LTH-1318

The fate of discrete torsion on resolved heterotic
$\boldsymbol{\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ orbifolds using (0,2)
GLSMs

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

### Abstract

This paper aims to shed light on what becomes of discrete torsion within
heterotic orbifolds when they are resolved to smooth geometries. Gauged Linear
Sigma Models (GLSMs) possessing (0,2) worldsheet supersymmetry are employed as
interpolations between them. This question is addressed for resolutions of the
non–compact $\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ and the
compact $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds to keep track
of local and global aspects. The GLSMs associated with the non–compact
orbifold with or without torsion are to a large extent equivalent: only when
expressed in the same superfield basis, a field redefinition anomaly arises
among them, which in the orbifold limit reproduces the discrete torsion
phases. Previously unknown, novel resolution GLSMs for
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ are constructed. The GLSM
associated with the torsional compact orbifold suffers from mixed gauge
anomalies, which need to be cancelled by appropriate logarithmic superfield
dependent FI–terms on the worldsheet, signalling $H$–flux due to NS5–branes
supported at the exceptional cycles.

###### Contents

  1. 1 Introduction
    1. 1.1 Main paper objectives
    2. 1.2 Paper organisation
  2. 2 Properties of $\boldsymbol{\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ orbifolds
    1. 2.1 Orbifold twists and gauge shift vectors
    2. 2.2 Discrete torsion phase
    3. 2.3 Orbifold spectra with(out) torsion
  3. 3 Geometries and bundles from $\boldsymbol{(0,2)}$ gauged (linear) sigma models
    1. 3.1 (0,2) Superfields
    2. 3.2 Emergent effective geometry
    3. 3.3 Emergent effective vector bundle
    4. 3.4 The (2,2) locus
    5. 3.5 Worldsheet instantons and flux quantisation
    6. 3.6 Anomaly consistency conditions
    7. 3.7 Worldsheet Green–Schwarz mechanism: Torsion and NS5–branes
    8. 3.8 Orbifold resolution GLSMs
  4. 4 Non–compact $\boldsymbol{\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ resolution GLSMs
    1. 4.1 Geometrical interpretation
    2. 4.2 Pairs of GLSMs associated to torsion related orbifolds
  5. 5 GLSMs for resolutions of $\boldsymbol{T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$
    1. 5.1 Construction of resolution GLSMs for compact ${\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ orbifolds
    2. 5.2 Two–tori GLSM with ${\mathbbm{Z}_{2}}$ symmetries
    3. 5.3 Maximal full resolution GLSM
    4. 5.4 Minimal full resolution GLSM
    5. 5.5 Gauge background on the minimal full resolution of the non–torsional orbifold
    6. 5.6 Gauge background on the minimal full resolution of the torsional orbifold
  6. 6 Conclusions
  7. A Elements of $\boldsymbol{(0,2)}$ sigma models
    1. A.1 $\boldsymbol{(0,2)}$ superspace
    2. A.2 $\boldsymbol{(0,2)}$ superfields
    3. A.3 Super conformal transformations and scaling dimensions
    4. A.4 Scale invariant matter actions
    5. A.5 None scale invariant actions
    6. A.6 (0,2) non–linear sigma models
  8. B Anomalies in two dimensional GLSMs
    1. B.1 Chiral anomaly
    2. B.2 Super gauge anomalies
  9. C Charge matrices

##  1 Introduction

Given the current astrophysical, collider and cosmological data, the standard
cosmological and particle physics models may provide viable parameterisation
of all observational data up the Planck scale. Obtaining further insight into
the basic origin of these parameters necessitates the synthesis of the gauge
quantum field theories with gravity. The most developed contemporary
mathematical framework to explore the gauge–gravity unification is string
theory.

The consistency conditions of string theory require the existence of a finite
number of degrees of freedom beyond those observed in contemporary
experiments. These degrees of freedom may appear in different guises. They may
be interpreted as extra target space dimensions with vector bundles, or as two
dimensional fields propagating on the string worldsheet. Ultimately, the
different representations may describe the same physical objects and it is
vital to extract the physical characteristics, irrespective of the particular
language used.

The $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds of six dimensional
toroidal compactifications are among the most studied string constructions to
date. They have been used to derive phenomenological string models and to
study how the parameters of the Standard Model may be derived from string
theory, using their free fermionic [1, 2, 3, 4, 5, 6] and orbifold [7, 8, 9,
10] realisations, and their smooth resolutions [11]. Other phenomenological
interesting smooth compactifications have been investigated in e.g. [12, 13,
14, 15, 16, 17, 18]. These phenomenological studies encompass supersymmetric
and non–supersymmetric string vacua [19, 20, 21, 22, 23] with symmetric and
asymmetric boundary conditions [24, 25] and the
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolding can enable the fixing of
all of the untwisted geometrical moduli [26].

The relation between worldsheet string models and their effective field theory
geometrical limits presently occupies much of the discourse in string
phenomenology in the form of the so–called “swampland program”. This program
aims to address the question when does an effective field theory model of
quantum gravity have an ultra–violet complete embedding in string theory, and
hence can be viewed as a bottom–up approach to the study of this relation. An
alternative top–down approach seeks to find the imprint of worldsheet
symmetries in the effective field theory target space models. Notable examples
of this approach include mirror symmetry [27] and spinor–vector duality [28,
29, 30, 31, 32, 33, 34].

The worldsheet constructions of string vacua consist of a perturbative
expansion in string amplitudes. They are constrained to preserve the classical
symmetries of reparameterisation and Weyl invariance, i.e. they are invariant
under modular transformations of the worldsheet parameter, and are encoded in
the one–loop partition function. The requirement of modular invariance entails
that the partition function is a sum over different sectors that combine to
form a modular invariant object. While most of the signs in this sum are
dictated by modular invariance, some other may be arbitrary and play a vital
role in determining the physical properties of the string models. In
particular, the origins of mirror symmetry and spinor–vector dualities may be
traced back to (generalised) discrete torsions. Discrete torsions typically
arise in the worldsheet constructions as a result of multiple modding out
operations. For example, we may mod out by several twists of the internal
dimensions; or by identifications by translations of points in the internal
compactified space; or we may combine actions of these shifts and twists.
Additionally, in the heterotic–string these may be combined with an action on
the gauge bundles, which results in a reduction of the gauge symmetry. The
spinor–vector duality, for example, arises due to the action of Wilson lines
on the gauge bundles.

The interpretation of (generalised) discrete torsions from the geometrical
effective field theory point of view is obscured as one does not have an exact
partition function description in which these discrete torsion phases are
present. It is therefore of interest to elucidate the manifestation of the
discrete torsions in the effective field theory limit. If there is a discrete
action on the target space, this can be accompanied with discrete torsion in
the form of some non–trivial action on the $B$–field [35, 36, 37]. However, in
this paper we however wondered what happens to the discrete torsion between
orbifold twists, if one fully resolves the orbifold so that no discrete
symmetries are left on the smooth target space. We aim to investigate this
manifestation using the Gauged Linear Sigma Model (GLSM) representation of
string vacua. GLSMs provide a particularly appealing framework to explore this
question, as they provide a single framework in which one can interpolate
between different regimes, like the singular orbifold limit and smooth
compactifications.

###  1.1 Main paper objectives

One of the central objectives of this paper is to systematically study the
discrete torsion phases in smooth string compactifications using the GLSM
language to bridge the gap between the orbifold CFT formulations and the
effective field theory descriptions for smooth target spaces. Concretely, this
program is considered for $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds of
free CFTs where the discrete torsion is known as the Vafa–Witten phase.

First resolutions of the non–compact
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold are considered
in the GLSM language. To have a particular simple context the focus is on line
bundle resolutions generated by physical blowup modes, twisted string states
without oscillator excitations. The precise identification of such resolution
GLSMs from this data was worked out in the past [38]. Since only the standard
embedding bundles would allow for a $(2,2)$ worldsheet description, the
incorporation of line bundles requires a $(0,2)$ GLSM language. For both
orbifold CFTs without and with torsion the corresponding resolution GLSMs are
constructed. In order to compare them at the Lagrangian level on the
worldsheet, one has to ensure that one uses the same superfield basis. (In the
path integral formulation it only make sense to compare theories using their
classical actions when the same integration field variables are employed.)
Hence, as the charges of the superfields in the GLSMs of the non–torsion and
the torsion orbifolds do not agree, superfield redefinitions are needed before
this comparison is possible. As a cross check of the applied methods the GLSMs
are considered in the deep orbifold regime to investigate how the torsion
phases may be recovered.

The study of compact models with torsion is particularly intriguing since
certain fluxes cannot be pushed to infinity and thereby out of the realm of
the used description. Hence, the second part of the paper focusses on
resolutions of compact $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds
without or with discrete torsion switched on. Before, a careful study of the
imprints of discrete torsion can be investigated, first GLSMs for resolutions
of $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ have to be set up. In the past
GLSMs for compact orbifold resolutions were worked out in [39]. Even though
the necessary techniques were developed there, GLSM resolutions of
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ were not considered explicitly.
Moreover, that paper used the $(2,2)$ language throughout. However, to match
up with the considerations of the non–compact cases, it is necessary to
describe resolutions of $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ here
using $(0,2)$ GLSM terminology. Having fixed the geometrical aspects in the
GLSM description, similar blowups are considered induced by non–oscillator
twisted states as in the non–compact context. However, for the compact GLSM
resolutions this leads to more complicated bundle constructions which take
features of standard embedding bundles on the underlying torus cycles mixed
with line bundles on the resolved $\mathbbm{Z}_{2}$–singularities. With all
this in place, the resolution GLSMs of the compact orbifolds without and with
torsion can be investigated.

###  1.2 Paper organisation

The main part of the paper starts with a short review in Section 2 of some
features of $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds to provide the
necessary foundation for the subsequent investigations. Section 3 summarises
some essential prerequisites about $(0,2)$ GLSMs without which the remainder
of this manuscript might be a bit hard to follow for non–experts. Further
technical details on this topic are diverted to Appendix A. Next, Section 4
focusses on GLSM resolutions of non–compact
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ without and with
torsion. Some properties described there rely on charge matrices which are
collected in Appendix C not to interrupt the main flow of this section.
Section 5 repeats these exercises for compact
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ GLSM resolutions focussing on the
additional features and complications that compactness brings. Appendix B
derives gauge anomalies in two dimensions and provide $(0,2)$ superspace
expressions for them which are used frequently in Sections 4 and 5.

##  2 Properties of $\boldsymbol{\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$
orbifolds

The purpose of the present section is to recall some crucial information about
heterotic $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds to understand their
resolutions using GLSM methods that are laid out in subsequent sections.
Hence, it does not aim to give a complete review of heterotic orbifolds (for
more comprehensive discussions see e.g. [40, 41, 42, 43, 44]). In particular,
properties of $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds may be found in
e.g. [45, 46, 47, 48, 10]. A crucial feature of
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ is that they may posses discrete
torsion [49, 27]. As is recalled here this feature determines which twisted
states survive the orbifold projections.

###  2.1 Orbifold twists and gauge shift vectors

The bosonic description of the $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold
starts with the introduction of two twist vectors

$\displaystyle
v_{1}=\big{(}0,0,\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$}\big{)}~{},\qquad
v_{2}=\big{(}0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$}\big{)}~{},$
(2.1)

which act on the complex coordinate fields $z_{u}$ with $u=0,1,2,3$. Here
$z_{0}$ denotes the four dimensional non–compact directions in light–cone
gauge. (Since the main interest is on the internal coordinates, $u$ is taken
to label the internal coordinates and then runs over $u=1,2,3$ only.) Thus the
first entries of the twist vectors indicate that the twists act trivially on
the four dimensional Minkowski space. For the non–compact orbifold
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ the coordinates
$z_{u}\in\mathbbm{C}$ parametrise three complex planes. While for the compact
orbifold $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ they parametrise the
three underlying two–tori $T^{2}$. An arbitrary element $g$ of the
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold point group then corresponds
to the twist vector

$\displaystyle v_{g}=t_{1}\,v_{1}+t_{2}\,v_{2}~{},$ (2.2)

where $t_{1},t_{2}=0,1$ label its four elements.

To complete the definition of the orbifold actions gauge shift vectors have to
be given. In the orbifold standard embedding the gauge shift vectors are taken
to be equal to these twist vectors augmented with the appropriate number of
zero entries:

$\displaystyle
V_{1}=\big{(}0,\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}~{},\qquad
V_{2}=\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}~{},$
(2.3)

and define the gauge shift embedding

$\displaystyle V_{g}=t_{1}\,V_{1}+t_{2}\,V_{2}~{},$ (2.4)

for each of the four orbifold point group elements. As the notation of the
shift vectors suggest, this paper uses the $E_{8}\times E_{8}$ heterotic
string for concreteness. In addition a heterotic orbifold might feature a
number of discrete Wilson lines. In this paper the consequences of them are
not considered.

###  2.2 Discrete torsion phase

At the one loop level it is conventional to distinguish between constructing
elements $g,h$ of the orbifold group, which define the different orbifold
sectors of the theory, and the projecting elements $g^{\prime},h^{\prime}$,
which implement the appropriate orbifold projections. Hence, on the one loop
worldsheet torus a heterotic orbifold model is defined uniquely by the
properties introduced above up to a possible discrete torsion phase [49, 27]

$\displaystyle\Phi^{\\!\times}{}^{t_{1},t_{2}}_{t_{1}^{\prime},t_{2}^{\prime}}=e^{\pi
i\,\varepsilon^{\\!\times}(t_{1}^{\phantom{\prime}}t_{2}^{\prime}-t_{2}^{\phantom{\prime}}t_{1}^{\prime})}~{}$
(2.5)

in its one loop partition function [49]. The possible torsion phase leads to a
specific interplay between the constructing and projecting orbifold group
elements. Clearly, if $\varepsilon^{\\!\times}=0$ there is no torsion as the
torsion phase is equal to unity, but if $\varepsilon^{\\!\times}=1$ the model
possesses discrete torsion as the phase is non–trivial.

An alternative equivalent way that discrete torsion can be introduced is by
so–called brother models, i.e. models with gauge shift vectors that differ
from the original ones by appropriate lattice vectors [50]. In particular, for
the model (2.3) the brother model has gauge shift vectors

$\displaystyle
V^{\\!\times}_{1}=-V_{1}=\big{(}0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}~{},\qquad
V^{\\!\times}_{2}=-V_{2}=\big{(}\mbox{$\frac{1}{2}$},0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}~{},$
(2.6)

so that their differences are indeed lattice vectors.

###  2.3 Orbifold spectra with(out) torsion

Any state in the orbifold spectrum may be characterised by two shifted momenta

$\displaystyle p_{g}=p+v_{g}~{},\qquad P_{g}=P+V_{g}~{},$ (2.7)

where the vector $p$ is an element of the lattice $V_{4}\oplus S_{4}$ and $P$
of $(O_{8}\oplus S_{8})\otimes(O_{8}\oplus S_{8})$. The shifted momenta of
level matched massless states are subject to the following two conditions

$\displaystyle\frac{1}{2}\,p_{g}^{2}=\frac{1}{2}-\delta
c_{g}~{},\qquad\frac{1}{2}\,P_{g}^{2}=1-\delta
c_{g}-\omega_{g}\cdot\widetilde{N}_{g}-{\bar{\omega}}_{g}\cdot\overline{\widetilde{N}}_{g}~{},$
(2.8)

where the orbifold vacuum shift

$\displaystyle\delta c_{g}=\frac{1}{2}\sum_{u}\omega_{g,u}(1-\omega_{g,u})$
(2.9)

is defined in terms of $\omega_{g,u}\equiv(v_{g})_{u}$ and
${\bar{\omega}}_{g,u}\equiv-(v_{g})_{u}$ which satisfy the inequalities:
$0<\omega_{g,u},{\bar{\omega}}_{g,u}\leq 1$. Finally,
$(\widetilde{N}_{g})_{u}$ and $(\overline{\widetilde{N}}_{g})_{u}$ are the
number operators that count the number of right–moving oscillators act on the
state. Only the states that survive the orbifold projection conditions,

$\displaystyle P_{g}\cdot V_{g^{\prime}}-p_{g}\cdot
v_{g^{\prime}}\equiv\frac{1}{2}\big{(}V_{g}\cdot V_{g^{\prime}}-v_{g}\cdot
v_{g^{\prime}}\big{)}+\big{(}\overline{\widetilde{N}}_{g}-\widetilde{N}_{g}\big{)}\cdot
v_{g^{\prime}}+\frac{\varepsilon^{\\!\times}}{2}\big{(}t_{1}^{\phantom{\prime}}t_{2}^{\prime}-t_{2}^{\phantom{\prime}}t_{1}^{\prime}\big{)}~{},$
(2.10)

are part of the physical orbifold spectrum. The last term in these projection
conditions encodes the consequences of discrete torsion on the massless
spectrum. Consequently, the discrete torsion phases only affect the twisted
sectors. The resulting orbifold spectrum is conventionally divided in a number
of sectors:

#### Untwisted sector

The untwisted sector is identified by $(t_{1},t_{2})=(0,0)$. This sector
corresponds to so–called bulk states which live everywhere within the internal
geometry. It contains the metric, the anti–symmetric tensor and the dilaton
degrees of freedom as well as the target space gauge fields and all their
superpartners in ten dimensions. The non–Abelian unbroken gauge group in four
dimensions is $E_{6}\times E_{8}$. In addition, there are three copies of
charged matter in the $(27)+(\overline{27})$ of $E_{6}$ independently of
whether torsion is switched on or not.

#### Twisted sectors

There are three twisted sectors with $t=(t_{1},t_{2}):1=(1,0),2=(0,1)$ and
$3=(1,1)$ which only posses ${\cal N}=1$ supersymmetry in six
dimensions444Also sometimes referred to as ${\cal N}=2$ sectors from the four
dimensional point of view.: On the non–compact orbifold
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ the corresponding
twisted states are localised at the three complex codimension two
singularities of the three non–trivial orbifold twists. Each twisted sector is
supported on 16 fixed two–tori within the compact orbifold
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$. Half of these states are
projected out by the orbifold action of the second orbifold element. Which
half depends on whether torsion is switched on, see Table 1 which gives the
twisted states without twisted oscillator excitations.

