V4O2O2O describe vector zero-modes, aM, while C4C2C2C2 is a spinor Ma' , where a denotes an SO(4) spinor index of positive chirality,

whereas (---) denote the SO(2)3 internal chiralities. Analogously, S4S2S2S2 are fermionic

zero modes ka' . Notice that in the one-instanton p = 1 sector, k is projected out by the orientifold projection.

The charged instanton spectrum is obtained from strings stretched between the E5 instanton and the D9 background branes. The corresponding cylinder amplitude is

AE5-D9 =

Np f dt n2 2 v^

-JT^2 n ç

# [ a+l/2 ]2


( —1)m1 P1

¿[ßMß+W 4ß—1/2]

+ (—1)m2 P2-

U] HßMß+rn]


+ (—1)m3 P3-

#UV] 4ß—l/J *[ß]

1 4n2 n2 #2

The massless states are described by the contributions aE05)-D9 = Np(S4Ü2O2O2 - O4C2C2C2).

In particular, the state S4O2O2O2, coming from the NS sector, has a spinorial SO(4) index ma, whereas O4C2C2C2, coming from the R sector, is an SO(4) scalar with a spinorial SO(6) index or, which is the same, a fundamental SU(4) index \xa.

6.2. E1 instantons

The case of the E1 instantons is more subtle. Indeed, they wrap one internal torus while they are orthogonal to the two remaining ones, thus feeling the non-trivial effects of the freely-acting operations. The explicit discussion can be limited to the case of the E11 instantons, the other two cases E12,3 being obviously completely similar. It is useful to separately discuss the two distinct possibilities:

(i) the E1i instantons sit at one of the fixed points (tori) of the g orbifold generator in the

y1, ...,y6 directions;

(ii) the E11 instantons are located off the fixed points (tori) of the g orbifold generator in the y1 ,...,y6 directions.

It is worth to stress that, strictly speaking, the freely action g has no fixed tori, due, of course, to the shift along T1. However, since the instanton E11 wraps T1, while it is localized in the (T2,T3) directions, it is convenient to analyze the orbifold action in the space perpendicular to the instanton world-volume.

In the following, we discuss the first configurations with the instantons on the fixed tori, which are the relevant ones for matching the dual heterotic threshold corrections. Since the freely-acting operations (f, h) identify points in the internal space perpendicular to the instanton world-volume, they enforce the presence of doublets of E11 instantons, in complete analogy with similar phenomena happening in the case of background D5 branes in [18,19]. Indeed, the g-operation is the only one acting in a non-trivial way on the instantons. The doublet nature of the E11 instantons can be explicitly figured out in the following geometric way. Let the location of the E11 instanton be fixed at a point of the (y3,y4,y5,y6) space, which is left invariant by the g-operation. For instance, |E11) = |0, 0,nR5/2,0). Then, the f and h operations both map the point |E11) into |E11) = |nR3,0, 3nR5/2, 0), so that an orbifold invariant instanton state is provided by the combination ("doublet")

— [|0, 0,nR5/2, 0) + |nR3, 0, 3nR5/2, 0)].

The corresponding open strings can be stretched between fixed points and/or images, and can be described by the following amplitudes

Ae1-e1 =

q2 r dt 1 32 J Ttf-

2W3 + W"+1/2W"+1/2,

+ (-1)m1 P1

»[!]»[ a


ME1-O9 = -

q f dt 4n2 2 v-^

16/T»;n \

C afi-

» 2 »2

fi+1/2^ -1/2]

(-1)m1 P1W2W3


fi+1/J 1 4n

n3 n2 "' '

Since only the Z2 g-operation acts non-trivially on the characters, it is convenient in this case to use covariant SO(4) x SO(2) x SO(4) characters in order to describe the massless instanton zero-modes. Due to the doublet nature of the instantons, particle interpretation asks for a rescaling of the "charge" q = 2Q, meaning that the tension of the elementary instanton is twice the tension of the standard D1-brane. The result is

Q(Q + 1).



-(V4O2O4 - C4C2S4)

Q(Q -1) + —-(O4V2 04 - S4S2S4).

