LTH-1269

Taming Triangulation Dependence of
$\boldsymbol{T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ Resolutions

A.E. Faraggia,111 E-mail<EMAIL_ADDRESS>S. Groot
Nibbelinkb,222 E-mail<EMAIL_ADDRESS>M. Hurtado Herediaa,333
E-mail<EMAIL_ADDRESS>

a Department of Mathematical Sciences, University of Liverpool, Liverpool L69
7ZL, UK
b School of Engineering and Applied Sciences, Rotterdam University of Applied
Sciences,
G.J. de Jonghweg 4 - 6, 3015 GG Rotterdam, the Netherlands
Research Centre Innovations in Care, Rotterdam University of Applied Sciences,
Postbus 25035, 3001 HA Rotterdam, the Netherlands
School of Education, Rotterdam University of Applied Sciences,
Museumpark 40, 3015 CX Rotterdam, the Netherlands

### Abstract

Resolutions of certain toroidal orbifolds, like
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$, are far from unique, due to
triangulation dependence of their resolved local singularities. This leads to
an explosion of the number of topologically distinct smooth geometries
associated to a single orbifold. By introducing a parameterisation to keep
track of the triangulations used at all resolved singularities simultaneously,
(self–)intersection numbers and integrated Chern classes can be determined for
any triangulation configuration. Using this method the consistency conditions
of line bundle models and the resulting chiral spectra can be worked out for
any choice of triangulation. Moreover, by superimposing the Bianchi identities
for all triangulation options much simpler though stronger conditions are
uncovered. When these are satisfied, flop–transitions between all different
triangulations are admissible. Various methods are exemplified by a number of
concrete models on resolutions of the
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold.

##  1 Introduction

String theory provides a perturbatively consistent approach to quantum
gravity. An important advantage of string theory is that its consistency
requirements mandate the existence of the gauge and matter structures that
form the bedrock of the Standard Model of particle physics. As such, it
enables the construction of phenomenological models, which in turn can be used
to explore the theory and its possible relevance to observational data. Its
internal consistency predicts the existence of a specific number of extra
quantum fields propagating on a two dimensional string worldsheet, which in
some guise can be interpreted as extra spacetime dimensions beyond those
observed in the physical world. Therefore, it has been suggested that these
extra dimensions are compactified and are made sufficiently small to evade
detection in contemporary experiments.

Phenomenological string models can be constructed by using exact worldsheet
formulations of string theory in four dimensions, as well as target space
tools that describe the effective field theory limit of string
compactifications. Ultimately, a viable string theory model should have a low
energy effective field theory description. Conversely, an effective field
theory representation, which is compatible with string quantum gravity, should
have a consistent ultra–violet embedding in string theory. However, at present
the relation between these different regimes is poorly understood. The study
of the consistency constraints on effective field theories of quantum gravity
is a subject of intense contemporary research in the so–called “Swampland
program” (for review and references see e.g. [1]).

An alternative route is to explore the effective field theory limit of exact
string solutions. This is hampered by the poor understanding of the moduli
spaces of generic string compactifications. Exact string solutions are
typically studied by constructing the one–loop partition function and
requiring it to be invariant under modular transformations. A plausible way
forward is therefore to seek the imprint of the modular properties of the
partition function in the effective field theory limit and their
phenomenological consequences in string models.
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds of a six dimensional torus
$T^{6}$ within the compactified heterotic–string are probably the most
frequently studied examples of this route. Such compactifications have been
analysed by using the free fermionic formulation [2, 3, 4] and the free
bosonic formulation [5, 6] of the heterotic–string in four dimensions. These
free bosonic and fermionic worldsheet constructions are merely different
languages to study the same physical object; a detailed dictionary can be
employed to translate the models between the two descriptions [7]. Both
languages were used to construct models that mimic the structure of the
Minimal Supersymmetric Standard Model, e.g. [8, 9, 10, 11, 12, 13] provide
examples of free fermionic models and [14, 15, 16, 17] of free bosonic
constructions. Even though orbifolds are singular spaces, many quantities,
like the full partition function, can be computed exactly at the one–loop
level (and partially beyond) because of the power of the underlying modular
symmetries.

However, these free worldsheet descriptions typically only apply to very
specific points in the string moduli space or, at best, only parameterise a
very small portion of the entire moduli space. In particular, to exploit the
richness of the moduli space beyond the target space singularities, these
singularities need to be deformed and/or resolved to form smooth Calabi–Yau
compactifications with vector bundles [18]. A variety of effective field
theory and cohomology methods have been developed to study the resulting
theories [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. In particular, methods
to resolve orbifold singularities using well–established toric geometry
methodology have been worked out in many cases [30, 31, 32, 33, 34, 35, 36,
37].

The analysis of the effective field theory limit of
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ heterotic–string orbifolds and their
resolutions is therefore well motivated from the phenomenological as well as
the mathematical point of views. The analysis proceeds by the construction of
toroidal $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ heterotic–string
orbifolds and resolving the orbifold singularities using these
well–established methodologies. However, a problematic caveat is the enormous
number of possibilities that this opens up [35, 36, 37]: The
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold has 64
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ singularities where
$\mathbbm{Z}_{2}$–fixed tori intersect, which all need to be resolved to
obtain a smooth geometry. Each
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ singularity can be
blown up in four topologically distinct ways encoded by four triangulations of
the toric diagram of the resolved singularity. This results in a total of
$4^{64}$ a priori distinct possibilities. While the symmetry structure of the
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold can be used to reduce this
number by some factor, it still leaves a huge number (of the order of
$10^{33}$) genuinely distinct choices. This is not a minor complication, as
many physical properties of the resulting effective field theories are
sensitively dependent on the triangulation chosen. These range from the
spectra of massless states in the low energy effective theory to the structure
and strength of interactions among them. The only way to overcome this
complication was by side stepping it: one simply makes some choice for the
triangulation of all these resolved singularities and analyses the resulting
physics in that particular case. This led to some insights in the structure of
the theory in a somewhat larger part of the moduli space, but it seemed
hopeless to extract any meaningful generic information about the properties of
resolved $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds.

A way forward is therefore to develop a formalism which allows computations
for any choice of the triangulation of the 64 resolved
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ singularities. This is the task that we
undertake in this paper. Moreover, having established such a method opens up
the possibility to study some properties of resolved
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds which are independent
of triangulation choices or that hold in all possible triangulations
simultaneously. To this end the paper has been structured as follows:

#### Outline

Section 2 lays the foundation of this work by first recalling some basic facts
of resolutions of the $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold
and line bundle backgrounds on them. After that notation is developed to
parameterise the triangulation choice at each of the 64 resolved
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ singularities, in terms of which the
fundamental (self–)intersection numbers and the Chern classes are expressed.
This allows to obtain relatively compact expressions for the volumes of
curves, divisors and the manifold as a whole. Moreover, the flux quantisation
conditions, the Bianchi identities and the multiplicity operator to determine
the chiral spectrum can all be written down for any triangulation choice.

In Section 3 it is argued that the flux quantisation conditions are, in fact,
triangulation independent: if satisfied in a particular choice of
triangulation, it holds for all. In addition, having written down Bianchi
identities for any possible choice of triangulation of all 64 resolved
singularities, one may wonder what requirements are obtained if one insists
that these conditions hold for all triangulation choices simultaneously.
Surprisingly, it can be shown that the resulting conditions are much simpler
than those in any particular triangulation.

The following two sections provide various examples of the general results of
the preceding two. In Section 4 models are considered without any Wilson lines
so that all 64 resolved singularities may be treated in the same way. In
particular, it stresses that the flux quantisation conditions are essential:
when violated, the difference between the local multiplicities is not
integral. Finally, Section 5 revisits the so–called resolved Blaszczyk GUT
model [17, 37]. A model inspired by this GUT model is considered, which is
consistent for any possible choice of triangulation.

The paper is completed with a summary and an outlook. The Appendix A provides
some useful identities for second and third Chern classes for manifolds with
vanishing first Chern class.

##  2 Resolutions of $\boldsymbol{T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$

This section is devoted to develop some of the topological and geometrical
properties of resolutions of the toroidal orbifold
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$. In fact, there are various
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds [38, 39, 40, 7]: here
we focus exclusively on the orbifold with Hodge numbers (51,3). Techniques to
determine resolutions of toroidal orbifolds have been well–studied [35]; here,
in particular, the methods exploited in [37] are used. Also the resolutions of
this orbifold have been considered before, however in the past one always had
to make some assumptions which triangulation(s) to be considered, as the total
number of choices (naively $4^{64}$) is a daunting number. This section
provides a brief review of this literature, but the main purpose is to develop
a formalism to treat all of these possible triangulations simultaneously.

###  2.1 The $\boldsymbol{T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$
orbifold

The orbifold geometry will be taken to be factorisable of $T^{6}$ on the
simplest rectangular lattice. The six torus coordinates are grouped into three
complex ones on which two order–two orbifold reflections $R_{1}$, $R_{2}$ and
their product $R_{3}=R_{1}R_{2}$ act. They are representations of
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ with non–trivial elements

$\displaystyle\text{diag}(R_{1})=(1,-1,-1)~{},\quad\text{diag}(R_{2})=(-1,1,-1)~{},\quad\text{diag}(R_{3})=\text{diag}(R_{1}R_{2})=(-1,-1,1)~{}.$
(1)

Each reflection, $R_{1}$, $R_{2}$ and $R_{3}$, has $4\cdot 4=16$ fixed points:
$f^{1}_{\beta\gamma}$, $f^{2}_{\alpha\gamma}$ and $f^{3}_{\alpha\beta}$. These
singularities are conveniently labeled by
$\mu,\nu,\alpha,\beta,\gamma=1,2,3,4=00,01,10,11;$ i.e. interpreting them as
binary multi–indices $\alpha=\left(\alpha_{1},\alpha_{2}\right)$ is reserved
for the first two–torus, $\beta=\left(\beta_{3},\beta_{4}\right)$ for the
second and $\gamma=\left(\gamma_{5},\gamma_{6}\right)$ for the third, with the
entries take the values
$\alpha_{1},\alpha_{2},\beta_{3},\beta_{4},\gamma_{5},\gamma_{6}=0,1$. The
translation between both conventions read: $\alpha=2\alpha_{1}+\alpha_{2}+1$,
$\beta=2\beta_{3}+\beta_{4}+1$ and $\gamma=2\gamma_{5}+\gamma_{6}+1$,
respectively. (The (multi–)indices $\mu,\nu$ are used to label the fixed
points in any of the three two–tori in order to write compact expressions.)

Assuming that the tori have unit length, the fixed points may be represented
as

$\displaystyle
f^{1}_{\beta\gamma}=\Big{(}0,\tfrac{\beta_{1}+\beta_{2}\,i}{2},\tfrac{\gamma_{1}+\gamma_{2}\,i}{2}\Big{)}~{},\quad
f^{2}_{\alpha\gamma}=\Big{(}\tfrac{\alpha_{1}+\alpha_{2}\,i}{2},0,\tfrac{\gamma_{1}+\gamma_{2}\,i}{2}\Big{)}~{},\quad
f^{3}_{\alpha\beta}=\Big{(}\tfrac{\alpha_{1}+\alpha_{2}\,i}{2},\tfrac{\beta_{1}+\beta_{2}\,i}{2},0\Big{)}~{}.$
(2)

The fixed set of each reflection has the topology of a torus orbifolded by the
action of the other orbifold actions which leads to four fixed points on a
fixed tori. Hence, in total the $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$
orbifold possesses 64 $\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$
singularities,

$\displaystyle
f_{\alpha\beta\gamma}=\Big{(}\tfrac{\alpha_{1}+\alpha_{2}\,i}{2},\tfrac{\beta_{1}+\beta_{2}\,i}{2},\tfrac{\gamma_{1}+\gamma_{2}\,i}{2}\Big{)}~{},$
(3)

coming from every combination of the four fixed points in each of the three
complex planes.

###  2.2 Geometry of the
$\boldsymbol{T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ Resolutions

The geometry of the resulting resolved orbifolds are characterised by the set
of four-cycles (divsors), which are obtained by setting one complex coordinate
used in the resolution to zero. There are three classes of divisors [35, 37]:
6 inherited divisors $R_{i}:=\left\\{u_{i}=0\right\\}$ and
$R_{i}^{\prime}:=\left\\{v_{i}=0\right\\}$ that descend from each of the three
torus of the orbifold ($u_{i}$ and $v_{i}$, $i=1,2,3$ are the coordinates of
the elliptic curves describing the two–dimensional tori that make up $T^{6}$),
12 ordinary divisors
$D_{1,\alpha}:=\left\\{z_{1,\alpha}=0\right\\},D_{2,\beta}:=\left\\{z_{2,\beta}=0\right\\}$,
and $D_{3,\gamma}:=$ $\left\\{z_{3,\gamma}=0\right\\}$ ($z_{i,\mu}$ $i=1,2,3$
are the coordinates of the covering space) and finally 48 exceptional divisors
$E_{1,\beta\gamma}:=\left\\{x_{1,\beta\gamma}=0\right\\},E_{2,\alpha\gamma}:=\left\\{x_{2,\alpha\gamma}=0\right\\}$,
and $E_{3,\alpha\beta}:=\left\\{x_{3,\alpha\beta}=0\right\\}$ ($x_{i,\mu\nu}$
are extra coordinates used for the resolution) that appear in the blow–up
process.

Not all these divisors are independent; there are a number of linear relations
among them, namely:

$\displaystyle\begin{array}[]{ll}2D_{1,\alpha}\sim
R_{1}-\sum\limits_{\gamma}E_{2,\alpha\gamma}-\sum\limits_{\beta}E_{3,\alpha\beta}~{},&2D_{2,\beta}\sim
R_{2}-\sum\limits_{\gamma}E_{1,\beta\gamma}-\sum\limits_{\alpha}E_{3,\alpha\beta}\\\\[8.61108pt]
2D_{3,\gamma}\sim
R_{3}-\sum\limits_{\beta}E_{1,\beta\gamma}-\sum\limits_{\alpha}E_{2,\alpha\gamma}~{},&R_{i}^{\prime}\sim
R_{i}\end{array}$ (6)

Here $\sim$ means that these divisors interpreted as $(1,1)$–forms differ by
exact forms. So in the end 3 $R_{i}$ and 48 $E_{r}$ provide via the Poincaré
duality a basis of the real cohomology group, i.e. of the $(1,1)$–forms, on
the resolved manifold.

$D_{2}$$E_{3}$$D_{1}$$E_{2}$$D_{3}$$E_{1}$$D_{2}$$E_{3}$$D_{1}$$E_{2}$$D_{3}$$E_{1}$$D_{2}$$E_{3}$$D_{1}$$E_{2}$$D_{3}$$E_{1}$
Triangulation $S$ Triangulation $E_{1}$ Triangulation $E_{2}$ Triangulation
$E_{3}$ $D_{2}$$E_{3}$$D_{1}$$E_{2}$$D_{3}$$E_{1}$ Figure 1:  The four
different triangulation, the $E_{1}$–, $E_{2}$–, $E_{3}$– and
$S$–triangulation, of the projected toric diagram are given of the resolved
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$. The left–right–arrows
indicate the possible flop–transition between different triangulations, which
shows that any flop–transition always involves the $S$–triangulation.

###  2.3 Triangulation Dependence and Flop–Transitions

To complete the description of the geometry of a resolved orbifold, the
intersection numbers of these divisors have to be specified. A major
complication to specify the intersection numbers of the resolved
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold is that there is an
indeterminacy, because of the triangulation dependence: each resolved
$\mathbb{C}^{3}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ admits four inequivalent
resolutions encoded by four different triangulations of the toric diagram of
the $\mathbb{C}^{3}/\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ singularity. The local
projected toric diagrams are given in figure 1. There are three
triangulations, $E_{1}$, $E_{2}$ and $E_{3}$, where are all curves, that go
through the interior of the projected toric diagram, connect to one of these
exceptional divisors. For example in triangulation $E_{1}$ the curves
$E_{1}E_{2}$, $E_{1}E_{3}$ and $E_{1}D_{1}$ all exist. In the final
triangulation, dubbed the $S$–triangulation, all the exceptional divisors
intersect since the curves $E_{1}E_{2}$, $E_{2}E_{3}$ and $E_{3}E_{1}$ all
exist.

The four triangulations of the projected toric diagram given in figure 1 are
related to each other via flop–transitions. From this figure it can be
inferred, that the $E_{1}$, $E_{2}$ and $E_{3}$–triangulations are all related
via a single flop to the $S$–triangulation. For example, during the
flop–transition from the $E_{1}$–triangulation to the $S$–triangulation, the
curve $E_{1}D_{1}$ shrinks to zero size and disappears while the curve
$E_{1}E_{2}$ appears. To go from one $E$–triangulation to another one always
has to go through the $S$–triangulation. For example, for the transition from
triangulation $E_{1}$ to $E_{2}$, first the curve $E_{1}D_{1}$ is replaced by
the curve $E_{2}E_{3}$ to form the $S$–triangulation and after that the curve
$E_{1}E_{3}$ is replaced by the curve $E_{2}D_{2}$ to arrive in the
$E_{2}$–triangulation. This shows that the special role the $S$–triangulation
plays in flop–transitions.

During a flop–transition some curve shrinks to zero size. This means that in
this process the effective field theory approximation in the supergravity
regime breaks down and stringy corrections could become important. Since, this
work only makes use of effective field theory and geometrical methods, the
flop–transitions themselves are beyond our description. But the geometries and
the spectra on both sides of flops can be determined.

###  2.4 Parameterising Triangulations

$\begin{array}[]{|c||cccc|ccc|ccc|}\hline\cr\text{Triangl.}&\delta^{E_{1}}_{\alpha\beta\gamma}&\delta^{E_{2}}_{\alpha\beta\gamma}&\delta^{E_{3}}_{\alpha\beta\gamma}&\delta^{S}_{\alpha\beta\gamma}&\Delta^{1}_{\alpha\beta\gamma}&\Delta^{2}_{\alpha\beta\gamma}&\Delta^{3}_{\alpha\beta\gamma}&1-\Delta^{1}_{\alpha\beta\gamma}&1-\Delta^{2}_{\alpha\beta\gamma}&1-\Delta^{3}_{\alpha\beta\gamma}\\\
\hline\cr\hline\cr E_{1}&1&0&0&0&-1&1&1&2&0&0\\\ E_{2}&0&1&0&0&1&-1&1&0&2&0\\\
E_{3}&0&0&1&0&1&1&-1&0&0&2\\\ S&0&0&0&1&0&0&0&1&1&1\\\ \hline\cr\end{array}$
Table 1:  The values of the step functions $\delta^{T}_{\alpha\beta\gamma}$
and their variations $\Delta^{i}_{\alpha\beta\gamma}$, defined in (7) and
(12), resp., for the different triangulations are given.

Given that there are four triangulation for each
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ and 64
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ singularities, this gives a naively
total number of $4^{64}$ possibilities (up to some permutation symmetries)
[37]. As important topological data such as the intersection numbers of the
divisors varies for each triangulation, it is particularly useful to develop
some formalism to study spectra and the consistency conditions (such as
Bianchi identities) for all triangulation choices simultaneously. Next, a
formalism will be laid out that is capable of doing just that.

