HDTHEP089
SIASCMTP081
CPHTRR002.0108
LPTORSAY0816
Compact heterotic orbifolds in blow–up
[.5cm] Stefan Groot Nibbelink, Denis Klevers, Felix Plöger,
Michele Trapletti, Patrick K.S. Vaudrevange
[.5cm] Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16 und 19, D69120 Heidelberg, Germany & Shanghai Institute for Advanced Study, University of Science and Technology of China, 99 Xiupu Rd, Pudong, Shanghai 201315, P.R. China
[.3cm] Physikalisches Institut der Universität Bonn, Nussallee 12, D53115 Bonn, Germany
[.3cm] Institut für Chemie und Dynamik der Geosphäre, ICG1: Stratosphäre, Forschungszentrum Jülich GmbH, D52425 Jülich, Germany
[.3cm] Laboratoire de Physique Theorique, Bat. 210, Université de Paris–Sud, F91405 Orsay, France & Centre de Physique Théorique, École Polytechnique, F91128 Palaiseau, France
Abstract
We compare heterotic string models on orbifolds with supergravity models on smooth compact spaces, obtained by resolving the orbifold singularities. Our main focus is on heterotic models on the resolution of the compact orbifold with Wilson lines. We explain how different gauge fluxes at various resolved fixed points can be interpreted in blow down as Wilson lines. Even when such Wilson lines are trivial from the orbifold perspective, they can still lead to additional symmetry breaking in blow–up. Full agreement is achieved between orbifold and resolved models, at the level of gauge interactions, massless spectrum and anomaly cancellation. In this matching the blow–up modes are of crucial importance: they play the role of model–dependent axions involved in the cancellation of multiple anomalous U(1)’s on the resolution. We illustrate various aspects by investigating blow–ups of a MSSM model with two Wilson lines: if all its fixed points are resolved simultaneously, the SM gauge group is necessarily broken. Finally, we explore in detail the anomaly cancellation on the complex two dimensional resolution of .
1 Introduction
Orbifold compactification of the heterotic string [1] has been one of the first approaches to string phenomenology. In the past, vast scans of possible 4D models were undertaken with the aim of reproducing the spectrum and the interactions of the Standard Model of particle physics or of a supersymmetric extension of it (MSSM), see e.g. [2]. The interest in this approach has been recently revived with the initial goal to obtain “orbifold GUTs” [3] from string compactifications [4, 5, 6, 7, 8]. With this technology in mind, the original aim of building a 4D MSSM model was re–established, leading to many successful constructions [7, 9] that are nowadays some of the best known string models (for other constructions see e.g. [10, 11, 12, 13]).
The orbifold constructions have proven to be one of the most successful approaches to string phenomenology, yet this approach has a severe limitation: exact string quantization is only possible on the orbifold, as it is constructed by combining free conformal field theories (CFTs). This means that “calculability” is limited to a single point in the moduli space of the model. This does not mean that away from the orbifold point one has no control over the resulting 4D model: we can describe a model away from the orbifold point by giving vevs to some twisted states. Nevertheless, this extrapolation away from the orbifold point in moduli space is sensible only if these vevs are sufficiently small. Otherwise, the standard truncation to the 4D supergravity Lagrangian cannot be trusted. However, there are good reasons to consider big deformations of the orbifold model: having access to only a limited region in moduli space makes it virtually impossible to achieve an efficient moduli stabilization mechanism or to study supersymmetry breaking vacua.
In order to overcome this obstacle, it would be crucial to combine the model building power of orbifold constructions with an approach able to characterize realistic models away from the orbifold point in moduli space, i.e. when the orbifold singularities are resolved [14]. This is the main intention of this paper. To do so, we build on the results of [15], where the resolutions of singularities were considered. The freedom in the embedding of the U(1) bundle on these resolution spaces into the SO(32) gauge group of 10D heterotic supergravity allowed for the construction of a range of resolved models in 4D and 6D. These models could be matched with the corresponding singular orbifold models built by quantizing the heterotic string on , with , using the standard CFT techniques. In this matching it was crucial to “blow–up” the orbifold model by giving a vev to a certain twisted scalar which defines the “blow–up mode” [16]. This matching was refined in [17], where the issues of multiple anomalous U(1)’s and generalized Green–Schwarz mechanisms were addressed. Using toric geometry [18] this program can be carried out to a much wider class of orbifold singularities.
