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 CalabiYau threefolds up to 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, Dterms, higher order perturbative contributions to the superpotential as well as nonperturbative and threshold effects. We then show that the inclusion of 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 . 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 threeform 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. Lowenergy supersymmetry could instead be obtained by focusing on particular CalabiYau constructions where the gauge bundle is holomorphic only at a pointlike sublocus 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 noncomplex 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.
1 Introduction
String theory is a candidate for a quantum theory of gravity with full unification 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 spacetime. The multitude of possible compactification choices together with a plethora of massless 4D ‘moduli’ fields 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 compactifications on both smooth CalabiYau (CY) manifolds [3] and their singular limits in moduli space, orbifolds [4, 5, 6, 7, 8], provided constructions of 4D lowenergy effective field 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 compactifications lack a well controlled description of moduli stabilisation, and consequently, of inflationary cosmology as well.^{1}^{1}1However, for some recent attempts see e.g. [9, 10].
As weakly coupled heterotic CY compactifications lack both Dbranes and a part of the threeform flux available in type IIB, historically moduli stabilisation in the heterotic context focused mostly on the moduli dependence of 4D nonperturbative contributions to the effective action from gaugino condensation [11, 12, 13]. While this produced models of partial stabilisation of the dilaton and some Kähler moduli [9, 14, 15, 16], this route generically failed at describing controlled and explicit stabilisation of the complex structure moduli of a given CY. Moreover, the resulting vacua tend to yield values of the compactification radius and string coupling (given by the dilaton) at the boundary of validity of the supergravity approximation and the weak coupling regime.
The works [17, 18, 19] proposed to include the threeform flux to stabilise the complex structure moduli in combination with hidden sector gaugino condensation for supersymmetric dilaton stabilisation. The inclusion of fluxes in the heterotic string was originally studied by Strominger [20] who showed that, by demanding supersymmetry, the classical 10D equations of motion imply where is the fundamental form on the internal space. Hence a nonvanishing threeform flux breaks the Kähler condition . Note that this is the case of compactifications which allow for MSSMlike model building and the generation of worldsheet instantons, since in the nonstandard embedding the ChernSimons term gives a nonzero contribution to the threeform flux . However this contribution is at order , implying that the CalabiYau condition is preserved at treelevel and broken only at order . Moreover, in the heterotic case, due to the absence of RamondRamond threeform fluxes, there is generically no freedom to tune the superpotential small enough to fix the dilaton at weak coupling. However, a sufficiently small superpotential could be obtained by considering fractional ChernSimons invariants (such as discrete Wilson lines) [17]. Note that it is natural to take these effects into account for compactifications which are the most relevant for both model building and moduli stabilisation since, as we pointed out above, they feature a nonvanishing ChernSimons contribution to , regardless of the presence of fractional ChernSimons invariants.^{2}^{2}2As we shall describe in section 3.1.1, the coexact piece of the ChernSimons term is responsible for the breaking of the Kähler condition while the generation of fractional invariants is controlled by the harmonic piece of the ChernSimons term.
Supersymmetric vacua with all geometric moduli stabilised could be achieved by fixing the Kähler 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 difficulty [19]. However, none of these vacua break supersymmetry, resulting in unrealistic antide 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 Kähler moduli directions. Let us summarise our main results:

We identify two small parameters, one loopsuppressed and the other volumesuppressed, which allow us to expand the scalar potential in a leading and a subleading piece. This separation of scales allows us to perform moduli stabilisation in two steps.

The leading scalar potential is generated by Dterms, quantised background fluxes (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 fixed supersymmetrically at leading order.