Define the following four functions:

$\displaystyle\delta^{T}_{\alpha\beta\gamma}=\begin{cases}1&\text{if
triangulation }T\text{ is used,}\\\\[4.30554pt] 0&\text{if other triangulation
is used,}\end{cases}$ (7)

of $(\alpha,\beta,\gamma)$ for the four possible triangulations $T=S$,
$E_{1}$, $E_{2}$ and $E_{3}$. Since at any of the 64 singularity resolutions
one of the four triangulations has to be used, it follows that

$\displaystyle\delta^{E_{1}}_{\alpha\beta\gamma}+\delta^{E_{2}}_{\alpha\beta\gamma}+\delta^{E_{3}}_{\alpha\beta\gamma}+\delta^{S}_{\alpha\beta\gamma}=1~{}.$
(8)

Thus, say, $\delta^{S}_{\alpha\beta\gamma}$ is a function of the others. The
following combinations of the remaining three independent functions prove
particularly useful:

$\displaystyle\begin{array}[]{rcl}\Delta^{1}_{\alpha\beta\gamma}&=&-\delta^{E_{1}}_{\alpha\beta\gamma}+\delta^{E_{2}}_{\alpha\beta\gamma}+\delta^{E_{3}}_{\alpha\beta\gamma}~{},\\\\[8.61108pt]
\Delta^{2}_{\alpha\beta\gamma}&=&\phantom{-}\delta^{E_{1}}_{\alpha\beta\gamma}-\delta^{E_{2}}_{\alpha\beta\gamma}+\delta^{E_{3}}_{\alpha\beta\gamma}~{},\\\\[8.61108pt]
\Delta^{3}_{\alpha\beta\gamma}&=&\phantom{-}\delta^{E_{1}}_{\alpha\beta\gamma}+\delta^{E_{2}}_{\alpha\beta\gamma}-\delta^{E_{3}}_{\alpha\beta\gamma}~{}.\end{array}$
(12)

For example, this means that $\Delta^{1}_{\alpha\beta\gamma}$ equals $-1$ if
singularity $f_{\alpha\beta\gamma}$ is resolved using triangulation $E_{1}$,
$1$ if $E_{2}$ and $E_{3}$ and $0$ if $S$. The values that these functions
take can be easily read off from the Table 1. It follows immediately that

$\displaystyle
1-\Delta^{1}_{\alpha\beta\gamma}-\Delta^{2}_{\alpha\beta\gamma}-\Delta^{3}_{\alpha\beta\gamma}=\delta^{S}_{\alpha\beta\gamma}~{},\qquad
1-\Delta^{i}_{\alpha\beta\gamma}=2\,\delta^{E_{i}}_{\alpha\beta\gamma}+\delta^{S}_{\alpha\beta\gamma}~{}.$
(13a) and
$\displaystyle\Delta^{2}_{\alpha\beta\gamma}+\Delta^{3}_{\alpha\beta\gamma}=2\,\delta^{E_{1}}_{\alpha\beta\gamma}~{},\qquad\Delta^{1}_{\alpha\beta\gamma}+\Delta^{3}_{\alpha\beta\gamma}=2\,\delta^{E_{2}}_{\alpha\beta\gamma}~{},\qquad\Delta^{1}_{\alpha\beta\gamma}+\Delta^{2}_{\alpha\beta\gamma}=2\,\delta^{E_{3}}_{\alpha\beta\gamma}~{}.$
(13b)

###  2.5 Triangulation Dependence of (Self–)Intersections and Chern Classes

The fundamental (self–)intersection numbers of the basis of divisors read:

$\displaystyle\begin{array}[]{ll}R_{1}E_{1,\beta\gamma}^{2}=R_{2}E_{2,\alpha\gamma}^{2}=R_{3}E_{3,\alpha\beta}^{2}=-2~{},&R_{1}R_{2}R_{3}=2~{},\\\\[8.61108pt]
E_{1,\beta\gamma}E_{2,\alpha\gamma}^{2}=E_{1,\beta\gamma}E_{3,\beta\gamma}^{2}=-1+\Delta^{1}_{\alpha\beta\gamma}~{},&E_{1,\beta\gamma}^{3}=\sum\limits_{\alpha}\big{(}1+\Delta^{1}_{\alpha\beta\gamma}\big{)}~{},\\\\[8.61108pt]
E_{2,\alpha\gamma}E_{1,\beta\gamma}^{2}=E_{2,\alpha\gamma}E_{3,\beta\gamma}^{2}=-1+\Delta^{2}_{\alpha\beta\gamma}~{},&E_{2,\alpha\gamma}^{3}=\sum\limits_{\beta}\big{(}1+\Delta^{2}_{\alpha\beta\gamma}\big{)}~{},\\\\[8.61108pt]
E_{3,\alpha\beta}E_{1,\beta\gamma}^{2}=E_{3,\alpha\beta}E_{2,\alpha\gamma}^{2}=-1+\Delta^{3}_{\alpha\beta\gamma}~{},&E_{3,\alpha\beta}^{3}=\sum\limits_{\gamma}\big{(}1+\Delta^{3}_{\alpha\beta\gamma}\big{)}~{},\\\\[8.61108pt]
E_{1,\beta\gamma}E_{2,\alpha\gamma}E_{3,\beta\gamma}=1-\Delta^{1}_{\alpha\beta\gamma}-\Delta^{2}_{\alpha\beta\gamma}-\Delta^{3}_{\alpha\beta\gamma}~{}.\end{array}$
(19)

and all others are always zero. These (self–)intersection numbers can be
partially inferred from the results in ref. [37] as follows: as observed in
that paper the (partially self–)intersection numbers involving the ordinary
divisors $R_{i}$ are triangulation independent. The (partial
self–)intersection numbers involving all three labels $\alpha,\beta$ and
$\gamma$ are fully local, i.e. defined only at the resolution of the single
singularity $f_{\alpha\beta\gamma}$. Thus the intersection numbers for these
(partial self–)intersections can be directly read off from Table 4 of ref.
[37]. (Using the functions $\Delta^{i}_{\alpha\beta\gamma}$ precisely the
local intersection numbers of the four different triangulations of that table
are reproduced.) This leaves the cubic self–intersection numbers
$E_{1,\beta\gamma}^{3}$, $E_{2,\alpha\gamma}^{3}$ and $E_{3,\alpha\beta}^{3}$.
But these can be determined using the linear equivalence relations (6). For
example, since the divisors $D_{1,\alpha}$, $D_{3,\gamma}$ and
$E_{2,\alpha\gamma}$ lie on a straight line in the toric diagram, their
intersection vanishes: $D_{1,\alpha}E_{2,\alpha\gamma}D_{3,\gamma}=0$.
Inserting the linear equivalence relations then leads to the identity

$\displaystyle
E_{2,\alpha\gamma}^{3}=-\sum_{\beta}\Big{\\{}E_{1,\beta\gamma}E_{2,\alpha\gamma}^{2}+E_{3,\alpha\beta}E_{2,\alpha\gamma}^{2}+E_{1,\beta\gamma}E_{2,\alpha\gamma}E_{3,\alpha\beta}\Big{\\}}=\sum_{\beta}\big{(}1+\Delta^{2}_{\alpha\beta\gamma}\big{)}~{}.$
(20)

This expresses $E_{2,\alpha\gamma}^{3}$ in fully local (partial
self–)intersection numbers just determined. Inserting those leads to the final
expression in this equation. The other two cubic self–intersections are
computed in an analogous fashion.

With the fundamental (self–)intersection numbers fixed for any choice of
triangulation of all of the 64 resolved $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$
singularities, all kind of other quantities can be computed. For example, the
second Chern classes integrated over the basis of divisors can be determined
to be given by

$\displaystyle\begin{array}[]{ll}c_{2}R_{1}=c_{2}R_{2}=c_{2}R_{3}=24~{},&c_{2}E_{1,\beta\gamma}=\sum\limits_{\alpha}\big{(}1-2\Delta^{1}_{\alpha\beta\gamma}\big{)}~{},\\\\[8.61108pt]
c_{2}E_{2,\alpha\gamma}=\sum\limits_{\beta}\big{(}1-2\Delta^{2}_{\alpha\beta\gamma}\big{)}~{},&c_{2}E_{3,\alpha\beta}=\sum\limits_{\gamma}\big{(}1-2\Delta^{3}_{\alpha\beta\gamma}\big{)}~{}.\end{array}$
(23)

The third Chern class can be evaluated as

$\displaystyle c_{3}=\dfrac{1}{3}\sum_{u}(-)^{u}S_{u}^{3}~{},$ (24)

using (A.8) given that the first Chern class vanishes. Since the inherited
torus divisors $R_{i},R_{i}^{\prime}$ have vanishing triple
self–intersections, this expression reduces to a sum over all ordinary and
exceptional divisors

$\displaystyle
c_{3}=\dfrac{1}{3}\sum_{\alpha}D_{1,\alpha}^{3}+\dfrac{1}{3}\sum_{\beta}D_{2,\beta}^{3}+\dfrac{1}{3}\sum_{\gamma}D_{3,\gamma}^{3}+\dfrac{1}{3}\sum_{\beta.\gamma}E_{1,\beta\gamma}^{3}+\dfrac{1}{3}\sum_{\alpha,\gamma}E_{2,\alpha\gamma}^{3}+\dfrac{1}{3}\sum_{\alpha,\beta}E_{3,\alpha\beta}^{3}~{}.$
(25)

The first term can be written as

$\displaystyle\sum_{\alpha}D_{1,\alpha}^{3}=-\dfrac{1}{8}\sum_{\alpha,\gamma}E_{2,\alpha\gamma}^{3}-\dfrac{1}{8}\sum_{\alpha,\beta}E_{3,\alpha\beta}^{3}-\dfrac{3}{8}\sum_{\alpha,\beta,\gamma}\Big{(}E_{2,\alpha\gamma}^{2}E_{3,\alpha\beta}+E_{2,\alpha\gamma}E_{3,\alpha\beta}^{2}\Big{)}~{},$
(26)

using that there are no non–vanishing intersections of $R_{1}$ with
$E_{2,\alpha\gamma}$ or $E_{3,\alpha\beta}$. Adding similar expressions
involving $D_{2,\beta}$ and $D_{3,\gamma}$, one can show that

$\displaystyle
c_{3}=-\dfrac{1}{8}\sum_{\alpha,\beta,\gamma}\Big{\\{}E_{1,\beta\gamma}\big{(}E_{2,\alpha\gamma}^{2}+E_{3,\alpha\beta}^{2}\big{)}+E_{2,\alpha\gamma}\big{(}E_{1,\beta\gamma}^{2}+E_{3,\alpha\beta}^{2}\big{)}+E_{3,\alpha\beta}\big{(}E_{1,\beta\gamma}^{2}+E_{2,\alpha\gamma}^{2}\big{)}\Big{\\}}$
$\displaystyle+\dfrac{1}{4}\sum_{\beta.\gamma}E_{1,\beta\gamma}^{3}+\dfrac{1}{4}\sum_{\alpha,\gamma}E_{2,\alpha\gamma}^{3}+\dfrac{1}{4}\sum_{\alpha,\beta}E_{3,\alpha\beta}^{3}~{}.$
(27)

Finally, inserting the triangulation dependent intersection numbers (19),
gives

$\displaystyle
c_{3}=\dfrac{1}{4}\sum_{i,\alpha,\beta,\gamma}\Big{(}1+\Delta_{\alpha\beta\gamma}^{i}\Big{)}-\dfrac{1}{4}\sum_{i,\alpha,\beta,\gamma}\Big{(}-1+\Delta_{\alpha\beta\gamma}^{i}\Big{)}=96~{}.$
(28)

Note, in particular, that all the triangulation dependence in the form of the
functions $\Delta^{i}_{\alpha\beta\gamma}$ drops out and the final result
equals the well–known Euler number 96.

###  2.6 Line Bundle Backgrounds

The line bundle backgrounds considered in this paper only have flux supported
on the exceptional cycles:

$\displaystyle\frac{{\cal
F}}{2\pi}=\sum_{i,\mu,\nu}E_{i,\mu\nu}\,\mathsf{H}_{i,\mu\nu}~{},\qquad\mathsf{H}_{i,\mu\nu}=\sum_{I}{\cal
V}_{i,\mu\nu}^{I}\,\mathsf{H}_{I}~{}.$ (29)

Here the Cartan generators $\mathsf{H}_{I}$ are anti–Hermitian and therefore
so is the field strength ${\cal F}$. The entries of the line bundle vectors
${\cal V}_{i,\mu\nu}$ are subject to flux quantisation conditions which are
triangulation dependent:

$\displaystyle\int_{C}\frac{{\cal F}}{2\pi}=L^{I}\,\mathsf{H}_{I}~{},\qquad
L\cong 0~{},$ (30)

where $\cong$ means equal up to $E_{8}\times E_{8}$ lattice vectors, for any
$C$ inside the resolved orbifold. The resulting conditions for any choice of
triangulation are listed in Table 2.

$\begin{array}[]{|c|c||c|c|}\hline\cr\vrule\lx@intercol\hfil\text{Flux
quantisation conditions for arbitrary
triangulations}\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr\hline\cr
R_{1}E_{1,\beta\gamma}&2\,{\cal V}_{1,\beta\gamma}\cong
0&D_{1,\alpha}E_{1,\beta\gamma}&\big{(}{\cal V}_{1,\beta\gamma}-{\cal
V}_{2,\alpha\gamma}-{\cal
V}_{3,\alpha\beta}\big{)}\delta^{E_{1}}_{\alpha\beta\gamma}\cong 0\\\
\hline\cr R_{2}E_{2,\alpha\gamma}&2\,{\cal V}_{2,\alpha\gamma}\cong
0&D_{2,\beta}E_{2,\alpha\gamma}&\big{(}{\cal V}_{2,\alpha\gamma}-{\cal
V}_{1,\beta\gamma}-{\cal
V}_{3,\alpha\beta}\big{)}\delta^{E_{2}}_{\alpha\beta\gamma}\cong 0\\\
\hline\cr R_{3}E_{3,\alpha\beta}&2\,{\cal V}_{3,\alpha\beta}\cong
0&D_{3,\gamma}E_{3,\alpha\beta}&\big{(}{\cal V}_{3,\alpha\beta}-{\cal
V}_{1,\beta\gamma}-{\cal
V}_{2,\alpha\gamma}\big{)}\delta^{E_{3}}_{\alpha\beta\gamma}\cong 0\\\
\hline\cr\hline\cr R_{1}D_{2,\beta}&-\sum\limits_{\gamma}{\cal
V}_{1,\beta\gamma}\cong
0&D_{1,\alpha}E_{2,\alpha\gamma}&-\sum\limits_{\beta}\Big{\\{}{\cal
V}_{3,\alpha\beta}+\big{(}{\cal V}_{1,\beta\gamma}-{\cal
V}_{2,\alpha\gamma}-{\cal
V}_{3,\alpha\beta}\big{)}\delta^{E_{1}}_{\alpha\beta\gamma}\Big{\\}}\cong 0\\\
\hline\cr R_{1}D_{3,\gamma}&-\sum\limits_{\beta}{\cal V}_{1,\beta\gamma}\cong
0&D_{1,\alpha}E_{3,\alpha\beta}&-\sum\limits_{\gamma}\Big{\\{}{\cal
V}_{2,\alpha\gamma}+\big{(}{\cal V}_{1,\beta\gamma}-{\cal
V}_{2,\alpha\gamma}-{\cal
V}_{3,\alpha\beta}\big{)}\delta^{E_{1}}_{\alpha\beta\gamma}\Big{\\}}\cong 0\\\
\hline\cr R_{2}D_{1,\alpha}&-\sum\limits_{\gamma}{\cal
V}_{2,\alpha\gamma}\cong
0&D_{2,\beta}E_{1,\beta\gamma}&-\sum\limits_{\alpha}\Big{\\{}{\cal
V}_{3,\alpha\beta}+\big{(}{\cal V}_{2,\alpha\gamma}-{\cal
V}_{1,\beta\gamma}-{\cal
V}_{3,\alpha\beta}\big{)}\delta^{E_{2}}_{\alpha\beta\gamma}\Big{\\}}\cong 0\\\
\hline\cr R_{2}D_{3,\gamma}&-\sum\limits_{\alpha}{\cal
V}_{2,\alpha\gamma}\cong
0&D_{2,\beta}E_{3,\alpha\beta}&-\sum\limits_{\gamma}\Big{\\{}{\cal
V}_{1,\beta\gamma}+\big{(}{\cal V}_{2,\alpha\gamma}-{\cal
V}_{1,\beta\gamma}-{\cal
V}_{3,\alpha\beta}\big{)}\delta^{E_{2}}_{\alpha\beta\gamma}\Big{\\}}\cong 0\\\
\hline\cr R_{3}D_{1,\alpha}&-\sum\limits_{\beta}{\cal V}_{3,\alpha\beta}\cong
0&D_{3,\gamma}E_{1,\beta\gamma}&-\sum\limits_{\alpha}\Big{\\{}{\cal
V}_{2,\alpha\gamma}+\big{(}{\cal V}_{3,\alpha\beta}-{\cal
V}_{1,\beta\gamma}-{\cal
V}_{2,\alpha\gamma}\big{)}\delta^{E_{3}}_{\alpha\beta\gamma}\Big{\\}}\cong
0\\\ \hline\cr R_{3}D_{2,\beta}&-\sum\limits_{\alpha}{\cal
V}_{3,\alpha\beta}\cong
0&D_{3,\gamma}E_{2,\alpha\gamma}&-\sum\limits_{\beta}\Big{\\{}{\cal
V}_{1,\beta\gamma}+\big{(}{\cal V}_{3,\alpha\beta}-{\cal
V}_{1,\beta\gamma}-{\cal
V}_{2,\alpha\gamma}\big{)}\delta^{E_{3}}_{\alpha\beta\gamma}\Big{\\}}\cong
0\\\ \hline\cr\hline\cr E_{1,\beta\gamma}E_{2,\alpha\gamma}&\lx@intercol\hfil
2\,{\cal V}_{2,\alpha\gamma}\,\delta^{E_{1}}_{\alpha\beta\gamma}+2\,{\cal
V}_{1,\beta\gamma}\,\delta^{E_{2}}_{\alpha\beta\gamma}+\big{(}{\cal
V}_{1,\beta\gamma}+{\cal V}_{2,\alpha\gamma}-{\cal
V}_{3,\alpha\beta}\big{)}\delta^{S}_{\alpha\beta\gamma}\cong
0\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr
E_{1,\beta\gamma}E_{3,\alpha\beta}&\lx@intercol\hfil 2\,{\cal
V}_{3,\alpha\beta}\,\delta^{E_{1}}_{\alpha\beta\gamma}+2\,{\cal
V}_{1,\beta\gamma}\,\delta^{E_{3}}_{\alpha\beta\gamma}+\big{(}{\cal
V}_{1,\beta\gamma}+{\cal V}_{3,\alpha\beta}-{\cal
V}_{2,\alpha\gamma}\big{)}\delta^{S}_{\alpha\beta\gamma}\cong
0\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr
E_{2,\alpha\gamma}E_{3,\alpha\beta}&\lx@intercol\hfil 2\,{\cal
V}_{3,\alpha\beta}\,\delta^{E_{2}}_{\alpha\beta\gamma}+2\,{\cal
V}_{2,\alpha\gamma}\,\delta^{E_{3}}_{\alpha\beta\gamma}+\big{(}{\cal
V}_{2,\alpha\gamma}+{\cal V}_{3,\alpha\beta}-{\cal
V}_{1,\beta\gamma}\big{)}\delta^{S}_{\alpha\beta\gamma}\cong
0\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr\end{array}$ Table 2: The
flux quantisation conditions on the line bundle vectors ${\cal V}_{i,\mu\nu}$
the resolved orbifold $X$ using arbitrary triangulation at the 64
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\\!\times\\!\mathbbm{Z}_{2}$ resolutions.

###  2.7 General Bianchi Identities

Consistency of the effective field theory description demands that the
integrated Bianchi identity

$\displaystyle\int_{D}\Big{\\{}\text{tr}{\cal F}^{2}-\text{tr}{\cal
R}^{2}\Big{\\}}=0$ (31)

over any divisor $D$ vanishes. Here ${\cal R}$ denotes the anti–Hermitian
curvature two–form. (When non–perturbative contributions of heterotic
five–branes are taken into account this condition can be weakened somewhat
[41].) By considering the basis of divisors spanned by the ordinary divisors
$R_{i}$ and the exceptional divisors $E_{1,\beta\gamma}$, $E_{2,\alpha\gamma}$
and $E_{3,\alpha\beta}$ the complete set of integrated Bianchi identities is
obtained.