$\begin{array}[]{|c||c|c|c|c|}\hline\cr\text{Sector}&\text{Shifted momentum
}P_{g}&\text{Repr.}&\varepsilon^{\\!\times}=0&\varepsilon^{\\!\times}=1\\\
\hline\cr\hline\cr&\big{(}1,{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}&(1)&\hbox{\multirowsetup\text{in}}&\hbox{\multirowsetup\text{out}}\\\
\cline{2-3}\cr
1=&\big{(}{\,\mbox{-}}1,{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$}0^{5}\big{)}\big{(}0^{8}\big{)};\big{(}0,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},\underline{\pm
1,0^{4}}\big{)}\big{(}0^{8}\big{)};\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,0,\underline{{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{e},\mbox{$\frac{1}{2}$}^{5-e}}\big{)}\big{(}0^{8}\big{)}&(27)&&\\\
\cline{2-5}\cr(1,0)&\big{(}{\,\mbox{-}}1,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}&(\overline{1})&\hbox{\multirowsetup\text{out}}&\hbox{\multirowsetup\text{in}}\\\
\cline{2-3}\cr&\big{(}1,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)};\big{(}0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},\underline{\pm
1,0^{4}}\big{)}\big{(}0^{8}\big{)};\big{(}\mbox{$\frac{1}{2}$},0,0,\underline{{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{o},\mbox{$\frac{1}{2}$}^{5-o}}\big{)}\big{(}0^{8}\big{)}&(\overline{27})&&\\\
\hline\cr\hline\cr&\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},1,{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}&(1)&\hbox{\multirowsetup\text{in}}&\hbox{\multirowsetup\text{out}}\\\
\cline{2-3}\cr
2=&\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}1,{\,\mbox{-}}\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)};\big{(}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$},\underline{\pm
1,0^{4}}\big{)}\big{(}0^{8}\big{)};\big{(}0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,\underline{{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{e},\mbox{$\frac{1}{2}$}^{5-e}}\big{)}\big{(}0^{8}\big{)}&(27)&&\\\
\cline{2-5}\cr(0,1)&\big{(}\mbox{$\frac{1}{2}$},{\,\mbox{-}}1,\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)}&(\overline{1})&\hbox{\multirowsetup\text{out}}&\hbox{\multirowsetup\text{in}}\\\
\cline{2-3}\cr&\big{(}\mbox{$\frac{1}{2}$},1,\mbox{$\frac{1}{2}$},0^{5}\big{)}\big{(}0^{8}\big{)};\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},\underline{\pm
1,0^{4}}\big{)}\big{(}0^{8}\big{)};\big{(}0,\mbox{$\frac{1}{2}$},0,\underline{{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{o},\mbox{$\frac{1}{2}$}^{5-o}}\big{)}\big{(}0^{8}\big{)}&(\overline{27})&&\\\
\hline\cr\hline\cr&\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},1,0^{5}\big{)}\big{(}0^{0}\big{)}&(1)&\hbox{\multirowsetup\text{in}}&\hbox{\multirowsetup\text{out}}\\\
\cline{2-3}\cr
3=&\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}1,0^{5}\big{)}\big{(}0^{8}\big{)};\big{(}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0,\underline{\pm
1,0^{4}}\big{)}\big{(}0^{8}\big{)};\big{(}0,0,{\,\mbox{-}}\mbox{$\frac{1}{2}$},\underline{{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{e},\mbox{$\frac{1}{2}$}^{5-e}}\big{)}\big{(}0^{8}\big{)}&(27)&&\\\
\cline{2-5}\cr(1,1)&\big{(}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}1,0^{5}\big{)}\big{(}0^{8}\big{)}&(\overline{1})&\hbox{\multirowsetup\text{out}}&\hbox{\multirowsetup\text{in}}\\\
\cline{2-3}\cr&\big{(}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},1,0^{5}\big{)}\big{(}0^{8}\big{)};\big{(}{\,\mbox{-}}\mbox{$\frac{1}{2}$},{\,\mbox{-}}\mbox{$\frac{1}{2}$},0,\underline{\pm
1,0^{4}}\big{)}\big{(}0^{8}\big{)};\big{(}0,0,\mbox{$\frac{1}{2}$},\underline{{\,\mbox{-}}\mbox{$\frac{1}{2}$}^{o},\mbox{$\frac{1}{2}$}^{5-o}}\big{)}\big{(}0^{8}\big{)}&(\overline{27})&&\\\
\hline\cr\end{array}$ Table 1:  This table lists the twisted sector spectra
obtained from non–oscillator excitation states and indicates whether they are
in the physical spectrum without or with torsion, $\epsilon^{\\!\times}=0$ or
$1$, respectively.

##  3 Geometries and bundles from $\boldsymbol{(0,2)}$ gauged (linear) sigma
models

$\begin{array}[]{|c||c|c|c
||c|c||c|c||c|c|c||c|c|}\hline\cr\text{Superfield}&\partial&\bar{\partial}&D_{+}&\Phi^{a}&\Lambda^{m}&\Psi^{A}&\Gamma^{M}&V_{i}&A_{i}&F_{i}&\Sigma_{I}&\Upsilon_{I}\\\
\hline\cr\text{Phys.\,Comp.}&&&&(z^{a},\phi^{a})&(\lambda^{m},h^{m})&(y^{A},\psi^{A})&(\gamma^{M},f^{M})&\lx@intercol\hfil(A_{\sigma}^{i},A_{\bar{\sigma}}^{i},\varphi^{i},D^{i})\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil(s^{I},\chi^{I})\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\\#&&&&\mathrm{N}_{\Phi}&\mathrm{N}_{\Lambda}&\mathrm{N}_{\Psi}&\mathrm{N}_{\Gamma}&\lx@intercol\hfil\mathrm{N}_{V}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\mathrm{N}_{\Sigma}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr\hline\cr{\cal
L}&0&1&0&0&\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}&0&1&1&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
\hline\cr{\cal
R}&1&0&\mbox{$\frac{1}{2}$}&0&0&0&0&0&0&\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
\hline\cr\text{R}&0&0&-1&0&0&1&1&0&0&1&0&1\\\ \hline\cr\hline\cr{\cal
Q}&0&0&0&(q_{i})^{a}&(Q_{i})^{m}&(\textsf{q}_{i})^{A}&(\textsf{Q}_{i})^{M}&\text{n.l.}&\text{n.l.}&0&0&0\\\
\hline\cr\end{array}$ Table 2:  This table specifies the left– and right–Weyl
dimensions, ${\cal L}$ and ${\cal R}$, the R–charge and the gauge charges
${\cal Q}_{i}$ of the operators $\partial,\bar{\partial},D_{\pm}$ and the
superfields which may be used in a (0,2) GLSM. The physical components of
these multiplets are indicated as well as the indices that label them; the
third line gives the total number of these multiplets.

###  3.1 (0,2) Superfields

Two dimensional theories with $(0,2)$ supersymmetry admit a number of
different types of superfields (or multiplets). Appendix A gives a short
review of $(0,2)$ superfields on superspace and sets notations and conventions
used in this work. Gauged sigma models are a special class of $(0,2)$ theories
with bosonic and possibly also fermionic gaugings. The superfields used in
this work are summarised in Table 2 and the labels used to enumerate them are
indicated there. In addition, their gauge charges, left– and right–Weyl
dimensions and R–charges (defined in Appendix A.3) are given.

The most important matter superfields are chiral and chiral Fermi multiplets.
A chiral multiplet $\Phi=(z,\phi)$ contain a complex scalar $z$ and a
right–moving fermion $\phi$. A chiral Fermi multiplet $\Lambda=(\lambda,h)$
consists of a left–moving fermion $\lambda$ and an auxiliary scalar field $h$.
In addition, there are chiral multiplets $\Psi=(y,\psi)$ and chiral Fermi
multlplets $\Gamma=(\gamma,f)$. The distinction between these chiral and
chiral Fermi superfields is made by their R–symmetry charge: $\Phi$ and
$\Lambda$ are neutral while $\Psi$ and $\Gamma$ carry charge 1. The last line
of this table gives the gauge charges and dictates the super gauge
transformations of these matter superfields.

For the corresponding bosonic gaugings vector multiplets have to be introduced
consisting of two real bosonic superfields $V$ and $A$ from which gauge
invariant super field strengths $F$ can be constructed

$\displaystyle
F=-\mbox{$\frac{1}{2}$}{\overline{D}}_{+}\big{(}A-i\bar{\partial}V\big{)}~{}.$
(3.1)

The physical components of these multiplets are the gauge field $A_{\sigma}$,
$A_{\bar{\sigma}}$ with field strength
$F_{\sigma{\bar{\sigma}}}=\partial_{\sigma}A_{\bar{\sigma}}-\partial_{\bar{\sigma}}A_{\sigma}$
and a right–moving fermion $\varphi$ and a real auxiliary field $D$.

On the chiral Fermi multiplets fermionic gauge transformations

$\displaystyle\Lambda\rightarrow\Lambda+U(\Phi)\\!\cdot\\!\Xi~{},\qquad\Gamma\rightarrow\Gamma+\Psi\,W(\Phi)\\!\cdot\\!\Xi$
(3.2)

may act with chiral Fermi super gauge parameters. To obtain invariant action
under these transformation, Fermi gauge multiplets $\Sigma$ need to be
introduced with super field strengths

$\displaystyle\Upsilon={\overline{D}}_{+}\Sigma~{}.$ (3.3)

Their physical components are complex scalars $s$ and left–moving fermions
$\chi$.

A few comments are in order. The theories that are studied here do not define
proper string theories as their worldsheet actions are not fully conformal. In
particular, dynamical gauge fields on the worldsheet are not scale invariant
as their gauge coupling is dimensionfull. Nevertheless it is useful to use
characterisations, like the left– and right–moving Weyl dimensions, as in the
scale invariant limit the corresponding superconformal symmetries are
recovered. Moreover, the “linear” in GLSMs signifies that only kinetic terms
quadratic in the fields are considered, while in non–linear sigma models this
restriction is lifted for chiral superfields.

The main reason why GLSMs are of interest for string theory is that they can
provide interesting insights in how geometries and vector bundles on them can
arise:

###  3.2 Emergent effective geometry

The scalar part of GLSMs can be associated to target space geometries like
weighted projective spaces, complete intersection Calabi–Yaus and many
generalisations of these as was realised by the pioneering paper [51]. The
scalar components $z$ of the chiral multiplets $\Phi$ can be interpreted as
the homogeneous coordinates of projected spaces, where the
$\mathbbm{C}^{*}$–scalings are encoded by the scalar part of the super gauge
transformations:

$\displaystyle z\rightarrow
e^{q_{i}\cdot\theta}\,z~{},\qquad\theta=\mbox{$\frac{1}{2}$}\,a-i\,\alpha\in\mathbbm{C}^{N_{V}}~{}.$
(3.4)

In the Wess–Zumino gauge the sizes of these projective spaces are set by the
D–term equations

$\displaystyle\sum_{a}(q_{i})^{a}|z^{a}|^{2}=r_{i}~{},$ (3.5)

for each $i=1,\ldots,N_{V}$. (In principle there is a second sum over the
scalars $y^{A}$ here, but they are typically all forced to zero as discussed
below.) Here the parameters $r$ are the real parts of the Fayet–Iliopoulos
(FI) coefficients $\rho(\Phi)$ which define superpotentials involving the
super gauge field strengths

$\displaystyle
W_{\text{FI}}=\rho(\Phi)\\!\cdot\\!F~{},\qquad\rho(z)=\mbox{$\frac{1}{2}$}\,r+i\,\beta\in\mathbbm{C}^{N_{V}}~{}.$
(3.6)

This is gauge invariant if the functions $\rho(\Phi)$ are neutral. The target
space interpretation of $r$ are moduli, that set the radii of certain cycles,
and $\beta$ may be interpreted as axions in the effective geometry.

String backgrounds, like Calabi–Yaus, are often defined as hypersurfaces in
such projected spaces. In the GLSM language this can be encoded in a $(0,2)$
superpotential

$\displaystyle P_{\text{geom}}=\Gamma\,P(\Phi)~{}.$ (3.7)

In the conformal limit, the scalar components of the algebraic equations of
motion of chiral Fermi superfields $\Gamma^{M}$ lead to F–term equations:

$\displaystyle P_{M}(z)=0~{},$ (3.8)

for $M=1,\ldots,N_{\Gamma}$, which precisely cut out such hypersurfaces.
Consequently, the dimension of the resulting target space manifold ${\cal M}$
equals:

$\displaystyle\text{dim}_{\mathbbm{C}}({\cal
M})=\mathrm{N}_{\Phi}-\mathrm{N}_{V}-\mathrm{N}_{\Gamma}~{}.$ (3.9)

This should be equal to $2$ or $3$ if one only considers the internal manifold
of complex dimension $2$ or $3$; or $4$ if the complete spacetime in
light–cone gauge is described by the GLSM.

In addition, the GLSM description can be used to determine an atlas of
coordinate patches: in a given phase one or multiplet set(s) of scalar fields
are necessarily non–zero. Hence, by analysing the combined D–term and F–term
equations, (3.5) and (3.8), all the coordinate patches within a phase of the
GLSM can be determined.

###  3.3 Emergent effective vector bundle

The part of $(0,2)$ GLSMs that involve the chiral Fermi multiplets can be
interpreted as vector bundles (or as sheafs if they are not fully regular)
[51, 52, 53]. The fermionic components $\lambda$ of the Fermi multiplets
$\Lambda$ are line bundle sections on this manifold as their
$\mathbbm{C}^{*}$–scalings read

$\displaystyle\lambda\rightarrow e^{Q_{i}\cdot\theta}\,\lambda~{}.$ (3.10)

If there are no fermionic super gauge transformations and no chiral
superfields $\Psi$ in the model, then the target space gauge background is
simply a collection of line bundles.

However, in general, they describe a more complicated vector bundle ${\cal V}$
which is derived from a complex (generalisation of a monad construction),
since they have to satisfy the constraints

$\displaystyle M(z)\lambda=0~{},$ (3.11)

due to the lowest components of the algebraic equations of motion of $\Psi$
that follows from the bundle superpotential

$\displaystyle P_{\text{bundle}}=\Psi\,M(\Phi)\,\Lambda$ (3.12)

and are subject to gauge transformations

$\displaystyle\lambda\rightarrow\lambda+U(z)\\!\cdot\\!\xi~{},$ (3.13)

which are the lowest components of the fermionic super gauge transformations
(3.2). Combined the equations (3.11) and (3.13) imply that a vector bundle
${\cal V}=\text{Ker}(U)/\text{Im}(M)$ is constructed from the complex

$\displaystyle 0\rightarrow{\cal
O}^{\mathrm{N}_{\Sigma}}\stackrel{{\scriptstyle
U}}{{\longrightarrow}}\bigoplus\limits^{\widetilde{\mathrm{N}}_{\Lambda}}_{m=1}{\cal
O}(Q^{m})\stackrel{{\scriptstyle
M}}{{\longrightarrow}}\bigoplus\limits^{\mathrm{N}_{\Psi}}_{A=1}{\cal
O}(-\textsf{q}^{A})\rightarrow 0~{}.$ (3.14)

Here $\widetilde{\mathrm{N}}_{\Lambda}\leq{\mathrm{N}}_{\Lambda}$ denotes the
number of interacting Fermi multiplets in the GLSM. (The numbers in the ${\cal
O}$s of such complexes are conventionally integers. But in the normalisations
used in this paper they might be fractional (like $1/2$), hence they should
then be multiplied by an appropriate common factor. In addition, the charges
of the chiral superfields $\Psi$ are negative in the conventions used in this
work and they set the degrees of the constraints (3.11) on the fermions.) The
dimensionality of the fibers of resulting vector bundle ${\cal V}$ is given by

$\displaystyle\text{dim}_{\mathbbm{C}}({\cal
V})=\widetilde{\mathrm{N}}_{\Lambda}-\mathrm{N}_{\Sigma}-\mathrm{N}_{\Psi}~{},$
(3.15)

provided that $M(z)$ and $U(z)$ have maximal ranks
$\mathrm{N}_{\Psi}\leq\mathrm{N}_{\Lambda}$ and
$\mathrm{N}_{\Sigma}\leq\mathrm{N}_{\Lambda}$, respectively [52]. (If this is
not everywhere the case, this indicates that there are singularities in the
bundle instead.) In order that this bundle can be embedded in the gauge
degrees of freedom of the heterotic string $\text{dim}_{\mathbbm{C}}({\cal
V})$ should less than eight so as to fit within an $E_{8}$–factor. (The bundle
might also fill up part of both $E_{8}$–factors, but then it has to split
accordingly.) Since the full rank of $E_{8}\times E_{8}$ is 16, the total
number of Fermi multiplets is given by
${\mathrm{N}}_{\Lambda}=16+\mathrm{N}_{\Sigma}+\mathrm{N}_{\Psi}$. Hence,
there are a number of spectator (non–interacting and neutral) Fermi multiplets
$\Lambda_{n}$,
$n=1,\ldots,{\mathrm{N}}_{\Lambda}-\widetilde{\mathrm{N}}_{\Lambda}$, which
lead to the unbroken gauge degrees of freedom in target space.

The superpotential (3.12) has another important consequence: If $M(z)$ has
maximal rank, the equations of motion of $\Lambda$ induced by the bundle
superpotential (3.12) imply that all $y^{A}=0$. This was implicitly assumed
when (3.5) were written down, since, in general, also contributions from the
scalars $y^{A}$ should be present in these equations.

The fermionic gauge transformations (3.2) only leaves the superpotentials
(3.7) and (3.12) combined inert when the following compatibility conditions
hold

$\displaystyle W_{A}{}^{IM}(\Phi)P_{M}(\Phi)+M_{Am}(\Phi)U^{mI}(\Phi)=0~{}.$
(3.16)

In general, it is not so straightforward to find functions such that these
conditions are fulfilled. However, when the superpotentials and the fermionic
gaugings are taken to lie on the $(2,2)$ locus discussed below, these
conditions are automatically satisfied.

###  3.4 The (2,2) locus

The interacting part of $(0,2)$ GLSMs (or at least the part that involves
fermionic gaugings) might possess a higher amount of supersymmetry. For this
to happen the (0,2) multiplets need to be able to pair up. This means in
particular, that there are the following relations between the number of
interacting multiplets:

$\displaystyle\widetilde{\mathrm{N}}_{\Lambda}=\mathrm{N}_{\Phi}~{},\qquad\mathrm{N}_{\Gamma}=\mathrm{N}_{\Psi}~{},\qquad\mathrm{N}_{\Sigma}=\mathrm{N}_{V}~{},$
(3.17)

This allows to identify various indices: $m=a$, $M=A$ and $I=i$; we use the
latter indices for each type of indices. Furthermore, the gauge charges of
chiral and Fermi multiplets need to line up:

$\displaystyle Q_{i}=q_{i}~{},\qquad\textsf{Q}_{i}=\textsf{q}_{i}~{}.$ (3.18)

When some of these relations are not satisfied it is impossible to deform the
interactions of the $(0,2)$ GLSM to become $(2,2)$. If this is possible, then
the $(0,2)$ theory is said to be on the $(2,2)$ locus.

