CERN-TH-2023-114

LTH-1342

June 2023

$D$–Term Uplifts
in Non–Supersymmetric Heterotic String Models

Alonzo R. Diaz Avalos 1***E-mail address<EMAIL_ADDRESS>
Alon E. Faraggi1,2†††E-mail address<EMAIL_ADDRESS>Viktor G.
Matyas1‡‡‡E-mail address<EMAIL_ADDRESS>and Benjamin
Percival1§§§E-mail address<EMAIL_ADDRESS>
1 Dept. of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL,
UK

2 CERN, Theoretical Physics Department, CH–1211 Geneva 23, Switzerland

Recently, we proposed that the one–loop tadpole diagram in perturbative
non–supersymmetric heterotic string vacua that contain an anomalous $U(1)$
symmetry, leads to an analog of the Fayet–Iliopoulos $D$–term in
$\mathcal{N}=1$ supersymmetric models, and may uplift the vacuum energy from
negative to positive value. In this paper, we extend this analysis to new
types of vacua, including those with Stringy Scherk–Schwarz (SSS) spontaneous
supersymmetry breaking versus those with explicit breaking. We develop a
criteria that facilitates the extraction of vacua with Scherk–Schwarz
breaking. We develop systematic tools to analyse the T–duality property of
some of the vacua and demonstrate them in several examples. The extraction of
the anomalous $U(1)$ $D$–terms is obtained in two ways. The first utilises the
calculation of the $U(1)$–charges from the partition function, whereas the
second utilises the free fermionic classification methodology to classify
large spaces of vacua and analyse the properties of the massless spectrum. The
systematic classification method also ensures that the models are free from
physical tachyons. We provide a systematic tool to relate the free fermionic
basis vectors and one–loop Generalised GSO phases that define the string
models, to the one–loop partition function in the orbifold representation. We
argue that a $D$–term uplift, while rare, is possible for both the SSS class
of models, as well as in those with explicit breaking. We discuss the steps
needed to further develop the arguments presented here.

##  1 Introduction

The Standard Model (SM) of particle physics provides an effective
parametrisation for all observational sub–atomic data to date. It is even
possible that it remains viable up to the Grand Unified Theory (GUT) scale, or
Planck scale, where gravitational effects become prominent. In this case,
gaining insight into the fundamental origin of the SM parameters can only be
gleaned by fusing it with gravity. String theory provides the most advanced
contemporary framework to pursue the synthesis of gravity with the sub–atomic
gauge interactions. For that purpose, Standard–like Models were constructed in
the free fermionic formulation of the heterotic string and provide a
laboratory to study how the Standard Model parameters may arise in a theory of
quantum gravity [1, 2, 3, 4, 5, 6, 7]. The free fermionic heterotic string
models are $\mathbb{Z}_{2}\\!\times\\!\mathbb{Z}_{2}$ orbifolds of six
dimensional tori at enhanced symmetry points in the moduli space [8, 9, 10,
11, 12].

While the majority of phenomenological string models constructed to date
possess $\mathcal{N}=1$ spacetime supersymmetry (SUSY), absence of SUSY at
observable energy scales mandates that it has to be broken. Spacetime SUSY in
string models can be broken by non–perturbative effects in the effective field
theory limit of the string vacua, or it may be broken directly at the string
scale. In the string constructions, we may distinguish between explicit and
spontaneous breaking, where, in the former, the remaining gravitino is
projected from the spectrum, whereas spontaneous breaking can arise through
the Scherk–Schwarz mechanism [13, 14, 15, 16, 17], in which case the gravitino
mass is proportional to the inverse of an internal radius of the six
dimensional compactified torus.

It is clear that addressing many of the questions in string phenomenology
mandates the breaking of SUSY. In particular when it comes to the cosmological
evolution and string dynamics near the Planck scale. The non–SUSY string vacua
typically contain physical tachyons in their spectra that indicate that they
are unstable. However, also those configurations that are free of physical
tachyons, in general have non–vanishing tadpoles and vacuum energy that in
general lead to instability.

One of the recurring features in supersymmetric string derived models is the
existence of an anomalous $U(1)$ ($U(1)_{A}$) symmetry. The $U(1)_{A}$ is
cancelled by an analogue of the Green–Schwarz mechanism, which generates a
Fayet–Iliopoulos (FI) $D$–term that breaks SUSY near the Planck scale [18,
19]. The non–vanishing $D$–term gives rise to a non–vanishing vacuum energy at
two–loop [20]. SUSY can be restored by assigning non–vanishing Vacuum
Expectation Values (VEV) to some Standard Model singlet fields along $F$– and
$D$–flat directions. The anomalous $U(1)$ symmetry in string construction
plays a pivotal role in many of the phenomenological studies of string
compactifications [21, 22, 23, 24].

An anomalous $U(1)$ symmetry is a recurring feature also in non–SUSY string
vacua. The same diagram in string perturbation theory that generates the FI
term in the SUSY configurations is also present in the non–SUSY
configurations, i.e. both in those with explicit SUSY breaking, as well as
those in which it is broken by the SSS mechanism. Similarly, the two–loop
diagram contributing to the vacuum energy is also present in the case of
non–SUSY vacua, either with explicit or SSS SUSY breaking. Thus, it imperative
to take into account this contribution to the vacuum energy also in these
cases.

This contribution to the vacuum energy is particularly pertinent to the
question of the existence of a de–Sitter vacuum in string theory.
Astrophysical and cosmological data indicate that the universe is
accelerating. The existence of a positive vacuum energy is one of the possible
explanations. However, the existence of string vacua with positive vacuum
energy and stable moduli is currently under intense scrutiny and doubt. For
instance, for the non–SUSY heterotic constructions examined through effective
field theory methods in ref. [25] only AdS vacua are found to be possible.
However, through exact worldsheet evaluation of the one–loop potentials of
non–SUSY heterotic string orbifolds, Florakis and Rizos demonstrated the
existence of string vacua with positive vacuum energy [26]. However, many open
issues remain in the study of string vacua without SUSY and with respect to
(related) issues around moduli stabilisation. Additionally, it remains
important for this analysis to be extended to models in which more
phenomenological criteria are satisfied.

One direction in which progress can be made to the evaluation of vacuum energy
for non–SUSY heterotic string vacua is through exploring the contribution of
the would–be FI $D$–term. Since this contribution is positive definite it may
uplift an a priori negative vacuum energy to a positive one, an idea discussed
in [27]. Recently, we demonstrated this possibility in a particular string
vacuum [28]. The analysis utilises the free fermionic classification
methodology to extract tachyon free non–SUSY string vacua. It then calculates
the traces of the $U(1)$ gauge symmetries and extracts the tachyon free vacua
with a $U(1)_{A}$. The one–loop vacuum amplitude was analysed in comparison to
the $U(1)_{A}$ would–be $D$–term contribution. Following refs [26, 29, 30], a
numerical analysis of the potential and its dependence on the moduli in
specific string models was performed in the neighbourhood of the a local
minimum. It was then found that a $D$–term uplift to a positive value may
indeed be possible in a model with explicit supersymmetry breaking.

In this paper we extend the analysis of [28]. We develop a criteria to
distinguish the models with SSS supersymmetry breaking that allow for the
vacuum energy to be exponentially suppressed provided that the number of
massless bosons and fermions is equal. We then proceed to analyse the vacuum
energy and potential of a range of models with both explicit and SSS SUSY
breaking. We demonstrate that a $D$–term uplift may indeed be possible in
models with SSS breaking as well as in models with explicit breaking. We
further provide some statistical measure for the frequency of tachyon free
models with SSS breaking, and provide further examples of cases where a
minimum of potential is not obtained for finite value of the moduli.

Our paper is organised as follows: in section 2 we review some general aspects
of the free fermion construction that are particularly relevant for the
analysis in this paper, and refer to the literature for more details. In
section 3 we review the calculation of the Fayet–Iliopoulos term in
$\mathcal{N}=1$ supersymmetric string vacua and discuss its adaptation to the
$\mathcal{N}=0$ case. In section 4 we discuss the analysis of the one–loop
partition function and potential, as well as the derivation of the anomalous
$U(1)$ from the partition function that serves as a counter–check on its
derivation from the massless spectrum. In section 5 we elaborate on explicit
SUSY breaking versus spontaneous SUSY breaking by the SSS–mechanism. We derive
conditions that facilitate the extraction of the string vacua that utilise the
SSS–mechanism and provide examples that demonstrate their utilisation in
Appendix B. Similarly, in section 5 we identify the conditions on the
worldsheet phases that exhibit the T–duality property of the string vacua and
supplement these with examples in Appendix B. In section 6 we discuss the
conditions for the extraction of tachyon free configurations in the space of
vacua, and section 7 elaborates on the analysis of the chiral sectors and
extraction of the anomalous $U(1)_{A}$ symmetry from the massless spectrum.
Section 8 presents our results that include examples of uplift models with SSS
supersymmetry breaking as well as explicit breaking. Section 9 contains our
conclusions and discussion on further steps that can be taken to improve the
rigour of the analysis presented in this paper as well as its predictability.
In Appendix A we discuss in detail how to relate a free fermion model which is
specified in term of the set of boundary condition basis vectors and one–loop
GGSO phases, to the one–loop partition function in a bosonic representation.
The art in this regard is in the translation of the GGSO projection
coefficients to the modular invariant phase that appears in the one–loop
partition function. This tool therefore facilitates the writing of the
partition function, which provides access to the entire string spectrum, for
any string model, which is specified in terms of the boundary condition basis
vectors and one–loop phases.

##  2 Model building in the free fermionic formulation

In the free fermionic formulation of [31, 32, 33], all degrees of freedom are
realised as free fermions propagating on the string worldsheet. For the
heterotic string in four dimensions, we consider holomorphic fields that
realise a supersymmetric $c=10$ conformal algebra and antiholomorphic fields
that realise a non–supersymmetric conformal algebra. Along with the spacetime
bosons $X^{\mu}(z,\bar{z})$ we denote the fermionic fields as

$\displaystyle\begin{split}\text{Holomorphic}:\ \ \
&\psi^{\mu=1,2},\chi^{1,...,6},y^{1,...,6},w^{1,..,6}\ (z)\\\
\text{Antiholomorphic}:\ \ \
&\bar{y}^{1,...,6},\bar{w}^{1,..,6},\bar{\psi}^{1,...,5},\bar{\eta}^{1,2,3},\bar{\phi}^{1,...,8}\
(\bar{z}),\end{split}$ (2.1)

where:

  * •

$\psi^{\mu}$ is the superpartner of the bosonic spacetime field $X^{\mu}(z)$
in the lightcone gauge.

  * •

$\chi^{1,...,6}$ are the superpartners of the six compact directions of the
bosonic coordinate fields.

  * •

$\\{y^{i},w^{i}\ |\ \bar{y}^{i},\bar{w}^{i}\\}$ realise the conformal field
theory associated to the six dimensional compact geometry.

  * •

Currents associated to $\\{\bar{\psi}^{1,...,5},\bar{\eta}^{1,2,3}\\}$ can
realise a gauge $c=8$ conformal algebra containing an $SO(10)$ GUT from the
$\bar{\psi}^{1,...,5}$ that may contain Standard Model like gauge fields.

  * •

Currents associated to $\\{\bar{\phi}^{1,...,8}\\}$ can be associated to a
gauge $c=8$ conformal algebra relating to the hidden sector of the theory.

Consistent model building requires that a $N=1$ superconformal algebra on the
string worldsheet is realised among the holomorphic degrees of freedom. This
can be achieved via the worldsheet supercurrent

$T_{F}(z)=i\psi^{\mu}\partial X^{\mu}(z)+i\sum^{6}_{I=1}\chi^{I}y^{I}w^{I},$
(2.2)

which has conformal weight $(3/2,0)$. This results in a local enhanced
symmetry group $SU(2)^{6}$, the adjoint representation of which is given by
the six $SU(2)$–triplets from $\\{\chi^{I},y^{I},w^{I}\\}$.

Models in the free fermionic formalism are then defined by considering a
one–loop torus and defining a set of $N$ boundary condition basis vectors,
$\bm{v_{i}}\in\mathcal{B}$, specifying how each free fermion, $f$, propagates
around the two non–contractible loops of the torus. An element ${\bm{\beta}}$
of the space $\Xi=\text{span}\\{\mathcal{B}\\}$ can then be written as

$\displaystyle\begin{split}{\bm{\beta}}=\\{&\beta(\psi^{\mu}),\beta(\chi^{12}),\beta(\chi^{34}),\beta(\chi^{56}),\beta(y^{1}),...\beta(w^{6})\
|\ \\\ &\beta({\bar{y}}^{1}),...,\beta({\bar{w}}^{6});\
\beta({\bar{\psi}}^{1}),...,\beta({\bar{\psi}}^{5}),\beta({\bar{\eta}}^{1}),\beta({\bar{\eta}}^{2}),\beta({\bar{\eta}}^{3}),\beta({\bar{\phi}}^{1}),...,\beta({\bar{\phi}}^{8})\\},\end{split}$
(2.3)

such that $\beta(f)\in(-1,1]$ and Ramond (R) boundary conditions correspond to
$\beta(f)=1$, while Neveu–Schwarz (NS) is given by $\beta(f)=0$.

The partition function in the fermionic formulation can be written as

$Z=Z_{B}\sum_{\bm{\alpha},\bm{\beta}\in\Xi}C{\bm{\alpha}\atopwithdelims[
]\bm{\beta}}Z{\bm{\alpha}\atopwithdelims[ ]\bm{\beta}},$ (2.4)

where $Z_{B}=1/\eta^{2}{\bar{\eta}}^{2}$ is the bosonic partition function and
$C{\bm{\alpha}\atopwithdelims[ ]\bm{\beta}}$ are Generalised GSO (GGSO) phases
which respect modular invariance. The $Z{\bm{\alpha}\atopwithdelims[
]\bm{\beta}}$ represent the worldsheet fermions and are thus products of
Jacobi theta functions. The partition function for the models we explore in
this work is discussed in section 4.

Aside from the partition function we can also view the spectrum through the
modular invariant Hilbert space, $\mathcal{H}$, of states,
$\ket{S_{{\bm{\beta}}}}$. This is constructed through implementing the
one–loop GGSO projections on each sector according to:

$\mathcal{H}=\bigoplus_{{\bm{\beta}}\in\Xi}\prod^{N}_{i=1}\left\\{e^{i\pi\bm{v_{i}}\cdot
F_{{\bm{\beta}}}}\ket{S_{{\bm{\beta}}}}=\delta_{{\bm{\beta}}}C{{\bm{\beta}}\atopwithdelims[
]\bm{v_{i}}}^{*}\ket{S_{{\bm{\beta}}}}\right\\}\mathcal{H}_{{\bm{\beta}}},$
(2.5)

where $F_{{\bm{\beta}}}$ is the fermion number operator and
$\delta_{{\bm{\beta}}}$ is the spin–statistics index.

The sectors, ${\bm{\beta}}$, in the model can be characterised according to
their holomorphic (H) and antiholomorphic (A) moving vacuum separately

$\displaystyle\begin{split}M_{H}^{2}&=-\frac{1}{2}+\frac{{\bm{\beta}}_{H}\cdot{\bm{\beta}}_{H}}{8}+N_{H}\\\
M_{A}^{2}&=-1+\frac{{\bm{\beta}}_{A}\cdot{\bm{\beta}}_{A}}{8}+N_{A},\end{split}$
(2.6)

where $N_{H}$ and $N_{A}$ are sums over left and right moving oscillator
frequencies, respectively

$\displaystyle N_{H}$
$\displaystyle=\sum_{\lambda}\nu_{\lambda}+\sum_{\lambda^{*}}\nu_{\lambda^{*}}$
(2.7) $\displaystyle N_{A}$
$\displaystyle=\sum_{\bar{\lambda}}\nu_{\bar{\lambda}}+\sum_{\bar{\lambda}^{*}}\nu_{\bar{\lambda}^{*}},$
(2.8)

where $\lambda$ is a holomophic oscillator and $\bar{\lambda}$ is an
antiholomorphic oscillator and the frequency is defined through the boundary
condition in the sector ${\bm{\beta}}$

$\nu_{\lambda}=\frac{1+\beta(\lambda)}{2},\ \ \
\nu_{\lambda^{*}}=\frac{1-\beta(\lambda)}{2}.$ (2.9)

Physical states must satisfy the Virasoro matching condition,
$M_{H}^{2}=M_{A}^{2}$, such that massless states are those with
$M_{H}^{2}=M_{A}^{2}=0$ and on–shell tachyons arise for sectors with
$M_{H}^{2}=M_{A}^{2}<0$.