The three Bianchi identities on the three ordinary divisors, $R_{1}$, $R_{2}$
and $R_{3}$ are the ones one expects on K3 surfaces:

$\sum_{\beta,\gamma}{\cal
V}_{1,\beta\gamma}^{2}=24~{},\quad\sum_{\alpha,\gamma}{\cal
V}_{2,\alpha\gamma}^{2}=24~{},\quad\sum_{\alpha,\beta}{\cal
V}_{3,\alpha\beta}^{2}=24~{},$ (32a) and do not depend on the triangulations
chosen. In contrast the Bianchi identities on the exceptional divisors are
very sensitive to the triangulations used in the local resolutions. The
sixteen Bianchi identities on $E_{1,\beta\gamma}$ take the form
$\displaystyle\sum\limits_{\alpha}\Big{[}(1+\Delta^{1}_{\alpha\beta\gamma}){\cal
V}^{2}_{1,\beta\gamma}+(-1+\Delta^{1}_{\alpha\beta\gamma})({\cal
V}^{2}_{2,\alpha\gamma}+{\cal
V}^{2}_{3,\alpha\beta})+2(1-\Delta^{1}_{\alpha\beta\gamma}-\Delta^{2}_{\alpha\beta\gamma}-\Delta^{3}_{\alpha\beta\gamma}){\cal
V}_{2,\alpha\gamma}\cdot{\cal V}_{3,\alpha\beta}$
$\displaystyle+2(-1+\Delta^{2}_{\alpha\beta\gamma}){\cal
V}_{1,\beta\gamma}\cdot{\cal
V}_{2,\alpha\gamma}+2(-1+\Delta^{3}_{\alpha\beta\gamma}){\cal
V}_{1,\beta\gamma}\cdot{\cal
V}_{3,\alpha\beta}\Big{]}=\sum\limits_{\alpha}\Big{[}-2+4\,\Delta^{1}_{\alpha\beta\gamma}\Big{]}~{}.$
(32b) The sixteen Bianchi identities on $E_{2,\alpha\gamma}$ take the form
$\displaystyle\sum\limits_{\beta}\Big{[}(1+\Delta^{2}_{\alpha\beta\gamma}){\cal
V}^{2}_{2,\alpha\gamma}+(-1+\Delta^{2}_{\alpha\beta\gamma})({\cal
V}^{2}_{1,\beta\gamma}+{\cal
V}^{2}_{3,\alpha\gamma})+2(1-\Delta^{1}_{\alpha\beta\gamma}-\Delta^{2}_{\alpha\beta\gamma}-\Delta^{3}_{\alpha\beta\gamma}){\cal
V}_{1,\beta\gamma}\cdot{\cal V}_{3,\alpha\beta}$
$\displaystyle+2(-1+\Delta^{1}_{\alpha\beta\gamma}){\cal
V}_{2,\alpha\gamma}\cdot{\cal
V}_{1,\beta\gamma}+2(-1+\Delta^{3}_{\alpha\beta\gamma}){\cal
V}_{2,\alpha\gamma}\cdot{\cal
V}_{2,\alpha\gamma}\Big{]}=\sum\limits_{\beta}\Big{[}-2+4\,\Delta^{2}_{\alpha\beta\gamma}\Big{]}~{}.$
(32c) And finally, the sixteen Bianchi identities on $E_{3,\alpha\beta}$ take
the form
$\displaystyle\sum\limits_{\gamma}\Big{[}(1+\Delta^{3}_{\alpha\beta\gamma}){\cal
V}^{2}_{3,\alpha\beta}+(-1+\Delta^{3}_{\alpha\beta\gamma})({\cal
V}^{2}_{1,\beta\gamma}+{\cal
V}^{2}_{2,\alpha\gamma})+2(1-\Delta^{1}_{\alpha\beta\gamma}-\Delta^{2}_{\alpha\beta\gamma}-\Delta^{3}_{\alpha\beta\gamma}){\cal
V}_{1,\beta\gamma}\cdot{\cal V}_{2,\alpha\gamma}$
$\displaystyle+2(-1+\Delta^{1}_{\alpha\beta\gamma}){\cal
V}_{3,\alpha\beta}\cdot{\cal
V}_{1,\beta\gamma}+2(-1+\Delta^{2}_{\alpha\beta\gamma}){\cal
V}_{3,\alpha\beta}\cdot{\cal
V}_{2,\alpha\gamma}\Big{]}=\sum\limits_{\gamma}\Big{[}-2+4\,\Delta^{3}_{\alpha\beta\gamma}\Big{]}~{}.$
(32d)

###  2.8 Multiplicity Operators

A convenient tool to compute the chiral spectrum on a resolution with a line
bundle background is the multiplicity operator $\mathsf{N}$. It reads [31,
32]:

$\displaystyle\mathsf{N}=\int_{X}\Big{\\{}\dfrac{1}{6}\Big{(}\dfrac{{\cal
F}}{2\pi}\Big{)}^{2}-\dfrac{1}{24}\Big{(}\dfrac{{\cal
R}}{2\pi}\Big{)}^{2}\Big{\\}}\dfrac{{\cal F}}{2\pi}$ (33)

and may be thought of as a representation dependent index. Hence, on all
states it should be integral provided that the fundamental consistency
conditions, flux quantisation and the integrated Bianchi identities, are
fulfilled.

On the $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ resolutions the
multiplicity operator can be evaluated to be equal to:

$\displaystyle\begin{array}[]{rl}\mathsf{N}=\sum\limits_{\alpha,\beta,\gamma}\Big{[}&\mathsf{H}_{1,\beta\gamma}\Big{\\{}\tfrac{1}{3}(\mathsf{H}_{1,\beta\gamma}^{2}-\tfrac{1}{4})-\big{(}1-\Delta^{1}_{\alpha\beta\gamma}\big{)}\Big{(}\tfrac{1}{6}(\mathsf{H}_{1,\beta\gamma}^{2}-1)+\tfrac{1}{2}(\mathsf{H}_{2,\alpha\gamma}-\mathsf{H}_{3,\alpha\beta})^{2}\Big{)}\Big{\\}}\\\\[8.61108pt]
&\\!\\!\\!\\!\\!+\mathsf{H}_{2,\alpha\gamma}\Big{\\{}\tfrac{1}{3}(\mathsf{H}_{2,\alpha\gamma}^{2}-\tfrac{1}{4})-\big{(}1-\Delta^{2}_{\alpha\beta\gamma}\big{)}\Big{(}\tfrac{1}{6}(\mathsf{H}_{2,\alpha\gamma}^{2}-1)+\tfrac{1}{2}(\mathsf{H}_{1,\beta\gamma}-\mathsf{H}_{3,\alpha\beta})^{2}\Big{)}\Big{\\}}\\\\[8.61108pt]
&\\!\\!\\!\\!\\!+\mathsf{H}_{3,\alpha\beta}\Big{\\{}\tfrac{1}{3}(\mathsf{H}_{3,\alpha\beta}^{2}-\tfrac{1}{4})-\big{(}1-\Delta^{3}_{\alpha\beta\gamma}\big{)}\Big{(}\tfrac{1}{6}(\mathsf{H}_{3,\alpha\beta}^{2}-1)+\tfrac{1}{2}(\mathsf{H}_{1,\beta\gamma}-\mathsf{H}_{2,\alpha\gamma})^{2}\Big{)}\Big{\\}}\\\\[8.61108pt]
&\\!\\!\\!\\!\\!-2\mathsf{H}_{1,\beta\gamma}\mathsf{H}_{2,\alpha\gamma}\mathsf{H}_{3,\alpha\beta}~{}~{}\Big{]}~{}.\end{array}$
(38)

The triangulation dependance is isolated to the second terms on the first
three lines of this expression. From Table 1 it may be inferred that only the
terms in the first line are switched on (with a multiplicative factor of 2) if
triangulation $E_{1}$ is chosen, the second for $E_{2}$ and the third for
$E_{3}$; all three are switched on (with a factor 1) for triangulation $S$.

Using the constraint (8) another representation of this operator can be
obtained

$\displaystyle\mathsf{N}=\sum_{\alpha,\beta,\gamma}\Big{[}\delta^{E_{1}}_{\alpha\beta\gamma}\,\mathsf{N}^{E_{1}}_{\alpha\beta\gamma}+\delta^{E_{2}}_{\alpha\beta\gamma}\,\mathsf{N}^{E_{2}}_{\alpha\beta\gamma}+\delta^{E_{3}}_{\alpha\beta\gamma}\,\mathsf{N}^{E_{3}}_{\alpha\beta\gamma}+\delta^{S}_{\alpha\beta\gamma}\,\mathsf{N}^{S}_{\alpha\beta\gamma}\Big{]}~{},$
(39)

where

$\displaystyle\mathsf{N}^{E_{1}}_{\alpha\beta\gamma}$
$\displaystyle=\tfrac{1}{4}\,\mathsf{H}_{1,\beta\gamma}+\tfrac{1}{12}\,\mathsf{H}_{2,\alpha\gamma}\big{(}4\mathsf{H}_{2,\alpha\gamma}^{2}-1\big{)}+\tfrac{1}{12}\,\mathsf{H}_{3,\alpha\beta}\big{(}4\mathsf{H}_{3,\alpha\beta}^{2}-1\big{)}-\mathsf{H}_{1,\beta\gamma}\big{(}\mathsf{H}_{2,\alpha\gamma}^{2}+\mathsf{H}_{3,\alpha\beta}^{2}\big{)}~{},$
(40a) $\displaystyle\mathsf{N}^{E_{2}}_{\alpha\beta\gamma}$
$\displaystyle=\tfrac{1}{4}\,\mathsf{H}_{2,\alpha\gamma}+\tfrac{1}{12}\,\mathsf{H}_{1,\beta\gamma}\big{(}4\mathsf{H}_{1,\beta\gamma}^{2}-1\big{)}+\tfrac{1}{12}\,\mathsf{H}_{3,\alpha\beta}\big{(}4\mathsf{H}_{3,\alpha\beta}^{2}-1\big{)}-\mathsf{H}_{2,\alpha\gamma}\big{(}\mathsf{H}_{1,\beta\gamma}^{2}+\mathsf{H}_{3,\alpha\beta}^{2}\big{)}~{},$
(40b) $\displaystyle\mathsf{N}^{E_{3}}_{\alpha\beta\gamma}$
$\displaystyle=\tfrac{1}{4}\,\mathsf{H}_{3,\alpha\beta}+\tfrac{1}{12}\,\mathsf{H}_{1,\alpha\beta}\big{(}4\mathsf{H}_{1,\alpha\beta}^{2}-1\big{)}+\tfrac{1}{12}\,\mathsf{H}_{2,\alpha\gamma}\big{(}4\mathsf{H}_{2,\alpha\gamma}^{2}-1\big{)}-\mathsf{H}_{3,\alpha\beta}\big{(}\mathsf{H}_{1,\beta\gamma}^{2}+\mathsf{H}_{2,\alpha\gamma}^{2}\big{)}~{},$
(40c) $\displaystyle\mathsf{N}^{S}_{\alpha\beta\gamma}$
$\displaystyle=\tfrac{1}{12}\,\mathsf{H}_{1,\beta\gamma}\big{(}2\mathsf{H}_{1,\beta\gamma}^{2}+1)+\tfrac{1}{12}\,\mathsf{H}_{2,\alpha\gamma}\big{(}2\mathsf{H}_{2,\alpha\gamma}^{2}+1)+\tfrac{1}{12}\,\mathsf{H}_{3,\alpha\beta}\big{(}2\mathsf{H}_{3,\alpha\beta}^{2}+1)+\mathsf{H}_{1,\beta\gamma}\mathsf{H}_{2,\alpha\gamma}\mathsf{H}_{3,\alpha\beta}$
$\displaystyle\phantom{=}-\tfrac{1}{2}\,\mathsf{H}_{1,\beta\gamma}\big{(}\mathsf{H}_{2,\alpha\gamma}^{2}+\mathsf{H}_{3,\alpha\beta}^{2}\big{)}-\tfrac{1}{2}\,\mathsf{H}_{2,\alpha\gamma}\big{(}\mathsf{H}_{1,\beta\gamma}^{2}+\mathsf{H}_{3,\alpha\beta}^{2}\big{)}-\tfrac{1}{2}\,\mathsf{H}_{3,\alpha\beta}\big{(}\mathsf{H}_{1,\beta\gamma}^{2}+\mathsf{H}_{2,\alpha\gamma}^{2}\big{)}~{}.$
(40d)

These operators can be thought of as the local resolution multiplicities at
the resolved singularity $(\alpha,\beta,\gamma)$ using one of the four
triangulations. In particular, when taking the same triangulation at all fixed
points, the expressions (56) and (58) of ref. [37] are obtained from (39). In
general, (39) implies that the spectrum in any triangulation can be determined
from the local resolution operators (40) times the functions that indicate
which triangulation has been used at each of the 64
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ resolved singularities.
It should be emphasised that these local multiplicity operators
$\mathsf{N}^{T}_{\alpha\beta\gamma}$ for a given triangulation $T$ are not
necessarily all integral; only their combination in (39) in general is.

###  2.9 Jumping Spectra due to Flop–Transitions

For a flop–transition to be possible it is necessary that all fundamental
consistency conditions, like flux quantisation and the Bianchi identities,
have to hold for both triangulation choices before and after the flop. Note
that this implies, that if at some resolved singularity
$f_{\alpha\beta\gamma}$ some of these fundamental consistency conditions are
not fulfilled for triangulation $S$, then no flop–transitions can occur and
resolution is frozen in one of the three triangulations $E_{1},E_{2}$ or
$E_{3}$. Moreover, if at all resolved $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$
singularities triangulation $S$ is not admissible, no flop–transition is
possible at all!

Assuming that at a resolved singularity $f_{\alpha\beta\gamma}$ a
flop–transition can occur between triangulations $S$ to $E_{i}$, the
difference multiplicity

$\displaystyle\Delta\mathsf{N}^{i}_{\alpha\beta\gamma}=\mathsf{N}^{E_{i}}_{\alpha\beta\gamma}-\mathsf{N}^{S}_{\alpha\beta\gamma}$
(41)

measures the jump in the spectra when the flop–transition goes from
triangulation $S$ to $E_{i}$; $-\Delta\mathsf{N}^{i}_{\alpha\beta\gamma}$ the
spectra jump in the opposite direction. This difference multiplicity operator
has to be integral because the multiplicity operator (38) before and after the
flop–transition is integral by an index theorem (since the fundamental
consistency conditions are assumed to be fulfilled) and this operator is
simply the difference of the spectra in the two cases.

###  2.10 Volumes and the DUY equations

$\begin{array}[]{|c|c||c|c|}\hline\cr\vrule\lx@intercol\hfil\text{Curves}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr
R_{1}R_{2}&2\,a_{3}&D_{1,\alpha}E_{1,\beta\gamma}&\big{(}b_{1,\beta\gamma}-b_{2,\alpha\gamma}-b_{3,\alpha\beta}\big{)}\delta^{E_{1}}_{\alpha\beta\gamma}\\\
\hline\cr
R_{1}E_{1,\beta\gamma}&2\,b_{1,\beta\gamma}&D_{1,\alpha}E_{2,\alpha\gamma}&a_{2}-\sum\limits_{\beta}\Big{\\{}b_{3,\alpha\beta}+\big{(}b_{1,\beta\gamma}-b_{2,\alpha\gamma}-b_{3,\alpha\beta}\big{)}\delta^{E_{1}}_{\alpha\beta\gamma}\Big{\\}}\\\
\hline\cr
R_{1}D_{2,\beta}&a_{3}-\sum\limits_{\gamma}b_{1,\beta\gamma}&E_{1,\beta\gamma}E_{2,\alpha\gamma}&2\,b_{2,\alpha\gamma}\,\delta^{E_{1}}_{\alpha\beta\gamma}+2\,b_{1,\beta\gamma}\,\delta^{E_{2}}_{\alpha\beta\gamma}+\big{(}b_{1,\beta\gamma}+b_{2,\alpha\gamma}-b_{3,\alpha\beta}\big{)}\delta^{S}_{\alpha\beta\gamma}\\\
\hline\cr\hline\cr\vrule\lx@intercol\hfil\text{Divisors}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr R_{1}&\lx@intercol\hfil
2\,a_{2}a_{3}-\sum\limits_{\beta,\gamma}b_{1,\beta\gamma}^{2}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr D_{1,\alpha}&\lx@intercol\hfil
a_{2}a_{3}-\sum\limits_{\gamma}a_{2}b_{2,\alpha\gamma}-\sum\limits_{\beta}a_{3}b_{3,\alpha\beta}+\sum\limits_{\beta,\gamma}\big{(}1-\delta^{E_{1}}_{\alpha\beta\gamma}\big{)}b_{2,\alpha\gamma}b_{3,\alpha\beta}\hfil\lx@intercol\vrule\lx@intercol\\\
&\lx@intercol\hfil+\sum\limits_{\beta,\gamma}\delta^{E_{1}}_{\alpha\beta\gamma}\Big{\\{}b_{1,\beta\gamma}\big{(}b_{2,\alpha\gamma}+b_{3,\alpha\beta}\big{)}-\frac{1}{2}\big{(}b_{1,\beta\gamma}^{2}+b_{2,\alpha\gamma}^{2}+b_{3,\alpha\beta}^{2}\big{)}\Big{\\}}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr E_{1,\beta\gamma}&\lx@intercol\hfil
2\,a_{1}b_{1,\beta\gamma}+\sum\limits_{\alpha}\Big{\\{}\frac{1}{2}\big{(}1+\Delta^{1}_{\alpha\beta\gamma}\big{)}b_{1,\beta\gamma}^{2}+\big{(}1-\Delta^{1}_{\alpha\beta\gamma}-\Delta^{2}_{\alpha\beta\gamma}-\Delta^{3}_{\alpha\beta\gamma}\big{)}b_{2,\alpha\gamma}b_{3,\alpha\beta}\hfil\lx@intercol\vrule\lx@intercol\\\
&\lx@intercol\hfil-\frac{1}{2}\big{(}1-\Delta^{1}_{\alpha\beta\gamma}\big{)}\big{(}b_{2,\alpha\gamma}^{2}+b_{3,\alpha\beta}^{2}\big{)}-\big{(}1-\Delta^{2}_{\alpha\beta\gamma}\big{)}b_{1,\beta\gamma}b_{2,\alpha\gamma}-\big{(}1-\Delta^{3}_{\alpha\beta\gamma}\big{)}b_{1,\beta\gamma}b_{3,\alpha\beta}\Big{\\}}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\vrule\lx@intercol\hfil\text{Full
manifold}\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr\hline\cr
X&\lx@intercol\hfil
2\,a_{1}a_{2}a_{3}-\sum\limits_{\beta,\gamma}a_{1}b_{1,\beta\gamma}^{2}-\sum\limits_{\alpha,\gamma}a_{2}b_{2,\alpha\gamma}^{2}-\sum\limits_{\alpha,\beta}a_{3}b_{3,\alpha\beta}^{2}-\sum\limits_{\alpha,\beta,\gamma}\Big{\\{}\frac{1}{2}\big{(}\Delta^{1}_{\alpha\beta\gamma}-1\big{)}b_{1,\beta\gamma}\big{(}b_{2,\alpha\gamma}^{2}+b_{3,\alpha\beta}^{2}\big{)}\hfil\lx@intercol\vrule\lx@intercol\\\
&\lx@intercol\hfil+\frac{1}{2}\big{(}\Delta^{2}_{\alpha\beta\gamma}-1\big{)}b_{2,\alpha\gamma}\big{(}b_{1,\beta\gamma}^{2}+b_{3,\alpha\beta}^{2}\big{)}+\frac{1}{2}\big{(}\Delta^{3}_{\alpha\beta\gamma}-1\big{)}b_{3,\alpha\beta}\big{(}b_{1,\beta\gamma}^{2}+b_{2,\alpha\gamma}^{2}\big{)}+\frac{1}{6}\big{(}1+\Delta^{1}_{\alpha\beta\gamma}\big{)}b_{1,\beta\gamma}^{3}\hfil\lx@intercol\vrule\lx@intercol\\\
&\lx@intercol\hfil+\frac{1}{6}\big{(}1+\Delta^{2}_{\alpha\beta\gamma}\big{)}b_{2,\alpha\gamma}^{3}+\frac{1}{6}\big{(}1+\Delta^{3}_{\alpha\beta\gamma}\big{)}b_{3,\alpha\gamma}^{3}+\big{(}1-\Delta^{1}_{\alpha\beta\gamma}-\Delta^{2}_{\alpha\beta\gamma}-\Delta^{3}_{\alpha\beta\gamma}\big{)}b_{1,\beta\gamma}b_{2,\alpha\gamma}b_{3,\alpha\beta}\Big{\\}}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\end{array}$ Table 3: Volume of a collection of possibly existing
curves, divisors and the resolved orbifold $X$ as a whole using arbitrary
triangulation at the 64
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\\!\times\\!\mathbbm{Z}_{2}$ resolutions.
Similar expression of the other curves and divisors can be obtained by
permutations.