In the present paper we want to consider the heterotic string on the compact orbifold . (For a discussion of the heterotic SO(32) string on such an orbifold, see e.g. [19].) The compactness of this orbifold is very relevant for phenomenology since, apart from a finite 4D Planck mass, it allows us to include discrete Wilson lines, which are crucial for model building. To prepare for our study of this compact orbifold, we extend the results of [15] to the heterotic string on in Section 2. We first recall the resolved geometry and the form of U(1) bundles on it. After this we consider all possible embeddings of the U(1) flux and describe the resulting five inequivalent resolved models. We relate each resolution model to a known heterotic orbifold model by switching on a certain blow–up mode. We check that the vev of this twisted state is essentially compatible with F– and D–flatness. Finally, we explain how one can use field redefinitions to understand that their spectra agree in detail.
In Section 3 we construct the resolution of the compact orbifold by cutting a local patch around each singularity and replacing it by the resolved space with U(1) bundles, as described in Section 2. The matching in the absence of discrete Wilson lines seems to be a straightforward extension of the results of Section 2. However, two minor complications arise: firstly, the superpotential in the compact case is generically more complicated than the non–compact one, hence F–flatness needs to be rechecked. Secondly, it is possible that there is a trivial Wilson line between two orbifold fixed points, which in blow–up nevertheless leads to a further symmetry breaking.
From the resolution perspective, we interpret discrete Wilson lines as the possibility of wrapping different fluxes around each resolved singularity. In other words, this freedom can be understood as non–trivial transition functions for the gauge backgrounds going from one resolved singularity to the other. We study the consistency conditions for the transition functions. Furthermore, we show that the presence of these transition functions affects the computation of the unbroken 4D gauge group and of the localized (twisted) and delocalized (untwisted) matter spectrum. We conclude this section with two examples: the first example illustrates how to embed a discrete Wilson line on the resolution of and exemplifies the possibility of having multiple anomalous U(1)’s in compact blow–ups. The second example demonstrates some of the potential consequences of blowing up all singularities for semi–realistic MSSM–like models: contrary to the orbifold theory, in a full resolution of the model in exam the hypercharge U(1) is necessarily broken.
In Section 4 we pass to the study of resolutions of the orbifold and extend the purely topological approach to the resolution of singularities, as considered in [20]: there it was noted that the 6D anomaly polynomials of the heterotic orbifold and of the related smooth models are not the same. We analyze this problem in the same spirit of Section 2, matching the models at the level of the gauge interactions and spectra after giving a vev to a suitable blow–up mode. We show that the matching of the anomaly cancellations requires carefully considering the consequences of the field redefinitions that make the U(1) charges of the models match.
The paper is concluded by Section 5, which summarizes our main findings.
2 Heterotic orbifold and resolution models
We consider the heterotic string quantized on the singular space and on its resolution. We start by giving the geometrical details of the singularity. Then we show how to resolve it and how to construct gauge fluxes on the resolution. After this study of the geometry, we consider the heterotic string on the singular space and on the resolution, leading to 4D heterotic orbifold and resolved models, respectively. Finally, we investigate how the two classes of models match with particular care for the anomaly cancellation: we show that, on the orbifold side, the standard Green–Schwarz mechanism, involving one single universal axion, is combined with a Higgs mechanism giving rise to the blow–up. On the resolution, this combination is mapped into a Green–Schwarz mechanism involving two axions. These are mixtures of the orbifold axion and of the blow–up mode. This identification is completed by the observation that the new Fayet–Iliopoulos term produced on the resolution is nothing else than the (tree–level) D–term due to the non–vanishing vev of the blow–up mode.