These zero-modes describe the positions xM of the E1 instantons in spacetime, scalars yl along the torus wrapped by theinstanton and fermions &a,-,a, 0a,+,a. The charged E11-D9 instanton spectrum is obtained from strings stretched between the E1 instantons and the D9 background branes. The corresponding cylinder amplitude is

NqJ 4%1/2]2 L^m n2

<»2 n ^ n2 »[a ]1 ^ n3 »2

Hmzffl »[£1/2] n2

+ (-1)»1 p L/-1/2J I (6.10)

The surviving massless states are now described by

aE0)_D9 = NQ(-O4S2O4), (6.11)

and correspond to the surviving "would be" world-sheet current algebra fermionic modes in the "heterotic string" interpretation (with Q = 1 and N = 32 [11,35]).

The second configuration, where the E1i instantons are off the fixed points (tori) of the g orbifold generator in y1 ...y6, for instance |E11) = |0,0, 0, 0>, can be worked out as well. In this case a quartet structure of instantons is present, in a situation again similar to the ones described in [18,19]. Indeed, g produces the image g: |0, 0, 0,0) ^ |0,0,nR5,0), while f and h produce two other images f: |0, 0, 0,0) ^ |nR3, 0,0, 0), h: |0,0,0, 0) ^ |nR3, 0,nR5,0). In conclusion, the orbifold-invariant linear superposition of the instanton images is now the combination

2[|0, 0, 0, 0) + |0, 0,nR5, 0) + |nR3, 0, 0, 0) + |nR3, 0,nR5, 0)]. (6.12)

For a given number of "bulk" E1 instantons, they have twice the number of neutral (uncharged) fermionic zero modes as compared to their "fractional" instantons cousins (6.9), whose minimal number of uncharged zero modes is four. On the other hand, their tension is twice bigger. If n "fractional" E1 instanton doublets wrap the torus Tl, one expects a contribution proportional to e-4nnTi, whereas if they wrap half of the internal torus, consistently with the shift identification, the contributions should be proportional to e-4n(n+1/2)Ti. These considerations are perfectly in agreement with the N = 2 nature of the threshold corrections appearing in the heterotic computation (5.23), (5.29) and (5.31). On the other hand, the quartet structure of the "bulk" instantons is probably incompatible with them. It should be also noticed that the absence of N = 1 sectors contributing to the threshold corrections (moduli-independent threshold corrections) on the heterotic side reflects the fact that only the f and h action create instanton images.

A similar analysis to the one carried out in this section can be performed for the more general type I SO(no) ® SO(ng) ® SO(nf) ® SO(nh) model presented in Section 3.1. However, we do not find any remarkable difference in nature between different choices of no, ng, nf and nh, contrary to what the heterotic dual model seems to suggest. It would be interesting to clarify this issue and to understand why type I models differing only in the Chan-Paton charges lead to so different models in the heterotic dual side.

7. Fluxes and moduli stabilization

7.1. Z2 x Z2 freely-acting orbifolds of twisted tori

Background fluxes for the RR and NSNS fields have been shown to be relevant for lifting some of the flat directions of the closed string moduli space. From the four-dimensional effective field theory perspective, the lift can be properly understood in terms of a non-trivial superpotential encoding the topological properties of the background. Many models based on ordinary Abelian orientifolds of string theory have appeared in the literature (for recent reviews and references see for instance [36]). Here we would like to extend this construction to the case of orientifolds with a free action. The motivation is two-fold. First, in these models the twisted sector modes are massive, as has been previously shown. The same happens for the open string moduli transforming in the adjoint. Second, we have enough control over the non-perturbative regime, so that this model provides us with a laboratory on which to explicitly test the combined effect of fluxes and non-perturbative effects.

For the particular type I (heterotic) orbifolds considered here, the orientifold projection kills a possible constant H3 (F3) background, so that the only possibilities left, apart from non-geometric deformations, are RR (NSNS) 3-form fluxes and metric fluxes [37-39]. The latter correspond to twists of the cohomology of the internal manifold M,

dm = Mja j + Nij^j,

where is a basis of harmonic 2-forms in M, and (ai,fij) a symplectic basis of harmonic 3-forms. The resulting manifold M is in general no longer Calabi-Yau, but rather it possesses S^(3)-structure [38,40]. Duality arguments show, however, that the light modes of the compact-ification in M can be suitably described in terms of a compactification in M, together with a non-trivial superpotential Wtwist accounting for the different moduli spaces.