Using the (self–)intersections (19) various volumes can be computed using the
Kähler form

$\displaystyle J=\sum_{i}a_{i}R_{i}-\sum_{r}b_{r}E_{r}~{},$ (42)

involving the Kähler parameters $a_{i}$ and $b_{r}$. The volumes of a curve
$C$, a divisor $D$ and the orbifold resolution $X$ are given by

$\displaystyle\text{Vol}(C)=\int_{C}J~{},\qquad\text{Vol}(D)=\int_{D}\frac{1}{2}\,J^{2}~{},\qquad\text{Vol}(X)=\int_{X}\frac{1}{3!}\,J^{3}~{},$
(43)

respectively. The resulting expressions for any choice of triangulation are
given in Table 3.

The volumes of the divisors are constrained by the DUY equations [42, 43]. The
one–loop corrections to these equations are given by [24, 36]

$\displaystyle\int\dfrac{1}{2}J^{2}\,\dfrac{{\cal
F}}{2\pi}=\dfrac{e^{2\phi}}{16\pi}\int\Big{\\{}\text{tr}\Big{(}\dfrac{{\cal
F}^{\prime}}{2\pi}\Big{)}^{2}-\dfrac{1}{2}\text{tr}\Big{(}\dfrac{{\cal
R}}{2\pi}\Big{)}^{2}\Big{\\}}\dfrac{{\cal
F}^{\prime}}{2\pi}+(^{\prime}\rightarrow^{\prime\prime})~{},$ (44)

where ${\cal F}^{\prime}$ and ${\cal F}^{\prime\prime}$ denote the Abelian
gauge fluxes in the first and second factor of the $E_{8}\times E_{8}$ group,
respectively, so that ${\cal F}={\cal F}^{\prime}+{\cal F}^{\prime\prime}$.
This equation thus links the Kähler moduli $a_{i},b_{r}$ and the dilaton
$\phi$ in general.

If the gauge background is embedded in just a single, say first $E_{8}$, or if
one considers the heterotic $SO(32)$ theory instead, this equation may be
rewritten as

$\displaystyle\int\dfrac{1}{2}J^{2}\,\dfrac{{\cal
F}}{2\pi}=\dfrac{e^{2\phi}}{32\pi}\int\text{tr}\Big{(}\dfrac{{\cal
R}}{2\pi}\Big{)}^{2}\dfrac{{\cal F}}{2\pi}=-\dfrac{e^{2\phi}}{16\pi}\int
c_{2}\dfrac{{\cal F}}{2\pi}~{},$ (45)

as ${\cal F}={\cal F}^{\prime}$ and ${\cal F}^{\prime\prime}=0$ using the
integrate Bianchi identities (31). Inserting the expansion for the gauge flux
in terms of the exceptional divisors $E_{r}$ and using the integrated second
Chern classes (23), leads to

$\displaystyle\sum_{\beta,\gamma}\Big{\\{}\text{Vol}(E_{1,\beta\gamma})-\dfrac{e^{2\phi}}{16\pi}\sum_{\alpha}(1-2\Delta^{1}_{\alpha\beta\gamma})\Big{\\}}{\cal
V}^{I}_{1,\beta\gamma}+\sum_{\alpha,\gamma}\Big{\\{}\text{Vol}(E_{2,\alpha\gamma})-\dfrac{e^{2\phi}}{16\pi}\sum_{\beta}(1-2\Delta^{2}_{\alpha\beta\gamma})\Big{\\}}{\cal
V}^{I}_{2,\alpha\gamma}+$
$\displaystyle+\sum_{\alpha,\beta}\Big{\\{}\text{Vol}(E_{3,\alpha\beta})-\dfrac{e^{2\phi}}{16\pi}\sum_{\gamma}(1-2\Delta^{3}_{\alpha\beta\gamma})\Big{\\}}{\cal
V}^{I}_{3,\alpha\beta}=0~{}.$ (46)

If the gauge background lie in both $E_{8}$ factors simultaneously, then the
$\text{tr}{\cal R}^{2}$ term can be eliminated using the Bianchi identities
(31) instead. Moreover, since both $E_{8}$ factors are independent, the DUY
equation may be split into two equations; one for each $E_{8}$ factor:

$\displaystyle\begin{array}[]{rcl}\displaystyle\int\dfrac{1}{2}J^{2}\,\dfrac{{\cal
F}^{\prime}}{2\pi}&=&\displaystyle\phantom{-}\dfrac{e^{2\phi}}{32\pi}\int\Big{\\{}\text{tr}\Big{(}\dfrac{{\cal
F}^{\prime}}{2\pi}\Big{)}^{2}-\text{tr}\Big{(}\dfrac{{\cal
F}^{\prime\prime}}{2\pi}\Big{)}^{2}\Big{\\}}\dfrac{{\cal
F}^{\prime}}{2\pi}~{},\\\\[8.61108pt]
\displaystyle\int\dfrac{1}{2}J^{2}\,\dfrac{{\cal
F}^{\prime\prime}}{2\pi}&=&\displaystyle-\dfrac{e^{2\phi}}{32\pi}\int\Big{\\{}\text{tr}\Big{(}\dfrac{{\cal
F}^{\prime}}{2\pi}\Big{)}^{2}-\text{tr}\Big{(}\dfrac{{\cal
F}^{\prime\prime}}{2\pi}\Big{)}^{2}\Big{\\}}\dfrac{{\cal
F}^{\prime\prime}}{2\pi}~{}.\end{array}$ (49)

Notice the relative sign difference between the otherwise very similar
expressions in both $E_{8}$’s. Evaluating these expressions further by
inserting the intersection numbers (19) leads to rather lengthy and not very
illuminating expressions. For this reason we refrain from stating them here.

##  3 Triangulation Independence

The results obtained in the previous section hold for any particular choice of
the triangulation of each of the 64 resolved
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ singularities. The aim
of this section is to obtain results that hold for all choices of
triangulation simultaneously: such results can be uncovered by superimposing
the conditions for all the different choices of triangulation. It should be
emphasised that we do not wish to imply that it is necessary that such results
apply in all triangulations from the supergravity perspective. But
surprisingly, superimposing consistency conditions leads to a huge reduction
of the complexity of these equations. However, if all consistency conditions
are satisfied in any triangulation, then arbitrary flop–transitions are
admissible which opens up the possibility to study the resulting transitions
in the massless spectra.

###  3.1 Flux Quantisation

Even though the flux quantisation conditions might seem to be dependent on the
choice of the triangulations at the local singularities, in fact, they are all
equivalent to

$\displaystyle 2\,{\cal V}_{i,\mu\nu}\cong 0~{},\quad\sum_{\rho}{\cal
V}_{i,\rho\nu}\cong 0~{},\quad\sum_{\rho}{\cal V}_{i,\mu\rho}\cong
0~{},\quad{\cal V}_{1,\beta\gamma}+{\cal V}_{2,\alpha\gamma}+{\cal
V}_{3,\alpha\beta}\cong 0$ (50)

independently of the local triangulations chosen. To see this, notice first of
all that the first three relations derived from curves that exist for any
triangulation, see Table 2. Now, if triangulation $E_{1}$ has been chosen at
the resolution of $f_{\alpha\beta\gamma}$, one has to impose the condition
associated to curve $D_{1,\alpha}E_{1,\beta\gamma}$, if triangulation $E_{2}$
the condition associated to curve $D_{2,\beta}E_{2,\alpha,\gamma}$ and if
triangulation $E_{3}$ the condition associated to curve
$D_{3,\gamma}E_{3,\alpha\beta}$, respectively, while if trangulation $S$ is
used all the resulting three conditions have to be superimposed. However, all
three of them are equivalent to the last condition in (50) using the first
condition in this line which basically says that the signs of the bundle
vectors in the flux quantisation conditions are irrelevant modulo 2. In other
words, if the flux quantisation is satisfied for a single triangulation choice
at all the 64 resolved $\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$
singularities, the fluxes are properly quantised for any triangulation choice.

###  3.2 Reduction of Bianchi Identities

To determine the set of equations which guarantee that for any choice of
triangulation of the 64 $\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$
resolutions, the Bianchi identities are solved, we can treat the triangulation
dependent functions, $\Delta^{1}_{\alpha\beta\gamma}$,
$\Delta^{2}_{\alpha\beta\gamma}$ and $\Delta^{3}_{\alpha\beta\gamma}$, as
arbitrary functions. Hence, to solve the Bianchi identities for all choices,
the coefficients in front of these functions need to cancel among themselves
as well as the remaining contributions which do not multiply any of them. This
leads to four set of equations for each set of sixteen Bianchi identities on
each of the exceptional cycles. For the sixteen Bianchi identities on
$E_{1,\beta\gamma}$ they read:

$\begin{array}[]{c}\sum\limits_{\alpha}\Big{[}{\cal
V}_{2,\alpha\gamma}^{2}+{\cal V}_{3,\alpha\beta}^{2}-{\cal
V}_{1,\beta\gamma}^{2}+2\,{\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{2,\alpha\gamma}+2\,{\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{3,\alpha\beta}-2\,{\cal V}_{2,\alpha\gamma}\cdot{\cal
V}_{3,\alpha\beta}\Big{]}=-8~{},\\\\[12.91663pt] {\cal
V}_{1,\beta\gamma}^{2}+{\cal V}_{2,\alpha\gamma}^{2}+{\cal
V}_{3,\alpha\beta}^{2}-2\,{\cal V}_{2,\alpha\gamma}\cdot{\cal
V}_{3,\alpha\beta}=4~{},\quad{\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{2,\alpha\gamma}={\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{3,\alpha\beta}={\cal V}_{2,\alpha\gamma}\cdot{\cal
V}_{3,\alpha\beta}~{}.\end{array}$ (51)

For the sixteen Bianchi identities on $E_{2,\alpha\gamma}$ they read:

$\begin{array}[]{c}\sum\limits_{\beta}\Big{[}{\cal
V}_{1,\beta\gamma}^{2}+{\cal V}_{3,\alpha\beta}^{2}-{\cal
V}_{2,\alpha\gamma}^{2}+2\,{\cal V}_{2,\alpha\gamma}\cdot{\cal
V}_{1,\beta\gamma}+2\,{\cal V}_{2,\alpha\gamma}\cdot{\cal
V}_{3,\alpha\beta}-2\,{\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{3,\alpha\beta}\Big{]}=-8~{},\\\\[12.91663pt] {\cal
V}_{1,\beta\gamma}^{2}+{\cal V}_{2,\alpha\gamma}^{2}+{\cal
V}_{3,\alpha\beta}^{2}-2\,{\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{3,\alpha\beta}=4~{},\quad{\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{2,\alpha\gamma}={\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{3,\alpha\beta}={\cal V}_{2,\alpha\gamma}\cdot{\cal
V}_{3,\alpha\beta}~{}.\end{array}$ (52)

And, finally, for the sixteen Bianchi identities on $E_{3,\alpha\beta}$ they
read:

$\begin{array}[]{c}\sum\limits_{\gamma}\Big{[}{\cal
V}_{1,\beta\gamma}^{2}+{\cal V}_{2,\alpha\gamma}^{2}-{\cal
V}_{3,\alpha\beta}^{2}+2\,{\cal V}_{3,\alpha\beta}\cdot{\cal
V}_{1,\beta\gamma}+2\,{\cal V}_{3,\alpha\beta}\cdot{\cal
V}_{2,\alpha\gamma}-2\,{\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{2,\alpha\gamma}\Big{]}=-8~{},\\\\[12.91663pt] {\cal
V}_{1,\beta\gamma}^{2}+{\cal V}_{2,\alpha\gamma}^{2}+{\cal
V}_{3,\alpha\beta}^{2}-2\,{\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{2,\alpha\gamma}=4~{},\quad{\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{2,\alpha\gamma}={\cal V}_{1,\beta\gamma}\cdot{\cal
V}_{3,\alpha\beta}={\cal V}_{2,\alpha\gamma}\cdot{\cal
V}_{3,\alpha\beta}~{}.\end{array}$ (53)

Note that every time the top equations have a sum over one of the fixed point
labels, while the lower three do not. Fortunately, many of these equations are
redundant. The lower three relations for all three exceptional divisors have
the same content: for any choice of $(\alpha,\beta,\gamma)$ the three inner
products are constraint to:

${\cal V}_{1,\beta\gamma}\cdot{\cal V}_{2,\alpha\gamma}={\cal
V}_{1,\beta\gamma}\cdot{\cal V}_{3,\alpha\beta}={\cal
V}_{2,\alpha\gamma}\cdot{\cal V}_{3,\alpha\beta}=\tfrac{1}{2}\big{(}{\cal
V}_{1,\beta\gamma}^{2}+{\cal V}_{2,\alpha\gamma}^{2}+{\cal
V}_{3,\alpha\beta}^{2}\big{)}-2~{}.$ (54)

Inserting these in the top equations with the sums results in $3\cdot 16$
equations

$\sum\limits_{\alpha}\Big{[}{\cal V}_{2,\alpha\gamma}^{2}+{\cal
V}_{3,\alpha\beta}^{2}\Big{]}=12~{},\quad\sum\limits_{\beta}\Big{[}{\cal
V}_{1,\beta\gamma}^{2}+{\cal
V}_{3,\alpha\beta}^{2}\Big{]}=12~{},\quad\sum\limits_{\alpha}\Big{[}{\cal
V}_{1,\beta\gamma}^{2}+{\cal V}_{2,\alpha\gamma}^{2}\Big{]}=12~{}.$ (55)

If these equations are satisfied, then also the three Bianchi identities on
the inherited divisors are automatically satisfied. Indeed, if one sums each
of these sets of equations over the other two labels and then add two and
subtract a third, the inherited Bianchi identities are recovered. But in fact
these equations can be reduced even further by a similar procedure: Sum over
one of the two free labels in these equations. One of the two terms is then
precisely of the form of one of the inherited Bianchi identities equal to 24.
Inserting that and rewriting leads to three sets of $2\cdot 4=8$ (hence 24 in
total) even simpler equations:

$\sum\limits_{\beta}{\cal V}_{1,\beta\gamma}^{2}=\sum\limits_{\gamma}{\cal
V}_{1,\beta\gamma}^{2}=6~{},\quad\sum\limits_{\alpha}{\cal
V}_{2,\alpha\gamma}^{2}=\sum\limits_{\gamma}{\cal
V}_{2,\alpha\gamma}^{2}=6~{},\quad\sum\limits_{\alpha}{\cal
V}_{3,\alpha\beta}^{2}=\sum\limits_{\beta}{\cal V}_{3,\alpha\beta}^{2}=6~{}.$
(56)

Hence, if the equations (54) and (56) are simultaneously satisfied, then a
solution is obtained of the 51 Bianchi identities that holds in any
triangulation. In fact, in each of the three sets of 8 equations there is one
linear dependence, since summing over the free indices in both (four)
equations in each set, leads to the same equation.

###  3.3 Blowup Modes Without Oscillator Excitations

Assuming that all line bundle vectors can be associated to twisted states
without oscillators, then they all square to:

${\cal V}_{a,\mu\nu}^{2}=\tfrac{3}{2}\qquad\Rightarrow\qquad{\cal
V}_{a,\mu\nu}\cdot{\cal V}_{b,\rho\sigma}=\tfrac{1}{4}~{};$ (57)

the equation after the implication sign follows upon using (54), with
$\nu=\sigma$ for $a=1$ and $b=2$, $\mu=\sigma$ for $a=1$ and $b=3$, $\mu=\rho$
for $a=2$ and $b=3$, respectively. Since this solves all equations (54) and
(56), such choices solve all Bianchi identities in any triangulation
simultaneously.

###  3.4 Consequences of Triangulation Independence for the Multiplicity
Operator

Contrary to the fundamental consistency conditions, the multiplicity operator
does not simplify in any particular way, when line bundle resolutions models
are considered that are admissible in any choice of triangulation of the 64
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ resolved singularities. However, it can
be brought in a specific form. Since the $S$–triangulation plays a special
role in flop–transitions as any flop involves the $S$–triangulation, the
$S$–triangulation can be taken to be the reference triangulation at all the 64
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ resolved singularities. Using this the
total multiplicity operator $\mathsf{N}$ can be written as

$\displaystyle\mathsf{N}=\mathsf{N}^{S}+\sum_{\alpha,\beta,\gamma}\Big{[}\delta^{E_{1}}_{\alpha\beta\gamma}\,\Delta\mathsf{N}^{1}_{\alpha\beta\gamma}+\delta^{E_{2}}_{\alpha\beta\gamma}\,\Delta\mathsf{N}^{2}_{\alpha\beta\gamma}+\delta^{E_{3}}_{\alpha\beta\gamma}\,\Delta\mathsf{N}^{3}_{\alpha\beta\gamma}\Big{]}~{},\qquad\mathsf{N}^{S}=\sum_{\alpha,\beta,\gamma}\mathsf{N}^{S}_{\alpha\beta\gamma}$
(58)

and $\Delta\mathsf{N}^{i}_{\alpha\beta\gamma}$ defined in (41). Both
$\mathsf{N}^{S}$ and $\Delta\mathsf{N}^{i}_{\alpha\beta\gamma}$ are always
integer: $\mathsf{N}^{S}$ is the multiplicity operator when at all 64 resolved
singularities triangulation $S$ is taken, hence it has to be integral on all
chiral states in the spectrum; for the triangulation difference multiplicities
$\Delta\mathsf{N}^{i}_{\alpha\beta\gamma}$ it was already established in
Subsection 2.9 that they are always integral. This means that in the most
general case one can define $3\cdot 4^{3}+1=193$ multiplicity operators
($\Delta\mathsf{N}^{i}_{\alpha\beta\gamma}$ for $i=1,2,3$,
$\alpha,\beta,\gamma=1,2,3,4$ and $\mathsf{N}^{S}$) that all have to be
integral on any $E_{8}\times E_{8}$ root.

##  4 Models Without Wilson Lines

This section is devoted to a number of simple line bundle models to illustrate
the main ideas about dealing with the triangulation dependence. The focus is
on demonstrating that the approach to parameterise all triangulations in the
way discussed in the preceding sections always lead to sensible, e.g. integral
spectra for any triangulation chosen provided that the consistency conditions
have been solved for all triangulations simultaneously. However, these models
should not be considered as fully realistic models. In particular, the
consequences of the DUY equations will be mostly ignored.

###  4.1 $\boldsymbol{T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ Orbifold
Models

From the orbifold perspective these are models without Wilson lines, this
means that such orbifold models are characterised by just two gauge shifts
$V_{a}$. They satisfy

$\displaystyle V_{a}\cdot V_{b}-v_{a}\cdot v_{b}\equiv 0~{},$ (59)

for $a,b=1,2$. Here $v_{a}$ denote the two independent four–component
geometrical $\mathbbm{Z}_{2}$ orbifold twists and $V_{a}$ the
sixteen–component shift embedding on the gauge lattice taken to be either the
weight lattice of $E_{8}\times E_{8}$ or $Spin(32)/\mathbbm{Z}_{2}$.
Furthermore, it is often convenient to introduce $v_{3}\cong v_{1}+v_{2}$ and
$V_{3}\cong V_{1}+V_{2}$.

