LMU-ASC 03/09

Heterotic MSSM Orbifolds in Blowup


Stefan Groot Nibbelink111    E-mail: grootnib@thphys.uni-heidelberg.de, Johannes Held222    E-mail: johannes@tphys.uni-heidelberg.de, Fabian Ruehle333    E-mail: fabian@tphys.uni-heidelberg.de,

Michele Trapletti444    E-mail: michele.trapletti@cpht.polytechnique.fr and Patrick K.S. Vaudrevange555    E-mail: Patrick.Vaudrevange@physik.uni-muenchen.de

[0pt] Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16 und 19, D-69120 Heidelberg, Germany

[1ex] Shanghai Institute for Advanced Study, University of Science and Technology of China, 99 Xiupu Rd, Pudong, Shanghai 201315, P.R. China

Laboratoire de Physique Theorique, Univ, Paris-Sud and CNRS, F-91405 Orsay, France

[1ex] CPhT, École Polytechnique, CNRS, F-91128 Palaiseau, France

Arnold-Sommerfeld-Center for Theoretical Physics, Department für Physik, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München, Germany


Heterotic orbifolds provide promising constructions of MSSM–like models in string theory. We investigate the connection of such orbifold models with smooth Calabi-Yau compactifications by examining resolutions of the orbifold (which are far from unique) with Abelian gauge fluxes. These gauge backgrounds are topologically characterized by weight vectors of twisted states; one per fixed point or fixed line. The VEV’s of these states generate the blowup from the orbifold perspective, and they reappear as axions on the blowup. We explain methods to solve the 24 resolution dependent Bianchi identities and present an explicit solution. Despite that a solution may contain the MSSM particle spectrum, the hypercharge turns out to be anomalous: Since all heterotic MSSM orbifolds analyzed so far have fixed points where only SM charged states appear, its gauge group can only be preserved provided that those singularities are not blown up. Going beyond the comparison of purely topological quantities (e.g. anomalous U(1) masses) may be hampered by the fact that in the orbifold limit the supergravity approximation to lowest order in is breaking down.

1 Introduction

One of the central tasks of string phenomenology is to build models which make contact with the observations of the real world. A basic step towards this goal is the construction of models in which gauge interactions and chiral matter are those of a (Minimal) Supersymmetric extension of the Standard Model of Particle Physics (MSSM). In the resulting framework one may hope to comprehend the nature of supersymmetry breaking, and recover the properties of the particle masses and couplings as part of the Standard Model. In this approach we implicitly assume that we can disentangle the problem of finding the correct matter spectrum from the issue of breaking four dimensional supersymmetry in string theory.

This basic step of obtaining MSSM–like models from string theory has been faced in the past from many different perspectives with some remarkable successes: Among the others, we would like to mention interesting findings based on purely Conformal Field Theory (CFT) constructions, like the so–called free–fermionic formulation [1], the Gepner models [2], and the rational conformal field theory models [3]. Most of the other approaches are geometrical in nature. Among these we would like to remind the reader of the works of [4] in the intersecting D–brane context (see also references therein for models including chiral exotics), those of [5] for what concerns local constructions with D3 branes at singularities in Type IIB string theory, those of [6, 7, 8, 9] for similar constructions in a local F–theory language, and those of [10] for globally consistent GUT models from intersecting D7-branes. Finally, there has been recent progress in heterotic model building by [11] on smooth (elliptically fibered) Calabi Yau spaces that resulted in interesting constructions [12, 13, 14, 15, 16]. The results of [17, 18] on heterotic orbifold model building were further exploited by [19, 20].

Each construction has peculiar properties and shows a certain amount of complementarity: Models can be global or only local. They may be obtained via elaborate computer scans or in a more geometric/constructive perspective, and they may or may not incorporate issues such as moduli stabilization, decoupling of exotics, Yukawa textures, etc. Comparing these diverse approaches can have severe impacts, as one might be able to use the good features of a given construction to overcome the limitations of others. Bringing these different approaches together can be achieved by using the duality properties of string theory (e.g. S–duality linking heterotic strings to type I strings, or T–duality linking IIB with IIA). Often this requires to overcome a language dichotomy by attaining some dictionary between the different terminologies.