On the $(2,2)$ locus of the space of $(0,2)$ GLSM, exact $(2,2)$ models
possess various interactions encoded in the various functions introduced that
need to be of a very specific form. The relations given here are subject to
specific normalizations; but the implied proportionalities are essential.
First of all, the functions $U(\Phi)$ and $W(\Phi)$ that describe the Fermi
gauge transformations now read

$\displaystyle U^{ai}(\Phi)=(q_{i})^{a}\Phi^{a}~{},\qquad
W_{A}{}^{iB}=(\textsf{q}_{i})^{B}\delta^{B}_{A}~{}.$ (3.19)

They are fully dictated by the index structure and the gauge charges
$(q_{i})^{a}$ and $(\textsf{q}_{i})^{A}$. The functions $M(\Phi)$ are
determined as the derivatives of $P(\Phi)$:

$\displaystyle M_{Aa}(\Phi)=P_{A,a}(\Phi)~{},$ (3.20)

where $F_{,a}(\Phi)$ denotes the partial derivative of $F(\Phi)$ with respect
to $\Phi^{a}$. Consequently, the invariance of the superpotential action under
fermionic gauge transformations (3.16) reduces to the gauge invariance of the
superpotential:

$\displaystyle(\textsf{q}_{i})^{A}P_{A}(\Phi)+P_{A,a}(\Phi)\Phi^{a}(q_{i})^{a}=0~{}.$
(3.21)

###  3.5 Worldsheet instantons and flux quantisation

It is possible that on the worldsheet non–trivial gauge configurations, like
instantons, are realised. The involved gauge fluxes need to be properly
quantised [54]:

$\displaystyle\sum_{j}\left(q_{j}\right)^{a}\int\frac{F_{E_{2}}^{j}}{2\pi}\in\mathbbm{Z}~{},\qquad\sum_{j}\left(\mathsf{q}_{j}\right)^{A}\int\frac{F_{E_{2}}^{j}}{2\pi}\in\mathbbm{Z}$
(3.22)

for all charged chiral superfields $\Phi^{a}$ and $\Psi^{A}$. Here the
subscript $E$ indicates that the gauge fluxes are computed in the Euclidean
theory.

###  3.6 Anomaly consistency conditions

On a GLSM there are a number of requirements in order that the theory is both
consistent as a quantum theory and that it is likely to have the right
properties in the conformal limit.

First of all, like any gauge theory, the GLSM has to be free of gauge
anomalies. With the gauge charges given in Table 2, this amounts to the
following conditions

$\displaystyle{\cal
A}_{ij}=-\sum_{a}(q_{i})^{a}(q_{j})^{a}-\sum_{A}(\textsf{q}_{i})^{A}(\textsf{q}_{j})^{A}+\sum_{m}(Q_{i})^{m}(Q_{j})^{m}+\sum_{M}(\textsf{Q}_{i})^{M}(\textsf{Q}_{j})^{M}\stackrel{{\scriptstyle!}}{{=}}0~{},$
(3.23)

for all $i,j=1,\ldots,N_{V}$. The signs in these equations are determined by
whether the fermions in the matter multiplets are right– or left–moving. For
$j=i$ this corresponds to pure and for $j\neq i$ to mixed gauge anomalies.

The left–, right–Weyl dimensions and R–charge correspond to bosonic parts of
super conformal symmetries in the scale invariant limit of the GLSM. For this
limit not to be obstructed the mixed left– and right–Weyl gauge anomalies
should vanish. In detail, from Table 2 it follows that the left–Weyl – gauge
anomalies vanish provided that

$\displaystyle\sum_{m}(Q_{i})^{m}+\sum_{M}(\textsf{Q}_{i})^{M}\stackrel{{\scriptstyle!}}{{=}}0~{},$
(3.24)

for all $i$, since the only charged superfields that carry ${\cal L}$–charge
are $\Lambda$ and $\Gamma$. These conditions can be summarised by the demand
that the sum of the charges of all chiral Fermi superfields need to vanish for
each gauge symmetry separately.

In addition, the charged right–moving fermions $\phi$ and $\gamma$ are
obtained by hitting chiral multiplets $\Phi$ and $\Psi$ with $D_{+}$, hence
the right–Weyl – gauge anomalies are absent when

$\displaystyle\sum_{A}(\textsf{q}_{i})^{a}+\sum_{A}(\textsf{q}_{i})^{A}\stackrel{{\scriptstyle!}}{{=}}0~{},$
(3.25)

for all $i$. Thus, these conditions say that the sum of the charges of all
chiral superfields need to vanish for each gauge symmetry separately. At the
same time these conditions ensure that the FI–parameters (3.6) do not
renormalise. If this isn’t the case, it would not be possible to interpret
them to set the scales of target space cycles as they would always run off to
zero or infinity.

Finally, the R–symmetry survives quantisation provided that

$\displaystyle\sum_{a}(q_{i})^{a}+\sum_{M}(\textsf{Q}_{i})^{M}\stackrel{{\scriptstyle!}}{{=}}0~{},$
(3.26)

for all $i$, since the right–moving fermions $\phi$ and the left–moving
fermions $\gamma$ have R–charges $-1$ and $+1$, respectively, and opposite
chiralities. When these equations are combined with (3.25), they can be stated
as the sum of the charges of the chiral Fermi superfields $\Gamma$ have to be
equal to that of the chiral superfields $\Psi$.

###  3.7 Worldsheet Green–Schwarz mechanism: Torsion and NS5–branes

When the gauge anomalies do not vanish, i.e. not all ${\cal A}_{ij}$ in (3.23)
vanish, the GLSM is anomalous. It is sometimes possible that certain field
dependent none gauge invariant FI–terms (3.6) are precisely able to cancel
these gauge anomalies [55, 56]. The FI–term coefficients $\rho(\Phi)$ then
need to transform as a shift under the anomalous gauge symmetries. This can be
viewed as a Green–Schwarz mechanism on the worldsheet and might have some far
reaching consequences for the geometry and the interpretation of the theory.

To understand how this comes about, note that in the naive conformal limit,
the kinetic terms of the vector multiplets $V,A$ can be set to zero and their
equations of motion become non–dynamical. In particular, the superfields $A$
appear linear in the actions of the chiral multiplets (A.26) and the FI–terms
(A.30), hence their equation of motion lead to superfield constraints:

$\displaystyle{\overline{\Phi}}\,e^{2q\cdot
V}\\!q_{i}\,\Phi=\rho_{i}(\Phi)+{\bar{\rho}}_{i}({\overline{\Phi}})~{}.$
(3.27)

Thus after enforcing the equations of motion of $A$, the vector multiplets $V$
become (implicit) functions of the chiral superfields $\Phi$ and their
conjugates ${\overline{\Phi}}$. In the Wess–Zumino gauge the lowest component
of these equations are the $D$–term constraints (3.5). However, in any gauge
from (3.27) it can be inferred which (scalars of the) chiral multiplets are
necessarily non–zero in a given phase with a certain choice of the
FI–parameters. Hence, a unitary gauge can be chosen such that all chiral
superfields, that are necessarily non–zero, are set to such values that the
solution for the vector superfields $V$ are all zero when all of the remaining
chiral superfields are vanishing555In the remainder of this paper for
presentational simplicity, the D–term equations (3.5) are given in the
Wess–Zumino gauge, while for the analysis of the torsional effects (3.27) the
unitary gauges, as defined here, are used implicitly..

Non–constant FI–terms (3.6) modify the target space geometry and generically
introduces torsion onto it in the form of non–vanishing $H$–flux [57, 54, 58].
Indeed, since by (3.27) the vector superfields $V$ become (implicit) functions
of the chiral multiplets. Inserting them in the kinetic terms of the chiral
multiplets shows that the torsion tensor, the three–form $H$,

$\displaystyle
H_{ab{\underline{c}}}\sim\rho_{,[a}\\!\cdot\\!V_{,b]{\underline{c}}}~{},$
(3.28)

is non–zero in general, see Appendix A.6 or ref. [56] for a derivation. (It
reads here in general, because if both $\rho_{i}$ and $V_{i}$ only depend on a
single chiral superfield this expression still anti–symmetrises to zero.)
Since typically, the GLSM only contains chiral superfields $\Phi$, that are
linearly charged under the gauge symmetries, the required FI–coefficients can
only be made by taking logarithms of combinations of them. As was argued in
[55, 56, 59] such logarithmic singularities can be viewed as the imprints of
non–perturbative physics in the form of NS5–branes on the worldsheet as the
target space exterior derivative of (3.28) lead to delta–function–like sources
in the Bianchi identity of the three–form666In addition, the inclusion of
log–dependent FI–terms may lead to a back reaction to the geometry [56, 59];
in this paper these consequences are not studied in detail..

###  3.8 Orbifold resolution GLSMs

Even though this section so far described properties of GLSMs in general, the
main focus of this work is on GLSMs which are associated to (toroidal)
orbifold resolutions. The study of resolution of singularities using $(0,2)$
GLSMs have a long history. Some pioneering works are [52, 53]. A GLSM orbifold
resolution construction has the advantage over other methods to match the
singular orbifold situations for which exact CFT descriptions exists with
smooth compactifications using effective field theory methods. Within a single
GLSM framework one has both access to the orbifold phase as well as completely
resolved (and potentially many other) phases. The trade off here is that a
GLSM is not (yet) a full blown CFT description.

A fully complete correspondence between orbifold CFTs and GLSMs does not
exists, but two methods have been uncovered in the past which apply to
partially overlapping situations:

  1. A

Twisted shifted momenta as (0,2) GLSM charges [38]:
As was recalled in Section 2.3, twisted states are uniquely identified by
their shifted right– and left–moving momenta (2.7). In particular, the right–
and left-moving shifted momenta of non–oscillator massless twisted states
automatically satisfy the pure anomaly cancellation conditions when they are
interpreted as GLSM gauge charges of chiral and chiral Fermi superfields,
respectively. In target space these configurations may have the interpretation
of line bundles on the resolved local singularities.

  2. B

(2,2) GLSMs for toroidal orbifold resolutions [39]:
Contrary, full global orbifold resolutions in the standard embedding can be
obtained in $(2,2)$ GLSMs. The underlying two–tori are described using
(variants of) the Weierstrass models. On some of their homogeneous coordinates
additional (exceptional) gaugings are implemented. For certain ranges of their
FI–parameters the fixed point structure of toroidal orbifolds, while for
others resolved compact Calabi–Yaus emerge.

In the next section method A is employed, while in Section 5 method A is
combined with a partial (0,2) reduction of method B for the case of
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold resolutions that were
not discussed in the literature before explicitly.

##  4 Non–compact
$\boldsymbol{\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ resolution
GLSMs

$\begin{array}[]{|c||c|c|c||c|c|c||c||c|c|c|}\hline\cr\text{Superfield}&\Phi_{1}&\Phi_{2}&\Phi_{3}&\Phi_{1}^{\prime}&\Phi_{2}^{\prime}&\Phi_{3}^{\prime}&\Lambda=(\Lambda^{1},\dots,\Lambda^{16})&{\Omega_{1}}&{\Omega_{2}}&{\Omega_{3}}\\\
\hline\cr\text{U(1)
charge}&z_{1}&z_{2}&z_{3}&x_{1}&x_{2}&x_{3}&\lambda=(\lambda^{1},\ldots,\lambda^{16})&{\omega_{1}}&{\omega_{2}}&{\omega_{3}}\\\
\hline\cr\hline\cr
E_{1}&0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&-1&0&0&Q_{1}=(Q_{1}^{1},\ldots,Q_{1}^{16})&1&0&0\\\
\hline\cr
E_{2}&\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}&0&-1&0&Q_{2}=(Q_{2}^{1},\ldots,Q_{2}^{16})&0&1&0\\\
\hline\cr
E_{3}&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0&0&0&-1&Q_{3}=(Q_{3}^{1},\ldots,Q_{3}^{16})&0&0&1\\\
\hline\cr\end{array}$ Table 3:  Superfield charge table for resolutions of the
non–compact $\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold.

This section focus on heterotic resolutions of the non–compact
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ using (0,2) GLSMs.
(Some ingredients of the present discussion are inspired by ref. [38].) The
three complex coordinates $z_{u}$, $u=1,2,3$, of $\mathbbm{C}^{3}$ augmented
with three exceptional coordinates $x_{r}$, $r=1,2,3$, to describe the
resolution. These coordinates become part of the chiral superfields $\Phi_{u}$
and $\Phi_{r}^{\prime}$ on which three $U(1)$ gauge symmetries $E_{r}$ act
according to the charge table 3. In this table the unit charged chiral
superfields $\Omega_{r}$ are composite, i.e. functions of the fundamental
superfields $\Phi_{u}$ and $\Phi_{r}^{\prime}$.

###  4.1 Geometrical interpretation

The analysis of the geometrical interpretation of this GLSM starts with
writing down the D–term equations

$\displaystyle\frac{1}{2}\left|z_{2}\right|^{2}+\frac{1}{2}\left|z_{3}\right|^{2}=b_{1}+\left|x_{1}\right|^{2}~{},$
(4.1a)
$\displaystyle\frac{1}{2}\left|z_{1}\right|^{2}+\frac{1}{2}\left|z_{3}\right|^{2}=b_{2}+\left|x_{2}\right|^{2}~{},$
(4.1b)
$\displaystyle\frac{1}{2}\left|z_{1}\right|^{2}+\frac{1}{2}\left|z_{2}\right|^{2}=b_{3}+\left|x_{3}\right|^{2}~{}.$
(4.1c)

Here the three parameters $b_{r}$ are the real parts of the three
FI–parameters $\rho_{r}$ associated with the three gaugings $E_{r}$ which are
assumed to be constant. An equivalent but useful representation of these
equations are obtained by adding two of them and subtracting the third:

$\displaystyle\left|z_{1}\right|^{2}+\left|x_{1}\right|^{2}=b_{2}+b_{3}-b_{1}+\left|x_{2}\right|^{2}+\left|x_{3}\right|^{2}~{},$
(4.2a)
$\displaystyle\left|z_{2}\right|^{2}+\left|x_{2}\right|^{2}=b_{1}+b_{3}-b_{2}+\left|x_{1}\right|^{2}+\left|x_{3}\right|^{2}~{},$
(4.2b)
$\displaystyle\left|z_{3}\right|^{2}+\left|x_{3}\right|^{2}=b_{1}+b_{2}-b_{3}+\left|x_{1}\right|^{2}+\left|x_{2}\right|^{2}~{}.$
(4.2c)

Depending on the relative values of the three FI–parameters the model can be
in a number of phases which have different geometrical interpretations [38].
Here not all of them are listed and discussed, instead, the focus is on a
number of particular interesting phases: the orbifold phase and the three full
resolved phases which are characterised by having all three FI–parameters
negative or positive, respectively. Other phases, in which some FI–parameters
are positive while others are negative, correspond to partial blowups and are
ignored here. (In ref. [38] some aspects of these other phases were
investigated.)

Some topological properties of the effective geometries in the various phases
can be determined. The divisors in the effective geometry can be identified by
setting one of the complex coordinates to zero while satisfying all the D–term
equations. The ordinary divisors are defined by $D_{u}:=\\{z_{u}=0\\}$ and the
exceptional ones by $E_{r}:=\\{x_{r}=0\\}$. The results of this analysis are
summarised in Table 4.

For each set of non–vanishing fields
$Z_{(P)}=(Z_{(P)}^{1},Z_{(P)}^{2},Z_{(P)}^{3})$, that defines a coordinate
patch within a phase of the resolution GLSM, the other the complement set of
fields
$\\{{\widetilde{Z}}_{(P)}^{1},{\widetilde{Z}}_{(P)}^{2},{\widetilde{Z}}_{(P)}^{3}\\}\in\mathbbm{R}^{3}$
then define a coordinate patch. The resulting patches are also given in Table
4. A gauge can be chosen such that the phases of these non–zero fields
$Z_{(P)}$ are all trivial, i.e. multiplets of $2\pi i$. This only leaves
residual discrete gauge transformations in each of these patches:

$\displaystyle Z_{(P)}^{a}\rightarrow e^{i({\cal
Q}_{(P)})^{a}{}_{r}\,\alpha^{p}}\,Z_{(P)}^{a}\stackrel{{\scriptstyle!}}{{=}}e^{2\pi
i\,m^{a}}\,Z_{(P)}^{a}~{},$ (4.3)

where $Z_{(P)}^{a}$, $a=1,2,3$, are the three scalar fields that do not vanish
in patch $(P)$ with charges $({\cal Q}_{(P)})^{a}{}_{r}$ and $m^{a}$ are
integers. For the coordinate patches under investigation the charge matrices
are given in (C.1). Hence, the gauge parameters of the residual gauge
transformations read:

$\displaystyle\alpha^{T}=2\pi\,m^{T}\,{\cal Q}_{(P)}^{-T}~{}.$ (4.4)

with $\alpha=\big{(}\alpha_{1},\alpha_{2},\alpha_{3}\big{)}$ and
$m^{T}=\big{(}m^{1},m^{2},m^{3}\big{)}$. This induces residual gauge
transformation on the coordinates of the coordinate patch $(P)$ transform

$\displaystyle{\widetilde{Z}}_{(P)}^{a}\rightarrow e^{i(\widetilde{{\cal
Q}}_{(P)})^{a}{}_{r}\,\alpha^{p}}\,{\widetilde{Z}}_{(P)}^{a}=e^{2\pi i\,({\cal
R}m)^{a}}\,{\widetilde{Z}}_{(P)}^{a}~{},\qquad{\cal R}_{(P)}=\widetilde{{\cal
Q}}_{(P)}{\cal Q}_{(P)}^{-1}$ (4.5)

where ${\widetilde{Q}}_{(P)}$ are the charges of the coordinates of the patch
which are given in (C.3). Thus if ${\cal R}_{(P)}$ is integral, the residual
gauge transformations are trivial.