$\begin{array}[]{|c||l|l|l|}\hline\cr\text{Model}&\lx@intercol\hfil\text{Twists
/ Gauge Shift Embeddings}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr
I&v_{1}=(0,0,\tfrac{1}{2},-\tfrac{1}{2})~{},&v_{2}=(0,-\tfrac{1}{2},0,\tfrac{1}{2})~{},&v_{3}=(0,\tfrac{1}{2},-\tfrac{1}{2},0)~{}.\\\
\hline\cr
I.a&V_{1}=(0,\tfrac{1}{2},-\tfrac{1}{2},0^{5})(0^{8})&V_{2}=(-\tfrac{1}{2},0,\tfrac{1}{2},0^{5})(0^{8})&V_{3}=(\tfrac{1}{2},-\tfrac{1}{2},0,0^{5})(0^{8})\\\
I.b&V_{1}=(0,\tfrac{1}{2},-\tfrac{1}{2},0^{5})(1,0^{7})&V_{2}=(-\tfrac{1}{2},0,\tfrac{1}{2},0^{5})(-1,0^{7})&V_{3}=(\tfrac{1}{2},-\tfrac{1}{2},0,0^{5})(0,0^{7})\\\
\hline\cr\hline\cr
II&v_{1}=(0,0,\tfrac{1}{2},\tfrac{1}{2})~{},&v_{2}=(0,\tfrac{1}{2},0,\tfrac{1}{2})~{},&v_{3}=(0,-\tfrac{1}{2},-\tfrac{1}{2},1)~{},\\\
\hline\cr
II.a&V_{1}=(0,\tfrac{1}{2},\tfrac{1}{2},0^{5})(0^{8})&V_{2}=(\tfrac{1}{2},0,\tfrac{1}{2},0^{5})(0^{8})&V_{3}=(-\tfrac{1}{2},-\tfrac{1}{2},1,0^{5})(0^{8})\\\
II.b&V_{1}=(0,\tfrac{1}{2},\tfrac{1}{2},0^{5})(1,0^{7})&V_{2}=(\tfrac{1}{2},\tfrac{1}{2},0,0^{5})(-1,0^{7})&V_{3}=(-\tfrac{1}{2},-\tfrac{1}{2},1,0^{5})(0,0^{7})\\\
\hline\cr\end{array}$ Table 4:  This table gives two different choices of the
orbifold twist vectors $v_{a}$ and two associated inequivalent gauge shift
embeddings $V_{a}$ for each choice.

The geometrical twists $v_{1}$ and $v_{2}$ are conventionally chosen such as
to preserve ${\cal N}=1$ target space supersymmetry. On the level of the
orbifold theory there are various equivalent choices for them. The most
commonly used choice $I.$ is

$v_{1}=(0,0,\tfrac{1}{2},-\tfrac{1}{2})~{},\qquad
v_{2}=(0,-\tfrac{1}{2},0,\tfrac{1}{2})~{},\qquad
v_{3}=v_{1}+v_{2}=(0,-\tfrac{1}{2},\tfrac{1}{2},0)\cong(0,\tfrac{1}{2},-\tfrac{1}{2},0)~{}.$
(60a) The first entry of these vectors corresponds to the four–dimensional
Minkowski space in light–cone gauge; the other three components to the twist
actions on the three two–torus that make up the $T^{6}$. The final expression
for $v_{3}$ obtained by adding a lattice vector $(0,1,-1,0)$ making a
permutation symmetry between the entries of $v_{1},v_{2}$ and $v_{3}$
manifest. Note that with this form of $v_{3}$ $\displaystyle v_{a}\cdot
v_{b}=-\tfrac{1}{4}+\tfrac{3}{4}\,\delta_{ab}~{}.$ (60b)

A second choice $II.$ is given by

$v_{1}=(0,0,\tfrac{1}{2},\tfrac{1}{2})~{},\qquad
v_{2}=(0,\tfrac{1}{2},0,\tfrac{1}{2})~{},\qquad
v_{3}=v_{1}+v_{2}=(0,\tfrac{1}{2},\tfrac{1}{2},1)\cong(0,-\tfrac{1}{2},-\tfrac{1}{2},1)~{},$
(61a) where the latter form of $v_{3}$ in this case is obtained by adding
$(0,-1,-1,0)$. With this form of $v_{3}$ the vectors $v_{a}$ satisfy
$\displaystyle v_{a}\cdot
v_{b}=\tfrac{1}{4}\big{(}1+\delta_{ab}\big{)}+\delta_{a3}\,\delta_{b3}~{}.$
(61b)

On the level of the orbifold theory both choices are equivalent. For both
these forms there are two inequivalent choices for the gauge embedding,
denoted by $a$ and $b$. This leads to four possible gauge shift embeddings
referred to as $I.a$, $I.b$, $II.a$ and $II.b$ in Table 4. On the level of the
orbifold only the choices $a$ or $b$ lead to physically different models; as
is shown below on the level of the resolution the choice of twist $I.$ or
$II.$ is of significance as only one of the two choices can be associated to a
line bundle model.

###  4.2 Models with Three Independent Line Bundles

First some general facts about associated blowup models are given. In this
section the line bundle vectors are taken to be independent of the labels
$\alpha,\beta,\gamma$, i.e.:

$\displaystyle{\cal V}_{1,\beta\gamma}={\cal V}_{1}~{},\quad{\cal
V}_{2,\alpha\gamma}={\cal V}_{2}~{},\quad{\cal V}_{3,\alpha\beta}={\cal
V}_{3}~{}.$ (62)

Consequently the triangulation independent flux quantisation conditions (50)
reduce to

$\displaystyle 2\,{\cal V}_{1}\cong 2\,{\cal V}_{2}\cong 2\,{\cal
V}_{3}\cong{\cal V}_{1}+{\cal V}_{2}+{\cal V}_{3}\cong 0~{}.$ (63)

Such bundle vectors, ${\cal V}_{1},{\cal V}_{2}$ and ${\cal V}_{3}$, can be
obtained from the orbifold gauge shift vectors $V_{1},V_{2}$ and $V_{3}$, by
adding appropriate lattice vectors $L_{1,\beta\gamma},L_{2,\alpha\gamma}$ and
$L_{3,\alpha\beta}$. In this section they are chosen such that the bundle
vectors ${\cal V}_{1},{\cal V}_{2}$ and ${\cal V}_{3}$ can be associated with
twisted states without oscillators satisfying the conditions (57).

The number $N_{T}$ of times, that triangulation $T=E_{1}$, $E_{2}$, $E_{3}$,
$S$ has been chosen at the 64 resolved
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ singularities, can be
determined by summing the functions $\delta^{T}_{\alpha\beta\gamma}$ over all
of them, e.g.:

$\displaystyle
N_{T}=\sum_{\alpha,\beta,\gamma}\delta^{T}_{\alpha\beta\gamma}~{},$ (64)

hence, in particularly, for $i=1,2,3$:

$\displaystyle\sum_{\alpha,\beta,\gamma}\Big{(}1-\Delta^{i}_{\alpha\beta\gamma}\Big{)}=2\,N_{E_{i}}+N_{S}~{},\qquad
N_{E}+N_{S}=64~{},\qquad N_{E}=N_{E_{1}}+N_{E_{2}}+N_{E_{3}}~{}.$ (65)

Then, if also the Cartan operators are abbreviated as

$\displaystyle\mathsf{H}_{1}=\mathsf{H}_{1,\beta\gamma}~{},\qquad\mathsf{H}_{2}=\mathsf{H}_{2,\alpha\gamma}~{},\qquad\mathsf{H}_{3}=\mathsf{H}_{3,\alpha\beta}~{},$
(66)

the multiplicity operator (39) simplifies to

$\displaystyle\mathsf{N}=N_{E_{1}}\,\mathsf{N}^{1}+N_{E_{2}}\,\mathsf{N}^{2}+N_{E_{3}}\,\mathsf{N}^{3}+N_{S}\,\mathsf{N}^{S}~{},$
(67)

expressed in terms of four multiplicity operators for each of the four
triangulations

$\displaystyle\mathsf{N}^{1}$
$\displaystyle=\tfrac{1}{4}\,\mathsf{H}_{1}+\tfrac{1}{12}\,\mathsf{H}_{2}\big{(}4\mathsf{H}_{2}^{2}-1\big{)}+\tfrac{1}{12}\,\mathsf{H}_{3}\big{(}4\mathsf{H}_{3}^{2}-1\big{)}-\mathsf{H}_{1}\big{(}\mathsf{H}_{2}^{2}+\mathsf{H}_{3}^{2}\big{)}~{},$
(68a) $\displaystyle\mathsf{N}^{2}$
$\displaystyle=\tfrac{1}{4}\,\mathsf{H}_{2}+\tfrac{1}{12}\,\mathsf{H}_{1}\big{(}4\mathsf{H}_{1}^{2}-1\big{)}+\tfrac{1}{12}\,\mathsf{H}_{3}\big{(}4\mathsf{H}_{3}^{2}-1\big{)}-\mathsf{H}_{2}\big{(}\mathsf{H}_{1}^{2}+\mathsf{H}_{3}^{2}\big{)}~{},$
(68b) $\displaystyle\mathsf{N}^{3}$
$\displaystyle=\tfrac{1}{4}\,\mathsf{H}_{3}+\tfrac{1}{12}\,\mathsf{H}_{1}\big{(}4\mathsf{H}_{1}^{2}-1\big{)}+\tfrac{1}{12}\,\mathsf{H}_{2}\big{(}4\mathsf{H}_{2}^{2}-1\big{)}-\mathsf{H}_{3}\big{(}\mathsf{H}_{1}^{2}+\mathsf{H}_{2}^{2}\big{)}~{},$
(68c) $\displaystyle\mathsf{N}^{S}$
$\displaystyle=\tfrac{1}{12}\,\mathsf{H}_{1}\big{(}2\mathsf{H}_{1}^{2}+1)+\tfrac{1}{12}\,\mathsf{H}_{2}\big{(}2\mathsf{H}_{2}^{2}+1)+\tfrac{1}{12}\,\mathsf{H}_{3}\big{(}2\mathsf{H}_{3}^{2}+1)+\mathsf{H}_{1}\mathsf{H}_{2}\mathsf{H}_{3}$
$\displaystyle\phantom{=}-\tfrac{1}{2}\,\mathsf{H}_{1}\big{(}\mathsf{H}_{2}^{2}+\mathsf{H}_{3}^{2}\big{)}-\tfrac{1}{2}\,\mathsf{H}_{2}\big{(}\mathsf{H}_{1}^{2}+\mathsf{H}_{3}^{2}\big{)}-\tfrac{1}{2}\,\mathsf{H}_{3}\big{(}\mathsf{H}_{1}^{2}+\mathsf{H}_{2}^{2}\big{)}~{}.$
(68d)

Since $N_{E_{1}}$, $N_{E_{2}}$, $N_{E_{3}}$ and $N_{S}$ are arbitrary
non–negative integers subject to (65), it follows that if we substitute one of
them away in (67), the resulting expression has to be integral on all
$E_{8}\times E_{8}$ weights for any choice of the remaining numbers. In
particular, taking the triangulation $S$ again as reference, i.e. solving
$N_{S}$ from (65) and substituting this in (67), gives

$\displaystyle\mathsf{N}=N_{E_{1}}\,\Delta\mathsf{N}^{1}+N_{E_{2}}\,\Delta\mathsf{N}^{2}+N_{E_{3}}\,\Delta\mathsf{N}^{3}+64\,\mathsf{N}^{S}~{},\qquad\Delta\mathsf{N}^{i}=\mathsf{N}^{i}-\mathsf{N}^{S}~{},$
(69)

for $i=1,2,3$. In line with the general observation in Section 3.4, this
expression should always be integral. Hence, in particular, the operators
$\Delta\mathsf{N}^{i}$ have to be integral on any state.

###  4.3 $\boldsymbol{SO(10)\times SO(12)}$ Line Bundle Models

Starting from the orbifold gauge embeddings $II.b$ of the classification in
Table 4 a set of three line bundle vectors can be obtained

$\displaystyle{\cal
V}_{1}=V_{1}+L_{1}=(0,\tfrac{1}{2},\tfrac{1}{2},0^{5})(1,0,0^{6})~{},$
$\displaystyle L_{1}=(0^{8})(0^{8})$ (70a) $\displaystyle{\cal
V}_{2}=V_{2}+L_{2}=(\tfrac{1}{2},0,\tfrac{1}{2},0^{5})(0,1,0^{6})~{},$
$\displaystyle L_{2}=(0^{8})(-1,1,0^{6})~{},$ (70b) $\displaystyle{\cal
V}_{3}=V_{3}+L_{3}=(-\tfrac{1}{2},-\tfrac{1}{2},1,0^{5})(0,0,0^{6})~{},$
$\displaystyle L_{3}=(0^{8})(0^{8})~{}.$ (70c)

The bundle vectors satisfy the flux quantisation (63) for arbitrary
triangulations. Note that, these bundle vectors cannot be obtained from
orbifold model $I.b$ . Thus equivalent choices on the orbifold level might
lead to inequivalent choices from the smooth resolved perspective. The
unbroken non–Abelian gauge group is $SO(10)\times SO(12)$.

$\begin{array}[]{|c||ccc|cccc|ccc|c|}\hline\cr\text{weight}&\mathsf{H}_{1}&\mathsf{H}_{2}&\mathsf{H}_{3}&\mathsf{N}^{1}&\mathsf{N}^{2}&\mathsf{N}^{3}&\mathsf{N}^{S}&\Delta\mathsf{N}^{1}&\Delta\mathsf{N}^{2}&\Delta\mathsf{N}^{3}&\text{repr.}\\\
\hline\cr\hline\cr(1,0,0,{\underline{\pm
1,0^{4}}})(0^{8})&0&\tfrac{1}{2}&-\tfrac{1}{2}&0&0&0&0&0&0&0&(\mathbf{10})(\mathbf{1})\\\
(0,1,0,{\underline{\pm
1,0^{4}}})(0^{8})&\tfrac{1}{2}&0&-\tfrac{1}{2}&0&0&0&0&0&0&0&(\mathbf{10})(\mathbf{1})\\\
(0,0,1,{\underline{\pm
1,0^{4}}})(0^{8})&\tfrac{1}{2}&\tfrac{1}{2}&1&-\tfrac{1}{4}&-\tfrac{1}{4}&-\tfrac{1}{4}&-\tfrac{1}{4}&0&0&0&(\mathbf{10})(\mathbf{1})\\\
\hline\cr(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{e},\tfrac{1}{2}^{5-e}}})(0^{8})&\tfrac{1}{2}&\tfrac{1}{2}&0&0&0&0&0&0&0&0&(\mathbf{16})(\mathbf{1})\\\
(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{5-o}}})(0^{8})&\tfrac{1}{2}&0&\tfrac{1}{2}&0&0&0&0&0&0&0&(\overline{\mathbf{16}})(\mathbf{1})\\\
(\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{5-o}}})(0^{8})&0&\tfrac{1}{2}&\tfrac{1}{2}&0&0&0&0&0&0&0&(\overline{\mathbf{16}})(\mathbf{1})\\\
(\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{5-o}}})(0^{8})&0&0&-1&-\tfrac{1}{4}&-\tfrac{1}{4}&-\tfrac{1}{4}&-\tfrac{1}{4}&0&0&0&(\overline{\mathbf{16}})(\mathbf{1})\\\
\hline\cr(1,1,0,0^{5})(0^{8})&\tfrac{1}{2}&\tfrac{1}{2}&-1&-\tfrac{3}{4}&-\tfrac{3}{4}&\tfrac{1}{4}&-\tfrac{3}{4}&0&0&1&(\mathbf{1})(\mathbf{1})\\\
(1,0,1,0^{5})(0^{8})&\tfrac{1}{2}&1&\tfrac{1}{2}&-\tfrac{1}{4}&-\tfrac{1}{4}&-\tfrac{1}{4}&-\tfrac{1}{4}&0&0&0&(\mathbf{1})(\mathbf{1})\\\
(0,1,1,0^{5})(0^{8})&1&\tfrac{1}{2}&\tfrac{1}{2}&-\tfrac{1}{4}&-\tfrac{1}{4}&-\tfrac{1}{4}&-\tfrac{1}{4}&0&0&0&(\mathbf{1})(\mathbf{1})\\\
\hline\cr(1,-1,0,0^{5})(0^{8})&-\tfrac{1}{2}&\tfrac{1}{2}&0&0&0&0&0&0&0&0&(\mathbf{1})(\mathbf{1})\\\
(1,0,-1,0^{5})(0^{8})&-\tfrac{1}{2}&0&-\tfrac{3}{2}&0&-1&0&0&0&-1&0&(\mathbf{1})(\mathbf{1})\\\
(0,1,-1,0^{5})(0^{8})&0&-\tfrac{1}{2}&-\tfrac{3}{2}&-1&0&0&0&-1&0&0&(\mathbf{1})(\mathbf{1})\\\
\hline\cr\hline\cr(0^{8})(1,0,{\underline{\pm
1,0^{5}}})&1&0&0&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1})(\mathbf{12})\\\
(0^{8})(0,1,{\underline{\pm
1,0^{5}}})&0&1&0&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1})(\mathbf{12})\\\
\hline\cr(0^{8})(\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{e},\tfrac{1}{2}^{6-e}}})&\tfrac{1}{2}&\tfrac{1}{2}&0&0&0&0&0&0&0&0&(\mathbf{1})(\mathbf{32})\\\
(0^{8})(\tfrac{1}{2},-\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{6-o}}})&\tfrac{1}{2}&-\tfrac{1}{2}&0&0&0&0&0&0&0&0&(\mathbf{1})(\overline{\mathbf{32}})\\\
\hline\cr(0^{8})(1,1,0^{6})&1&1&0&-\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{2}&-\tfrac{1}{2}&0&0&1&(\mathbf{1})(\mathbf{1})\\\
(0^{8})(1,-1,0^{6})&1&-1&0&-1&1&0&0&-1&1&0&(\mathbf{1})(\mathbf{1})\\\
\hline\cr\end{array}$
Table 5:  The line bundle charges $\mathsf{H}_{i}$, the triangulation
multiplicities $\mathsf{N}^{i}$, $\mathsf{N}^{S}$ and the difference
multiplicities $\Delta\mathsf{N}^{i}$ are given for all the $E_{8}\times
E_{8}$ roots charged under the line bundle background defined by (70). The
underline of some of the entries in these roots denote all possible
permutations of them. Notice, that these difference multiplicities, that
measure jumps in the spectrum when going through a flop–transition, are always
integral.

The line bundle charges $\mathsf{H}_{i}$, the triangulation multiplicities
$\mathsf{N}^{i}$, $\mathsf{N}^{S}$ and the triangulation difference
multiplicities $\Delta\mathsf{N}^{i}$ of all the $E_{8}\times E_{8}$ roots are
given in Table 5. The triangulation multiplicities, and $\mathsf{N}^{S}$ in
particular, are not integrally quantised. This might seem problematic, but it
is not: triangulation $S$ can be taken to be the reference triangulation at
all 64 resolved singularities. Hence, if triangulation $S$ is chosen at all
resolved singularities, the spectrum is 64 times the triangulation
multiplicity $\mathsf{N}^{S}$ and all states come in multiples of 16. Now, if
at a certain resolved singularities one of the exceptional triangulations is
used then the spectrum always changes by an integral amount as the
triangulation difference multiplicities $\Delta\mathsf{N}^{i}$ are integral,
see Table 5. Indeed, using this table the full spectrum in any triangulation
can be determined to be:

$\displaystyle
16\big{\\{}(\mathbf{10})(\mathbf{1})_{0,0,{\,\mbox{-}}2;0}+(\mathbf{16})(\mathbf{1})_{{\,\mbox{-}}1,{\,\mbox{-}}1,1;0}+(\mathbf{1})(\mathbf{12})_{0;2,0}+(\mathbf{1})(\mathbf{12})_{0;0,2}+(\mathbf{1})(\mathbf{1})_{{\,\mbox{-}}2,0,{\,\mbox{-}}2;0}+(\mathbf{1})(\mathbf{1})_{0,{\,\mbox{-}}2,{\,\mbox{-}}2;0}\big{\\}}+$
$\displaystyle
48\,(\mathbf{1})(\mathbf{1})_{{\,\mbox{-}}2,{\,\mbox{-}}2,0;0}+N_{E_{3}}\,(\mathbf{1})(\mathbf{1})_{2,2,0;0}+N_{E_{2}}\,(\mathbf{1})(\mathbf{1})_{{\,\mbox{-}}2,0,2;0}+N_{E_{1}}\,(\mathbf{1})(\mathbf{1})_{0,{\,\mbox{-}}2,2;0}+$
(71) $\displaystyle
32\,(\mathbf{1})(\mathbf{1})_{0;{\,\mbox{-}}2,{\,\mbox{-}}2}+N_{E_{3}}\,(\mathbf{1})(\mathbf{1})_{0;2,2}+N_{E_{1}}\,(\mathbf{1})(\mathbf{1})_{0;{\,\mbox{-}}2,2}+N_{E_{2}}\,(\mathbf{1})(\mathbf{1})_{0;2,{\,\mbox{-}}2}~{}.$

The five $U(1)$ charges given here are two times the first three weight
entries of the observable $E_{8}$ and the first two of the hidden $E_{8}$.
They can be used to distinguish otherwise vector–like states. When
triangulation $S$ is chosen at all fixed points, e.g.
$N_{E_{1}}=N_{E_{2}}=N_{E_{3}}=0$ , the spectrum does not contain any
vector–like pairs. For most other choices vector–like pairs do arise, but they
presumably acquire a mass at some stage in the effective field theory
description.