###  2.1 Symmetric $\mathbb{Z}_{2}\\!\times\\!\mathbb{Z}_{2}$ $SO(10)$ models

For this work we explore the one–loop cosmological constant and $U(1)_{A}$
tadpole calculations for models defined through the basis set

$\displaystyle{\mathds{1}}$ $\displaystyle=\\{\psi^{\mu},\
\chi^{1,\dots,6},y^{1,\dots,6},\omega^{1,\dots,6}\
|~{}~{}~{}\overline{y}^{1,\dots,6},\overline{\omega}^{1,\dots,6},\overline{\eta}^{1,2,3},\overline{\psi}^{1,\dots,5},\overline{\phi}^{1,\dots,8}\\},$
$\displaystyle\bm{S}$ $\displaystyle=\\{{\psi^{\mu}},\chi^{1,\dots,6}\\},$
$\displaystyle\bm{e_{i}}$
$\displaystyle=\\{y^{i},w^{i}\;|\;\overline{y}^{i},\overline{w}^{i}\\},\
i=1,\dots,6,$ $\displaystyle\bm{b_{1}}$
$\displaystyle=\\{\chi^{34},\chi^{56},y^{34},y^{56}\;|\;\overline{y}^{34},\overline{y}^{56},\overline{\psi}^{1,\dots,5},\overline{\eta}^{1}\\},$
(2.10) $\displaystyle\bm{b_{2}}$
$\displaystyle=\\{\chi^{12},\chi^{56},y^{12},y^{56}\;|\;\overline{y}^{12},\overline{y}^{56},\overline{\psi}^{1,\dots,5},\overline{\eta}^{2}\\},$
$\displaystyle\bm{z_{1}}$ $\displaystyle=\\{\overline{\phi}^{1,\dots,4}\\},$
$\displaystyle\bm{z_{2}}$ $\displaystyle=\\{\overline{\phi}^{5,\dots,8}\\}.$

Such a basis can be associated with symmetric
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orbifolds [8, 9, 10, 11, 12] extensively
classified in previous works [7, 34, 35, 36, 37, 38, 39, 40, 41]. The NS
sector of the models associated to this basis produce spacetime vector bosons
generating the gauge group

$SO(10)\times U(1)_{1}\times U(1)_{2}\times U(1)_{3}\times SO(8)^{2},$ (2.11)

where we note that $U(1)_{1,2,3}$ are generated by the antiholomorphic
currents $\bar{\eta}^{k}\bar{\eta}^{k*}$. With respect to the basis (2.1) it
is useful to identify the important linear combination

$\bm{x}=\mathds{1}+\bm{S}+\sum_{i=1}^{6}\bm{e_{i}}+\bm{z_{1}}+\bm{z_{2}}=\\{\bar{\eta}^{1,2,3},\bar{\psi}^{1,...,5}\\}$
(2.12)

and
$\bm{b_{3}}=\bm{b_{1}}+\bm{b_{2}}+\bm{x}=\\{\chi^{12},\chi^{34},y^{12},y^{34}\;|\;\overline{y}^{12},\overline{y}^{34},\overline{\psi}^{1,\dots,5},\overline{\eta}^{3}\\}$,
which spans the third twisted plane of the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orbifold and facilitates the analysis of
the observable spinorial and vectorial representations as first developed in
[34, 35].

Models may then be defined through the choice of GGSO phases
$C{\bm{v_{i}}\atopwithdelims[ ]\bm{v_{j}}}$. There are 66 free phases for this
basis, with all others specified by modular invariance. The full space of
models is thus of size $2^{66}\sim 10^{19.9}$. The $\mathcal{N}=1$
supersymmetric subset of which is defined by those satisfying

$C{\bm{S}\atopwithdelims[ ]\bm{e_{i}}}=C{\bm{S}\atopwithdelims[
]\bm{z_{1}}}=C{\bm{S}\atopwithdelims[ ]\bm{z_{2}}}=-1,$ (2.13)

in order to preserve one gravitino. Furthermore, we note that the phases
$C{\mathds{1}\atopwithdelims[ ]\bm{S}}$ and $C{\bm{S}\atopwithdelims[
]\bm{b_{k}}}$, $k=1,2,3$, determine the chirality of the degenerate Ramond
vacuum $\ket{\bm{S}}$ and the gravitino is retained so long as

$C{\mathds{1}\atopwithdelims[ ]\bm{S}}=C{\bm{S}\atopwithdelims[
]\bm{b_{1}}}C{\bm{S}\atopwithdelims[ ]\bm{b_{2}}}C{\bm{S}\atopwithdelims[
]\bm{b_{3}}},$ (2.14)

which can, without loss of generality, be fixed to

$C{\mathds{1}\atopwithdelims[ ]\bm{S}}=C{\bm{S}\atopwithdelims[
]\bm{b_{1}}}=C{\bm{S}\atopwithdelims[ ]\bm{b_{2}}}=-1,$ (2.15)

for a scan of $\mathcal{N}=1$ vacua.

Since we are interested in non–SUSY vacua in this work we will be considering
the complement to this space of $\mathcal{N}=1$ vacua. In previous work on
non–SUSY heterotic string vacua from $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
orbifolds [42, 43, 44] tachyon free configurations satisfying various
phenomenological requirements and their one–loop cosmological constants are
explored. In section 6 we will detail how we ensure that only those models
free from physical tachyons are explored.

In the next section we show how the $U(1)_{A}$ gauge transformation manifests
in a 4 dimensional theory and how the Green–Schwarz mechanism deals with it.
We then compute the FI term through a string theory computation at 1–loop and
we conjecture how this term can lift the vacua from an anti–de Sitter to a de
Sitter.

##  3 Fayet–Iliopoulos $D$–term calculation

Anomalies arise whenever one, or some, of the classical symmetries are broken
by quantum effects. Some global symmetries need to be broken by anomalies in
order to reproduce observable phenomenology, however a breakdown of a local
symmetry indicates a symmetry current is no longer conserved and longitudinal,
non–physical modes of the gauge fields no longer may decouple from the
S–matrix. This can result in the loss of unitarity and appearance of
unphysical divergences.

In a four dimensional heterotic string theory the anomalies come from the
one–loop triangle diagram, where the external lines can be gauge fields,
gravitons or a mixture of each. In the following, for simplicity, we will
consider purely $U(1)$ gauge anomalies.
We start from the four dimensional effective action in the Einstein frame

$S=\frac{1}{2\kappa^{2}}\int
d^{4}x\sqrt{-G}\left[-\frac{\kappa^{2}e^{-2\Phi}}{2g^{2}}F_{\mu\nu}F^{\mu\nu}-\frac{e^{-4\Phi}}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}\right]$
(3.1)

with $H$ the field strength of the $B$ field such that

$H=dB-\frac{\kappa^{2}}{g^{2}}\Omega_{3}^{YM}=dB-\frac{\kappa^{2}}{g^{2}}\Tr\left[A\wedge
dA-\frac{2i}{3}A\wedge A\wedge A\right].$ (3.2)

Under a Gauge transformation, the gauge fields transform as
$A_{\mu}\longrightarrow A_{\mu}+\partial_{\mu}\Lambda$ and the effective
action varies as follows

$\delta S_{one-loop}=\frac{1}{4}\frac{1}{96\pi^{2}}\int
d^{4}x\Tr[Q^{3}]\epsilon^{\mu\nu\rho\sigma}\Lambda F_{\mu\nu}F_{\rho\sigma}.$
(3.3)

When the sum of these $U(1)$ charges is not zero this anomalous triangle
diagram contribution will be present, with massless particles circulating in
the loop.
The Green–Schwarz mechanism [45] provides a way to cancel these one loop
anomalies through the introduction of an antisymmetric $2$–form coupled at one
loop to the $U(1)$ $2$–form field strength in the effective Lagrangian

$-\frac{\zeta}{2}\epsilon^{\mu\nu\rho\sigma}B_{\mu\nu}F_{\rho\sigma},$ (3.4)

imposing the $B$ field to vary under the $U(1)$ Gauge transformation as
$\delta B=\frac{\kappa^{2}}{g^{2}}\Lambda F$, with the condition
$4\Tr[Q^{3}]=\Tr[Q]$, such that it compensates the anomalous triangle diagram.
The action can also be written in terms of the axion field, $a$, dual to the
antisymmetric $B$ field. The axion field is introduced as a Lagrange
multiplier term into the Lagrangian

$\displaystyle\begin{split}\frac{1}{\sqrt{-G}}\mathcal{L}=&-\frac{e^{-2\Phi}}{4g^{2}}F_{\mu\nu}F^{\mu\nu}-\frac{e^{-4\Phi}}{24\kappa^{2}}H_{\mu\nu\rho}H^{\mu\nu\rho}-\frac{\zeta}{2}\epsilon^{\mu\nu\rho\sigma}B_{\mu\nu}F_{\rho\sigma}\\\
&+\frac{a}{6}\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}H_{\nu\rho\sigma}+\frac{a}{4}\frac{\kappa^{2}}{g^{2}}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma},\end{split}$
(3.5)

such that its equations of motion give the definition of the $H$ tensor

$\displaystyle\begin{split}&\frac{1}{6}\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}H_{\nu\rho\sigma}+\frac{\kappa^{2}}{4g^{2}}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}=0\\\
\implies&H=dB-\frac{\kappa^{2}}{g^{2}}A\wedge F.\end{split}$ (3.6)

Using this result and integrating out the $H$ tensor in the Lagrangian (3.5)
gives

$\frac{1}{\sqrt{-G}}\mathcal{L}=-\frac{e^{-4\Phi}}{4g^{2}}F_{\mu\nu}F^{\mu\nu}-\kappa^{2}e^{4\Phi}\left(\partial_{\mu}a+2\zeta
A_{\mu}\right)^{2}+\frac{a}{4}\frac{\kappa^{2}}{g^{2}}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma},$
(3.7)

while the gauge transformation of the axion field can be found through the
gauge invariance of $H$ to give

$\delta a=-2\zeta\Lambda,$ (3.8)

such that when $\zeta\neq 0$ the anomalous gauge field acquires a mass and the
gauge symmetry is spontaneously broken.

In order to compute the FI coefficient $\zeta$ we have to evaluate the 2–point
function of the antisymmetric $B$ field and the $U(1)_{A}$ gauge boson at one
loop, where only chiral fermions with odd spin structure give a non–vanishing
contribution.

For the ghosts $b,c$ their zero modes are saturated by inserting
$\left<b\bar{b}c\bar{c}\right>$ into the path integral. For the ghost
superpartners $\beta,\gamma$ in order to project out of integration their zero
modes one of the vertex operators has to be put in the $0$ picture, the other
in the $-1$ and an insertion of the picture changing operator $e^{\phi}T_{F}$
is needed, where the scalar $\phi$ arises by the bosonization of the
$\beta-\gamma$ fields

$\beta=e^{-\phi}\partial\xi,\qquad\gamma=e^{\phi}\eta.$ (3.9)

The amplitude then reads

$\int\frac{d\tau_{1}d\tau_{2}}{2\tau_{2}}\int
d^{2}zd^{2}w\>\left<b(0)\bar{b}(0)c(0)\bar{c}(0)\>e^{\phi}T_{F}(0)\>V_{A,0}^{\mu}(z)V_{B,-1}^{\nu\rho}(w)\right>$
(3.10)

with the denominator needed to fix the translation Killing symmetries of the
torus and the residual discrete symmetry $z\longrightarrow-z$. The vertex
operators for the gauge boson and the antisymmetric field in the $-1$ and $0$
pictures are the following

$\begin{split}V_{A,-1}^{\mu,i}&=g_{c}\hat{k}^{-1/2}\psi^{\mu}\bar{J}^{i}e^{ik\cdot
X}\\\
V_{A,0}^{\mu}&=\sqrt{\frac{2}{\alpha^{\prime}}}g_{c}\hat{k}^{-1/2}\left(i\partial
X^{\mu}+\frac{\alpha^{\prime}}{2}k\cdot\psi\psi^{\mu}\right)\bar{J}e^{ik\cdot
X}\\\
V_{B,-1}^{\mu\nu}&=i\sqrt{\frac{2}{\alpha^{\prime}}}g_{c}\bar{\partial}X^{\mu}e^{-\phi}\psi^{\nu}e^{ip\cdot
X}\\\
V_{B,0}^{\mu\nu}&=\frac{2i}{\alpha^{\prime}}g_{c}\bar{\partial}X^{\mu}\left(i\partial
X^{\nu}+\frac{\alpha^{\prime}}{2}p\cdot\psi\psi^{\nu}\right)e^{ik\cdot
X},\end{split}$ (3.11)

with $g_{c}=\kappa/2\pi$ and $\hat{k}=1/2$. These normalisations are chosen
such that the string amplitudes match with the field theory calculation and
with the vertex operators in the two pictures related via
$V_{q+1}(z)=\lim\limits_{w\to z}e^{\phi}T_{F}(w)V_{q}(z)$, where the matter
supercurrent takes the form

$T_{F}=i\sqrt{\frac{2}{\alpha^{\prime}}}\psi^{\mu}\partial X_{\mu}.$ (3.12)

We note that we included the $\alpha^{\prime}$ dependence and that the
supercurrent is composed of only the uncompactified fields since vertex
operators do not involve internal lattice excitations.
When $\mathcal{N}\geq 2$ the amplitude vanishes due to the fermion zero modes.
However, for $\mathcal{N}=1$ they can be soaked up by the fermion correlator.
To see this, we can take the $\mathcal{O}(p)$ linear approximation of the
amplitude

$-\frac{k_{\alpha}}{\alpha^{{}^{\prime}1/2}}g_{c}^{2}\int\frac{d\tau_{1}d\tau_{2}}{\tau_{2}}\int
d^{2}zd^{2}w\>\langle
b\bar{b}c\bar{c}\rangle\>\langle\psi^{\sigma}\psi^{\alpha}\psi^{\mu}\psi^{\rho}\rangle\>\langle\bar{J}^{i}\rangle\>\langle\partial
X^{\gamma}\bar{\partial}X^{\nu}\rangle\>\langle
e^{\phi}e^{-\phi}\rangle\eta_{\sigma\gamma},$ (3.13)

where the fermion correlator can be written in terms of the fermion current
correlator

$\langle\psi^{i}\psi^{j}\rangle=\langle
J^{ij}\rangle=\frac{\epsilon^{ij}}{2\pi
i}\partial_{\nu}\Tr\left[(-1)^{F}e^{2\pi i\nu J^{ij}}\right]_{\nu=0},$ (3.14)

such that the $\langle\psi^{\sigma}\psi^{\alpha}\psi^{\mu}\psi^{\rho}\rangle$
term acts as follows on the four non–compact fermions

$\langle\psi^{\sigma}\psi^{\alpha}\psi^{\mu}\psi^{\rho}\rangle=\epsilon^{\sigma\alpha\mu\rho}\frac{\partial_{\nu}}{2\pi
i}\frac{\partial_{\omega}}{2\pi
i}\left(-\frac{1}{2}\frac{\vartheta_{1}(\nu)}{\eta^{2}}\frac{\vartheta_{1}(\omega)}{\eta}\right)=\frac{1}{2}\epsilon^{\sigma\alpha\mu\rho}\eta^{4},$
(3.15)

making use of $\partial_{\nu}\vartheta_{1}(\nu)=2\pi\eta^{3}$. The other
one–loop correlators are the following

$\begin{split}\langle b\bar{b}c\bar{c}\rangle&=\eta^{2}\bar{\eta}^{2}\\\
\langle
X^{\gamma}(w,\bar{w})X^{\nu}(0)\rangle&=\left(-\frac{\alpha^{\prime}}{2}\ln\vartheta_{1}\left(\frac{w}{2\pi}\right)\bar{\vartheta}_{1}\left(\frac{\bar{w}}{2\pi}\right)+\frac{\alpha^{\prime}}{4\pi\tau_{2}}\left(\text{Im}\
z\right)^{2}\right)\eta^{\gamma\nu}\\\ \langle
e^{\phi}e^{-\phi}\rangle&=\frac{1}{\eta^{2}},\end{split}$ (3.16)

and for the current

$\langle\bar{J}^{i}\rangle=\frac{\partial_{\nu}}{2\pi
i}\Tr\left[\left(-1\right)^{F}q^{H}\bar{q}^{\bar{H}}e^{2\pi
i\nu\bar{J}^{i}}\right]\Big{|}_{\nu=0}=\bar{\eta}^{2}\Tr\left[\left(-1\right)^{F}q^{H}\bar{q}^{\bar{H}-\frac{1}{12}}q_{i}\right],$
(3.17)

where the derivative acts on the the states charged under $U(1)_{i}$,
$i=1,2,3$. In the massless limit only chiral fermions contribute with
$\left(H,\bar{H}\right)=\left(0,\frac{1}{12}\right)$ so we get