2.1 Orbifold and blow–up geometry
We start from parameterized by the three complex coordinates (), on which the orbifold rotation acts as
(1) 
The non–compact orbifold is obtained by identifying those points in that are mapped into each other by . Such a space is singular in the fixed point , and is naturally equipped with a Kähler potential, inherited from ,
(2) 
We can cover {0} by means of three coordinate patches, defined as
(3) 
It is convenient to choose new coordinates on the orbifold, which allow for a systematic construction of a resolution of the singularity as a line bundle over . In the language of toric geometry [18, 21], the is called an exceptional divisor, and it replaces the singularity in the resolution of . When its volume shrinks to zero, the singularity is recovered, and the space approaches (blow–down). Thus the blowing–up/down procedure is controlled by the size of the exceptional divisor. To make this more explicit we consider the patch , where , and define for . To remove the deficit angle of we perform the coordinate transformation . In this way the Kähler potential becomes
(4) 
This change of variables trades the deficit angle for a non–analyticity in .
A resolution of the orbifold is given by considering the open patches introduced above, equipped with a new Kähler potential [15]
(5) 
that is Ricci–flat and matches the orbifold Kähler potential in the limit. In this limit the curvature vanishes for points where , whereas for it diverges. Moreover, it vanishes for any value of when . Therefore, blowing up means that the orbifold singularity is replaced by the smooth compact that shrinks to zero as (the situation is illustrated in Figure 1).
2.2 Gauge fluxes wrapped on the orbifold and the resolution
When defining the heterotic string on , the 10D gauge group ^{1}^{1}1We restrict to this case as the SO(32) theory was considered in [15, 17]. is broken by the orbifolding procedure. We can understand this breaking from an effective field theory perspective: let be the one–form gauge field, with values in , and let be its field strength. Moreover, define , for , as the basis elements of the Cartan subalgebra of . In a given coordinate patch with local coordinates , the orbifold action is realized as . On the orbifold there can be non–trivial orbifold boundary conditions for
(6) 
where and is a vector in the root lattice. These boundary conditions induce a gauge symmetry breaking, precisely to those algebra elements with root vectors such that . Our normalization of the orbifold gauge shift vector differs by a factor from the common one; our convention avoids such an additional factor when we make identification with gauge bundle fluxes below.
The non–trivial orbifold boundary conditions can be reformulated in terms of fields with trivial ones, but having a non–zero constant gauge background. The existence of this non–vanishing gauge flux, localized at the singularity, should become “visible” on the resolution. To obtain a matching of orbifold models with models built on the resolved space, we consider the possibility of a gauge bundle wrapped around the resolution. In general such a bundle has structure group embedded into . This embedding breaks the 10D gauge group to the maximal subgroup that commutes with . We therefore expand the 10D gauge field strength around the internal background , living in the algebra of , in terms of the 4D gauge field strength , taking values in the algebra of . To preserve supersymmetry in four dimensions, the bundle field strength has to satisfy the Hermitian Yang–Mills equations [22]^{2}^{2}2Here we ignore loop corrections to these equations, discussed in [23]. We will return to this point later in the paper.
(7) 
where denotes the inverse Hermitian metric of . One further (topological) consistency requirement follows from the integrated Bianchi–identity of the two–form of the supergravity multiplet:
(8) 
for all compact four–cycles of the resolution and denotes the curvature of the internal space . This condition is crucial to ensure that the effective four dimensional theory is free of non–Abelian anomalies [24]. The resolution space only contains a single compact four–cycle, the at the resolved singularity, leading to a single consistency condition.
We give two examples of bundles on the resolution that satisfy (7) and (8). The simplest construction of such a bundle is the standard embedding (to which we refer as “AS”) with the gauge connection taken to be equal to the spin connection [22]. In terms of the curvature this means . Since SU(3), this describes an SU(3) bundle, embedded into , leading to the 4D gauge group . However, in this paper we mainly focus on the U(1) gauge bundle with field strength
(9) 
see [15] for notational conventions. Such a bundle can be embedded into as
(10) 
where we use the notation . Since the bundle is only well–defined if its first Chern class, integrated over all compact two–cycles, is integral, an extra consistency requirement arises for the vector . For the two–cycle at ,
(11) 
must be integral for all roots. This implies that has to be an root lattice vector itself. The two–form is regular everywhere for . In the blow–down limit , it is zero for and it diverges for , in such a way that the integral above remains constant. This means that the bundle is “visible” as a two–form only in the blow–up, but in the blow–down its physical effect is not lost because the gauge flux gets localized in the fixed point. In this sense, this bundle is exactly the counterpart of the orbifold boundary conditions discussed above.