Phase | Non–zero fields | Patches | Curves | Intersection
---|---|---|---|---
Orbifold | $x_{1},x_{2},x_{3}\neq 0$ | $(O):=\\{z_{1},z_{2},z_{3}\\}$ |  $D_{1}D_{2}$, $D_{2}D_{3}$, $D_{3}D_{1}$ | $D_{1}D_{2}D_{3}$
S–triangulation | $z_{1},z_{2},z_{3}\neq 0$ | $(S):=\\{x_{1},x_{2},x_{3}\\}$ |  $E_{1}E_{2}$, $E_{2}E_{3}$, $E_{3}E_{1}$ | $E_{1}E_{2}E_{3}$
| $z_{1},z_{2},x_{3}\neq 0$ | $(33):=\\{x_{1},x_{2},z_{3}\\}$ |  $E_{1}E_{2}$, $E_{2}D_{3}$, $D_{3}E_{1}$ | $E_{1}E_{2}D_{3}$
| $z_{1},x_{2},z_{3}\neq 0$ | $(22):=\\{x_{1},x_{3},z_{2}\\}$ |  $E_{1}E_{3}$, $E_{1}D_{2}$, $D_{2}E_{3}$ | $E_{1}E_{3}D_{2}$
| $x_{1},z_{2},z_{3}\neq 0$ | $(11):=\\{x_{2},x_{3},z_{1}\\}$ |  $E_{2}E_{3}$, $E_{2}D_{1}$, $D_{1}E_{3}$ | $E_{2}E_{3}D_{1}$
E1–triangulation | $z_{2},z_{3},x_{3}\neq 0$ | $(31):=\\{x_{1},x_{2},z_{1}\\}$ |  $E_{1}E_{2}$, $E_{2}D_{1}$, $D_{1}E_{1}$ | $E_{1}E_{2}D_{1}$
| $z_{1},z_{2},x_{3}\neq 0$ | $(33):=\\{x_{1},x_{2},z_{3}\\}$ |  $E_{1}E_{2}$, $E_{2}D_{3}$, $D_{3}E_{1}$ | $E_{1}E_{2}D_{3}$
| $z_{2},z_{3},x_{2}\neq 0$ | $(21):=\\{x_{1},x_{3},z_{1}\\}$ |  $E_{1}E_{3}$, $E_{3}D_{1}$, $D_{1}E_{1}$ | $E_{1}E_{3}D_{1}$
| $z_{1},z_{3},x_{2}\neq 0$ | $(22):=\\{x_{1},x_{3},z_{2}\\}$ |  $E_{1}E_{3}$, $E_{3}D_{2}$, $D_{2}E_{1}$ | $E_{1}E_{3}D_{2}$
E2–triangulation | $z_{1},z_{3},x_{3}\neq 0$ | $(32):=\\{x_{1},x_{2},z_{2}\\}$ |  $E_{1}E_{2}$, $E_{2}D_{2}$, $D_{2}E_{1}$ | $E_{1}E_{2}D_{2}$
| $z_{1},z_{2},x_{3}\neq 0$ | $(33):=\\{x_{1},x_{2},z_{3}\\}$ |  $E_{1}E_{2}$, $E_{2}D_{3}$, $D_{3}E_{1}$ | $E_{1}E_{2}D_{3}$
| $z_{2},z_{3},x_{1}\neq 0$ | $(11):=\\{x_{2},x_{3},z_{1}\\}$ |  $E_{2}E_{3}$, $E_{3}D_{1}$, $D_{1}E_{2}$ | $E_{2}E_{3}D_{1}$
| $z_{1},z_{3},x_{1}\neq 0$ | $(12):=\\{x_{2},x_{3},z_{2}\\}$ |  $E_{2}E_{3}$, $E_{3}D_{2}$, $D_{2}E_{2}$ | $E_{2}E_{3}D_{2}$
E3–triangulation | $z_{2},z_{3},x_{1}\neq 0$ | $(11):=\\{x_{2},x_{3},z_{1}\\}$ |  $E_{2}E_{3}$, $E_{3}D_{1}$, $D_{1}E_{2}$ | $E_{2}E_{3}D_{1}$
| $z_{1},z_{2},x_{1}\neq 0$ | $(13):=\\{x_{2},x_{3},z_{3}\\}$ |  $E_{2}E_{3}$, $E_{3}D_{3}$, $D_{3}E_{2}$ | $E_{2}E_{3}D_{3}$
| $z_{1},z_{2},x_{2}\neq 0$ | $(23):=\\{x_{1},x_{3},z_{3}\\}$ |  $E_{1}E_{3}$, $E_{3}D_{3}$, $D_{3}E_{1}$ | $E_{1}E_{3}D_{3}$
| $z_{1},z_{3},x_{2}\neq 0$ | $(22):=\\{x_{1},x_{3},z_{2}\\}$ |  $E_{1}E_{3}$, $E_{3}D_{2}$, $D_{2}E_{1}$ | $E_{1}E_{3}D_{2}$
Table 4:  This table indicates which combination of fields are necessarily
non–vanishing in the orbifold and the three full resolution phases. This in
turn determines the coordinate patches of the phases and hence the curves and
intersections that exist within the patches. The notation $(ru)$ of the
patches of the fully resolved geometries signify that the coordinates $x_{r}$
and $z_{v\neq u}$ are non–zero.

#### Orbifold phase

In the orbifold regime all three Kähler parameters are negative:
$b_{1},b_{2},b_{3}<0$. The D–term equations (4.1) then imply that all three
exceptional coordinates are non–vanishing:

$\displaystyle\left|x_{1}\right|^{2}=-b_{1}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}>0~{},$
(4.6a)
$\displaystyle\left|x_{2}\right|^{2}=-b_{2}+\left|z_{1}\right|^{2}+\left|z_{3}\right|^{2}>0~{},$
(4.6b)
$\displaystyle\left|x_{3}\right|^{2}=-b_{3}+\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}>0~{},$
(4.6c)

hence there is a single coordinate patch: $\\{z_{1},z_{2},z_{3}\\}$. In
particular, the D–term equations allow to set all these three coordinates to
zero at the same time. Moreover, it is clear that none of the exceptional
divisors $E_{r}$ exist in this phase. Instead the intersection of
$D_{1}D_{2}D_{3}$ exists.

By exploiting the gauge symmetries it is possible to fix the phases of
$x_{1},x_{2},x_{3}$ some arbitrary values (which are typically taken to be
zero for simplicity). However, these gauge fixings do not fix the gauges
completely, since the matrix (4.5) in this case,

$\displaystyle{\cal R}_{(O)}=\widetilde{{\cal Q}}_{(O)}{\cal
Q}_{(O)}^{-1}=-\widetilde{{\cal
Q}}_{(O)}=\begin{pmatrix}~{}0~{}&~{}\mbox{$\frac{1}{2}$}~{}&~{}\mbox{$\frac{1}{2}$}~{}\\\
\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\end{pmatrix}~{},$ (4.7)

is non–integer, therefore, there are non–trivial residual $\mathbbm{Z}_{2}$
gauge transformations which act as

$\displaystyle
E_{1}~{}:~{}(z_{1},z_{2},z_{3})\rightarrow(z_{1},-z_{2},-z_{3})~{},$ (4.8a)
$\displaystyle
E_{2}~{}:~{}(z_{1},z_{2},z_{3})\rightarrow(-z_{1},z_{2},-z_{3})~{},$ (4.8b)
$\displaystyle
E_{3}~{}:~{}(z_{1},z_{2},z_{3})\rightarrow(-z_{1},-z_{2},z_{3})\phantom{~{},}$
(4.8c)

on the remaining coordinates. The first two are precisely the transformations
that defined the $\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$
orbifold and the third one is simply the combination of the first two and
hence redundant in the orbifold phase.

#### S–triangulation full resolution phase

In the S–triangulation the Kähler parameters satisfy the following
inequalities:

$\displaystyle 0<b_{3}<b_{1}+b_{2}~{},\quad 0<b_{2}<b_{1}+b_{3}~{},\quad
0<b_{1}<b_{2}+b_{3}~{}.$ (4.9)

From (4.1) it follows that at least two of the three $z_{u}$ are non–zero.
Hence, there is one coordinate patch $\\{x_{1},x_{2},x_{3}\\}$ when all three
$z_{u}$ are non–vanishing. Taking (4.2) into account, there are, in addition,
three coordinate patches $\\{z_{u},x_{p\neq u}\\}$ for $u=1,2,3$ when $x_{u}$
and $z_{p\neq u}$ are non–zero.

There is no non–trivial residual gauge transformation on the coordinate patch
$(S):=\\{x_{1},x_{2},x_{3}\\}$, since fixing the phases of all three $z_{u}$
fixes all gauge parameters $\theta_{r}$ up to multiples of $2\pi i$, hence the
actions on the coordinates $x_{r}$ are trivial. For the coordinate patch
$(33):=\\{z_{1},x_{2},x_{3}\\}$ the non–vanishing coordinates of which the
phases can be set to unity are $x_{1},z_{2},z_{3}$, consequently, the gauge
parameters $\theta_{2,3}$ are fixed modulo multiples of $4\pi i$ and
$\theta_{1}$ modulo multiplets of $2\pi i$. But the residual gauge
transformations on coordinates

$\displaystyle z_{1}\rightarrow
e^{\mbox{$\frac{1}{2}$}\,\theta_{2}+\mbox{$\frac{1}{2}$}\,\theta_{3}}z_{1}~{},\quad
x_{2}\rightarrow e^{-\theta_{2}}x_{2}~{},\quad x_{3}\rightarrow
e^{-\theta_{3}}x_{3}$ (4.10)

in the patch $(33)$ only involve the gauge parameters $\theta_{2,3}$, and
hence these phase transformations are trivial. Similar arguments can be
provided for the other patches $(22):=\\{x_{1},x_{3},z_{2}\\}$ and
$(11):=\\{x_{2},x_{3},z_{1}\\}$. The fact that all the coordinate patches of
this triangulation are regular can also be verified by showing that the
matrices ${\cal R}_{(P)}$ defined in (4.5) are all integral.

It follows that in the S–triangulation all the divisors $D_{u}$ and $E_{r}$
exist, though not in all coordinate patches. Aside from the curves
$E_{r}D_{u\neq r}$, all three exceptional curves $E_{1}E_{2}$, $E_{2}E_{3}$
and $E_{3}E_{1}$ exist. In particular, the intersections

$\displaystyle
E_{1}E_{2}E_{3}=E_{2}E_{3}D_{1}=E_{1}E_{3}D_{2}=E_{1}E_{2}D_{3}=1$ (4.11)

are all equal to unity as there is just a single solution to the D–term
equations and there is no residual gauge transformation acting on the
coordinates in any given coordinate patch. All this information is encoded in
the toric diagram for the S–triangulation:

$D_{2}$$E_{3}$$D_{1}$$E_{2}$$D_{3}$$E_{1}$

Indeed, all the divisors are indicated as dots. The existing curves are
represented as lines between two adjacent dots and the unit intersections are
the smallest triangles in the diagram. At the same time these smallest
triangles also indicate the four coordinate patches.

#### E1–triangulation full resolution phase

In the E1–triangulation the Kähler parameters satisfy the conditions

$\displaystyle 0<b_{2}+b_{3}<b_{1}~{},\quad 0<b_{3}<b_{1}+b_{2}~{},\quad
0<b_{2}<b_{1}+b_{3}~{}.$ (4.12)

Again at least two of the three $z_{u}$ are non–zero. In light of the first
inequality above, it is convenient to write the equation (4.2a) as

$\displaystyle\left|x_{2}\right|^{2}+\left|x_{3}\right|^{2}=b_{1}-b_{2}-b_{3}+\left|z_{1}\right|^{2}+\left|x_{1}\right|^{2}~{}.$
(4.13)

Hence either $x_{2}$ or $x_{3}$ is non–zero. If $x_{2}\neq 0$ then (4.2c)
implies that $z_{3}$ is non–vanish as there needs to at least two $z_{u}\neq
0$. Similarly, if $x_{3}\neq 0$ then (4.2b) says that $z_{2}$ is
non–vanishing. Therefore, in total there are four coordinate patches:
$\\{x_{1},x_{2},z_{1}\\}$, $\\{x_{1},x_{2},z_{3}\\}$,
$\\{x_{1},x_{3},z_{1}\\}$, $\\{x_{1},x_{3},z_{2}\\}$. Again all these patches
are regular; there is no residual orbifold action on them.

In this phase the exceptional curves $E_{1}E_{2}$ and $E_{1}E_{3}$ exist but
$E_{2}E_{3}$ does not. Instead the curve $D_{1}E_{1}$ is allowed by the D–term
equations. The following intersections

$\displaystyle
E_{1}E_{2}D_{3}=E_{1}E_{3}D_{2}=E_{1}E_{2}D_{1}=E_{1}E_{3}D_{1}=1$ (4.14)

are all equal to unity. All this information is encoded in the toric diagram
for the E1–triangulation:

$D_{2}$$E_{3}$$D_{1}$$E_{2}$$D_{3}$$E_{1}$

A similar analysis can be performed for the other two full resolution phases
corresponding to the triangulations E2 and E3. A summary of the results are
given in table 4.

###  4.2 Pairs of GLSMs associated to torsion related orbifolds

The charges of the Fermi superfields are kept arbitrary in table 3. In order
that the GLSM is free of gauge anomalies these charge vectors are subject to
the conditions [38]

$\displaystyle Q_{1}^{2}=Q_{2}^{2}=Q_{3}^{2}=\frac{3}{2}~{},\qquad Q_{1}\cdot
Q_{2}=Q_{2}\cdot Q_{3}=Q_{3}\cdot Q_{1}=\frac{1}{4}$ (4.15)

and the sum of charges for each of the three gaugings vanishes, see Subsection
3.6. The first three equations indicates that consistent choices for the
charge vectors are given by the shifted momenta of the three twisted sectors
without oscillators, see table 1, since they all square to $3/2$. The latter
three equations can be satisfied by taking the shifted momenta

$\displaystyle
Q_{1}=\big{(}0,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}1,0,0,0^{2}\big{)}\big{(}0^{8}\big{)}~{};~{}~{}Q_{2}=\big{(}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$},0,{\,\mbox{-}}1,0,0^{2}\big{)}\big{(}0^{8}\big{)}~{};~{}~{}Q_{3}=\big{(}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0,0,0,{\,\mbox{-}}1,0^{2}\big{)}\big{(}0^{8}\big{)}$
(4.16)

out of the three twisted sectors of the orbifold model without discrete
torsion or by

$\displaystyle
Q_{1}^{\\!\times}={\,\mbox{-}}\big{(}0,\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},{\,\mbox{-}}1,0,0,0^{2}\big{)}\big{(}0^{8}\big{)};\,Q_{2}^{\\!\times}={\,\mbox{-}}\big{(}\mbox{$\frac{1}{2}$},0,\mbox{$\frac{1}{2}$},0,{\,\mbox{-}}1,0,0^{2}\big{)}\big{(}0^{8}\big{)};\,Q_{3}^{\\!\times}={\,\mbox{-}}\big{(}\mbox{$\frac{1}{2}$},\mbox{$\frac{1}{2}$},0,0,0,{\,\mbox{-}}1,0^{2}\big{)}\big{(}0^{8}\big{)}$
(4.17)

of the orbifold model with torsion. Notice that this is precisely how the
brother gauge shift vectors were related to the original ones as discussed in
Subsection 2.2.

These certainly do not represent unique choices, but for any choice of anomaly
free charge vectors from shifted momenta of the physical twisted states
without oscillators in the orbifold model without torsion, the choice of the
same charge vectors but all with the opposite sign, is an anomaly free choice
with torsion. Hence, switching torsion on or off corresponds to the mapping

$\displaystyle Q_{1}\leftrightarrow Q_{1}^{\\!\times}=-Q_{1}~{},\qquad
Q_{2}\leftrightarrow Q_{2}^{\\!\times}=-Q_{2}~{},\qquad Q_{3}\leftrightarrow
Q_{3}^{\\!\times}=-Q_{3}$ (4.18)

of all the charges in the two associated resolution GLSMs simultaneous. This
suggests that there is a field redefinition from the Fermi superfields
$\Lambda$ in the non–torsion model to the Fermi superfields
$\Lambda^{\\!\\!\times}$ in the torsion model. Formally, in terms of the
chiral superfields $\Omega_{r}$ defined in table 3 this superfield
redefinition can be stated as

$\displaystyle\Lambda\rightarrow\Lambda^{\\!\\!\times}=e^{-2\log\Omega\cdot
Q}\,\Lambda~{},$ (4.19)

since this precisely reverses all the charges of $\Lambda$. In order that this
field redefinition is well–defined $\Omega_{r}$ should be non–singular. Given
that in various coordinate patches within the phases of the theory, there are
always three superfields non–vanishing they can be used in this field
redefinition. Table 5 summarises the choices for $\Omega_{r}$ in the patches
under investigation here.

Phase | Patch | Non–singular superfield representation of
---|---|---
| (P) | ${\Omega_{1}}$ | ${\Omega_{2}}$ | ${\Omega_{3}}$
Orbifold | $(O)$ |  $\Phi_{1}^{\prime-1}$ |  $\Phi_{2}^{\prime-1}$ |  $\Phi_{3}^{\prime-1}$
S–triangulation | $(S)$ | $\Phi_{2}\Phi_{3}\Phi_{1}^{-1}$ | $\Phi_{1}\Phi_{3}\Phi_{2}^{-1}$ | $\Phi_{1}\Phi_{2}\Phi_{3}^{-1}$
| $(33)$ | $\Phi_{2}^{2}\Phi_{3}^{\prime}$ | $\Phi_{1}^{2}\Phi_{3}^{\prime}$ | $\Phi_{3}^{\prime-1}$
| $(22)$ | $\Phi_{3}^{2}\Phi_{2}^{\prime}$ | $\Phi_{2}^{\prime-1}$ | $\Phi_{1}^{2}\Phi_{2}^{\prime}$
| $(11)$ | $\Phi_{1}^{\prime-1}$ | $\Phi_{3}^{2}\Phi_{1}^{\prime}$ | $\Phi_{2}^{2}\Phi_{1}^{\prime}$
E1–triangulation | $(31)$ | $\Phi_{2}^{2}\Phi_{3}^{\prime}$ | $\Phi_{3}^{2}\Phi_{2}^{-2}\Phi_{3}^{\prime-1}$ | $\Phi_{3}^{\prime-1}$
| $(33)$ | $\Phi_{2}^{2}\Phi_{3}^{\prime}$ | $\Phi_{1}^{2}\Phi_{3}^{\prime}$ | $\Phi_{3}^{\prime-1}$
| $(21)$ | $\Phi_{3}^{2}\Phi_{2}^{\prime}$ | $\Phi_{2}^{\prime-1}$ | $\Phi_{2}^{2}\Phi_{3}^{-2}\Phi_{2}^{\prime-1}$
| $(22)$ | $\Phi_{3}^{2}\Phi_{2}^{\prime}$ | $\Phi_{2}^{\prime-1}$ | $\Phi_{1}^{2}\Phi_{2}^{\prime}$
E2–triangulation | $(32)$ | $\Phi_{1}^{-2}\Phi_{3}^{2}\Phi_{3}^{\prime-1}$ | $\Phi_{1}^{2}\Phi_{3}^{\prime}$ | $\Phi_{3}^{\prime-1}$
| $(33)$ | $\Phi_{2}^{2}\Phi_{3}^{\prime}$ | $\Phi_{1}^{2}\Phi_{3}^{\prime}$ | $\Phi_{3}^{\prime-1}$
| $(11)$ | $\Phi_{1}^{\prime-1}$ | $\Phi_{3}^{2}\Phi_{1}^{\prime}$ | $\Phi_{2}^{2}\Phi_{1}^{\prime}$
| $(12)$ | $\Phi_{1}^{\prime-1}$ | $\Phi_{3}^{2}\Phi_{1}^{\prime}$ | $\Phi_{1}^{2}\Phi_{3}^{-2}\Phi_{1}^{\prime-1}$
E3–triangulation | $(11)$ | $\Phi_{1}^{\prime-1}$ | $\Phi_{3}^{2}\Phi_{1}^{\prime}$ | $\Phi_{2}^{2}\Phi_{1}^{\prime}$
| $(13)$ | $\Phi_{1}^{\prime-1}$ | $\Phi_{2}^{2}\Phi_{3}^{-2}\Phi_{1}^{\prime-1}$ | $\Phi_{3}^{2}\Phi_{1}^{\prime}$
| $(23)$ | $\Phi_{1}^{-2}\Phi_{2}^{2}\Phi_{2}^{\prime-1}$ | $\Phi_{2}^{\prime-1}$ | $\Phi_{1}^{2}\Phi_{2}^{\prime}$
| $(22)$ | $\Phi_{3}^{2}\Phi_{2}^{\prime}$ | $\Phi_{2}^{\prime-1}$ | $\Phi_{1}^{2}\Phi_{2}^{\prime}$
Table 5:  This table gives the explicite non–singular forms of $\Omega_{r}$
the orbifold and the full resolution patches in the three triangulations.