$\langle\bar{J}^{i}\rangle=\bar{\eta}^{2}\Tr\left[\left(-1\right)^{F}q_{i}\right]=\bar{\eta}^{2}\Tr
Q_{i},$ (3.18)

with $\Tr Q_{i}=\sum n\,q_{i}\,h$ summed over all states in the spectrum such
that $n$, $q_{i}$ and $h$ are the number of massless fermions, their
$U(1)_{i}$ charges and their chirality, respectively.
Note that the anomalous $U(1)_{A}$ charge is given as combination of the three
$U(1)_{1,2,3}$ produced by the world–sheet currents
$:\bar{\eta}^{i*}\bar{\eta}^{i}:$, $i=1,2,3$, according to

$U(1)_{A}=\sum_{i=1}^{3}\frac{aU_{1}+bU_{2}+cU_{3}}{\sqrt{a^{2}+b^{2}+c^{2}}}$
(3.19)

where $(a,b,c)=\frac{1}{k}(\Tr U_{1},\Tr U_{2},\Tr U_{3})$ and
$k=\text{gcd}(\Tr U_{1},\Tr U_{2},\Tr U_{3})$.
Adding the contribution of the four non–compact bosons from the partition
function and their zero modes allows us to write the amplitude as

$\frac{ig_{c}^{2}}{256\pi^{5}\alpha^{{}^{\prime}3/2}}k_{\alpha}\epsilon^{\nu\alpha\mu\rho}\Tr
Q_{A}\int\frac{d\tau_{1}d\tau_{2}}{\tau_{2}^{4}}\int
d^{2}zd^{2}w=\frac{ig_{c}^{2}}{12\alpha^{{}^{\prime}3/2}}k_{\alpha}\epsilon^{\nu\alpha\mu\rho}\Tr
Q_{A},$ (3.20)

using that $\int d^{2}z=2\left(2\pi\right)^{2}\tau_{2}$ and $\pi/3$ is the
volume of the fundamental domain. Reducing the constants and doing further
contractions of this $\mathcal{N}=1$ amplitude allows us to finally write the
FI term as

$\zeta=M_{s}^{2}\frac{\Tr Q_{A}}{192\pi^{2}},$ (3.21)

such that, when $\Tr Q_{A}\neq 0$, there is an additional positive
contribution in the effective $D$–term potential

$V_{D}=\frac{1}{2}g_{s}^{2}\zeta^{2}$ (3.22)

that corresponds to a two–loop dilaton tadpole. This additional term was
originally computed both from the low energy energy effective action in ref.
[18], as well as through explicit two–loop string calculations [20].

As stated above, only for $\mathcal{N}=1$ unbroken supersymmetry is the FI
term non–vanishing. In the free fermionic models, the breaking
$\mathcal{N}=4\rightarrow\mathcal{N}=1$ is achieved by the introduction of the
$\bm{b_{1}}$ and $\bm{b_{2}}$ vectors (2.1) associated to
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orbifold twists. The breaking of the last
supersymmetry $\mathcal{N}=1\rightarrow\mathcal{N}=0$ is achieved by setting
properly the GGSO phases as delineated in section 2. We will discuss the
contribution of the FI $D$–term associated with $\Tr U(1)_{A}$ in these
$\mathcal{N}=0$ models in the following sections.

Our statement is that once the last supersymmetry is broken, either
spontaneously or explicitly, the $D$–term contribution (3.22) will still be
present. The same tadpole diagram leading to the FI term in the
$\mathcal{N}=1$ supersymmetric case is also present in the $\mathcal{N}=0$
case. The analysis outlined above follows through irrespective of whether the
model has $\mathcal{N}=1$ or $\mathcal{N}=0$ supersymmetry. The supersymmetric
case does guarantee a measure of stability whereas the non–supersymmetric case
is fraught with further uncertainties. For example, non–supersymmetric string
vacua contain additional tadpole diagrams for the dilaton, which indicate that
the string equations of motion are not satisfied in Minkowski four dimensional
spacetime with constant dilaton, and the computational stability from higher
loops is not preserved. Furthermore, in the following, analysis of the
potential is performed with respect to a single internal moduli and all other
internal moduli are set at the free fermionic point but are not fixed. One
direction of improvement on the analysis that we present here is to use the
Kiritsis–Kounnas modular invariant regularisation scheme [46] that can
regulate the infrared divergences. We note that these caveats are relevant in
general in non–supersymmetric string vacua that have been of some old and
recent interest in the literature [47, 48, 49, 50, 51, 52, 53, 54, 55]. For
our purposes here we note that the contribution of eq. (3.22), when non–zero,
which has the same mass dimension, $M_{s}^{4}$, as the cosmological constant,
will destabilize the vacua adding a positive contribution and lift the minima
of the one–loop potential.

For the purpose of getting numerical results we fix the string coupling
$\mathcal{O}(g_{s})\sim 1$, which corresponds to
$\alpha^{\prime}=g_{s}/4\pi\sim 0.1$. This order of magnitude can be justified
by reference to work on gauge coupling unification from string model building,
for example in ref. [56]. We observe that the $D$–term contribution goes with
$g_{s}^{2}$ so smaller values will quickly make an uplift less likely. In
section 8, we will see that we obtain an uplift from AdS to dS only very
rarely in our setup and so choosing this order of magnitude for the string
coupling, rather than a smaller one, helps provide a proof of concept.

In the next sections we study some heterotic string models through the
analysis of the partition function and its potential behavior and we
explicitly show how the FI term is used in order to uplift the minima.

##  4 Partition function and one–loop potential

The generic form of the partition function is given in eq. (2.4) applied to
the models defined through the basis (2.1). Using the techniques developed in
Appendix A, the partition function written in the free fermionic construction
can then be written in the following form

$\displaystyle Z=$
$\displaystyle\frac{1}{\eta^{10}\bar{\eta}^{22}}\,\frac{1}{2^{2}}\sum_{\begin{subarray}{c}a,k\\\
b,l\end{subarray}}\;\frac{1}{2^{6}}\,\sum_{\begin{subarray}{c}H_{i}\\\
G_{i}\end{subarray}}\;\frac{1}{2^{4}}\sum_{\begin{subarray}{c}h_{1},h_{2},P_{i}\\\
g_{1},g_{2},Q_{i}\end{subarray}}(-1)^{a+b+P_{1}Q_{1}+P_{2}Q_{2}+\Phi\left[\begin{smallmatrix}a&k&H_{i}&h_{1}&h_{2}&P_{i}\\\
b&l&G_{i}&g_{1}&g_{2}&Q_{i}\end{smallmatrix}\right]}$ (4.1)
$\displaystyle\times\vartheta\left[\begin{smallmatrix}a\\\
b\end{smallmatrix}\right]_{\psi^{\mu}}\vartheta\left[\begin{smallmatrix}a+h_{1}\\\
b+g_{1}\end{smallmatrix}\right]_{\chi^{12}}\vartheta\left[\begin{smallmatrix}a+h_{2}\\\
b+g_{2}\end{smallmatrix}\right]_{\chi^{34}}\vartheta\left[\begin{smallmatrix}a-h_{1}-h_{2}\\\
b-g_{1}-g_{2}\end{smallmatrix}\right]_{\chi^{56}}$
$\displaystyle\times\Gamma^{(1)}_{2,2}\left[\begin{smallmatrix}H_{1}&H_{2}\\\
G_{1}&G_{2}\end{smallmatrix}\middle|\begin{smallmatrix}h_{1}\\\
g_{1}\end{smallmatrix}\right](T^{(1)},U^{(1)})$
$\displaystyle\times\Gamma^{(2)}_{2,2}\left[\begin{smallmatrix}H_{3}&H_{4}\\\
G_{3}&G_{4}\end{smallmatrix}\middle|\begin{smallmatrix}h_{2}\\\
g_{2}\end{smallmatrix}\right](T^{(2)},U^{(2)})$
$\displaystyle\times\Gamma^{(3)}_{2,2}\left[\begin{smallmatrix}H_{5}&H_{6}\\\
G_{5}&G_{6}\end{smallmatrix}\middle|\begin{smallmatrix}h_{1}+h_{2}\\\
g_{1}+g_{2}\end{smallmatrix}\right](T^{(3)},U^{(3)})$
$\displaystyle\times\bar{\vartheta}\left[\begin{smallmatrix}k\\\
l\end{smallmatrix}\right]^{5}_{\bar{\psi}^{1-5}}\bar{\vartheta}\left[\begin{smallmatrix}k+h_{1}\\\
l+g_{1}\end{smallmatrix}\right]_{\bar{\eta}^{1}}\bar{\vartheta}\left[\begin{smallmatrix}k+h_{2}\\\
l+g_{2}\end{smallmatrix}\right]_{\bar{\eta}^{2}}\bar{\vartheta}\left[\begin{smallmatrix}k-h_{1}-h_{2}\\\
l-g_{1}-g_{2}\end{smallmatrix}\right]_{\bar{\eta}^{3}}\bar{\vartheta}\left[\begin{smallmatrix}k+P_{1}\\\
l+Q_{1}\end{smallmatrix}\right]^{4}_{\bar{\phi}^{1-4}}\bar{\vartheta}\left[\begin{smallmatrix}k+P_{2}\\\
l+Q_{2}\end{smallmatrix}\right]^{4}_{\bar{\phi}^{5-8}}.$

For symmetric $\mathbb{Z}_{2}\\!\times\\!\mathbb{Z}_{2}$ orbifolds the moduli
space is generally parametrised by three complex structure and three Kähler
moduli, one for each torus associated to the $\Gamma_{2,2}$ lattices. The
moduli space is then $SO(2,2)/SO(2)\times SO(2)$. At the maximally symmetric
(free fermionic) point $(T_{*},U_{*})$, at which bosonic degrees of freedom
can be fermionised, the lattices admit a factorised form which can be written
entirely in terms of theta functions

$\displaystyle\Gamma^{(1)}_{2,2}\left[\begin{smallmatrix}H_{1}&H_{2}\\\
G_{1}&G_{2}\end{smallmatrix}\middle|\begin{smallmatrix}h_{1}\\\
g_{1}\end{smallmatrix}\right](T^{(1)}_{*},U^{(1)}_{*})=\left|\vartheta\left[\begin{smallmatrix}H_{1}\\\
G_{1}\end{smallmatrix}\right]\vartheta\left[\begin{smallmatrix}H_{1}+h_{1}\\\
G_{1}+g_{1}\end{smallmatrix}\right]\vartheta\left[\begin{smallmatrix}H_{2}\\\
G_{2}\end{smallmatrix}\right]\vartheta\left[\begin{smallmatrix}H_{2}+h_{1}\\\
G_{2}+g_{1}\end{smallmatrix}\right]\right|$
$\displaystyle\Gamma^{(2)}_{2,2}\left[\begin{smallmatrix}H_{3}&H_{4}\\\
G_{3}&G_{4}\end{smallmatrix}\middle|\begin{smallmatrix}h_{2}\\\
g_{2}\end{smallmatrix}\right](T^{(2)}_{*},U^{(2)}_{*})=\left|\vartheta\left[\begin{smallmatrix}H_{3}\\\
G_{3}\end{smallmatrix}\right]\vartheta\left[\begin{smallmatrix}H_{3}+h_{2}\\\
G_{3}+g_{2}\end{smallmatrix}\right]\vartheta\left[\begin{smallmatrix}H_{4}\\\
G_{4}\end{smallmatrix}\right]\vartheta\left[\begin{smallmatrix}H_{4}+h_{2}\\\
G_{4}+g_{2}\end{smallmatrix}\right]\right|$ (4.2)
$\displaystyle\Gamma^{(3)}_{2,2}\left[\begin{smallmatrix}H_{5}&H_{6}\\\
G_{5}&G_{6}\end{smallmatrix}\middle|\begin{smallmatrix}h_{1}+h_{2}\\\
g_{1}+g_{2}\end{smallmatrix}\right](T^{(3)}_{*},U^{(3)}_{*})=\left|\vartheta\left[\begin{smallmatrix}H_{5}\\\
G_{5}\end{smallmatrix}\right]\vartheta\left[\begin{smallmatrix}H_{5}-h_{1}-h_{2}\\\
G_{5}-h_{1}-h_{2}\end{smallmatrix}\right]\vartheta\left[\begin{smallmatrix}H_{6}\\\
G_{6}\end{smallmatrix}\right]\vartheta\left[\begin{smallmatrix}H_{6}-h_{1}-h_{2}\\\
G_{6}-h_{1}-h_{2}\end{smallmatrix}\right]\right|.$

We furthermore note that the modular invariant phase
$\Phi\left[\begin{smallmatrix}a&k&H_{i}&h_{1}&h_{2}&P_{i}\\\
b&l&G_{i}&g_{1}&g_{2}&Q_{i}\end{smallmatrix}\right]$ in (4.1) implements the
various GGSO projections. A choice of phase is equivalent to a choice of GGSO
matrix and hence there is a unique one–to–one map between them. The factor of
$a+b$ ensures correct spin statistics, while the explicit inclusion of the
extra phase $P_{1}Q_{1}+P_{2}Q_{2}$ means that $\Phi=0$ is a valid modular
invariant choice.

The summation indices used to write the fermionic partition function (4.1)
correspond to various features of the model. The indices $a,b$ correspond to
the spin structures of the spacetime fermions $\psi^{\mu}$, while $k,l$ are
associated to the 16 right–moving complex fermions giving the gauge degrees of
freedom of the heterotic string. The non–freely acting
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ orbifold twists are associated to the
parameters $h_{1},g_{1}$ and $h_{2},g_{2}$. One of the key features of models
defined by the basis (2.1) is the inclusion of the basis vectors $\bm{e_{i}}$
which generate freely acting orbifold shifts in the internal dimensions of the
compact torus. In the partition function, these are realised by the indices
$H_{i},G_{i}$ parametrising each of the six independent shifts. The additional
twists $P_{i},Q_{i}$ correspond to the basis vectors $\bm{z_{1}}$ and
$\bm{z_{2}}$ acting on the hidden sector of our model.

The moduli–dependent form of the twisted/shifted lattice requires closer
attention. We know that all dependence on the geometric moduli is contained in
the untwisted sector of the model and hence

$\Gamma_{2,2}\left[\begin{smallmatrix}H_{1}&H_{2}\\\
G_{1}&G_{2}\end{smallmatrix}\middle|\begin{smallmatrix}h\\\
g\end{smallmatrix}\right](T,U)\Big{|}_{h,g\neq
0}=\Gamma_{2,2}\left[\begin{smallmatrix}H_{1}&H_{2}\\\
G_{1}&G_{2}\end{smallmatrix}\middle|\begin{smallmatrix}h\\\
g\end{smallmatrix}\right](T_{*},U_{*}).$ (4.3)

This means that for nonzero twists the lattice is precisely given by its
factorised form in (4). In the case of the untwisted sector, the shifted
lattice can be written in a Poisson resummed Hamiltonian form as

$\Gamma_{2,2}\left[\begin{smallmatrix}H_{1}&H_{2}\\\
G_{1}&G_{2}\end{smallmatrix}\middle|\begin{smallmatrix}0\\\
0\end{smallmatrix}\right](T,U)=\sum_{m_{i},n_{i}\in\mathbb{Z}}q^{\frac{1}{2}|\mathcal{P}_{L}(T,U)|^{2}}\bar{q}^{\frac{1}{2}|\mathcal{P}_{R}(T,U)|^{2}}e^{i\pi\sum_{i}((m_{i}+n_{i}+H_{i})G_{i}},$
(4.4)

where the left and right–moving momenta are

$\displaystyle\mathcal{P}_{L}=$
$\displaystyle\frac{1}{\sqrt{2T_{2}U_{2}}}\left[\frac{U}{2}(m_{1}+n_{1})-\frac{1}{2}(m_{2}+n_{2})+T(m_{1}-n_{1}+H_{1})+TU(m_{2}-n_{2}+H_{2})\right]$
(4.5) $\displaystyle\mathcal{P}_{R}=$
$\displaystyle\frac{1}{\sqrt{2T_{2}U_{2}}}\left[\frac{U}{2}(m_{1}+n_{1})-\frac{1}{2}(m_{2}+n_{2})+\bar{T}(m_{1}-n_{1}+H_{1})-TU(m_{2}-n_{2}+H_{2})\right].$

Written in this form, it is easy to extract the $q$–expansion of the partition
function at any given point in the moduli space which is crucial for
calculating the one–loop potential. It can be shown that the twisted/shifted
lattice sums (4.3) and (4.4) evaluated at the special point
$(T_{*},U_{*})=(i/2,i)$ indeed reproduce the free fermionic form of the
partition function (4).