###  4.4 A “swampland” $\boldsymbol{SO(10)\times SO(10)}$ models

A seemingly closely related model with three independent bundle vectors is
given by

$\displaystyle{\cal V}_{1,\beta\gamma}={\cal
V}_{1}=(0,\tfrac{1}{2},\tfrac{1}{2},0^{5})(-1,0,0,0^{5})~{},$ (72a)
$\displaystyle{\cal V}_{2,\alpha\gamma}={\cal
V}_{2}=(\tfrac{1}{2},0,\tfrac{1}{2},0^{5})(0,-1,0,0^{5})~{},$ (72b)
$\displaystyle{\cal V}_{3,\alpha\beta}={\cal
V}_{3}=(\tfrac{1}{2},\tfrac{1}{2},0,0^{5})(0,0,-1,0^{5})~{}.$ (72c)

This leads to a gauge group $SO(10)\times SO(10)$. The unbroken roots are
given by $(0^{3},{\underline{\pm 1,\pm 1,0^{3}}})(0^{8})$ and
$(0^{8})(0^{3},{\underline{\pm 1,\pm 1,0^{3}}})$. At first sight this seems to
be a valid choice as well, but this model has a number of issues:

First of all, even thought the bundle vectors clearly satisfy the strong
conditions (57), this model cannot be obtained as the blowup of any orbifold
model. The first two bundle vectors are identical to the model discussed in
the previous subsection and can be obtained from orbifold gauge shift vectors
given there. But the third one does not differ by a lattice vector from
$V_{1}+V_{2}$:

$\displaystyle{\cal
V}_{3}-V_{1}-V_{2}=(\tfrac{1}{2},\tfrac{1}{2},0,0^{5})(0,0,-1,0^{5})-(\tfrac{1}{2},\tfrac{1}{2},1,0^{5})(0,0,0,0^{5})=(0,0,-1,0^{5})(0,0,-1,0^{5})~{}.$
(73)

(If both $-1$–entries would have lain in the same $E_{8}$, this would be a
lattice vector, but they don’t.)

Moreover, this choice of line bundle vectors does not satisfy the final flux
quantisation condition in (63). As a consequence, the spectrum is not integral
for a generic choice of triangulation at the 64
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ resolutions. This can
be inferred from the appearance of multiplicities $\pm 1/16$ and $-5/16$ for
the $\Delta\mathsf{N}^{1}$, $\Delta\mathsf{N}^{2}$ and $\Delta\mathsf{N}^{3}$
in Table 6 when the states are distinguished by their (implicitly given)
$U(1)$ charges. Even if one ignores the $U(1)$ charges, the spectrum combined
is not necessarily integral:

$\displaystyle
16\,(\overline{\mathbf{16}})(\mathbf{1})+48\,(\mathbf{1})(\mathbf{10})+4\,(\mathbf{1})(\mathbf{16})+36\,(\mathbf{1})(\overline{\mathbf{16}})+\tfrac{1}{8}N_{E}\big{\\{}(\mathbf{16})(\mathbf{1})+(\overline{\mathbf{16}})(\mathbf{1})+4\,(\mathbf{1})(\mathbf{16})\big{\\}}+\text{singlets}~{}.$
(74)

Note, that if the same triangulation is chosen at all 64 resolved
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ singularities, the
spectrum would be integral. But any single flop–transition would then lead to
an inconsistent spectrum. This demonstrates that satisfying the flux
quantisation conditions in any triangulation is essential for the difference
multiplicities $\Delta\mathsf{N}^{i}$ to be always integral.

$\begin{array}[]{|c||ccc|cccc|ccc|c|}\hline\cr\text{weight}&\mathsf{H}_{1}&\mathsf{H}_{2}&\mathsf{H}_{3}&\mathsf{N}^{1}&\mathsf{N}^{2}&\mathsf{N}^{3}&\mathsf{N}^{S}&\Delta\mathsf{N}^{1}&\Delta\mathsf{N}^{2}&\Delta\mathsf{N}^{3}&\text{repr.}\\\
\hline\cr\hline\cr(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{5-o}}})(0^{8})&\tfrac{1}{2}&\tfrac{1}{2}&0&\tfrac{1}{8}&\tfrac{1}{8}&\tfrac{1}{8}&\tfrac{1}{16}&\tfrac{1}{16}&\tfrac{1}{16}&\tfrac{1}{16}&(\overline{\mathbf{16}})(\mathbf{1})\\\
(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{5-o}}})(0^{8})&\tfrac{1}{2}&0&0&\tfrac{1}{8}&0&0&\tfrac{1}{16}&\tfrac{1}{16}&-\tfrac{1}{16}&-\tfrac{1}{16}&(\overline{\mathbf{16}})(\mathbf{1})\\\
(\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{5-o}}})(0^{8})&0&\tfrac{1}{2}&0&0&\tfrac{1}{8}&0&\tfrac{1}{16}&-\tfrac{1}{16}&\tfrac{1}{16}&-\tfrac{1}{16}&(\overline{\mathbf{16}})(\mathbf{1})\\\
(\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{5-o}}})(0^{8})&0&0&\tfrac{1}{2}&0&0&\tfrac{1}{8}&\tfrac{1}{16}&-\tfrac{1}{16}&-\tfrac{1}{16}&\tfrac{1}{16}&(\overline{\mathbf{16}})(\mathbf{1})\\\
\hline\cr(-1,-1,0,0^{5})(0^{8})&-\tfrac{1}{2}&-\tfrac{1}{2}&-1&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1})(\mathbf{1})\\\
(-1,0,-1,0^{5})(0^{8})&-\tfrac{1}{2}&-1&-\tfrac{1}{2}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1})(\mathbf{1})\\\
(0,-1,-1,0^{5})(0^{8})&-1&-\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1})(\mathbf{1})\\\
\hline\cr\hline\cr(0^{8})(-1,0,0,{\underline{\pm
1,0^{4}}})&1&0&0&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1})(\mathbf{10})\\\
(0^{8})(0,-1,0,{\underline{\pm
1,0^{4}}})&0&1&0&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1})(\mathbf{10})\\\
(0^{8})(0,0,-1,{\underline{\pm
1,0^{4}}})&0&0&1&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1})(\mathbf{10})\\\
\hline\cr(0^{8})(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{e},\tfrac{1}{2}^{5-e}}})&-\tfrac{1}{2}&-\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{8}&\tfrac{1}{8}&\tfrac{1}{8}&\tfrac{1}{16}&\tfrac{1}{16}&\tfrac{1}{16}&\tfrac{1}{16}&(\mathbf{1})(\mathbf{16})\\\
\hline\cr(0^{8})(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{5-o}}})&\tfrac{1}{2}&-\tfrac{1}{2}&-\tfrac{1}{2}&-\tfrac{1}{8}&\tfrac{1}{8}&\tfrac{1}{8}&\tfrac{3}{16}&-\tfrac{5}{16}&-\tfrac{1}{16}&-\tfrac{1}{16}&(\mathbf{1})(\overline{\mathbf{16}})\\\
0^{8})(\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{5-o}}})&-\tfrac{1}{2}&\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{8}&-\tfrac{1}{8}&\tfrac{1}{8}&\tfrac{3}{16}&-\tfrac{1}{16}&-\tfrac{5}{16}&-\tfrac{1}{16}&(\mathbf{1})(\overline{\mathbf{16}})\\\
0^{8})(\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},{\underline{-\tfrac{1}{2}^{o},\tfrac{1}{2}^{5-o}}})&\tfrac{1}{2}&-\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{8}&\tfrac{1}{8}&-\tfrac{1}{8}&\tfrac{3}{16}&-\tfrac{1}{16}&-\tfrac{1}{16}&-\tfrac{5}{16}&(\mathbf{1})(\overline{\mathbf{16}})\\\
\hline\cr(0^{8})(0,1,1,0^{5})&0&-1&-1&-\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&-1&0&0&(\mathbf{1})(\mathbf{1})\\\
(0^{8})(1,0,1,0^{5})&-1&0&-1&\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&0&-1&0&(\mathbf{1})(\mathbf{1})\\\
(0^{8})(1,1,0,0^{5})&-1&-1&0&\tfrac{1}{2}&\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{2}&0&0&-1&(\mathbf{1})(\mathbf{1})\\\
\hline\cr(0^{8})(1,-1,0,0^{5})&-1&1&0&1&-1&0&0&1&-1&0&(\mathbf{1})(\mathbf{1})\\\
(0^{8})(-1,0,1,0^{5})&1&0&-1&-1&0&1&0&-1&0&1&(\mathbf{1})(\mathbf{1})\\\
(0^{8})(0,1,-1,0^{5})&0&1&-1&0&-1&1&0&0&-1&1&(\mathbf{1})(\mathbf{1})\\\
\hline\cr\end{array}$ Table 6:  The line bundle charges $\mathsf{H}_{i}$ and
the triangulation multiplicities $\mathsf{N}^{i}$, $\mathsf{N}^{S}$ are given
for all the $E_{8}\times E_{8}$ roots charged under the line bundle background
defined by (72). Note that for this model many of the triangulation difference
multiplicities $\Delta\mathsf{N}^{i}$ are non–integral signifying that this
model is not fully consistent.

###  4.5 Blaszczyk’s $\boldsymbol{SU(3)\times SU(2)}$ Line Bundle Models

An example with very similar line bundle vectors, but where all their
non–trivial entries lie in the observable $E_{8}$ can be obtained from the
orbifold gauge embeddings $I.a$ of Table 4. (But these bundle vectors cannot
be obtained from orbifold model $II.a$ .) The defining set of three line
bundle vectors are given by:

$\displaystyle{\cal
V}_{1}=V_{1}+L_{1}=(0,\tfrac{1}{2},\tfrac{1}{2},-1,0,0,0^{2})(0^{8})~{},$
$\displaystyle L_{1}=(0,0,1,-1,0,0,0^{2})(0^{8})$ (75a) $\displaystyle{\cal
V}_{2}=V_{2}+L_{2}=(\tfrac{1}{2},0,\tfrac{1}{2},0,-1,0,0^{2})(0^{8})~{},$
$\displaystyle L_{2}=(0,1,0,0,-1,0,0^{2})(0^{8})~{},$ (75b)
$\displaystyle{\cal
V}_{3}=V_{3}+L_{3}=(\tfrac{1}{2},\tfrac{1}{2},0,0,0,-1,0^{2})(0^{8})~{},$
$\displaystyle L_{3}=(1,0,0,0,0,-1,0^{2})(0^{8})~{},$ (75c)

These bundle vectors were considered in Section 4.3 of ref. [37] before. In
that work the spectra were obtained when at all 64 resolved
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ singularities one of
the four possible triangulations were chosen. However, they satisfy the very
restrictive conditions (57) that ensures that the Bianchi identities are
satisfied and the flux quantisation conditions (63) for all triangulations
simultaneously. Hence, this set of bundle vectors do not suffer from the flaws
encountered in the section above.

Besides all the hidden $E_{8}$ roots there are six unbroken $SU(3)$ roots
$\pm(0^{6},1^{2})$ and $\pm(\tfrac{1}{2}^{6},\pm\tfrac{1}{2}^{2})(0^{8})$ and
two unbroken $SU(2)$ roots $\pm(0^{6},1,-1)$, consequently the unbroken
non–Abelian gauge group is $SU(3)\times SU(2)\times E_{8}$. The Cartan
generators of $SU(3)$ are $h_{1}=(\tfrac{1}{2}^{6},-\tfrac{1}{2}^{2})$ and
$h_{2}=(0^{6},1^{2})$ and of $SU(2)$ $h=(0^{6},1,-1)$.

The triangulation multiplicities evaluated on all observable $E_{8}$ roots are
given in Table 7. If the same triangulation is used at all resolved
singularities then the spectra given in Table 10 of ref. [37] are reproduced.
But with the formalism laid out in this paper an arbitrary triangulation of
each of the resolved fixed points can be considered. As the triangulation
difference multiplicities $\Delta\mathsf{N}^{i}$ are all integral and the
states come in multiples of 16 if triangulation $S$ is used at all resolved
singularities, the spectrum is integral for any choice of local
triangulations. Indeed, ignoring all $U(1)$ charges, the full charged
$SU(3)\times SU(2)$ spectrum from the observable $E_{8}$ can be compactly
summarised as

$\displaystyle
48\,(\mathbf{3},\mathbf{2})+96\,(\overline{\mathbf{3}},\mathbf{1})+(96+N_{E})\,\big{\\{}(\overline{\mathbf{3}},\mathbf{1})+(\mathbf{3},\mathbf{1})\big{\\}}+(176-2\,N_{E})\,(\mathbf{1},\mathbf{2})+(144+N_{E})\,(\mathbf{1},\mathbf{1})~{}.$
(76)

It can be easily confirmed from this spectrum that $SU(3)$ cubed anomaly
cancels for any $N_{E}$ and the $SU(2)$ Witten anomaly is always absent since
the number of $SU(2)$ doublets is always even.

$\begin{array}[t]{cc}\begin{array}[]{|c||ccc|cccc|ccc|c|}\hline\cr\text{weight}&\mathsf{H}_{1}&\mathsf{H}_{2}&\mathsf{H}_{3}&\mathsf{N}^{1}&\mathsf{N}^{2}&\mathsf{N}^{3}&\mathsf{N}^{S}&\Delta\mathsf{N}^{1}&\Delta\mathsf{N}^{2}&\Delta\mathsf{N}^{3}&\text{repr.}\\\
\hline\cr\hline\cr(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2},\tfrac{1}{2}}})&1&0&0&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{3},\mathbf{2})\\\
(0,0,0,-1,0,0,{\underline{\pm 1,0}})&&&&&&&&&&&\\\
\hdashline(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2},\tfrac{1}{2}}})&0&1&0&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{3},\mathbf{2})\\\
(0,0,0,0,-1,0,{\underline{\pm 1,0}})&&&&&&&&&&&\\\
\hdashline(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},{\underline{-\tfrac{1}{2},\tfrac{1}{2}}})&0&0&1&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{3},\mathbf{2})\\\
(0,0,0,0,0,-1,{\underline{\pm 1,0}})&&&&&&&&&&&\\\
\hline\cr(-1,-1,0,0,0,0,0^{2})&-\tfrac{1}{2}&-\tfrac{1}{2}&-1&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\overline{\mathbf{3}},\mathbf{1})\\\
(-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\pm\tfrac{1}{2}^{2})&&&&&&&&&&&\\\
\hdashline(-1,0,-1,0,0,0,0^{2})&-\tfrac{1}{2}&-1&-\tfrac{1}{2}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\overline{\mathbf{3}},\mathbf{1})\\\
(-\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\pm\tfrac{1}{2}^{2})&&&&&&&&&&&\\\
\hdashline(0,-1,-1,0,0,0,0^{2})&-1&-\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\overline{\mathbf{3}},\mathbf{1})\\\
(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\pm\tfrac{1}{2}^{2})&&&&&&&&&&&\\\
\hline\cr(-1,0,0,-1,0,0,0^{2})&1&-\tfrac{1}{2}&-\tfrac{1}{2}&-\tfrac{1}{4}&\tfrac{3}{4}&\tfrac{3}{4}&\tfrac{3}{4}&-1&0&0&(\overline{\mathbf{3}},\mathbf{1})\\\
(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\pm\tfrac{1}{2}^{2})&&&&&&&&&&&\\\
\hdashline(0,-1,0,0,-1,0,0^{2})&-\tfrac{1}{2}&1&-\tfrac{1}{2}&\tfrac{3}{4}&-\tfrac{1}{4}&\tfrac{3}{4}&\tfrac{3}{4}&0&-1&0&(\overline{\mathbf{3}},\mathbf{1})\\\
(\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\pm\tfrac{1}{2}^{2})&&&&&&&&&&&\\\
\hdashline(0,0,-1,0,0,-1,0^{2})&-\tfrac{1}{2}&-\tfrac{1}{2}&1&\tfrac{3}{4}&\tfrac{3}{4}&-\tfrac{1}{4}&\tfrac{3}{4}&0&0&-1&(\overline{\mathbf{3}},\mathbf{1})\\\
(\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\pm\tfrac{1}{2}^{2})&&&&&&&&&&&\\\
\hline\cr(0,0,0,0,1,1,0^{2})&0&-1&-1&-\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&-1&0&0&(\mathbf{3},\mathbf{1})\\\
(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\pm\tfrac{1}{2}^{2})&&&&&&&&&&&\\\
\hdashline(0,0,0,1,0,1,0^{2})&-1&0&-1&\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&0&-1&0&(\mathbf{3},\mathbf{1})\\\
(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\pm\tfrac{1}{2}^{2})&&&&&&&&&&&\\\
\hline\cr\end{array}&\begin{array}[]{|c||ccc|cccc|ccc|c|}\hline\cr\text{weight}&\mathsf{H}_{1}&\mathsf{H}_{2}&\mathsf{H}_{3}&\mathsf{N}^{1}&\mathsf{N}^{2}&\mathsf{N}^{3}&\mathsf{N}^{S}&\Delta\mathsf{N}^{1}&\Delta\mathsf{N}^{2}&\Delta\mathsf{N}^{3}&\text{repr.}\\\
\hline\cr\hline\cr(0,0,0,1,1,0,0^{2})&-1&-1&0&\tfrac{1}{2}&\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{2}&0&0&-1&(\mathbf{3},\mathbf{1})\\\
(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\pm\tfrac{1}{2}^{2})&&&&&&&&&&&\\\
\hline\cr(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2},\tfrac{1}{2}}})&-1&-1&-1&\tfrac{5}{4}&\tfrac{5}{4}&\tfrac{5}{4}&\tfrac{5}{4}&0&0&0&(\mathbf{1},\mathbf{2})\\\
\hline\cr(-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2},\tfrac{1}{2}}})&\tfrac{1}{2}&-\tfrac{1}{2}&-1&-\tfrac{3}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&-1&0&0&(\mathbf{1},\mathbf{2})\\\
\hdashline(-\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2},\tfrac{1}{2}}})&\tfrac{1}{2}&-1&-\tfrac{1}{2}&-\tfrac{3}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&-1&0&0&(\mathbf{1},\mathbf{2})\\\
\hdashline(-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2},\tfrac{1}{2}}})&-\tfrac{1}{2}&\tfrac{1}{2}&-1&\tfrac{1}{4}&-\tfrac{3}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&-1&0&(\mathbf{1},\mathbf{2})\\\
\hdashline(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},{\underline{-\tfrac{1}{2},\tfrac{1}{2}}})&-1&\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{4}&-\tfrac{3}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&-1&0&(\mathbf{1},\mathbf{2})\\\
\hdashline(-\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},{\underline{-\tfrac{1}{2},\tfrac{1}{2}}})&-\tfrac{1}{2}&-1&\tfrac{1}{2}&\tfrac{1}{4}&\tfrac{1}{4}&-\tfrac{3}{4}&\tfrac{1}{4}&0&0&-1&(\mathbf{1},\mathbf{2})\\\
\hdashline(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2},{\underline{-\tfrac{1}{2},\tfrac{1}{2}}})&-1&-\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{4}&\tfrac{1}{4}&-\tfrac{3}{4}&\tfrac{1}{4}&0&0&-1&(\mathbf{1},\mathbf{2})\\\
\hline\cr(-1,0,0,1,0,0,0^{2})&-1&-\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1},\mathbf{1})\\\
\hdashline(0,-1,0,0,1,0,0^{2})&-\tfrac{1}{2}&-1&-\tfrac{1}{2}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1},\mathbf{1})\\\
\hdashline(0,0,-1,0,0,1,0^{2})&-\tfrac{1}{2}&-\tfrac{1}{2}&-1&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&\tfrac{1}{4}&0&0&0&(\mathbf{1},\mathbf{1})\\\
\hline\cr(1,0,0,0,-1,0,0^{2})&0&\tfrac{3}{2}&\tfrac{1}{2}&1&0&0&0&1&0&0&(\mathbf{1},\mathbf{1})\\\
\hdashline(1,0,0,0,0,-1,0^{2})&0&\tfrac{1}{2}&\tfrac{3}{2}&1&0&0&0&1&0&0&(\mathbf{1},\mathbf{1})\\\
\hdashline(0,1,0,-1,0,0,0^{2})&\tfrac{3}{2}&0&\tfrac{1}{2}&0&1&0&0&0&1&0&(\mathbf{1},\mathbf{1})\\\
\hdashline(0,1,0,0,0,-1,0^{2})&\tfrac{1}{2}&0&\tfrac{3}{2}&0&1&0&0&0&1&0&(\mathbf{1},\mathbf{1})\\\
\hdashline(0,0,1,-1,0,0,0^{2})&\tfrac{3}{2}&\tfrac{1}{2}&0&0&0&1&0&0&0&1&(\mathbf{1},\mathbf{1})\\\
\hdashline(0,0,1,0,0,-1,0^{2})&\tfrac{1}{2}&\tfrac{3}{2}&0&0&0&1&0&0&0&1&(\mathbf{1},\mathbf{1})\\\
\hline\cr(0,0,0,0,1,-1,0^{2})&0&-1&-1&-\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&-1&0&0&(\mathbf{1},\mathbf{1})\\\
\hdashline(0,0,0,1,0,-1,0^{2})&-1&0&-1&\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&0&-1&0&(\mathbf{1},\mathbf{1})\\\
\hdashline(0,0,0,1,-1,0,0^{2})&-1&-1&0&\tfrac{1}{2}&\tfrac{1}{2}&-\tfrac{1}{2}&\tfrac{1}{2}&0&0&-1&(\mathbf{1},\mathbf{1})\\\
\hline\cr\vrule\lx@intercol\hfil\phantom{xxx}\hfil\lx<EMAIL_ADDRESS>\hline\cr\end{array}\end{array}$