gauge group  

EE  
ESU(3)E  A  
ESU(3)  B  
ESO(14)U(1)  C  
SU(9)SO(14)U(1)  D 
2.3 Classifying orbifold and smooth line bundle models
The heterotic string on the is specified by the orbifold gauge shift vector defined in (6). The freedom in the choice of is constrained by modular invariance of the string partition function:
(12) 
There are only five inequivalent shift vectors, each of them giving rise to a different orbifold model. In Table 1 we list the possible together with the gauge groups surviving the orbifold projection. Using the standard CFT procedure it is possible to compute the spectra of these models. They are listed in the second column of Table 3. The spectra are given with the multiplicity numbers with which the various states contribute to the 4D anomaly polynomial localized in the singularity. Thus, these numbers can be fractional if the corresponding states are not localized in the singularity. The untwisted states have multiplicities that are multiples of , because the compact orbifold has 27 singularities and untwisted states come with multiplicity three. On the other hand, these multiplicities are integers for localized (i.e. twisted) states.
The blow–up model is completely specified by the way how the gauge flux is embedded in , i.e. by the vector . The Bianchi identity integrated over yields the consistency condition
(13) 
and enormously constrains the number of possible models. All solutions to (13) together with the corresponding gauge groups are given in Table 2. The chiral matter content is determined by the Dirac index theorem that for U(1) bundles takes the form of a multiplicity operator
(14) 
see [15] for details. The computation of the spectra for each of the U(1) embeddings shows that there are in fact only five inequivalent models amongst them. We distinguish them by their chiral spectra, which are given in the third column of Table 3.
bundle vector  gauge group  label  
AI  
BI  
CI  
CII  


DI  
2.4 Matching orbifold and blow–up models
We now want to investigate the matching between the heterotic orbifold and the blow–up models discussed in the previous section. This matching can be considered at various levels and we begin with some simple observations before entering more subtle issues.
The first basic observation was made in (6): the embedding of the orbifold rotation in the gauge bundle (via the shift ) can be seen as the presence of a gauge flux localized in the singularity. On the resolution, such a gauge flux appears, and it is immediate to identify
(15) 
The integration above is made on the variable defined in (6) in such a way that the integral can be immediately read as a contour integral of around the phase of or, in other words, precisely as the Wilson line associated with in (6).
This basic observation is corroborated by the fact that any blow–up shift , listed in Table 2, is modular invariant, because mod . At first sight the reverse, any orbifold shift , classified in Table 1 corresponds to a blow–up, does not seem to be true. However, we should take into account that two orbifold shift vectors are equivalent, i.e. lead to the same model, if: i) they differ by where is any element of the root lattice of , ii) they differ by sign flips of an even number of entries, or iii) are related by Weyl reflections. By suitable combining these operations one can show that all blow–up vectors of Table 2 can be obtained from the orbifold shifts in Table 1. (Only the first model in Table 1, characterized by the zero vector , does not have a blow–up counterpart in Table 2.) This leads to a direct matching between orbifold and blow–up models. Using the notation from the Table 1 and 2, we match model B with BI, model D with DI. We also see that even though CI and CII are different blow–up models, they correspond to the same orbifold theory C. The same applies to the U(1) bundle model AI and the standard embedding model AS (introduced in Section 2.2): They are both related to orbifold model A.
Given the matching at the level of the gauge bundles, we can pass to checks at the level of the 4D gauge groups. A quick glance over the Table 1 and 2 shows that their gauge groups are never the same. This is easily explained from the orbifold perspective: the blow–up is generated by a non–vanishing vev of some twisted state, the so–called blow–up mode [25]. As all twisted states are charged, this vev induces a Higgs mechanism accompanied with gauge symmetry breaking and mass terms. It is not difficult to see from these tables that all non–Abelian blow–up gauge groups can be obtained from the orbifold gauge groups by switching on suitable vevs of twisted states.
Even after taking symmetry breaking, i.e. the branching of the representations of the orbifold state, into account the spectra of the orbifold models still do not agree with the ones of the resolved models: singlets w.r.t. non–Abelian blow–up groups, and some vector–like states are missing. Moreover, the U(1) charges of localized states do not coincide with the ones expected from the branchings. This can be confirmed from Table 3: for each model we give the orbifold spectrum (second column) and the resolution spectrum (third column).