Given the fermionic partition function (4.1), the one–loop potential is
evaluated by summing over all inequivalent worldsheet tori via the modular
invariant integral

$V_{\text{one--
loop}}(T^{(i)},U^{(i)})=-\frac{1}{2}\frac{M_{s}^{4}}{(2\pi)^{4}}\int_{\mathcal{F}}\frac{d^{2}\tau}{\tau_{2}^{2}}\,Z(\tau,\bar{\tau},T^{(i)},U^{(i)}),$
(4.6)

where in $Z(\tau,\bar{\tau},T^{(i)},U^{(i)})$ we include the spacetime bosonic
degrees of freedom arising from the worldsheet. In models with a $U(1)_{A}$,
an additional contribution to the potential $V_{D}$ is generated as discussed
in Section 3. Since this term is independent of the geometric moduli it
provides a constant shift of the potential throughout moduli space. Hence we
use this to write the vacuum energy as

$V_{\text{total}}=V_{\text{one--loop}}(T^{(i)},U^{(i)})+V_{D},$ (4.7)

where $V_{D}$ is given in term of the trace of the anomalous $U(1)_{A}$ via
(3.22).

In order to calculate the one–loop potential of our models we must be able to
move away from this special point in the moduli space. The details of the
translation of a free fermionic model into a $\mathbb{Z}_{2}^{N}$ orbifold was
developed in [57] and used in [26, 29] to calculate one–loop potentials. Some
details of this translation are given in Appendix A.

The motivation for this procedure is that it enables us to move away from the
free fermionic point in the moduli space. Although being fixed at this point
allows for a generic analysis of many important features of a string model,
for issues such as SUSY breaking and one–loop stability, analysis across the
moduli space is required.

In general, we can perturb away from the free fermionic point using marginal
operators given by the Thirring Interactions [58] which take the form
$J^{i}(z){\overline{J}}^{j}(\bar{z})=:y^{i}w^{i}::{\bar{y}}^{j}{\bar{y}}^{j}:$.
Writing these currents in bosonised form we identify the geometric moduli
$J^{i}(z){\overline{J}}^{j}({\bar{z}})=\partial X^{i}\overline{\partial
X}^{j}$. For symmetric $\mathbb{Z}_{2}\\!\times\\!\mathbb{Z}_{2}$ orbifolds we
parametrise the moduli space by a complex structure and Kähler modulus for
each torus associated to the $\Gamma_{2,2}$ lattices: $(T^{(1)},U^{(1)})$,
$(T^{(2)},U^{(2)})$ and $(T^{(3)},U^{(3)})$. These moduli span the familiar
$SO(2,2)/SO(2)\times SO(2)$ moduli space of
$\mathbb{Z}_{2}\\!\times\\!\mathbb{Z}_{2}$ symmetric orbifolds.

Once we have installed this moduli dependence and followed the translation
procedure we can calculate the one–loop potential numerically at specific
moduli values using

$V_{\text{one--
loop}}(T^{(i)},U^{(i)})=-\frac{1}{2}\frac{M_{s}^{4}}{(2\pi)^{4}}\int_{\mathcal{F}}\frac{d^{2}\tau}{\tau_{2}^{2}}\,Z(\tau,\bar{\tau},T^{(i)},U^{(i)})$
(4.8)

where we integrate over the fundamental domain

$\mathcal{F}=\\{\tau\in\mathbb{C}\;|\;|\tau|>1\,,\,|\tau_{1}|<1/2\\}.$ (4.9)

Calculating the one–loop potential is then an exercise in solving modular
integrals.

One important observation is that we have 6 complex moduli inside this
integral rendering an analysis of the potential in all directions impractical.
A logical approach used in [26, 29], is to take the volume of the first torus
associated to $\text{Im}(T^{(1)})=T_{2}$ and analyse the potential solely in
this direction, with the other moduli all fixed at their values at the free
fermionic point. This choice is somewhat arbitrary except in the case of an
SSS breaking (discussed further in the section 5) where an internal shift in
the first torus means that $T_{2}$ parameterises the SUSY–breaking and SUSY
will be restored in the large volume limit $T_{2}\rightarrow\infty$.

###  4.1 Calculating $\Tr U(1)_{A}$ from the partition function

In section 3 we discussed how the FI $D$–term contributed at 2–loop to the
potential of our model. Its magnitude depended on the trace of the fields
charged under the $U(1)_{A}$ in our model, that propagate in the anomalous
triangle diagram in four dimensions. There are two equivalent ways to
calculate this trace. One way is to extract those states of the massless
spectrum charged under $U(1)_{A}$ and add up their charges. This approach
utilises free fermionic classification tools that are easily computerised. The
details of this approach are given in Section 7. The second way to calculate
$\Tr U(1)_{A}$ is directly from the partition function, which will be
explained in this subsection.
In order to perform this calculation it helps to rewrite the partition
function (4.1) as follows

$\displaystyle\tilde{Z}=$
$\displaystyle\frac{1}{\eta^{11}\bar{\eta}^{22}}\,\frac{1}{2^{2}}\sum_{\begin{subarray}{c}a,k\\\
b,l\end{subarray}}\;\frac{1}{2^{6}}\,\sum_{\begin{subarray}{c}H_{i}\\\
G_{i}\end{subarray}}\;\frac{1}{2^{4}}\sum_{\begin{subarray}{c}h_{1},h_{2},P_{i}\\\
g_{1},g_{2},Q_{i}\end{subarray}}(-1)^{a+b+P_{1}Q_{1}+P_{2}Q_{2}+\Phi\left[\begin{smallmatrix}a&k&H_{i}&h_{1}&h_{2}&P_{i}\\\
b&l&G_{i}&g_{1}&g_{2}&Q_{i}\end{smallmatrix}\right]}$ (4.10)
$\displaystyle\times\vartheta\left[\begin{smallmatrix}a\\\
b\end{smallmatrix}\right](\nu)_{\psi^{\mu}}\vartheta\left[\begin{smallmatrix}a\\\
b\end{smallmatrix}\right](\omega)_{\psi^{\mu}}\left(\frac{1}{\eta^{2}}\right)\vartheta\left[\begin{smallmatrix}a+h_{1}\\\
b+g_{1}\end{smallmatrix}\right]_{\chi^{12}}\vartheta\left[\begin{smallmatrix}a+h_{2}\\\
b+g_{2}\end{smallmatrix}\right]_{\chi^{34}}\vartheta\left[\begin{smallmatrix}a-h_{1}-h_{2}\\\
b-g_{1}-g_{2}\end{smallmatrix}\right]_{\chi^{56}}$
$\displaystyle\times\Gamma^{(1)}_{2,2}\left[\begin{smallmatrix}H_{1}&H_{2}\\\
G_{1}&G_{2}\end{smallmatrix}\middle|\begin{smallmatrix}h_{1}\\\
g_{1}\end{smallmatrix}\right](T^{(1)},U^{(1)})$
$\displaystyle\times\Gamma^{(2)}_{2,2}\left[\begin{smallmatrix}H_{3}&H_{4}\\\
G_{3}&G_{4}\end{smallmatrix}\middle|\begin{smallmatrix}h_{2}\\\
g_{2}\end{smallmatrix}\right](T^{(2)},U^{(2)})$
$\displaystyle\times\Gamma^{(3)}_{2,2}\left[\begin{smallmatrix}H_{5}&H_{6}\\\
G_{5}&G_{6}\end{smallmatrix}\middle|\begin{smallmatrix}h_{1}+h_{2}\\\
g_{1}+g_{2}\end{smallmatrix}\right](T^{(3)},U^{(3)})$
$\displaystyle\times\bar{\vartheta}\left[\begin{smallmatrix}k\\\
l\end{smallmatrix}\right]^{5}_{\bar{\psi}^{1-5}}\bar{\vartheta}\left[\begin{smallmatrix}k+h_{1}\\\
l+g_{1}\end{smallmatrix}\right](u_{1})_{\bar{\eta}^{1}}\bar{\vartheta}\left[\begin{smallmatrix}k+h_{2}\\\
l+g_{2}\end{smallmatrix}\right](u_{2})_{\bar{\eta}^{2}}\bar{\vartheta}\left[\begin{smallmatrix}k-h_{1}-h_{2}\\\
l-g_{1}-g_{2}\end{smallmatrix}\right](u_{3})_{\bar{\eta}^{3}}\bar{\vartheta}\left[\begin{smallmatrix}k+P_{1}\\\
l+Q_{1}\end{smallmatrix}\right]^{4}_{\bar{\phi}^{1-4}}\bar{\vartheta}\left[\begin{smallmatrix}k+P_{2}\\\
l+Q_{2}\end{smallmatrix}\right]^{4}_{\bar{\phi}^{5-8}}.$

where we have reinserted the orthogonal component of $\psi^{\mu}$ and its
ghost contribution and the theta functions corresponding to the
$\bar{\eta}^{i}$ fields acquire a non–zero argument $u_{i}$. The total
$U(1)_{i=1,2,3}$ traces will then be given as

$\Tr
U(1)_{i}=\frac{\partial_{\nu}}{2\pi}\frac{\partial_{\omega}}{2\pi}\frac{\partial_{u_{i}}}{2\pi}\tilde{Z}\Big{|}_{\nu,\omega,u_{i}=0}$
(4.11)

where the derivatives $\partial_{\nu}\partial_{\omega}$ correspond to the
correlator $\langle\psi\psi\psi\psi\rangle$, while $\partial_{u_{i}}$
corresponds to the $U(1)_{i}$ contribution of $\langle\bar{J}^{i}\rangle$ in
(3.13) acting on the partition function. The total anomalous $U(1)_{A}$ will
be the combination of the three $U(1)_{i}$ according to (3.19).

##  5 Explicit vs spontaneous SUSY breaking

The SUSY breaking $\mathcal{N}=1\rightarrow 0$ can happen in two ways:
explicit or spontaneous (SSS) breaking. In the latter case, the GGSO phases
can be set such that the gravitino acquires a mass and supersymmetry is broken
spontaneously. The requirement is that the partition function, which will be
non–zero in any point in the $T_{2}$ moduli space, vanishes when
$T_{2}\rightarrow\infty$ such that the potential in that limit vanishes and
supersymmetry is restored. Spontaneous breaking certainly has attractive
features compared with the explicit case. As discussed in various works [13,
14, 15, 59, 54, 60, 61], when accompanied by massless Bose–Fermi degeneracy at
some point in the moduli space, $N_{b}^{0}=N_{f}^{0}$, we have the so–called
‘Super No–Scale’ models in which the cosmological constant is exponentially
suppressed according to [62]

$\Lambda\propto(N^{0}_{b}-N^{0}_{f})\frac{1}{T_{2}^{2}}+\mathcal{O}(e^{-c\sqrt{T_{2}}}),$
(5.1)

We note that without the $N_{b}^{0}=N_{f}^{0}$ condition, SSS models have
polynomial, rather than exponential, suppression of their one–loop
cosmological constant. However, as noted in [26, 29, 30] such ‘super no–scale’
conditions are merely necessary, not sufficient, conditions on the global
structure of the effective potential, which will be crucially dependent on the
full mass tower of states, including the non level–matched ones around special
self–dual points in moduli space.

Despite the attractive features of such ‘super no–scale’ models, we note that
the cosmological constant problem remains an issue, also in these models. Even
in the case of spontaneous SUSY–breaking, there are sectors in the additive
group $\Xi$ generated by the basis vectors that produce equal numbers of
bosons and fermions [53]. The states from these sectors do not reside in
super–multiplets as supersymmetry is broken. As usual the respective bosonic
and fermionic states arise from the generic sectors, e.g. $\alpha\in\Xi$ and
$S+\alpha\in\Xi$, and the bosonic and fermionic states from these sectors
differ in some of their $U(1)$ charges, reflecting the fact that supersymmetry
is broken. However, the phenomenological requirement still demands that the
bosonic states from these sectors, that may, for example, correspond to the
would be superpartners of the chiral generations, receive mass of the order of
$1{\rm TeV}$. Generating this mass splitting between will produce a
cosmological constant. Similarly, the other mass scales in the Standard Model,
e.g. the QCD scale, will contribute to the cosmological constant. The
cosmological constant problem is therefore much more profound, indicating a
fundamental dichotomy between Quantum Field Theories expectations and
gravitational observations, and it is naive to expect that the suppression
observed in eq. (5.1) can address the problem. We would ideally also consider
higher loop contributions, e.g. the two–loop cosmological constant that should
ideally be incorporated into this analysis. The fact is that the cosmological
constant problem remains regardless of the SUSY–breaking mechanism. For this
reason, we also explore some models with explicitly broken supersymmetry in
our analysis. However, we also detail in the following how to identify the SSS
models, which have their distinct phenomenological characteristics.

In the SSS SUSY–breaking the gravitino acquires a mass proportional to
$\frac{1}{T_{2}}$, such that SUSY is restored at the border of the moduli
space when $T_{2}\rightarrow\infty$ and the partition function vanishes. For
this to happen, we require that in that limit the modular block in the
partition function relating to the
$\\{\psi^{\mu},\chi^{12},\chi^{34},\chi^{56}\\}$ fermions gives rise to the
Jacobi identity. To explore when this can happen, we set
$\left(h_{i},g_{i}\right)$ indices to zero since for all cases when SUSY is
restored from $\mathcal{N}=0$ to $\mathcal{N}\rightarrow 1,2,4$, the partition
function $Z$ will vanish.

The momenta in (4.5), through a redefinition of the summation variables
$n_{i}\rightarrow n_{i}+m_{i}+H_{i}$ and fixing $T_{1}$, $U_{1}$ and $U_{2}$
at the free fermionic point, can be written as follows

$\displaystyle\mathcal{P}_{L}=$
$\displaystyle\frac{1}{\sqrt{2T_{2}}}\left[\left(m_{1}+\frac{n_{1}}{2}+\frac{H_{1}}{2}-n_{1}T_{2}\right)+i\left(m_{2}+\frac{n_{2}}{2}+\frac{H_{2}}{2}-n_{2}T_{2}\right)\right]$
(5.2) $\displaystyle\mathcal{P}_{R}=$
$\displaystyle\frac{1}{\sqrt{2T_{2}}}\left[\left(m_{1}+\frac{n_{1}}{2}+\frac{H_{1}}{2}+n_{1}T_{2}\right)+i\left(m_{2}+\frac{n_{2}}{2}+\frac{H_{2}}{2}+n_{2}T_{2}\right)\right].$

We can now observe that when $T_{2}\rightarrow\infty$ the only non–zero term
in the moduli–dependent lattice sum (4.4) is for $n_{i}=0$ and in this limit
it contributes according to

$\Gamma^{(1)}_{2,2}\left[\begin{smallmatrix}H_{1}&H_{2}\\\
G_{1}&G_{2}\end{smallmatrix}\middle|\begin{smallmatrix}h_{1}\\\
g_{1}\end{smallmatrix}\right](T_{2}^{(1)}\rightarrow\infty,T_{1*}^{(1)},U^{(1)}_{*})\rightarrow\sum_{m_{i}\in\mathbb{Z}}1,$
(5.3)

where the sum, up to a normalization that will not affect our discussion, can
be set to $1$. Then the partition function (4.1) will factorize as

$\displaystyle Z$
$\displaystyle(T_{2}^{(1)}\rightarrow\infty,T_{1*}^{(1)},U^{(1)}_{*})=\,\frac{1}{\eta^{10}\bar{\eta}^{22}}\,\frac{1}{2^{12}}\sum_{\begin{subarray}{c}k,H_{i},P_{i}\\\
l,G_{i},Q_{i}\end{subarray}}(-1)^{P_{1}Q_{1}+P_{2}Q_{2}}\,\bar{\vartheta}\left[\begin{smallmatrix}k\\\
l\end{smallmatrix}\right]^{8}\bar{\vartheta}\left[\begin{smallmatrix}k+P_{1}\\\
l+Q_{1}\end{smallmatrix}\right]^{4}\bar{\vartheta}\left[\begin{smallmatrix}k+P_{2}\\\
l+Q_{2}\end{smallmatrix}\right]^{4}$ (5.4)
$\displaystyle\times\;\Gamma^{(2)}_{2,2}\left[\begin{smallmatrix}H_{3}&H_{4}\\\
G_{3}&G_{4}\end{smallmatrix}\middle|\begin{smallmatrix}0\\\
0\end{smallmatrix}\right](T^{(2)}_{*},U^{(2)}_{*})\;\Gamma^{(3)}_{2,2}\left[\begin{smallmatrix}H_{5}&H_{6}\\\
G_{5}&G_{6}\end{smallmatrix}\middle|\begin{smallmatrix}0\\\
0\end{smallmatrix}\right](T^{(3)}_{*},U^{(3)}_{*})$
$\displaystyle\times\biggl{(}(-1)^{\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}\;\vartheta_{3}^{4}+(-1)^{1+\Phi\left[\begin{smallmatrix}1&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}\;\vartheta_{2}^{4}+(-1)^{1+\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
1&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}\;\vartheta_{4}^{4}\biggl{)}.$

The SSS condition requires

$Z(T_{2}^{(1)}\rightarrow\infty,T_{1*}^{(1)},U^{(1)}_{*})=0$ (5.5)

In order to vanish the phase has to satisfy

$\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}=\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}1&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}=\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
1&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}$ (5.6)

which translates into a set of intricate conditions and relations between the
GGSO phases. In Appendix B we will show explicitly two examples, one
satisfying the SSS conditions and the other with explicit SUSY breaking.