The dichotomy between CFT construction of heterotic strings on orbifolds and the corresponding supergravity compactifications on smooth Calabi–Yau manifolds will be one of the central themes of the current paper. Heterotic orbifolds allow for a systematic computer assisted search that can be very effective: In e.g. [17, 18, 19, 20], based on the embedding in string theory of the orbifold-GUT picture (see e.g. [21]), more than two hundred MSSM–like models have been assembled on the orbifold . However, the CFT construction of heterotic orbifold models are only valid at very specific (orbifold) points of the string moduli space. This hinders the introduction of simple moduli stabilization mechanisms such as those due to flux compactifications [22]. Moreover, the generic presence of an anomalous U(1) in orbifold models induces Fayet–Iliopoulos terms driving them out of the orbifold points, which might shed uneasiness on consistency of the orbifold construction. Obtaining good models by compactifying the heterotic supergravity on smooth Calabi–Yau manifolds is a very challenging mathematical problem, and only a handful of such models have been uncovered so far. Establishing a more and more detailed glossary between heterotic orbifold and Calabi–Yau compactifications has been one of the essential challenge pursued in the papers [23, 24, 25, 26] for heterotic strings on simple (mostly non–compact) orbifolds and their supergravity counterpart on their explicit blowups and topological resolutions. Our aim is to extend these results to the heterotic orbifold that has been the spring of the largest set of MSSM–like models constructed from strings to date.

In this paper we outline how to construct smooth Calabi–Yau manifolds from the orbifold , and how to identify the supergravity analog of the heterotic models. These smooth Calabi–Yau’s are compiled in steps: The local orbifold singularities are resolved using techniques of toric geometry, and they are subsequently glued together according to the prescriptions presented in [27]. During the local resolution process we are able to detect the “exceptional divisors”: the four–cycles (compact hyper surfaces) hidden in the orbifold singularities. The local orbifold singularity is blown up once the volumes of the exceptional divisors become non–zero. The compact orbifold in addition has “inherited cycles”, that are four dimensional sub–tori of . Combining the knowledge of the exceptional and inherited cycles we come in the possession of a complete description of the set of two– and four–cycles/forms of the orbifold resolutions, including their intersection ring (i.e. all their intersection numbers). Let us stress that the single orbifold has a very large number of topologically distinct resolutions. Depending on one’s perspective this means that out of this orbifold many Calabi–Yaus are constructed, or that the corresponding Calabi–Yau has a large number of phases related by so–called flop transitions.

The description of cycles is perfectly compatible with the supergravity language, and thus we can consider compactifications of ten dimensional heterotic supergravity on the resolved spaces. By embedding U(1) gauge fluxes on the hidden exceptional cycles we are able to obtain the gauge symmetry breaking and the chiral matter localized on the resolved singularities, that are the supergravity counterparts of the action of the orbifold rotation on the gauge degrees of freedom (and Wilson lines), and the twisted states, respectively. In this way we determine the relationship between the CFT data of heterotic orbifold and supergravity and super Yang–Mills on its resolutions.

Following this procedure we can potentially describe resolutions of every heterotic orbifold model in the supergravity language. To investigate the properties of such resolution models, we apply our approach to a specific MSSM model (“benchmark model 2” of [19, 20]) as a concrete testing case. This example illustrates a number of generic features of such blowups: we can identify a number of generic features of such blowups: We uncover an intimate relation between the specifications of the flux background and the twisted states that generate the blowup from the orbifold point of view. The Standard Model hypercharge turns out to be always broken in complete blowup. This is due to the fact that the full blowup requires non–vanishing VEVs for twisted states at all fixed points, and some fixed points only have states charged under the Standard Model, hence at least the hypercharge is always lost. We stress that this does not depend on the specific choice we make for the gauge bundle. We comment in the conclusions about possible phenomenological consequences of this result as well as about how to avoid it.

