Heterotic moduli stabilisation

Abstract

We perform a systematic analysis of moduli stabilisation for weakly coupled heterotic string theory compactified on internal manifolds which are smooth Calabi-Yau three-folds up to [alpha]^sup '^ effects. We first review how to stabilise all the geometric and gauge bundle moduli in a supersymmetric way by including fractional fluxes, the requirement of a holomorphic gauge bundle, D-terms, higher order perturbative contributions to the superpotential as well as non-perturbative and threshold effects. We then show that the inclusion of [alpha]^sup '^ corrections to the Kähler potential leads to new stable Minkowski (or de Sitter) vacua where the complex structure moduli and the dilaton are fixed supersymmetrically at leading order, while the stabilisation of the Kähler moduli at a lower scale leads to spontaneous breaking supersymmetry. The minimum lies at moderately large volumes of all the geometric moduli, at perturbative values of the string coupling and at the right phenomenological value of the GUT gauge coupling. We also provide a dynamical derivation of anisotropic compactifications with stabilised moduli which allow for perturbative gauge coupling unification around 10^sup 16^ GeV. The value of the gravitino mass can be anywhere between the GUT and the TeV scale depending on the stabilisation of the complex structure moduli. In general, these are fixed by turning on background fluxes, leading to a gravitino mass around the GUT scale since the heterotic three-form flux does not contain enough freedom to tune the superpotential to small values. Moreover accommodating the observed value of the cosmological constant is a challenge. Low-energy supersymmetry could instead be obtained by focusing on particular Calabi-Yau constructions where the gauge bundle is holomorphic only at a point-like sub-locus of complex structure moduli space, or situations with a small number of complex structure moduli (like orbifold models), since in these cases one may fix all the moduli without turning on any quantised background flux. However obtaining the right value of the cosmological constant is even more of a challenge in these cases. Another option would be to focus on compactifications on non-complex manifolds, since these allow for new geometric fluxes which could be used to tune the superpotential as well as the cosmological constant, even if the moduli space of these manifolds is presently only poorly understood.

Published for SISSA by Springer

Received: August 1, 2013

Revised: September 17, 2013

Accepted: October 11, 2013

Published: October 29, 2013

M. Cicoli,a,b,c S. de Alwisc,d and A. Westphale

aDipartimento di Fisica ed Astronomia, Universit di Bologna,

Bologna, Italy

bINFN, Sezione di Bologna,

Bologna, Italy

cAdbus Salam ICTP,

Strada Costiera 11, Trieste 34014, Italy

dUCB 390 Physics Dept. University of Colorado,

Boulder CO 80309, U.S.A.

eDeutsches Elektronen-Synchrotron DESY, Theory Group,

D-22603 Hamburg, Germany

E-mail: mailto:[email protected]

Web End [email protected] , [email protected], mailto:[email protected]

Web End [email protected]

Abstract: We perform a systematic analysis of moduli stabilisation for weakly coupled heterotic string theory compactied on internal manifolds which are smooth Calabi-Yau three-folds up to e ects. We rst review how to stabilise all the geometric and gauge bundle moduli in a supersymmetric way by including fractional uxes, the requirement of a holomorphic gauge bundle, D-terms, higher order perturbative contributions to the superpotential as well as non-perturbative and threshold e ects. We then show that the inclusion of corrections to the Kahler potential leads to new stable Minkowski (or de

Sitter) vacua where the complex structure moduli and the dilaton are xed supersymmetrically at leading order, while the stabilisation of the Kahler moduli at a lower scale leads to spontaneous breaking supersymmetry. The minimum lies at moderately large volumes of all the geometric moduli, at perturbative values of the string coupling and at the right phenomenological value of the GUT gauge coupling. We also provide a dynamical derivation of anisotropic compactications with stabilised moduli which allow for perturbative gauge coupling unication around 1016 GeV. The value of the gravitino mass can be anywhere between the GUT and the TeV scale depending on the stabilisation of the complex structure moduli. In general, these are xed by turning on background uxes, leading to a gravitino mass around the GUT scale since the heterotic three-form ux does not contain enough freedom to tune the superpotential to small values. Moreover accommodating the observed value of the cosmological constant is a challenge. Low-energy supersymmetry

Open Access doi:http://dx.doi.org/10.1007/JHEP10(2013)199

Web End =10.1007/JHEP10(2013)199

Heterotic moduli stabilisation

JHEP10(2013)199

could instead be obtained by focusing on particular Calabi-Yau constructions where the gauge bundle is holomorphic only at a point-like sub-locus of complex structure moduli space, or situations with a small number of complex structure moduli (like orbifold models), since in these cases one may x all the moduli without turning on any quantised background ux. However obtaining the right value of the cosmological constant is even more of a challenge in these cases. Another option would be to focus on compactications on non-complex manifolds, since these allow for new geometric uxes which could be used to tune the superpotential as well as the cosmological constant, even if the moduli space of these manifolds is presently only poorly understood.

Keywords: Flux compactications, dS vacua in string theory, Superstrings and Heterotic Strings, Superstring Vacua

ArXiv ePrint: 1304.1809

JHEP10(2013)199

Contents

1 Introduction 2

2 Heterotic framework 72.1 Tree-level expressions 82.2 Corrections beyond leading order 102.2.1 Higher derivative e ects 112.2.2 Loop e ects 132.2.3 Non-perturbative e ects 132.3 Moduli-dependent Fayet-Iliopoulos terms 14

3 Supersymmetric vacua 143.1 Tree-level scalar potential 153.1.1 Chern-Simons action and gauge bundle moduli 193.2 Corrections beyond tree-level 223.2.1 Step 1: Z and S stabilisation by gaugino condensation 233.2.2 Step 2: T stabilisation by worldsheet instantons and threshold e ects 243.2.3 Tuning the Calabi-Yau condition 263.3 Flux vacua counting 26

4 Supersymmetry breaking vacua 274.1 Fluxes, non-perturbative e ects and threshold corrections 284.1.1 Derivation of the F-term potential 284.1.2 Moduli stabilisation 304.2 Inclusion of e ects 314.3 Minkowski solutions 344.4 D-term potential 364.4.1 D + F-term stabilisation 36

5 Moduli mass spectrum, supersymmetry breaking and soft terms 37

6 Anisotropic solutions 40

7 Conclusions 41

A Dimensional reduction of 10D heterotic action 44

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1 Introduction

String theory is a candidate for a quantum theory of gravity with full unication of the forces of nature. As such it should be able to describe the patterns of the Standard Models (SMs) of particle physics and cosmology. For this description of 4D physics, string theory needs to compactify its ambient 10D space-time. The multitude of possible compactication choices together with a plethora of massless 4D moduli elds originating from the deformation modes of the extra dimensions, leads to vacuum degeneracy and moduli problems. Recent progress in achieving moduli stabilisation points to the possibility of an exponentially large set of cosmologically distinct de Sitter (dS) solutions of string theory with positive but tiny cosmological constant, the landscape (for reviews see [1, 2]).

These results need to be combined with string constructions of viable particle physics. One fruitful region of the string landscape for this purpose is weakly coupled heterotic string theory. Recent works on heterotic compactications on both smooth Calabi-Yau (CY) manifolds [3] and their singular limits in moduli space, orbifolds [48], provided constructions of 4D low-energy e ective eld theories matching the minimal supersymmetric version of the SM (MSSM) almost perfectly. However, in contrast to the understanding achieved in type IIB string theory, heterotic CY or orbifold compactications lack a well controlled description of moduli stabilisation, and consequently, of inationary cosmology as well.1

As weakly coupled heterotic CY compactications lack both D-branes and a part of the three-form ux available in type IIB, historically moduli stabilisation in the heterotic context focused mostly on the moduli dependence of 4D non-perturbative contributions to the e ective action from gaugino condensation [1113]. While this produced models of partial stabilisation of the dilaton and some Kahler moduli [9, 1416], this route generically failed at describing controlled and explicit stabilisation of the O(100) complex structure

moduli of a given CY. Moreover, the resulting vacua tend to yield values of the compact-ication radius and string coupling (given by the dilaton) at the boundary of validity of the supergravity approximation and the weak coupling regime.

The works [1719] proposed to include the three-form ux H to stabilise the complex structure moduli in combination with hidden sector gaugino condensation for supersym-metric dilaton stabilisation. The inclusion of uxes in the heterotic string was originally studied by Strominger [20] who showed that, by demanding N = 1 supersymmetry, the

classical 10D equations of motion imply H = i2(

)J where J is the fundamental (1, 1)-form on the internal space. Hence a non-vanishing three-form ux breaks the Kahler condition dJ = 0. Note that this is the case of (0, 2)-compactications which allow for MSSM-like model building and the generation of worldsheet instantons, since in the nonstandard embedding the Chern-Simons term gives a non-zero contribution to the three-form ux H. However this contribution is at order , implying that the Calabi-Yau condition is preserved at tree-level and broken only at order . Moreover, in the heterotic case, due to the absence of Ramond-Ramond three-form uxes, there is generically no freedom to tune the superpotential small enough to x the dilaton at weak coupling. However, a su ciently small superpotential could be obtained by considering fractional Chern-Simons invariants

1However, for some recent attempts see e.g. [9, 10].

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(such as discrete Wilson lines) [17]. Note that it is natural to take these e ects into account for (0, 2)-compactications which are the most relevant for both model building and moduli stabilisation since, as we pointed out above, they feature a non-vanishing Chern-Simons contribution to H, regardless of the presence of fractional Chern-Simons invariants.2

Supersymmetric vacua with all geometric moduli stabilised could be achieved by xing the Kahler moduli via contributions from threshold corrections to the gauge kinetic function [21, 22]. However this minimum cannot be trusted since it resides in a strong coupling regime where the gauge coupling is even driven into negative values [17]. The inclusion of a single worldsheet instanton contribution can resolve this di culty [19]. However, none of these vacua break supersymmetry, resulting in unrealistic anti-de Sitter (AdS) solutions.

In this paper, we shall present new stable Minkowski (or de Sitter) vacua where all geometric moduli are stabilised and supersymmetry is broken spontaneously along the Kahler moduli directions. Let us summarise our main results:

We identify two small parameters, one loop-suppressed and the other volume-

suppressed, which allow us to expand the scalar potential in a leading and a sub-leading piece. This separation of scales allows us to perform moduli stabilisation in two steps.

The leading scalar potential is generated by D-terms, quantised background uxes

(if needed for the stabilisation of the complex structure moduli), perturbative contributions to the superpotential and gaugino condensation. This potential depends on the gauge bundle moduli, the complex structure moduli and the dilaton which are all xed supersymmetrically at leading order.

The subleading scalar potential depends on the Kahler moduli and it is generated

by threshold corrections to the gauge kinetic function, worldsheet instantons and

O(2) [23], and O(3) [24, 25] corrections to the Kahler potential. These e ects

give rise to new Minkowski vacua (assuming the ne-tuning problem can be solved) which break supersymmetry spontaneously along the Kahler moduli directions. The dilaton is stabilised at a value Re(S) 2 in a way compatible with gauge coupling

unication, while the compactication volume is xed at V 20 which is the upper

limit compatible with string perturbativity. These new minima represent a heterotic version of the type IIB LARGE Volume Scenario (LVS) [26, 27].

By focusing on CY manifolds with K3- or T 4-bres over a

P1 base, we shall also show that this LVS-like moduli stabilisation mechanism allows for anisotropic constructions where the overall volume is controlled by two larger extra dimensions while the remaining four extra dimensions remain smaller. This anisotropic setup is particularly interesting phenomenologically, as it allows one to match the e ective string scale to the GUT scale of gauge coupling unication [28, 29], and ts very well

2As we shall describe in section 3.1.1, the co-exact piece of the Chern-Simons term is responsible for the breaking of the Kahler condition dJ = 0 while the generation of fractional invariants is controlled by the harmonic piece of the Chern-Simons term.

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with the picture of intermediate 6D orbifold GUTs emerging from heterotic MSSM orbifolds [29, 30].

The soft terms generated by gravity mediation feature universal scalar masses, A-

terms and /B-term of order the gravitino mass, m3/2 = |W |MP /

p2Re(S)V, and suppressed gaugino masses at the %-level. In turn, the value of the supersymmetry breaking scale depends on the stabilisation of the complex structure moduli:

1. If the complex structure moduli are xed by turning on quantised background uxes, due to the lack of tuning freedom in the heterotic three-form ux, |W | can

at most be made of order |W | O(0.10.01) by turning on only Chern-Simons

fractional uxes. Hence the gravitino mass becomes of order MGUT 1016 GeV for Re(S) 2 and V 20, leading to high scale supersymmetry breaking.

2. If the complex structure moduli are xed without turning on quantised background uxes, the main contribution to |W | can come from higher order pertur

bative operators or gaugino condensation. Hence |W | can acquire an exponen

tially small value, leading to low-scale supersymmetry [31, 32].

Let us discuss the stabilisation of the complex structure moduli in more detail. In a series of recent papers [3335], it has been shown that in particular examples one could be able to x all the complex structure moduli without the need to turn on any quantised background ux. Note that, as we explained above, if one focuses on (0, 2)-compactications, this observation is not important for preserving the CY condition (since this is broken at order regardless of the presence of a harmonic quantised ux) but it is instead crucial to understand the order of magnitude of the superpotential which sets the gravitino mass scale once supersymmetry is broken. Following the original observation of Witten [36], the authors of [3335] proved that, once the gauge bundle is required to satisfy the Hermitian Yang-Mills equations, the combined space of gauge bundle and complex structure moduli is not a simple direct product but acquires a cross-structure. Denoting the gauge bundle moduli as Ci, i = 1, . . . , N, and the complex structure moduli as Z , = 1, . . . , h1,2, this observation implies that the dimensionality of the gauge bundle moduli space is actually a function of the complex structure moduli, i.e. N = N(Z), and viceversa the number of massless Z-elds actually depends on the value of the gauge bundle moduli. As a simple intuitive example, consider a case with just one gauge bundle modulus and a leading order scalar potential which looks like:

V =

The form of this potential implies that:

If C is xed by some stabilisation mechanism (like D-terms combined with higher

order C-dependent terms in the superpotential) at hCi 6

= 0, then h1,2x complex structure moduli are xed at hZ i = 0 = 1, . . . , h1,2x. Hence the number of Z-moduli

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h1,2x

X =1

|Z |2

|C|2 . (1.1)

left at is given by h1,2hol = h1,2 h1,2x, which is also the dimensionality of the sub-locus

in complex structure moduli space for C 6= 0 where the gauge bundle is holomorphic.