###  5.1 T–duality

For our choice of models specified by the basis set (2.1) T–duality is not
always preserved. In particular, the symmetric shifts represented by the
$\bm{e_{i}}$ basis vectors may spoil the original $SL(2;\mathbb{Z})_{T}$
symmetry associated to the moduli $T$. We will now show how, for an SSS model,
T–duality can be broken. As already specified in section 4, we will vary
$T_{2}$ only associated to the first 2–torus.

The left and right momenta of the shifted lattice in (4.5) are left invariant
under the following transformation

$T\rightarrow-\frac{1}{4T}\;\;\;\leftrightarrow\;\;\;T_{2}\rightarrow\frac{1}{4T_{2}}.$
(5.7)

However, the phase in the lattice (4.4) gets an additional term according to

$e^{i\pi\sum_{i}((m_{i}+n_{i}+H_{i})G_{i}}\rightarrow
e^{i\pi\sum_{i}((m_{i}+n_{i}+H_{i})G_{i}}\times
e^{i\pi\left(H_{1}G_{1}+H_{2}G_{2}\right)},$ (5.8)

which will generally break T–duality. The same result can also be obtained
following the discussion of section 5. In the $T_{2}\rightarrow 0$ limit, with
the other moduli fixed, the only non–zero contributions from the lattice (4.4)
are $1$, setting $m_{i}=n_{i}=H_{i}=0$ in the lattice sum, $\sum_{m_{i}}1$,
setting $n_{i}=-2m_{i},H_{i}=0$, and
$e^{i\pi\left(H_{1}G_{1}+H_{2}G_{2}\right)}$ with $m_{i}=0,n_{i}=-H_{i}$. The
first two terms correspond, up to a normalization constant, to (5.3) which for
an SSS model give a vanishing contribution. Meanwhile, the third contribution
generates an additional phase

$\Gamma^{(1)}_{2,2}\left[\begin{smallmatrix}H_{1}&H_{2}\\\
G_{1}&G_{2}\end{smallmatrix}\middle|\begin{smallmatrix}h_{1}\\\
g_{1}\end{smallmatrix}\right](T_{2}^{(1)}\rightarrow
0,T_{1*}^{(1)},U^{(1)}_{*})\rightarrow
e^{i\pi\left(H_{1}G_{1}+H_{2}G_{2}\right)},$ (5.9)

In order to impose T–duality (5.7), in addition to (5.5) for the SSS
condition, we must then require

$Z(T_{2}^{(1)}\rightarrow\infty,T_{1*}^{(1)},U^{(1)}_{*})=Z(T_{2}^{(1)}\rightarrow
0,T_{1*}^{(1)},U^{(1)}_{*})=0,$ (5.10)

with

$\displaystyle Z$ $\displaystyle(T_{2}^{(1)}\rightarrow
0,T_{1*}^{(1)},U^{(1)}_{*})=\,\frac{1}{\eta^{10}\bar{\eta}^{22}}\,\frac{1}{2^{12}}\sum_{\begin{subarray}{c}k,H_{i},P_{i}\\\
l,G_{i},Q_{i}\end{subarray}}(-1)^{P_{1}Q_{1}+P_{2}Q_{2}+H_{1}G_{1}+H_{2}G_{2}}$
(5.11) $\displaystyle\times\bar{\vartheta}\left[\begin{smallmatrix}k\\\
l\end{smallmatrix}\right]^{8}\bar{\vartheta}\left[\begin{smallmatrix}k+P_{1}\\\
l+Q_{1}\end{smallmatrix}\right]^{4}\bar{\vartheta}\left[\begin{smallmatrix}k+P_{2}\\\
l+Q_{2}\end{smallmatrix}\right]^{4}\;\Gamma^{(2)}_{2,2}\left[\begin{smallmatrix}H_{3}&H_{4}\\\
G_{3}&G_{4}\end{smallmatrix}\middle|\begin{smallmatrix}0\\\
0\end{smallmatrix}\right](T^{(2)}_{*},U^{(2)}_{*})\;\Gamma^{(3)}_{2,2}\left[\begin{smallmatrix}H_{5}&H_{6}\\\
G_{5}&G_{6}\end{smallmatrix}\middle|\begin{smallmatrix}0\\\
0\end{smallmatrix}\right](T^{(3)}_{*},U^{(3)}_{*})$
$\displaystyle\times\biggl{(}(-1)^{\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}\;\vartheta_{3}^{4}+(-1)^{1+\Phi\left[\begin{smallmatrix}1&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}\;\vartheta_{2}^{4}+(-1)^{1+\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
1&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}\;\vartheta_{4}^{4}\biggl{)}.$

As in (5.6), T–duality now requires

$\displaystyle\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]+H_{1}G_{1}+H_{2}G_{2}}$
$\displaystyle=\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}1&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]+H_{1}G_{1}+H_{2}G_{2}}$ (5.12)
$\displaystyle=\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
1&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]+H_{1}G_{1}+H_{2}G_{2}},$

which again will correspond to specific constraints on the GGSO phases.

Models which satisfy (5.10) will then exhibit a SSS SUSY breaking with
unbroken T–duality and the one–loop potential will then have the following
behavior

$V_{\text{one--loop}}(T_{2})=V_{\text{one--
loop}}\left(\frac{1}{4T_{2}}\right)$ (5.13)

In particular the extrema of the potential, either a maximum or a minimum,
will lie at the self–dual point $T_{2}=\frac{1}{2}$.

We note that if instead of having $\bm{e_{i}}$ in our basis, we had
$\bm{T_{j}}=\bm{e_{2j-1}}+\bm{e_{2j}}$, $j=1,2,3$, as used in [26, 29], the
additional phase in (5.8) is modified according to
$e^{i\pi(H_{1}G_{1}+H_{2}G_{2})}\rightarrow
e^{i\pi(H_{1}G_{1}+H_{1}G_{1})}=1$. Thus for these models, with the indices
$H_{i},G_{i}\in\mathbb{Z}$, T–duality (5.7) will always be satisfied. In
Appendix B we will show how for an SSS model the T–duality conditions can be
implemented.

##  6 Tachyon projection

The tachyonic sectors in the models defined by the basis (2.1) and their
projection conditions are much the same as detailed in refs. [42, 43]. The
sectors and their mass levels are summarised in Table 1.

Mass Level | Vectorials | Spinorials
---|---|---
$(-1/2,-1/2)$ | $\\{\bar{\lambda}^{m}\\}\ket{0}$ |  $\bm{z_{1}},\ \bm{z_{2}}$
$(-3/8,-3/8)$ | $\\{\bar{\lambda}^{m}\\}\bm{e_{i}}$ |  $\bm{e_{i}}+\bm{z_{1}},\ \bm{e_{i}}+\bm{z_{2}}$
$(-1/4,-1/4)$ | $\\{\bar{\lambda}^{m}\\}\bm{e_{i}}+\bm{e_{j}}$ | $\bm{e_{i}}+\bm{e_{j}}+\bm{z_{1}},\ \bm{e_{i}}+\bm{e_{j}}+\bm{z_{2}}$
$(-1/8,-1/8)$ | $\\{\bar{\lambda}^{m}\\}\bm{e_{i}}+\bm{e_{j}}+\bm{e_{k}}$ | $\bm{e_{i}}+\bm{e_{j}}+\bm{e_{k}}+\bm{z_{1}},\ \bm{e_{i}}+\bm{e_{j}}+\bm{e_{k}}+\bm{z_{2}}$
Table 1:  _Level-matched tachyonic sectors and their mass level, where $i\neq
j\neq k=1,...,6$ and $\bar{\lambda}^{m}$ is any right-moving complex fermion
with NS boundary condition for the relevant tachyonic sector._

In order to determine whether a sector survives the GGSO projections and
remains in the spectrum we can construct a projector. For example, taking a
sector with no oscillators, $\ket{{\bm{\beta}}}$, the survival/projection
condition is encapsulated in the generalised projector

$\mathbb{P}_{{\bm{\beta}}}=\prod_{\bm{\xi}\in\Upsilon({\bm{\beta}})}\frac{1}{2}\left(1+\delta_{{\bm{\beta}}}C{{\bm{\beta}}\atopwithdelims[
]\bm{\xi}}\right),$ (6.1)

where

$\delta_{{\bm{\beta}}}=\begin{cases}+1\ \ \text{if }\beta(\psi^{\mu})=0\ \
\iff\ \ \text{sector is bosonic}\\\ -1\ \ \text{if }\beta(\psi^{\mu})=1\ \
\iff\ \ \text{sector is fermionic}.\end{cases}$ (6.2)

The $\Upsilon({\bm{\beta}})$ is defined as a minimal linearly independent set
of vectors $\bm{\xi}$ such that $\bm{\xi}\cap{\bm{\beta}}=\emptyset$. To check
whether the sector ${\bm{\beta}}$ is projected simply amounts to checking
$\mathbb{P}_{{\bm{\beta}}}=0$.

In the presence of a single right–moving oscillator $\bar{\lambda}$ with
$\nu_{f}=\frac{1}{2}$, the generalised projector is modified to

$\mathbb{P}_{{\bm{\beta}}}=\prod_{\bm{\xi}\in\Upsilon({\bm{\beta}})}\frac{1}{2}\left(1+\delta_{{\bm{\beta}}}\delta^{\bar{\lambda}}_{\bm{\xi}}C{{\bm{\beta}}\atopwithdelims[
]\bm{\xi}}\right),$ (6.3)

such that

$\delta^{\bar{\lambda}}_{\bm{\xi}}=\begin{cases}+1\ \ \text{if }\
\bar{\lambda}\in\bm{\xi}\\\ -1\ \ \text{if }\
\bar{\lambda}\notin\bm{\xi}\end{cases}.$ (6.4)

In order to build tachyon–free models we simply require that
$\mathbb{P}_{\bm{t}}=0$ for all tachyonic sectors, $\bm{t}$. This requires
defining the projection sets $\Upsilon(\bm{t})$ for each tachyonic sector of
Table 1. For example, the tachyonic states from $\bm{z_{1}}$ have the
projection set

$\displaystyle\Upsilon(\bm{z_{1}})$
$\displaystyle=\\{\bm{S},\bm{e_{1}},\bm{e_{2}},\bm{e_{3}},\bm{e_{4}},\bm{e_{5}},\bm{e_{6}},\bm{x},\bm{z_{2}}\\}.$
(6.5)

Checking for the absence of tachyonic sectors then amounts to checking the
GGSO phases associated to such projection sets for all tachyonic sectors.

##  7 Chiral sector analysis

Having explained how to get $\Tr U(1)_{A}$ directly from the partition
function in subsection (4.1) we now explain how this can be calculated more
efficiently by analysis of the sectors that produce massless states that can
be charged under $U(1)_{A}$. This is equivalent to inspecting chiral sectors
giving rise to states that are charged under the complex
${\bar{\eta}}^{1,2,3}$ worldsheet fermion fields, where we recall that the
charge of a free fermion is given by

$Q(f)=\frac{1}{2}\alpha(f)+F(f),$ (7.1)

and the action of the fermion number operator is

$F:\begin{cases}f\ket{0}_{NS}=+1\\\ f^{*}\ket{0}_{NS}=-1\\\ \ket{+}=0\\\
\ket{-}=-1\end{cases},$ (7.2)

where we write the two helicities of the degenerate Ramond vacuum as
$\ket{\pm}$.

The first sectors we can identify with non–trivial chirality come from

$\displaystyle\begin{split}\bm{F}^{1}_{pqrs}=&\bm{S}+\bm{b_{1}}+p\bm{e_{3}}+q\bm{e_{4}}+r\bm{e_{5}}+s\bm{e_{6}}\\\
=&\\{\psi^{\mu},\chi^{1,2},(1-p)y^{3}\bar{y}^{3},pw^{3}\bar{w}^{3},(1-q)y^{4}\bar{y}^{4},qw^{4}\bar{w}^{4}\\\
&(1-r)y^{5}\bar{y}^{5},rw^{5}\bar{w}^{5},(1-s)y^{6}\bar{y}^{6},sw^{6}\bar{w}^{6},\bar{\eta}^{1},\bar{\psi}^{1,\ldots,5}\\}\\\
\bm{F}^{2}_{pqrs}=&\bm{S}+\bm{b_{2}}+p\bm{e_{1}}+q\bm{e_{2}}+r\bm{e_{5}}+s\bm{e_{6}}\\\
\bm{F}^{3}_{pqrs}=&\bm{S}+\bm{b_{3}}+p\bm{e_{1}}+q\bm{e_{2}}+r\bm{e_{3}}+s\bm{e_{4}},\end{split}$
(7.3)

which we note are the sectors that generate the spinorial
$\mathbf{16}/\overline{\mathbf{16}}$’s of our $SO(10)$ GUT, although we will
not be interested in this aspect of our models in this work.

Associated to these sectors are the projection sets

$\displaystyle\begin{split}\Upsilon(\bm{F}^{1}_{pqrs})&=\\{\bm{z_{1}},\bm{z_{2}},\bm{e_{1}},\bm{e_{2}}\\}\\\
\Upsilon(\bm{F}^{2}_{pqrs})&=\\{\bm{z_{1}},\bm{z_{2}},\bm{e_{3}},\bm{e_{4}}\\}\\\
\Upsilon(\bm{F}^{3}_{pqrs})&=\\{\bm{z_{1}},\bm{z_{2}},\bm{e_{5}},\bm{e_{6}}\\},\end{split}$
(7.4)

which are used to determine whether a sector remains in the Hilbert space of
the model, just as explained for the tachyonic sectors in the previous
section.

Once we have checked the survival of a particular sector, we can then
determine the chirality of the ${\bar{\eta}}^{1,2,3}$ for the resultant states
through the chirality projections defined for the three orbifold planes as
follows

$\displaystyle\begin{split}&\chi(\bm{F}^{1}_{pqrs})=\text{ch}(\bar{\eta}^{1})=-\text{ch}(\psi^{\mu})\
C{\bm{F}^{1}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{2}}+\bm{x}+(1-r)\bm{e_{5}}+(1-s)\bm{e_{6}}}^{*}\\\
&\chi(\bm{F}^{2}_{pqrs})=\text{ch}(\bar{\eta}^{2})=-\text{ch}(\psi^{\mu})\
C{\bm{F}^{2}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{1}}+\bm{x}+(1-r)\bm{e_{5}}+(1-s)\bm{e_{6}}}^{*}\\\
&\chi(\bm{F}^{3}_{pqrs})=\text{ch}(\bar{\eta}^{3})=-\text{ch}(\psi^{\mu})\
C{\bm{F}^{3}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{1}}+\bm{x}+(1-r)\bm{e_{3}}+(1-s)\bm{e_{4}}}^{*}.\end{split}$
(7.5)

Without loss of generality we can choose ch$(\psi^{\mu})=\ket{+}$ since the
CPT–conjugates are necessarily present with the opposite chirality choice.
This then fully determines the charges under the $U(1)_{1,2,3}$ for these
sectors.