Here we want to take a further step in the models of the previous sections and to consider geometries which go beyond the toroidal one by adding metric fluxes to the original torus. In terms of the global 1-forms of the torus, the cohomology twist reads,

de' = 1 fjkej A ek,

the resulting manifold being a group manifold M = G/r with structure constants fjk and r a discrete subgroup of G. Modding (7.2) by the orbifold action (2.8)-(2.10) will in general put restrictions on the structure constants fjk and the lattice r. More concretely, the surviving structure constants are

f62 f24'

/k^ h2

' — f35 f425

f2 f 36

— f46 f316

— f5l

f4 f61

— f6

f6 f23 f6 J14

f5 f154

f5 f23 f6 — f24

b11 b12 b13

b21 b22 b23

b31 b32 b33

b_11 b12 b13

b_21 b22 b23

b31 b32 b33

as in an ordinary Z2 x Z2 orbifold. The Jacobi identity of the algebra G requires in addition f[jkfo\i = 0 [22,37]. The set of metric fluxes transforms trivially under S-duality, so one can build heterotic-type I dual pairs by simply exchanging F3 ^ H3.

The low energy physics of the x r x (Z2 x Z2)] compactification can be then suitably

described in terms of a T6/[fip x (Z2 x Z2)] compactification, with Z2 x Z2 being the freely-acting orbifold action described in Section 2, together with a superpotential [41],

Wtwist = E Ti

-ill i + J2 bjiUj + ibiiU2U3 + b2iUiU3 j=i

+ ib3iUiU2 - hiUiU2U3

Notice that the freely-acting Z2 x Z2 orbifold of the full ten-dimensional picture will in general differ from the freely-acting Z2 x Z2 orbifold of the effective description. For illustration, consider the following simple example given by,

de1 = bne3 A e5, de2 = de3 = de4 = de5 = de6 = 0. (7.4)

We may integrate these equations as,

e1 = dy1 + b11y3 dy5, ei = dyi for i = 1, (7.5)

so that G is a fibration of y5 over y1. The lattice r is then suitably chosen as,

r: |y3 ^ y3 + 1, y1 ^ y1 - b11y5, (7 6)

y ^ y' + 1 for i = 3,

with b11 e Z so that the vielbein vectors remain invariant under r transformations. Acting now with the orbifold generators (2.8)-(2.10), it is not difficult to convince oneself that in order the vielbein vectors to transform covariantly, the orbifold generators have to be replaced by some new ones (g , f , h} defined as,

(y1 ,y2,y3,y4,y5,y6) -i (y1 + 1/2,y2, -y3, -y4, -y5 + 1/2, -y6),

(y1 ,y2,y3,y4,y5,y6) -i (-y1 + 1/2 + bny5/2, -y2,y3 + 1/2,y4, -y5, -y6),

(y1 ,y2,y3,y4,y5,y6) -- (-y1 - bny5/2, -y2, -y3 + 1/2, -y4,y5 + 1/2,y6). (7.7)

The generators (g , f , h} still define a Z2 x Z2 discrete group. Indeed, requiring the quantization condition bn e 2Z, one can prove that g2 = h2 = f2 = 1 and gf = f g = h, gh = hg = f, hf = f h = g, up to discrete transformations of the lattice r. Hence, the light modes of the 5^(3)-structure orientifold defined by the group manifold (7.5), together with the lattice (7.6) and the orbifold generators (7.7), can be consistently described by a T6 compactification with an orbifold action given by Eqs. (2.8) and a superpotential term,

Wtwist = ibnT1U2U3. (7.8)

7.2. Moduli stabilization in an S3 x T3/(Z2 x Z2) orbifold

To illustrate the interplay between non-perturbative effects and metric fluxes we consider in this section the following one-parameter family of twists,

de1 = ae4 A e6, de2 = ae4 A e6,

de3 = ae6 A e2, de4 = ae6 A e2, de5 = ae2 A e4, de6 = ae2 A e4. The particular solution to these equations

e1 = dy1 + e2, e2 = sin (ay6) dy4 + cos(ay6) cos(ay4) dy2, e3 = dy3 + e4, e4 = — cos(ay6) dy4 + sin (ay6) cos(ay4) dy2, e5 = dy5 + e6, e6 = dy6 + sin(ay4) dy2,

is corresponding to a product of a 3-sphere and a 3-torus. Consistency requires a to be multiple of 2n. On the other hand, in this particular case the orbifold action remains unaffected by the fluxes and is still given by (2.8)-(2.10).

We will also add a possible RR 3-form flux along the 3-sphere,

F3 = me2 A e4 A e6. (7.9)

One may easily check that this flux, together with the above twists, does not give rise to tadpole contributions.

The model can be effectively described by a T6/(Z2 x Z2) compactification with Kahler potential and superpotential,

K = -log(S + S*) -J2log(Ui + U*) -J2log(Ti + T*), (7.10)

i=1 i=1

W = m + a J] Tj(-i + Uj) + Wnp(S, T1,T2,T3,U1,U2,U3), (7.11)

where we have introduced a generic non-perturbative superpotential possibly depending on all moduli, as shown in the previous sections.12

For Re Ti > 1 and Re Ui > 1, the dependence of the non-perturbative superpotential on the Kahler and complex structure moduli can be neglected, dUi Wnp ~ d Wnp ~ 0, and the above superpotential has a perturbative vacuum given by

ImUi ~ 1, Re Wnp + m ~ a(Re Ti)(ReUi),

Im Ti ~ 0, Im Wnp ~ 0, DsW = 0, (7.12)

with DsW = dsW - (S + S*)-1 W, as usual. Then, for Wnp the racetrack superpotential (5.20), one may stabilize S at a reasonably not too big coupling.

The model can be viewed in the S-dual heterotic side as an asymmetric Z2 x Z2 orbifold of some Freedman-Gibbons electrovac solution [43,44].13 In particular, the full string ground state includes an SU(2) Wess-Zumino-Witten model describing the radial stabilization of the 3-sphere by m units of H3 flux, provided by F3 ^ H3 in (7.9). In terms of the radii Ri, i = 1,..., 6, Eqs. (7.12) lead to

2 2 2 Re Wnp + m (R2)2 = (R4)2 = (R6)2 ^---, (7.13)

12 Perturbative corrections to the Kahler potential could also play a role in the moduli stabilization. We restrict here to the tree-level form of the Kahler potential, for the possible effect of a' or quantum corrections to it, see, e.g., [42].

13 We thank E. Kiritsis for pointing out to us this connection.

whereas the radii of the 3-torus, R1, R3, R5, remain as flat directions. Having ReT' > 1 and Re U' > 1 then requires the volume of the 3-sphere to be much bigger than the volume of the 3-torus, i.e., m/a > 1.


We would like to thank C. Angelantonj, C. Bachas, M. Bianchi, E. Kiritsis, J.F. Morales and A. Sagnotti for discussions. E.D. thanks CERN-TH and G.P. would like to thank CPhT-Ecole Polytechnique for the kind invitation and hospitality during the completion of this work. G.P. would also like to thank P. Anastasopoulos and F. Fucito for interesting discussions. P.G.C. also thanks A. Font for discussions on related topics. This work was also partially supported by INFN, by the INTAS contract 03-51-6346, by the EU contracts MRTN-CT-2004-005104 and MRTN-CT-2004-503369, by the CNRS PICS #2530, 3059 and 3747, by the MIUR-PRIN contract 2003023852, by a European Union Excellence Grant, MEXT-CT-2003-509661 and by the NATO grant PST.CLG.978785.

Appendix A. Normalization of string amplitudes

For sake of brevity, throughout the paper we ignored the overall factors coming from integrating over the non-compact momenta. For arbitrary string tension a', the complete string amplitudes T, K, A, M are related to the ones used in the main text by

=-^—r T, KL =-~—r- K,

(4n 2a')2 (8n 2a')2

A = (snwA, M = (WM (A.1)

Appendix B. Characters for Z2 x Z2 orbifolds

In the light-cone RNS formalism, the vacuum amplitudes involve the following characters Too = V2I2I2I2 + I2V2 V2V2 - S2S2S2S2 - C2C2C2C2,

Tog = I2V2I2I2 + V2I2V2V2 - C2C2S2S2 - S2S2C2C2, Toh = I2I2I2V2 + V2V2V2I2 - C2S2S2C2 - S2C2C2S2, Tof = I2I2V2I2 + V2V2I2V2 - C2S2C2S2 - S2C2S2C2, Tgo = V2I2S2C2 + I2V2C2S2 - S2S2V2I2 - C2C2I2V2,