Notice that (4.19) precisely looks like a super gauge transformation (A.28)
but with the super gauge parameters $\Theta$ replaced by $-2\log\Omega$. Since
only the Fermi multiplet are involved in this superfield redefinition, it is
anomalous. Because this superfield redefinition is of the same form as a super
gauge transformation, the form of the anomaly is known to be

$\displaystyle W_{\text{sf\, redef\, anom}}=-\frac{1}{2\pi}\,\sum_{r,s}{\cal
A}_{rs}\,\log\Omega^{r}F^{s}=-\frac{1}{2\pi}\,\Big{\\{}\frac{3}{2}\sum_{r}\log\Omega_{r}\,F^{r}+\frac{1}{4}\sum_{s\neq
r}\log\Omega_{s}\,F^{r}\Big{\\}}~{},$ (4.20)

using the general form of the super gauge anomaly (B.14). The latter form is
obtained by using the explicit expression (4.15) of the anomaly matrix ${\cal
A}_{rs}=Q_{r}\cdot Q_{s}$ given by

$\displaystyle{\cal
A}=\begin{pmatrix}~{}\mbox{$\frac{3}{2}$}~{}&~{}\mbox{$\frac{1}{4}$}~{}&~{}\mbox{$\frac{1}{4}$}~{}\\\
\mbox{$\frac{1}{4}$}&\mbox{$\frac{3}{2}$}&\mbox{$\frac{1}{4}$}\\\
\mbox{$\frac{1}{4}$}&\mbox{$\frac{1}{4}$}&\mbox{$\frac{3}{2}$}\end{pmatrix}~{}.$
(4.21)

The superfield anomaly (4.20) is of the form of superfield dependent
FI–actions (A.30) but with the FI–parameters $\rho_{r}$ replaced by

$\displaystyle\rho^{T}\rightarrow\rho^{\\!\times\,T}=\rho^{T}-\frac{1}{2\pi}\,\log\Omega^{T}\,{\cal
A}~{}.$ (4.22)

where $\rho^{T}=\big{(}\rho_{1},\rho_{2},\rho_{3}\big{)}$ and
$\log\Omega^{T}=\big{(}\log\Omega_{1},\log\Omega_{2},\log\Omega_{3}\big{)}$.

The field redefinition anomaly (4.20) is not gauge invariant: it gives a phase
in the Euclidean path integral

$\displaystyle\delta_{\Theta}S_{\text{sf\,redef\,anom}}\supset-i\int\alpha_{r}\,{\cal
A}_{rs}\ \frac{F^{s}_{E2}}{2\pi}~{}.$ (4.23)

However, since it is only obtained under the assumption that the field
redefinition (4.19) is non–singular, it only receives discrete phase
contributions from the scalar fields in Table 4 that do not vanish.

The flux quantisation conditions (3.22) for the present GLSM read

$\displaystyle\int\frac{F_{E_{2}}^{1}}{2\pi}\in\mathbb{Z},\qquad\int\frac{F_{E_{2}}^{2}}{2\pi}\in\mathbb{Z},\qquad\int\frac{F_{E_{2}}^{3}}{2\pi}\in\mathbb{Z}~{},$
(4.24a)
$\displaystyle\frac{1}{2}\,\int\frac{F_{E_{2}}^{2}}{2\pi}+\frac{1}{2}\,\int\frac{F_{E_{2}}^{3}}{2\pi}\in\mathbb{Z},\quad\frac{1}{2}\,\int\frac{F_{E_{2}}^{1}}{2\pi}+\frac{1}{2}\,\int\frac{F_{E_{2}}^{3}}{2\pi}\in\mathbb{Z},\quad\frac{1}{2}\,\int\frac{F_{E_{2}}^{1}}{2\pi}+\frac{1}{2}\,\int\frac{F_{E_{2}}^{2}}{2\pi}\in\mathbb{Z}~{}.$
(4.24b)

The first three conditions follow from the charges of the chiral superfields
$\Phi_{r}^{\prime}$ and the latter three from those of $\Phi_{u}$. Thus all
gauge fluxes are integers and the sums of two gauge fluxes are even integers.
The latter quantisation conditions are solved by adding two equations and
subtracting the third:

$\displaystyle\frac{1}{2\pi}\int\begin{pmatrix}F_{E2}^{1}\\\ F_{E2}^{2}\\\
F_{E2}^{3}\end{pmatrix}={\cal F}\,n~{},\qquad{\cal F}={\cal Q}_{(S)}^{-1}~{},$
(4.25)

in terms of three integers $n^{T}=\big{(}n_{1},n_{2},n_{3}\big{)}$. Here,
${\cal Q}_{(S)}$ is one of the charge matrices defined in (C.1) of Appendix C
and their inverse transposed forms in (C.2). As can be seen from there, ${\cal
F}$ is an integral matrix, the first three quantisation conditions are
fulfilled as well. As was argued in [54] possible vacuum phases in (orbifold)
partition functions may be recovered in the GLSM as non–invariances of the
path integral encoded in

$\displaystyle\delta_{\Theta}S_{\text{sf\, redef\, anom}}\supset-2\pi
i\,m^{T}{\cal M}_{(P)}\,n~{}~{},\qquad{\cal M}_{(P)}={\cal Q}_{r}^{-T}{\cal
A}{\cal F}={\cal Q}_{(P)}^{-T}{\cal A}{\cal Q}_{(S)}^{-1}~{}.$ (4.26)

Hence, the path integral is invariant if ${\cal M}_{(P)}$ is an integral
matrix. By explicit matrix multiplications it may be confirmed that ${\cal
M}_{(P)}$ is indeed integral for all charge matrices (C.1) that correspond to
any of the patches of the three fully resolved phases. On the contrary in the
orbifold phase one finds:

$\displaystyle{\cal
M}_{(O)}=\begin{pmatrix}~{}1~{}&\,{\,\mbox{-}}\mbox{$\frac{3}{2}$}~{}&\,{\,\mbox{-}}\mbox{$\frac{3}{2}$}~{}\\\
\,{\,\mbox{-}}\mbox{$\frac{3}{2}$}~{}&~{}1~{}&\,{\,\mbox{-}}\mbox{$\frac{3}{2}$}~{}\\\
\,{\,\mbox{-}}\mbox{$\frac{3}{2}$}~{}&\,{\,\mbox{-}}\mbox{$\frac{3}{2}$}~{}&~{}1~{}\end{pmatrix}\equiv\begin{pmatrix}0&\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&0&\mbox{$\frac{1}{2}$}\\\
\mbox{$\frac{1}{2}$}&\mbox{$\frac{1}{2}$}&0\end{pmatrix}~{}.$ (4.27)

The final expression is obtained modulo integral matrices. This shows that in
the orbifold phase the discrete torsion phases are reproduced by the residual
gauge transformations of the field redefinition anomaly.

To summarise, the two non–compact resolution GLSMs associated to the orbifold
theories with and without torsions are both free of gauge anomalies and hence
consistent models. The effect of discrete torsion between the two models is
recovered in their orbifold phases, if both models are expressed in the same
field basis (i.e. with chiral Fermi multiplets with the same gauge charges in
both models) because of a field redefinition anomaly (4.20). Even though in
this expression there are logs of chiral superfields, these are non–singular,
because the superfields which appear in the field redefinition (4.19) do not
vanish in the patch where the particular field redefinition is defined (see
Table 5). In particular this does not signify that the geometry has torsion or
should be augmented with NS5–branes, since in the unitary gauge the FI–terms
are constants in each patch, hence the three–form flux (3.28) vanishes.

##  5 GLSMs for resolutions of
$\boldsymbol{T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$

The study of $(0,2)$ resolution GLSMs of the toroidal orbifold
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ is more involved than those for
the non–compact orbifold
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ considered in the
previous section. First of all, additional ingredients are needed to describe
the geometry as the orbifold is compact. And partially because of this also
the description of possible gauge backgrounds is more complicated. Only with
these aspects understood, the consequences of discrete torsion in the
underlying orbifold model can be properly investigated. Therefore, first
Subsections 5.1 to 5.4 are used to develop a both accurate and manageable
description of resolution GLSMs associated with the singular
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ geometry dubbed a minimal full
resolution model. Subsection 5.5 then gives the GLSM for a particular gauge
background using the same blowup modes as in the non–compact model studied in
the previous section. Finally, Subsection 5.6 the GLSM for the compact
orbifold model with discrete torsion is studied.

###  5.1 Construction of resolution GLSMs for compact
$\boldsymbol{\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}}$ orbifolds

To construct GLSMs that describe resolutions of toroidal orbifold geometries,
the following steps need to be taken [39]:

  1. 1.

Give GLSM descriptions for each of the three underlying two–tori compatible
with the orbifold symmetries.

  2. 2.

Add so–called exceptional gaugings to introduce the orbifold actions and
define the exceptional cycles.

  3. 3.

Confirm that there is a regime where the GLSM description can be interpreted
as the orbifold geometry under consideration.

  4. 4.

Determine the regimes in which the GLSM description can be interpreted as
resolved geometries.

This program was discussed in [39] for $(2,2)$ models, but these steps can
equally well be executed in the $(0,2)$ language, which is used throughout
this work.

It is important to realise that there are a number of different
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds depending on their
underlying six–torus lattice, see e.g. [48, 60, 61]. The construction here is
aimed to resolve the particular one with Hodge numbers $(51,3)$. Moreover, one
single orbifold geometry may be associated to many different GLSMs, even if
the target space gauge configurations are not considered. The descriptions
differ in the number of exceptional gaugings. Descriptions in which for all
exceptional cycles of the resolved geometry there are exceptional gaugings,
were dubbed maximal full resolution GLSMs in ref. [39]. On the other end there
are GLSMs descriptions with the least number of exceptional gaugings such that
still the effective geometry in appropriate regimes corresponds to fully
resolved orbifold resolutions. Such models were called minimal full resolution
GLSMs. Between these two extremes there is a whole variety of GLSMs. Some of
these models cannot describe fully resolved geometries; while others do [39].
The focus in this paper is on full resolution GLSMs only. Such resolution
GLSMs might possess many different phases. Only the orbifold phase and fully
smooth resolution phases are investigated in this work in detail, while all
kinds of interesting other phases will be ignored.

Maximal full resolution GLSMs are the most complete in the sense that all the
Kähler parameters associated to the volumes of the exceptional cycles are made
explicit. On the down side, this means that such models typically contain a
large number of $U(1)$ gauge symmetries. As is discussed below the maximal
full resolution GLSM for the toroidal orbifold
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ contains 51 $U(1)$ gaugings: for
each of the 51 Kähler parameters there is a dedicated gauging available. The
minimal full resolution GLSM for this orbifold only requires six $U(1)$
gaugings: The radii of the three two–tori and collective volumes of the three
types of exceptional cycles are explicit in that description.

Below, first the GLSMs description of a two–torus with $\mathbbm{Z}_{2}$
symmetries is recalled. After that the basic ingredients of the maximal full
resolution GLSM are laid out. Details of the resulting geometry and the
consequences of the discrete torsion of the orbifold model are investigated in
the minimal full resolution GLSM only for simplicity.

###  5.2 Two–tori GLSM with $\boldsymbol{\mathbbm{Z}_{2}}$ symmetries

$\begin{array}[]{|c||cccc||cc|}\hline\cr\text{Superfield}&\Phi_{u\,1}&\Phi_{u\,2}&\Phi_{u\,3}&\Phi_{u\,4}&\Gamma^{\phantom{\prime}}_{u}&\Gamma_{u}^{\prime}\\\
\hline\cr\text{U(1)
charges}&z_{u\,1}&z_{u\,2}&z_{u\,3}&z_{u\,4}&\gamma^{\phantom{\prime}}_{u}&\gamma_{u}^{\prime}\\\
\hline\cr\hline\cr
R_{u^{\prime}}&\mbox{$\frac{1}{2}$}\delta_{u^{\prime}u}&\mbox{$\frac{1}{2}$}\delta_{u^{\prime}u}&\mbox{$\frac{1}{2}$}\delta_{u^{\prime}u}&\mbox{$\frac{1}{2}$}\delta_{u^{\prime}u}&-\delta_{u^{\prime}u}&-\delta_{u^{\prime}u}\\\
\hline\cr\end{array}$ Table 6:  Superfield charge table for the GLSM for three
two–tori admiting $\mathbbm{Z}_{2}$ symmetries.

In ref. [39] it was argued that a convenient description of two–tori that
admit $\mathbbm{Z}_{2}$ involutions are given by the superfields given in
Table 6 with the superpotential

$\displaystyle P_{\text{three two--
tori}}=\sum_{u}\Big{(}\kappa_{u}\,\Phi_{u\,1}^{2}+\Phi_{u\,2}^{2}+\Phi_{u\,3}^{2}\Big{)}\Gamma^{\phantom{\prime}}_{u}+\Big{(}\Phi_{u\,1}^{2}+\Phi_{u\,2}^{2}+\Phi_{u\,4}^{2}\Big{)}\Gamma_{u}^{\prime}~{},$
(5.1)

where

$\displaystyle\kappa_{u}=\frac{{\cal
P}_{\tau_{u}}(\mbox{$\frac{\tau_{u}}{2}$})-{\cal
P}_{\tau_{u}}(\mbox{$\frac{1}{2}$})}{{\cal
P}_{\tau_{u}}(\mbox{$\frac{1+\tau_{u}}{2}$})-{\cal
P}_{\tau_{u}}(\mbox{$\frac{1}{2}$})}$ (5.2)

parameterise the complex structures $\tau_{u}$ of the three two–tori in terms
of the Weierstrass ${\cal P}$ function. This description was obtained as a
rewriting of the well–known Weierstrass model of an eliptic curve. On each of
the four chiral superfields $\Phi_{u\,x}$ of two–torus $T^{2}_{u}$ a separate
$\mathbbm{Z}_{2}$ reflection symmetry $\Phi_{u\,x}\rightarrow-\Phi_{u\,x}$ can
act leaving the superpotential invariant. In addition, there are two
involutions per two–torus which can be identified with $\mathbbm{Z}_{2}$
translation on the two–torus lattice [39]. The Kähler structures of the
two–tori are encoded in the GLSM description as the FI–parameter $a_{u}$
associated with the gauging $R_{u}$. The resulting D– and F–term equations in
the conformal limit read:

$\displaystyle|z_{u\,1}|^{2}+|z_{u\,2}|^{2}+|z_{u\,3}|^{2}+|z_{u\,4}|^{2}=a_{u}~{},$
(5.3a)
$\displaystyle\kappa_{u}\,z_{u\,1}^{2}+z_{u\,2}^{2}+z_{u\,3}^{2}=0~{},\quad
z_{u\,1}^{2}+z_{u\,2}^{2}+z_{u\,4}^{2}=0~{}.$ (5.3b)

Because $\kappa_{u}\neq 1$ the two F–term conditions can never combined to an
equation with just two terms. Together with the $U(1)$ gaugings, which can
remove a phase per $u$, shows that each set of $z_{u\,1},\ldots,z_{u\,4}$
coordinates for a given $u$ describes a geometry of real dimension two.

###  5.3 Maximal full resolution GLSM

$\begin{array}[]{|c||c|c|c||cc|cc|cc||c|c|c|}\hline\cr\text{Superfield}&\Phi_{1\,x}&\Phi_{2\,y}&\Phi_{3\,z}&\Gamma_{1}&\Gamma_{1}^{\prime}&\Gamma_{2}&\Gamma_{2}^{\prime}&\Gamma_{3}&\Gamma_{3}^{\prime}&\Phi_{1\,yz}^{\prime}&\Phi_{2\,xz}^{\prime}&\Phi_{3\,xy}^{\prime}\\\
\hline\cr\text{U(1)
charge}&z_{1\,x}&z_{2\,y}&z_{3\,z}&\gamma_{1}&\gamma_{1}^{\prime}&\gamma_{2}&\gamma_{2}^{\prime}&\gamma_{3}&\gamma_{3}^{\prime}&x_{1\,yz}&x_{2\,xz}&x_{3\,xy}\\\
\hline\cr\hline\cr R_{1}&\mbox{$\frac{1}{2}$}&0&0&-1&-1&0&0&0&0&0&0&0\\\
\hline\cr R_{2}&0&\mbox{$\frac{1}{2}$}&0&0&0&-1&-1&0&0&0&0&0\\\ \hline\cr
R_{3}&0&0&\mbox{$\frac{1}{2}$}&0&0&0&0&-1&-1&0&0&0\\\ \hline\cr\hline\cr
E_{1\,y^{\prime}z^{\prime}}&0&\mbox{$\frac{1}{2}$}\delta_{y^{\prime}y}&\mbox{$\frac{1}{2}$}\delta_{z^{\prime}z}&0&0&0&0&0&0&-\delta_{y^{\prime}y}\delta_{z^{\prime}z}&0&0\\\
\hline\cr
E_{2\,x^{\prime}z^{\prime}}&\mbox{$\frac{1}{2}$}\delta_{x^{\prime}x}&0&\mbox{$\frac{1}{2}$}\delta_{z^{\prime}z}&0&0&0&0&0&0&0&-\delta_{x^{\prime}x}\delta_{z^{\prime}z}&0\\\
\hline\cr
E_{3\,x^{\prime}y^{\prime}}&\mbox{$\frac{1}{2}$}\delta_{z^{\prime}z}&\mbox{$\frac{1}{2}$}\delta_{y^{\prime}y}&0&0&0&0&0&0&0&0&0&-\delta_{x^{\prime}x}\delta_{y^{\prime}y}\\\
\hline\cr\end{array}$ Table 7:  Superfield charge table that determines the
geometry of maximal full resolution of
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$.