The paper has been organized as follows:

Section 2 briefly reviews the heterotic orbifolds, specifying the details necessary to understand the orbifold of the heterotic EE string. As a particular example of a MSSM–like model the “benchmark model 2” of [19, 20] is recalled.

Section 3 explains how to resolve the orbifold using toric geometry and gluing procedures presented in [27]. We first describe the three different possible singularities present in the orbifold, namely , and . The first two singularities are resolved in a unique way. Contrary to this, a singularity has five possible distinct resolutions. Since the orbifold contains 12 singularities, the number of topologically different resolutions is huge: The most naive estimate would be ; the number of resolutions that lead to distinct models is close to two million.

Section 4 considers ten dimensional heterotic supergravity on a generic resolution of . Following the procedure of [25] we introduce U(1) gauge fluxes wrapped on the exceptional divisors. We describe how to single out the gauge fluxes such that they correspond to the embedding of the orbifold rotation and the Wilson lines in the gauge degrees of freedom in the heterotic orbifold theory. The Bianchi identity leads to a set of 24 coupled consistency conditions on the fluxes which depend on the local resolutions chosen. Solving them almost seems to be a mission impossible. However, by identifying the localized axions on the blowup with the twisted states of orbifold theory, that generate the blowup via their VEV’s, shows that the U(1) fluxes are in one–to–one correspondence to the defining gauge lattice momenta of these states. The massless chiral spectrum of the model is computed by integrating the ten dimensional gaugino anomaly polynomial and turns out to suffer from a multitude of anomalous U(1)’s, among them the hypercharge.

Section 5 illustrates our general findings on resolutions of heterotic MSSM–like orbifolds, by specializing to the study of the blowup of the MSSM orbifold model “benchmark model 2”. We outline how solutions to the 24 coupled Bianchi identities can be updated, and illustrate that the line bundle vectors correspond to twisted states. In particular, we illustrate that the hypercharge is broken in full blowup.

Finally, Section 6 contains our conclusions, and additional technical details have been collected in the appendices.

2 Heterotic MSSM models

2.1 Orbifold geometry

First we want to give some general properties of orbifolds, as given for example in [28, 29] or [30]. Later we will examine in detail the orbifold on , where we use the conventions of [20].

General description of orbifolds

A orbifold is produced by identifying the points of a six–dimensional torus under the action of a discrete symmetry . Using complex coordinates (), the action of the –twist is


The order of the symmetry constrains the orbifold twist vector ,


Furthermore, the twist must fulfill the Calabi–Yau condition


One can also consider an orbifold as being produced by modding out its space group from . is defined as a combination of twists and torus shifts . Here (summation over ), where the define a basis of the torus lattice of . The space group yields an equivalence relation,


on . The elements of fulfill the simple multiplication rule . In this picture, the torus is produced by dividing by the basis vectors , and one can take as the covering space of the orbifold.

The space group does not act freely, i.e. there are fixed points. A (non-trivial) space group element specifies a fixed point up to shifts by the torus vectors:


If one now takes the fundamental domain of the torus as the cover for the orbifold, the fixed points in this domain will have different space group elements with a one–to–one correspondence between them.

If the twist acts trivially in one complex plane, i.e. for one , one obtains a two dimensional fixed subspace. On the cover, such a space looks like a torus and is often referred to as a fixed torus. However, on the orbifold the topology is not necessarily that of a torus, but it can also be a two dimensional orbifold. Since in any way one complex coordinate is not affected, we also call those subspaces fixed (complex) lines.

torus basis vectors on
on G , ,
on SU(3) , ,
on SO(4) , ,
Table 1: The basis vectors of the root lattice , in real and complex notation.