Tgg = I2V2S2C2 + V2I2C2S2 - S2S2I2V2 - C2C2V2I2, Tgh = I2I2S2S2 + V2 V2C2C2 - C2S2 V2 V2 - S2C2I2I2, Tgf = I2I2C2C2 + V2V2S2S2 - S2C2V2V2 - C2S2I2I2, Tho = V2S2C2I2 + I2C2S2V2 - C2I2V2C2 - S2V2I2S2, Thg = I2C2C2I2 + V2S2S2V2 - C2I2I2S2 - S2V2V2C2, Thh = I2S2C2V2 + V2C2S2I2 - S2I2V2S2 - C2V2I2C2, Thf = I2S2S2I2 + V2C2C2 V2 - C2 V2 V2S2 - S2I2I2C2, Tfo = V2S2I2C2 + I2C2V2S2 - S2V2S2I2 - C2I2C2V2,

Tfg = I2C2I2C2 + V2S2V2S2 — C2I2S212 — S2V2C2V2, Tfh = I2S2I2S2 + V2C2 V2C2 — C2V2S2V2 — S2I2C2I2,

Tff = I2S2V2C2 + V2C2I2S2 — C2V2C2I2 — S2I2S2V2, (B.1)

where each term is a tensor product of the characters of the vector representation (V2), the scalar representation (I2), the spinor representation (S2) and the conjugate-spinor representation (C2) of the four SO(2) factors that enter the light-cone restriction of the ten-dimensional Lorentz algebra.

Appendix C. Details on the threshold correction computations

C.1. Threshold corrections in the type I SO(no) ® SO(ng) ® SO(nf) ® SO(nh) models

In order to implement the background field method, it is convenient to express the orbifold characters in terms of the corresponding &-functions:

Too + + Toh + Tof = —j # - < - ^24 - O, (C.1)

Too + Tog - Toh - T0f = (#f # 2 + # 1 #2 - #4# + #32#2). (C.2)

Too - Tog + Toh - Tof = ^ (# 1#f # 1 + #2# ?#2 + #3#42#3 - #4^4), (C.3)

Making use of the expansion (valid for even spin structure a)

Too - Tog - Toh + Tof = —4 (#2#1#2#1 + #1#2#1#2 - #4#3#4#3 + #3#4#3#4)- (C.4)

#a(eT|t) 1 #a eT

——— =--a +---a +----, (C.5)

#1(eTIt) 2neT n3 4n n3

and the modular identities (D.2) and (D.3) in Appendix D, the expansions of the characters in terms of the (small) magnetic field or, equivalently, in terms of the e of Eq. (5.8), are

(Too + Tog + Toh + Tof)(eT, T) ~ -—J(#3'#33 - #4#3 - #2'#23) = 0,

(Too + Tog - Toh - Tof)(eT, T) = (Too - Tog - Toh + T0f)(€T, T)

= (Too - Tog + Toh - Tof)(eT, t)

- — ^^ ( — &4&4&32 + &3 &3&2) = -r Ttf2!^' (C.6)

The one-loop threshold corrections on any of the gauge group factors can therefore be written in the form

^2 = 16n2 J d {[2Tr(Q2) - Tr(Kg) Tr(Kg Q2)] (- 1)m1 A

+ [2Tr(Q2) - Tr(Kf) Tr(Yf Q2)] (-1)m2 P2

+ [2Tr(Q2) - Tr(Yh)Tr(YhQ2)](-1)m3P3}, (C.7)

where the action induced by the orbifold on the CP matrices, defined in (3.10), has been used. The last step is to compute the momentum sums (-1)m P .To this end, it is useful to reexpress (3.7) as

[ dt !t{


n(Re T) 4t(Re U)

E exp[—n(m — b)TA(m — b)\,

m — b =

iReTI x m 2t(Re U) \ ,/ , i(Re T)(Im U)

t / |U|2 ImU (Re T)(Re U)\ Im U 1

2t(Re U) /

Making use of the Poisson summation formula (D.1) and redefining t — 1/1 in order to move to the transverse channel picture, one gets

r = (Re T)

fdt £

nl(Re T)

n1 + ! — n2 Im U) + (n2 Re U)2

. (C.10)