All these differences can be understood by more carefully taking into account the possible consequences of a twisted state’s vev . After branching, this field is a singlet of the non–Abelian gauge group. In the quantum theory this means that the corresponding chiral superfield with charge under the broken U(1) never vanishes. Hence, it can be redefined as , where is a new chiral superfield taking unconstraint values. As it transforms as an axion
(16) 
under a U(1) transformation with parameter , it is neutral. Hence, it is not part of the charged chiral spectrum computed using the Dirac index (14) on the resolution. In addition, we can use this axion chiral superfield to redefine the charges of other twisted states (see the last column of Table 3) so that all U(1) charges of the twisted states agree with the ones of the localized resolution fields. For models CII and DI one needs in addition to change the U(1) basis when identifying the orbifold and blow–up states, if one enforces that the field getting a vev is only charged under the first blow–up U(1) factor.
orbifold model  resolution model  field redefinitions  










Finally, the remaining states that are missing in blow–up (referred to in Table 3 with a superscript ) have Yukawa couplings with the blow–up mode, so that they get a mass term in the blow–up. Taking all these blow–up effects into account shows that the spectra of the blown–up orbifold and resolution models become perfectly identical.
2.5 F– and D– flatness of the blow–up mode
In the matching of heterotic orbifold models with their resolved counterparts we assumed that a single twisted field of the orbifold model was responsible for generating the blow–up. No other twisted or untwisted states attained non–vanishing vevs. However, in order to obtain a supersymmetric configuration, we have to pay attention to possible non–vanishing F–terms arising from the non–zero vev. The analysis of the F–flatness for a superpotential is rather involved in the context of heterotic orbifold model building, because in principle it contains an infinite set of terms with coefficients determined by complicated string amplitudes. In practice string selection rules can be used to argue that a large class of these coefficients vanishes identically, while the others are taken to be arbitrary [26, 27, 28].
Our assumption above that only a single twisted superfield has a non–vanishing vev greatly simplifies the F–flatness analysis: non–vanishing F–terms can only arise from terms in the superpotential that are at most linear in fields having zero vevs. As in most of the cases the vanishing vev fields form non–Abelian representations gauge invariance of the superpotential implies that they cannot appear linearly. This means that the complicated analysis of the superpotential involving many superfields often reduces to the analysis of a complex function of a single variable. In what follows we show that all the blow–ups described previously are F–flat and therefore constitute viable resolutions of orbifold models.
Non–vanishing D–terms can only arise under the following conditions [29]: let be the scalar component of the only superfield that acquires a vev as discussed above. The D–terms are proportional to , where are the generators of the orbifold gauge group . Therefore, certainly all D–terms corresponding to the generators that annihilate vanish. They generate the little group of gauge symmetries unbroken by the vev. Consequently, non–vanishing D–terms are only possible for the generators of the coset . Under an infinitesimal gauge transformation with parameter the D–terms transform as . This means that for all generators which do not commute with all generators of , we can find a gauge such that the ’s associated to them vanish. But since defines a gauge invariant object, all these have to vanish in any gauge. The only possibly non–vanishing D–terms correspond to the Abelian subgroup of the coset . As we explain in the next subsection, precisely those D–terms, which are associated with anomalous U(1)’s on the resolution , are non–vanishing. Apart from this subtle issue, D–flatness is automatically guaranteed.
Matching AAS by a vev of
We begin our analysis with the standard embedding defined by a gauge bundle with . Because both transform by conjugation, and under gauge and internal local Lorentz (holonomy) transformations, respectively, we know that these gauge transformations are identified: . This fact will help us to identify the blow–up field in the following.
To reconstruct the corresponding blow–up mode we need a field that transforms under both of these transformations. In orbifold model A the only candidates for this are the three triplets . The multiplicity three is due to internal oscillator excitations of these states, i.e. these states form a triplet under the R–symmetry group SU(3) as it commutes with the orbifold holonomy, which in turn is proportional to the identity. A more precise way of referring to these states is therefore: ; we can view them collectively as a 33 matrix , where is the SU(3) index and the SU(3) index^{3}^{3}3We use the subscript notation to indicate that it is in the complex conjugate representation.. The SU(3) gauge and SU(3) R–symmetry groups act on it as . Since any complex matrix can be written as a product of a unitary matrix and a Hermitian matrix , which in turn can be diagonalized by another unitary matrix as with a real diagonal matrix, we can use the gauge and R–symmetry transformations to bring in a real diagonal form. If we choose the vev matrix to be proportional to the identity: , we find that only a diagonal gauge and R–symmetry transformation preserves this vev. This means that in the blow–up the symmetry is broken to the diagonal subgroup (with ). Comparing to the standard embedding on the resolution (with ), the vev of changed the orbifold holonomy to the full SU(3) holonomy of a Calabi–Yau.
To understand whether such a vev for is possible, we need to analyze the superpotential of the theory. Because the action is proportional to the identity, the orbifold theory is left invariant by any unitary transformation U(3) of the internal coordinates, there is no superpotential involving only : the SU(3)SU(3) invariance requires such superpotential to be a function of , but that is not invariant under U(1). (Similar arguments for the compact allow a cubic superpotential for in that case [16].) Hence, any vev for defines a F–flat direction. However, it can be seen that D–flatness requires it to be of the form above [16]: The SU(3) D–terms correspond to the traceless part of the matrix . In the diagonal form the matrix has the vevs , and as its diagonal elements. This means that is a diagonal matrix with entries , and . But this has a non–vanishing traceless part unless all vevs are equal.
Matching AAI by a vev of
In this case the blow–up mode is in the twisted state , and the relevant part of the superpotential are formed from its cubic E invariant
(17) 
Here, the notation indicates that we only give the lowest order gauge invariant structure ignoring its (order one) coefficient, and means that there is a whole power series of this invariant, restricted by some string selection rules. Such a superpotential is always F–flat. To see this, consider the branching of the twisted state due to its own vev that breaks . In terms of –representations the invariant reads as
(18) 
Since the represents the blow–up mode, and hence by definition the and have zero vev, F–flatness is automatically guaranteed. It is clear from this decomposition of the cubic invariant that the becomes massive and decouples, while the stays strictly massless. This is in agreement with the blow–up spectrum given in Table 3.
Matching BBI by a vev of
In orbifold model B the only twisted state is a –plet, hence it is the only possible blow–up mode. Like the blow–up mode in the case of standard embedding AS discussed above, this blow–up mode defines a 33 matrix denoted by . Gauge transformations with SU(3) and SU(3) act via left and right multiplication . The relevant part of the superpotential is therefore also very similar
(19) 
As the two SU(3)’s are independent, we can again assume that the matrix is diagonalized. To obtain the appropriate symmetry breaking of both SU(3)s, only one of the three diagonal elements has a non–vanishing vev. This is a very different orientation of the vev as compared to the standard embedding. Expanding the superpotential around this vev, shows that the state becomes massive.
Matching CCI by a vev of
In orbifold model C we can construct gauge invariant structures for the superpotential only by combining the states , and
(20) 
Since the CI blow–up is realized by giving a vev to the orbifold state , which is always coupled to pairs of ’s in the superpotential, this vev defines a flat direction of the potential and a mass term for the is generated, provided that we perform the field redefinition indicated in Table 3. Hence, this state decouples.
Matching CCII by a vev of
The CII blow–up is obtained when gets a vev. Naively one expects that a vev for this state would lead to a symmetry breaking , but this is not in agreement with Table 3. To understand what is happening we have to consider the possible orientations of such a vev , where denotes the SO(14) vector index. Since all states are chiral multiplets, we cannot use the real group SO(14) to put the vev in a single component. Indeed, writing where and are real, we see that one can use a SO(14) transformation to obtain . This orientation is left invariant by SO(13) subgroup. This subgroup can be used to bring to the form . Hence, for generic values of and only the SO(12) subgroup is left unbroken, as Table 3 implies. Furthermore, the superpotential contains again the coupling (20). To have the auxiliary component of the superfield vanishing in extremum, the vev of the SO(14) invariant
(21) 
has to vanish. The only non–vanishing solution has: and . This vev induces a mass by pairing up one of the singlets from the branching of with the singlet already present in the orbifold spectrum, see Table 3.
Notice that there is a third field in model that could have a non–zero vev, the R–symmetry triplet . This superfield cannot appear in any superpotential by itself, this means that any vev for this superfield leads to a supersymmetric configuration. Nevertheless, we do not have any candidate for a gauge configuration on the resolution that corresponds to this vev.
Matching DDI by a vev of
Finally, we consider the orbifold model D. As it has only one charged twisted state it is not possible, due to the string selection rules, to write down any superpotential with terms at most linear in the other fields. Thus, it can attain any vev leading to the symmetry breaking as described in Table 3.
Other gauge bundles on the resolution?
The list of possible vevs of twisted states of a given heterotic orbifold model is exhausted only for the last case, model D. The other models allow other blow–ups in principle:
First of all, model also has an SU(3) triplet of scalars, there is no obvious reason why one of them cannot have a non–vanishing vev. Model B allows for other possible orientations for the vev of the state, because it defines a –matrix with three eigenvalues. Thus, in general one should allow for blow–ups defined by a multitude of vevs for possibly all the twisted states that a given orbifold model possesses.
Since our classification of Abelian gauge bundle models on the resolution of is complete, and we have identified the blow–up modes in the various heterotic theories leading to these models, we conclude that other (multiple field) vevs correspond to non–Abelian bundles on the resolution. Aside from the standard embedding, model AS, their classification is beyond the scope of this paper.
2.6 Multiple anomalous U(1)’s on the blow–up
In this section we investigate the anomaly cancellation and D–flatness on the resolution . We find that there can be at most two anomalous U(1)’s, and that their cancellation involves two axions [30, 31], the model–independent and a model–dependent one.^{4}^{4}4For a recent review on axions from string theory see [32]. We show that the counterpart of such an anomaly cancellation, from the orbifold perspective, is a mixture of the standard orbifold Green–Schwarz mechanism and the Higgs mechanism related to the blow–up mode. From this we deduce relations between the two axions and their orbifold counterparts, namely, the universal axion of heterotic orbifold models and a second field related to the blow–up mode. Finally we discuss the issue of D–flatness of the resolution. We show that the blow–up is not along a D–flat direction. Rather, in the blow–up a constant D–term is produced, which is matched, from the resolution perspective, with the appearance of a new Fayet–Iliopoulos term due to the presence of two, rather then one anomalous U(1)’s.
Anomalous U(1)’s on the resolution: the axions
We deduce the 4D anomaly polynomial from dimensional reduction of the 10D one, . For notational convenience we absorb some factors in the definition of the anomaly polynomial: . The anomaly polynomial factorizes as , where [33, 34]
(22) 
(23) 
with denoting the 10D curvature. The trace tr in the “fundamental” of is formally defined via , Tr being the standard trace in the adjoint representation. From the 4D anomaly polynomial can be derived via an integration over the resolution manifold. The integration will be performed after inserting the expansions and and splitting the forms and according to
(24) 
where we read as an form with indices in the 6D internal and indices in the 4D Minkowski space. Since the backgrounds are such that the Bianchi Identity is fulfilled, the 4D anomaly polynomial is written as the factorized sum
(25) 
Inserting the expressions for and in terms of the field strengths and rearranging the terms on the right hand side yields
(26) 
where and
(27) 
Now the integration is performed and results in
(28) 
(29) 
These equations describe how the 4D anomaly can be written as a sum of two factorized parts, a universal part and a non–universal part . Since both parts are proportional to , they are non–vanishing only for anomalous U(1)–factors. As we started from an anomaly free theory in 10D the 4D Green–Schwarz mechanism will cancel the two summands in by two axions. The universal anomaly is canceled by the anomalous variation of the model–independent axion and the non–universal part by the model–dependent axion, as shown in the following. Therefore on the resolution of there can be at most two anomalous U(1)’s. In Table 4, we give the anomaly terms for each of the models.
The 4D anomaly must be canceled by the anomalous variation of the 10D twoform , which can be expanded as
(30) 
The Kähler form , obtained from the Kähler potential, and the U(1) gauge bundle field strength are harmonic two–forms on the resolution . Thus and are 4D massless scalars. In addition, is a two–form in Minkowski space, the 4D B–field. The gauge transformation of the two–form and the scalar are determined by the expansion of the three–form field strength [17]