The maximal full resolution GLSM for the toroidal orbifold
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ has three ordinary gaugings
$R_{1}$, $R_{2}$ and $R_{3}$ to define three two–tori and $3\cdot 16=48$
exceptional gaugings $E_{1,yz}$, $E_{2,xz}$ and $E_{3,xy}$ associated to the
exceptional cycles. The full charge table is given in Table 7. The fermi
superfields $\Gamma_{1},\Gamma_{1}^{\prime}$, $\Gamma_{2},\Gamma_{2}^{\prime}$
and $\Gamma_{3},\Gamma_{3}^{\prime}$ feature in the superpotential to define
the three underlying two–tori, see (5.1). Because the exceptional gaugings the
superpotential has to be extended to

$\displaystyle\begin{array}[]{rl}P_{\text{max\,res}}&=\Big{(}\kappa_{1}\,\Phi_{1\,1}^{2}\prod\limits_{z}\Phi_{2\,1z}^{\prime}\prod\limits_{y}\Phi_{3\,1y}^{\prime}+\Phi_{1\,2}^{2}\prod\limits_{z}\Phi_{2\,2z}^{\prime}\prod\limits_{y}\Phi_{3\,2y}^{\prime}+\Phi_{1\,3}^{2}\prod\limits_{z}\Phi_{2\,3z}^{\prime}\prod\limits_{y}\Phi_{3\,3y}^{\prime}\Big{)}\Gamma_{1}\\\\[4.30554pt]
&\phantom{\kappa_{1}}+\,\Big{(}\Phi_{1\,1}^{2}\prod\limits_{z}\Phi_{2\,1z}^{\prime}\prod\limits_{y}\Phi_{3\,1y}^{\prime}+\Phi_{1\,2}^{2}\prod\limits_{z}\Phi_{2\,2z}^{\prime}\prod\limits_{y}\Phi_{3\,2y}^{\prime}+\Phi_{1\,4}^{2}\prod\limits_{z}\Phi_{2\,4z}^{\prime}\prod\limits_{y}\Phi_{3\,4y}^{\prime}\Big{)}\Gamma_{1}^{\prime}\\\\[8.61108pt]
&\,+\,\Big{(}\kappa_{2}\,\Phi_{2\,1}^{2}\prod\limits_{x}\Phi_{1\,1x}^{\prime}\prod\limits_{z}\Phi_{3\,1z}^{\prime}+\Phi_{2\,2}^{2}\prod\limits_{x}\Phi_{1\,2x}^{\prime}\prod\limits_{z}\Phi_{3\,2z}^{\prime}+\Phi_{2\,3}^{2}\prod\limits_{x}\Phi_{1\,3x}^{\prime}\prod\limits_{z}\Phi_{3\,3z}^{\prime}\Big{)}\Gamma_{2}\\\\[4.30554pt]
&\phantom{\kappa_{2}}+\,\Big{(}\Phi_{2\,1}^{2}\prod\limits_{x}\Phi_{1\,1x}^{\prime}\prod\limits_{z}\Phi_{3\,1z}^{\prime}+\Phi_{2\,2}^{2}\prod\limits_{x}\Phi_{1\,2x}^{\prime}\prod\limits_{z}\Phi_{3\,2z}^{\prime}+\Phi_{2\,4}^{2}\prod\limits_{x}\Phi_{1\,4x}^{\prime}\prod\limits_{z}\Phi_{3\,4z}^{\prime}\Big{)}\Gamma_{2}^{\prime}\\\\[8.61108pt]
&\,+\,\Big{(}\kappa_{3}\,\Phi_{3\,1}^{2}\prod\limits_{x}\Phi_{1\,1x}^{\prime}\prod\limits_{y}\Phi_{2\,1y}^{\prime}+\Phi_{3\,2}^{2}\prod\limits_{x}\Phi_{1\,2x}^{\prime}\prod\limits_{y}\Phi_{2\,2y}^{\prime}+\Phi_{3\,3}^{2}\prod\limits_{x}\Phi_{1\,3x}^{\prime}\prod\limits_{y}\Phi_{2\,3y}^{\prime}\Big{)}\Gamma_{3}\\\\[4.30554pt]
&\phantom{\kappa_{3}}+\,\Big{(}\Phi_{3\,1}^{2}\prod\limits_{x}\Phi_{1\,1x}^{\prime}\prod\limits_{y}\Phi_{2\,1y}^{\prime}+\Phi_{3\,2}^{2}\prod\limits_{x}\Phi_{1\,2x}^{\prime}\prod\limits_{y}\Phi_{2\,2y}^{\prime}+\Phi_{3\,4}^{2}\prod\limits_{x}\Phi_{1\,4x}^{\prime}\prod\limits_{y}\Phi_{2\,4y}^{\prime}\Big{)}\Gamma_{3}^{\prime}\end{array}$
(5.10)

in order to make it gauge invariant under all exceptional gaugings. The
resulting D– and F–term conditions are rather involved and not particularly
illuminating. For this reason we refrain from giving them here and turn to the
more transparant minimal full resolution model.

###  5.4 Minimal full resolution GLSM

$\begin{array}[]{|c||c|c|c||cc|cc|cc||c|c|c|}\hline\cr\text{Superfield}&\Phi_{1x}&\Phi_{2y}&\Phi_{3z}&\Gamma_{1}&\Gamma_{1}^{\prime}&\Gamma_{2}&\Gamma_{2}^{\prime}&\Gamma_{3}&\Gamma_{3}^{\prime}&\Phi_{1}^{\prime}&\Phi_{2}^{\prime}&\Phi_{3}^{\prime}\\\
\hline\cr\text{U(1)
charge}&z_{1x}&z_{2y}&z_{3z}&\gamma_{1}&\gamma_{1}^{\prime}&\gamma_{2}&\gamma_{2}^{\prime}&\gamma_{3}&\gamma_{3}^{\prime}&x_{1}&x_{2}&x_{3}\\\
\hline\cr\hline\cr R_{1}&\mbox{$\frac{1}{2}$}&0&0&-1&-1&0&0&0&0&0&0&0\\\
\hline\cr R_{2}&0&\mbox{$\frac{1}{2}$}&0&0&0&-1&-1&0&0&0&0&0\\\ \hline\cr
R_{3}&0&0&\mbox{$\frac{1}{2}$}&0&0&0&0&-1&-1&0&0&0\\\ \hline\cr\hline\cr
E_{1}&0&\mbox{$\frac{1}{2}$}\delta_{y1}&\mbox{$\frac{1}{2}$}\delta_{z1}&0&0&0&0&0&0&-1&0&0\\\
\hline\cr
E_{2}&\mbox{$\frac{1}{2}$}\delta_{x1}&0&\mbox{$\frac{1}{2}$}\delta_{z1}&0&0&0&0&0&0&0&-1&0\\\
\hline\cr
E_{3}&\mbox{$\frac{1}{2}$}\delta_{x1}&\mbox{$\frac{1}{2}$}\delta_{y1}&0&0&0&0&0&0&0&0&0&-1\\\
\hline\cr\end{array}$ Table 8:  A choice for a superfield charge table that
determines the geometry of a minimal full resolution of
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$.

The minimal full resolution GLSM has three ordinary and three exceptional
gaugings. Contrary to the maximal full resolution GLSM, the charge assignments
of minimal full resolution models are not unique as for each of the three
exceptional gaugings there are $4\cdot 4=16$ choices, which of the homogeneous
coordinate superfields to be gauged.

Here only gaugings of the superfields $\Phi_{1\,1}$, $\Phi_{2\,1}$ and
$\Phi_{3\,1}$ are considered777Other choices would be equally well justified,
however we expect that the physical understanding does not depend much on
this, even though the detailed description will., as can be seen in Table 8.
Consequently, the superpotential for the geometry reduces to

$\displaystyle
P_{\text{min\,res}}=\sum\limits_{u=1}^{3}\Gamma_{u}\Big{(}\kappa_{u}\,\Phi_{u\,1}^{2}\prod\limits_{r\neq
u}\Phi_{r}^{\prime}+\Phi_{u\,2}^{2}+\Phi_{u\,3}^{2}\Big{)}\,+\,\sum\limits_{u=1}^{3}\Gamma_{u}^{\prime}\Big{(}\Phi_{u\,1}^{2}\prod\limits_{r\neq
u}\Phi_{2}^{\prime}\Phi_{3}^{\prime}+\Phi_{u\,2}^{2}+\Phi_{u\,4}^{2}\Big{)}~{}.$
(5.11)

The effective target space geometries are determined by six D– and six F–term
equations. The six resulting D–term conditions read

$\displaystyle\begin{array}[]{lcl}|z_{1\,1}|^{2}+|z_{1\,2}|^{2}+|z_{1\,3}|^{2}+|z_{1\,4}|^{2}=a_{1}~{},&&|z_{2\,1}|^{2}+|z_{3\,1}|^{2}-2\,|x_{1}|^{2}=2\,b_{1}~{},\\\\[4.30554pt]
|z_{2\,1}|^{2}+|z_{2\,2}|^{2}+|z_{2\,3}|^{2}+|z_{2\,4}|^{2}=a_{2}~{},&&|z_{1\,1}|^{2}+|z_{3\,1}|^{2}-2\,|x_{2}|^{2}=2\,b_{2}~{},\\\\[4.30554pt]
|z_{3\,1}|^{2}+|z_{3\,2}|^{2}+|z_{3\,3}|^{2}+|z_{3\,4}|^{2}=a_{3}~{},&&|z_{1\,1}|^{2}+|z_{2\,1}|^{2}-2\,|x_{3}|^{2}=2\,b_{3}\end{array}$
(5.15)

and the six F–term conditions

$\displaystyle\begin{array}[]{lcl}\kappa_{1}\,z_{1\,1}^{2}\,x_{2}x_{3}+z_{1\,2}^{2}+z_{1\,3}^{2}=0~{},&&z_{1\,1}^{2}\,x_{2}x_{3}+z_{1\,2}^{2}+z_{1\,4}^{2}=0~{},\\\\[4.30554pt]
\kappa_{2}\,z_{2\,1}^{2}\,x_{1}x_{3}+z_{2\,2}^{2}+z_{2\,3}^{2}=0~{},&&z_{2\,1}^{2}\,x_{1}x_{3}+z_{2\,2}^{2}+z_{2\,4}^{2}=0~{},\\\\[4.30554pt]
\kappa_{3}\,z_{3\,1}^{2}\,x_{1}x_{2}+z_{3\,2}^{2}+z_{3\,3}^{2}=0~{},&&z_{3\,1}^{2}\,x_{1}x_{2}+z_{3\,2}^{2}+z_{3\,4}^{2}=0~{}.\end{array}$
(5.19)

The properties of the resulting geometries depend crucially on the values of
the Kähler parameters. As can be seen from the three D–term conditions on the
left in (5.15) the parameters $a_{1},a_{2},a_{3}$ all need to be positive
(since we have assumed that all $y^{A}=0$). The other Kähler parameters
$b_{1},b_{2},b_{3}$ may in principle have either sign.

#### Orbifold phase

Consider the phase in which all three parameters $b_{1},b_{2},b_{3}$ are
negative while the parameters $a_{1},a_{2},a_{3}$ all positive. It follows
that all three coordinates $x_{1},x_{2},x_{3}$ are necessarily non–zero so
that their phases can be fixed to some preset values. This does not fix the
gauge symmetries completely, as there are residual $\mathbbm{Z}_{2}$ actions
left over:

$\displaystyle\begin{array}[]{l}\mathbbm{Z}_{2}:\quad z_{2\,1}\rightarrow-
z_{2\,1}~{},\quad z_{3\,1}\rightarrow-z_{3\,1}~{},\\\\[4.30554pt]
\mathbbm{Z}_{2}:\quad z_{1\,1}\rightarrow-z_{1\,1}~{},\quad
z_{3\,1}\rightarrow-z_{3\,1}~{},\\\\[4.30554pt] \mathbbm{Z}_{2}:\quad
z_{1\,1}\rightarrow-z_{1\,1}~{},\quad z_{2\,1}\rightarrow-
z_{2\,1}~{}.\end{array}$ (5.23)

For concreteness, focus on the first of these three $\mathbbm{Z}_{2}$ actions.
The fixed set of this action is given by $z_{2\,1}=z_{3\,1}=0$. In the target
space geometry this does not correspond to a single fixed set, but a
collection of disjoint fixed sets. Indeed, inserting this in the second and
third equations in (5.19) gives the equations:

$\displaystyle z_{2\,2}^{2}+z_{2\,3}^{2}=z_{2\,2}^{2}+z_{2\,4}^{2}=0~{},\qquad
z_{3\,2}^{2}+z_{3\,3}^{2}=z_{3\,2}^{2}+z_{3\,4}^{2}=0~{}.$ (5.24)

Each of these equations are quadratic with two roots:

$\displaystyle z_{2\,3}=\pm i\,z_{2\,2}~{},\quad z_{2\,4}=\pm
i\,z_{2\,2}~{},\qquad z_{3\,3}=\pm i\,z_{3\,2}~{},\quad z_{3\,4}=\pm
i\,z_{3\,2}~{},$ (5.25)

where all the signs are independent, hence there are $2^{4}=16$ solutions in
total. Each of these fixed sets have the topology of a two–torus: The
equations for the homogeneous coordinates $z_{1\,x}$ are those of the deformed
two–torus used in Subsection 5.2 since the absolute values of the coordinates
$x_{2}$ and $x_{3}$ are determined by the second and third equation on the
right hand side in (5.15). This argumentation may be repeated for the second
and third $\mathbbm{Z}_{2}$ actions in (5.23). Hence one has in total $3\cdot
16=48$ fixed two–tori; precisely the number of fixed two–tori to be expected
in the $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold.

The coordinate patches suggested by the minimal full resolution model for the
orbifold geometry can be extracted from the D– and F–term equations (5.15) and
(5.19). Since all blowup parameters $b_{1},b_{2},b_{3}$ are negative, the
three D–term equations on the right–hand–side of (5.15) imply that
$x_{1},x_{2},x_{3}\neq 0$. Each of the other three D–term equations imply that
at least one coordinate in each is non–zero. But then the F–term equations
(5.19) imply that two other coordinates are non–zero. Hence, three out of four
$z_{1x}$, $z_{2y}$ and $z_{3z}$ coordinates are non–zero. This leads to
$4^{3}=64$ coordinate patches; the same number of coordinate patches as the
$(T^{2})^{3}$ torus GLSM would have.

#### Full resolution phases

In the full resolution phases all parameters $b_{1},b_{2},b_{3}$ are positive
but parametrically much smaller than the parameters $a_{1},a_{2},a_{3}$. (If
this is not the case, the GLSM might develop more exotic phases, like
critical– and over–blowup phases [39].) In the full resolution phases it is
useful to reshuffle the three D–term equations on the right hand side of
(5.15) in the following fashion:

$\displaystyle\begin{array}[]{l}|z_{1\,1}|^{2}+|x_{1}|^{2}=b_{2}+b_{3}-b_{1}+|x_{2}|^{2}+|x_{3}|^{2}~{}.\\\\[4.30554pt]
|z_{2\,1}|^{2}+|x_{2}|^{2}=b_{1}+b_{3}-b_{2}+|x_{1}|^{2}+|x_{3}|^{2}~{},\\\\[4.30554pt]
|z_{3\,1}|^{2}+|x_{3}|^{2}=b_{1}+b_{2}-b_{3}+|x_{1}|^{2}+|x_{2}|^{2}~{}.\end{array}$
(5.29)

These equations contain important information as they decide which coordinate
fields are necessarily non–zero. For example, if the sign of the combination
$b_{2}+b_{3}-b_{1}$ is positive either $z_{1\,1}$ or $x_{1}$ is necessarily
non–zero, while if this combination is negative either $x_{2}$ or $x_{3}$ is
necessarily non–zero.

The following divisors can be easily defined by setting one of the homogeneous
coordinates to zero: the exceptional divisors $E_{r}:=\\{x_{r}=0\\}$ and the
ordinary divisors $D_{u}:=\\{z_{u\,1}=0\\}$. The exceptional divisors consists
of $2^{4}=16$ disjoint components and the ordinary divisors of $2^{2}=4$
disjoint components. As was observed in [39] the inherited torus divisors
$R_{u}$ and $R_{u}^{\prime}$ can be identified with the polynomials
multiplying the chiral Fermi superfields $\Gamma_{u}$ and
$\Gamma_{u}^{\prime}$ in the superpotential (5.11).

#### S–triangulation full resolution phase

In the S–triangulation phase of the GLSM the three Kähler parameters are of
similar size in the sense that each one is smaller than the sum of the other
two, e.g. the following three inequalities

$\displaystyle 0<b_{1}<b_{2}+b_{3}~{},\quad 0<b_{2}<b_{1}+b_{3}~{},\quad
0<b_{3}<b_{1}+b_{2}~{},$ (5.30)

hold simultaneously. In this phase the intersection $E_{1}E_{2}E_{3}$ exists
because it is possible to satisfy all the D– and F–term equations while
setting $x_{1}=x_{2}=x_{3}=0$. In fact, in this case the F–term equations have
$2^{6}=64$ solutions. This number comes as no surprise, since the
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ has 64
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ fixed points. When all resolved using
the S–triangulation, one 64 times the intersection of these three exceptional
divisors. Note that the first equation in (5.29) implies that not both
$z_{1\,1}$ and $x_{1}$ can be zero at the same time, hence, in particular, the
curve $D_{1}E_{1}$ does not exist. All this is in accordance with the
topological properties of the S–triangulation of the resolved
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold.

#### E1–triangulation full resolution phase

In the E1–triangulation phase of the GLSM the Kähler parameter $b_{1}$ is much
larger than the sum of the other two:

$\displaystyle 0<b_{2}<b_{1}+b_{3}~{},\quad 0<b_{3}<b_{1}+b_{2}~{},\quad
b_{2}+b_{3}<b_{1}~{}.$ (5.31)

Then (5.29) implies that not both $x_{2}$ and $x_{3}$ can be zero at the same
time, hence, in particular, the curve $E_{2}E_{3}$ and the intersection
$E_{1}E_{2}E_{3}$ do not exist in this phase. Contrary, in this phase the
curve $D_{1}E_{1}$ does exist. All this is, again, in accordance with the
topological properties of the E1–triangulation of the resolved
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold. The transition from the
S– to the E1–triangulation phase thus provides the GLSM description of the
flop transition. Note that in the GLSM there is nothing singular at the
transition $b_{1}=b_{2}+b_{3}$ even though the target space geometry is
singular there.

The other two full resolutions phases, the E2– and $E_{3}$–triangulations may
be defined in an analogous fashion.