We consider the torus obtained by dividing out by the root lattice of . Since the lattice factorizes in three two dimensional parts, the same will be true for the torus. Therefore can be depicted by three parallelograms spanned by the simple root vectors of , as given in Table 1. The orbifold twist vector for is


where the –th entry is included for later use. Therefore, a single twist acts as a counterclockwise rotation of and on the first and second torus and as a (clockwise) rotation of on the third. The general structure of singularities, appearing after modding out the action, is shown in Figure 1. The numbers denote the locations of the orbifold singularities. Singularities in the covering space (i.e. the torus) that are identified on the orbifold are labeled by the same number.

Figure 1: The general fixed point structure of the orbifold. For each complex plane, equal numbers denote singularities that are mapped to the same point of the orbifold.
torus shifts in the –sector
torus shifts in the –sector
Figure 2: Upper Figure: the fixed points in the – and –sector. They are labeled by , and . Lower table: the corresponding torus shifts , see equation (5). For example, the space group element associated to the fixed point and in the –sector reads .

In order to obtain the detailed fixed point structure we look at every twist –sector separately. For the twist (and its inverse ) one obtains the full order of the group . The fixed points are shown in Figure 2. They are labeled by in the first torus, by in the second and by in the third. The lattice shifts needed to bring the points back after a rotation are given in the table of Figure 2. Since in the first and fifth sector, the fixed points are determined by and . Next we consider the fixed points in the – and –sector with twists and , respectively. The order of these twists is and they act trivially on the third torus. Thus, concentrating solely on the – and –sector, the compactification can be described as a orbifold resulting in a six–dimensional theory. The fixed lines of the orbifold are shown in Figure 3. By comparing with Figure 1 we see that the points and correspond to the same point on the orbifold as they are mapped onto each other by further twists. Hence, there are six independent fixed lines, labeled by and . The corresponding lattice shifts are given in the table of Figure 3. At last we examine the –sector. Here, the twist leaves the second torus invariant and acts with order two. In this case one obtains fixed lines, depicted in Figure 4. Again one notes by comparing with Figure 1 that the points , and are mapped onto each other by further twists and correspond to one point on the orbifold. Hence there are eight independent fixed lines, labeled by and . The lattice shifts for this sector are given in the table of Figure 4.

torus shifts in the –sector
torus shifts in the –sector
Figure 3: Upper Figure: the fixed lines in the – and –sector. They are labeled by and , where the points and are identified on the orbifold. Lower table: the corresponding torus shifts .
torus shifts in the –sector
Figure 4: Upper Figure: the fixed lines in the –sector. They are labeled by and , where the points , and are identified on the orbifold. Lower table: the corresponding torus shifts .

2.2 Heterotic orbifold models

Next, we review some technical details of the compactification of the heterotic string on orbifolds. The starting point of our discussion is the consideration of boundary conditions for closed strings. On orbifolds, there are new boundary conditions associated to non–trivial elements of the space group, i.e. defines a boundary condition for the six compactified dimensions of the string. If is not freely–acting (i.e. it has a fixed point), the string is attached to the fixed point and is called the constructing element of a so–called twisted string. On the other hand, strings with a constructing element correspond to the ordinary strings of the ten–dimensional heterotic string theory (being the supergravity and the gauge multiplets). They are henceforth referred to as untwisted strings.

However, the geometrical action of the space group is not sufficient to define a consistent compactification. One needs to accompany the geometrical action of by some action in the 16 gauge degrees of freedom, in our case in . In the case of shift embedding, the most general embedding of the space group is


That is, whenever a rotation by and a translation by is performed in the six compact dimensions of the orbifold, the 16 gauge degrees of freedom are shifted by , summation over . is called the shift vector and are (up to six) Wilson lines. They are constrained to lie in the root lattice as follows:


no summation over . The order of the Wilson line is determined by the action of the twist in the direction of the Wilson line. In addition, Wilson lines have to be constrained due to further geometrical considerations. In the case of the orbifold this results in three independent Wilson lines, (of order 3) and , (both of order 2) with the identifications


where , and are introduced for later use.