As expected, the integral contains infrared (IR) divergences as I — 0, corresponding to loops of massless modes. It can be regularized introducing an IR regulator / via a factor F/ = (1 -e-l// ). Performing the integral in I the result is

r = lim

n nfn2V (n1 + 1 — n2Im U)2 + (n2 Re U)2

(m + 1 — n2 Im U)2 + (n2Re U)2 + ^RUf


Finally, using the Dixon, Kaplunovsky and Louis (DKL) formula [33] to evaluate the sum over n1, the expression become

r= —E

1 ( qn2 — 1 + qn2 — 1 +

n2\ qn2 + 1 q n2 + 1

n2 + (1/n(Re U)(Re Fp2)-1


with q = exp[—2nU] and where we have taken p2 ^ 1 (in string units). A Taylor expansion (using Eq. (D.19)) produces

r = £ (2 —

<A n2 ^n2 + (1/n(Re U)(Re T)p2)


(—1)" n2

( — 1 )m + 2 y ' ( qmn2


-n (2—

cA n2 ^n22 + (1 /n(Re U)(Re T)p2)

- 2 log(1 - q2m) + 2 log(1 - q2m-1) + c.c.


Taking the / 2 - 0 limit and at the same time subtracting the finite14 and the cut-off dependent parts, in terms of the modular functions (D.17) and (D.16) one gets

(-1)mF/P = log

- n Re U - log[(Re U)(Re T)/2].


C.2. Threshold corrections in the type I racetrack models

The procedure for the racetrack models is completely analogous to the one in the previous section. Plugging (C.6) into (3.11) and (3.12) one gets

A%p = 16n2Tr(Q2)J d [[(2 - p)P1 - q(Pm,+1 Pm1 )](-1) 0

+ (2 - p)P2(-1)m2 + (2 - p)P3(-1)m3],


where the Q generator has been taken in the SO(p) factor. In this case there is a new lattice summation to compute, namely

P = / ( 1)m Pm'+ 2 Pm

f dfJ2(-Dm exp

n m,m!

m + — - iUm

JT exp

n(Re T) 4t(Re U)

(Re T)(Re U) E exp[-n(m - b)TA(m - b)\,

where now mb

m- '(Re T) m 2t(Re U) , '(Re T)(Im U) . 1 r 2t(Re U) + 2

t /|U |2 Im U (Re T)(Re U^ Im U 1

Thus, the integration in the transverse channel gives Re U ^ (-1)"2

r ' = ■

n n^n2 (m + 1 - »2Im U)2 + («2 Re U)2'

Using again the (DKL) formula, after some algebra, the r' can be written

U q«2 -1 q2n2 -1

^ n2\ qn2 + 1 q2n2 + 1

«2 >0

+ c.c.,





14 The finite term can be actually reabsorbed into the value of the gauge coupling at the compactification scale.

with q = exp[-2n^]. It should be noticed that in this case there is no need of an IR regulator for this sum. In terms of modular functions the integral becomes

T(-r)mPm+ 1 Pm = log

- n Re U

and the moduli dependent part of the gauge coupling threshold corrections is

Mp = -i6n 2Tr(e2)

(2 - Re U

+ log[(ReUj)(Re Tj)^2] - log


+ d log

(2iU i )

(4iU i )

+ n Re Uij

with a p-function coefficient,

bp = -3(p - 2), that can be easily extracted from the previous expression.

C.3. Threshold corrections in the heterotic models




We consider separately the contributions from left- and right-mover oscillators in (5.22). The left-mover contributions read

Aleft=¿3 e

$3\ — *

.m; + 2 ,ni + 2

+ (d^)*3*2 - d^(tnr)^32)A?++1

* 2 *4

.(*)**2 - a/*4

Making use of the identities (D.4)-(D.8), we get after some small algebra

ni + 2 4n2 2n

* 2 *3

Aleft=p-6 [(-i)mi- z?+1 ,n-+2

- ( - i )

Ji *3 *4 H +ni Zmi +1 ,ni +1 *2*21




where the toroidal lattice sums Zi = |n|4Ai are provided by (5.24)-(5.25), after Poisson resum-mation in m1 and m2.

Regarding the contributions from the right-mover fermionic oscillators, we get

Aright = I Q,


1 -[ b ]—b ]15 i „ -[ b ]16

=--Ô b b---V ■ (C.25)

8n2 n16 8nt2^ n16

Making use of relations (D.5)-(D.12), these terms can be rearranged in the very compact expression

bright =-4^16-' (C26)

E4(EE4E2 - E6) 12jj!

corresponding to the modular covariant derivative of £8.