#### Full resolution coordinate patches

To understand the coordinate patches in the full resolution phases, first
observe that the three D–term equations on the right–hand–side of (5.15) lead
to the same options for non–vanishing coordinates
$z_{1\,1},z_{2\,1},z_{3\,1},x_{1},x_{2},x_{3}$ as obtained in the non–compact
case summarised in Table 4. Hence, in the $S$–, $E_{1}$–, $E_{2}$– or
$E_{3}$–triangulation the following coordinate combinations

$\displaystyle S:~{}~{}z_{1\,1}z_{2\,1}z_{3\,1}\neq
0~{},~{}~{}z_{1\,1}z_{2\,1}x_{3}\neq 0~{},~{}~{}z_{1\,1}z_{3\,1}x_{2}\neq
0~{}~{}\text{or}~{}~{}z_{2\,1}z_{3\,1}x_{1}\neq 0~{},$ (5.32a) $\displaystyle
E_{1}:~{}~{}z_{2\,1}z_{3\,1}x_{3}\neq 0~{},~{}~{}z_{1\,1}z_{2\,1}x_{3}\neq
0~{},~{}~{}z_{2\,1}z_{3\,1}x_{2}\neq
0~{}~{}\text{or}~{}~{}z_{1\,1}z_{3\,1}x_{2}\neq 0~{},$ (5.32b) $\displaystyle
E_{2}:~{}~{}z_{1\,1}z_{3\,1}x_{3}\neq 0~{},~{}~{}z_{1\,1}z_{2\,1}x_{3}\neq
0~{},~{}~{}z_{2\,1}z_{3\,1}x_{1}\neq
0~{}~{}\text{or}~{}~{}z_{1\,1}z_{3\,1}x_{1}\neq 0~{},$ (5.32c) $\displaystyle
E_{3}:~{}~{}z_{2\,1}z_{3\,1}x_{1}\neq 0~{},~{}~{}z_{1\,1}z_{2\,1}x_{1}\neq
0~{},~{}~{}z_{1\,1}z_{2\,1}x_{2}\neq
0~{}~{}\text{or}~{}~{}z_{1\,1}z_{3\,1}x_{2}\neq 0~{}.$ (5.32d)

are non–zero, respectively. Next, observe that the first D–term equation on
the left–hand–side of (5.15) implies that at least $z_{1\,x}$ is non–zero. If
this happens to be $z_{1\,1}$ then the latter two D–term equations on the
right–hand–side of (5.15) imply that $x_{2}$ and $x_{3}$ are also non–zero
because $a_{1}$ is parametrically larger than the parameters
$b_{1},b_{2},b_{3}$ so that cancellations are never possible. But the the two
top F–term equations (5.19) imply that two other $z_{1x}$, $x\neq 1$ are
non–zero. There are three options for this to happen. Finally, it is possible
that all three $z_{1\,x}$, $x\neq 1$ are non–zero. In total this gives four
non–vanishing coordinate combinations for the first lines of the D– and F–term
equations. A similar analysis can be performed for the second and third lines
of these equations, leading to the following combinations of non–vanishing
coordinates

$\displaystyle z_{1\,2}z_{1\,3}z_{1\,4}\neq
0~{},~{}~{}z_{1\,1}z_{1\,3}z_{1\,4}x_{2}x_{3}\neq
0~{},~{}~{}z_{1\,1}z_{1\,2}z_{1\,4}x_{2}x_{3}\neq
0~{}~{}\text{or}~{}~{}z_{1\,1}z_{1\,2}z_{1\,3}x_{2}x_{3}\neq 0~{},$ (5.33a)
$\displaystyle z_{2\,2}z_{2\,3}z_{2\,4}\neq
0~{},~{}~{}z_{2\,1}z_{2\,3}z_{2\,4}x_{1}x_{3}\neq
0~{},~{}~{}z_{2\,1}z_{2\,2}z_{2\,4}x_{1}x_{3}\neq
0~{}~{}\text{or}~{}~{}z_{2\,1}z_{2\,2}z_{2\,3}x_{1}x_{3}\neq 0~{},$ (5.33b)
$\displaystyle z_{3\,2}z_{3\,3}z_{3\,4}\neq
0~{},~{}~{}z_{3\,1}z_{3\,3}z_{3\,4}x_{1}x_{2}\neq
0~{},~{}~{}z_{3\,1}z_{3\,2}z_{3\,4}x_{1}x_{2}\neq
0~{}~{}\text{or}~{}~{}z_{3\,1}z_{3\,2}z_{3\,3}x_{1}x_{2}\neq 0~{}.$ (5.33c)

Coordinate patches can now be composed by taking one out of four equations on
each line of (5.33) combined with one out of the four equations from the line
in (5.32) corresponding to the chosen triangulation. Not all combinations are
valid however, in total there should be 12 non–vanishing coordinates out of
the 15 original ones, so that the coordinate patch has complex dimension
three.

The results of this analysis are summarised in Table 9. The GLSM description
leads to 76 coordinate patches for each of the full resolution phases. There
are 72 universal coordinate patches which exist independently of which
triangulation is chosen: for each triangulation choice in (5.32) there is at
least one combination of non–vanishing fields which is contained in the
non–vanishing set coordinates of that patch to the extent that precisely 12
coordinates are non–zero. 54 of those patches do not involve any of the
exceptional coordinates and therefore coincide with the coordinate patches of
the orbifold discussed above. These coordinate patches are indicated above the
line that splits the universal patches in Table 9. In addition, to the 72
universal coordinate patches there are four patches that depend on the
triangulation. The GLSM therefore dictates a gluing procedure in which ten of
the coordinate patches of the orbifold are replaced by 22 patches for the full
resolutions.

Phase | $\\#$ | Non–zero fields | Patches | Conditions
---|---|---|---|---
Universal | $54$ | $z_{1\,x^{\prime}\neq x}\,z_{2\,y^{\prime}\neq y}\,z_{z^{\prime}\neq z}\,x_{1}\,x_{2}\,x_{3}\neq 0$ | $\\{z_{1\,x},z_{2\,y},z_{3\,z}\\}$ | $x,y,z\neq 1$
|  | $z_{1\,x^{\prime}\neq 1}\,z_{2\,y^{\prime}\neq y}\,z_{z^{\prime}\neq z}\,x_{1}\,x_{2}\,x_{3}\neq 0$ | $\\{z_{1\,1},z_{2\,y},z_{3\,z}\\}$ | $y,z\neq 1$
|  | $z_{1\,x^{\prime}\neq x}\,z_{2\,y^{\prime}\neq 1}\,z_{z^{\prime}\neq z}\,x_{1}\,x_{2}\,x_{3}\neq 0$ | $\\{z_{1\,x},z_{2\,1},z_{3\,z}\\}$ | $x,z\neq 1$
|  | $z_{1\,x^{\prime}\neq x}\,z_{2\,y^{\prime}\neq y}\,z_{z^{\prime}\neq 1}\,x_{1}\,x_{2}\,x_{3}\neq 0$ | $\\{z_{1\,x},z_{2\,y},z_{3\,1}\\}$ | $x,y\neq 1$
| $18$ | $z_{u^{\prime}\neq u\,1}z_{1\,x^{\prime}\neq 1}\,z_{2\,y^{\prime}\neq 1}\,z_{z^{\prime}\neq z}\,x_{1}\,x_{2}\neq 0$ | $\\{z_{u\,1},z_{3\,z},x_{3}\\}$ | $u^{\prime},u=1,2;z\neq 1$
|  | $z_{u^{\prime}\neq u\,1}z_{1\,x^{\prime}\neq 1}\,z_{2\,y^{\prime}\neq y}\,z_{z^{\prime}\neq 1}\,x_{1}\,x_{3}\neq 0$ | $\\{z_{u\,1},z_{2\,y},x_{2}\\}$ | $u^{\prime},u\neq 1,3;y\neq 1$
|  | $z_{u^{\prime}\neq u\,1}z_{1\,x^{\prime}\neq x}\,z_{2\,y^{\prime}\neq 1}\,z_{z^{\prime}\neq 1}\,x_{2}\,x_{3}\neq 0$ | $\\{z_{u\,1},z_{1\,x},x_{1}\\}$ | $u^{\prime},u=2,3;x\neq 1$
S–triang. | $4$ | $z_{u\,x}\neq 0$ | $\\{x_{1},x_{2},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 3\,1}x_{3}\neq 0$ | $\\{z_{3\,1},x_{1},x_{2}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 2\,1}x_{2}\neq 0$ | $\\{z_{2\,1},x_{1},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 1\,1}x_{1}\neq 0$ | $\\{z_{1\,1},x_{2},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
E1–triang. | $4$ | $z_{u\,x\neq 1\,1}x_{3}\neq 0$ | $\\{z_{1\,1},x_{1},x_{2}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 3\,1}x_{3}\neq 0$ | $\\{z_{3\,1},x_{1},x_{2}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 1\,1}x_{2}\neq 0$ | $\\{z_{1\,1},x_{1},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 2\,1}x_{2}\neq 0$ | $\\{z_{2\,1},x_{1},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
E2–triang. | $4$ | $z_{u\,x\neq 2\,1}x_{3}\neq 0$ | $\\{z_{2\,1},x_{1},x_{2}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 3\,1}x_{3}\neq 0$ | $\\{z_{3\,1},x_{1},x_{2}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 1\,1}x_{1}\neq 0$ | $\\{z_{1\,1},x_{2},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 2\,1}x_{1}\neq 0$ | $\\{z_{2\,1},x_{2},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
E3–triang. | $4$ | $z_{u\,x\neq 1\,1}x_{1}\neq 0$ | $\\{z_{1\,1},x_{2},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 3\,1}x_{1}\neq 0$ | $\\{z_{3\,1},x_{2},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 3\,1}x_{2}\neq 0$ | $\\{z_{3\,1},x_{1},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
|  | $z_{u\,x\neq 2\,1}x_{2}\neq 0$ | $\\{z_{2\,1},x_{1},x_{3}\\}$ | $u=1,2,3;x=1,..,4$
Table 9:  The 76 coordinate patches of the full resolution phases of the
minimal full resolution GLSM. There are 72 universal coordinate patches which
are the same for each of the full resolution phases. In addition, there are
four coordinate patches which are specific for the triangulation chosen.

###  5.5 Gauge background on the minimal full resolution of the non–torsional
orbifold

The gauge charges of the Fermi and chiral multiplets that define a simple
gauge bundle on the minimal full resolution model is given in Table 10. This
gauge bundle is quite closely related to the standard embedding on the
two–tori. The exceptional $E$–gauge charges are identical to those indicated
in (4.15) of the non–compact resolution model. In order to avoid any of the
four types of anomalies mentioned in Subsection 3.6, additional chiral
multiplets $\Psi_{u},\Psi_{u}^{\prime}$ are introduced with identical charges
as $\Gamma_{u},\Gamma_{u}^{\prime}$ and the sum of charges of all chiral
superfields and all chiral Fermi superfields vanish separately.

In total there are $3\cdot 4+3=15$ Fermi multiplets involved in the gauge
bundle subject to $3\cdot 2=6$ constraints enforced by the chiral multiplets
$\Psi_{u},\Psi_{u}^{\prime}$. This leave nine Fermi multiplets part of the
gauge background which cannot be fitted into a single $E_{8}$ factor. Hence a
number of fermionic gaugings are needed. If all six gaugings in the minimal
resolution model are accompanied by fermionic gaugings, (a deformation of) the
standard embedding is obtained. To make contact with the non–torsion line
bundle model that was discussed in Section 4.2, only the inherited
$R_{u}$–gaugings are accompanied with fermionic gaugings with parameters
$\Xi_{u}$, while the exceptional $E_{r}$–gaugings are not. With this choice of
Fermionic gaugings $9-3=6$ gauge bundle directions are left over, exactly
matching the number in the non–compact resolution of the non–torsion orbifold
model.

In target space this gauge background does not correspond to the standard
embedding as there are no fermionic gaugings associated to the exceptional
$E_{r}$–gaugings. Neither can this background be interpreted as line bundles
only because of the presence of the chiral multiplets
$\Psi_{u},\Psi_{u}^{\prime}$ that enforce constraints on the bundle degrees of
freedom as well as the fermionic gaugings $\Xi_{u}$.

$\begin{array}[]{|c||c|c|c||cc|cc|cc||c|c|c||c|}\hline\cr\text{Superfield}&\Lambda_{1x}&\Lambda_{2y}&\Lambda_{3z}&\Psi_{1}&\Psi_{1}^{\prime}&\Psi_{2}&\Psi_{2}^{\prime}&\Psi_{3}&\Psi_{3}^{\prime}&\Lambda_{1}^{\prime}&\Lambda_{2}^{\prime}&\Lambda_{3}^{\prime}&\Lambda_{n}\\\
\hline\cr\text{U(1)
charge}&\lambda_{1x}&\lambda_{2y}&\lambda_{3z}&\psi_{1}&\psi_{1}^{\prime}&\psi_{2}&\psi_{2}^{\prime}&\psi_{3}&\psi_{3}^{\prime}&\lambda_{1}^{\prime}&\lambda_{2}^{\prime}&\lambda_{3}^{\prime}&\lambda_{n}\\\
\hline\cr\hline\cr R_{1}&\mbox{$\frac{1}{2}$}&0&0&-1&-1&0&0&0&0&0&0&0&0\\\
\hline\cr R_{2}&0&\mbox{$\frac{1}{2}$}&0&0&0&-1&-1&0&0&0&0&0&0\\\ \hline\cr
R_{3}&0&0&\mbox{$\frac{1}{2}$}&0&0&0&0&-1&-1&0&0&0&0\\\ \hline\cr\hline\cr
E_{1}&0&\mbox{$\frac{1}{2}$}\delta_{y1}&\mbox{$\frac{1}{2}$}\delta_{z1}&0&0&0&0&0&0&-1&0&0&0\\\
\hline\cr
E_{2}&\mbox{$\frac{1}{2}$}\delta_{x1}&0&\mbox{$\frac{1}{2}$}\delta_{z1}&0&0&0&0&0&0&0&-1&0&0\\\
\hline\cr
E_{3}&\mbox{$\frac{1}{2}$}\delta_{x1}&\mbox{$\frac{1}{2}$}\delta_{y1}&0&0&0&0&0&0&0&0&0&-1&0\\\
\hline\cr\end{array}$ Table 10:  A choice for a charge table of the
superfields that determine a gauge bundle on the minimal full resolution of
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$. The Fermi multiplets
$\Lambda_{n}$, $n=1,\ldots,18$, are spectators and generate the broken gauge
group.

Given the charges of Table 10 the following superpotential can be written
down:

$\displaystyle\begin{array}[]{rl}P_{\text{min\,res\,bundle}}&=\sum\limits_{u=1}^{3}\Psi_{u}\Big{(}2\kappa_{u}\,\Phi_{u\,1}\prod\limits_{r\neq
u}\Phi_{r}^{\prime}\,\Lambda_{u\,1}+\kappa_{u}\,\Phi_{u\,1}^{2}\prod\limits_{r\neq
s\neq
u}\Phi_{r}^{\prime}\Lambda_{s}^{\prime}+2\,\Phi_{u\,2}\Lambda_{u\,2}+2\,\Phi_{u\,3}\Lambda_{u\,3}\Big{)}\\\\[8.61108pt]
&\,+\,\sum\limits_{u=1}^{3}\Psi_{u}^{\prime}\Big{(}2\,\Phi_{u\,1}\prod\limits_{r\neq
u}\Phi_{r}^{\prime}\,\Lambda_{u\,1}+\Phi_{u\,1}^{2}\prod\limits_{r\neq s\neq
u}\Phi_{r}^{\prime}\Lambda_{s}^{\prime}+2\,\Phi_{u\,2}\Lambda_{u\,2}+2\,\Phi_{u\,4}\Lambda_{u\,4}\Big{)}~{}.\end{array}$
(5.36)

This specific form of a general expression for this superpotential is inspired
by the standard embedding following (3.20).

In the model under investigation only the $R_{u}$–gaugings are associated to
fermionic gauge transformations, hence the only non–zero fermionic gauge
transformations are:

$\displaystyle\delta\Lambda_{u\,x}=\mbox{$\frac{1}{2}$}\,\Phi_{u\,x}\,\Xi_{u}~{},\qquad\delta\Gamma_{u}=-\Psi_{u}\,\Xi_{u}~{},\qquad\delta\Gamma_{u}^{\prime}=-\Psi_{u}^{\prime}\,\Xi_{u}~{}.$
(5.37)

The specific form, given here, is obtained by requiring that the fermionic
gauges are on the (2,2)–locus. In this case is follows automatically that
(5.11) and (5.36) combined are inert under these fermionic transformations.

This construction leads to a regular bundle as for each of the three fermionic
gaugings in (5.37) not all coefficients vanish simultaneously. The same goes
for the six constraints coming from (5.36). It is straightforward to check
this for all coordinate patches given in Table 9 for all four fully resolved
phases of this GLSM. This should not come as a surprise as the fermionic
gaugings (5.37) and the bundle superpotential (5.36) are precisely those that
are dictated by the (2,2) locus, see Subsection 3.4.

###  5.6 Gauge background on the minimal full resolution of the torsional
orbifold

In section 2.3 it was explained that the twisted states that survive the
orbifold projections are precisely opposite when torsion is switched on to
when it is absent. Since the shifted momenta of the twisted states without
oscillators dictated the exceptional $E_{1},E_{2},E_{3}$–charges in the GLSM
of the Fermi multiplet $\Lambda$. Hence the charge Table 8, which determines
the geometry, remains unchanged when torsion is switched on, but the charge
table for the vector bundle is modified to Table 11: the $R_{i}$–charges
remain the same while the $E_{r}$–charges are all sign–flipped as compared to
those in Table 10.

$\begin{array}[]{|c||c|c|c||cc|cc|cc||c|c|c||c|}\hline\cr\text{Superfield}&\Lambda_{1x}^{\\!\\!\times}&\Lambda_{2y}^{\\!\\!\times}&\Lambda_{3z}^{\\!\\!\times}&\Psi_{1}&\Psi_{1}^{\prime}&\Psi_{2}&\Psi_{2}^{\prime}&\Psi_{3}&\Psi_{3}^{\prime}&\Lambda_{1}^{\\!\\!\times\,\prime}&\Lambda_{2}^{\\!\\!\times\,\prime}&\Lambda_{3}^{\\!\\!\times\,\prime}&\Lambda_{n}\\\
\hline\cr\text{U(1)
charge}&\lambda_{1x}&\lambda_{2y}&\lambda_{3z}&\psi_{1}&\psi_{1}^{\prime}&\psi_{2}&\psi_{2}^{\prime}&\psi_{3}&\psi_{3}^{\prime}&\lambda_{1}^{\prime}&\lambda_{2}^{\prime}&\lambda_{3}^{\prime}&\lambda_{n}\\\
\hline\cr\hline\cr R_{1}&\mbox{$\frac{1}{2}$}&0&0&-1&-1&0&0&0&0&0&0&0&0\\\
\hline\cr R_{2}&0&\mbox{$\frac{1}{2}$}&0&0&0&-1&-1&0&0&0&0&0&0\\\ \hline\cr
R_{3}&0&0&\mbox{$\frac{1}{2}$}&0&0&0&0&-1&-1&0&0&0&0\\\ \hline\cr\hline\cr
E_{1}&0&-\mbox{$\frac{1}{2}$}\delta_{y1}&-\mbox{$\frac{1}{2}$}\delta_{z1}&0&0&0&0&0&0&+1&0&0&0\\\
\hline\cr
E_{2}&-\mbox{$\frac{1}{2}$}\delta_{x1}&0&-\mbox{$\frac{1}{2}$}\delta_{z1}&0&0&0&0&0&0&0&+1&0&0\\\
\hline\cr
E_{3}&-\mbox{$\frac{1}{2}$}\delta_{x1}&-\mbox{$\frac{1}{2}$}\delta_{y1}&0&0&0&0&0&0&0&0&0&+1&0\\\
\hline\cr\end{array}$ Table 11:  A choice for a charge table of the
superfields that determine a gauge bundle on the minimal full resolution of
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ with torsion.