The subleading scalar potential depends on the Kähler moduli and it is generated by threshold corrections to the gauge kinetic function, worldsheet instantons and [23], and [24, 25] corrections to the Kähler potential. These effects give rise to new Minkowski vacua (assuming the finetuning problem can be solved) which break supersymmetry spontaneously along the Kähler moduli directions. The dilaton is stabilised at a value in a way compatible with gauge coupling unification, while the compactification volume is fixed at 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 fibres over a base, we shall also show that this LVSlike 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 effective string scale to the GUT scale of gauge coupling unification [28, 29], and fits very well 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, Aterms and term of order the gravitino mass, , 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:

If the complex structure moduli are fixed by turning on quantised background fluxes, due to the lack of tuning freedom in the heterotic threeform flux, can at most be made of order by turning on only ChernSimons fractional fluxes. Hence the gravitino mass becomes of order GeV for and , leading to high scale supersymmetry breaking.

Let us discuss the stabilisation of the complex structure moduli in more detail. In a series of recent papers [33, 34, 35], it has been shown that in particular examples one could be able to fix all the complex structure moduli without the need to turn on any quantised background flux. Note that, as we explained above, if one focuses on compactifications, this observation is not important for preserving the CY condition (since this is broken at order regardless of the presence of a harmonic quantised flux) 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 [33, 34, 35] proved that, once the gauge bundle is required to satisfy the Hermitian YangMills equations, the combined space of gauge bundle and complex structure moduli is not a simple direct product but acquires a ‘crossstructure’. Denoting the gauge bundle moduli as , , and the complex structure moduli as , , this observation implies that the dimensionality of the gauge bundle moduli space is actually a function of the complex structure moduli, i.e. , and viceversa the number of massless fields 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:
(1) 
The form of this potential implies that:

If is fixed by some stabilisation mechanism (like Dterms combined with higher order dependent terms in the superpotential) at , then complex structure moduli are fixed at . Hence the number of moduli left flat is given by , which is also the dimensionality of the sublocus in complex structure moduli space for where the gauge bundle is holomorphic. Hence the best case scenario is when this sublocus is just a point, i.e. .

If the moduli are fixed by some stabilisation mechanism (like by turning on background quantised fluxes) at values different from zero, then the gauge bundle modulus is fixed at .^{3}^{3}3See also [37] for a mathematical discussion of this issue which basically comes to the same conclusion that gauge bundle moduli are generically absent.
However this stabilisation mechanism generically does not lead to the fixing of all complex structure moduli due to the difficulty of finding examples with , i.e. with a pointlike sublocus in complex structure moduli space where the gauge bundle is holomorphic. In fact, there is so far no explicit example in the literature where can be obtained without having a singular CY even if there has been recently some progress in understanding how to resolve these singular pointlike subloci [38]. Moreover, let us stress that even if one finds a nonsingular CY example with (there is in principle no obstruction to the existence of this best case scenario), all the complex structure moduli are fixed only if , since for the directions would still be flat. As we pointed out above, could be guaranteed by the interplay of Dterms and higher order terms in the superpotential, but in the case when the number of moduli is large, one should carefully check that all of them are fixed at nonzero values (for example, one might like to have some of them to be fixed at zero in order to preserve some symmetries relevant for phenomenology like ). Thus the requirement of a holomorphic gauge bundle generically fixes some complex structure moduli but not all of them. Note also that these solutions are not guaranteed to survive for a nonvanishing superpotential, since one would then need to solve a set of nonholomorphic equations.
Let us therefore analyse the general case where some moduli are left flat 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 nonstandard embedding where the flux already gets a nonvanishing contribution from the coexact piece of the ChernSimons term, we consider quite natural the option to turn on also a harmonic ChernSimons piece that could yield a fractional dependent superpotential that lifts the remaining complex structure moduli [17].

If , as in the case of compactifications, 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’ [39].

The inclusion of quantised background fluxes cannot fix the remaining complex structure moduli in a supersymmetric way with, at the same time, a vanishing flux superpotential . In fact, setting the Fterms of the moduli to zero corresponds to setting the component of to zero, whereas setting implies a vanishing component of . As a consequence, given that the flux is real, the entire harmonic flux is zero, and so the moduli are still flat.^{4}^{4}4This statement is also implicit in [18]. Note that this would not be the case in type IIB where the threeform flux is complex (because of the presence of also RamondRamond fluxes) [40].