Table 7:  The line bundle charges $\mathsf{H}_{i}$, the triangulation
multiplicities $\mathsf{N}^{i}$, $\mathsf{N}^{S}$ and the triangulation
difference multiplicities $\Delta\mathsf{N}^{i}$ are given for the
$SU(3)\times SU(2)$ charged and singlet states obtained from the line bundle
background defined by (75).

Since in this model all the gauge flux is located in a single $E_{8}$, the
loop–corrected DUY equations in the form (46) can be used. Since for this
model there are only three bundle vectors ${\cal V}_{1}$, ${\cal V}_{2}$ and
${\cal V}_{3}$ which are clearly independent, the DUY equations reduce to
three equations:

$\displaystyle\begin{array}[]{rcl}\sum\limits_{\beta\gamma}e^{-2\phi}\,\text{Vol}(E_{1,\beta\gamma})&=&\dfrac{1}{16\pi}\big{(}64+2\,N_{E_{1}}-2\,N_{E_{2}}-2\,N_{E_{3}}\big{)}~{},\\\\[8.61108pt]
\sum\limits_{\alpha\gamma}e^{-2\phi}\,\text{Vol}(E_{2,\alpha\gamma})&=&\dfrac{1}{16\pi}\big{(}64+2\,N_{E_{2}}-2\,N_{E_{1}}-2\,N_{E_{3}}\big{)}~{},\\\\[8.61108pt]
\sum\limits_{\beta\gamma}e^{-2\phi}\,\text{Vol}(E_{3,\alpha\beta})&=&\dfrac{1}{16\pi}\big{(}64+2\,N_{E_{3}}-2\,N_{E_{1}}-2\,N_{E_{2}}\big{)}~{}.\end{array}$
(80)

Since, the sum of volumes all need to be non–negative, the right–hand–sides of
these equations all have to be positive. This leads to the conditions on the
number of times the exceptional triangulations may be chosen:

$\displaystyle N_{E_{2}}+N_{E_{3}}-N_{E_{1}}\leq 32~{},\quad
N_{E_{1}}+N_{E_{3}}-N_{E_{2}}\leq 32~{},\quad
N_{E_{1}}+N_{E_{2}}-N_{E_{3}}\leq 32~{}.$ (81)

Adding two of these three conditions shows that $N_{E_{i}}\leq 32$. In
addition, (8) implies that

$\displaystyle N_{E_{1}}+N_{E_{2}}+N_{E_{3}}\leq 64~{}.$ (82)

Thus apparently, one can only choose the $S$–triangulation at all 64 resolved
singularities, but not one of the exceptional triangulations. However, one can
choose to use exceptional triangulations at all resolved singularities, but
not at all of them the same one. For example, the choice,
$N_{E_{1}}=N_{E_{2}}=16$ and $N_{E_{3}}=32$, would be allowed by the DUY
conditions.

##  5 Jumping Spectra in a Blasczcyk–like GUT model

In ref.[37] a semi–realistic MSSM model line bundle model on a resolution of
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ was constructed with gauge group
$\mathrm{SU}(5)\times\mathrm{SU}^{\prime}(3)\times\mathrm{SU}^{\prime}(2)$.
This model possessed an freely acting involution that reduced the gauge
symmetry to the standard model gauge group. For this model the
$E_{1}$–triangulation was chosen at all 64 resolved
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$. In this section models
similar to the Blasczcyk’s GUT model are considered. The emphasis is not so
much on finding a phenomenologically satisfactory model but rather on
illustrating the effects of flop–transitions on the spectrum.

###  5.1 Generalities of Blasczcyk–like GUT models

Models like the Blaszczyk’s GUT model are particular resolution of an orbifold
theory with, in addition to two shifts $V_{1}$ and $V_{2}$ associated to the
twists $v_{1}$ and $v_{2}$, up to five Wilson lines in all torus directions
are switched on. The Wilson lines in the second, fourth and sixth torus
directions are all taken equal: $W_{2}=W_{4}=W_{6}$ and independent of the two
remaining Wilson lines $W_{3}$ and $W_{5}$. The resulting line bundle vectors
are given by

$\displaystyle\begin{array}[]{rcl}{\cal V}_{1,\beta\gamma}={\cal
V}_{1\beta_{3}\gamma_{5}(\beta_{4}+\gamma_{6})}&=&V_{1}+\beta_{3}\,W_{3}+\gamma_{5}\,W_{5}+(\beta_{4}+\gamma_{6})W_{2}+L_{1\beta_{3}\gamma_{5}(\beta_{4}+\gamma_{6})}\\\\[4.30554pt]
{\cal V}_{2,\alpha\gamma}={\cal
V}_{2\\_\gamma_{5}(\alpha_{2}+\gamma_{6})}&=&V_{1}+\gamma_{5}\,W_{5}+(\alpha_{2}+\gamma_{6})W_{2}+L_{2\gamma_{5}(\alpha_{2}+\gamma_{6})}\\\\[4.30554pt]
{\cal V}_{3,\alpha\beta}={\cal
V}_{3\beta_{3}\\_(\alpha_{2}+\beta_{4})}&=&V_{3}+\beta_{3}\,W_{3}+(\alpha_{2}+\beta_{4})W_{2}+L_{3\gamma_{5}(\alpha_{2}+\beta_{4})}\end{array}$
(86)

using the binary multi–index notation introduced in Subsection 2.1. Here
$L_{1\beta_{3}\gamma_{5}(\beta_{4}+\gamma_{6})}$,
$L_{2\gamma_{5}(\alpha_{2}+\gamma_{6})}$ and
$L_{3\gamma_{5}(\alpha_{2}+\beta_{4})}$ are appropriately chosen $E_{8}\times
E_{8}$ lattice vectors. The sum in between brackets is defined modulo two
(since two times a Wilson line is a lattice vector which can be absorbed in
one of the $L$’s). Thus, in total these kind of blowup models are defined by
$8+4+4=16$ line bundle vectors and the 64 resolved fixed points are
distinguished in 32 bunches of two fixed points as the index $\alpha_{1}=0,1$
still parameterises a twofold degeneracy. In addition, there is a freely
acting symmetry in such models: if one simultaneously adds 1 to the three
indices $\alpha_{2},\beta_{4},\gamma_{6}$ modulo two:

$\displaystyle(\alpha_{2},\beta_{4},\gamma_{6})\mapsto(\alpha_{2}+1,\beta_{4}+1,\gamma_{6}+1)~{},$
(87)

all bundle vectors are identical. This isometry was used in ref.[37] introduce
a freely acting Wilson line to break the $SU(5)$ GUT to the standard model.
This step won’t be considered here.

$\begin{array}[]{|c|c|}\hline\cr\vrule\lx@intercol\hfil\text{Bundle
vectors}\hfil\lx@intercol\vrule\lx@intercol\\\ \hline\cr\hline\cr{\cal
V}_{1000}={\cal
V}_{1010}&(-\mbox{$\frac{1}{2}$},-\mbox{$\frac{1}{2}$},1,0,0,0,0,0)(0,0,0,0,0,0,0,0)\\\
{\cal V}_{1100}={\cal
V}_{1110}&(0,1,0,0,0,0,0,0)(0,0,0,0,0,0,-\mbox{$\frac{1}{2}$},-\mbox{$\frac{1}{2}$})\\\
{\cal V}_{1001}={\cal
V}_{1011}&(\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{3}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$})(-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$})\\\
{\cal V}_{1101}={\cal
V}_{1111}&(\mbox{$\frac{1}{4}$},-\mbox{$\frac{3}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$})(-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$})\\\
\hline\cr{\cal V}_{2\\_00}={\cal
V}_{2\\_10}&(\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$})(0,0,0,0,0,0,0,-1)\\\
{\cal V}_{2\\_01}={\cal
V}_{2\\_11}&(\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$},0,0,0,0,0)(\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},-\mbox{$\frac{3}{4}$},\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$})\\\
\hline\cr{\cal
V}_{30\\_0}&(-\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$})(0,0,0,0,0,0,-1,0)\\\
{\cal
V}_{31\\_0}&(-\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{3}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$})(0,0,0,0,0,0,\mbox{$\frac{1}{2}$},-\mbox{$\frac{1}{2}$})\\\
{\cal
V}_{30\\_1}&(0,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0,0,0,0,0)(\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},-\mbox{$\frac{3}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$})\\\
{\cal
V}_{31\\_1}&(-\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$},0,0,0,0,0)(-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$},\mbox{$\frac{3}{4}$},-\mbox{$\frac{1}{4}$},-\mbox{$\frac{1}{4}$})\\\
\hline\cr\end{array}$ Table 8: A set of bundle vectors associated to two
shifts and four Wilson lines that satisfy the flux quantisation conditions and
the Bianchi identities in all triangulations.

###  5.2 Triangulation independent Blaszczyk–like GUT models

The aim of this section is to engineer a modification of the Blaszczyk’s GUT
model such that it fulfils the Bianchi identities in an arbitrary
triangulation. As this turned out to be a very difficult, here only models are
considered in which the Wilson lines $W_{2}=W_{4}=W_{6}$ and $W_{3}$ are
switched on. Concretely the orbifold data of the model under consideration
here is given by:

$\displaystyle\begin{array}[]{c}V_{1}=({\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},1,0^{5})(0^{8})~{},\qquad
V_{2}=(\mbox{$\frac{1}{4}$},{\,\mbox{-}}\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},{\,\mbox{-}}\mbox{$\frac{1}{4}$}^{5})(0^{6},0,{\,\mbox{-}}1)~{},\\\\[8.61108pt]
W_{3}=(0,0,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$}^{5})(0^{6},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$})~{},\qquad
W_{2}=W_{4}=W_{6}=(\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$}^{5})({\,\mbox{-}}\mbox{$\frac{1}{4}$}^{6},\mbox{$\frac{1}{4}$},\mbox{$\frac{1}{4}$})~{}.\end{array}$
(90)

Using the freedom to add lattice vectors in (86) it is possible to obtain a
set of bundle vectors that satisfy the strong conditions (50)and (57), which
guarantee that the flux quantisation conditions and the Bianchi identities are
satisfied in any triangulation. Such a set is given in Table 8.

$\begin{array}[]{|cc|cc||cc|cc|}\hline\cr\vrule\lx@intercol\hfil\text{Observable
}E_{8}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\text{Hidden
}E_{8}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr\vrule\lx@intercol\hfil\text{$SU(5)$--adjoint}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\text{Singlets}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\text{$SU(4)$--adjoint}\hfil\lx@intercol\vrule\lx@intercol&\mathbf{1}_{4}&(0^{4},0,1,0,1)\\\
\mathbf{24}&(0^{3},{\underline{1,-1,0^{3}}})&\mathbf{1}_{1}&(1,1,0,0^{5})&\mathbf{15}&({\underline{1,-1,0^{2}}},0^{4})&\mathbf{1}_{5}&(0^{4},0,0,1,1)\\\
\cline{1-2}\cr\cline{5-6}\cr\vrule\lx@intercol\hfil\text{5--
plets}\hfil\lx@intercol\vrule\lx@intercol&\mathbf{1}_{2}&(1,0,1,0^{5})&\lx@intercol\hfil\text{4
--
plets}\hfil\lx@intercol\vrule\lx@intercol&\mathbf{1}_{6}&(0^{4},1,{\,\mbox{-}}1,0,0)\\\
\mathbf{5}_{1}&(0,1,0,{\underline{1,0^{4}}})&\mathbf{1}_{3}&(0,1,1,0^{5})&\mathbf{4}_{1}&({\underline{1,0^{3}}},0,0,1,0)&\mathbf{1}_{7}&(0^{4},1,0,{\,\mbox{-}}1,0)\\\
\mathbf{5}_{2}&(0,0,1,{\underline{1,0^{4}}})&\mathbf{1}_{4}&(1,{\,\mbox{-}}1,0,0^{5})&\mathbf{4}_{2}&({\underline{1,0^{3}}},0,0,0,1)&\mathbf{1}_{8}&(0^{4},1,0,0,{\,\mbox{-}}1)\\\
\mathbf{5}_{3}&(0,{\,\mbox{-}}1,0,{\underline{1,0^{4}}})&\mathbf{1}_{5}&(1,0,{\,\mbox{-}}1,0^{5})&\mathbf{4}_{3}&({\underline{1,0^{3}}},{\,\mbox{-}}1,0,0,0)&\mathbf{1}_{8}&(0^{4},1,0,0,{\,\mbox{-}}1)\\\
\mathbf{5}_{4}&(0,0,{\,\mbox{-}}1,{\underline{1,0^{4}}})&\mathbf{1}_{6}&(0,1,{\,\mbox{-}}1,0^{5})&\mathbf{4}_{4}&({\underline{1,0^{3}}},0,{\,\mbox{-}}1,0,0)&\mathbf{1}_{10}&(0^{4},0,1,0,{\,\mbox{-}}1)\\\
\mathbf{5}_{5}&({\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\underline{\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{4}}})&\mathbf{1}_{7}&({\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$}^{5})&\mathbf{4}_{5}&({\underline{1,0^{3}}},0,0,{\,\mbox{-}}1,0)&\mathbf{1}_{11}&(0^{4},0,0,1,{\,\mbox{-}}1)\\\
\mathbf{5}_{6}&({\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\underline{\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{4}}})&\mathbf{1}_{8}&({\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$}^{5})&\mathbf{4}_{6}&({\underline{1,0^{3}}},0,0,0,{\,\mbox{-}}1)&\mathbf{1}_{12}&(\mbox{$\frac{1}{2}$}^{4},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$})\\\
\mathbf{5}_{7}&(\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\underline{\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{4}}})&\mathbf{1}_{9}&(\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$}^{5})&\mathbf{4}_{7}&({\underline{\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{3}}},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$})&\mathbf{1}_{13}&(\mbox{$\frac{1}{2}$}^{4},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$})\\\
\mathbf{5}_{8}&(\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\underline{\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{4}}})&\mathbf{1}_{10}&(\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$}^{5})&\mathbf{4}_{8}&({\underline{\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{3}}},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$})&\mathbf{1}_{14}&(\mbox{$\frac{1}{2}$}^{4},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$})\\\
\cline{1-4}\cr\vrule\lx@intercol\hfil\text{10--
plet}\hfil\lx@intercol&\mathbf{10}&(\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\underline{\mbox{$\frac{1}{2}$}^{2},{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{3}}})&\mathbf{4}_{9}&({\underline{\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{3}}},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$})&\mathbf{1}_{15}&(\mbox{$\frac{1}{2}$}^{4},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$})\\\
\cline{1-6}\cr\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\text{Singlets}\hfil\lx@intercol\vrule\lx@intercol&\mathbf{1}_{16}&(\mbox{$\frac{1}{2}$}^{4},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$})\\\
\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\mathbf{1}_{1}&(0^{4},1,0,1,0)&\mathbf{1}_{17}&(\mbox{$\frac{1}{2}$}^{4},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$})\\\
\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\mathbf{1}_{2}&(0^{4},1,0,0,1)&\mathbf{1}_{18}&(\mbox{$\frac{1}{2}$}^{4},\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$})\\\
\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\mathbf{1}_{3}&(0^{4},0,1,1,0)&\mathbf{1}_{19}&(\mbox{$\frac{1}{2}$}^{4},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$})\\\
\cline{5-8}\cr\end{array}$

Table 9: The identification between the roots and the states in the spectrum
in both the observable and hidden sectors. States in the same non–Abelian
representation but with different $U(1)$–charges are enumerated.