The flipping of the $E_{r}$–gauge charges has various consequences. First of
all, the fermionic gauge transformations (5.37) are not gauge covariant any
more. This is easily alleviated by inserting appropriate factors of
$\Phi_{r}^{\prime}$ in the first column of fermionic gauge transformations of
$\Lambda_{u1}$:

$\displaystyle\delta\Lambda_{u\,1}^{\\!\\!\times}=\mbox{$\frac{1}{2}$}\,\prod_{r\neq
u}\Phi_{r}^{\prime}\,\Phi_{u\,1}\,\Xi_{u}~{},\qquad\delta\Lambda_{u\,x}^{\\!\\!\times}=\mbox{$\frac{1}{2}$}\,\Phi_{u\,x}\,\Xi_{u}~{},\qquad\delta\Gamma_{u}=-\Psi_{u}\,\Xi_{u}~{},\qquad\delta\Gamma_{u}^{\prime}=-\Psi_{u}^{\prime}\,\Xi_{u}~{},$
(5.38)

for $x\neq 1$. Secondly, the bundle superpotential (5.36) has to be modified
to

$\displaystyle\begin{array}[]{rl}P_{\text{min\,res\,bundle}}&=\sum\limits_{u=1}^{3}\Psi_{u}\Big{(}2\kappa_{u}\,\Phi_{u\,1}\,\Lambda_{u\,1}^{\\!\\!\times}+\kappa_{u}\,\Phi_{u\,1}^{2}\prod\limits_{r\neq
s\neq u}\Phi_{r}^{\prime}\Phi_{s}^{\prime
2}\Lambda_{s}^{\\!\\!\times\,\prime}+2\,\Phi_{u\,2}\Lambda_{u\,2}^{\\!\\!\times}+2\,\Phi_{u\,3}\Lambda_{u\,3}^{\\!\\!\times}\Big{)}\\\\[8.61108pt]
&\,+\,\sum\limits_{u=1}^{3}\Psi_{u}^{\prime}\Big{(}2\,\Phi_{u\,1}\,\Lambda_{u\,1}^{\\!\\!\times}+\Phi_{u\,1}^{2}\prod\limits_{r\neq
s\neq u}\Phi_{r}^{\prime}\Phi_{s}^{\prime
2}\Lambda_{s}^{\\!\\!\times\,\prime}+2\,\Phi_{u\,2}\Lambda_{u\,2}^{\\!\\!\times}+2\,\Phi_{u\,4}\Lambda_{u\,4}^{\\!\\!\times}\Big{)}\end{array}$
(5.41)

by making the following replacements

$\displaystyle\Lambda_{u1}\rightarrow\Phi_{r}^{\prime-1}\Phi_{s}^{\prime-1}\Lambda_{u1}^{\\!\\!\times}~{},\quad\Lambda_{ux}\rightarrow\Lambda_{ux}^{\\!\\!\times}~{},\quad\Lambda_{r}^{\prime}\rightarrow\Phi_{r}^{\prime
2}\Lambda_{r}^{\\!\\!\times\,\prime}~{},$ (5.42)

with $x\neq 1$ and $r\neq s\neq u$, to ensure that it is gauge invariant
again. With these modifications of the fermionic gauge transformations and the
bundle superpotential, it is not difficult to see that the full superpotential
including the part for the geometry (5.10) is invariant under the fermionic
gauge transformations.

The replacements (5.42) are the same as the field redefinitions (4.19) in the
non–compact case with the chiral superfields $\Omega_{r}$ given by the ones in
the orbifold case ($O$) of Table 5. It should be stressed that in the present
case the replacements (5.42) in the bundle superpotential apply to the GLSM
theory as a whole globally, not just to a particular (coordinate patch within
a) phase of the theory. Moreover, it is unique in the sense that other
factors, that would have the same charges (like the other combinations in
Table 5), would always involve some powers of $\Psi_{u}$ or
$\Psi_{u}^{\prime}$, but that is forbidden because they are only allowed to
appear linearly in the superpotential because of the R–symmetry, as was
emphasised below (3.16).

#### Mixed anomalies and worldsheet Green–Schwarz mechanism

The flipped $E_{r}$–gauge charges in Table 11 is irrelevant for most anomalies
which still vanish identically as can be verified using (3.23) through (3.25).
Only mixed $R_{u}E_{r\neq u}$–anomalies are now non–zero:

$\displaystyle{\cal A}_{ur}={\cal
A}_{ru}=\mbox{$\frac{1}{2}$}\cdot\mbox{$\frac{1}{2}$}-\mbox{$\frac{1}{2}$}\cdot(-\mbox{$\frac{1}{2}$})=\mbox{$\frac{1}{2}$}~{},$
(5.43)

$u\neq r$. Hence, contrary to the GLSMs associated to the non–compact orbifold
models, the GLSMs associated to the compact orbifold models without or with
torsion are genuinely physically distinct.

These mixed anomalies need to be cancelled by field dependent FI–terms of the
form

$\displaystyle
W_{\text{FI\,anom}}=\frac{1}{4\pi}\sum_{u,r}\frac{c_{ru}}{2}\,\log(N^{r})\,F^{u}+\frac{1}{4\pi}\sum_{u,r}\frac{1-c_{ru}}{2}\,\log(N^{u})\,F^{r}~{},$
(5.44)

where the composite $N^{r}$ and $N^{u}$ have negative unit charge under the
$R_{u}$– and $E_{r}$–gaugings, respectively, and all other gauge charges zero.
The arbitrary coefficients $c_{ru}$ arise as it is possible by counter terms
to shift two dimensional mixed anomalies around. The choice $c_{ru}=1/2$ would
treat all mixed anomalies symmetrically. (See e.g. ref. [55] for a more
extensive discussion.) The composite chiral superfields $N^{u}$ and $N^{r}$
can be realised as rational functions of (fractional) powers of the chiral
superfields. They may be expressed as

$\displaystyle N^{r}=\Phi_{r}^{\prime}~{},\quad N^{u}=\sum_{x,y\neq
1}n_{uxy}\,\Phi_{ux}^{-1}\Phi_{uy}^{-1}+n_{u11}\Phi_{u1}^{-2}\prod_{r\neq
u}\Phi_{r}^{\prime-1}+\sum_{x\neq
1}n_{u1x}\,\Phi_{u1}^{-1}\Phi_{ux}^{-1}\prod_{r\neq
u}\Phi_{r}^{\prime-\mbox{$\frac{1}{2}$}}~{},$ (5.45)

with some generically non–zero parameters $n_{uxy}$, $n_{u11}$ and $n_{u1x}$.
Since the chiral superfields $\Psi_{u}$ and $\Psi_{u}^{\prime}$ cannot appear
here as they would break R–symmetry, the possible forms in these expressions
are restricted.

The superfield dependent FI–terms (5.44) are defined on the level of the
definition of the model and are singular independently of how the coefficients
$c_{ru}$ and $n_{uxy}$, $n_{u11}$ and $n_{u1x}$ are chosen, hence they signify
the presence of NS5–branes [55, 56]. The interpretation of the coefficients
$c_{ru}$ is not so clear. However, if they are all set to zero: $c_{ru}=1$,
then the expressions of $N^{u}$ become irrelevant. The NS5–branes are then
located on the resolved exceptional cycles $E_{r}$ and they would disappear
inside the orbifold singularities in the blow down limit. Maybe other values
of $c_{ru}$ could be interpreted that the NS5–branes are moved around the
resolved orbifold geometry and for $c_{ru}=0$ they are pushed fully off the
resolved singularities onto the two–torus cycles. This seems to signify that
the NS5–branes can move around on the resolved geometry without losing their
influx effects on the worldsheet. This interpretation may be more transparent
in another parameterisation

$\displaystyle W_{\text{FI
anom}}=\frac{1}{8\pi}\sum_{u,r}c_{ru}\log\Phi_{r}^{\prime}F^{u}-\frac{1}{4\pi}\sum_{u,r}\\!\left[\sum_{x\neq
1}c_{uxr}\log\Phi_{u\,x}+c_{u1r}\Big{(}\log\Phi_{u\,1}\\!-\\!\sum_{r^{\prime}\neq
u}\mbox{$\frac{1}{2}$}\log\Phi_{r^{\prime}}^{\prime}\Big{)}\\!\right]\\!\\!F^{r}$
(5.46)

of (5.44), since the coefficients determining the position of the NS5–branes
are subject to the constraint $c_{ru}+\sum\limits_{x}c_{uxr}=1\,.$

#### Comparing the pair of torsion related GLSMs

Just like in the non–compact case, it is instructive to compare the resoluton
GLSMs of the orbifold theories without and with torsion with each other by
working in the same superfield basis. By interpreting (5.42) as a superfield
redefinition (4.20), but now both $R_{u}$– and $E_{u}$–transformations are
involved, the anomaly matrix ${\cal A}$ extends to

$\displaystyle{\cal
A}=\begin{pmatrix}~{}\mbox{$\frac{1}{4}$}~{}&0&0&0&~{}\mbox{$\frac{1}{4}$}~{}&~{}\mbox{$\frac{1}{4}$}~{}\\\
0&~{}\mbox{$\frac{1}{4}$}~{}&0&~{}\mbox{$\frac{1}{4}$}~{}&0&~{}\mbox{$\frac{1}{4}$}~{}\\\
0&0&~{}\mbox{$\frac{1}{4}$}~{}&~{}\mbox{$\frac{1}{4}$}~{}&~{}\mbox{$\frac{1}{4}$}~{}&0\\\
0&~{}\mbox{$\frac{1}{4}$}~{}&~{}\mbox{$\frac{1}{4}$}~{}&~{}\mbox{$\frac{3}{2}$}~{}&~{}\mbox{$\frac{1}{4}$}~{}&~{}\mbox{$\frac{1}{4}$}~{}\\\
~{}\mbox{$\frac{1}{4}$}~{}&0&~{}\mbox{$\frac{1}{4}$}~{}&\mbox{$\frac{1}{4}$}&\mbox{$\frac{3}{2}$}&\mbox{$\frac{1}{4}$}\\\
~{}\mbox{$\frac{1}{4}$}~{}&~{}\mbox{$\frac{1}{4}$}~{}&0&\mbox{$\frac{1}{4}$}&\mbox{$\frac{1}{4}$}&\mbox{$\frac{3}{2}$}\end{pmatrix}~{}.$
(5.47)

Notice that the lower $3\times 3$–block is identical to (4.21). Since in the
replacements (5.42) only the superfields $\Phi_{r}^{\prime}$ feature, the
superfield redefintion anomalies reads

$\displaystyle W_{\text{field\, redef\,
anom}}=-\frac{1}{2\pi}\,\Big{\\{}\frac{1}{4}\sum_{u\neq
r}\log\Phi_{r}^{\prime}\,F^{u}+\frac{3}{2}\sum_{r}\log\Phi_{r}^{\prime}\,F^{r}+\frac{1}{4}\sum_{r^{\prime}\neq
r}\log\Phi_{r}^{\prime}\,F^{r^{\prime}}\Big{\\}}~{}.$ (5.48)

The first contributions coincides with the general expression (5.44) provided
that $c_{ur}=0$ and $N^{r}=\Phi_{r}^{\prime}$, hence they cancel each other
exactly. The latter two contributions were also obtained in the non–compact
situation (4.20). Hence, the analysis performed in Subsection (4.2) can be
repeated here as well. In particular in the orbifold phase, that analysis
recovers the discrete torsion phases.

##  6 Conclusions

Discrete torsion within the $\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifolds
correspond to particular additional phases between the sum of partition
functions of different sectors corresponding to different boundary conditions
on the worldsheet torus. Smooth geometries are typically described by NLSMs
which cannot be exactly quantised and the path integral cannot be represented
as a sum over similar sectors as the orbifold theory. It is therefore unclear
how to include effects of discrete torsion for smooth geometries. The main aim
of this paper was to understand where discrete torsion goes when orbifolds
have been resolved to fully smooth geometries. This question was addressed
both for resolutions of the non–compact orbifold
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ as well as the compact
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold with Hodge numbers
$(51,3)$ to understand both local and global aspects.

GLSMs were chosen as the framework for this investigation, as they can both
make contact with the orbifolds as well as with fully resolved smooth
geometries within the same description. From an effective field theory point
of view orbifold resolutions correspond to giving VEVs to twisted states
defining the blowup modes. Unless very particular blowup modes are selected,
this leads to $(0,2)$ compactifications in which the gauge backgrounds are not
dictated by the standard embedding. Therefore, in this work $(0,2)$ GLSMs were
used for the interpolation between singular orbifolds and smooth
compactifications.

The non–compact resolution GLSM of the
$\mathbbm{C}^{3}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ geometry had already
given in the literature, the same goes for the resulting line bundle
backgrounds obtained by using non–oscillator blowup modes on the three
$\mathbbm{C}^{2}/\mathbbm{Z}_{2}$ singularities. The GLSM gauge charges of the
chiral Fermi multiplets under the resulting three exceptional gauge symmetries
are given as the shifted left–moving momenta of these blowup modes. The effect
of discrete torsion on the orbifold is that the twisted states with the
opposite left–moving shifted momenta survive the orbifold projections.
Consequently, the chiral Fermi multiplets in resolution GLSM for the torsional
orbifold has the opposite worldsheet gauge charges as the non–torsional case.
The GLSM associated to the torsional orbifold is equally well defined as the
non–torsional model in the sense that all (gauge) anomalies vanish. In many
respects the two models look identical. However, if one wants to express the
physics of the GLSM associated with the torsional orbifold in terms of the
superfield basis of the non–torsional GLSM, one has to perform anomalous
superfield redefinitions. Since, these superfield redefinitions have to be
well defined in each patch where they are performed, the expression of the
anomaly is harmless within the smooth resolution phases. But in the orbifold
phase this anomaly turns out not to be invariant under residual discrete
$\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ gauge transformations, precisely
reproducing the torsion phases of the orbifold theory.

The story for the compact case is more involved. GLSMs for resolutions of the
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ orbifold have not explicitly
appeared in the literature. Moreover, GLSMs for other compact orbifold
resolutions have only been studied in the $(2,2)$ context. Therefore, before
the question about discrete torsion on compact orbifold resolutions could be
addressed, first resolution GLSMs for
$T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ had to be constructed. Contrary
to the existing literature on compact orbifold resolutions, this was done
immediately in the $(0,2)$ language. The simplest version of such a GLSM
involves six gaugings on the worldsheet: three to define modified Weierstrass
models to describe the underlying two–tori of the $T^{6}$ and three
exceptional gaugings associated with the blowup process. In order to make
comparisons with the non–compact situations most transparant, the same blowup
modes were chosen as in the non–compact case, i.e. non–oscillatory twisted
states. To pass all consistency conditions this resulted in a more complicated
bundle that shares both features of line bundles on the resolved fixed
two–tori as well as the standard embedding on the underlying two–tori of the
$T^{6}$.

The resolution GLSM of the $T^{6}/\mathbbm{Z}_{2}\times\mathbbm{Z}_{2}$ with
discrete torsion was obtained in a similar fashion as its non–compact analog:
the exceptional gauge charges were flipped, while the other three gauge
charges remained unchanged. As a consequence the resolution GLSM associated
with the torsional orbifold suffers from mixed gauge anomalies. These
anomalies can be cancelled by superfield dependent FI–terms in the GLSM
globally. This signifies that the target space geometry has torsion in the
sense that the three–form $H$–flux is non–zero. Moreover, given the GLSM
chiral superfield content, the field dependent FI–terms need to involve logs
of chiral superfields. As argued in the past, this signals that there are
NS5–branes in the system. The structure of these logs can be taken such that
these NS5–branes are located at the resolved exceptional cycles. In the
orbifold limit they would disappear inside these singularities.

It is striking to see the differences of the effect of discrete torsion in the
resolution process for non–compact and compact orbifolds. In the non–compact
case apart from a physically irrelevant flip of charge conjugated states the
non–torsional and torsional orbifold resolutions are to a large extent
indistinguishable: only the relative signs of the gauge charges of the chiral
and chiral Fermi multiplets distinguish them. In the compact case the GLSM
associated to the torsional orbifold is really physically different from the
non–torsional one as the mixed gauge anomalies and the related NS5–branes
signify. These differences may be explained by realising that in the
non–compact case the effect of flux can be pushed off to infinity while in the
compact case this is impossible.

#### Outlook

The work presented here can be extended in a number of ways.

First of all, it would be interesting if it is possible by other means to show
that the emerged picture that NS5–branes are located at the resolved
singularities of the resolved torsional orbifold can be corroborate. And it
would be interesting to confirm the interpretation of the coefficients that
allow to shift mixed gauge anomalies around as moving around the NS5–branes of
the resolved geometry. In addition, it would be interesting to investigate
what the geometrical consequences are of the back reaction induced by the
log–dependent FI–terms.

In this paper the focus was on only very simple bundles (line bundles combined
with bundles that are on the (2,2) locus, hence closely related to the
standard embedding). However, the procedures used here could be applied to
other gauge backgrounds as well. In particular, by choosing other blowup
modes, for example, those with oscillator excitations, see e.g. [38].

Moreover, in this work only the discrete torsion between two orbifold twists
was considered. For possible applications of the spinor–vector duality on
smooth geometries other generalised discrete torsion phases would be of
interest. First attempts in this direction were performed using effective
field theory techniques in [34]. Such phases are between orbifold twists and
torus translations and associated Wilson lines or among two different torus
translations. This requires that within the GLSM distinctions between the
various (resolved) fixed tori can be made. Clearly, this is possible in the
maximal full resolution model, which treats all 48 (resolved) fixed tori
independently, or in certain full resolution GLSM that have a certain number
of additional gauging so that at least some fixed two–tori in certain
directions can be distinguished. The effect of the Wilson lines would then be
that the exceptional GLSM charges (dictated by the shifted twisted state
momenta) are not the same at the different fixed tori. Then, just like in the
models considered here, the effect of generalised discrete torsion is that
particular states are projected out or in, leading to different charge
assignments for the Fermi multiplets. Presumably, the consequences of these
differences could then be analysed in much the same fashion as done in the
current work.

#### Acknowledgements

SGN would like to thank the University of Liverpool for the kind hospitality
during the completion of this work. We have benefitted from enlightening
discussions with Sav Sethi during the String Phenomenology 2022 conference at
Liverpool and subsequent email exchanges. We would also like to thank Eric