The remaining moduli can be fixed only if but this would lead to a runaway for the dilaton if is not finetuned to exponentially small values to balance the dilatondependent contribution from gaugino condensation. However, due to the absence of RamondRamond fluxes, the heterotic flux does not contain enough freedom to tune to small values, since it is used mostly to stabilise the complex structure moduli in a controlled vacuum. There are then two options:

Models with either accidentally cancelling integer flux quanta or only ChernSimons fractional fluxes where the flux superpotential could be small enough to compete with gaugino condensation, even if this case would lead to supersymmetry breaking around the GUT scale;

Compactifications on nonKähler manifolds which do not admit a closed holomorphic form, since these cases allow for new geometric fluxes which could play a similar rôle as type IIB RamondRamond fluxes, and could be used to tune to small values [39, 41, 42, 43, 44]. 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 effects.

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

Models where the requirement of a holomorphic gauge bundle fixes all complex structure moduli without inducing singularities (so that the supergravity approximation is reliable), i.e. models with [33, 34, 35]. The dilaton could then be fixed in a supersymmetric way by using a double gaugino condensate while the Kähler moduli could be fixed following our LVSlike method by including worldsheet instantons, threshold and effects. This global minimum would break supersymmetry spontaneously along the Kähler moduli directions. The gravitino mass could then be around the TeV scale because of the exponential suppression from gaugino condensation.

Simple models with a very small number of complex structure moduli, like Abelian orbifolds with a few untwisted moduli, or even nonAbelian orbifolds with no complex structure moduli at all. In fact, in this case gauge singlets could be fixed at nonzero values via Dterms induced by anomalous factors and higher order terms in the superpotential [4, 5, 6, 7, 8], so resulting in cases where all the moduli become massive by the holomorphicity of the gauge bundle. The dilaton could then be fixed 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 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 finally note that accommodating our observed cosmological constant, which is a challenge even with fluxes and complex structures, is even more of a challenge in cases without quantised fluxes.
This paper is organised as follows. In Section 2 we introduce the general framework of heterotic CY compactifications [45, 46], reviewing the form of the treelevel effective action and then presenting a systematic discussion of quantum corrections from nonperturbative effects [11, 12, 13], string loops [47, 48, 49], and higherderivative corrections [23, 24, 25] 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 finally present our conclusions in Section 7.
2 Heterotic framework
Let us focus on weakly coupled heterotic string theory compactified on a smooth CY threefold . The 4D effective supergravity theory involves several moduli: complex structure moduli , ; the dilaton and Kähler moduli , (besides several gauge bundle moduli).
The real part of is set by the 4D dilaton (see appendix A for the correct normalisation):
(2) 
where is the 10D dilaton whose VEV gives the string coupling . The imaginary part of is given by the universal axion which is the 4D dual of . On the other hand, the real part of the Kähler moduli, , measures the volume of internal twocycles in units of the string length . The imaginary part of is given by the reduction of along the basis form dual to the divisor .
We shall focus on general nonstandard embeddings with possible factors in the visible sector. Hence the gauge bundle in the visible takes the form where is a nonAbelian bundle whereas the are line bundles. On the other hand the vector bundle in the hidden involves just a nonAbelian factor . We shall not allow line bundles in the hidden sector since, just for simplicity, we shall not consider matter fields charged under anomalous s. In fact, if we want to generate a superpotential from gaugino condensation in the hidden sector in order to fix the moduli, all the anomalous s have to reside in the visible sector otherwise, as we shall explain later on, the superpotential would not be gauge invariant.
2.1 Treelevel expressions
The treelevel Kähler potential takes the form:
(3) 
where is the CY volume measured in string units, while is the holomorphic form of that depends implicitly on the moduli. The internal volume depends on the moduli since it looks like:
(4) 
where are the triple intersection numbers of .
The treelevel holomorphic gauge kinetic function for both the visible and hidden sector is given by the dilaton:
(5) 
The treelevel superpotential is generated by the threeform flux and it reads:
(6) 
with the correct definition of including effects:
(7) 
where is the ChernSimons threeform for the gauge connection :
(8) 
and is the gravitational equivalent for the spin connection .
The VEV of the treelevel superpotential, , is of crucial importance. Due to the difference with type IIB where one has two threeform fluxes, which can give rise to cancellations among themselves leading to small values of , in the heterotic case is generically of order unity. Hence one experiences two problems:

Contrary to type IIB, the heterotic dilaton is not fixed by the flux superpotential, resulting in a supergravity theory which is not of noscale type. More precisely, the Fterm scalar potential:
(9) derived from (3) and (6) simplifies to:
(10) since and . Setting , the scalar potential (10) reduces to:
(11) yielding a runaway for both and if . Given that generically , it is very hard to balance this treelevel runaway against dependent nonperturbative effects which are exponentially suppressed in . One could try to do it by considering small values of but this would involve a strong coupling limit where control over moduli stabilisation is lost. A possible way to lower was proposed in [17] where the authors derived the topological conditions to have fractional ChernSimons invariants.

If , even if it is fractional, one cannot obtain lowenergy supersymmetry. In fact, the gravitino mass is given by , and so the invariant quantity has to be of order to have TeVscale supersymmetry. As we have seen, the 4D gauge coupling is given by , and so a huge value of the internal volume would lead to a hyperweak GUT coupling. Note that a very large value of cannot be compensated by a very small value of since we do not want to violate string perturbation theory.
Let us briefly mention that in some particular cases one could have an accidental cancellation among the flux quanta which yields a small as suggested in [18]. We stress that in the heterotic case, contrary to type IIB, this cancellation is highly nongeneric, and so it is not very appealing to rely on it to lower . Hence it would seem that the most promising way to get lowenergy supersymmetry is to consider the case where and generate an exponentially small superpotential only at subleading nonperturbative level. This case was considered in [34], where the authors argued that, at treelevel, one can in principle obtain a Minkowski supersymmetric vacuum with all complex structure moduli stabilised and flat directions corresponding to the dilaton and the Kähler moduli. As explained in Section 1, this corresponds to the best case scenario where the gauge bundle is holomorphic only at a nonsingular pointlike sublocus in complex structure moduli space.
If instead one focuses on the more general case where moduli are left flat after imposing the requirement of a holomorphic gauge bundle, as we shall show in section 3, the conditions and imply that no quantised flux is turned on, resulting in the impossibility to stabilise the remaining moduli. This result implies that it is impossible to stabilise the remaining complex structure moduli and the dilaton in two steps with a moduli stabilisation at treelevel and a dilaton stabilisation at subleading nonperturbative level. In this case there are two possible wayouts:

Focus on the case and so that can be nontrivial. In this case one has however a dilaton runaway, implying that no moduli can be fixed at treelevel. One needs therefore to add dependent nonperturbative effects which have to be balanced against the treelevel superpotential to lift the runaway. A small could be obtained either considering fractional ChernSimons invariants or advocating accidental cancellations among the flux quanta.