Along with these 3 groups of 16 sectors we have a further 12 such groups
$\bm{F}^{4,5,6}_{pqrs},\bm{F}^{7,8,9}_{pqrs}$ and $\bm{V}^{1,2,3}_{pqrs}$ but
we relegate the details of how to extract their charge contributions to
Appendix C. As with the analysis of the tachyonic sectors, the projection
conditions that determine the overall $\Tr U(1)_{A}$ can then be computerised
to allow for efficient scans of large spaces of different GGSO phase
configurations. The results from such a scan is presented in the next section
along with the results of our analysis of the $D$–term uplifted potentials.

As mentioned above, in this work we will not consider extra phenomenological
issues in our models such as the number of spinorial $\mathbf{16}$’s or
vectorial $\mathbf{10}$’s classified in previous works for these symmetric
$\mathbb{Z}_{2}\\!\times\\!\mathbb{Z}_{2}$ models [7, 34, 35, 36, 37, 38, 39,
40, 41]. One may wonder about the relationship between such characteristics
and the value of $\Tr U(1)_{A}$. Since viable $SO(10)$ phenomenology requires
a condition such as $N_{\mathbf{16}}-N_{\mathbf{\overline{16}}}\geq 6$ we
could expect some relationship between the trace values and the presence of an
appropriate number of these representations. However, since
$\bm{F}^{4,5,6,7,8,9}_{pqrs}$ are hidden sectors and the vectorials
$\bm{V}^{1,2,3}_{pqrs}$ would only be constrained by the presence of at least
vectorial $\mathbf{10}$ we don’t expect there to be any significant change to
whether we can find $D$–term uplifted models once we incorporate such
phenomenological considerations into the analysis.

##  8 Results

The methodology we use for the extraction of $D$–term uplifted models defined
through the basis (2.1) follows the 5 step procedure:

  1. 1.

Extract $\mathcal{N}=0$ models, by checking whether eq (2.13) and /or eq.
(2.14) are violated. Subsequently, we check that the models are free from
physical tachyons by checking the projection conditions outlined in section
(6).

  2. 2.

Compute the values of $\Tr U(1)_{A}$ using the efficient analysis of chiral
sectors explained in section 7 and appendix C for these $\mathcal{N}=0$
tachyon–free models.

  3. 3.

Extract out those models with larger $\Tr U(1)_{A}$ values and satisfying the
SSS SUSY breaking conditions discussed in section 5.

  4. 4.

Perform the numerical analysis of the one–loop potentials and check for an
uplift from AdS to dS for these SSS models with large $\Tr U(1)_{A}$.

  5. 5.

The models with explicit SUSY breaking but large $\Tr U(1)_{A}$ can then be
analysed and checked for an uplift.

We note that the key bottleneck in this methodology is performing the
numerical analysis of the one–loop potential integral(s). Depending on how
many points with respect to $T_{2}$ are evaluated, the number of models we can
analyse the potentials for in reasonable computing time is approximately only
$\mathcal{O}(10^{3})$. This helps to motivate the 5 step procedure above that
seeks to maximise the probability we find an uplifted model, with or without
SSS breaking.

###  8.1 Distribution of $\Tr U(1)_{A}$ for $\mathcal{N}=0$ models

For our purpose of finding an uplifted model it was sufficient in step 1. to
take a random scan of $10^{9}$ $\mathcal{N}=0$ GGSO configurations and
checking them for the absence of physical tachyons using the conditions
explained in section 6. This scan resulted in $\sim 1.6\times 10^{7}$
tachyon–free models. Following step 2. of the methodology, we then calculated
the values of $\Tr U(1)_{A}$ for these models. The results for the
distribution of these values are shown in Figure 1.

Figure 1: _The distribution of Tr $U(1)_{A}$_

###  8.2 Example Scherk–Schwarz models

Once the $\Tr U(1)_{A}$’s for tachyon–free models were founds, we moved to
step 3. to begin the analysis of the one–loop potential starting with those
models that had a larger value for $\Tr U(1)_{A}$ and also satisfied the SSS
condition derived in section 5. Then we perform step 4. to search for the
$D$–term uplift for these larger $\Tr U(1)_{A}$ SSS models.

In a scan of approximately $10^{3}$ such models we did indeed find a single
model that exhibited the desired uplift. This model is defined through the
following set of GGSO phases

$\small C{\bm{v_{i}}\atopwithdelims[
]\bm{v_{j}}}=\blockarray{cccccccccccccc}&\mathbf{1}\bm{S}\bm{e_{1}}\bm{e_{2}}\bm{e_{3}}\bm{e_{4}}\bm{e_{5}}\bm{e_{6}}\bm{b_{1}}\bm{b_{2}}\bm{z_{1}}\bm{z_{2}}\\\
\block{c(rrrrrrrrrrrrr)}\mathbf{1}-1-1-11-11-11-1-1-11\ \\\
\bm{S}-1-111-1-1-1-11-11-1\ \\\ \bm{e_{1}}-111-1-1-1111-1-1-1\ \\\
\bm{e_{2}}11-1-111111-1-11\ \\\ \bm{e_{3}}-1-1-11111-11111\ \\\
\bm{e_{4}}1-1-111-11-11111\ \\\ \bm{e_{5}}-1-11111111111\ \\\
\bm{e_{6}}1-111-1-11-1-1-111\ \\\ \bm{b_{1}}-1-111111-1-1-1-11\ \\\
\bm{b_{2}}-11-1-1111-1-1-1-11\ \\\ \bm{z_{1}}-11-1-11111-1-1-11\ \\\
\bm{z_{2}}1-1-1111111111\ \\\ $ (8.1)

and has a one–loop cosmological constant value of $\Lambda=-0.000212496$ at
the free fermionic point and $\Tr U(1)_{A}=72\sqrt{2}$, with a FI contribution
of $0.00144365$. Performing the numerical analysis allowed us to graph the
potential and demonstrate its uplift, as shown in Figure 2.

Figure 2: _One-loop Potential of SSS Example Model with Uplift_

Through an analysis of $\mathcal{O}(10^{3})$ GGSO configurations, we also
evaluated the one–loop potential shapes for SSS broken models. We summarise
these possibilities in Figures 3-8. We find two examples of SSS models with
unbroken T–duality, see Figures 3 and 4, and two with broken T–duality, see
Figures 5 and 6, where the difference can be clearly seen from the behavior of
the potential when the modulus approaches zero. Figures 7 and 8 instead show
two SSS models with broken T–duality and with no minima or maxima.

The models we are especially interested in for our purpose are the ones with a
negative minima. In these cases the additional $D$–term contribution (3.22)
typically is not sufficient to uplift the minima. It is only in very rare,
finely tuned, examples that this uplift is observed. Finally, we note that
although we projected out physical tachyons at the free fermionic point, in
general there may be tachyons at other points in the moduli space. However,
through analysis across the range of the modulus $T_{2}$ we find that none of
our potentials suffer tachyonic instabilities.

Figure 3: _One-loop Scherk-Schwarz Potential with local minimum and unbroken
T-duality_

Figure 4: _One-loop Scherk-Schwarz Potential with local maximum and unbroken
T-duality_

Figure 5: _One-loop Scherk-Schwarz Potential with local minimum and broken
T-duality_

Figure 6: _One-loop Scherk-Schwarz Potential with local maximum and broken
T-duality_

Figure 7: _One-loop Scherk-Schwarz Potential without any extreme point and
broken T-duality_

Figure 8: _One-loop Scherk-Schwarz Potential without any extreme point and
broken T-duality_

###  8.3 Explicitly broken model examples

Performing a similar scan of large $\Tr U(1)_{A}$ models but with explicit
breaking in step 5., we also find models in which the FI $D$–term uplifts the
one–loop potential. An example of such a model with explicit SUSY–breaking is
given by the GGSO phases

$\small C{\bm{v_{i}}\atopwithdelims[
]\bm{v_{j}}}=\blockarray{cccccccccccccc}&\mathbf{1}\bm{S}\bm{e_{1}}\bm{e_{2}}\bm{e_{3}}\bm{e_{4}}\bm{e_{5}}\bm{e_{6}}\bm{b_{1}}\bm{b_{2}}\bm{z_{1}}\bm{z_{2}}\\\
\block{c(rrrrrrrrrrrrr)}\mathbf{1}-1-111-11-1111-11\ \\\
\bm{S}-1-1-1-1-1-1-1-11-1-11\ \\\ \bm{e_{1}}1-1-1-1-111111-1-1\ \\\
\bm{e_{2}}1-1-1-1-1-1-1-11-111\ \\\ \bm{e_{3}}-1-1-1-11-1-1-1-1-1-1-1\ \\\
\bm{e_{4}}1-11-1-1-111-1-1-1-1\ \\\ \bm{e_{5}}-1-11-1-111-11-1-11\ \\\
\bm{e_{6}}1-11-1-11-1-11-1-1-1\ \\\ \bm{b_{1}}1-111-1-111111-1\ \\\
\bm{b_{2}}111-1-1-1-1-1111-1\ \\\ \bm{z_{1}}-1-1-11-1-1-1-111-1-1\ \\\
\bm{z_{2}}11-11-1-11-1-1-1-11\ \\\ $ (8.2)

corresponding to a model with one–loop cosmological constant
$\Lambda=-0.000785598$ and $\Tr U(1)_{A}=72\sqrt{2}$ that generates a FI
contribution of $0.00144365$ to the one–loop potential, ensuring a positive
minimum as depicted in Figure 9.

Figure 9: _One-loop Potential of Example Model with explicitly broken SUSY
with Uplift_

Also in this case, through analysis of $\mathcal{O}(10^{3})$ models with
explicitly broken SUSY, we find only certain possibilities for the shapes of
the potential. We summarise these possibilities in Figures 10-15. In all these
graphs, when the moduli $T_{2}\rightarrow\infty$, the potential diverges so
SUSY is broken explicitly.
In particular in Figures 10 and 12 the minima is positive while in Figure 11
only a negative maxima is present. In Figures 13 and 15 instead there are no
extremal points at all. These are not the cases we are interested in. Only the
shapes of the potential as in Figure 13 present a negative minima required for
the uplift.

Figure 10: _One-loop Potential with explicitly broken SUSY and local minimum_

Figure 11: _One-loop Potential with explicitly broken SUSY and local maximum_

Figure 12: _One-loop Potential with explicitly broken SUSY and local minima
and maximum_

Figure 13: _One-loop Potential with explicitly broken SUSY and local minimum
and maxima_

Figure 14: _One-loop Potential with explicitly broken SUSY without any extreme
point_

Figure 15: _One-loop Potential with explicitly broken SUSY without any extreme
point_

##  9 Discussion & Conclusions

One important, and generic, issue with non–SUSY theories is the issue of a
non–vanishing dilaton tadpole within such models with a non–zero cosmological
constant. This means that the string equations of motion are not satisfied for
our theory on a Minkowskian 4D spacetime with a constant dilaton. If we want
to find the true perturbative quantum vacuum, we would need to solve the
string equations of motion to all loop levels. However, for non–SUSY strings
we generically lose computational control at higher orders in the string loop
expansion.

We note that in that respect the analysis performed in this paper is rather
heuristic. The dilaton VEV was fixed by hand and most of the moduli are set at
the free fermionic point but are not fixed, i.e. they can be varied away from
the free fermionic point. Similarly, the two–loop contribution to the vacuum
energy was adopted from the SUSY cases and not carried our explicitly. We
remark that while our arguments here are, in these respects, not conclusive,
the analysis can be made more rigorous by adopting the Kiritsis–Kounnas
modular invariant regularisation method of one–loop string amplitudes [46]. In
that scheme, the four dimensional spacetime curvature is used as a infrared
regulator, and is achieved by compactifying the four dimensional spacetime on
a Wess–Zumino–Witten current algebra sigma model. The simplest solution is
obtained with the conformal theory $W_{k}^{(4)}=SU(2)_{k}\times U(1)$, which
has the asymptotic (large $k$) geometry $S^{3}\times R^{+}$. The heterotic
string constructions with a mass gap for the massless spectrum is then
constructed by substituting the worldsheet coordinate and the spin–fields of
the uncompactified Minkowski spacetime conformal and superconformal blocks of
appropriate worldsheet supersymmetry and central charge. It should be noted
that while this regularisation methodology was developed for vacua with
$\mathcal{N}=1$ spacetime supersymmetry, its adaptation to non–SUSY string
vacua should be possible. This scheme therefore provide the tools to construct
non–SUSY string solutions with regulated infrared divergences. Similarly,
while perhaps presenting a substantial technical challenge, the two–loop
costribution to the vacuum energy can in principle be calculated directly in
string theory [20, 61]. Thus, the somewhat heuristic arguments made here can,
in principle, be put on firmer grounds.

We also remark that the question of moduli stabilisation can also be partially
addressed directly in the heterotic string worldsheet constructions. String
theory allows for asymmetric boundary conditions of the left– and right–moving
worldsheet internal fermions, which the corresponding bosonic representation
entail an asymmetric action on the internal coordinates of the compactified
six dimensional space. As a consequence, some or even all of the moduli fields
that parameterise the properties of the internal manifold can be projected out
[63] and be frozen at a specific value, typically at the self–dual point in
the moduli space. In that case, the space of unfixed moduli is substantially
reduced. One can envision that stabilisation of the remaining unfixed moduli
and extraction of the global minimum is possible. The question whether it is
possible to obtain a model with positive vacuum energy and fixed moduli with
or without a $D$–term uplift may be further investigated.

In this paper we explored the contribution of the would–be FI $D$–term to the
vacuum energy in non–SUSY heterotic string vacua. This contribution can uplift
the vacuum energy from a negative to positive value and give rise to a
positive constant cosmological background. We distinguished in our analysis
between string vacua with explicit SUSY breaking versus vacua in which SUSY is
broken spontaneously by the Scherk–Schwarz mechanism. We found that while
rare, a $D$–term uplift to a positive cosmological constant might indeed be
possible.

## Acknowledgments

AEF would like to that the CERN theory division for hospitality and support.
The work of ARDA is supported in part by EPSRC grant EP/T517975/1. The work of
BP is supported by EPSRC grant EP/W522399/1.

##  Appendix A Translation of Fermionic Partition Function

The goal of this appendix is to find a one–to–one correspondence between the
partition functions written in the free fermionic construction, (2.4), and in
the orbifold language. An outline of this procedure can be found in [57] and
is also presented in [64].

The aim of this procedure is to find an equality of the form

$Z_{F}=\frac{1}{2^{N}}\sum_{\bm{\alpha},\bm{\beta}}C{\bm{\alpha}\atopwithdelims[
]\bm{\beta}}Z\left[\begin{matrix}\bm{\alpha}\\\
\bm{\beta}\end{matrix}\right]=\sum_{\begin{subarray}{c}a,k,\cdots\\\
b,l,\cdots\end{subarray}}(-1)^{\Psi\left[\begin{smallmatrix}a&k&\cdots\\\
b&l&\cdots\end{smallmatrix}\right]}\;\,Z\left[\begin{matrix}a&k&\cdots\\\
b&l&\cdots\end{matrix}\right],$ (A.1)

where the product over the fermions is now implicit and contained within
$Z\left[\begin{smallmatrix}\bm{\alpha}\\\ \bm{\beta}\end{smallmatrix}\right]$.
The right–hand side requires further comments. The term
$Z\left[\begin{smallmatrix}a&k&\cdots\\\ b&l&\cdots\end{smallmatrix}\right]$
represents the theta functions in terms of the summation indices
$a,b,k,l,\dots$. The phase $(-1)^{\Psi\left[\begin{smallmatrix}a&k&\cdots\\\
b&l&\cdots\end{smallmatrix}\right]}$ is the analogue of the GGSO phase in this
new formulation.