Putting all together we then arrive to the final expression for the gauge kinetic threshold corrections to the SO(32) heterotic model,

i f d t

A 2 = -— -A left Aright

= -55/ d2T È [(-Dmi - ZZ ri+1n+2 -22-2

F 2 i=1

- (-i)Mj +nizm+1n+1 ^ E4(ee - E6) (C.27)

In the limit of large volume, Re Ti > 1, or equivalently q ^ 0 and ni = 0, only degenerate orbits consisting of A matrices (5.25) with zero determinant in the sector (h, g) = (1, 0) contribute to the toroidal lattice sums. Following [33], then we can pick an element A0 in each orbit and to integrate its contribution over the image under V of the fundamental domain, for all V e SL(2) yielding A0 V = A0. The representatives can be chosen to be,

A0 =(0 ' + , (C.28) enforcing the identification

(j, P) - (-j - 1, -p). (C.29) With this representation, A0 V' = A0 V" if and only if

v'=(0 7) V

Therefore, the contributions are integrated over (r2 > 0, |r1| < 1}, and the double covering is taking into account by summing over all p and j,

/f dT2 ( n Re T

-j E -p(- T2RT7

_1 0 2 J,p

j + ^ + iUiP


This is exactly the same expression as (C.10), so the contributions of the degenerate orbits perfectly match the perturbative type I threshold corrections,

I- = log

- nReUi - log[(ReU)(Re Ti)\ (C.31)

Analogously, in the limit q ^ 0 but ni = 0 also the non-degenerate orbits in the sector (h, g) = (1, 0) contribute. The representative in this class can be chosen to have the form

j + 2 p


with k > j > 0,p = 0. For these, V' = V" implies A0V' = A0 V", and therefore these contributions must be integrated over the double cover of the upper half plane (t2 > 0),

Ind = 2 (Re T) Y. e2nTkP

c^j<k,p=c n Re T

kT + j + 2 + ipU

r2Re U

Evaluating the Gaussian integral over r1 and summing on j, one gets


Ind = 2

d (ReURT) (-i)k

0<k,p=C c

n Re T

:(kT2 + p Re U)2

t2 Re U

and the contribution of this sector becomes

Ind = log

-4 (2iT)

- n Re T.



It corresponds to E1 instanton corrections in the type I SO(32) dual model. Indeed, expanding the n-function in (C.35), Ind can be expressed as

Ind = -2j2(-1)n log(l -

e-2nnT) + c.c.,


which should correspond to a sum over the contributions of El-instantons wrapping n times the (1, 1)-cycle associated to T, a fact that would be very interesting to verify explicitly. Notice that the dependence on T perfectly agrees with the general arguments in [16] for the mirror type IIA picture.

Appendix D. Some useful formulae

Poisson summation formula:

VExpl"-n(m - b)TA(m - b)l = , ^Expl"-nnTA-1n + 2inbTn]. (D.l) ^ Vdet A ^

Modular identities:

—3—3 - -4-4 - -2-2 = o, -3-3-2 - -4-4-32 = -4n2n6-2,

-2-3-4 = 2n3,

(D.2) (D.3) (D.4)

#2' = 4ni dt$2 = -y #2 ( E2 + #34 + #4) - (D.5) n 2

#3' = 4nid# = --3-#3 (E2 + #24 - #44)- (D.6)

#4' = 4nidt#4 = --3-#4 (E2 - #24 - #34)• (D.7)

Eisenstein series:

E2 = £2 +-= —dr log n = 1 - 24q----, (D.8)

nT2 in

E4 = 2 (#28 + #38 + #48) = 1 + 240q + •••, (D.9)

E6 = 1 (#24 + #34)(#34 + #44)(#44 - #24) = 1 - 540q -•••, (D.10)

E8 = £2 = 1 (#216 + #416 + #316) = 1 + 480q + •••, (D.11)

E10 = E4E6 = -1 [#216(#34 + #44) + #316(#24 - #4) - #416(#24 + #34)]• (D.12) Series expansions:

log(1 - Q) = -V —, (D.13)

log(1 + Q) = -V (-1)n — , (D.14)

log #2 = log 2q1/8 + J2 log(1 - qn) + 2J2 log(1 + qn)- (D.15)

n=1 n=1

log #4 = J2 log(1 - qn) + ^log(1 - qn-1)- (D.16)

n=1 n=1

log n = log q1/24 + J2 log(1 - qn)- (D.17)

1+— = 1 + £ Qm, (D.18)

Q m = 1

Q+1 = -1 -2 £(-1)mQm. (D.19)


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