Additionally, modular invariance of one–loop amplitudes imposes strong conditions on the shifts and Wilson lines. In orbifolds, the order shift and the twist must fulfill [29, 31]


In the presence of Wilson lines, there are additional conditions


where denotes the greatest common divisor of and  [32]666In the case of two order 2 Wilson lines in an torus, can be replaced by ..

The spectrum

The coordinates of a string can be split into left– and right–movers, i.e. on–shell. After quantization, a string is described by a state of the form . Here, denotes the momentum of the (bosonized) right–mover (describing the space–time properties of the string) and labels the left–moving momentum of the 16 gauge degrees of freedom (describing the strings representation under gauge transformations). Furthermore, denotes possible oscillator excitations. In general, physical states have to satisfy the mass–shell conditions for left– and right–movers, i.e.


and the so–called level–matching condition . Here, denotes the local shift (7) corresponding to the constructing element of the (twisted) string. Analogously, is called the local twist. Furthermore, yields a change in the zero–point energy and is given by , where such that . It is convenient to define the shifted momentum , as twisted strings transform according to their weight under gauge transformations.

If the local twist is non–trivial, i.e. for , the compact space is six–dimensional resulting in an effective four dimensional theory. Furthermore, the –th component of the solution to the right–moving mass–shell condition (12) defines four dimensional chirality, being in this case. This corresponds to a chiral multiplet of supersymmetry (and its CPT conjugate). For , this is the case for the / –sector, which therefore contains only chiral multiplets of supersymmetry in four dimensions. On the other hand, if the twist acts trivially in one complex plane, i.e. for , the compact space is first of all only four dimensional resulting in an effective theory in six dimensions. The massless states are then hyper multiplets of supersymmetry in six dimensions. For , this is the case for the higher –sectors, . However, as we will see in the following, these hyper multiplets are decomposed into chiral multiplets of four dimensional supersymmetry when forming orbifold invariant states.

Orbifold invariant states

The general idea is that orbifolded strings have to be compatible with the underlying orbifold space. To ensure this one has to analyze the action of the space group on the string states, i.e. under the action of some element , the state with constructing element transforms with a phase


The transformation phase reads in detail


is called the vacuum phase; for simplicity we assume that it can be set to in this Subsection. Furthermore, in order to summarize the transformation properties of and of the oscillators we have introduced the so–called R–charge777These R–charges correspond to discrete R–symmetries of the superpotential in the context of string selection rules for allowed interactions.


and , , are integer oscillator numbers, counting the number of left–moving oscillators and , and , acting on the ground state , respectively. In detail, they are given by splitting the eigenvalues of the number operator according to , where and such that .

In general, the transformation phase (14) has to be trivial in order for a string to be compatible with the orbifold background. In other words, strings with have to be removed from the spectrum. However, for a given string with constructing element we do not need to consider the action of all elements . It is useful to distinguish two cases for :

Case 1:

In the first case, and commute (). This condition can be interpreted as a string located at the fixed point of but having still some freedom to move, especially in the direction of (e.g. when is from the –sector of the orbifold, it has a fixed torus in the , direction. Then, corresponds to loops on which the string can move around). In this case the transformation phase (14) has to be trivial, i.e.


In other words, the total vertex operator of the state with boundary condition has to be single–valued when transported along if is an allowed loop, .

For , this projection acts for example on the higher –sectors with in two ways: 1) by Wilson lines in the fixed torus and 2) by a projection on . We concentrate on the second case. For example, for and in the –sector, the constructing element obviously commutes with , see Figure 4. This induces the condition . In general, this kind of conditions can remove parts of the localized spectrum, or in some cases even the complete massless localized matter of some fixed lines.

Example for Case 1: Breaking of

One further important example of equation (16) is the breaking of the ten dimensional gauge group by the orbifold compactification. Gauge bosons are untwisted strings (with constructing element ). Hence, all elements of the space group commute and induce projection conditions. As for the gauge bosons, this leads to the following conditions on the roots (with ) of the unbroken gauge group


Case 2:

In the second case, and do not commute (). Then, maps the fixed point of to an equivalent one, which corresponds to the space–group element . In other words, a string located at cannot move along the direction of . But still, the state corresponding to has to be invariant under the action of . Therefore, one has to build linear combinations of states located at equivalent fixed points. These equivalent fixed points are distinguishable only in the covering space of the orbifold (for example, for , states from the –sector located at the two fixed points have to be combined, since maps the corresponding fixed points to each other, see Figure 3). These linear combinations can in general involve relative phases , i.e.


where denotes the localization of the state at the fixed point of and . The geometrical part of the linear combination transforms non–trivially under


Now, has to act as the identity on the linear combination. Consequently, we have to impose the following condition using the equations (14), (18) and (19) for non–commuting elements:


However, given some solution to the mass equations (12) one can always choose an appropriate to fulfill this condition. In this sense, equation (20) does not remove states from the spectrum and is hence not a projection condition.


Using the material discussed so far, one can construct consistent heterotic orbifold models. One way to check their consistency is to analyze whether all gauge anomalies of the massless spectrum vanish. For example, for a gauge factor there are several possible anomalies:


where denotes a non–Abelian gauge group factor (like ) and is another factor. We denote the 16–dim. vector that generates a by and the associated charge by . Then, a state with left–moving momentum carries a charge . However, it is known that in heterotic compactifications one factor can seem to be anomalous, where we denote its generator by and its charge by . Then, the anomalous has to satisfy the following conditions [33, 34]


in order to be canceled by the universal Green–Schwarz mechanism, i.e. by a cancelation induced from the anomalous transformation of the axion . Here, is the Dynkin index888The Dynkin index of some representation is defined by , using the generator of in the representation . The conventions are such that for and for . with respect to the non–Abelian gauge group factor . Since all other anomalies vanish this results in an anomaly–free theory.

# irrep. label # irrep. label
3 3
7 4
8 5
47 26
20 20
2 2
4 4
2 9
Table 2: The massless spectrum of the benchmark model 2 contains three generations of quarks and leptons plus vector–like exotics. The representations (irrep.) with respect to are shown, where the hypercharge is given as a subscript.

2.3 Example: Benchmark model 2

The so–called “benchmark model 2” [19, 35, 20] is defined by the shift and two non–trivial Wilson lines and , i.e.


and the Wilson line corresponding to the direction is set to zero, 999The shift and the Wilson lines are given here in a different, but equivalent form compared to [19]. These vectors satisfy the modular invariance conditions (10), (11). The gauge group of the four dimensional theory is


and originate from the first and second , respectively. A hypercharge generator can be defined by


such that the observable sector only contains the Standard Model gauge group times some factors, while the hidden sector contains further non–Abelian gauge factors.

The massless matter spectrum is given in Table 2. It contains three generations of quarks and leptons plus vector–like exotics. It turns out that one , generated by


is anomalous with . Obviously, the generator mixes hidden and observable sectors. However, the hypercharge is non–anomalous because its generator is orthogonal to the anomalous one, i.e. . Furthermore, as expected, the anomaly fulfills the universality condition (22) and consequently can be canceled by the Green–Schwarz mechanism.

Finally, we briefly review the conditions for a supersymmetric vacuum of the benchmark model 2. Due to the anomalous , the corresponding D–term contains the so–called Fayet–Iliopoulos (FI) term, i.e.


Thus, a supersymmetric vacuum with forces some fields (with negative anomalous charge ) to obtain VEVs. In [20] it is shown that there are non–trivial solutions in which the Standard Model gauge group is left unbroken while all additional factors are broken and, furthermore, in which the vector–like exotics get massive and decouple from the low energy effective theory. In these configurations there are some fixed points where more than one twisted state acquires a VEV. In addition, there are also fixed points where no twisted state has a non–trivial VEV, e.g. the fixed point in the –sector with and .

3 Resolutions of

Since it is crucial for the derivation of the main results of this paper, we want to give a comprehensive review of the techniques needed to resolve compact orbifolds. This is mainly based on [36, 27, 37, 38, 25]. Mathematical fundamentals can be found in [39, 40, 41].

Before going into details, we want to outline the general strategy. The main step is to subdivide the problem of resolving a compact orbifold into the easier problem of resolving several non–compact orbifolds. This is done by considering every fixed point separately in the sense that it is “far away” from other fixed points and can be locally considered as the fixed point of a non–compact orbifold. Then one can identify the group of this orbifold, which is a subgroup of the group acting in the compact case. This provides all the information needed to resolve the singularities locally.

To obtain the resolution of the compact orbifold, one has to combine the local information in a proper way. This procedure is referred to as “gluing” and can be achieved by considering global information coming from the torus . The final result of this procedure will be topological informations about the resolved orbifold, which is needed in later computations.

3.1 Local resolutions

First we determine which subgroup of acts on which kind of fixed objects. As was stated in Section 2.1 one obtains 12 fixed points under the full action of with the labels where runs from to and form to (compare also with Figure 2). Furthermore, there are 6 independent fixed lines out of which 3 are simply fixed lines () and 3 are the combination of two equivalent fixed lines (; the fixed lines denoted by and in Figure 3 are identified on the orbifold). At last there are 8 independent fixed lines that are subdivided in a similar way: the ones with are just fixed lines and the ones with are a combination of the three equivalent lines that are denoted by , , and in Figure 4. Therefore we obtain locally three different types of orbifolds that we have to resolve: for the fixed points, for the fixed lines and for the fixed lines.

How to resolve non–compact orbifolds is a well–known problem in toric geometry. A mathematical introduction to toric geometry is given in [41]. The orbifold case is covered in [27, 38, 25]. The main tool in the resolving procedure is the toric diagram of the orbifold, which is constructed in the following way. The orbifold group acts in the d-dimensional complex space like


We can define -invariant monomials () by fixing a condition on the vectors :


From the Calabi–Yau condition (3) one knows that . Due to this, we can choose the last component of every vector to be equal to , which means that the endpoints of all vectors lie in a plane. The toric diagram of the orbifold is obtained by connecting all those points.

A further statement of toric geometry is that every such vector can be associated with a codimension one hypersurface denoted by . These hypersurfaces are called ordinary divisors. Since for each divisor there exists a holomorphic scalar transition function on the orbifold, a holomorphic line bundle can be associated to each divisor, whose first Chern class gives the Poincare dual form of the cycle . For a holomorphic line bundle this will be a –form. In what follows, the cycle as well as the form is denoted by , since the context should make clear which object is meant.

To resolve the orbifold one introduces a new class of divisors, called exceptional divisors . In principle one has to introduce one exceptional divisor for every non–trivial twist . This is the case for orbifolds. In the toric diagram (which is a line in this case) the exceptional divisors are placed in such a way that the distances between two divisors are distributed equally. For orbifolds a more thorough examination yields the following condition for exceptional divisors, as described in [42]:
If the twist in the –th sector acts like


an exceptional divisor will be placed in the toric diagram at


The toric diagrams of the resolved orbifolds , and are shown in Figure 5. For the orbifolds the toric diagram is the line that connects the endpoints of the vectors. There is one exceptional divisor for the orbifold, two for and four for . The divisors of the orbifolds are named in a way convenient for the gluing procedure.

(a) Toric diagram of . One exceptional divisor is needed for the resolution.
(b) Toric diagram of . Two exceptional divisors are needed for the resolution.