### The Modular Invariant Phase

To see all possible choices of indices, which in turn fix the form of
$Z\left[\begin{smallmatrix}a&k&\cdots\\\ b&l&\cdots\end{smallmatrix}\right]$,
we note that to represent a partition function of a model with $N$ basis
vectors requires the use of $N$ summation indices. This can be seen by
matching the number of terms on each side of (A.1). Thus, the translation of
the form of the partition function is uniquely determined by the choice of a
change of basis matrix, $S$, which encodes the correspondence between the
basis vectors and the summation indices. For our choice of models specified by
the basis vectors (2.1) and the summation indices in the partition function as
in (4.1), the $S$ matrix is given by

$\small S=\blockarray{cccccccccccccc}\block{c(rrrrrrrrrrrrr)}&\ 111111110000\
\\\ \ 100000000000\ \\\ \ 001000000000\ \\\ \ 000100000000\ \\\ \
000010000000\ \\\ \ 000001000000\ \\\ \ 000000100000\ \\\ \ 000000010000\ \\\
\ 010000000111\ \\\ \ 010000001011\ \\\ \ 000000000010\ \\\ \ 000000000001\
\\\ $ (A.2)

All invertible $N\times N$ matrices whose entries take values in
$\mathbb{Z}_{2}$ are valid choices. However, the above choice is the one which
best illuminates the geometry of the underlying compactification. Choosing
$S=I_{N}$, the $N$–dimensional identity matrix, would render the translation
trivial and the form of $Z\left[\begin{smallmatrix}a&k&\cdots\\\
b&l&\cdots\end{smallmatrix}\right]$ and
$(-1)^{\Psi\left[\begin{smallmatrix}a&k&\cdots\\\
b&l&\cdots\end{smallmatrix}\right]}$ would match that of
$Z\left[\begin{smallmatrix}\bm{\alpha}\\\ \bm{\beta}\end{smallmatrix}\right]$
and $C\left(\begin{smallmatrix}\bm{\alpha}\\\
\bm{\beta}\end{smallmatrix}\right)$, respectively, modulo some subtleties we
discuss in the following section.

Once $S$ is specified, the partition function is written in its index form and
we can start making the connection between the GGSO Phases
$C{\bm{\alpha}\atopwithdelims[ ]\bm{\beta}}$ and the modular invariant phase
$\Psi$. We assume that $\Psi$ can be expressed as a polynomial in the
summation variables. Then, two–loop modular invariance imposed on the GGSO
phases via the rule

$C{\bm{v_{i}}\atopwithdelims[
]\bm{v_{j}}+\bm{v_{k}}}=\delta_{\bm{v_{i}}}C{\bm{v_{i}}\atopwithdelims[
]\bm{v_{j}}}C{\bm{v_{i}}\atopwithdelims[ ]\bm{v_{k}}},$ (A.3)

implies that $\Psi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]$ is at most second order in its
variables. Moreover, the presence of $\delta_{\bm{v_{i}}}$ restricts the
first–order terms. That is $\Psi$ must include a term $a$ and cannot contain
other terms like it. More precisely, (A.3) implies

$\displaystyle\begin{cases}\Psi\ni a,\\\ \Psi\not\owns
k,h_{i},H_{i},P_{i},\end{cases}$ (A.4)

where we take "$\in$" to mean a term in the sum. These conditions can be
implemented in a compact form by requiring the phase to be of the form

$\Psi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]=a+\beta_{i}\Delta_{i}+\Gamma_{i}\Omega_{ij}\Delta_{j},$
(A.5)

where we defined

$\displaystyle\Gamma$ $\displaystyle=(a,\ k,\ H_{1},\ H_{2},\ H_{3},\ H_{4},\
H_{5},\ H_{6},\ h_{1},\ h_{2},\ P_{1},\ P_{2}),$ (A.6) $\displaystyle\Delta$
$\displaystyle=(b,\ l,\ G_{1},\ G_{2},\ G_{3},\ G_{4},\ G_{5},\ G_{6},\
g_{1},\ g_{2},\ Q_{1},\ Q_{2}),$

to be the vectors containing top and bottom indices respectively.

We now impose one–loop modular invariance by requiring that the partition
function (A.1) remains invariant under $S$ and $T$–transformations, under
which the theta functions transform as

$\displaystyle S:$ $\displaystyle\;\;\;\vartheta\left[\begin{smallmatrix}a\\\
b\end{smallmatrix}\right]\;\longrightarrow\;e^{i\pi
ab/2}\;\vartheta\left[\begin{smallmatrix}b\\\ -a\end{smallmatrix}\right],$
(A.7) $\displaystyle T:$
$\displaystyle\;\;\;\vartheta\left[\begin{smallmatrix}a\\\
b\end{smallmatrix}\right]\;\longrightarrow\;e^{i\pi
a(a-2)/4}\;\vartheta\left[\begin{smallmatrix}a\\\
a+b-1\end{smallmatrix}\right].$

By using a compact notation for the theta and eta function terms as in (A.1),
i.e.

$Z_{F}=\frac{1}{2^{2}}\sum_{\begin{subarray}{c}a,k\\\
b,l\end{subarray}}\;\frac{1}{2^{10}}\sum_{\begin{subarray}{c}H_{i},h_{i},P_{i}\\\
G_{i},g_{i},Q_{i}\end{subarray}}(-1)^{\Psi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]}Z\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right],$ (A.8)

we can express the modular transformations more readily. In particular, under
modular transformations

$\displaystyle Z\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]\overset{S}{\longrightarrow}Z\left[\begin{smallmatrix}b&l&G_{i}&g_{i}&Q_{i}\\\
-a&-k&-H_{i}&-h_{i}&-P_{i}\end{smallmatrix}\right],$ $\displaystyle
Z\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]\overset{T}{\longrightarrow}(-1)^{1+a+P_{1}+P_{2}}\,Z\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
a+b-1&k+l-1&H_{i}+G_{i}-1&h_{i}+g_{i}&P_{i}+Q_{i}\end{smallmatrix}\right],$

where the extra factor of $-1$ in the $T$–transformation comes from the
$\eta$–functions. By noting that the phase $\Psi$ transforms trivially, as it
is just a constant factor, we can conclude that to be modular invariant the
phase must satisfy

$\displaystyle\Psi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]$
$\displaystyle\overset{S}{=}\Psi\left[\begin{smallmatrix}b&l&G_{i}&g_{i}&Q_{i}\\\
-a&-k&-H_{i}&-h_{i}&-P_{i}\end{smallmatrix}\right],$ (A.9)
$\displaystyle\Psi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]$
$\displaystyle\overset{T}{=}1+a+P_{1}+P_{2}+\Psi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
a+b-1&k+l-1&H_{i}+G_{i}-1&h_{i}+g_{i}&P_{i}+Q_{i}\end{smallmatrix}\right].$

The first equation, i.e. $S$–invariance, shows that $\Psi$ must be symmetric
under the exchange of lower and upper indices, which together with (A.5),
implies that

$\Psi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]=a+b+\Gamma_{i}\Omega_{ij}\Delta_{j},$
(A.10)

with $\Omega_{ij}=\Omega_{ji}$. Implementing the condition for $T$–invariance
in (A.9) further restricts the form of $\Omega$ imposing the conditions on its
elements

$\displaystyle\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{8}\Omega_{ij}$ $\displaystyle=0\quad$
$\displaystyle\text{for}\quad i=2,\dots,8,$ (A.11)
$\displaystyle\sum_{j=1}^{8}\Omega_{ij}$ $\displaystyle=\Omega_{ii}\quad$
$\displaystyle\text{for}\quad i=9,10,$
$\displaystyle\sum_{j=1}^{8}\Omega_{ij}$ $\displaystyle=1+\Omega_{ii}\quad$
$\displaystyle\text{for}\quad i=11,12,$

where all equalities are understood modulo 2. These fix a further 11
components of $\Omega_{ij}$. Together with the condition from $S$–invariance
$\Omega_{ij}=\Omega_{ji}$, we are left with $(12^{2}/2+12/2)-11=67$
independent choices for the $\Omega_{ij}$. This precisely matches the number
of independent GGSO phases for a 12 basis vector model¶¶¶We can either count
66 or 67 independent GGSO phases depending on whether we specify that the
unimportant phase $C{\mathds{1}\atopwithdelims[ ]\mathds{1}}$, generating an
overall chirality, is fixed or not..

What we achieved here is precisely the derivation of the modular invariance
conditions, for the phase $\Psi$. These are analogous to the well–known
conditions on the GGSO coefficients in the fermionic formulation. All
remaining independent components of $\Omega$ can be freely chosen as
$\Omega_{ij}\in\\{0,1\\}$ with each choice giving a new consistent model.

### The Translation

We have found a consistent modular invariant way of representing a model in
terms of a phase $\Psi$, what remains is to find a translation between the
GGSO phases and $\Psi$ as set out in (A.1). With the above setup, this means
finding a correspondence between the independent GGSO phases
$C{\bm{\alpha}\atopwithdelims[ ]\bm{\beta}}$ and the matrix elements
$\Omega_{ij}$. We have already established that the number of these elements
is in agreement on both sides and both quantities perform the same role so
such a translation should be possible in principle.

To make the connection, one has to notice that the forms of the theta
functions on the left and right–hand sides of (A.1) do not match. In
particular the expression

$\sum_{\begin{subarray}{c}a,k,\cdots\\\
b,l,\cdots\end{subarray}}(-1)^{\Psi\left[\begin{smallmatrix}a&k&\cdots\\\
b&l&\cdots\end{smallmatrix}\right]}\;\,Z\left[\begin{matrix}a&k&\cdots\\\
b&l&\cdots\end{matrix}\right]$ (A.12)

involves theta functions which may take arguments such as
$\vartheta\left[\begin{smallmatrix}1\\\
-1\end{smallmatrix}\right],\vartheta\left[\begin{smallmatrix}3\\\
0\end{smallmatrix}\right],...$ not permitted on the free fermionic side where
the arguments are either 0 or 1. We can, however, use the periodicity
properties of the theta functions

$\displaystyle\vartheta\left[\begin{smallmatrix}a+2\\\
b\end{smallmatrix}\right]=$
$\displaystyle\,\vartheta\left[\begin{smallmatrix}a\\\
b\end{smallmatrix}\right]$ (A.13)
$\displaystyle\vartheta\left[\begin{smallmatrix}a\\\
b+2\end{smallmatrix}\right]=$ $\displaystyle\,e^{i\pi
a}\vartheta\left[\begin{smallmatrix}a\\\ b\end{smallmatrix}\right],$

to rewrite (A.12) in terms of the standard theta functions. This will allow
for consistent term–by–term matching. By denoting the ‘fundamental’ form of
the theta functions as

$\vartheta_{f}\left[\begin{smallmatrix}a\\\
b\end{smallmatrix}\right]\equiv\vartheta\left[\begin{smallmatrix}a\mod 2\\\
b\mod 2\end{smallmatrix}\right],$ (A.14)

we can find equations using (A.13) that help bring all theta functions to this
reduced form, e.g.

$\displaystyle\vartheta\left[\begin{smallmatrix}a+h_{1}\\\
b+g_{1}\end{smallmatrix}\right]$
$\displaystyle=(-1)^{(a+h_{1})bg_{1}}\;\vartheta_{f}\left[\begin{smallmatrix}a+h_{1}\\\
b+g_{1}\end{smallmatrix}\right]$ (A.15)
$\displaystyle\vartheta\left[\begin{smallmatrix}a-h_{1}-h_{2}\\\
b-g_{1}-g_{2}\end{smallmatrix}\right]$
$\displaystyle=(-1)^{(a-h_{1}-h_{2})(g_{1}+g_{2}+bg_{1}+bg_{2}+g_{1}g_{2})}\;\vartheta_{f}\left[\begin{smallmatrix}a-h_{1}-h_{2}\\\
b-g_{1}-g_{2}\end{smallmatrix}\right].$

These relations can always be found by writing
$\vartheta\left[\begin{smallmatrix}a&\cdots\\\
b&\cdots\end{smallmatrix}\right]=(-1)^{F(a,b,\cdots)}\vartheta_{f}\left[\begin{smallmatrix}a&\cdots\\\
b&\cdots\end{smallmatrix}\right]$, with $F(a,b,\cdots)$ a suitably general
polynomial, and restricting the form of $F$ by requiring (A.13) to hold.

Utilising these expressions, we can rewrite the right–hand side of (A.1),
fully in terms of the ‘reduced’ theta functions as

$Z_{F}=\sum_{\begin{subarray}{c}a,k,\cdots\\\
b,l,\cdots\end{subarray}}(-1)^{\chi\left[\begin{smallmatrix}a&k&\cdots\\\
b&l&\cdots\end{smallmatrix}\right]+\Psi\left[\begin{smallmatrix}a&k&\cdots\\\
b&l&\cdots\end{smallmatrix}\right]}\;\,Z_{f}\left[\begin{matrix}a&k&\cdots\\\
b&l&\cdots\end{matrix}\right],$ (A.16)

where we defined the compensating phase factor $\chi$. For our specific model
it is given by

$\chi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]=(a+k)(g_{1}+g_{2}+g_{1}g_{2})+(b+l)(h_{1}g_{2}+h_{2}g_{1}),$
(A.17)

which enforces the rules (A.15). Here, by $Z_{f}$ we denote that all theta
functions have been brought to their mod 2 form as written in (A.14). This
compensating phase is crucial for the matching of the partition functions.

We are now ready to make the connection between the two formalisms. To compare
the two sides of (A.1) we must re–express the GGSO matrix $C$ in the form

$C_{ij}=(-1)^{G_{ij}},$ (A.18)

this allows for a direct comparison of $\Psi$ and $G$. Furthermore, it will be
convenient to separate $\Psi$ into its first and second order terms, that is
we define

$\Psi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]=a+b+\Gamma_{i}\Omega_{ij}\Delta_{j}\coloneqq
a+b+\Phi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right].$ (A.19)

We can now express the factor of $a+b+\chi$ in the basis formed by the basis
vectors (2.1) as a matrix $P$ whose elements are

$P_{ij}=\\{a+b+\chi\left[\begin{smallmatrix}a&k&\cdots\\\
b&l&\cdots\end{smallmatrix}\right]\,|\;\Gamma_{k}=S_{ik}\text{ and
}\Delta_{k}=S_{jk}\\}.$ (A.20)

All that remains is to express $\Phi$, i.e. $\Omega$, in the same basis so we
can equate the two. We can do this by noticing that

$\\{\Phi\left[\begin{smallmatrix}a&k&\cdots\\\
b&l&\cdots\end{smallmatrix}\right]\,|\;\Gamma_{k}=S_{ik}\text{ and
}\Delta_{k}=S_{jk}\\}=S_{ik}\,\Omega_{kl}\,S_{jl}=S\,\Omega\,S^{T},$ (A.21)

and so $\tilde{\Omega}=S\,\Omega\,S^{T}$ is the phase expressed in the basis
formed by the basis vectors of the free fermionic model. Since all quantities
are now expressed in the same basis we can write down the equality which
implements the translation, namely

$G+P=S\,\Omega\,S^{T},$ (A.22)

where the equality is understood modulo 2. Solving the above equation means
finding values for all $\Omega_{ij}$ and so fixing $\Omega$. Once the solution
is found to the linear system, the final phase can be expressed using (A.19),
that is

$\Psi\left[\begin{smallmatrix}a&k&H_{i}&h_{i}&P_{i}\\\
b&l&G_{i}&g_{i}&Q_{i}\end{smallmatrix}\right]=a+b+\Gamma\,\Omega\,\Delta.$
(A.23)

This gives a precise one–to–one correspondence between the modular invariant
phase and the GGSO matrix.

It is important to note that the above methods only cover the case for real
boundary conditions, i.e. ‘$\mathbb{Z}_{2}$–models’ such that the fermions are
either R or NS. This, in turn, implies that all GGSO phases are real. It is,
however, possible to generalise this construction to allow for more general
choices of boundary condition vectors and GGSO matrices.

##  Appendix B Scherk–Schwarz and T–duality Conditions

In this section we will explicitly show how the techniques developed in
sections 5 and 5.1 can be applied in order to check for SSS and T–duality. In
particular, we will use the two models of section 8, specified by the GGSO
phase configurations (8.1) and (8.2).