Focus on the case with trivial so that no scalar potential is generated at treelevel. The dilaton and the complex structure moduli could then be fixed at nonperturbative level via a racetrack superpotential generating an exponentially small which could lead to lowenergy supersymmetry. Note that even though for models, it is still possible to have since only the harmonic part of the flux 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 nonperturbative terms. Since this is generically not the case for heterotic compactifications, this avenue will not lead to supersymmetric moduli stabilisation except perhaps for a few specific cases. Note that in this case to get a Minkowski supersymmetric vacuum one would have to finetune the prefactors of the two (or more) condensates so that at the minimum. Then one would have (under the conditions mentioned above) a set of holomorphic equations for the moduli which will always have a solution. However once supersymmetry is broken this option is nolonger available since now one needs to have at the minimum if one is to have any hope of finetuning the cosmological constant to zero. However now the equations for the moduli are a set of real nonlinear equations which are not guaranteed to have a solution.
2.2 Corrections beyond leading order
As explained in the previous section, in smooth heterotic compactifications with complex structure moduli not fixed by the holomorphicity of the gauge bundle, these moduli cannot be frozen at treelevel by turning on a quantised background flux since this stabilisation would need 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 effects to be taken into account (perturbative and nonperturbative in both and ), let us mention two wellknown control issues in heterotic constructions:

Tension between weak coupling and large volume: In order to have full control over the effective field theory, one would like to stabilise the moduli in a region of field space where both perturbative and higher derivative corrections are small, i.e. respectively for and . However, as we have already pointed out, this can be the case only if the 4D coupling is hyperweak, in contrast with phenomenological observations. In fact, we have:
(12) and so if we require , the CY volume cannot be very large, , implying that one has never a solid parametric control over the approximations used to fix the moduli.