$\begin{array}[]{|c||c|c|c|c|}\hline\cr\text{Resolved}&\text{Spectra in S--
triangulation}&\lx@intercol\hfil\text{Spectrum jumps due to flop--
transitions}\hfil\lx@intercol\vrule\lx@intercol\\\ \text{fixed
points}&4\times\mathsf{N}^{S}&\Delta\mathsf{N}^{1}&\Delta\mathsf{N}^{2}&\quad\qquad\Delta\mathsf{N}^{3}\\\
\hline\cr\hline\cr
f_{\alpha_{1}0\,00\,\gamma_{5}0}\,,&\mathbf{5}_{2}+\mathbf{5}_{4}+\overline{\mathbf{5}}_{5}+\overline{\mathbf{10}}+\overline{\mathbf{1}}_{1}+3\,\mathbf{1}_{7}+\mathbf{1}_{8}+\mathbf{1}_{9}&\overline{\mathbf{1}}_{7}&\overline{\mathbf{1}}_{5}&\mathbf{1}_{6}\\\
f_{\alpha_{1}1\,01\,\gamma_{5}1}&\pagecolor{gray!20}\overline{\mathbf{4}}_{1}+\overline{\mathbf{4}}_{2}+\mathbf{4}_{5}+\mathbf{4}_{6}+\overline{\mathbf{1}}_{1}+\overline{\mathbf{1}}_{2}+\overline{\mathbf{1}}_{3}+\overline{\mathbf{1}}_{4}+2\,\mathbf{1}_{5}+\mathbf{1}_{7}+\mathbf{1}_{8}+&\pagecolor{gray!20}\overline{\mathbf{1}}_{5}&\pagecolor{gray!20}\overline{\mathbf{1}}_{11}&\pagecolor{gray!20}\mathbf{1}_{11}\\\
&\pagecolor{gray!20}\mathbf{1}_{9}+\mathbf{1}_{10}&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\hfil\\\
\hline\cr
f_{\alpha_{1}0\,10\,\gamma_{5}0}\,,{}&\overline{\mathbf{5}}_{1}+\mathbf{5}_{2}+\overline{\mathbf{5}}_{3}+\mathbf{5}_{7}+\mathbf{1}_{1}+2\,\overline{\mathbf{1}}_{3}+\mathbf{1}_{4}+\overline{\mathbf{1}}_{5}+3\,\mathbf{1}_{6}+3\,\mathbf{1}_{7}+&\overline{\mathbf{1}}_{6}&\mathbf{5}_{1}+\overline{\mathbf{5}}_{7}+\mathbf{1}_{3}+\overline{\mathbf{1}}_{4}+&\overline{\mathbf{1}}_{7}\\\
f_{\alpha_{1}1\,11\,\gamma_{5}1}&3\,\overline{\mathbf{1}}_{8}+\overline{\mathbf{1}}_{10}&&\mathbf{1}_{8}+\mathbf{1}_{10}&\\\
&\pagecolor{gray!20}\mathbf{4}_{2}+\overline{\mathbf{4}}_{6}+\mathbf{1}_{2}+\mathbf{1}_{4}+2\,\mathbf{1}_{5}+\overline{\mathbf{1}}_{8}+\overline{\mathbf{1}}_{10}+2\,\overline{\mathbf{1}}_{11}&\pagecolor{gray!20}\mathbf{1}_{11}&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\overline{\mathbf{1}}_{5}\\\
\hline\cr
f_{\alpha_{1}1\,00\,\gamma_{5}0}\,,&\overline{\mathbf{5}}_{2}+\mathbf{5}_{4}+\mathbf{5}_{5}+\overline{\mathbf{10}}+3\,\overline{\mathbf{1}}_{1}+\overline{\mathbf{1}}_{2}+\overline{\mathbf{1}}_{3}+\mathbf{1}_{7}&\mathbf{1}_{1}&\overline{\mathbf{1}}_{5}&\overline{\mathbf{1}}_{6}\\\
f_{\alpha_{1}0\,01\,\gamma_{5}1}&\pagecolor{gray!20}\mathbf{4}_{3}+\mathbf{4}_{4}+\overline{\mathbf{4}}_{8}+\overline{\mathbf{4}}_{9}+\overline{\mathbf{1}}_{1}+\overline{\mathbf{1}}_{2}+\overline{\mathbf{1}}_{3}+\overline{\mathbf{1}}_{4}+\mathbf{1}_{14}+\mathbf{1}_{15}+\mathbf{1}_{16}+&\pagecolor{gray!20}\mathbf{1}_{19}&\pagecolor{gray!20}\mathbf{1}_{6}&\pagecolor{gray!20}\overline{\mathbf{1}}_{6}\\\
&\pagecolor{gray!20}\mathbf{1}_{17}+2\,\overline{\mathbf{1}}_{19}&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\hfil\\\
\hline\cr
f_{\alpha_{1}1\,10\,\gamma_{5}0}\,,&\mathbf{5}_{1}+\overline{\mathbf{5}}_{3}+\overline{\mathbf{1}}_{1}+\mathbf{1}_{2}+\overline{\mathbf{1}}_{3}+\mathbf{1}_{4}+\overline{\mathbf{1}}_{5}+3\,\mathbf{1}_{6}&\overline{\mathbf{1}}_{6}&\overline{\mathbf{1}}_{4}&\mathbf{1}_{1}\\\
f_{\alpha_{1}0\,11\,\gamma_{5}1}&\pagecolor{gray!20}\mathbf{4}_{3}+\overline{\mathbf{4}}_{4}+\overline{\mathbf{4}}_{7}+\mathbf{4}_{9}+\mathbf{1}_{1}+\mathbf{1}_{2}+\mathbf{1}_{5}+2\,\mathbf{1}_{6}+\overline{\mathbf{1}}_{9}+\overline{\mathbf{1}}_{10}+\mathbf{1}_{13}+&\pagecolor{gray!20}\overline{\mathbf{1}}_{6}&\pagecolor{gray!20}\overline{\mathbf{1}}_{13}&\pagecolor{gray!20}\mathbf{1}_{19}\\\
&\pagecolor{gray!20}\mathbf{1}_{14}+\mathbf{1}_{15}+\overline{\mathbf{1}}_{16}+\overline{\mathbf{1}}_{17}+\overline{\mathbf{1}}_{19}&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\hfil\\\
\hline\cr
f_{\alpha_{1}0\,01\,\gamma_{5}0}\,,&\mathbf{5}_{4}+\overline{\mathbf{5}}_{8}+\overline{\mathbf{1}}_{2}+2\,\overline{\mathbf{1}}_{3}+2\,\mathbf{1}_{8}+\mathbf{1}_{9}&&\mathbf{1}_{3}&\overline{\mathbf{1}}_{8}\\\
f_{\alpha_{1}1\,00\,\gamma_{5}1}&\pagecolor{gray!20}\overline{\mathbf{4}}_{2}+\mathbf{4}_{4}+\overline{\mathbf{4}}_{6}+\mathbf{4}_{8}+\overline{\mathbf{1}}_{2}+\overline{\mathbf{1}}_{3}+2\,\mathbf{1}_{4}+\mathbf{1}_{5}+\mathbf{1}_{6}+2\,\overline{\mathbf{1}}_{8}+3\,\mathbf{1}_{10}+&\pagecolor{gray!20}\mathbf{4}_{6}+\overline{\mathbf{4}}_{8}+\overline{\mathbf{1}}_{4}+\overline{\mathbf{1}}_{5}+&\pagecolor{gray!20}\overline{\mathbf{1}}_{10}&\pagecolor{gray!20}\overline{\mathbf{1}}_{16}\\\
&\pagecolor{gray!20}\mathbf{1}_{11}+3\,\mathbf{1}_{16}+\overline{\mathbf{1}}_{17}+3\,\overline{\mathbf{1}}_{18}+\overline{\mathbf{1}}_{19}&\pagecolor{gray!20}\mathbf{1}_{8}+\mathbf{1}_{17}+\mathbf{1}_{18}+\mathbf{1}_{19}&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\hfil\\\
\hline\cr
f_{\alpha_{1}0\,11\,\gamma_{5}0}\,,&\mathbf{5}_{3}+\overline{\mathbf{5}}_{6}+\overline{\mathbf{1}}_{4}+\overline{\mathbf{1}}_{5}+\mathbf{1}_{6}+\overline{\mathbf{1}}_{7}+\overline{\mathbf{1}}_{8}+3\,\mathbf{1}_{9}&\overline{\mathbf{1}}_{9}&\mathbf{1}_{7}&\mathbf{1}_{4}\\\
f_{\alpha_{1}1\,10\,\gamma_{5}1}&\pagecolor{gray!20}\mathbf{4}_{2}+\overline{\mathbf{4}}_{4}+\mathbf{4}_{6}+\overline{\mathbf{4}}_{7}+\mathbf{1}_{2}+\mathbf{1}_{4}+\mathbf{1}_{5}+\overline{\mathbf{1}}_{6}+\mathbf{1}_{8}+\mathbf{1}_{9}+2\,\overline{\mathbf{1}}_{10}+&\pagecolor{gray!20}\mathbf{1}_{10}&\pagecolor{gray!20}\overline{\mathbf{1}}_{17}&\pagecolor{gray!20}\overline{\mathbf{1}}_{4}\\\
&\pagecolor{gray!20}\mathbf{1}_{11}+\mathbf{1}_{12}+\mathbf{1}_{13}+\mathbf{1}_{16}+\mathbf{1}_{17}&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\hfil\\\
\hline\cr
f_{\alpha_{1}0\,00\,\gamma_{5}1}\,,&\mathbf{5}_{4}+\overline{\mathbf{5}}_{8}+2\,\overline{\mathbf{1}}_{2}+\overline{\mathbf{1}}_{3}+\mathbf{1}_{8}+2\,\mathbf{1}_{9}&&\overline{\mathbf{1}}_{9}&\mathbf{1}_{2}\\\
f_{\alpha_{1}1\,01\,\gamma_{5}0}&\pagecolor{gray!20}\overline{\mathbf{4}}_{1}+\mathbf{4}_{3}+\overline{\mathbf{4}}_{5}+\mathbf{4}_{9}+2\,\mathbf{1}_{1}+\overline{\mathbf{1}}_{2}+\overline{\mathbf{1}}_{3}+\mathbf{1}_{5}+\overline{\mathbf{1}}_{6}+3\,\mathbf{1}_{7}+\overline{\mathbf{1}}_{9}+&\pagecolor{gray!20}\mathbf{4}_{5}+\overline{\mathbf{4}}_{9}+\overline{\mathbf{1}}_{1}+\overline{\mathbf{1}}_{5}+&\pagecolor{gray!20}\overline{\mathbf{1}}_{15}&\pagecolor{gray!20}\overline{\mathbf{1}}_{7}\\\
&\pagecolor{gray!20}\overline{\mathbf{1}}_{11}+\overline{\mathbf{1}}_{14}+3\,\mathbf{1}_{15}+3\,\overline{\mathbf{1}}_{18}+\overline{\mathbf{1}}_{19}&\pagecolor{gray!20}\mathbf{1}_{9}+\mathbf{1}_{14}+\mathbf{1}_{18}+\mathbf{1}_{19}&\pagecolor{gray!20}\hfil&\pagecolor{gray!20}\hfil\\\
\hline\cr
f_{\alpha_{1}0\,10\,\gamma_{5}1}\,,&\overline{\mathbf{5}}_{2}+\mathbf{5}_{3}+\mathbf{5}_{6}+\overline{\mathbf{5}}_{7}+\overline{\mathbf{1}}_{2}+\overline{\mathbf{1}}_{3}+\overline{\mathbf{1}}_{4}+\overline{\mathbf{1}}_{5}+\mathbf{1}_{6}+2\,\overline{\mathbf{1}}_{7}+\mathbf{1}_{9}+\overline{\mathbf{1}}_{10}&\mathbf{1}_{3}&\mathbf{1}_{7}&\mathbf{1}_{4}\\\
f_{\alpha_{1}1\,11\,\gamma_{5}0}&\pagecolor{gray!20}\mathbf{4}_{3}+\overline{\mathbf{4}}_{9}+\mathbf{1}_{1}+\mathbf{1}_{2}+\overline{\mathbf{1}}_{6}+\mathbf{1}_{11}+\overline{\mathbf{1}}_{12}+3\,\mathbf{1}_{14}+\overline{\mathbf{1}}_{15}+\mathbf{1}_{19}&\pagecolor{gray!20}\mathbf{1}_{15}&\pagecolor{gray!20}\overline{\mathbf{1}}_{14}&\pagecolor{gray!20}\overline{\mathbf{1}}_{1}\\\
\hline\cr\end{array}$

Table 10:  Each big row corresponds to two sets of four resolved
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ fixed points labelled
by $\alpha_{1},\gamma_{5}=0,1$ (because their local bundle vectors are
identical and thus so are their local spectra). The lines with the white
background give the observable spectra resulting from the first $E_{8}$ and
the lines with grey background the hidden spectrum from the second $E_{8}$.
The charge states are labeled in Table 9. (Since all singlet are charged it
make sense to talk about a singlet state or its conjugate.) The second column
gives the contributions at the four local resolved singularities using the
S–triangulation combined. The columns $\Delta\mathsf{N}^{1}$,
$\Delta\mathsf{N}^{2}$ and $\Delta\mathsf{N}^{3}$ indicate the jumps in the
spectra for a single resolved fixed point out of these sets of four
singularities.

The resulting spectra are given in Table 10. The states used in that table are
defined in Table 9 from the roots of both $E_{8}$–factors. Notice, that not
all $E_{8}$–roots (up to conjugation) appear here; only the states, that have
a non–vanishing multiplicity in the models defined here, are listed. The
subscripts are used to distinguish states that have the same non–Abelian
representation but different $U(1)$ charges. The second column gives the
spectra from the local resolved singularities when the $S$–triangulation is
used at all 64 of them. Since the labels $\alpha_{1},\gamma_{5}=0,1$ are
arbitrary, there will be a fourfold degeneracy in the spectrum, this is
already taking into account in the table by multiplying the spectra in the
$S$–triangulation by 4. The additional two–fold degeneracy due to the freely
action symmetry is made apparent by giving two sets of four resolved
singularities. It is not difficult to see that the full spectrum using the
$S$–triangulation is free of non–Abelian anomalies.

The final three columns of Table 10 displays the jumps in the spectra when at
a given singularity the $S$–triangulation is flopped to the triangulation
$E_{1}$, $E_{2}$ or $E_{3}$. These are the jumps at a single resolved fixed
point. It can be seen that in accordance with our general findings this jumps
are always integral. Most jumps that occur in the spectra involve singlets
only. At the resolved fixed points $f_{\alpha_{1}0\,10\,\alpha_{5}0}$ and
$f_{\alpha_{1}1\,11\,\gamma_{5}0}$ a $\mathbf{5}$ and $\overline{\mathbf{5}}$
pair appears during a flop from the $S$ to the $E_{2}$–triangulation.
Similarly, a $\mathbf{4}$ and $\overline{\mathbf{4}}$ pair appears at resolved
fixed points $f_{\alpha_{1}0\,01\,\gamma_{5}0}$ and
$f_{\alpha_{1}0\,00\,\gamma_{5}1}$. Thus, at most only non–Abelian vector–like
pairs can arise during a flop transition.

##  6 Conclusion

### Summary

This paper has been devoted to a specific problem which occurs in resolutions
of certain toroidal orbifolds, namely that the resolutions of the local
singularities is not unique at the topological level and therefore leads to an
explosion of topologically distinct smooth geometries all associated to one
and the same orbifold. As a concrete working example the focus was on the
resolutions of a $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold which
contains 64 $\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$
singularities, each of which admits four distinct resolutions encoded by
different triangulations of their toric diagram.

The key idea to overcome this complication is to use a parameterisation to
keep track of the triangulations chosen at all resolved fixed points
simultaneously. It turned out not to be very cumbersome to express the
fundamental (self–)intersection numbers of the divisors of the resolution in
terms of this data. Once the (self–)intersection numbers were determined, many
derived objects can be computed without much more difficulty as determining
them within a specific triangulation. In particular, we checked our procedure
by computing the integrated third Chern class directly and confirmed that it
equals 96 independently of any triangulation choice. We obtained expressions
for the volumes of curves, divisors and the manifold as a whole for any
possible choice of the triangulation of the 64
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ singularities. In addition, we worked
out some of the fundamental consistency conditions of line bundle models on
the resolutions of the $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ like the
flux quantisation conditions and the integrated Bianchi identities (which for
simplicity were only considered without five branes). Even a tool which is
often used to compute the chiral part of the spectrum, the multiplicity
operator, could be determined once and for all for any choice of
triangulation.

Having written down the fundamental consistency conditions for any possible
choice of triangulation, allowed for posing the question what conditions have
to be enforced to ensure that they are satisfied for all possible
triangulations simultaneously. It turned out that if the flux quantisation
conditions are satisfied for a given specific choice of triangulation, they
are, in fact, fulfilled for any configuration of triangulations: the flux
quantisation conditions turned out to be triangulation independent. The
superimposed integrated Bianchi identities reduced to much simpler
requirements than those within any particular choice of triangulation.
Moreover, they are quite reminiscent of some of the properties of shifted
momenta of the blowup modes that induce the resolution from the orbifold
perspective.

These ideas and results were illustrated by a number of examples in the
remainder of the paper. For simplicity, first line bundle models were
considered, where the 48 line bundle vectors were chosen to be determined by
three defining vectors. By computing spectra in all triangulations explicitly,
it was confirmed that the full chiral spectra are always integral. We take
this as a very strong crosscheck of the procedure outlined in this paper to
parameterise all possible triangulations of the resolved singularities of the
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold. This was also checked
explicitly in a variant of the Blasczczyk’s GUT model with four Wilson lines
of which three were set equal. The full spectrum computed in the
$S$–triangulation everywhere is integral and free of non–Abelian anomalies.
But also all the local difference multiplicities measuring the jumps in the
local spectra at specific resolved singularities are always integral and free
of non–Abelian anomalies (as the jumping spectra were all vector–like in this
particular example).

### Outlook

This paper focussed on one particular
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold, it is to be expected
that this procedure can also be applied to the other
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds. In fact, applications
do not stop there, for any orbifold for which the resolution of some of the
local singularities is not unique, it may be applied. Table 11 gives an
overview of some toroidal orbifolds for which the triangulations of their
local singularities are not unique and a naive estimate of the number of
resolved geometries which therefore can be associated to that orbifold. (The
numbers quoted in this table are upper limits: these orbifolds can be defined
on different lattices on which the number of fixed points may be lower than
the numbers indicated here.) Moreover, triangulation ambiguities do not only
show up in toroidal orbifolds resolutions, also in other Calabi–Yau
constructions they might be present. For example, some Calabi–Yaus in the
Kreuzer–Skarke list obtained as hypersurfaces in toric varieties are not
unique due to different triangulation choices [44, 45]. One may therefore
speculate whether similar methods may also be applied there.

Toroidal | Number of | Triangulations | Naive number
---|---|---|---
orbifold | fixed points | per fixed point | of resolutions
$T^{6}/\mathbbm{Z}_{\text{6--II}}$ | $12$ | $5$ | $5^{12}\sim 10^{8}$
$T^{6}/\mathbbm{Z}_{\text{2}}\times\mathbbm{Z}_{\text{2}}$ | $64$ | $4$ | $4^{64}\sim 10^{38}$
$T^{6}/\mathbbm{Z}_{\text{2}}\times\mathbbm{Z}_{\text{4}}$ | $24$ | $16$ | $16^{24}\sim 10^{28}$
$T^{6}/\mathbbm{Z}_{\text{3}}\times\mathbbm{Z}_{\text{3}}$ | $27$ | $79$ | $79^{27}\sim 10^{51}$
Table 11:  Triangulation dependence and the naive number of resulting
resolutions of toroidal orbifolds as can be inferred from the data in ref.
[35].

Another direction of research where we could imagine that the results of this
paper might be beneficial are investigations of the spinor–vector duality on
smooth geometries. The spinor–vector duality is a symmetry akin to mirror
symmetry in the space of $(2,0)$ heterotic–string compactifications [46, 47,
48, 49, 50, 51, 52]. It arises due to the exchange of Wilson line moduli
rather than moduli of the internal compactified space and operates separately
on each of the twisted sectors in $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$
orbifolds [47, 50] and hence can be studied in vacua with a single
$\mathbbm{Z}_{2}$ twist of the internal space and an additional freely acting
$\mathbbm{Z}_{2}$ that operates as a Wilson line. Similar to mirror symmetry
[53, 54], which can be realised as an exchange of discrete torsion in a
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ toroidal orbifold [55], the
spinor–vector duality can be realised in terms of certain generalised discrete
torsions [46, 49, 50]. Moreover, like the imprint of mirror symmetry on
Calabi–Yau manifolds, the spinor–vector duality imprints can be explored in
the effective field theory limit of smooth compactifications as was
investigated in refs. [52] and [51] in six and five dimensions, respectively.
To take these studies further to the resolutions of
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds the present work is likely to
be instrumental as it allows to study the required resolutions in general and
not be hampered by focusing on a particular triangulation from the very
beginning.

##  Appendix A Some Chern Class Identities

The total Chern class $c$ of a matrix $A$ is given by

$\displaystyle c=\det(\text{\small 1}\text{1}+A)~{}.$ (A.1)

If $x_{i}$ denote the eigenvalues of $A$, then the total Chern class can be
written as

$\displaystyle c=\prod(1+x_{i})~{}.$ (A.2)

Expanding the total Chern class $c$ in different powers of $x_{i}$ can be used
to define the first, second and third Chern classes, $c_{1}$, $c_{2}$ and
$c_{3}$, respectively. These Chern classes may be expressed as

$\displaystyle c_{1}=\sum_{i}x_{i}~{},\quad
c_{2}=\sum_{i<j}x_{i}x_{j}~{},\quad c_{3}=\sum_{i<j<k}x_{i}x_{j}x_{k}~{}.$
(A.3)