Hence the best case scenario is when this sub-locus is just a point, i.e. h1,2hol = 0.

If the Z-moduli are xed by some stabilisation mechanism (like by turning on back

ground quantised uxes) at values di erent from zero, then the gauge bundle modulus C is xed at hCi = 0.3

However this stabilisation mechanism generically does not lead to the xing of all complex structure moduli due to the di culty of nding examples with h1,2hol = 0, i.e. with a point-like sub-locus in complex structure moduli space where the gauge bundle is holomorphic. In fact, there is so far no explicit example in the literature where h1,2hol = 0 can be obtained without having a singular CY even if there has been recently some progress in understanding how to resolve these singular point-like sub-loci [39]. Moreover, let us stress that even if one nds a non-singular CY example with h1,2hol = 0 (there is in principle no obstruction to the existence of this best case scenario), all the complex structure moduli are xed only if C 6= 0, since for C = 0 the Z-directions would still be at. As we pointed

out above, C 6= 0 could be guaranteed by the interplay of D-terms and higher order terms

in the superpotential, but in the case when the number of C-moduli is large, one should carefully check that all of them are xed at non-zero values (for example, one might like to have some of them to be xed at zero in order to preserve some symmetries relevant for phenomenology like U(1)BL). Thus the requirement of a holomorphic gauge bundle

generically xes some complex structure moduli but not all of them. Note also that these solutions are not guaranteed to survive for a non-vanishing superpotential, since one would then need to solve a set of non-holomorphic equations.

Let us therefore analyse the general case where some Z-moduli are left at after the requirement of a holomorphic gauge bundle, and summarise our results for their stabilisation:

Given that promising phenomenological model building requires us to focus on the

non-standard embedding where the H-ux already gets a non-vanishing contribution from the co-exact piece of the Chern-Simons term, we consider quite natural the option to turn on also a harmonic Chern-Simons piece that could yield a fractional Z-dependent superpotential that lifts the remaining complex structure moduli [17].

If H 6= 0, as in the case of (0, 2)-compactications, both the dilaton and the warp

factor could depend on the internal coordinates. For simplicity, we shall however restrict to the solutions where both of them are constant, corresponding to the case of special Hermitian manifolds [40].

The inclusion of quantised background uxes cannot x the remaining h1,2hol > 0 complex structure moduli in a supersymmetric way with, at the same time, a vanishing ux superpotential W0. In fact, setting the F-terms of the Z-moduli to zero corresponds to setting the (1, 2)-component of H to zero, whereas setting W0 = 0 implies

3See also [37, 38] for a mathematical discussion of this issue which basically comes to the same conclusion that gauge bundle moduli are generically absent.

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a vanishing (3, 0)-component of H. As a consequence, given that the ux is real, the entire harmonic ux H is zero, and so the h1,2hol > 0 Z-moduli are still at.4 Note that this would not be the case in type IIB where the three-form ux is complex (because of the presence of also Ramond-Ramond uxes) [41].

The remaining h1,2hol > 0 Z-moduli can be xed only if W0 6= 0 but this would lead

to a runaway for the dilaton if W0 is not ne-tuned to exponentially small values to balance the dilaton-dependent contribution from gaugino condensation. However, due to the absence of Ramond-Ramond uxes, the heterotic H-ux does not contain enough freedom to tune W0 to small values, since it is used mostly to stabilise the complex structure moduli in a controlled vacuum. There are then two options:

1. Models with either accidentally cancelling integer ux quanta or only Chern-Simons fractional uxes where the ux superpotential could be small enough to compete with gaugino condensation, even if this case would lead to supersymmetry breaking around the GUT scale;

2. Compactications on non-Kahler manifolds which do not admit a closed holomorphic (3, 0)-form, since these cases allow for new geometric uxes which could play a similar rle as type IIB Ramond-Ramond uxes, and could be used to tune W0 to small values [40, 4245]. In this case one could lower the gravitino mass to the TeV scale and have enough freedom to tune the cosmological constant. However, the moduli space of these manifolds is at present only poorly understood.

In this paper, we shall not consider the second option given that we want to focus on cases, like special Hermitian manifolds, which represent the smallest departure from a CY due to e ects.

This analysis suggests that if one is interested in deriving vacua where our Kahler moduli stabilisation mechanism leads to spontaneous supersymmetry breaking around the TeV scale, one should focus on one of the two following situations:

1. Models where the requirement of a holomorphic gauge bundle xes all complex structure moduli without inducing singularities (so that the supergravity approximation is reliable), i.e. models with h1,2hol = 0 [3335]. The dilaton could then be xed in a supersymmetric way by using a double gaugino condensate while the Kahler moduli could be xed following our LVS-like method by including worldsheet instantons, threshold and e ects. This global minimum would break supersymmetry spontaneously along the Kahler moduli directions. The gravitino mass could then be around the TeV scale because of the exponential suppression from gaugino condensation.

2. Simple models with a very small number of complex structure moduli, like Abelian orbifolds with a few untwisted Z-moduli, or even non-Abelian orbifolds with no complex structure moduli at all. In fact, in this case gauge singlets could be xed at

4This statement is also implicit in [18].

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non-zero values via D-terms induced by anomalous U(1) factors and higher order terms in the superpotential [48], so resulting in cases where all the Z-moduli become massive by the holomorphicity of the gauge bundle. The dilaton could then be xed by balancing gaugino condensation with the contribution from a gauge bundle modulus (i.e. a continuous Wilson line in the orbifold language) which develops a small vacuum expectation value (VEV) because it comes from R-symmetry breaking higher order terms in the superpotential [31, 32]. A low gravitino mass could then be obtained due to this small VEV.

Let us nally note that accommodating our observed cosmological constant, which is a challenge even with uxes and O(100) complex structures, is even more of a challenge in

cases without quantised uxes.

This paper is organised as follows. In section 2 we introduce the general framework of heterotic CY compactications [46, 47], reviewing the form of the tree-level e ective action and then presenting a systematic discussion of quantum corrections from non-perturbative e ects [1113], string loops [4850], and higher-derivative -corrections [2325] according to their successive level of suppression by powers of the string coupling and inverse powers of the volume. Supersymmetric vacua are then discussed in section 3, while in section 4 we derive new global minima with spontaneous supersymmetry breaking which can even be Minkowski (or slightly de Sitter) if enough tuning freedom is available. After discussing in section 5 the resulting pattern of moduli and soft masses generated by gravity mediation, we derive anisotropic constructions in section 6. We nally present our conclusions in section 7.

2 Heterotic framework

Let us focus on weakly coupled heterotic string theory compactied on a smooth CY threefold X. The 4D e ective supergravity theory involves several moduli: h1,2(X) complex

structure moduli Z , = 1, . . . , h1,2(X); the dilaton S and h1,1 Kahler moduli Ti, i = 1, . . . , h1,1(X) (besides several gauge bundle moduli).

The real part of S is set by the 4D dilaton (see appendix A for the correct normalisation):

Re(S) s =

14 e24 =

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14 e2 V , (2.1)

where is the 10D dilaton whose VEV gives the string coupling ehi = gs. The imaginary

part of S is given by the universal axion a which is the 4D dual of B2. On the other hand, the real part of the Kahler moduli, ti = Re(Ti), measures the volume of internal two-cycles in units of the string length s = 2. The imaginary part of Ti is given by the reduction of B2 along the basis (1, 1)-form Di dual to the divisor Di.

We shall focus on general non-standard embeddings with possible U(1) factors in the visible sector. Hence the gauge bundle in the visible Evis8 takes the form Vvis = Uvis

L

L

where Uvis is a non-Abelian bundle whereas the L are line bundles. On the other hand

the vector bundle in the hidden Ehid8 involves just a non-Abelian factor Vhid = Uhid. We shall not allow line bundles in the hidden sector since, just for simplicity, we shall not consider matter elds charged under anomalous U(1)s. In fact, if we want to generate a

7

superpotential from gaugino condensation in the hidden sector in order to x the moduli, all the anomalous U(1)s have to reside in the visible sector otherwise, as we shall explain later on, the superpotential would not be gauge invariant.

2.1 Tree-level expressions

The tree-level Kahler potential takes the form:

Ktree = ln V ln(S + S) ln

i

ZX , (2.2)

where V is the CY volume measured in string units, while is the holomorphic (3, 0)-

form of X that depends implicitly on the Z-moduli. The internal volume depends on the T -moduli since it looks like:

V =

16 kijktitjtk =

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148 kijk Ti + T i

T

j + T j

T

k + T k

, (2.3)

where kijk =

RX Di Dj Dk are the triple intersection numbers of X.

The tree-level holomorphic gauge kinetic function for both the visible and hidden sector is given by the dilaton:

ftree = S Re(ftree) g24 = s . (2.4)

The tree-level superpotential is generated by the three-form ux H and it reads:

Wux =

ZX H , (2.5)

with the correct denition of H including e ects:

H = dB2

4 [CS(A) CS()] , (2.6)

where CS(A) is the Chern-Simons three-form for the gauge connection A:

CS(A) = Tr

A dA +23 A A A

, (2.7)

and CS() is the gravitational equivalent for the spin connection .

The VEV of the tree-level superpotential, W0, is of crucial importance. Due to the di erence with type IIB where one has two three-form uxes, which can give rise to cancellations among themselves leading to small values of W0, in the heterotic case W0 is generically of order unity. Hence one experiences two problems:

1. Contrary to type IIB, the heterotic dilaton is not xed by the ux superpotential, resulting in a supergravity theory which is not of no-scale type. More precisely, the F-term scalar potential:

VF = eK

KI JDIW D J W 3|W |2 , (2.8)

8

derived from (2.2) and (2.5) simplies to:

VF = eK"XZ

K D W D

W + KS SKSKS +

XTKjKiKj3! |

W |2#

= eK

XZK D W D W + |W |2!, (2.9)

since KS SKSKS = 1 and

PT KjKiKj 3 = 0. Setting D W = 0 = 1, . . . , h1,2(X), the scalar potential (2.9) reduces to:

VF = eK|W0|2 = |

W0|2

2sV

, (2.10)

yielding a run-away for both s and V if |W0| 6= 0. Given that generically |W0| O(1), it is very hard to balance this tree-level run-away against S-dependent nonperturbative e ects which are exponentially suppressed in S. One could try to do it by considering small values of s = g24D but this would involve a strong coupling limit where control over moduli stabilisation is lost. A possible way to lower |W0|

was proposed in [17] where the authors derived the topological conditions to have fractional Chern-Simons invariants.

2. If |W0| 6

= 0, even if it is fractional, one cannot obtain low-energy supersymmetry. In fact, the gravitino mass is given by m3/2 = eK/2|W0|MP , and so the invariant quantity eK/2|W0| = |W0|/(2sV) has to be of order 1015 to have TeV-scale supersymmetry.

As we have seen, the 4D gauge coupling is given by 1GUT = g2sV, and so a huge value of the internal volume would lead to a hyper-weak GUT coupling. Note that a very large value of V cannot be compensated by a very small value of g2s since we

do not want to violate string perturbation theory.

Let us briey mention that in some particular cases one could have an accidental cancellation among the ux quanta which yields a small |W0| as suggested in [18]. We

stress that in the heterotic case, contrary to type IIB, this cancellation is highly non-generic, and so it is not very appealing to rely on it to lower |W0|. Hence it would seem

that the most promising way to get low-energy supersymmetry is to consider the case where |W0| = 0 and generate an exponentially small superpotential only at sub-leading

non-perturbative level. This case was considered in [34], where the authors argued that, at tree-level, one can in principle obtain a Minkowski supersymmetric vacuum with all complex structure moduli stabilised and 2(h1,1 + 1) at directions corresponding to the dilaton and the Kahler moduli. As explained in section 1, this corresponds to the best case scenario where the gauge bundle is holomorphic only at a non-singular point-like sub-locus in complex structure moduli space.

If instead one focuses on the more general case where h1,2hol > 0 Z-moduli are left at after imposing the requirement of a holomorphic gauge bundle, as we shall show in section 3, the conditions DZ Wux = 0 = 1, . . . , h1,2hol and |W0| = 0 imply that no quantised H ux

is turned on, resulting in the impossibility to stabilise the remaining Z-moduli. This result

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implies that it is impossible to stabilise the remaining complex structure moduli and the dilaton in two steps with a Z-moduli stabilisation at tree-level and a dilaton stabilisation at sub-leading non-perturbative level. In this case there are two possible way-outs:

1. Focus on the case DZ W = 0 = 1, . . . , h1,2hol and |W0| 6= 0 so that H can be non

trivial. In this case one has however a dilaton run-away, implying that no moduli can be xed at tree-level. One needs therefore to add S-dependent non-perturbative e ects which have to be balanced against the tree-level superpotential to lift the runaway. A small |W0| could be obtained either considering fractional Chern-Simons

invariants or advocating accidental cancellations among the ux quanta.