Tension between GUT scale and large volume: In heterotic constructions, the unification scale is identified with the KaluzaKlein scale, , which cannot be lowered that much below the string scale for , resulting in a GUT scale which is generically higher than the value inferred from the 1loop running of the MSSM gauge couplings. In more detail, the string scale can be expressed in terms of the 4D Planck scale from dimensional reduction as (see appendix A for an explicit derivation):
(13) In the case of an isotropic compactification, the KaluzaKlein scale takes the form:
(14) which is clearly above the phenomenological value GeV. On the other hand, anisotropic compactifications with large dimensions of size with and small dimensions of string size , can lower the KaluzaKlein scale:
(15) For the case , one would get the encouraging result GeV.
2.2.1 Higher derivative effects
Let us start considering higher derivative effects, i.e. perturbative corrections to the Kähler potential. In the case of the standard embedding corresponding to worldsheet theories, the leading correction arises at [24] and depends on the CY Euler number . Its form can be derived by substituting the corrected volume into the treelevel expression (3) with . Given that , is of the order for ordinary CY threefolds with . Hence for , the ratio is a small number which justifies the expansion:
(16) 
As pointed out in [23] however, this is the leading order higher derivative effect only for the standard embedding since worldsheet theories admit corrections already at which deform the Kähler form as:
(17) 
where both and are modulidependent forms which are orthogonal to , i.e. . Plugging into the treelevel expression for (3) and then expanding, one finds that the correction vanishes because of the orthogonality between and whereas at one finds:^{5}^{5}5In looking at the derivation of the correction at in [23], one may wonder about the rôle of field redefinitions. The fact that the corrected Kähler potential can be written in terms of as a function of alone, just the same way as the treelevel in terms of , may imply that a field redefinition of the Kähler form may actually fully absorb the correction at . To this end, the observation in [23] that the generically nonvanishing string 1loop corrections in type IIB appearing at are Sdual to the heterotic correction, provides additional evidence for the existence of this term.
(18) 
Note that the correction (18) is generically leading with respect to (16) since (18) should be more correctly rewritten as:
(19) 
where is a homogeneous function of the Kähler moduli of degree 0 given that scales as and does not depend on . As an illustrative example, let us consider the simplest Swisscheese CY with one large twocycle and one small blowup mode so that and the volume reads:
(20) 
In the limit , the function then becomes (considering, without loss of generality, as moduliindependent):
(21) 
The sign of and can be constrained as follows. In the limit , reduces to . On the other hand, requiring that is semipositive definite for any point in Kähler moduli space one finds:
(22) 
where is a semipositive definite quantity.
2.2.2 Loop effects
Let us now focus on perturbative effects which can modify both the Kähler potential and the gauge kinetic function. The exact expression of the string loop corrections to the Kähler potential is not known due to the difficulty in computing string scattering amplitudes on CY backgrounds. However, in the case of type IIB compactifications, these corrections have been argued to be subleading compared to effects by considering the results for simple toroidal orientifolds [47] and trying to generalise them to arbitrary CY backgrounds [48, 49]. Following [49], we shall try to estimate the behaviour of string loop corrections to the scalar potential by demanding that these match the ColemanWeinberg potential:
(23) 
where we took the cutoff scale and we considered arbitrary large dimensions. Note that these effects are indeed subdominant with respect to the ones for large volume since the and corrections, (19) and (16), give respectively a contribution to the scalar potential of the order and , whereas the potential (23) scales as for the isotropic case with and for the anisotropic case with . Due to this subdominant behaviour of the string loop effects, we shall neglect them in what follows.
2.2.3 Nonperturbative effects
The 4D effective action receives also nonperturbative corrections in both and . The effects are worldsheet instantons wrapping an internal twocycle . These give a contribution to the superpotential of the form:
(26) 
Note that these contributions arise only for worldsheet theories whereas they are absent in the case of the standard embedding. On the other hand, nonperturbative effects include gaugino condensation and NS5 instantons. In the case of gaugino condensation in the hidden sector group, the resulting superpotential looks like:
(27) 
where in the absence of hidden sector factors, all the hidden sector gauge groups have the same gauge kinetic function. Finally, NS5 instantons wrapping the whole CY manifold would give a subleading nonperturbative superpotential suppressed by , and so we shall neglect them.
2.3 Modulidependent FayetIliopoulos terms
As already pointed out, we shall allow line bundles in the visible sector where we turn on a vector bundle of the form . The presence of anomalous factors induces charges for the moduli in order to cancel the anomalies and gives rise to modulidependent FayetIliopoulos (FI) terms. In particular, the charges of the Kähler moduli and the dilaton under the th anomalous read:
(28) 
so that the FIterms become [22]:
(29) 
Note that the dilatondependent term in the previous expression is a 1loop correction to the FIterms which at treelevel depend just on the Kähler moduli. The final Dterm potential takes the form:
(30) 
From the expressions (28) for the charges of the moduli, we can now check the invariance of the nonperturbative superpotentials (26) and (27). In the absence of charged matter fields, the only way to obtain a gauge invariant worldsheet instanton is to choose the gauge bundle such that all the appearing in are not charged, i.e. and . The superpotential generated by gaugino condensation is instead automatically invariant by construction since all the anomalous 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 :
(31) 
Before concluding this section, we recall that in supergravity the Dterms are proportional to the Fterms for . In fact, the total charge of the superpotential is given by , and so one can write:
(32) 
where the Fterms are defined as . Therefore if all the Fterms are vanishing with , the FIterms are also all automatically zero without giving rise to independent modulifixing relations.
3 Supersymmetric vacua
In this section, we shall perform a systematic discussion of heterotic supersymmetric vacua starting from an analysis of the treelevel scalar potential and then including corrections beyond the leading order expressions.
3.1 Treelevel scalar potential
In [20], Strominger analysed the 10D equations of motion and worked out the necessary and sufficient conditions to obtain supersymmetry in 4D assuming a 10D spacetime of the form where is a maximally symmetric 4D spacetime and is a compact 6D manifold:

is Minkowski;