The condition for SSS is that for $T_{2}\rightarrow\infty$ the partition
function vanishes so SUSY is restored, which can happen only if a Jacobi
identity is realised. This identity in (5.4) must hold for all indices fixed,
except for $H_{1},H_{2},G_{1},G_{2}$ which do not appear as arguments in the
theta functions, since the moduli–dependent lattice
$\Gamma^{(1)}_{2,2}\left[\begin{smallmatrix}H_{1}&H_{2}\\\
G_{1}&G_{2}\end{smallmatrix}\middle|\begin{smallmatrix}h_{1}\\\
g_{1}\end{smallmatrix}\right]$ is set to $1$.
Given the GGSO phases, the phase $\Phi$ in the partition function (4.1) can be
calculated using (A.23) following the discussion of Appendix A. For the model
(8.1) this phase is given by

$\displaystyle\begin{split}\Phi\left[\begin{smallmatrix}a&k&H_{i}&h_{1}&h_{2}&P_{i}\\\
b&l&G_{i}&g_{1}&g_{2}&Q_{i}\end{smallmatrix}\right]=&\;b(a+H_{1}+h_{2}+H_{2}+P_{1})\\\
&+l(h_{2}+k+P_{1})\\\ &+G_{1}(a+h_{1}+H_{2}+H_{3}+H_{4}+P_{1}+P_{2})\\\
&+G_{2}(a+H_{1}+h_{2}+H_{2}+P_{1})\\\ &+G_{3}(H_{1}+H_{6})\\\
&+G_{4}(H_{1}+H_{4}+H_{6})\\\ &+G_{6}(h_{1}+h_{2}+H_{3}+H_{4}+H_{6})\\\
&+g_{1}(H_{1}+H_{6}+P_{1})\\\ &+g_{2}(a+H_{2}+H_{6}+k+P_{1})\\\
&+Q_{1}(a+h_{1}+H_{1}+h_{2}+H_{2}+k)\\\ &+Q_{2}(H_{1}+P_{2}).\end{split}$
(B.1)

Then, as stated in section 5, the Jacobi identity holds only if the following
condition is satisfied

$\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}=\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}1&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}=\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
1&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]}.$ (B.2)

Actually it is sufficient to prove just the first equality, since due to
modular invariance the phase
$\Phi\left[\begin{smallmatrix}a&k&H_{i}&h_{1}&h_{2}&P_{i}\\\
b&l&G_{i}&g_{1}&g_{2}&Q_{i}\end{smallmatrix}\right]$ is the same by exchanging
lower and upper indices. Then, by performing the lattice sum, it is easy to
check that the equality holds.

Alternatively, it can also be check directly by inspecting the phase
$\Phi\left[\begin{smallmatrix}a&k&H_{1}&H_{2}&H_{i}&0&0&P_{i}\\\
b&l&G_{1}&G_{2}&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]$, where we have
distinguished $H_{1},H_{2}$ from the other $H_{i}$’s to render the discussion
clearer. For $\Phi\left[\begin{smallmatrix}0&k&H_{1}&H_{2}&H_{i}&0&0&P_{i}\\\
0&l&G_{1}&G_{2}&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]$ set the indices
$H_{1}$ and $H_{2}$ to zero. If there are some combinations of $H_{1}^{*}$ and
$H_{2}^{*}$ for
$\Phi\left[\begin{smallmatrix}1&k&H_{1}^{*}&H_{2}^{*}&H_{i}&0&0&P_{i}\\\
0&l&G_{1}&G_{2}&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]$ such that the
following equality holds

$\Phi\left[\begin{smallmatrix}0&k&0&0&H_{i}&0&0&P_{i}\\\
0&l&G_{1}&G_{2}&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]=\Phi\left[\begin{smallmatrix}1&k&H_{1}^{*}&H_{2}^{*}&H_{i}&0&0&P_{i}\\\
0&l&G_{1}&G_{2}&G_{i}&0&0&Q_{i}\end{smallmatrix}\right],$ (B.3)

then the condition (B.2) will be satisfied and the Jacobi identity will hold.
In particular, for the phase (B.1) it is easy to check that the equality is
satisfied setting $H_{1}^{*}=0$, $H_{2}^{*}=1$. Once this holds, for all other
combinations of $H_{1},H_{2}$ the equality will hold according to

$\Phi\left[\begin{smallmatrix}0&k&0+i&0+j&H_{i}&0&0&P_{i}\\\
0&l&G_{1}&G_{2}&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]=\Phi\left[\begin{smallmatrix}1&k&H_{1}^{*}+i&H_{2}^{*}+j&H_{i}&0&0&P_{i}\\\
0&l&G_{1}&G_{2}&G_{i}&0&0&Q_{i}\end{smallmatrix}\right].$ (B.4)

This second way of checking for SSS is particularly convenient in terms of
time efficiency, being easy to implement in a computational program.
We now check whether T–duality is preserved for this model, i.e. if condition
(5.12) holds. To do this, we check that

$\displaystyle\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]+H_{1}G_{1}+H_{2}G_{2}}$
$\displaystyle=\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}1&k&H_{i}&0&0&P_{i}\\\
0&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]+H_{1}G_{1}+H_{2}G_{2}}$ (B.5)
$\displaystyle=\sum_{\begin{subarray}{c}H_{1},H_{2}\\\
G_{1},G_{2}\end{subarray}}(-1)^{\Phi\left[\begin{smallmatrix}0&k&H_{i}&0&0&P_{i}\\\
1&l&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]+H_{1}G_{1}+H_{2}G_{2}}$

is satisfied. As discussed in subsection 5.1, due to modular invariance, just
the first equality is sufficient. Performing the sum, we can see that the
T–duality condition does not hold. Also using (B.3) modified with the
additional T–dual contribution

$\Phi\left[\begin{smallmatrix}0&k&0&0&H_{i}&0&0&P_{i}\\\
0&l&G_{1}&G_{2}&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]=\Phi\left[\begin{smallmatrix}1&k&H_{1}^{*}&H_{2}^{*}&H_{i}&0&0&P_{i}\\\
0&l&G_{1}&G_{2}&G_{i}&0&0&Q_{i}\end{smallmatrix}\right]+H_{1}^{*}G_{1}+H_{2}^{*}G_{2}$
(B.6)

it is easy to check that there is no possible combination of
$H_{1}^{*},H_{2}^{*}$ such that the equality is preserved.

We thus conclude that for this particular model defined by the GGSO phases
(8.1) the potential will exhibit an SSS breaking with broken T–duality, such
that $V(T_{2}\rightarrow\infty)\rightarrow 0$ and $V(T_{2}\rightarrow
0)\not\to 0$. This behavior of the potential is demonstrated in Figure 2.

For the second model specified by the GGSO phases (8.2), the phase $\Phi$ is
given by

$\displaystyle\begin{split}\Phi\left[\begin{smallmatrix}a&k&H_{i}&h1&h2&P_{i}\\\
b&l&G_{i}&g1&g2&Q_{i}\end{smallmatrix}\right]=&\;b(a+h_{2}+P_{2})\\\
&+l(H_{2}+H_{3}+H_{5}+H_{6}+P_{1})\\\
&+G_{1}(H_{1}+H_{2}+H_{3}+P_{1}+P_{2})\\\
&+G_{2}(H_{1}+h_{2}+H_{2}+H_{3}+H_{4}+H_{5}+H_{6}+k)\\\
&+G_{3}(H_{1}+H_{2}+H_{4}+H_{5}+H_{6}+k+P_{1}+P_{2})\\\
&+G_{4}(h_{1}+h_{2}+H_{2}+H_{3}+H_{4}+P_{1}+P_{2})\\\
&+G_{5}(h_{1}+H_{2}+H_{3}+H_{6}+k+P_{1})\\\
&+G_{6}(h_{2}+H_{2}+H_{3}+H_{5}+H_{6}+k+P_{1}+P_{2})\\\
&+g_{1}(h_{2}+H_{4}+H_{5}+P_{1})\\\
&+g_{2}(a+h_{1}+H_{2}+H_{4}+H_{6}+P_{1})\\\
&+Q_{1}(h_{1}+H_{1}+h_{2}+H_{3}+H_{4}+H_{5}+H_{6}+k+P_{2})\\\
&+Q_{2}(a+H_{1}+H_{3}+H_{4}+H_{6}+P_{1}+P_{2}).\end{split}$ (B.7)

For this model it can be checked that the SSS condition (B.2), or equivalently
condition (B.3), is not satisfied which implies that the potential diverges at
infinity, $V(T_{2}\rightarrow\infty)\rightarrow\pm\infty$. The shape of the
potential in Figure 9 shows what we expect.

##  Appendix C Chiral Sector Analysis

In addition to the sectors $\bm{F}^{1,2,3}_{pqrs}$ discussed in section 7, the
following sectors also give rise to massless states transforming under
spinorial representations with chirality under the $U(1)_{1,2,3}$ gauge
factors

$\displaystyle\begin{split}\bm{F}^{4}_{pqrs}&=\bm{S}+\bm{b_{1}}+\bm{x}+\bm{z_{1}}+p\bm{e_{3}}+q\bm{e_{4}}+r\bm{e_{5}}+s\bm{e_{6}}\\\
\bm{F}^{5}_{pqrs}&=\bm{S}+\bm{b_{2}}+\bm{x}+\bm{z_{1}}+p\bm{e_{1}}+q\bm{e_{2}}+r\bm{e_{5}}+s\bm{e_{6}}\\\
\bm{F}^{6}_{pqrs}&=\bm{S}+\bm{b_{3}}+\bm{x}+\bm{z_{1}}+p\bm{e_{1}}+q\bm{e_{2}}+r\bm{e_{3}}+s\bm{e_{4}}\\\
\bm{F}^{7}_{pqrs}&=\bm{S}+\bm{b_{1}}+\bm{x}+\bm{z_{2}}+p\bm{e_{3}}+q\bm{e_{4}}+r\bm{e_{5}}+s\bm{e_{6}}\\\
\bm{F}^{8}_{pqrs}&=\bm{S}+\bm{b_{2}}+\bm{x}+\bm{z_{2}}+p\bm{e_{1}}+q\bm{e_{2}}+r\bm{e_{5}}+s\bm{e_{6}}\\\
\bm{F}^{9}_{pqrs}&=\bm{S}+\bm{b_{3}}+\bm{x}+\bm{z_{2}}+p\bm{e_{1}}+q\bm{e_{2}}+r\bm{e_{3}}+s\bm{e_{4}},\end{split}$
(C.1)

which have the following projecting sets

$\displaystyle\begin{split}\Upsilon(\bm{F}^{4}_{pqrs})&=\\{\bm{z_{2}},\bm{e_{1}},\bm{e_{2}}\\}\\\
\Upsilon(\bm{F}^{5}_{pqrs})&=\\{\bm{z_{2}},\bm{e_{3}},\bm{e_{4}}\\}\\\
\Upsilon(\bm{F}^{6}_{pqrs})&=\\{\bm{z_{2}},\bm{e_{5}},\bm{e_{6}}\\}\\\
\Upsilon(\bm{F}^{7}_{pqrs})&=\\{\bm{z_{1}},\bm{e_{1}},\bm{e_{2}}\\}\\\
\Upsilon(\bm{F}^{8}_{pqrs})&=\\{\bm{z_{1}},\bm{e_{3}},\bm{e_{4}}\\}\\\
\Upsilon(\bm{F}^{9}_{pqrs})&=\\{\bm{z_{1}},\bm{e_{5}},\bm{e_{6}}\\},\end{split}$
(C.2)

and chirality operators

$\displaystyle\begin{split}\chi(\bm{F}^{4}_{pqrs})&=\text{ch}(\bar{\eta}^{2})+\text{ch}(\bar{\eta}^{3})=\\\
&=-\ C{\bm{F}^{4}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{2}}+(1-r)\bm{e_{5}}+(1-s)\bm{e_{6}}}^{*}-\
C{\bm{F}^{4}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{3}}+(1-p)\bm{e_{3}}+(1-q)\bm{e_{4}}}^{*}\\\
\chi(\bm{F}^{5}_{pqrs})&=\text{ch}(\bar{\eta}^{1})+\text{ch}(\bar{\eta}^{3})=\\\
&=-\ C{\bm{F}^{5}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{1}}+(1-r)\bm{e_{5}}+(1-s)\bm{e_{6}}}^{*}-\
C{\bm{F}^{5}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{3}}+(1-p)\bm{e_{1}}+(1-q)\bm{e_{2}}}^{*}\\\
\chi(\bm{F}^{6}_{pqrs})&=\text{ch}(\bar{\eta}^{1})+\text{ch}(\bar{\eta}^{2})=\\\
&=-\ C{\bm{F}^{6}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{1}}+(1-r)\bm{e_{3}}+(1-s)\bm{e_{4}}}^{*}-\
C{\bm{F}^{6}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{2}}+(1-p)\bm{e_{1}}+(1-q)\bm{e_{2}}}^{*}\\\
\chi(\bm{F}^{7}_{pqrs})&=\text{ch}(\bar{\eta}^{2})+\text{ch}(\bar{\eta}^{3})=\\\
&=-\ C{\bm{F}^{7}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{2}}+(1-r)\bm{e_{5}}+(1-s)\bm{e_{6}}}^{*}-\
C{\bm{F}^{7}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{3}}+(1-p)\bm{e_{3}}+(1-q)\bm{e_{4}}}^{*}\\\
\chi(\bm{F}^{8}_{pqrs})&=\text{ch}(\bar{\eta}^{1})+\text{ch}(\bar{\eta}^{3})=\\\
&=-\ C{\bm{F}^{8}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{1}}+(1-r)\bm{e_{5}}+(1-s)\bm{e_{6}}}^{*}-\
C{\bm{F}^{8}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{3}}+(1-p)\bm{e_{1}}+(1-q)\bm{e_{2}}}^{*}\\\
\chi(\bm{F}^{9}_{pqrs})&=\text{ch}(\bar{\eta}^{1})+\text{ch}(\bar{\eta}^{2})=\\\
&=-\ C{\bm{F}^{9}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{1}}+(1-r)\bm{e_{3}}+(1-s)\bm{e_{4}}}^{*}-\
C{\bm{F}^{9}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{2}}+(1-p)\bm{e_{1}}+(1-q)\bm{e_{2}}}^{*},\end{split}$ (C.3)

Further to these sectors, the following can give rise to massless states when
accompanied by one right–moving Neveu–Schwarz oscillator given by

$\displaystyle\begin{split}\bm{V}^{1}_{pqrs}&=\bm{S}+\bm{b_{1}}+\bm{x}+p\bm{e_{3}}+q\bm{e_{4}}+r\bm{e_{5}}+s\bm{e_{6}}\\\
\bm{V}^{2}_{pqrs}&=\bm{S}+\bm{b_{2}}+\bm{x}+p\bm{e_{1}}+q\bm{e_{2}}+r\bm{e_{5}}+s\bm{e_{6}}\\\
\bm{V}^{3}_{pqrs}&=\bm{S}+\bm{b_{3}}+\bm{x}+p\bm{e_{1}}+q\bm{e_{2}}+r\bm{e_{3}}+s\bm{e_{4}},\end{split}$
(C.4)

which have the following projecting sets

$\displaystyle\begin{split}\Upsilon(\bm{V}^{1}_{pqrs})&=\\{\bm{z_{1}},\bm{z_{2}},\bm{e_{1}},\bm{e_{2}}\\}\\\
\Upsilon(\bm{V}^{2}_{pqrs})&=\\{\bm{z_{1}},\bm{z_{2}},\bm{e_{3}},\bm{e_{4}}\\}\\\
\Upsilon(\bm{V}^{3}_{pqrs})&=\\{\bm{z_{1}},\bm{z_{2}},\bm{e_{5}},\bm{e_{6}}\\},\end{split}$
(C.5)

and chirality operators

$\displaystyle\begin{split}\chi(\bm{V}^{1}_{pqrs})&=\text{ch}(\bar{\eta}^{2})+\text{ch}(\bar{\eta}^{3})=\\\
&=-\ C{\bm{V}^{1}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{2}}+(1-r)\bm{e_{5}}+(1-s)\bm{e_{6}}}^{*}-\
C{\bm{V}^{1}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{3}}+(1-p)\bm{e_{3}}+(1-q)\bm{e_{4}}}^{*}\\\
\chi(\bm{V}^{2}_{pqrs})&=\text{ch}(\bar{\eta}^{1})+\text{ch}(\bar{\eta}^{3})=\\\
&=-\ C{\bm{V}^{2}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{1}}+(1-r)\bm{e_{5}}+(1-s)\bm{e_{6}}}^{*}-\
C{\bm{V}^{2}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{3}}+(1-p)\bm{e_{1}}+(1-q)\bm{e_{2}}}^{*}\\\
\chi(\bm{V}^{3}_{pqrs})&=\text{ch}(\bar{\eta}^{1})+\text{ch}(\bar{\eta}^{2})=\\\
&=-\ C{\bm{V}^{3}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{1}}+(1-r)\bm{e_{3}}+(1-s)\bm{e_{4}}}^{*}-\
C{\bm{V}^{3}_{pqrs}\atopwithdelims[
]\bm{S}+\bm{b_{2}}+(1-p)\bm{e_{1}}+(1-q)\bm{e_{2}}}^{*}.\end{split}$ (C.6)

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