2. Focus on the case with trivial H so that no scalar potential is generated at tree-level. The dilaton and the complex structure moduli could then be xed at nonperturbative level via a race-track superpotential generating an exponentially small W which could lead to low-energy supersymmetry. Note that even though dH = R R F F 6= 0 for (0, 2)-models, it is still possible to have |W0| = 0 since only

the harmonic part of the H-ux contributes to this superpotential (see discussion in section 3.1). Hence, moduli stabilisation would have to proceed via a racetrack mechanism involving at least two condensing gauge groups with all moduli appearing in the gauge kinetic functions and/or the prefactors of the non-perturbative terms. Since this is generically not the case for heterotic compactications, this avenue will not lead to supersymmetric moduli stabilisation except perhaps for a few specic cases. Note that in this case to get a Minkowski supersymmetric vacuum one would have to ne-tune the prefactors of the two (or more) condensates so that W = 0 at the minimum. Then one would have (under the conditions mentioned above) a set of holomorphic equations for the Z-moduli which will always have a solution. However once supersymmetry is broken this option is no-longer available since now one needs to have W 6= 0 at the minimum if one is to have any hope of ne-tuning

the cosmological constant to zero. However now the equations for the Z-moduli are a set of real non-linear equations which are not guaranteed to have a solution.

2.2 Corrections beyond leading order

As explained in the previous section, in smooth heterotic compactications with h1,2hol > 0 complex structure moduli not xed by the holomorphicity of the gauge bundle, these Z-moduli cannot be frozen at tree-level by turning on a quantised background ux since this stabilisation would need |W0| 6= 0 which, in turn, would induce a dilaton and volume

runaway. Thus, one has to look at any possible correction beyond the leading order expressions. Before presenting a brief summary of the various e ects to be taken into account (perturbative and non-perturbative in both and gs), let us mention two well-known control issues in heterotic constructions:

Tension between weak coupling and large volume: In order to have full control over the

e ective eld theory, one would like to stabilise the moduli in a region of eld space where both perturbative and higher derivative corrections are small, i.e. respectively

10

JHEP10(2013)199

for gs 1 and V 1. However, as we have already pointed out, this can be

the case only if the 4D coupling is hyper-weak, in contrast with phenomenological observations. In fact, we have:

g2s

V

= GUT

1

25 , (2.11)

and so if we require gs [lessorsimilar] 1, the CY volume cannot be very large, V [lessorsimilar] 25, implying

that one has never a solid parametric control over the approximations used to x the moduli.

Tension between GUT scale and large volume: In heterotic constructions, the uni

cation scale is identied with the Kaluza-Klein scale, MGUT = MKK, which cannot

be lowered that much below the string scale for V [lessorsimilar] 25, resulting in a GUT scale

which is generically higher than the value inferred from the 1-loop running of the MSSM gauge couplings. In more detail, the string scale Ms 1s can be expressed in terms of the 4D Planck scale from dimensional reduction as (see appendix A for an explicit derivation):

M2s = M2P41GUT

M2P100 1.35 1017 GeV

2 . (2.12)

In the case of an isotropic compactication, the Kaluza-Klein scale takes the form:

MGUT = MKK

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[greaterorsimilar] 8 1016 GeV for V [lessorsimilar] 25 , (2.13)

which is clearly above the phenomenological value MGUT 2.1 1016 GeV. On the other hand, anisotropic compactications with d large dimensions of size L = xs with x 1 and (6 d) small dimensions of string size l = s, can lower the Kaluza-

Klein scale:

Vol(X) = Ldl6d = xd6s = V 6s MGUT = MKK

Msx

Ms

V1/6

Ms

V1/d

. (2.14)

For the case d = 2, one would get the encouraging result MGUT = Ms

V

[greaterorsimilar] 2.71016 GeV.

2.2.1 Higher derivative e ects

Let us start considering higher derivative e ects, i.e. perturbative corrections to the Kahler potential. In the case of the standard embedding corresponding to (2, 2) worldsheet theories, the leading correction arises at O(3)R4 [24] and depends on the CY Euler

number (X) = 2 h1,1 h1,2

. Its form can be derived by substituting the corrected volume V V + /2 into the tree-level expression (2.2) with = (3)(X)/(2(2)3).

Given that (3) 1.2, is of the order h1,2 h1,1

/200 O(1) for ordinary CY

three-folds with h1,2 h1,1

O(100). Hence for V O(20), the ratio /(2V) O(1/40)

is a small number which justies the expansion:

K ln V

2V

K 3 =

2V

. (2.15)

11

As pointed out in [23] however, this is the leading order higher derivative e ect only for the standard embedding since (0, 2) worldsheet theories admit corrections already at O(2)

which deform the Kahler form J as:

J J = J + O()

+ O(2)

(2) + . . . , (2.16)

where both and(2) are moduli-dependent (1, 1)-forms which are orthogonal to J, i.e.

RX J

=

(2) = 0. Plugging J into the tree-level expression for K (2.2) and then expanding, one nds that the O() correction vanishes because of the orthogonality

between and J whereas at O(2) one nds:5

K 2 = 1

2V

RX

J

ZX = ||||22V. (2.17)

Note that the correction (2.17) is generically leading with respect to (2.15) since (2.17) should be more correctly rewritten as:

K 2 = g

V2/3

with g ||

0 , (2.18)

where g is a homogeneous function of the Kahler moduli of degree 0 given that J scales as J V1/3 and

does not depend on V. As an illustrative example, let us consider the

simplest Swiss-cheese CY X with one large two-cycle tb and one small blow-up mode ts so

that J = tb Db ts

Ds and the volume reads:

V = kbt3b kst3s > 0 for 0

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||2

2V1/3

=

1

2V1/3 ZX

J

tstb <

kb ks

1/3. (2.19)

In the limit kbt3b kst3s, the function g then becomes (considering, without loss of gener

ality, as moduli-independent):

g = cb + cs tstb 0 with cb =

1

2 k1/3b

ZXDb and cs = 12 k1/3b

ZXDs .

(2.20)

The sign of cb and cs can be constrained as follows. In the limit ts/tb 0, g reduces to

g = cb = |cb| 0. On the other hand, requiring that g is semi-positive denite for any

point in Kahler moduli space one nds:

cs = |cb|

ks kb

1/3+ || , (2.21)

where || is a semi-positive denite quantity.

5In looking at the derivation of the correction at O( 2) in [23], one may wonder about the rle of eld redenitions. The fact that the corrected Kahler potential K can be written in terms of J as a function of

[integraltext] J J J alone, just the same way as the tree-level K in terms of J, may imply that a eld redenition of the Kahler form may actually fully absorb the correction at O( 2). To this end, the observation in [23]

that the generically non-vanishing string 1-loop corrections in type IIB appearing at O( 2) are S-dual to the heterotic correction, provides additional evidence for the existence of this term.

12

2.2.2 Loop e ects

Let us now focus on gs perturbative e ects which can modify both the Kahler potential and the gauge kinetic function. The exact expression of the string loop corrections to the Kahler potential is not known due to the di culty in computing string scattering amplitudes on CY backgrounds. However, in the case of type IIB compactications, these corrections have been argued to be sub-leading compared to e ects by considering the results for simple toroidal orientifolds [48] and trying to generalise them to arbitrary CY backgrounds [49, 50]. Following [50], we shall try to estimate the behaviour of string loop corrections to the scalar potential by demanding that these match the Coleman-Weinberg potential:

Vgs 2 Str M2 m23/2 M2KK |

W |2

2s

M4P

V2(1+1/d)

JHEP10(2013)199

, (2.22)

where we took the cut-o scale = MKK and we considered d arbitrary large dimensions. Note that these e ects are indeed subdominant with respect to the ones for large volume since the O(2) and O(3) corrections, (2.18) and (2.15), give respectively

a contribution to the scalar potential of the order V 2 |W |2/V5/3 and V

3

|W |2/V2,

whereas the gs potential (2.22) scales as Vgs |W |2/V7/3 for the isotropic case with d = 6

and Vgs |W |2/V3 for the anisotropic case with d = 2. Due to this subdominant behaviour

of the string loop e ects, we shall neglect them in what follows.

String loops correct also the gauge kinetic function (2.4). The 1-loop correction has a di erent expression for the visible and hidden E8 sectors [22]:

fvis = S + i2 Ti , fhid = S

i

2 Ti , (2.23)

where:

i = 1

4

ZX (c2(Vvis) c2(Vhid)) Di . (2.24)

2.2.3 Non-perturbative e ects

The 4D e ective action receives also non-perturbative corrections in both and gs. The e ects are worldsheet instantons wrapping an internal two-cycle Ti. These give a contribution to the superpotential of the form:

Wwi =

XjBj ebijTi . (2.25)

Note that these contributions arise only for (0, 2) worldsheet theories whereas they are absent in the case of the standard embedding. On the other hand, gs non-perturbative e ects include gaugino condensation and NS5 instantons. In the case of gaugino condensation in the hidden sector group, the resulting superpotential looks like:

Wgc =

XjAj eaj fhid =

XjAj eaj[parenleftBig]

S i2 Ti[parenrightBig] , (2.26)

where in the absence of hidden sector U(1) factors, all the hidden sector gauge groups have the same gauge kinetic function. Finally, NS5 instantons wrapping the whole CY manifold would give a sub-leading non-perturbative superpotential suppressed by eV 1, and so we shall neglect them.

13

2.3 Moduli-dependent Fayet-Iliopoulos terms

As already pointed out, we shall allow line bundles in the visible sector where we turn on a vector bundle of the form Vvis = Uvis

L

L . The presence of anomalous U(1) factors induces U(1) charges for the moduli in order to cancel the anomalies and gives rise to moduli-dependent Fayet-Iliopoulos (FI) terms. In particular, the charges of the Kahler moduli and the dilaton under the -th anomalous U(1) read:

q( )Ti = 4 ci1(L ) and q( )s = 2 ( ) = 2 i ci1(L ) , (2.27) so that the FI-terms become [22]:

( ) = q( )Ti

. (2.29)

From the expressions (2.27) for the U(1)-charges of the moduli, we can now check the U(1)-invariance of the non-perturbative superpotentials (2.25) and (2.26). In the absence of charged matter elds, the only way to obtain a gauge invariant worldsheet instanton is to choose the gauge bundle such that all the Ti appearing in Wwi are not charged, i.e.

ci1(L ) = 0 and i. The superpotential generated by gaugino condensation is instead

automatically U(1)-invariant by construction since all the anomalous U(1)s are in the visible sector whereas gaugino condensation takes place in the hidden sector. Thus, the hidden sector gauge kinetic function is not charged under any anomalous U(1):

q( )fhid = q( )s

DiW

W = q( )i

KTi q( )s

K

S =

ci1(L )

V

kijktjtk + ( )s . (2.28)

Note that the dilaton-dependent term in the previous expression is a 1-loop correction to the FI-terms which at tree-level depend just on the Kahler moduli. The nal D-term potential takes the form:

VD =

X 2( )Re f( )

JHEP10(2013)199

i

2 q( )Ti = 2 ( ) ici1(L )

= 0 . (2.30)

Before concluding this section, we recall that in supergravity the D-terms are proportional to the F-terms for W 6= 0. In fact, the total U(1)-charge of the superpotential W is given

by q( )W = q( )iWi/W = 0, and so one can write:

( ) = q( )iKi = q( )i

j Fj , (2.31)

where the F-terms are dened as F i = eK/2KjDj W . Therefore if all the F-terms are vanishing with W 6= 0, the FI-terms are also all automatically zero without giving rise to

independent moduli-xing relations.

3 Supersymmetric vacua

In this section, we shall perform a systematic discussion of heterotic supersymmetric vacua starting from an analysis of the tree-level scalar potential and then including corrections beyond the leading order expressions.

14

eK/2

W Ki

3.1 Tree-level scalar potential

In [20], Strominger analysed the 10D equations of motion and worked out the necessary and su cient conditions to obtain N = 1 supersymmetry in 4D assuming a 10D space-time

of the form M X where M is a maximally symmetric 4D space-time and X is a compact

6D manifold:

1. M is Minkowski;

2. X is a complex manifold, i.e. the Nijenhuis tensor has to vanish;

3. There exists one globally dened holomorphic (3, 0)-form which is closed, i.e. d = 0, and whose norm is related to the complex structure (1, 1)-form J as (up to a constant):6

dJ = i(

i2(

4 [tr(F F ) tr(R R)] . (3.5)

Some of the conditions listed above can be reformulated also in terms of constraints on the ve torsional classes Wi, i = 1, . . . , 5 (for a review see [1, 40]). The second condition

corresponds to W1 = W2 = 0 implying that the torsional class belongs to the space

W3 W4 W5. This is the case of Hermitian manifolds. Moreover, the third

condition above gives W5 = 2W4 = d ln || || implying that both W4 and W5 are exact real

1-forms. We shall focus on the simplest solution to 2W4 + W5 = 0 which is W4 = W5 = 0

corresponding to the case of special-Hermitian manifolds where the dilaton and the warp factor are constant [40]. More general solutions involve a non-constant dilaton prole in the extra dimensions and Wi 6= 0 for i = 4, 5 but we shall not consider this option [40].

6The adjoint operator d can be dened from the inner product h!, i = [integraltext]X ! as h!p, d!p1i = hd!p, !p1i. For an even dimensional manifold, as we have here, d = d.

7We are writing the total metric as ds2 = e2A(y) g (x)dxdx + gij(y)dyidyj[parenrightbig].

15

) ln || || (3.1)

4. The background gauge eld F has to satisfy the Hermitian Yang-Mills equations:

F(0,2) = F(2,0) = 0 and gjFj = 0 (3.2)

5. The dilaton and the warp factor A have to satisfy (again up to a constant):7

(y) = A(y) = 18 ln || ||(y) (3.3)

6. The background three-form ux is given by:

H =

JHEP10(2013)199

)J , (3.4)

together with the Bianchi identity:

dH =

Let us comment on the implications of the last Strominger condition (3.4) which for constant dilaton can be rewritten as H = 12dJ. Using the Hodge decomposition theorem,

the three-form H can be expanded uniquely as:

H = Hharm + Hexact + Hcoexact , (3.6)

where Hharm is a harmonic form, Hexact is an exact form and Hcoexact is a co-exact form

which are all orthogonal to each other. Given that dJ = d J, (3.4) implies that H

is a co-exact form, and so Hharm = Hexact = 0. Moreover, since dJ is a (2, 1) + (1, 2) form, (3.4) implies that the (3, 0) + (0, 3) component of Hcoexact is zero while the (2, 1) +

(1, 2) component breaks the Kahler condition dJ = 0. However this happens only at O(). In fact, the general expression of the H-ux is:

H = Hux + dB2

4 [CS(A) CS()] , (3.7)

where Hux is a harmonic piece and the combination of Chern-Simons three-forms can also be decomposed as:

[CS(A) CS()] = CSharm + CSexact + CScoexact . (3.8)

Comparing the two expressions for H, (3.6) and (3.7), we have (due to the uniqueness of the Hodge decomposition):

Hharm = Hux

4 CSharm , Hexact = dB2

4 CSexact , Hcoexact =

4 CScoexact .

JHEP10(2013)199

Then the relation (3.4) takes the form:

2 CScoexact = dJ , (3.9)

showing exactly that the Kahler condition dJ = 0 is violated at O(). Note that this would be the case for the non-standard embedding where CScoexact 6= 0 contrary to the

less generic situation of the standard embedding where the Chern-Simons piece vanishes. Taking the exterior derivative of (3.9) we recover the Bianchi identity (3.5) which now looks like:

d dJ =

2 [tr(F F ) tr(R R)] . (3.10)

This 10D analysis can also be recast in terms of an e ective potential which can be written as a sum of BPS-like terms and whose minimisation reproduces the conditions above [36, 4244]. Furthermore, some of these conditions can be re-derived as F- or D-term equations of 4D supergravity, which could lead to the stabilisation of some of the moduli in a Minkowski vacuum. For example, it has been shown in [36, 44], that the second equation in (3.2) is equivalent to a D-term condition since:

61 gjFij = 12 F J J . (3.11)

16

This D-term condition holds for general non-Abelian gauge elds. If we restrict to Abelian uxes and integrate the above condition over the CY, this reproduces the tree-level expression for the Fayet-Iliopoulos terms given in (2.28). If we expand the Abelian uxes as F( ) = ci1(L )

Di together with J = tj Dj we obtain:

( ) = 1

V

which reproduces exactly the tree-level part of (2.28).

Regarding the F-terms, as we have seen in section 2.1, the starting point is the expression of the ux superpotential which has been inferred in [45] by comparing the dimensional reduction of the 10D coupling of H to the gravitino mass term in the 4D supergravity action. The nal result is:8

Wux =

ZX H = ZX

Hharm . (3.13)

Note that only the harmonic component of H contributes to Wux. The harmonic piece Hharm can be expanded in a basis of harmonic (3, 0)- and (1, 2)-forms as:

Hharm =( Z) (Z) + b (Z, Z) (Z, Z) + c.c. , = 1, . . . , h1,2(X) . (3.14)

The same Hharm, together with the holomorphic (3, 0)-form , can also be expanded in a symplectic basis of harmonic three-forms (p, q) such that

RX p q =

RX p q = 0

RX p q = qp with p, q = 0, . . . , h1,2(X):

Hharm = epp mqq and (Z) = Zpp Gq(Z)q , (3.15)

where Gq(Z) = Zq G(Z) with G(Z) a homogeneous function of degree 2. Note that p and q do not depend on the complex structure moduli Z which are dened by the expansion of in (3.15). If (Ap, Bq) is the dual symplectic basis of 3-cycles such that

Ap Aq = Bp Bq = 0 and Ap Bq = qp, we have (choosing units such that 2 = 1):

ZBp Hharm = ZX

Hharm p = ZX(err mqq) p = ep , (3.16)

and similarly mq =

RAq Hharm. The quantities ep and mq are integer ux quanta. The expansion of the ux superpotential (3.13) is then given by:

Wux(Z) =

ZX Hharm = a(Z) ZX ( Z) (Z)

= ia(Z) = mqZq epGp(Z) , (3.17)

8In [42] and [44] it is suggested that the complete expression for W should more appropriately be W = [integraltext]X (H + i2 dJ) , similarly to the type IIB case where one has the RR ux in addition to the

H-ux. Integrating by parts, the new piece can be rewritten as [integraltext]X J d which clearly vanishes since d = 0. However, if one considers the case where d 6= 0, i.e. where supersymmetry is broken directly at the 10D level, this integral would still be zero if the internal manifold is complex since dJ is of Hodge type (2, 1) + (1, 2) while is (3, 0). Thus this term can play a useful rle only for non-complex manifolds with broken supersymmetry. Due to the di culty to study this case in a controlled way, we shall not consider it and neglect this additional piece.

17

ZX F( ) J J = ci1(L )

V

kijktjtk , (3.12)

JHEP10(2013)199

and

where we normalised

RX = i and used the fact that = i and the orthogonality of the di erent Hodge components of H.

Let us now evaluate the complex structure F-terms DZ Wux = Z Wux+WuxZ K.

Using the fact that (see for example [51]):

Z = k (Z, Z) + , Z K = k (Z,

and:

ZX ,

and expanding a generic element of the basis of harmonic (2, 1)-forms as (Z, Z) = fp (Z, Z)p gq, (Z,

Z)q, we nd:

DZ Wux =

Z) , (3.18)

where we used again the orthogonality of the di erent Hodge components of H and the fact that = i . On the other hand, the dilaton and Kahler moduli F-terms look like:

DSWux = WuxSK = i

a2s and DTiWux = WuxTiK = i

XZ

K b b + |a|2!. (3.20)

Let us now set all the F-terms to zero and see what they correspond to:

DZ

Z) ,

K

Z

Z K =

RX

RX = i

ZX Hharm = b (Z, Z)

ZX

= i b (Z, Z) = mqfq (Z, Z) epgp, (Z,

a 4V

kijktjtk .

(3.19)

Due to the no-scale cancellation, these F-terms give rise to a scalar potential which is positive denite and reads:

V = eK

XZ

K D W D W + |W |2!= 1 2sV

JHEP10(2013)199

Wux = 0 implies that the (2, 1) + (1, 2) component of Hharm is zero.

DSWux = 0 and DTiWux = 0 imply that the (3, 0) + (0, 3) component of Hharm

should also be zero, i.e. W0 hWuxi = 0, if one wants to avoid solutions with a

dilaton run-away (s ) or where the internal space decompacties (V ). Combining these two solutions, one has that the total harmonic piece of H should vanish and is of course consistent with the Strominger condition (3.4). An important question to ask now is whether these conditions allow for the xing of some moduli. The answer is no. Let us see why.

The rst condition DZ W = 0 appears to x the complex structure moduli supersymmetrically since one obtains as many equations, b (Z, Z) = 0, as the number of unknowns (assuming that the 2 h1,2 real equations have solutions for some sets of values of the 2 h1,2+2 uxes). The second condition W0 hWuxi = i a(Z) = 0 could then be satised by an

appropriate choice of ux quanta.

However the two conditions DZ W = 0 b (Z,

Z) = 0 and W0 = 0 a(Z) = 0 imply from (3.14) that Hharm = 0. Given that H does not depend on the complex structure

18

moduli, this implies that all ux quanta are zero. In turn, (3.17) and (3.18) are both identically zero, and so no potential for the Z-moduli is developed. Therefore no moduli, not even the complex structure ones, can be stabilised at tree-level by using quantised background uxes.9 In particular, this implies that one cannot perform a two-step stabilisation (similarly to type IIB) where at tree-level the Z-moduli are xed supersymmetrically while the S- and T -moduli are kept at by tuning W0 = 0, and then these remaining moduli are lifted by quantum corrections. As we have already pointed out in section 2.1, we shall avoid this problem by considering in the next section situations with non-zero ux quanta which allow to x the Z-moduli with W0 6= 0. The dilaton and volume runaway is then

prevented by scanning over integral and fractional uxes which give a value of W0 small enough to compete with non-perturbative e ects. Hence the system becomes stable only when non-perturbative e ects are included, implying that all the moduli get stabilised beyond tree-level.

3.1.1 Chern-Simons action and gauge bundle moduli

In this section, we shall show that also the rst equation in (3.2), i.e. F(0,2) = F(2,0) = 0, can be derived from an F-term condition in 4D supergravity. This requires a brief discussion of gauge bundle moduli. Let us focus on the Chern-Simons piece of the ux superpotential:

WCS[A] =

ZX Tr

A dA +23 A A A

JHEP10(2013)199

. (3.21)

In the previous expression A is a function of both x and y, i.e. non-compact and compact coordinates respectively, but the di erentiation is just d = dymm since we are only interested in the contribution to the 4D scalar potential. We shall now write the gauge potential as:

A(x, y) = A0(y) + Adef(x, y) , (3.22)

where A0 is a background contribution independent of x and Adef is a generic deformation which can be parameterised as:

Adef(x, y) =

XI=1CI(x)I(y) , (3.23)

where CI are 4D scalar elds and I are an innite set of 1-forms living on X and valued in the adjoint representation of the structure group of the gauge bundle dened by A0.10

The superpotential (3.21) then becomes the sum of a constant, a linear, a quadratic and a cubic term in the Cs:

WCS = WCS,(0) + W ICS,(1)CI + W IJCS,(2)CICJ + W IJKCS,(3)CICJCK , (3.24)

9The corresponding situation in type IIB is very di erent since there are two types of uxes and the e ective ux G3 is complex [41].

10We expect the set of 1-forms !I to be discrete since they will be solutions to an elliptic di erential equation on the compact manifold.

19

with (for notational simplicity we dropped the trace symbol):

WCS,(0) = WCS[A0] , W ICS,(1) = 2ZX I F0 , (3.25)

W IJCS,(2) =ZX I D0J , W IJKCS,(3) =2 3

ZX I J K , (3.26)

where the gauge covariant derivative D0 is dened as D0(0,p) = (0,p) + A0 (0,p)

(1)p(0,p) A0 for an arbitrary (0, p)-form (0,p). Note that in order to derive these

expressions we used d = 0, the anti-commutativity of d and 1-forms and the cyclicity of the trace. As we have argued earlier, classically the total superpotential W should be zero at the minimum (for all the moduli), and so the F-term equation for the bundle moduli CI is:

0 = FCI = WCS

CI = W ICS,(1) + 2W IJCS,(2)CJ + 2W IJKCS,(3)CJCK . (3.27)

If Adef is a small deformation of the background A0, i.e. CI(x) = I(x), then these F-term equations can be solved order by order in , obtaining:

At zeroth order FCI = 0 gives W ICS,(1) = 0 I which from (3.25) implies that the (0, 2)-

component of the unperturbed eld strength F0 has to vanish. Hence we recover the holomorphic Yang-Mills equation F0,(0,2) = 0 which determines (given a complex structure) A0 to be a at (0, 1) connection. This bundle, which we call Q0, then determines the exterior derivative operator D0.

At linear order FCI = 0 implies (see the expression of W IJCS,(2) in (3.26)):

W IJCS,(2)CJ = 0 I CJD0J = 0 . (3.28)

This equation has two possible solutions:

1. D0i = 0 Ci for i = 1, . . . , N2. D0 6= 0 C = 0 for = N + 1, . . . ,

The rst solution denes the gauge bundle moduli which parameterise all possible deformations of the background that keep the gauge bundle holomorphic. These rst order deformations correspond to i H1(End(Q0)) where N dim H1(End(Q0))

which is expected to be nite though it may change as one varies the complex structure since the equations determining the (0, 1)-forms i depend on the Z-moduli. Hence N is a function of the Z-moduli, i.e. N = N(Z). If N = 0 for Z = Z0, then if the complex structure moduli can be stabilised via the uxes exactly at Z = Z0, the

absence of any gauge bundle moduli is guaranteed (see [37, 38] for similar considerations). Conversely, the equation D0i(Z) = 0 could be used as a mechanism to reduce the number of complex structure moduli, or even to x all of them, if the Cis develop non-zero VEVs due to D-terms or higher order terms in W [3335]. We denoted as h1,2hol the number of Z-moduli unconstrained by the equation D0i(Z) = 0, which represents the dimensionality of the sub-locus in complex structure moduli space

20

JHEP10(2013)199

JHEP10(2013)199

Figure 1. Sketch of the leading order scalar potential V |CW |2 + . . . = |Z|2|C|2 + . . . as a function of the complex structure moduli (summarily denoted by Z) and the gauge bundle moduli (summarily denoted by C) as arising at the second order in W schematically as W = WCS,(2)C2

(Z + O(Z2)) C2.

where the gauge bundle is holomorphic. In the best case scenario where h1,2hol = 0 one does not need to turn on any harmonic ux to x the Z-moduli, whereas in the more general case where h1,2hol > 0 the remaining complex structure moduli can be xed only by turning on a quantised background ux. For a graphical sketch of the cross-structure of the combined complex structure and gauge bundle moduli space see gure 1.

The second solution of (3.28) implies that the forms are not closed under D0 and the index ranges over an innite set of values. Hence C are not at directions but correspond to massive deformations, namely the Kaluza-Klein modes. We can then easily realise that W CS,(2) gives the mass matrix for these Kaluza-Klein modes.

Focusing only on the massless modes, at quadratic order FCi = 0 implies:

W ijkCS,(3)CjCk = 0 i , (3.29)

showing that a possible obstruction to the presence of gauge bundle moduli can arise if the Yukawa couplings are di erent from zero, i.e. W ijkCS,(3) 6= 0 i. We stress again the fact that W ijkCS,(3) is a function of the Z-moduli, and so even if the equation D0i(Z) = 0 (or the ux stabilisation) gives a solution Z = Z such that N(Z) 6= 0,

one could still x all the C-moduli if W ijkCS,(3)(Z) 6= 0 i.

Having motivated both the background gauge ux and the nature of the leading deformation we can now work with an arbitrary deformation by separating the set {CI} =

21

{C(0)i} {CKK } with the rst set being the massless modes and the second the Kaluza-Klein modes. This corresponds to splitting the set of 1-forms as {I} = {i(0)} { KK}

where D0i(0) = 0 while D0 KK 6= 0. Then under the condition F0,(0,2) = 0, the F-term

equations (3.27) take the form:

0 = FC(0)

i = 2W iklCS,(3)C(0)kC(0)l + 4W ik CS,(3)C(0)kCKK + 2W i CS,(3)CKKCKK , (3.30)

and:

0 = FCKK

= 2W CS,(2)CKK + 2W klCS,(3)C(0)kC(0)l + 4W k CS,(3)C(0)kCKK + 2W CS,(3)CKKCKK .(3.31)

Note that W CS,(2) M KK is the mass matrix for the Kaluza-Klein modes which by de

nition is non-singular. So eq. (3.31) can be solved for the massive modes in terms of the massless modes giving a relation of the form:

CKK = [MKK]1 W mnCS,(3)C(0)mC(0)n + O(C30) . (3.32)

Using this in (3.30) we get the massless eld equation which is the generalisation of (3.29) for arbitrarily large deformations of the background gauge bundle:

2W iklCS,(3)C(0)kC(0)l + O(C3(0)) = 0 . (3.33)

These eld equations always admit the solution C(0)k = 0 for all gauge bundle moduli which leaves the complex structure moduli unxed in the absence of harmonic quantised ux. Moreover, this solution remains valid even in the presence of non-zero W since the additional term in DCW is proportional to C. However one could also have solutions with non-zero VEVs for the C-moduli which could be obtained by cancelling eld-dependent FI-terms associated with anomalous U(1) factors. By the cross structure of the combined moduli space [3335], this in turn implies stabilisation of at most h1,2 h1,2hol complex structure moduli. This situation is particularly relevant for the case of heterotic orbifold compactications which often have only a few untwisted Z-moduli. In this case it seems possible to stabilise all gauge bundle moduli and the small total number of untwisted complex structure moduli using only higher-order terms in (3.33) and a su cient number of D-terms from anomalous U(1) factors [48].

In the rest of the paper we will focus on the generic situation where this stabilisation procedure xes all the gauge bundle moduli and some, but not all, complex structure moduli, so that h1,2hol > 0 Z-moduli are still left at. Furthermore, even if h1,2hol = 0, it could still be that some C-moduli are xed at zero VEV, implying that the complex structure moduli could still be at (see gure 1).

3.2 Corrections beyond tree-level

Given that the remaining h1,2hol > 0 Z-moduli cannot be xed at tree-level by using quantised uxes (since |W0| 6= 0 would induce a runaway for both s and V), let us focus on

perturbative and non-perturbative corrections to the scalar potential. We shall proceed in

22

JHEP10(2013)199

two steps, showing rst how to x the complex structure moduli and the dilaton by the inclusion of an S-dependent gaugino condensate, and then explaining how to stabilise the Kahler moduli by an interplay of world-sheet instantons and threshold corrections to the gauge kinetic function. For the time being, we shall neglect perturbative corrections to the Kahler potential (either or gs) since these generically break supersymmetry, and so we shall include them only in section 4 where we shall study supersymmetry breaking vacua.

3.2.1 Step 1: Z and S stabilisation by gaugino condensation

Let us add a single S-dependent gaugino condensate to the superpotential and determine how this term modies the tree-level picture:

W = Wux + Wgc =

ZX H + A(Z) e S . (3.34)

The Kahler-covariant derivatives now become:

DZ W = i b (Z) + e S

A(Z) k (Z,Z)A(Z) , (3.35)

DSW =

hi a(Z) + (2s + 1)A(Z) e Si, (3.36)

kijktjtk . (3.37)

The potential is again of the no-scale type (i.e. given by the rst equality of (3.20)). At the minimum the complex structure moduli will be frozen at the solution to:

DZ W = 0 i b (Z) = e S k (Z, Z)A(Z) A(Z) , (3.38)

and now the dilaton is not forced anymore to run-away to innity:

DSW = 0 W0 i a(Z) = (2s + 1)A(Z) e S . (3.39)

The potential for the Kahler moduli is at, resulting in a Minkowski vacuum with broken supersymmetry since substituting (3.39) into (3.37) one nds:

DTiW =

82 N

1

2s

DTiW =

kijktjtk . (3.40)

The previous expression for W0 6= 0, nite volume and ti > 1 i, gives DTiW 6= 0 for a

generic point in moduli space.Let us comment now on the possibility to satisfy (3.39) at the physical point hsi 2

that corresponds to 1GUT 25. Setting A = 1 and = 82/N where N is the rank of the

SU(N) condensing gauge group, we have (xing the axion a at hai = ):

W0 =

162hsi

N + 1

i a(Z) + A(Z) e S

4V

JHEP10(2013)199

2s 2s + 1

W0

4V

hsi . (3.41)

As an illustrative example, for hsi 2 and N = 5, the previous expression would give

W0 1012, which for V 20 corresponds to a gravitino mass of the order m3/2 =

23

e

W0/(2sV) 330 TeV. On the other hand, for N = 30 (as in the case of E8), one would

obtain W0 0.06 corresponding to a GUT-scale gravitino mass: m3/2 1016 GeV. Due

to the absence of Ramond-Ramond uxes, there is in general no freedom to tune the heterotic ux superpotential W0 to values much smaller than unity, implying that heterotic

CY compactications generically predict a gravitino mass close to the GUT scale. As we already pointed out, low-energy supersymmetry could instead be obtained in the particular cases when h1,2hol = 0 so that one does not need to turn on W0 6= 0 to x all the Z-moduli,

in orbifold constructions or in compactications on non-complex manifolds.A possible way to obtain fractional values of W0 of order 0.1 0.01 has been described

in [17] where the authors considered a trivial B2 eld and a rigid 3-cycle 3 such that the integral of H over 3 (ignoring the contribution from the spin connection):

Z 3 H Z 3

CS(A) , (3.42)

gives rise to a fractional ux.11 Stabilisation of all complex structure moduli would then require scanning the three-form ux over all cycles to search for VEVs hZ i such that

the overall (0, 3)-contribution to the superpotential (3.13) is of the order of the fractional Chern-Simons contribution or smaller.12

3.2.2 Step 2: T stabilisation by worldsheet instantons and threshold e ects

The Kahler moduli can develop a potential either by loop corrections to the gauge kinetic function or via worldsheet instantons. Let us start considering the case with just threshold e ects.

Threshold e ects: The potential generated by gaugino condensation takes the form:

Wgc = A(Z) e [parenleftBig]

S i2 Ti[parenrightBig] , (3.43)

s

V

, (3.45)

11Note that these ux quanta are well-dened quantities even if H is not closed since a rigid homology class admits just one representative

12For the purpose of an explicit demonstration of such vacua one may rely on CYs arising in Greene-Plesser pairs of manifolds related by mirror symmetry [5254]. CY mirror pairs arising from the Greene-Plesser construction have their complex structure moduli space partitioned by a typically large discrete symmetry into an invariant subspace and its complement. One can then show that the periods of the invariant subspace depend at higher-order non-trivially on all the -non-invariant complex structure moduli. If the -invariant subspace is of low dimensionality (as is the case e.g. of the CY CP411169[18] as discussed in [55, 56]), then turning on the relatively few uxes on the invariant subspace is enough to stabilise all complex structure moduli at an isolated minimum [55, 56]. On such a CY manifold one can therefore stabilise all Z-moduli by just turning a few fractional Chern-Simons (0, 3)-type uxes on the cycles of the invariant subspace, which can serve to demonstrate the existence of such vacua.

24

JHEP10(2013)199

lifting the T -moduli and modifying (3.40) into:

DTiW =

W0 2(2s + 1)

This result, in turn, gives:

Re f1loophid

hi + s

V kijktjtk

i

= 0 i =

kijktjtk . (3.44)

=

i

2 ti =

s 2V

kijktitjtk = 3 s = 3 Re ftreehid

implying that perturbation theory in the hidden sector is not under control since the one-loop contribution is bigger than the tree-level one. Moreover the gauge kinetic function of the visible sector becomes negative:

Re (fvis) = g2vis = s +

i

2 ti = 2s < 0 , (3.46)

meaning that the positive tree-level contribution is driven to negative values by threshold e ects. Actually, before becoming negative, g2vis will vanish corresponding to a strong coupling transition whose understanding is not very clear [17]. Note that we neglected

D-terms since, due to the relation (2.31), if present, they would also cause the same problems. Let us see now how these control issues can be addressed by including worldsheet instantons [19].

Threshold e ects and worldsheet instantons: The new total non-perturbative superpotential reads:

Wnp = A(Z) e [parenleftBig]

S i2 Ti[parenrightBig] + B(Z) eT , (3.47)

where we included the contribution of a single worldsheet instanton dependent on T. In general, one could have more non-perturbative contributions, but we shall here show that just one worldsheet instanton is enough to overcome the previous problems. The new Kahler covariant derivatives become:

DZ W = i b (Z) + Wgc

A(Z)

A(Z) k (Z, Z)

JHEP10(2013)199

+ Wwi

B(Z)

B(Z) k (Z, Z)

, (3.48)

DSW =

1

2s [W0 + (2s + 1)Wgc + Wwi] , (3.49)

DTpW = p

2 Wgc

W0 + Wgc + Wwi

4V

kpjktjtk p 6= , (3.50)

DTW =

2 Wgc Wwi

W0 + Wgc + Wwi

4V

kjktjtk . (3.51)

The solutions describing supersymmetric vacua with vanishing F-terms are:

i b (Z) = Wgc

k (Z, Z) A(Z)A(Z)

+ Wwi

k (Z, Z) B(Z)

B(Z) , (3.52)

W0 = (2s + 1)Wgc Wwi , (3.53) p =

s

kpjktjtk p 6= , (3.54)

=

s

V

V kjktjtk + 2R , R

Wwi

Wgc . (3.55)

It is important to note that the total superpotential W = W0 +Wgc +Wwi 6= 0. Indeed if this were zero the dilaton would not be stabilised (see (3.53)). This of course means that the supersymmetric vacua are AdS in contrast to Stromingers classical analysis [20].

The hidden and visible sector gauge kinetic functions now improve their behaviour since they look like:

Re

f1loophid

=

i

2 ti = 3 s R t = 3 Re ftreehid

R t , (3.56)

25

and:

Re (fvis) = 2s + R t . (3.57)

Thus there is a regime where the hidden sector is weakly coupled and the real part of the gauge kinetic function of the visible sector (as well as that of the hidden sector) stays positive for:

2 s R t 4 s , (3.58)

which points towards values R t 3 s. In fact, in this regime, not only Re (fvis) > 0 and

Re (fhid) > 0, but also:

Re

JHEP10(2013)199

Re

f1loopvis

Re ftreehid

f1loophid

= Re ftreevis

3

R t

s

=

1 . (3.59)

3.2.3 Tuning the Calabi-Yau condition

As pointed out in [18], in the absence of worldsheet instantons and for A(Z) = 0,

eq. (3.48) reduces to:

i b = Wgc k (Z, Z) 6= 0 . (3.60)

This induces a (2, 1)-component of H (harmonic) that should vanish according to Stromingers analysis [20]. However from (3.52), one may speculate that the CY condition can be preserved by envisaging a situation where one tunes the ux quanta such that b = 0

= 1, . . . , h1,2hol corresponding to H2,1 = 0. The complex structure moduli would then be xed by:

DZ W = 0

Wwi

Wgc =

1

@ A(Z)

A(Z)k (Z, Z)

1

. (3.61)

However now we have 4 h1,2hol real equations determining 2 h1,2hol real complex structure moduli. Obviously the system has no solution unless we scan over the integer uxes. However there are only 2 h1,2 + 2 integer uxes. Thus we have only the freedom to scan over Q = 2

h1,2 h1,2hol + 1 integers while all 2 h1,2hol real complex structure moduli as well as

all but Q of the integers (i.e. 2 h1,2hol of them) must emerge as solutions to these non-linear equations. Thus we do not think that it is possible to have b = 0 in the presence of these non-perturbative terms. However, this condition emerges only on demanding a supersym-metric solution to the classical 10D equations, and so our 4D analysis cannot be expected to satisfy these classical conditions once non-perturbative e ects are included.

3.3 Flux vacua counting

Let us clarify here a crucial di erence between type IIB and heterotic string theory regarding complex structure stabilisation with three-form ux. The F-term equations (3.48) comprise 2 h1,2 conditions for 2 h1,2 real variables (setting now h1,2hol = h1,2 for ease of comparison with type IIB). A non-trivial H-ux yields exactly 2 h1,2 independent ux quanta (up to the two related to the overall scaling of (Z)) generically supplying the non-linear system of h1,2 complex F-term conditions for the 2 h1,2 complex structure moduli. However, the

26

@ B(Z)

B(Z)k (Z, Z)

existence of a nite number of isolated solutions for such non-linear systems with as many equations as variables (rendering the system well behaved) is not guaranteed. One expects therefore that most of the available freedom of choice among the 2 h1,2 H-uxes is used up to nd a relatively small number of isolated solutions for the complex structure moduli where all of them sit safely in the regime of large complex structure. Generically, this precludes the possibility of using the H-ux discretuum for tuning a very small VEV of Wux.

Note that this is di erent in the type IIB context. There, the availability of RR three-form ux F3 supplies an additional set of 2 h2,1 uxes for an overall discretuum made up from 4 h1,2 uxes. We have therefore an additional set of 2 h1,2 discrete parameters available for tuning Wux while keeping a given well-behaved complex structure moduli vacuum. Consequently, after having used 2 h1,2 ux parameters to construct a viable complex structure vacuum, we can use the additional 2 h1,2 ux quanta to construct a discrete 2 h1,2-parameter family of complex structure vacua, which allows for exponential tuning of Wux.

Finally we note that in the heterotic case the unavailability of any additional freedom in the ux choice after xing the Z-moduli, means that we have to depend on the far more restricted choices that are available in the solution space of the complex structure moduli. As mentioned before, one needs to scan over the H ux integers in order to nd 2 h1,2

acceptable (i.e. in the geometric regime) real solutions to the 2 h1,2 non-linear equations DZ W = 0. The size of the solution set that we get is likely to be much smaller than the size of the original set of ux integers. Thus even if we had started with, let us say, h1,2 = O(100) and let each ux scan over 1 to 10, the number of acceptable uxes are likely

to be far smaller than what is required to tune the cosmological constant. It should also be emphasised here that the only source of tuning that is available after all the low-energy contributions to the vacuum energy are included, has to come from these uxes.

4 Supersymmetry breaking vacua

In this section we shall show the existence of new Minkowski vacua with spontaneous supersymmetry breaking along the Kahler moduli directions. The strategy is to perform moduli stabilisation in two steps as follows:

Step 1: Fix at leading order some of the moduli supersymmetrically (all the h1,2hol > 0 complex structure moduli, the dilaton and some Kahler moduli) at a high scale.

Step 2: Stabilise the remaining light moduli at a lower scale breaking supersymmetry

mainly along the Kahler directions by the inclusion of corrections to the Kahler potential in a way similar to type IIB.

In subsection 4.1 we shall consider the contributions to the scalar potential generated by uxes, non-perturbative e ects and threshold corrections showing that there exist no supersymmetry breaking minimum which lies in the regime of validity of the e ective eld theory. However, in subsection 4.2 we shall describe how this situation improves by the inclusion of corrections to the Kahler potential which yield trustworthy Minkowski vacua

27

JHEP10(2013)199

(see subsection 4.3) where supersymmetry is spontaneously broken by the F-terms of the Kahler moduli.13 In subsection 4.4 we shall explain what is the rle played by D-terms in our stabilisation procedure. Let us nally stress that this new procedure to obtain supersymmetry breaking vacua is completely orthogonal to the way the complex structure moduli are xed, and so our results apply also to the case with h1,2hol = 0 where there is no need to turn on quantised background uxes to x the Z-moduli.

4.1 Fluxes, non-perturbative e ects and threshold corrections

In this section we shall derive the general expression for the scalar potential including uxes, non-perturbative e ects (both gaugino condensation and world-sheet instantons) and threshold corrections for a CY three-fold whose volume is given by:

V = kbt3b kst3s . (4.1)

The superpotential and the Kahler potential look like (neglecting a possible Z-dependence of A and B and setting for simplicity s = 0):

W = Wux(Z) + A e [parenleftBig]

S b2 Tb[parenrightBig] + B eTs , (4.2)

K = ln V ln S +

S

+ Kcs(Z, Z) . (4.3)

Performing the following eld redenition:

S

b

2 Tb , (4.4)

W and K take the form:

W = Wux(Z) + A e + B eTs , (4.5)

K = ln V ln

+ + b2 Tb +

JHEP10(2013)199

Tb

+ Kcs(Z, Z) . (4.6)

4.1.1 Derivation of the F-term potential

The F-term scalar potential turns out to be:

V = eK

"XZ

K D W D

W + K D W D

W

+ K TbK Tb + K TsK Ts

W D W + W D

W

+K Ts Ts W D W + KTs TsW D

W

+|W |2 XT

KjKiKj 3!

+ KTs TbK Tb + KTs TsK Ts

W TsW + W Ts W

+KTs TsTsW Ts W

i

.

13See [57] for another attempt to x the heterotic moduli via the inclusion of e ects.

28

Let us consider the limit:

Re

f1loophid

b

2 tb s , (4.7)

which implies (dening = + i):

b tb

Re ftreehid

2 =

b tb

2(s b2 tb)

=

1 1

2s b tb

b tb

2s 1 , (4.8)

together with:

tb O(10) > ts O(1) s

kst3s kbt3b

1 . (4.9)

We can then expand the relevant terms as:

K = 42 1 + 2 +

42

3 +

The no-scale structure gets broken by loop e ects:

XTKjKiKj 3 =2(1 + )2

1 + 76 + s 12

JHEP10(2013)199

s2

6

!, K Ts = KTs = 2ts , (4.10)

K TbK Tb + K TsK Ts =

2

1 +

1 + 43 + s 6

. (4.11)

. (4.12)

Note that one correctly recovers the no-scale cancellation for b = 0 = 0. Other relevant terms are:

KTs TbK Tb + KTs TsK Ts = 2 ts

1 + 3/2

1 +

, KTs Ts = 2t2s3s (1 + 2s) . (4.13)

We shall look for minima in the region V Wux ets implying that Wwi s Wux

Wux Wgc. The relevant derivatives scale as:Z W Wux , W Wgc Wux , TsW Wwi s Wux . (4.14)

Therefore the F-term scalar potential can be expanded in the small parameters and s as:

V = V0 + V1 + 2V2 + . . . (4.15)

where (dening = Wux + Wgc):

V0 = eK

XZK D D + 42D D

! O

eK|Wux|2

,

and:

V1 = eK

"XZ

K

D D Wwi + D WwiD

+ 42

D D Wwi + D WwiD

+82D D

+ 2

W D + D

+2|

|2 2 ts

W TsW + Ts W

+ 2t2s

3s TsW

Ts W

O

eK|Wux|2

,

29

and:

2V2 = eK

"XZ

K D WwiD

Wwi + 42D WwiD

Wwi + 82

D D Wwi + h.c.

+16

3 22D

D

+2 D Wwi+WwiD

+h.c.

+ 223

W D +h.c.

2ts

Ts W D + h.c. + 2

Wwi + h.c.

52

3 |

|2

2ts( WwiTsW +h.c.)ts

W TsW +h.c.

+ 4t2s3 TsW Ts W

O 2eK|Wux|2

.

JHEP10(2013)199

4.1.2 Moduli stabilisation

Let us perform moduli stabilisation in two steps.Step 1 : We stabilise the and Z-moduli by imposing DZ = D = 0 thus

minimising the leading order term in the potential. We then substitute this solution in the scalar potential obtaining V0 = 0 whereas the other contributions take the form:

V1 = eK 2| |2 2 ts

W TsW + Ts W

+ 2t2s

3s TsW

Ts W

,

and:

2V2 = eK

"XZ

K D WwiD

Wwi + 42D WwiD

Wwi

+2

W D Wwi + h.c.

+ 2

Wwi + h.c.

52

3 |

|2

2 ts WwiTsW + h.c.

ts

W TsW + h.c.

+ 4t2s

3 TsW

Ts W

.

Step 2 : We stabilise the T -moduli at order O() breaking supersymmetry. Writing

Ts = ts + ias, W e 0 = |W e 0|ei W and B = |B|ei B, and setting ehKcsi = 1, the explicit form

of the scalar potential at O() is:

V =

A1

V2/3

+ |A2|

|W e 0|

cos(B W as)

ts ets

V

+ |A3|

|W e 0|2

e2ts

ts

|W e 0|2

hi

, (4.16)

with:

A1

b 2hik1/3b

, |A2| 2 |B|, |A3|

2|B|2

3ks , (4.17)

where we have dened W e 0 h

i = hWux + Wgci. The axion as is minimised at hasi = B W so that (4.16) reduces to:

V =

A1

V2/3

|A2|

|W e 0|

ts ets

V

+ |A3|

|W e 0|2

e2ts

ts

|W e 0|2

hi

. (4.18)

Minimising with respect to ts one nds:

V = |

A2||W e 0|

|A3|

(ts 1)(2ts + 1) t2s ets

ts1

|A2||W e 0|

2|A3|

t2s ets = 3kst2s

|B| |

W e 0| ets , (4.19)

30

which implies:

ts = ln

V |0|

2 ln ts

tsO(1)

ln

V |0|

. (4.20)

Note that we can trust our e ective eld theory when ts 1, that is when x(V) = 2.

Substituting (4.19) and (4.20) in (4.18), we end up with:

V =

hA1 V4/3 |C0| x(V)3i

|W e 0|2

hi V2

x(V) with |0|

3ks|W e 0| |B|

, where |C0|

3ks

3 . (4.21)

The extrema of V are located at:V

V

= 0 A1V4/3 = 3|C0| x(V)2

JHEP10(2013)199

x(V)

3 2

, (4.22)

showing that A1 has to be positive, i.e. b > 0, if we want to have a minimum at large volume, i.e. x(V) 2. Evaluating the second derivative at these points one nds:

2V

V2

> 0 4x2 15x + 9 < 0 . (4.23)

Hence the scalar potential has a minimum only for:

3

4 < x(V) < 3 , (4.24)

provided one can nd values of 0 that satisfy (4.22) for this range of values for V. However

these minima are not trustworthy since the blow-up mode ts is xed below the string scale as htsi x(V)/(2) < 3/(2). Moreover, the above derivation assumed a regime x(V) > 2

but leads to a condition x(V) < 3 for a minimum to exist, demonstrating the absence of a

minimum for the T -moduli in a controlled region of the scalar potential. This is consistent with a numerical analysis of the scalar potential (4.18) which shows that in the range 3/4 < x < 3 the only critical point is a saddle point with one tachyonic direction.

4.2 Inclusion of e ects

Let us now try to improve this situation taking into account also corrections to the Kahler potential described in section 2.2.1. Including both O(2) and O(3) e ects, the

Kahler potential for the T -moduli receives the following corrections:

K ln V + |

cb|

V2/3

, with s |cb| k1/3s ||k1/3b , (4.25)

where we have used eq. (2.21) for the expression for cs. These higher-derivative corrections break the no-scale structure as (neglecting threshold e ects):

XTKjKiKj 3 2|cb|

V2/3+ 2sts + 3

V. (4.26)

The scalar potential (4.18) gets modied and reads:

V =

A1

V2/3

sts + /2

V

e2ts

|cb|

V5/3

|A2|

|W e 0|

ts

V

ets + |A3|

|W e 0|2

ts +

sts + 3/2

V2

|W e 0|2

hi

. (4.27)

31

Minimising with respect to ts we nd:

V = 1

s1 + 4|A3|s(2ts + 1) |A2|2t2s(ts 1)2

!

|A2||W e 0|t2s(ts 1)

2|A3|(2ts + 1)

ets

1 +

r1 + c t3s

3ks , (4.28)

where we focused only on the solution which for s = 0 correctly reduces to (4.19) since the other solution can be shown to give rise to a maximum along the ts direction. Note that we did not take an expansion for small c/t3s even if this quantity is suppressed by ts 1 since a large denominator might be compensated by a large value of the unknown

coe cient s. Performing the following approximation:

ts = ln

V ||

ts1

|A2||W e 0|

4|A3|

t2s ets with c = 8|A3|s |A2|2

= 2s

JHEP10(2013)199

2 ln ts

tsO(1)

ln

V ||

x(V) , (4.29)

with:

|| |

0| 2

1 +

r1 + c t3s

,

and substituting (4.28) in (4.27) we end up with (in the regime x(V) 1):

V

A1 V4/3 |cb| V1/3(1 x) |C| x3 +32

|W e 0|2

hiV2

, (4.30)

where we have dened:

|C| |

C0|

2 1 +

r1 + c3 x3

!

. (4.31)

Note that if we switch o the corrections by setting |cb| = c = = 0, the scalar

potential (4.30) correctly reduces to (4.21) since || |0| and |C| |C0|.

Before trying to minimise this scalar potential, let us show two important facts:

The quantity x is always smaller than unity since from (4.25) one nds that:

s |cb|k1/3s x

k1/3sts

, and

s |cb| V1/3

1/3s 1 . (4.32)

Therefore the term in (4.30) proportional to |cb| has always a positive sign.

If the condition |c|/t3s 1 is not satised, there is no minimum for realistic values

of the underlying parameters. In fact, in this case the term proportional to |C| is

always sub-leading with respect to the term proportional to cb since for c 0:

R |

C| x3

V1/3

|cb| V1/3

1 +

r1 + c t3s

t3sc , (4.33)

which for c/t3s O(1) reduces to R O(1), whereas for c/t3s 1 reduces to

R

t3s c

1/3s

x

1 +

r1 + c t3s

pt3s/c 1. On the other hand, for c < 0, one has |c|/t3s 1 but if |c|/t3s O(1),

32

the ratio R can be shown to reduce again to R < 1/3s/x 1. Therefore in this case

the leading order scalar potential is given by (neglecting the term proportional to ):

V

A1 V4/3 |cb| V1/3 +32

|W e 0|2

hiV2

, (4.34)

with:

|c| [greaterorsimilar] t3s |cb|

s k1/3s

= 3k2/3s

3k2/3s

2 t3s . (4.35)

2 c [greaterorsimilar]

However the potential (4.34) has a minimum only if:

> 5

12 |cb|V1/3 [greaterorsimilar]

58 kst3s

x1/3s

1 , (4.36)

which is never the case for ordinary CY three-folds with O(1). As an illustrative

example, for V = 20, ts = 1.5 and ks = n/6 with n

N, one nds [greaterorsimilar] 11 n2/3 11, corresponding to a CY with Euler number negative and very large in absolute value: || = 2(h1,2 h1,1) [greaterorsimilar] 4548, while most of the known CY manifolds have || [lessorsimilar] O(1000).

Hence we have shown that in order to have a trustable minimum we need to be in a region where |c| t3s. In this case, the scalar potential (4.30) simplies to:

V

A1 V4/3 |cb| V1/3(1 x) |C0| x3 +3 2

|W e 0|2

hiV2

JHEP10(2013)199

, (4.37)

where we have approximated |C| |C0|. Note that the sign of the numerical coe cient A1

is a priori undened and depends on the sign of the underlying parameter b.

The new extrema of V are located at:

A1V4/3 = 3|C0| x2

x

3 2

+ 5|cb|

2 V1/3

1

6 x

5 +

3 5

9

2 , (4.38)

and the second derivative at these points is positive if:

u(x) 12 5|cb| V1/3

1

8 x

5 + 2

2|C0|x(4x2 15x + 9) > 0 . (4.39)

Note that for |cb| = = = 0 (4.38) and (4.39) correctly reduce to (4.22) and (4.23)

respectively. However we shall now show that by including corrections we can nd a vacuum with x 1 where we can trust the e ective eld theory.

The value of the vacuum energy is:

hV i = |

W e 0|2 2hiV2

v(x) , (4.40)

v(x) 6 + |C0|x2 (4x 9) + 3|cb|V1/3

1

4 x

3 +

where:

. (4.41)

33

Let us perform the following tuning to get a Minkowski vacuum:

v(x) = 0 if 6 = 3|cb| V1/3

1

and substitute it in (4.39) obtaining:

u(x) |cb|V1/3 (1 4) + 12|C0|x

x

4 x

3 +

+ |C0|x2 (4x 9) , (4.42)

3 2

> 0 , (4.43)

which is automatically satised for 1 and x 1. Substituting (4.42) also in the

vanishing of the rst derivative (4.38), this simplies to:

4A1 V4/3 = |cb|V1/3 (1 3) + 9|C0|x2 , (4.44)

showing that if we want to have a Minkowski minimum A1 has to be positive, i.e. b > 0.

4.3 Minkowski solutions

Let us rst dene our use of the term Minkowski solutions. Owing to the lack of tuning freedom in the heterotic three-form ux superpotential, achieving vacua with exponentially small vacuum energy is a real challenge. Thus we shall use the terminology Minkowski vacuum to refer to a vacuum with a cosmological constant suppressed by at least a 1-loop factor 1/(82) 0.01 compared to the height of the barrier in the scalar potential (of order

m23/2M2P ) which protects the T -moduli from run-away.

The solutions depend on seven underlying parameters: ks, kb, b, |B|, |cb|, || and . We do not consider |W e 0| as a free variable at this stage since we x its value at |W e 0| = 0.06

by the phenomenological requirement of obtaining the right GUT coupling corresponding to hsi hi 2. Let us now describe a strategy to nd the values of these underlying

parameters which give Minkowski vacua for desired values of the moduli and within the regime of validity of all our approximations.

1. Choose the desired values for V and ts (so xing the value of x = 2 ts). Then work

out the value of |B| as a function of ks from (4.20).2. Choose the desired value of tb and work out the value of kb as a function of ks from (4.1).

3. Determine |cb| as a function of ks, and || from (4.42).4. Derive the value of b as a function of ks, and || from (4.44).5. Choose the values of ks, and || so that all our approximations are under control,

i.e. dened in (4.8) satises 1, s dened in (4.9) gives s 1, dened

in (4.31) satises 1 and

/(2V) 1. These values of ks, and || then

give the values of kb, |B|, |cb| and b knowing that this Minkowski vacuum is fully

consistent.

34

JHEP10(2013)199

V

0.0003

0.0002

0.0001

JHEP10(2013)199

n

50 100 150 200 250 300

Figure 2. V versus V assuming the parameters listed in the text which give rise to a near-Minkowski

vacuum with hVi = 20 and a cosmological constant of small magnitude compared to height of the

barrier set by m23/2M2P .

As an illustrative example, following this procedure we found a Minkowski vacuum (see gure 2) located at:

hi hsi = 2 , hVi = 20 , htbi 5 , htsi = 1.5 , (4.45)

for the following choice of the microscopic parameters:

kb = ks = 1/6 , b 0.035 , |W e 0| = 0.06 , cb = 0.75 , cs = 0.75 ,

B 3 , 1.49 , = 2 s 0.41 , = 0 . (4.46)

Note that one can get dS or AdS solutions by varying b either above or below its benchmark value. Moreover, our approximations are under control since:

0.043 , s 0.027 ,

0.037 , 0.032 . (4.47)

We stress that at the minimum these four quantities are all of the same size: s . This has to be the case since they weight the relative strengths of loops, non

perturbative and higher derivative e ects which all compete to give a minimum.

Moreover, we point out that there seem to be problems with the expansion since we managed to obtain a minimum by tuning the underlying parameters in order to have the O(2) term of the same order of magnitude of the O(3) term, and so higher order

corrections might not be negligible.

However this might not be a problem if at least one of the following is valid:

The O(2) corrections could be eliminated by a proper redenition of the moduli.

The coe cients of higher order corrections are not tuned larger than unity, result

ing in a expansion which is under control. In fact, the expansion parameter

35

is of order qV1/3 with q an unknown coe cient. Thus O(4) contributions to the

scalar potential can be estimated as:

V 4

V 3

2q

3V1/3

0.16 for q = 1 . (4.48)

4.4 D-term potential

So far only F-terms have been taken into account. This could be consistent since moduli-dependent D-terms might not be present in the absence of anomalous U(1)s, or they might be cancelled by giving suitable VEVs to charged matter elds.

However let us see how D-terms might change the previous picture in the presence of anomalous U(1)s but without introducing charged matter elds. Because of the U(1)-invariance of the superpotential (4.5), both and Ts have to be neutral. Therefore the only eld which can be charged under an anomalous U(1) is Tb with qTb = 4 cb1(L) 6= 0.

From (2.31), this induces an FI-term of the form:

= qTbKb = qTb

kb

V

JHEP10(2013)199

1/3, (4.49)

which gives the following D-term potential:

VD = 2Re(f)

p

V2/3

. (4.50)

This term has the same volume scaling as the rst term in (4.37) which is the contribution coming from threshold e ects. However the ratio between these two terms scales as:

Vthreshold

VD =

V1/3

with p

q2Tb k2/3b

|W e 0|2q2Tb k2/3b

1 , (4.51)

for 1 and qTb O(1). As an illustrative example, our explicit parameter choice would

give Vthreshold/VD 2 104 q2Tb, showing that VD is always dominant with respect to the F-term potential (4.37). In this case, VD would give a run-away for the volume direction and destroy our moduli stabilisation scenario.

As we have already pointed out, this might not be the case if there are no anomalous U(1)s or if the FI-term is cancelled by a matter eld VEV. There is however another wayout to this D-term problem which relies on the possibility to x all the moduli charged under anomalous U(1)s in a completely supersymmetric way, so ensuring the vanishing of the D-term potential. This requires qTb = 0 and the addition of a third Kahler modulus Tc which is charged under an anomalous U(1): qTc 6= 0. Let us describe this situation in the

next subsection.

4.4.1 D + F-term stabilisation

The Kahler and superpotential now read:

W = Wux(Z) + A e ( c2 Tc) + B eTs , (4.52)

K = ln

~V ln

+

+ b

2 Tb +

Tb

+ Kcs(Z) , (4.53)

36

with

~V = V kct3c = kbt3b kst3s kct3c . (4.54)

Note that now has to get charged under an anomalous U(1) so that the hidden sector gauge kinetic function fhid = c2 Tc becomes gauge invariant. In particular we will have

q = c2 qTc. From (2.31), the FI-term looks like:

= q

D W

W qTc

DTcW

W , (4.55)

implying that VD = 0 if both and Tc are xed supersymmetrically. However we have already seen that if all the Kahler moduli are xed supersymmetrically via threshold e ects, then perturbation theory breaks down in the hidden sector and the visible sector gauge kinetic function becomes negative. A way-out proposed in section 3.2.2 was to include worldsheet instantons but, given that we want to break supersymmetry at leading order along Tb and Ts, in order to follow this possibility we should include a fourth modulus with worldsheet instantons. Thus this case does not look very appealing since it requires at least four moduli.

A simpler solution can be found by noticing that the problems with Re f1loophid >

Re ftreehid

and Re (fvis) < 0 could be avoided if only some but not all of the Kahler moduli are xed supersymmetrically by threshold e ects. We shall now prove that this is indeed the case if the T -moduli xed in this way are blow-up modes like tc. In fact, the solution to DTcW = 0 gives:

c =

s

V

JHEP10(2013)199

kct2c . (4.56)

This result, in turn, gives hidden and visible sector gauge kinetic functions of the form:

Re f1loophid

Re ftreehid

=

btb

kcjktjtk =

6 s

V

2s

ctc

2s = + 3

kc t3c

V

= + 3s 1 ,

and:

Re (fvis) = s

1 + btb2s +ctc 2c

= s

1 + 3 kct3c

V

= s (1 + 3 s) s > 0 .

5 Moduli mass spectrum, supersymmetry breaking and soft terms

Expanding the e ective eld theory around the vacua found in the previous section, we can derive the moduli mass spectrum which turns out to be (see (4.29) and (4.31) for the denitions of x and ):

mts mas m3/2 x ,mZ m m3/2 ,mtb m3/2 ,mab 0 . (5.1)

37

Note that in the absence of Tb-dependent worldsheet instantons which would give ab a mass of order mab MP etb 10 TeV for tb 5, this axion might be a good QCD axion

candidate since it could remain a at direction until standard QCD non-perturbative e ects give it a tiny mass.

Moreover, the stabilisation procedure described in the previous sections leads to vacua which break supersymmetry spontaneously mainly along the Kahler moduli directions. In fact, from the general expression of the F-terms and the gravitino mass:

F i = eK/2KjDj W and m3/2 = eK/2|W | |

we nd that the Kahler moduli F-terms read:

F Tb

tb = 2 m3/2 and

F Ts

m3/2

tb m3/2 , (5.6)

showing that the gaugino masses are suppressed with respect to the gravitino mass by a factor of order 0.03.

Scalar masses: The canonically normalised scalar masses generated by gravity me

diation read:

m20, = m23/2 F i

Fjij ln ~K , (5.7)

where ~K is the Kahler metric for matter elds which we assumed to be diagonal. ~K is generically a function of all the moduli but we shall neglect its dependence on the dilaton and the complex structure moduli since they give only a sub-leading

14Assuming that there are no cancellations from shifts of the minimum due to sub-leading corrections.

38

W e 0|

V1/2

, (5.2)

JHEP10(2013)199

F Ts

ts

m3/2

x . (5.3)

On the other hand, the dilaton and the complex structure moduli are xed supersymmetrically at leading order. However, due to the fact that the prefactor of worldsheet instantons and e ects are expected to depend on these moduli, they would also break supersymmetry at sub-leading order developing F-terms whose magnitude can be estimated as:14

DZ , W DZ

, Wwi |W e 0| F Z , m3/2 . (5.4)

Thus we can see that supersymmetry is mainly broken along the tb-direction since:

F Tb

m3/2 tb

tsx

F Z ,

m3/2 . (5.5)

The goldstino is therefore mainly the Tb-modulino which is eaten up by the gravitino in the super-Higgs mechanism.

Soft supersymmetry breaking terms are generated in the visible sector via tree-level gravitational interactions due to moduli mediation. Let us now derive their expressions:

Gaugino masses: Their canonically normalised expression is given by:

M1/2 = 1

2Re(fvis) F iifvis

F

2 +

F Tb

contribution to supersymmetry breaking. Hence we shall consider a Kahler metric for matter elds of the form ~K tnss tnbb, where ns and nb are the so-called mod

ular weights. In the type IIB set-up, it is possible to determine the value of nb by

requiring physical Yukawa couplings which do not depend on the large cycle due to the localisation of the visible sector on one of the small cycles [58]. However, in the heterotic framework the situation is di erent. For instance, in CY compactications close to the orbifold point the visible sector typically is constructed from split multiplets which partially live in the bulk and partially arise as twisted sector states localised at orbifold xed points. The value of the modular weights for the di erent matter elds is then determined by the requirements of modular invariance. Hence, they cannot be constrained by using an argument similar to the one in [58]. We shall therefore leave them as undetermined parameters. The scalar masses turn out to be:

m20 = m23/2

1 nb ns 4x2

, (5.8)

showing that for x 1, the modular weight nb has to be nb 1 in order to avoid

tachyonic squarks and sleptons. If nb = 1, one has a leading order cancellation in the scalar masses which therefore get generated by the F-terms of the small cycle ts even if F Ts F Tb (in this case one would need ns < 0). This is indeed the case

in type IIB models because of the no-scale structure [59]. Given that the no-scale cancellation holds in the heterotic case as well, we expect a similar cancellation to occur in our case, i.e. nb = 1, with possibly the exception of twisted matter elds at orbifold xed points, i.e. nb < 1 for twisted states.

A-terms: The canonically normalised A-terms look like:

A = F i

hKi + i ln Y i ln ~K ~K ~K i, (5.9)

where Y are the canonically unnormalised Yukawa couplings which can in principle depend on all the moduli. Similarly to the Kahler metric for matter elds, we introduce two modular weights, pb and ps, and we write the Yukawa couplings as

Y tpbb tpss. Thus the A-terms take the form:

A = 3m3/2

1 + pb nb +ps2x ns2x 3x

. (5.10)

In the type IIB case, there is again a leading order cancellation (since nb = 1 and

pb = 0 given that the Yukawa couplings do not depend on the Kahler moduli due to the axionic shift-symmetry and the holomorphicity of W ) which is again due to the no-scale structure [59]. Similarly to the scalar masses, we expect this leading order cancellation also in the heterotic case for matter elds living in the bulk.

and B-term: The -term can be generated by a standard Giudice-Masiero

term in the Kahler potential K

~K(ts, tb)HuHd which gives again m3/2 and

B m23/2.

39

JHEP10(2013)199

Summarising, we obtained a very specic pattern of soft terms with scalars heavier than the gauginos and universal A-terms and /B-term of the order the gravitino mass:

m0 A B m3/2 M1/2 m3/2 . (5.11)

The soft masses scale with m3/2 and do not depend on the mechanism which stabilises the complex structure and bundle moduli. Hence one can obtain TeV-scale supersymmetry by considering either smooth CY models where all the complex structure moduli are xed by the holomorphicity of the gauge bundle or orbifold constructions with a small number of untwisted Z-moduli (or better with no untwisted Z-moduli at all as in the case of some non-Abelian orbifolds). On general Calabi-Yau manifolds, we expect the soft mass scale to be of order m3/2 MGUT due the fact that in the heterotic string there is not enough

freedom to tune the ux superpotential below values of O(0.1 0.01).

6 Anisotropic solutions

In this section we shall show how to generalise the previous results to obtain anisotropic compactications with 2 large and 4 small extra dimensions which allow for a right value of the GUT scale.15 For this purpose, we shall focus on CY three-folds whose volume is [63]:

V = kbtbt2f kst3s . (6.1)

This CY admits a 4D K3 or T 4 divisor of volume t2f bered over a 2D P1 base of volume tb with an additional del Pezzo divisor of size t2s. We shall now show how to x the moduli dynamically in the anisotropic region tb tf ts. We shall consider a hidden sector gauge

kinetic function of the form:

fhid = S

b

2 Tb

f

JHEP10(2013)199

2 Tf , with s = 0 . (6.2)

The superpotential looks exactly as the one in (4.5) whereas the Kahler potential reads:

K = ln V ln

+ + b2 Tb +

Tb

+ f

2 Tf +

Tf

+ Kcs(Z) . (6.3)

Focusing on the limit where 1-loop e ects are suppressed with respect to the tree-level expression of the gauge kinetic function:

b

b tb

2 1 and f

f tf

2 1 , (6.4)

the dilaton is again xed at leading order by requiring D W = 0. On the other hand the Kahler moduli develop a subdominant potential via non-perturbative contributions, corrections and threshold e ects which break the no-scale structure as:

XTKjKiKj 3 = 2 (b + f) + O(2) . (6.5)

15For anisotropic solutions in the type IIB case for the same kind of bred CY manifolds see [6062].

40

The scalar potential has therefore the same expression as (4.16) but with a di erent coefcient A1 which is now moduli-dependent and looks like:

A1(V, tf) = V2/32 hi

b kbt2f

+ f tf

V

!

, (6.6)

where we have traded tb for V. This is the only term which depends on tf since the

rest of the potential depends just on V and ts. Hence we can x tf just minimising

A1(V, tf) obtaining:

tf =

2b kbf

1/3V1/3 tf =2bf tb . (6.7)

Substituting this result in (6.6) we nd that A1 becomes:

A1 = 3b2hik1/3b

f 2b

JHEP10(2013)199

2/3, (6.8)

which is not moduli-dependent anymore and takes a form very similar to the one in (4.17). We can therefore follow the same stabilisation procedure described in the previous sections but now with the additional relation (6.7) which, allowing the moderate tuning f 20 b,

would give an anisotropic solution with tb 10 tf. For example for V 20 and kb = 1/2,

one would obtain tb 16 tf 1.6.

We nally mention that this kind of bred CY manifolds have been successfully used in type IIB for deriving inationary models from string theory where the inaton is the Kahler modulus controlling the volume of the bre [64]. It would be very interesting to investigate if similar cosmological applications could also be present in the heterotic case.

7 Conclusions

The heterotic string on a CY manifold (or its various limiting cases such as orbifolds and Gepner points) has been studied since the late eighties as a possible UV complete theory of gravity that can realise a unied version of the SM. In the last decade there has been much progress towards the goal of getting a realistic model with the correct spectrum. However the major problem in getting phenomenologically viable solutions for the heterotic string is that the gauge theory resides in the bulk, and so getting an acceptable model cannot be decoupled from the problem of moduli stabilisation. Unfortunately a complete and deep understanding of the mechanism which stabilises all the moduli in the heterotic string is still lacking.

In this paper we tried to perform a systematic analysis of all the e ects which can develop a potential for the various moduli for the case of (0, 2)-compactications which allow for MSSM-like model building and the generation of worldsheet instantons that are crucial e ects to x the Kahler moduli. According to the original Stromingers analysis [20], these compactications violate the Kahler condition dJ = 0 due to a non-zero H-ux at

O() since in the non-standard embedding the co-exact piece of the Chern-Simons term

41

in H does not cancel. We then considered solutions to the 10D equations of motion with constant dilaton and warp factor, corresponding to special Hermitian manifolds, which represent the smallest deviations from smooth CY manifolds at O() [40].

Let us summarise the various moduli stabilisation e ects that we have taken into account:

Holomorphicity of the gauge bundle, D-terms and higher order perturbative contri

butions to the superpotential: By demanding a supersymmetric gauge bundle, i.e. a gauge bundle which satises the Hermitian Yang-Mills equations, the combined space of complex structure and gauge bundle moduli reduces from a naive direct product to a cross-structure [3336]. Therefore if the gauge bundle moduli are xed at nonzero VEVs by D-terms combined with higher order perturbative contributions to the superpotential [48], the Z-moduli are automatically lifted. However, not all the complex structure moduli might get frozen by this mechanism since, in general, the sub-locus in complex structure moduli space where the gauge bundle is holomorphic turns out to have dimension h1,2hol > 0. Hence 0 < h1,2hol < h1,2 at Z-moduli are generically left over.

Fractional Chern-Simons invariants, gaugino condensation and threshold e ects: The

remaining at Z-directions could be lifted by turning on quantised background three-form uxes [1719]. However we showed that, contrary to type IIB, this cannot be done having at the same time a vanishing VEV of the tree-level ux superpotential W0 since setting the F-terms of the Z-moduli to zero corresponds to setting the (1, 2)-component of the H-ux to zero, while demanding W0 = 0 implies that also the (3, 0)-piece of H is vanishing. Hence, being real, the whole H-ux has to be zero, resulting in the impossibility of xing the remaining complex structure moduli. Thus one needs W0 6= 0 in order to x the Z-moduli. However, due to the absence of

Ramond-Ramond uxes, it is hard to tune W0 small enough to balance the exponentially suppressed contribution from gaugino condensation which introduces an explicit dependence on the dilaton [1113] unless one turns on fractional Chern-Simons invariants (i.e. discrete Wilson lines) [17]. In this way both the dilaton and the complex structure moduli can be stabilised supersymmetrically at non-perturbative level. The Kahler moduli could then be xed by the inclusion of threshold corrections to the gauge kinetic function [21, 22].

Worldsheet instantons: The supersymmetric minimum obtained by including thresh

old e ects is not in the weak coupling regime where one can trust the e ective eld theory. This problem can be avoided by considering also the contribution of T -dependent worldsheet instantons which can give rise to reliable supersymmetric AdS vacua [19].

Higher derivative and loop corrections to the Kahler potential: The last e ects to be

taken into account are corrections to the Kahler potential [2325], while string loop e ects can be estimated to give rise to negligible contributions to the scalar

42

JHEP10(2013)199

potential [4850]. These higher derivative corrections yield new stable vacua where supersymmetry is spontaneously broken by the stabilisation mechanism which induces non-zero F-terms for the Kahler moduli, in a way very similar to type IIB LARGE Volume Scenarios [26, 27]. These new vacua can be Minkowski due to the positive contribution from threshold e ects. However, due to the lack of tuning freedom in W0, it is very hard to achieve vacua with exponentially small vacuum energy. Thus we used the term Minkowski vacua to refer to solutions with a cosmological constant suppressed by at least a loop factor with respect to the height of the barrier in the scalar potential which prevents the Kahler moduli to run-away to innity. Moreover, this stabilisation mechanism allows for anisotropic compactications with two extra dimensions which are much larger than the other four. In this way, the unication scale can be lowered down to the observed phenomenological value [28, 29], tting very well with the picture of 6D orbifold GUTs [29, 30].

After showing the existence of this new kind of supersymmetry breaking vacua, we estimated the size of the soft terms generated by gravity mediation. Interestingly, they feature universal scalar masses, A-terms and /B-term of O(m3/2) and suppressed gaugino

masses at the %-level. Moreover, a potentially viable QCD axion candidate is given by the axionic partner of the large 2-cycle modulus. However, due to the lack of tuning freedom in the ux superpotential W0 O(0.1 0.01), the gravitino mass m3/2 = W0MP /

p2Re(S)V becomes of order MGUT 1016 GeV for Re(S) 2 and V 20. This is not a problem if one

does not believe in the solution of the hierarchy problem based on low-energy supersymmetry, but it represents a generic prediction of weakly coupled heterotic compactications on internal manifolds which are smooth CY three-folds up to e ects.

However, our stabilisation procedure for the Kahler moduli that leads to spontaneous supersymmetry breaking, is completely independent on the supersymmetric mechanism which is used to x the dilaton and the complex structure moduli. Hence, if one is instead interested in low-energy supersymmetry, our way to break supersymmetry along the Kahler moduli directions could still be used by focusing on di erent ways to freeze the S-and Z-moduli:

1. In some particular examples all the complex structure moduli could be stabilised by the requirement of a holomorphic gauge bundle [3335]. In this case one could have W0 = 0 and an exponentially small superpotential, leading to a TeV-scale gravitino mass, could be generated by gaugino condensation.

2. In Abelian orbifold models the number of untwisted complex structure moduli is very small. There are also some non-Abelian orbifolds with no Z-moduli at all. Hence in this case it is rather likely that all the Z-moduli could be xed by the holomorphicity of the gauge bundle once all the singlets are xed at non-zero VEVs by cancelling FI-terms or by the e ect of higher order terms in W [48]. Again, m3/2 could then be lowered to the TeV-scale due to the exponential suppression coming from gaugino condensation [31, 32].

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3. The ux superpotential could have enough tuning freedom in the presence of uxes which are the equivalent of type IIB Ramond-Ramond uxes. This is the case of non-complex manifolds with new geometric uxes where the H ux gets modied to

H = H + i dJ [40, 4245].16

We nally stress that, even if these models could give low-energy supersymmetry, the possibility to tune the cosmological constant to the observed value still remains a challenge, in particular in the cases without a large ux discretuum.

Acknowledgments

We would like to thank Lara Anderson, Lilia Anguelova, Arthur Hebecker, James Gray, Shamit Kachru, Oleg Lebedev, Andre Lukas, Anshuman Maharana, Hans Peter Nilles, Francisco Pedro, Fernando Quevedo, Callum Quigley, Stuart Raby, Marco Serone, Savdeep Sethi, Patrick Vaudrevange, Roberto Zucchini and especially Roberto Valandro for useful discussions and correspondence. We are grateful for the support of, and the pleasant environs provided by, the Abdus Salam International Center for Theoretical Physics. This work was supported by the Impuls und Vernetzungsfond of the Helmholtz Association of German Research Centres under grant HZ-NG-603, and German Science Foundation (DFG) within the Collaborative Research Center (CRC) 676 Particles, Strings and the Early Universe, and by the United States Department of Energy under grant DE-FG02-91-ER-40672. SdA would like to acknowledge the award of an CRC 676 fellowship from DESY/Hamburg University and a visiting professorship at Abdus Salam ICTP.

A Dimensional reduction of 10D heterotic action

The 10D heterotic supergravity action in string frame for energies below the mass of the rst excited string state Ms = 1s with s = 2 contains bosonic terms of the form:

S

d10xG e2 R

1

2g210

4M810 = 4M6s . (A.2)

Compactifying on a 6D CY three-fold X, the 4D Planck scale MP turns out to be:

M2P = e2hiM810Vol(X) = 4 g2sVM2s , (A.3) where we measured the internal volume in units of M1s as Vol(X) = V 6s and we explicitly

included factors of the string coupling gs = ehi. On the other hand, the 4D gauge coupling constant becomes:

1GUT = 4g24 = 4Vol(X)g210 e2hi = g2sV . (A.4)

16See also [65].

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1 (2)74

4 TrF 2

Z

d10xG e2TrF 2 . (A.1)

Comparing the rst with the second line in (A.1), we nd:

M810 = 2

(2)74 = 4M8s and g210 =

= M810 2

Z

Z

d10xG e2R

The tree-level expression of the gauge kinetic function f = S requires Re(S) = g24, implying the following normalisation of the denition of the dilaton eld:

S = 1

4

e2V + i a . (A.5)

From (A.4), we immediately realise that there is a tension between large volume and weak coupling for the physical value 1GUT 25:

V = g2s1GUT g2s25 [lessorsimilar] 25 for gs [lessorsimilar] 1 . (A.6)

On top of this problem, isotropic compactications cannot yield the right value of the GUT scale MGUT 2.1 1016 GeV which is given by the Kaluza-Klein scale MKK = Ms/V1/6. In

fact, combining (A.3) with (A.4), one nds that the string scale is xed to be very high:

M2s = M2P41GUT

M2P100 1.35 1017 GeV

2 . (A.7)

In turn, for V [lessorsimilar] 25, the GUT scale becomes too high: MGUT = MKK [greaterorsimilar] 8 1016 GeV. The situation can be improved by focusing on anisotropic compactications with d large extra dimensions of size L = xs with x 1 and (6 d) small dimensions of string size l = s.

The internal volume then becomes Vol(X) = Ldl(6d) = xd6s = V 6s, implying that the

Kaluza-Klein scale now becomes MKK = Ms/x = Ms/V1/d. Clearly, for the case d = 6,

we recover the isotropic situation. The case with d = 1 is not very interesting since CY manifolds do not admit non-trivial Wilson lines to perform the GUT breaking. We shall therefore focus on the case d = 2 where we get the promising result:

MGUT = MKK = Ms

V

JHEP10(2013)199

[greaterorsimilar] 2.7 1016 GeV. (A.8)

Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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