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

There exists one globally defined holomorphic form which is closed, i.e. , and whose norm is related to the complex structure form as (up to a constant):^{6}^{6}6The adjoint operator can be defined from the inner product as . For an even dimensional manifold, as we have here, .
(33) 
The background gauge field has to satisfy the Hermitian YangMills equations:
(34) 
The dilaton and the warp factor have to satisfy (again up to a constant):^{7}^{7}7We are writing the total metric as .
(35) 
The background threeform flux is given by:
(36) together with the Bianchi identity:
(37)
Some of the conditions listed above can be reformulated also in terms of constraints on the five torsional classes , (for a review see [1, 39]). The second condition corresponds to implying that the torsional class belongs to the space . This is the case of ‘Hermitian manifolds’. Moreover, the third condition above gives implying that both and are exact real 1forms. We shall focus on the simplest solution to which is corresponding to the case of ‘specialHermitian manifolds’ where the dilaton and the warp factor are constant [39]. More general solutions involve a nonconstant dilaton profile in the extra dimensions and for but we shall not consider this option [39].
Let us comment on the implications of the last Strominger condition (36) which for constant dilaton can be rewritten as . Using the Hodge decomposition theorem, the threeform can be expanded uniquely as:
(38) 
where is a harmonic form, is an exact form and is a coexact form which are all orthogonal to each other. Given that , (36) implies that is a coexact form, and so . Moreover, since is a form, (36) implies that the component of is zero while the component breaks the Kähler condition . However this happens only at . In fact, the general expression of the flux is:
(39) 
where is a harmonic piece and the combination of ChernSimons threeforms can also be decomposed as:
(40) 
Comparing the two expressions for , (38) and (39), we have (due to the uniqueness of the Hodge decomposition):
Then the relation (36) takes the form:
(41) 
showing exactly that the Kähler condition is violated at . Note that this would be the case for the nonstandard embedding where contrary to the less generic situation of the standard embedding where the ChernSimons piece vanishes. Taking the exterior derivative of (41) we recover the Bianchi identity (37) which now looks like:
(42) 
This 10D analysis can also be recast in terms of an effective potential which can be written as a sum of BPSlike terms and whose minimisation reproduces the conditions above [36, 41, 42, 43]. Furthermore, some of these conditions can be rederived as F or Dterm 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, 43], that the second equation in (34) is equivalent to a Dterm condition since:
(43) 
This Dterm condition holds for general nonAbelian gauge fields. If we restrict to Abelian fluxes and integrate the above condition over the CY, this reproduces the treelevel expression for the FayetIliopoulos terms given in (29). If we expand the Abelian fluxes as together with we obtain:
(44) 
which reproduces exactly the treelevel part of (29).
Regarding the Fterms, as we have seen in section 2.1, the starting point is the expression of the flux superpotential which has been inferred in [44] by comparing the dimensional reduction of the 10D coupling of to the gravitino mass term in the 4D supergravity action. The final result is:^{8}^{8}8In [41] and [43] it is suggested that the complete expression for should more appropriately be , similarly to the type IIB case where one has the RR flux in addition to the flux. Integrating by parts, the new piece can be rewritten as which clearly vanishes since . However, if one considers the case where , i.e. where supersymmetry is broken directly at the 10D level, this integral would still be zero if the internal manifold is complex since is of Hodge type while is . Thus this term can play a useful rôle only for noncomplex manifolds with broken supersymmetry. Due to the difficulty to study this case in a controlled way, we shall not consider it and neglect this additional piece.
(45) 
Note that only the harmonic component of contributes to . The harmonic piece can be expanded in a basis of harmonic  and forms as:
(46) 
The same , together with the holomorphic form , can also be expanded in a symplectic basis of harmonic threeforms () such that and with :
(47) 
where with a homogeneous function of degree 2. Note that and do not depend on the complex structure moduli which are defined by the expansion of in (47). If () is the dual symplectic basis of 3cycles such that and , we have (choosing units such that ):
(48) 
and similarly . The quantities and are integer flux quanta.
The expansion of the flux superpotential (45) is then given by:
(49)  
where we normalised and used the fact that and the orthogonality of the different Hodge components of .
Let us now evaluate the complex structure Fterms . Using the fact that (see for example [50]):
and:
and expanding a generic element of the basis of harmonic forms as , we find:
(50)  
where we used again the orthogonality of the different Hodge components of and the fact that . On the other hand, the dilaton and Kähler moduli Fterms look like: