#  KEK-TH-2503 New non-supersymmetric heterotic string theory with reduced
rank and exponential suppression of the cosmological constant

Sota Nakajima
Theory Center, IPNS, High Energy Accelerator Research Organization (KEK),
1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan e-mail<EMAIL_ADDRESS>
###### Abstract

We study the heterotic asymmetric orbifold model in which supersymmetry is
broken by the stringy Schark-Schwarz mechanism. This model is a natural non-
supersymmetric extension of CHL strings and can also be interpreted as the
interpolating model between the $E_{8}\times E^{\prime}_{8}$ theory and the
non-supersymmetric $E_{8}$ theory. The enhancement of gauge groups, of which
the rank is reduced to $8+d$, is explored. In particular, the enhancement to
non-simply-laced groups is possible with $d\geq 2$, as well as in the CHL
model. We also give the conditions that the massless matter spectrum must
satisfy. Moreover, the one-loop cosmological constant is evaluated in the
regime where supersymmetry is asymptotically restored, and we show that the
exponential suppression can occur unless $d=1$.

###### Contents

  1. 1 Introduction
  2. 2 Reveiw of CHL strings
    1. 2.1 Partition function and momentum lattice
    2. 2.2 Massless spectrum
  3. 3 Non-supersymmetric CHL strings
    1. 3.1 Partition function
    2. 3.2 Spectrum
      1. 3.2.1 Untwisted sector
      2. 3.2.2 Twisted sector
    3. 3.3 Relevance to CHL strings
    4. 3.4 $d=1$
    5. 3.5 $d=2$
      1. 3.5.1 Example: $C_{9}\times C_{1}$ gauge symmetry
      2. 3.5.2 Example: $A_{4}\times C_{2}\times C_{4}$ gauge symmetry
  4. 4 Cosmological constant
    1. 4.1 Suppression of $\Lambda^{(8)}$
    2. 4.2 Suppression of $\Lambda^{(7)}$
  5. 5 Summary and discussion
    1. 5.1 Moduli stabilization
  6. A Lattices and characters
  7. B Non-supersymmetric $E_{8}$ theory
  8. C Solutions in $d=2$
    1. C.1 Enhancement to $C_{9}\times C_{1}$
    2. C.2 Enhancement to $A_{4}\times C_{4}\times C_{2}$

##  1 Introduction

The landscape of non-supersymmetric (heterotic) string vacua is larger than
that of supersymmetric ones. In ten dimensions, for instance, there are only
two supersymmetric heterotic string theories: the $E_{8}\times E^{\prime}_{8}$
theory and the $SO(32)$ theory[1]. That follows from the fact that only two
types of Euclidean even self-dual lattices exist in $16$ dimensions, which are
the $E_{8}\times E^{\prime}_{8}$ lattice and the $Spin(32)/\mathbb{Z}_{2}$
lattice. On the other hand, by starting from supersymmetric theories and
$\mathbb{Z}_{2}$-orbifolding by freely acting shifts on $T_{L}^{16}$
associated with a $2\pi$ spatial rotation, various ten-dimensional heterotic
vacua without spacetime supersymmetry can be realized[2, 3]111This
construction can be generalized to lower dimensional cases just by extending
the shifts acting on $T_{L}^{16}$ to on $T_{L}^{16}\times T^{d}$[4].. In the
language of the fermionic formulation, this kind of construction corresponds
to introducing a discrete torsion[5]. Those non-supersymmetric theories are
classified into six types depending on the independent choices of shift
vectors of order $2$: the $SO(16)\times SO(16)$ model, the $SO(32)$ model, the
$SO(8)\times SO(24)$ model, the $U(16)$ model, the $SO(16)\times E_{8}$ model
and the $\left(E_{7}\times SU(2)\right)^{2}$ model. Among them, only the
$SO(16)\times SO(16)$ model is tachyon-free. That is a reason why most studies
about non-supersymmetric heterotic string phenomenology focus on the
$SO(16)\times SO(16)$ model (e.g. in refs. [6, 7, 8, 9, 10]), although it was
also proposed that phenomenologically viable models can be obtained by
starting from ten-dimensional tachyonic string vacua[11, 12, 13, 14, 15, 16].
By the construction, the $10$D non-supersymmetric theories are discretely
related to the 10D supersymmetric ones. They can also be connected smoothly
via lower dimensional non-supersymmetric models so-called interpolating models
[17, 18] which are constructed by the stringy Scherk-Schwarz
compactification[19, 20, 21]. The supersymmetry breaking scale $m_{3/2}$ is
controlled by a radius $R$ of the direction accompanied by the Scherk-Schwarz
mechanism as $m_{3/2}\sim 1/R$. Thus, the supersymmetry is restored as $R$
grows, while a non-supersymmetric string model in higher dimensions is
reproduced in the limit $R\to 0$. In that sense, these kinds of models
interpolate between supersymmetric and non-supersymmetric string vacua. Such
models are preferred when a top-down scenario with high-energy supersymmetry
breaking is considered (e.g. see [22, 23, 24, 25, 26, 27]). A reason for that
is that in the regime with supersymmetry restored, the asymptotic behavior of
one-loop cosmological constant takes the form[17, 28]222Through this paper, we
set $\alpha^{\prime}=1$.

$\displaystyle\Lambda^{(10-d)}\sim\frac{\xi}{R^{10-d}}\left(n_{F}-n_{B}\right)+\mathcal{O}\left(e^{-R}\right),$
(1.1)

where $\xi>0$ is a constant and $n_{F}$ ($n_{B}$) represents the degrees of
freedom of massless fermions (bosons). The formula (1.1) indicates that the
cosmological constant is exponentially suppressed if $n_{F}-n_{B}=0$. Namely,
interpolating models with Bose-Fermi degeneracy at the massless level can
avoid the instability arising from one-loop dilaton tadpoles. In fact, there
are some points in the moduli space where $n_{F}-n_{B}=0$ is realized not only
in closed string theories[29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40] but
also in open string theories[41, 42]. Note that whether this suppression is
achieved or not depends on the spectrum of only massless states. It is known
that it is possible to construct type II asymmetric orbifold models without
supersymmetry but with Bose-Fermi degeneracy at each mass level, and thus the
cosmological constants can strictly vanish at one-loop even in non-
supersymmetric cases[43, 44, 45, 46].

All the non-supersymmetric models mentioned above are constructed by
$\mathbb{Z}_{2}$ freely acting orbifolds, and the rank of (left-moving) gauge
groups is always $16+d$ with compact dimension $d$. By using the outer
automorphism of the $E_{8}\times E^{\prime}_{8}$ lattice (denoted by
$\Gamma_{E_{8}}\oplus\Gamma_{E^{\prime}_{8}}$) instead of freely acting
shifts, it is possible to present heterotic strings in which the rank of gauge
groups is reduced to $8+d$. One has to note that orbifolding by only this
automorphism, which we denote by $\bm{\mathrm{R}}$ in this paper, gives the
original $E_{8}\times E^{\prime}_{8}$ theory since the twisted sector
reproduces the spectrum projected out by $\bm{\mathrm{R}}$333This fact is
valid not only for $\bm{\mathrm{R}}$ but also for a $2\pi$ spatial rotation.
Twisting one of the two supersymmetric heterotic theories only by a freely
acting shift can give the other heterotic theory [47].. In order to obtain a
different model, hence, we need to define a $\mathbb{Z}_{2}$-orbifold twist by
combining $\bm{\mathrm{R}}$ with a nontrivial $\mathbb{Z}_{2}$ generator. For
instance, using a $2\pi$ spatial rotation as an additional $\mathbb{Z}_{2}$
generator, we obtain the reduced rank non-supersymmetric $E_{8}$ model in ten
dimensions [48], which we review in appendix B. In lower dimensions, one can
use a half-shift along one of the compact directions as an additional
$\mathbb{Z}_{2}$ generator and build a reduced rank model with all
supercharges preserved which is called the CHL string model [49], which is
originally built by using the fermionic construction [50, 48]. One of the
interesting features of CHL strings is that gauge groups can be enhanced to
non-simply-laced ones if the compact dimension $d$ is greater than or equal to
$2$. Recently, in ref. [51], the algorithms to explore the moduli space of the
CHL strings have been developed and the maximal enhancements of gauge groups
have been completely identified for $d=1,2$. In this paper, we investigate the
non-supersymmetric reduced rank models, which we call the non-supersymmetric
CHL models, constructed by the asymmetric orbifold with the
$\mathbb{Z}_{2}$-twist given by the product of the three $\mathbb{Z}_{2}$
generators: the outer automorphism $\bm{\mathrm{R}}$ of
$\Gamma_{E^{\prime}_{8}}\times\Gamma_{E^{\prime}_{8}}$, the half-shift
$\bm{\mathrm{T}}$ along a compatified direction and a $2\pi$ spatial rotation
$(-1)^{F}$444In ref. [52], another way to build reduced rank heterotic
orbifold models has been proposed by using the $SO(2n+1)$ characters.. We will
show that in the reduced rank model, the formula (1.1) is still valid, and the
cosmological constant can be exponentially suppressed.

As mentioned above, the ingredients that we need to construct the orbifold
models in this paper are only the following three types of the
$\mathbb{Z}_{2}$ generators:

  * •

outer automorphism of the $E_{8}\times E^{\prime}_{8}$ lattice, i.e.,
interchange of the two $E_{8}$ factors:

$\displaystyle\bm{\mathrm{R}}:~{}\Gamma_{E_{8}}\oplus\Gamma_{E^{\prime}_{8}}\to\Gamma_{E^{\prime}_{8}}\oplus\Gamma_{E_{8}}.$
(1.2)

  * •

half translation along a compact direction $x^{1}$ with a periodicity
$x^{1}\sim x^{1}+2\pi R$:

$\displaystyle\bm{\mathrm{T}}:~{}x^{1}\to x^{1}+\pi R.$ (1.3)

  * •

$2\pi$ spatial rotation: $(-1)^{F}$ where $F$ is the spacetime fermion number.

We here summarize reduced rank heterotic string models depending on the
combinations of the above $\mathbb{Z}_{2}$ generators:

  * •

$\bm{\mathrm{R}}\bm{\mathrm{T}}$: the CHL string (see section 2).

  * •

$\bm{\mathrm{R}}(-1)^{F}$: the non-supersymmetric $E_{8}$ string (see appendix
B).

  * •

$\bm{\mathrm{R}}\bm{\mathrm{T}}(-1)^{F}$: the non-supersymmetric CHL string
(see section 3).

This paper is organized as follows. In section 2, we review the CHL model,
including the partition function, the momentum lattice and the massless
spectrum. Those are useful when we study the non-supersymmetric CHL model. We
will see why the enhancement to non-simply-laced groups is possible in the
reduced rank model. In section 3, we move to consider the non-supersymmetric
CHL model. The partition function can be expressed in terms of the building
blocks of the CHL string. We show that this non-supersymmetric model
interpolates between the $E_{8}\times E^{\prime}_{8}$ theory and the $E_{8}$
theory. Comparing in the CHL model, moreover, we identify the conditions for
$U(1)_{L}^{16}\times U(1)_{l}^{d}$ to be enhanced a non-Abelian group and the
formulae that matter contents in the massless spectrum must satisfy. In
particular, some explicit examples of symmetry enhancement to non-simply-laced
algebra are given in the case $d=2$. We then evaluate the one-loop
cosmological constant of the non-supersymmetric CHL string in the regime $R\gg
1$ and show that the exponential suppression may occur for $d\geq 2$ in
section 4. Indeed, we give examples of the moduli that cause the suppression
in the cases $d=2$ and $d=3$. The summary and discussion are given in section
5.

##  2 Reveiw of CHL strings

In this section, we review the CHL string model, i.e., an asymmetric orbifold
model with $\bm{\mathrm{R}}\bm{\mathrm{T}}$. We will give the conditions on
the internal momenta of massless states and see that the symmetry enhancement
to non-simply-laced algebras is possible if $d\geq 2$ (see [51] for more
detail).

###  2.1 Partition function and momentum lattice

The starting point is the $E_{8}\times E^{\prime}_{8}$ heterotic model
compactified on $T^{d}$, in which we have $(16+d)\times d$ moduli: the metric
$G_{mn}$, the anti-symmetric tensor $B_{mn}$ and the Wilson lines $A_{m}^{A}$,
where the indices $m,n=1,\cdots,d$ and $A=1,\cdots,16$ represent the
directions along $T^{d}$ and
$T_{L}^{16}=\mathbb{R}^{16}/(\Gamma_{E_{8}}\oplus\Gamma_{E^{\prime}_{8}})$
respectively. The internal momenta, which are identified as elements of even
self-dual lattice with the signature $(16+d,d)$ so-called Narain lattice
(denoted by  $\mathbb{L}$ ), can be written as [53, 54]

$\displaystyle\ell_{L}^{A}$ $\displaystyle=\Pi^{A}-w^{m}A_{m}^{A},$ (2.1a)
$\displaystyle p_{L,a}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\Pi^{A}A_{m}^{A}+w^{n}\left(G-C\right)_{nm}+n_{m}\right)e^{*m}_{a},$
(2.1b) $\displaystyle p_{R,a}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\Pi^{A}A_{m}^{A}-w^{n}\left(G+C\right)_{nm}+n_{m}\right)e^{*m}_{a},$
(2.1c)

where $\Pi^{A}=\left(\pi^{I},\pi^{\prime
I}\right)\in\Gamma_{E_{8}}\oplus\Gamma_{E^{\prime}_{8}}$ with $I=1,\cdots,8$.
Here $C_{mn}:=-B_{mn}+\frac{1}{2}A^{A}_{m}A^{A}_{n}$ and $e^{a}_{m}$ is a
vierbein on $T^{d}$ satisfying

$\displaystyle
e^{a}_{m}e^{a}_{n}=G_{mn},~{}~{}~{}e^{a}_{m}e^{*n}_{a}=\delta_{m}^{n},~{}~{}~{}e^{*m}_{a}e^{*n}_{a}=(G^{-1})^{mn}.$
(2.2)

In order to be consistent with the orbifold symmetry $\bm{\mathrm{R}}$, the
components of the Wilson lines must satisfy

$\displaystyle A_{m}^{I}=A_{m}^{I+8}=a_{m}^{I}\in M_{d\times 16}(\mathbb{R}),$
(2.3)

where $M_{d\times d^{\prime}}(\mathbb{R})$ represents a set of $d\times
d^{\prime}$ matrices with real components. Since $\pi^{I}$ and $\pi^{\prime
I}$ are interchanged under $\bm{\mathrm{R}}$, it is convenient to change the
basis as

$\displaystyle\ell_{\pm}^{I}=\frac{1}{\sqrt{2}}\left(\ell_{L}^{I}\pm\ell_{L}^{I+8}\right).$
(2.4)

With the background (2.3), the internal momenta are then

$\displaystyle\ell_{-}$ $\displaystyle=\frac{1}{\sqrt{2}}\pi_{-},$ (2.5a)
$\displaystyle\ell_{+}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\pi_{+}-2wa\right),$ (2.5b)
$\displaystyle p_{L}$ $\displaystyle=\frac{1}{\sqrt{2}}\left(\pi_{+}\cdot
a+w\left(2G-E\right)+n\right)e^{-t},$ (2.5c) $\displaystyle p_{R}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\pi_{+}\cdot a-wE+n\right)e^{-t}.$
(2.5d)

In the above expression, we adopt the notations of the row vectors

$\displaystyle
w\in\mathbb{Z}^{d},~{}~{}~{}n\in\mathbb{Z}^{d},~{}~{}~{}\pi_{\pm}:=\pi\pm\pi^{\prime}\in\Gamma_{E_{8}},$
(2.6)

and the matrices555To be precise, the matrix $G$ is more restricted since it
must be positive definite.

$\displaystyle a\in M_{d\times 16}(\mathbb{R}),~{}~{}~{}G=G^{t}\in M_{d\times
d}(\mathbb{R}),~{}~{}~{}E:=G-B+aa^{t}\in M_{d\times d}(\mathbb{R}).$ (2.7)

Note that only the component $\ell_{-}$ is reflected under $\bm{\mathrm{R}}$:

$\displaystyle\bm{\mathrm{R}}:~{}\left(\ell_{-},\ell_{+},p_{L};p_{R}\right)\to\left(-\ell_{-},\ell_{+},p_{L};p_{R}\right).$
(2.8)

By introducing the generalized vierbein, we can express the momenta (2.5) more
simply:

$\displaystyle\left(\ell_{-},\ell_{+},p_{L};p_{R}\right)=\left(\pi_{-},\pi_{+},w,n\right)\mathcal{E}\left(e,B,A\right),$
(2.9)

where the generalized vierbein $\mathcal{E}$ is a $(16+2d)\times(16+2d)$
matrix

$\displaystyle\mathcal{E}\left(e,B,a\right)=\left(\begin{array}[]{cc}\frac{1}{\sqrt{2}}\bm{1}_{8}&0\\\
0&\mathcal{E}^{(d)}\left(e,B,a\right)\end{array}\right),$ (2.12)

with the $(8+2d)\times(8+2d)$ matrix $\mathcal{E}^{(d)}$ given by

$\displaystyle\mathcal{E}^{(d)}\left(e,B,a\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}\bm{1}_{8}&a^{t}e^{-t}&a^{t}e^{-t}\\\
-2a&\left(G-C\right)e^{-t}&-\left(G+C\right)e^{-t}\\\
0&e^{-t}&e^{-t}\end{array}\right).$ (2.16)

It is convenient to define a $(8+2d)$-dimensional lattice $\Gamma^{8+d,d}$ as

$\displaystyle\Gamma^{8+d,d}:=\left\\{P=\left(\pi_{+},w,n\right)\mathcal{E}^{(d)}\left(e,B,a\right)\left|~{}\pi_{+}\in\Gamma_{E_{8}},~{}w,n\in\mathbb{Z}^{d}\right.\right\\}.$
(2.17)

The Lorentzian inner product in $\Gamma^{8+d,d}$ is computed by the Narain-
like metric

$\displaystyle\mathcal{E}^{(d)}\left(e,B,a\right)\eta^{(8+d,d)}\mathcal{E}^{(d)t}\left(e,B,a\right)=\left(\begin{array}[]{ccc}\frac{1}{2}\bm{1}_{8}&0&0\\\
0&0&\bm{1}_{d}\\\ 0&\bm{1}_{d}&0\end{array}\right),$ (2.21)

where $\eta^{(8+d,d)}=\mathrm{diag}(\bm{1}_{8+d},-\bm{1}_{d})$. Note that
$\Gamma^{8+d,d}$ is not even lattice unlike  $\mathds{L}$ :

$\displaystyle\ell_{+}^{2}+p_{L}^{2}-p_{R}^{2}=\frac{1}{2}\pi_{+}^{2}+2wn^{t}\in\mathbb{Z}.$
(2.22)

The partition function of the CHL model is expressed as

$\displaystyle
Z_{\text{CHL}}^{(10-d)}=\frac{1}{2}Z_{B}^{(8-d)}\left(\bar{V}_{8}-\bar{S}_{8}\right)\sum_{a,b=0,1}Z(g^{a},g^{b}),$
(2.23)

where $Z_{B}^{(8-d)}:=\tau_{2}^{-(8-d)/2}(\eta\bar{\eta})^{-(8-d)}$ is the
contribution from the world-sheet bosonic fields propagating in the spacetime
dimensions, and $\bar{V}_{8}$ and $\bar{S}_{8}$ are the conjugacy classes of
the $SO(8)$ vector and spinor (see appendix A) which are the contribution from
the right-moving fermionic fields. Each of the building blocks
$Z(g^{a},g^{b})$ can be identified as (see appendix in ref. [51] for detail)

$\displaystyle Z(1,1)$
$\displaystyle=\frac{1}{\bar{\eta}^{d}\eta^{16+d}}\sum_{\left(\ell_{-},P\right)\in\reflectbox{\rotatebox[origin={c}]{180.0}{$\mathds{L}$}}}q^{\frac{1}{2}\left(\ell_{-}^{2}+P_{L}^{2}\right)}\bar{q}^{\frac{1}{2}p_{R}^{2}},$
(2.24a) $\displaystyle Z(1,g)$
$\displaystyle=\frac{1}{\bar{\eta}^{d}\eta^{16+d}}\left(\frac{2\eta^{3}}{\vartheta_{2}}\right)^{4}\sum_{P\in\mathbb{I}}q^{\frac{1}{2}P_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}e^{2\pi
iP\cdot V},$ (2.24b) $\displaystyle Z(g,1)$
$\displaystyle=\frac{1}{\bar{\eta}^{d}\eta^{16+d}}\left(\frac{\eta^{3}}{\vartheta_{4}}\right)^{4}\sum_{P\in\mathbb{I}^{*}}q^{\frac{1}{2}\left(P_{L}+V_{L}\right)^{2}}\bar{q}^{\frac{1}{2}\left(p_{R}+v_{R}\right)^{2}},$
(2.24c) $\displaystyle Z(g,g)$
$\displaystyle=\frac{1}{\bar{\eta}^{d}\eta^{16+d}}\left(\frac{e^{\frac{\pi
i}{4}}\eta^{3}}{\vartheta_{3}}\right)^{4}\sum_{P\in\mathbb{I}^{*}}q^{\frac{1}{2}\left(P_{L}+V_{L}\right)^{2}}\bar{q}^{\frac{1}{2}\left(p_{R}+v_{R}\right)^{2}}e^{\pi
i\left(P+V\right)^{2}}.$ (2.24d)

The first and second arguments of $Z(g^{a},g^{b})$ indicate the twists along
the spatial and time directions on the world-sheet respectively, and
$Z(g,1)+Z(g,g)$ is the contribution from the twisted sector which is necessary
for $Z_{\text{CHL}}^{(d)}$ to be modular invariant[55, 47]. Here $\mathbb{I}$
and $\mathbb{I}^{*}$ are the invariant lattice under $\bm{\mathrm{R}}$ and its
dual, which are identified below, and $V=\left(V_{L},v_{R}\right)$ is a shift
vector of order $2$ on $\Gamma^{8+d,d}$, which can be expressed as

$\displaystyle
V=\frac{1}{2}\left(\hat{\pi}_{+},\hat{w},\hat{n}\right)\mathcal{E}^{(d)}\left(e,B,A\right),~{}~{}~{}\text{for
}~{}\hat{\pi}_{+}\in\Gamma_{E_{8}},~{}\hat{w},\hat{n}\in\mathbb{Z}^{d}.$
(2.25)

In order for the total partition function to be modular invariant666To be
precise, $Z(g,1)$ must be invariant under $\tau\to\tau+2$., the shift vector
$V$ must be chosen such that

$\displaystyle\left(P+V\right)^{2}\in\mathbb{Z},~{}~{}~{}\text{for any
$P\in\mathbb{I}^{*}$}.$ (2.26)

In this paper, we choose $\hat{\pi}_{+}=0$, $\hat{n}=0$ and
$\hat{w}=(1,0^{d-1})$ so that $V$ generates the half-shift along $x^{1}$.

The invariant lattice $\mathbb{I}$ is defined by a subset of  $\mathds{L}$ in
which elements are invariant under the exchange
$\pi\leftrightarrow\pi^{\prime}$, i.e., $\pi_{-}=0$ and $\pi_{+}\in
2\Gamma_{E_{8}}$. Thus, we can express $\mathbb{I}$ in terms of the vierbein
$\mathcal{E}^{(d)}$:

$\displaystyle\mathbb{I}=\left\\{P=\left(\pi_{+},w,n\right)\mathcal{E}^{(d)}\left(e,B,a\right)\left|~{}\pi_{+}\in
2\Gamma_{E_{8}},~{}w,n\in\mathbb{Z}^{d}\right.\right\\}.$ (2.27)

Since the inner product in $\mathbb{I}$ is defined by the metric (2.21), we
can identify the dual lattice $\mathbb{I}^{*}$ as

$\displaystyle\mathbb{I}^{*}=\Gamma^{8+d,d}=\left\\{P=\left(\pi_{+},w,n\right)\mathcal{E}^{(d)}\left(e,B,a\right)\left|~{}\pi_{+}\in\Gamma_{E_{8}},~{}w,n\in\mathbb{Z}^{d}\right.\right\\}.$
(2.28)

The internal momenta of the twisted states are shifted by $V$, and the
momentum lattice in the twisted sector is given by

$\displaystyle\mathbb{I}^{*}+V=\left\\{P=\left(\pi_{+},w,n\right)\mathcal{E}^{(d)}\left(e,B,a\right)\left|~{}\pi_{+}\in\Gamma_{E_{8}},~{}w\in\mathbb{Z}^{d}+\frac{\hat{w}}{2},~{}n\in\mathbb{Z}^{d}\right.\right\\}.$
(2.29)

A more detailed study of the momentum lattice of CHL strings was done in ref.
[56].

###  2.2 Massless spectrum

Let us study the spectrum of the CHL string. As well as for the internal
momenta, it is convenient to define

$\displaystyle X_{\pm}^{I}:=\frac{1}{\sqrt{2}}\left(X_{L}^{I}\pm
X_{L}^{I+8}\right).$ (2.30)

The transformation law for the oscillators under $\bm{\mathrm{R}}$ is the same
as for the momenta $(\ell_{-},\ell_{+},p_{L};p_{R})$:

$\displaystyle\bm{\mathrm{R}}:~{}\left(\alpha_{-}^{I},\alpha_{+}^{I},\alpha^{\mu}\right)\to\left(-\alpha_{-}^{I},\alpha_{+}^{I},\alpha^{\mu}\right),$
(2.31)

where $\alpha_{\pm}^{I}$ denotes the oscillators for $X_{\pm}^{I}$ and
$\alpha^{\mu}$ for $X^{\mu}$ with $\mu=2,\cdots,9$ being the indices of the
spacetime transverse directions. The spectrum in the untwisted sector, which
is invariant under $\bm{\mathrm{R}}\bm{\mathrm{T}}$, consists of the following
combinations of states:

$\displaystyle\ket{\varphi}_{\text{untw}}=\frac{1}{\sqrt{2}}\left(\alpha\ket{w,n,\pi,\pi^{\prime}}+(-)^{n_{1}}g(\alpha)\ket{w,n,\pi^{\prime},\pi}\right),$
(2.32)

where $\alpha$ denotes any possible combinations of the oscillators and
$g(\alpha)$ its image under $\bm{\mathrm{R}}$, given by (2.31). Note that the
untwisted states without oscillator excitation and with $\pi=\pi^{\prime}$
must have $n_{1}$ even. As we will see later, this fact is the reason why the
gauge symmetry can not be enhanced to non-ADE algebras in the case $d=1$.

In the twisted sector, $X_{-}^{I}(\sigma+t)$ is anti-periodic under
$\sigma\to\sigma+2\pi$, up to elements of $\Gamma_{E_{8}}$ multiplied by
$1/\sqrt{2}$. So, the oscillator excitation for the $X_{-}^{I}$-directions is
given by $\alpha^{I}_{-,r}$ with $r\in\mathbb{Z}+1/2$. The orbifold projection
in the twisted sector can be figured out by the partition function:

$\displaystyle\frac{1}{2}\left(Z(g,1)+Z(g,g)\right)$
$\displaystyle=\frac{1}{\bar{\eta}^{d}\eta^{16+d}}\sum_{P\in\mathbb{I}^{*}+V}q^{\frac{1}{2}P_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}\frac{1}{2}\left(\left(\frac{\eta^{3}}{\vartheta_{4}}\right)^{4}-\left(\frac{\eta^{3}}{\vartheta_{3}}\right)^{4}e^{\pi
i\left(\frac{\pi_{+}^{2}}{2}+n_{1}\right)}\right)$
$\displaystyle=\frac{1}{\bar{\eta}^{d}\eta^{16+d}}\sum_{s=0,1}\sum_{\begin{subarray}{c}P\in\mathbb{I}^{*}+V\\\
\frac{\pi_{+}^{2}}{2}+n_{1}\in
2\mathbb{Z}+s\end{subarray}}q^{\frac{1}{2}P_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}F\left(q,s\right),$
(2.33)

where

$\displaystyle
F\left(q,s\right):=\frac{1}{2}\left(\left(\frac{\eta^{3}}{\vartheta_{4}}\right)^{4}-e^{\pi
is}\left(\frac{\eta^{3}}{\vartheta_{3}}\right)^{4}\right).$ (2.34)

Note that $s=0,1$ specifies whether $P^{2}$ is even or odd, i.e.,

$\displaystyle P^{2}=P_{L}^{2}-p_{R}^{2}\stackrel{{\scriptstyle
2}}{{=}}\frac{\pi_{+}^{2}}{2}+n_{1}\stackrel{{\scriptstyle 2}}{{=}}s,$ (2.35)

where the symbol “$\stackrel{{\scriptstyle n}}{{=}}$” indicates that the
equality holds for mod $n$. The $q$-expansion of $F(q,s)/\eta^{8}$, which
gives the contribution from the oscillators $\alpha_{-}^{I}$ with the orbifold
projection, is

$\displaystyle\frac{F(q,s)}{\eta^{8}}=\frac{1}{2}\left(\left(\prod_{n=1}^{\infty}\sum_{N_{n}=0}^{\infty}\left(q^{n-\frac{1}{2}}\right)^{N_{n}}\right)^{8}-e^{\pi
is}\left(\prod_{n=1}^{\infty}\sum_{N_{n}=0}^{\infty}\left(-q^{n-\frac{1}{2}}\right)^{N_{n}}\right)^{8}\right),$
(2.36)

and thus we find that the twisted states take the form

$\displaystyle\ket{\varphi}_{\text{tw}}=\frac{1}{\sqrt{2}}\left(\alpha\ket{\pi_{+},w,n}+(-1)^{s+1}g(\alpha)\ket{\pi_{+},w,n}\right),$
(2.37)

where $g(\alpha)$ is given by (2.31). Let $N_{-}$ be the total number of
$\alpha_{-,r}^{I}$ acting on $\ket{\pi_{+},w,n}$. The projection (2.37)
requires that $N_{-}$ must be even (odd) for $P^{2}$ odd (even), and as we
will see below, indeed, all states satisfying the level-matching condition
survive under the projection (2.37).

The mass formula is given by

$\displaystyle M_{R}^{2}$ $\displaystyle=p_{R}^{2}+2\left(N_{R}+a_{R}\right),$
(2.38a) $\displaystyle M_{L}^{2}$
$\displaystyle=\begin{cases}P_{L}^{2}+\ell_{-}^{2}+2\left(N_{L}-1\right)&(\text{untwisted
sector})\\\ P_{L}^{2}+2\left(N_{L}-\frac{1}{2}\right)&(\text{twisted
sector})\\\ \end{cases},$ (2.38b)

where $a_{R}$ is the zero-point energy for the right-moving states:
$a_{R}=-1/2~{}(0)$ for NS (R) sector. The level-matching condition in the
twisted sector reads

$\displaystyle
P_{L}^{2}-p_{R}^{2}+2\left(N_{L}-\frac{1}{2}-N_{R}-a_{R}\right)=0.$ (2.39)

The GSO projection for the right-moving states is performed such that
$N_{R}+a_{R}$ is an integer, and hence $P^{2}$ must be even (odd) for $N_{-}$
odd (even) in order to satisfy the condition (2.39). Then, the level-matching
condition demands that the twisted states should take the form (2.37).

Taking the GSO projection into account, $M_{R}=0$ is realized only when
$N_{R}+a_{R}=0$ and $p_{R}=0$. The latter is explicitly written as

$\displaystyle n=-\pi_{+}\cdot a+wE\in\mathbb{Z}^{d}.$ (2.40)

In order for a (left-moving) untwisted state to be massless, the excitation
number $N_{L}$ must be either $1$ or $0$;

  * •

$N_{L}=1$;
This state is massless if $P_{L}=\ell_{-}=0$, i.e., $\pi=\pi^{\prime}=w=n=0$.
By acting $\alpha_{-1}^{\mu}$ on the ground state, we obtain a gravity
multiplet and KK gauge bosons in NS sector and their fermionic partners in R
sector. As for the excitation by $\alpha_{-1}^{A}$, only the combinations

$\displaystyle\alpha_{-1}^{I}+\alpha_{-1}^{I+8}$ (2.41)

survive under the orbifold projection. The other independent eight Cartan
states are projected out, and then the rank of gauge groups is reduced to
$8+d$.

  * •

$N_{L}=0$;
With no oscillator excitation, by the mass formula (2.38b), it turns out that
$M_{L}^{2}=0$ requires

$\displaystyle\ell_{-}^{2}+P_{L}^{2}=2.$ (2.42)

Since $\ell_{-}^{2}$ is a non-negative integer, we have the following two
types of solutions777Note that $P_{L}=0$ ($\ell_{-}^{2}=2$) is excluded since
$P_{L}=p_{R}=0$ implies $\pi_{+}=w=n=0$, but $\ell_{-}^{2}\in 4\mathbb{Z}$
under $\pi_{+}=0$, which is inconsistent with $\ell_{-}^{2}=2$.:

$\displaystyle P_{L}^{2}\stackrel{{\scriptstyle\eqref{eq:
p_R=0}}}{{=}}\frac{1}{2}\pi_{+}^{2}+2wn=\begin{cases}1&\text{with
}\left(\pi-\pi^{\prime}\right)^{2}=1\\\ 2&\text{with
}~{}\pi=\pi^{\prime}\end{cases}.$ (2.43)

Note that in the second one, $\pi_{+}$ are restricted to $\pi_{+}\in
2\Gamma_{E_{8}}$ and thus $n_{1}$ must be even. (Recall that states with
$N_{L}=0$, $\pi=\pi^{\prime}$ are projected out if $n_{1}$ is odd.) A set of
states with $P_{L}^{2}=2$ corresponds to long roots of a non-simply-laced
algebra. Namely, the enhancement to a non-ADE gauge group is realized if there
exist solutions $\left(\pi_{+},w,n\right)$ of $P_{L}^{2}=2$ satisfying the
condition (2.40) for a given set of the moduli. One can see that in the case
$d=1$, the enhancement to non-simply-laced algebras is impossible since

$\displaystyle\frac{1}{2}\pi_{+}^{2}+2w^{1}n_{1}\in
4\mathbb{Z},~{}~{}~{}\text{for }\pi_{+}\in 2\Gamma_{E_{8}},~{}n_{1}\in
2\mathbb{Z},$ (2.44)

and $P_{L}^{2}$ can not be $2$. In the case $d\geq 2$, however, in eq. (2.43)
we have $2\sum_{m=2}^{d}w^{m}n_{m}$ in addition to (2.44), and solutions for
long roots can be found at appropriate points in the moduli space.

In the twisted sector, $P_{L}^{2}$ can not vanish due to
$w^{1}\in\mathbb{Z}+1/2$, and $M_{L}^{2}=0$ leads to $N_{L}=0$ and
$P_{L}^{2}=1$. Substituting $p_{R}=0$ and $P_{L}^{2}=1$ into eq. (2.22), we
obtain

$\displaystyle\frac{1}{2}\pi_{+}^{2}+2wn=1,~{}~{}~{}\text{for
}\pi_{+}\in\Gamma_{E_{8}},~{}w\in\mathbb{Z}^{d}+\frac{\hat{w}}{2},~{}n\in\mathbb{Z}^{d}.$
(2.45)

States with long roots $P_{L}^{2}=2$ never appear in the massless spectrum of
the twisted sector.

Noting $p_{L}=p_{R}+\sqrt{2}we$, one can find an alternative formula of (2.43)
or (2.45):

$\displaystyle
P_{L}^{2}=\frac{1}{2}\left(\pi_{+}-2wa\right)^{2}+2wGw^{t}=1\text{ or }2.$
(2.46)

For long roots, in particular, this formula is expressed as

$\displaystyle\left(\pi^{\prime}_{+}-wa\right)^{2}+wGw^{t}=1,~{}~{}~{}~{}\text{for
}\pi^{\prime}_{+}=\frac{\pi_{+}}{2}\in\Gamma_{E_{8}}.$ (2.47)

The formula (2.46) is more convenient in order to find a finite set of
$\left(\pi_{+},w,n\right)$ satisfying $p_{R}=0$ and $P_{L}^{2}=1,2$.

##  3 Non-supersymmetric CHL strings

In the following sections, we construct the asymmetric orbifold with
$\bm{\mathrm{R}}\bm{\mathrm{T}}(-1)^{F}$ and explore the non-supersymmetric
CHL model. A point that we have to note is that the $\mathbb{Z}_{2}$-twist
acts on bosonic states and fermionic states in different ways. Namely, bosonic
states in the untwisted sector are even under $\bm{\mathrm{R}}\bm{\mathrm{T}}$
as well in the CHL model, while fermionic states take the combinations such
that the eigenvalue under $\bm{\mathrm{R}}\bm{\mathrm{T}}$ is $-1$. Thus, the
bosonic spectrum in the untwisted sector is the same as in the CHL model, and
the fermionic spectrum is provided by the opposite projection. The twisted
sector in this case turns out to contain the scalar and conjugate spinor (co-
spinor) conjugacy class of the spacetime $SO(8)$ (see appendix A).

We provide the partition function which is not only useful to explore the
spectrum but also necessary to evaluate the cosmological constant in section
4. Then we study the massless spectrum of the non-supersymmetric CHL string.
The examples for $d=1,2$ are given, and we will explicitly see that the gauge
symmetries are enhanced to non-ADE algebras in the case $d=2$.

###  3.1 Partition function

The partition function of the non-supersymmetric CHL string is expressed in
terms of the building blocks $Z(g^{a},g^{b})$ of the CHL string, given by
(2.24). Since a $2\pi$ spatial rotation $(-1)^{F}$ assigns $\bar{V}_{8}$ and
$\bar{S}_{8}$ to $+1$ and $-1$ respectively, the partition function for the
untwisted sector takes the form

$\displaystyle\frac{1}{2}Z_{B}^{(8-d)}\left\\{\left(\bar{V}_{8}-\bar{S}_{8}\right)Z(1,1)+\left(\bar{V}_{8}+\bar{S}_{8}\right)Z(1,g)\right\\}.$
(3.1)

From the transformation law (A.17) of the $SO(2n)$ characters, we see under
$\tau\to-1/\tau$,

$\displaystyle\left(\bar{V}_{8}+\bar{S}_{8}\right)Z(1,g)\xrightarrow[]{\tau\to-\frac{1}{\tau}}\left(\bar{O}_{8}-\bar{C}_{8}\right)Z(g,1).$
(3.2)

Performing the transformation $\tau\to\tau+1$, we find

$\displaystyle\left(\bar{O}_{8}-\bar{C}_{8}\right)Z(1,g)\xrightarrow[]{\tau\to\tau+1}-\left(\bar{O}_{8}+\bar{C}_{8}\right)Z(g,g).$
(3.3)

The total partition function is then

$\displaystyle Z_{\mathcal{N}=0}^{(10-d)}$
$\displaystyle=\frac{1}{2}Z_{B}^{(8-d)}\left\\{\left(\bar{V}_{8}-\bar{S}_{8}\right)Z(1,1)+\left(\bar{V}_{8}+\bar{S}_{8}\right)Z(1,g)\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\left(\bar{O}_{8}-\bar{C}_{8}\right)Z(g,1)-\left(\bar{O}_{8}+\bar{C}_{8}\right)Z(g,g)\right\\}$
$\displaystyle=\frac{1}{2}Z_{B}^{(8-d)}\left\\{\bar{V}_{8}\left(Z(1,1)+Z(1,g)\right)-\bar{S}_{8}\left(Z(1,1)-Z(1,g)\right)\right.$
$\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\bar{O}_{8}\left(Z(g,1)-Z(g,g)\right)-\bar{C}_{8}\left(Z(g,1)+Z(g,g)\right)\right\\}.$
(3.4)

One can see that the contribution from $\alpha_{-}^{I}$ in the co-spinor
conjugacy class in the twisted sector is $F(q,s)/\eta^{8}$ as well as in the
CHL model, while in the scalar conjugacy class is $F(q,s+1)/\eta^{8}$.

In order to see that this model interpolates between the $E_{8}\times
E^{\prime}_{8}$ theory and the $E_{8}$ theory, let us restrict our attention
to the case $d=1$ without the Wilson line ($a=0$) for simplicity. In the limit
$R\to\infty$, the contribution with $w^{1}\neq 0$ to the partition function is
suppressed, which means that the twisted sector vanishes. One can also find
that the behavior of the untwisted sector in $R\to\infty$ is given by

$\displaystyle Z(1,1)$
$\displaystyle\sim\frac{1}{\bar{\eta}\eta^{17}}\sum_{\Pi\in\Gamma_{E_{8}}\oplus\Gamma_{E^{\prime}_{8}}}q^{\frac{1}{2}\Pi^{2}}\sum_{n\in\mathbb{Z}}e^{-\pi\tau_{2}\left(\frac{n}{R}\right)^{2}}\xrightarrow[]{R\to\infty}\frac{R}{\sqrt{\tau_{2}}}\frac{1}{\bar{\eta}\eta^{17}}\sum_{\Pi\in\Gamma_{E_{8}}\oplus\Gamma_{E^{\prime}_{8}}}q^{\frac{1}{2}\Pi^{2}},$
(3.5) $\displaystyle Z(1,g)$
$\displaystyle\sim\frac{1}{\bar{\eta}\eta^{17}}\sum_{\pi_{+}\in
2\Gamma_{E_{8}}}q^{\frac{1}{4}\pi_{+}^{2}}\left(\sum_{n~{}\text{even}}-\sum_{n~{}\text{odd}}\right)e^{-\pi\tau_{2}\left(\frac{n}{R}\right)^{2}}\xrightarrow[]{R\to\infty}0.$
(3.6)

Then, we see that $Z_{\mathcal{N}=0}^{(9)}$ reproduces the original partition
function, i.e., the $E_{8}\times E^{\prime}_{8}$ theory. In the limit $R\to
0$, the contributions with $n=0$ do not vanish, and we obtain the following
behaviors of $Z\left(g^{a},g^{b}\right)$:

$\displaystyle Z(1,1)$
$\displaystyle\sim\frac{1}{\bar{\eta}\eta^{17}}\sum_{\Pi\in\Gamma_{E_{8}}\oplus\Gamma_{E^{\prime}_{8}}}q^{\frac{1}{2}\Pi^{2}}\sum_{n\in\mathbb{Z}}e^{-\pi\tau_{2}\left(wR\right)^{2}}\xrightarrow[]{R\to
0}\frac{R^{-1}}{\sqrt{\tau_{2}}}\frac{1}{\bar{\eta}\eta^{17}}\sum_{\Pi\in\Gamma_{E_{8}}\oplus\Gamma_{E^{\prime}_{8}}}q^{\frac{1}{2}\Pi^{2}},$
(3.7) $\displaystyle Z(1,g)$
$\displaystyle\sim\frac{1}{\bar{\eta}\eta^{17}}\sum_{\pi_{+}\in
2\Gamma_{E_{8}}}q^{\frac{1}{4}\pi_{+}^{2}}\sum_{w\in\mathbb{Z}}e^{-\pi\tau_{2}\left(wR\right)^{2}}\xrightarrow[]{R\to
0}\frac{R^{-1}}{\sqrt{\tau_{2}}}\frac{1}{\bar{\eta}\eta^{17}}\sum_{\pi_{+}\in
2\Gamma_{E_{8}}}q^{\frac{1}{4}\pi_{+}^{2}},$ (3.8) $\displaystyle Z(g,1)$
$\displaystyle\sim\frac{1}{\bar{\eta}\eta^{17}}\sum_{\pi_{+}\in\Gamma_{E_{8}}}q^{\frac{1}{4}\pi_{+}^{2}}\sum_{w\in\mathbb{Z}}e^{-\pi\tau_{2}\left(wR\right)^{2}}\xrightarrow[]{R\to
0}\frac{R^{-1}}{\sqrt{\tau_{2}}}\frac{1}{\bar{\eta}\eta^{17}}\sum_{\pi_{+}\in\Gamma_{E_{8}}}q^{\frac{1}{4}\pi_{+}^{2}},$
(3.9) $\displaystyle Z(g,g)$
$\displaystyle\sim\frac{1}{\bar{\eta}\eta^{17}}\sum_{\pi_{+}\in\Gamma_{E_{8}}}e^{\pi
i\pi_{+}^{2}}q^{\frac{1}{4}\pi_{+}^{2}}\sum_{w\in\mathbb{Z}}e^{-\pi\tau_{2}\left(wR\right)^{2}}\xrightarrow[]{R\to
0}\frac{R^{-1}}{\sqrt{\tau_{2}}}\frac{1}{\bar{\eta}\eta^{17}}\sum_{\pi_{+}\in\Gamma_{E_{8}}}e^{\pi
i\pi_{+}^{2}}q^{\frac{1}{4}\pi_{+}^{2}}.$ (3.10)

We then check that $Z_{\mathcal{N}=0}^{(9)}$ in the limit $R\to 0$ provides
the partition function of the $E_{8}$ theory, given by (B).

###  3.2 Spectrum

The mass formula is given in the same form as (2.38), but the orbifold
projection and the GSO projection in the twisted sector are different from
those in the CHL string. As one can notice from the partition function (3.1),
the momentum lattice depends on the conjugacy classes of the spacetime
$SO(8)$.

####  3.2.1 Untwisted sector

A surviving state in the untwisted sector takes the form

$\displaystyle\ket{\varphi}_{\text{untw}}=\frac{1}{\sqrt{2}}\left(\alpha\ket{w,n,\pi,\pi^{\prime}}+(-1)^{n_{1}+F}g(\alpha)\ket{w,n,\pi^{\prime},\pi}\right).$
(3.11)

The bosonic spectrum is the same as in the untwisted sector of the CHL string,
while the fermionic one is made of the combination of states with the opposite
sign under the projection.

Let us first consider the case $N_{L}=1$ and $P_{L}=0$. In the vector
conjugacy class with $F=0$, we obtain the gravitational multiplet and gauge
bosons of $U(1)_{r}^{d}\times U(1)_{l}^{d}\times U(1)_{L}^{8}$ from
$\alpha_{-1}^{\mu}\ket{0}$ and $\alpha_{+,-1}^{I}\ket{0}$. In the spinor
conjugacy class with $F=1$, the left-moving states $\alpha_{-1}^{\mu}\ket{0}$
and $\alpha_{+,-1}^{I}\ket{0}$ are projected out, while there are eight
massless spinors which come from $\alpha_{-,-1}^{I}\ket{0}$.

With $N_{L}=0$, the following massless state survives under the projection:

$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{w,n,\pi,\pi^{\prime}}+(-)^{n_{1}+F}\ket{w,n,\pi^{\prime},\pi}\right),$
(3.12)

where $\left(w,n,\pi,\pi^{\prime}\right)$ is a solution of (2.40) and (2.46).
The number of bosonic and fermionic states with $P_{L}^{2}=1$ is always the
same since $\pi\neq\pi^{\prime}$, i.e, if one find a solution of (2.40) and
(2.46), then the following combinations always survive under the projection:

$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{w,n,\pi,\pi^{\prime}}+(-1)^{n_{1}}\ket{w,n,\pi^{\prime},\pi}\right),~{}~{}~{}\text{for
bosonic states},$ (3.13)
$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{w,n,\pi,\pi^{\prime}}-(-1)^{n_{1}}\ket{w,n,\pi^{\prime},\pi}\right),~{}~{}~{}\text{for
fermionic states}.$ (3.14)

On the other hand, for $P_{L}^{2}=2$, the solutions must satisfy
$\pi=\pi^{\prime}$, and hence the degrees of freedom of bosonic and fermionic
states are in general different; the solutions with $n_{1}$ even give gauge
bosons with long roots, while those with $n_{1}$ odd correspond to long
weights of a representation in which spinors transform.

To summarize, the procedure to identify the massless spectrum in the untwisted
sector is as follows:

  1. (1)

There are a gravitational multiplet, gauge bosons of $U(1)_{r}^{d}\times
U(1)_{l}^{d}\times U(1)_{L}^{8}$ and eight neutral spinors in the massless
spectrum at any point in the moduli space.

  2. (2)

For a given moduli $\left(E,a\right)$, find a set of solutions
$\left(w,n,\pi_{+}\right)$ for (2.40) and (2.46).

  3. (3)

For $P_{L}^{2}=1$, the solution is a short root element in both the vector and
spinor conjugacy classes such as in (3.13).

  4. (4)

For $P_{L}^{2}=2$, the solution is a long root (weight) element in the vector
(spinor) conjugacy class if $n_{1}$ is even (odd).

As well as in the CHL model, if $d=1$, there is no solution of $P_{L}^{2}=2$
with $n_{1}$ even and thus the symmetry enhancement to non-ADE algebras can
not be realized. On the other hand, we can obtain solutions of $P_{L}^{2}=2$
with $n_{1}$ odd, and thus massless spinors can transform in a representation
with long weights even if $d=1$.

####  3.2.2 Twisted sector

In the twisted sector, the GSO projection for right-moving states is opposite
to the untwisted sector. As was done in subsection 2.2, we can figure out the
orbifold projection for the twisted sector from the partition function. For
the co-spinor conjugacy class, the left-moving partition function is the same
as in the CHL model. As for the scalar conjugacy class, the partition function
indicates that the projection is performed to be the opposite sign to the
twisted sector in the CHL model. Then, the twisted states in the non-
supersymmetric CHL model can be expressed as

$\displaystyle\ket{\varphi}_{\text{tw}}=\alpha\ket{\pi_{+},w,n}+(-1)^{s+F}g(\alpha)\ket{\pi_{+},w,n}.$
(3.15)

As well as in the CHL model, all twisted states satisfying the level-matching
condition survive under the projection888In the scalar conjugacy class,
$N_{R}+a_{R}$ in (2.39) must be a half-integer in order to survive under the
GSO projection, and then the level-matching condition requires that $s$ be $0$
($1$) for $N_{-}$ even (odd)..

Let us next identify the conditions on $(P_{L},p_{R})$ for massless co-spinors
and scalars.

#### Co-spinor conjugacy class

Since the difference from the spinor conjugacy class is only the spin-
chirality, $M_{R}^{2}=0$ is satisfied only when $N_{R}=0$ and $p_{R}=0$. The
conditions on $(P_{L},p_{R})$ are hence the same as in the twisted sector of
the CHL string, which are given by (2.40) and (2.45).

#### Scalar conjugacy class

In the scalar conjugacy class, the situation is somewhat different from the
others because $M_{R}^{2}=0$ leads to $N_{R}=0$ and $p_{R}^{2}=1$, due to the
GSO projection on the right-moving spectrum. Thus, the conditions on
$(P_{L},p_{R})$ can not be written as simple as eqs.(2.40) and (2.46).
Substituting the expression (2.5d) into $p_{R}^{2}=1$, we find

$\displaystyle qG^{-1}q^{t}=2,$ (3.16)

where $q:=\sqrt{2}p_{R}e^{t}=\pi_{+}\cdot a-wE+n$. Note that the condition
$n\in\mathbb{Z}^{d}$ can be written in terms of $q$:

$\displaystyle q-\pi_{+}\cdot a+wE\in\mathbb{Z}^{d}.$ (3.17)

As for the left-movers, there are the following two possibilities of
$M_{L}^{2}=0$:

  * •

Neutral scalars999We here use “neutral scalar” to mean that the scalar is a
singlet of the non-Abelian enhanced group. The scalar can be charged under
non-enhanced $U(1)$’s.: $N_{L}=\frac{1}{2}$, $P_{L}^{2}=0$;
The condition $P_{L}^{2}=0$ (i.e. $p_{L}=\ell_{+}=0$) implies
$p_{R}=\sqrt{2}we$ and $\pi_{+}=2wa$, and we can simplify the conditions on
$\left(P_{L},p_{R}\right)$ as

$\displaystyle\pi_{+}=2wa,~{}~{}~{}wGw^{t}=\frac{1}{2},~{}~{}~{}n=-wE^{t}.$
(3.18)

One can show that these massless states are singlets of the non-Abelian gauge
group. In order to see that, let $(\pi_{+}^{(v)},w^{(v)},n^{(v)})$ be a
solution of (2.40) and (2.43), and $(\pi_{+}^{(o)},w^{(o)},n^{(o)})$ be that
of (3.18). Then we check that $(\pi_{+}^{(o)},w^{(o)},n^{(o)})$ is orthogonal
to $(\pi_{+}^{(v)},w^{(v)},n^{(v)})$:

$\displaystyle\frac{1}{2}\pi_{+}^{(v)}\cdot\pi_{+}^{(o)}+w^{(v)}n^{(o)t}+n^{(v)}w^{(o)t}=\left(\pi_{+}^{(v)}\cdot
a-w^{(v)}E+n^{(v)}\right)w^{(o)t}=0,$ (3.19)

where we used $\pi_{+}^{(o)}=2w^{(o)}a$ and $n^{(o)}=-w^{(o)}E^{t}$ in the
first equality.

  * •

Charged scalars: $N_{L}=0$, $P_{L}^{2}=1$;
The condition $P_{L}^{2}=1$ can be explicitly written as

$\displaystyle\left(\pi_{+}-2wa\right)^{2}+\left(q+2wG\right)G^{-1}\left(q+2wG\right)^{t}=2.$
(3.20)

We also notice that the norm of the internal momenta vanishes:

$\displaystyle P_{L}^{2}-p_{R}^{2}=\frac{1}{2}\pi_{+}^{2}+2wn^{t}=0.$ (3.21)

In the case without Wilson lines, in particular, the condition (3.20) requires
either $\pi_{+}=0$ or $\pi_{+}^{2}=2$. In the former case, we find $q=-wG$ by
substituting (3.16) into (3.20). Under $a=0$ and $q=-wG$, the conditions
(3.16) and (3.17) leads to $wGw^{t}=2$ and $wB\in\mathbb{Z}^{d}$ respectively.
In the latter case, the condition (3.20) yields $q=-2wG$, and then we find
$2wGw^{t}=1$ and $-w\left(G+B\right)\in\mathbb{Z}^{d}$ from (3.16) and (3.17).
These conditions are nothing but (3.18) with $a=0$, and thus we obtain scalars
in the adjoint representation of the $E_{8}$. Indeed, one can check that if
the moduli satisfy $a=0$ and (3.18) for a set of
$(\pi_{+};w,n)\in\Gamma_{E_{8}}\oplus(\mathbb{Z}^{d}+\hat{w}^{1}/2)\oplus\mathbb{Z}^{d}$,
then the gauge symmetry is enhanced to $E_{8}$.

From the mass formula (2.38), we see that tachyonic states appear if
$N_{L}=N_{R}=0$ and the following relation is satisfied.

$\displaystyle p_{R}^{2}=P_{L}^{2}<1,~{}~{}~{}~{}\text{for
}P\in\mathbb{I}^{*}+V.$ (3.22)

Explicitly writing, we obtain the bound on the moduli below which the theory
is tachyonic:

$\displaystyle
qG^{-1}q^{t}=\left(\pi_{+}-2wa\right)^{2}+\left(q+2wG\right)G^{-1}\left(q+2wG\right)^{t}<2,$
(3.23)

for
$(\pi_{+};w,n)\in\Gamma_{E_{8}}\oplus(\mathbb{Z}^{d}+\hat{w}^{1}/2)\oplus\mathbb{Z}^{d}$
that minimizes the quantity $qG^{-1}q^{t}$. Note that the moduli must also
satisfy eq.(3.17) for such an element $(\pi_{+};w,n)$. Assuming again $a=0$
which implies that the bound (3.23) holds only for $\pi_{+}=0$, it turns out
that the bound (3.23) can be rewritten as

$\displaystyle wGw^{t}<2.$ (3.24)

Recalling $w^{1}\in\mathbb{Z}+1/2$ and $w^{m\neq 1}\in\mathbb{Z}$, there are
tachyonic states in the region where $G_{11}<1$ and $B_{1m}\in 2\mathbb{Z}$.
Note that the constraint on $B$ comes from the condition (3.17) with $w=(\pm
1/2,0^{d-1})$.

###  3.3 Relevance to CHL strings

Let $G_{\text{CHL}}$ and $G_{\mathcal{N}=0}$ denote the gauge groups realized
at a point $(E,a)$ in the CHL model and non-supersymmetric CHL model
respectively. We also denote a set of the nonzero roots of a non-Abelian group
$G$ as $\Delta_{G}$. According to the result in subsection 2.2,
$\Delta_{G_{\text{CHL}}}$ is expressed as

$\displaystyle\Delta_{G_{\text{CHL}}}=\left\\{P\in\Gamma^{8+d,d}\left|~{}\eqref{eq:
p_R=0},~{}\eqref{eq: roots untwisted
CHL}\right.\right\\}+\left\\{P\in\mathbb{I}^{*}+V\left|~{}\eqref{eq:
p_R=0},~{}\eqref{eq: roots twisted CHL}\right.\right\\},$ (3.25)

where the first and second sets in r.h.s. come from the untwisted and twisted
sectors respectively. As we have seen in the previous subsection, in the non-
supersymmetric CHL model, spacetime vectors arise only from the untwisted
sector, and hence the first set in r.h.s. of (3.25) corresponds to
$\Delta_{G_{\mathcal{N}=0}}$. Namely, we can express
$\Delta_{G_{\mathcal{N}=0}}$ as a subset of $\Delta_{G_{\text{CHL}}}$ as
follows:

$\displaystyle\Delta_{G_{\mathcal{N}=0}}=\left\\{P\in\Delta_{G_{\text{CHL}}}\left|~{}w^{1}\in\mathbb{Z}\right.\right\\},$
(3.26)

It also turns out that the residual

$\displaystyle\Delta_{G_{\mathcal{N}=0}}\backslash\Delta_{G_{\text{CHL}}}=\left\\{P\in\Delta_{G_{\text{CHL}}}\left|~{}w^{1}\in\mathbb{Z}+\frac{1}{2}\right.\right\\}$
(3.27)

gives the representation of $G_{\mathcal{N}=0}$ in which massless co-spinors
transform. Note that the number of long roots is maintained under the process
of supersymmetry breaking.

###  3.4 $d=1$

As one of the simplest examples in the case $d=1$, let us consider the
following moduli:

$\displaystyle a=0,~{}~{}~{}~{}E=G=R^{2}.$ (3.28)

The condition $p_{R}=0$ with no Wilson line yields

$\displaystyle n=wR^{2}\in\mathbb{Z}.$ (3.29)

We next apply the procedure given in subsection 3.2 to the case $d=1$ with the
moduli (3.28) according to the spacetime $SO(8)$ conjugacy classes.

#### Vector

In the vector conjugacy class, we use the condition (2.46). As mentioned in
section 2, there is no solution with a long root in the case $d=1$. For short
roots, $\left(w,\pi_{+}\right)\in\mathbb{Z}\oplus\Gamma_{E_{8}}$ must satisfy

$\displaystyle\frac{1}{2}\pi_{+}^{2}+2\left(wR\right)^{2}=1.$ (3.30)

It is obvious that this condition (3.30) holds for either $\pi_{+}^{2}=2$
($w=0$) or $\pi_{+}=0$ ($(wR)^{2}=1/2$). In the former case, the other
condition (3.29) is always satisfied and we find the $240$ solutions, which
correspond to the nonzero roots of $E_{8}$:

$\displaystyle\Delta_{E_{8}}=\left\\{\left(\underline{\pm,\pm,0,0,0,0,0,0}\right),\frac{1}{2}\left(\underline{\pm,\pm,\pm,\pm,\pm,\pm,\pm,\pm}_{+}\right)\right\\},$
(3.31)

where the underline indicates all permutations of the slots and the subscript
$+$ implies that permutations with the number of $+1$ even are kept. In the
latter case, whereas, eq.(3.29) can not be satisfied for any $w\in\mathbb{Z}$
since inserting $(wR)^{2}=1/2$ into (3.29) yields $n=1/(2w)$. As a result,
with the moduli (3.28), we obtain the $E_{8}\times U(1)_{l}$ enhanced gauge
symmetry. Note that in the CHL model, $w$ can take a half-integer in the
twisted sector, and the $U(1)_{l}$ is enhanced to $SU(2)$ at $R=\sqrt{2}$.

#### Spinor

The conditions on short weight elements take the same form as in the vector
conjugacy class, and thus we find the massless spinors in $\bm{248}$ of the
$E_{8}$. Unlike in the vector, due to $n$ odd, one can also find the solutions
with long weights at a special radius:

$\displaystyle\frac{1}{2}\pi_{+}^{2}+2\left(wR\right)^{2}=2,~{}~{}~{}~{}\left(\pi_{+}\in
2\Gamma_{E_{8}}\right).$ (3.32)

This holds for only when $\pi_{+}=0$ and $(wR)^{2}=1$. Inserting $(wR)^{2}=1$
into eq. $(wR)^{2}=1$, we find $1/w\in\mathbb{Z}$ which means that there are
$E_{8}$-singlet spinors with $(w,n;\pi_{+})=(\pm 1,\pm 1;0)$ at $R=1$.

#### Co-spinor

The conditions for massless states in this class are given by the same form as
eqs.(3.29) and (3.30) but with $w\in\mathbb{Z}+1/2$. The second term in r.h.s.
of the condition (3.30) can not be zero, and thus we obtain $\pi_{+}=0$ and
$(wR)^{2}=1/2$ from eq.(3.30). Since the other condition (3.29) with
$(wR)^{2}=1/2$ leads to

$\displaystyle n=\frac{1}{w}\left(wR\right)^{2}=\frac{1}{2w}\in\mathbb{Z},$
(3.33)

it turns out that there exist $E_{8}$-singlet co-spinors with
$(w,n;\pi_{+})=(\pm 1/2,\pm 1;0)$ at $R=\sqrt{2}$. Note that in the CHL model,
the corresponding massless states are in the vector conjugacy class and
$U(1)_{l}$ is enhanced to $SU(2)$ at $R=\sqrt{2}$.

#### Scalar

In this class, we use either the condition (3.18) or (3.20), in addition to
(3.16) and (3.17). Inserting the moduli (3.28) into the conditions (3.16) and
(3.17), we get

$\displaystyle q^{2}=2R^{2},~{}~{}~{}~{}q+wR^{2}\in\mathbb{Z}.$ (3.34)

The conditions (3.18) for neutral scalars are written as

$\displaystyle\pi_{+}=0,~{}~{}~{}~{}2\left(wR\right)^{2}=1,~{}~{}~{}~{}n=-wR^{2}=-\frac{1}{2w}\in\mathbb{Z},$
(3.35)

We thus find $E_{8}$-singlet scalars with $(w,n;\pi_{+})=\left(\pm 1/2,\mp
1;0\right)$ at the special radius $R=\sqrt{2}$. Recall that these massless
states are excited by the oscillators $\alpha_{-,-\frac{1}{2}}^{I}$, and the
degree of the degeneracy is sixteen. As for charged scalars, the condition
(3.20) gives

$\displaystyle\pi_{+}^{2}+\frac{1}{R^{2}}\left(q+2wR^{2}\right)^{2}=2.$ (3.36)

If $\pi_{+}^{2}=2$, then eq.(3.36) gives $q=-2wR^{2}$. Substituting this into
eqs.(3.34), we get

$\displaystyle w^{2}R^{2}=\frac{1}{2},~{}~{}~{}~{}\frac{1}{2w}\in\mathbb{Z}.$
(3.37)

At $R=\sqrt{2}$, hence, there are $E_{8}$-charged massless scalars with
$(w,n;\pi_{+})=(\pm\frac{1}{2},\mp 1;\pi_{+}\in\Delta_{E_{8}})$. Combining
with the $E_{8}$-neutral scalars found above at the same radius, we obtain the
scalars transforming in the adjoint representation of the $E_{8}$. On the
other hand, if $\pi_{+}=0$, then we find from eq.(3.36)

$\displaystyle\left(q+2wR^{2}\right)^{2}=q^{2}+4wR^{2}\left(q+wR^{2}\right)=2R^{2}~{}\overset{q^{2}=2R^{2}}{\Longrightarrow}~{}q=-wR^{2}.$
(3.38)

It is clear that the second condition in (3.34) is satisfied for any
$w\in\mathbb{Z}+1/2$, due to $q+wR^{2}=0$. The first condition in (3.34) with
$q=-wR^{2}$ yields $w^{2}R^{2}=2$. Thus, at
$\sqrt{2}R^{-1}=|w|\in\mathbb{Z}_{\geq 0}+1/2$, we obtain the $E_{8}$-singlet
scalars with $(w,n;\pi_{+})=(\pm\sqrt{2}R^{-1},0;0)$. Note that the maximum
radius at which such $U(1)_{l}$-charged scalars appear is $R=2\sqrt{2}$. This
is indeed the critical radius where the theory is tachyon-free, as we will see
below. In Table 1, we summarize the above result of the additional massless
states and the corresponding special radii.

Let us next figure out the region where tachyonic states appear. As we are now
assuming $a=0$, such region is given by (3.24) which is expressed as follows
in the case $d=1$:

$\displaystyle R<\frac{\sqrt{2}}{|w|}\leq 2\sqrt{2}.$ (3.39)

This means that the massless neutral scalars we found above at
$R=\sqrt{2}/|w|$ become tachyonic in the region $R<\sqrt{2}/|w|$.

| $R=1$ | $R=\sqrt{2}$ |  $R=\frac{\sqrt{2}}{|w|}$ ($w\in\mathbb{Z}+\frac{1}{2}$)
---|---|---|---
space-time $SO(8)$ reps. | $\bm{8}_{S}$ | $\bm{8}_{C}$ | scalar | scalar
$E_{8}$ reps. | singlet | singlet | adjoint | singlet
$U(1)_{l}$-charge $(w,n)$ | $\left(\pm 1,\pm 1\right)$ | $\left(\pm\frac{1}{2},\pm 1\right)$ | $\left(\pm\frac{1}{2},\mp 1\right)$ | $\left(\pm\sqrt{2}R^{-1},0\right)$
Table 1: The extra massless states appearing at the special radii in the case
$d=1$ without the Wilson line.

###  3.5 $d=2$

The case $d=2$ is more interesting since we can see the enhancements to non-
simply-laced groups. In this subsection, we will give two examples of the
enhancements. The derivation of all solutions of the conditions for the
massless spectrum is straightforward but quite tedious. So, we only give the
results which are summarized in tables in appendix C. In this section, we
focus on the points where the maximal enhancements in the eight-dimensional
CHL model occur (see ref. [51] for more detail).

####  3.5.1 Example: $C_{9}\times C_{1}$ gauge symmetry

As a first example, let us consider the following point in the moduli space:

$\displaystyle E=\left(\begin{array}[]{cc}2&0\\\
-2&1\end{array}\right),~{}~{}~{}a_{1}=\frac{\omega_{1}}{3},~{}~{}~{}a_{2}=0,$
(3.42)

where

$\displaystyle\omega_{1}=-\frac{1}{2}\left(-1,1,1,1,1,1,1,-5\right).$ (3.43)

In the CHL model, at this point, $U(1)_{L}^{8}\times U(1)_{l}^{2}$ is enhanced
to $C_{10}$ [51]. The condition (2.40) with the above moduli becomes

$\displaystyle n_{1}=-\pi_{+}\cdot
a_{1}+2\left(w^{1}-w^{2}\right)\in\mathbb{Z},~{}~{}~{}n_{2}=w^{2}\in\mathbb{Z}.$
(3.44)

The metric $G$ and asymmetric tensor $B$ can be extracted from $E$ and $a$:

$\displaystyle G=\frac{1}{2}\left(E+E^{t}\right)-a\cdot
a=\left(\begin{array}[]{cc}\frac{10}{9}&-1\\\
-1&1\end{array}\right),~{}~{}~{}B=-\frac{1}{2}\left(E-E^{t}\right)=\left(\begin{array}[]{cc}0&-1\\\
1&0\end{array}\right).$ (3.49)

Inserting the metric $G$ into (2.46), we get the condition on short root
elements

$\displaystyle\left(\pi_{+}-2w^{1}a^{1}\right)+\frac{4}{9}\left(w^{1}\right)^{2}+4\left(w^{1}-w^{2}\right)^{2}=2,$
(3.50)

and on long root elements

$\displaystyle\left(\pi^{\prime}_{+}-w^{1}a^{1}\right)+\frac{1}{9}\left(w^{1}\right)^{2}+\left(w^{1}-w^{2}\right)^{2}=1.$
(3.51)

The condition (3.16) in the scalar conjugacy class is written as

$\displaystyle
9\left(q_{1}+q_{2}\right)^{2}+q_{2}^{2}=\tilde{q}_{1}^{2}+q_{2}^{2}=2,$ (3.52)

where $q_{1}=\pi_{+}\cdot a_{1}-2w^{1}+2w^{2}+n_{1}$ and $q_{2}=-w^{2}+n_{2}$,
and we define $\tilde{q}_{1}=3q_{1}+3q_{2}$. Note that both $\tilde{q}_{1}$
and $q_{2}$ are integers due to $3a_{1}\in\Gamma_{E_{8}}$, and thus we find
the four solutions of eq.(3.52)

$\displaystyle\left(\tilde{q}_{1},q_{2}\right)=\pm\left(1,1\right),\pm\left(1,-1\right).$
(3.53)

We can express the condition (3.20) for charged scalars as

$\displaystyle\left(\Pi_{+}-2W^{1}a_{1}\right)^{2}+\frac{4}{9}\left(W^{1}\right)^{2}+4\left(W^{1}-W^{2}\right)^{2}=2.$
(3.54)

Here, we define

$\displaystyle\Pi_{+}:=\pi_{+}+3\tilde{q}_{1}a_{1},~{}~{}~{}W^{1}:=w^{1}+\frac{3}{2}\tilde{q}_{1},~{}~{}~{}W^{2}:=w^{2}+\frac{1}{2}\left(3\tilde{q}_{1}-q_{2}\right).$
(3.55)

Since $\tilde{q}_{1}$ and $q_{2}$ are restricted as in eq.(3.53) and
$w^{1}\in\mathbb{Z}+1/2$, we notice $\Pi_{+}\in\Gamma_{E_{8}}$,
$W^{1}\in\mathbb{Z}$ and $W^{2}\in\mathbb{Z}$. The condition (3.17) can be
also written as

$\displaystyle
n_{1}=N_{1}+3\tilde{q}_{1}\in\mathbb{Z},~{}~{}~{}~{}n_{2}=N_{2}-\frac{1}{2}\left(3\tilde{q}_{1}-q_{2}\right)\in\mathbb{Z},$
(3.56)

where we define

$\displaystyle N_{1}:=-\Pi_{+}\cdot
a_{1}+2\left(W^{1}-W^{2}\right),~{}~{}~{}N_{2}:=W^{2}.$ (3.57)

The condition (3.56) clearly requires that $N_{1}$ and $N_{2}$ be integers.
Therefore it turns out that the solutions $(W^{1},W^{2};N_{1},N_{2};\Pi_{+})$
of (3.55) and (3.56) are nothing but those of (3.50) and (3.44). Namely, the
elements $(w^{1},w^{2};n_{1},n_{2};\pi_{+})$ on which charged massless scalars
lie are given by the short root solutions in the vector (or spinor) conjugacy
class shifted by the vector

$\displaystyle\left(-\frac{3}{2}\tilde{q}_{1},-\frac{1}{2}(\tilde{q}_{1}+q_{2});3\tilde{q}_{1},-\frac{1}{2}(3\tilde{q}_{1}-q_{2});-3\tilde{q}_{1}a_{1}\right).$
(3.58)

Furthermore, one can find that the shift vector (3.58) is a solution of the
conditions for neutral scalars given by (3.20). Since there are four
independent choices of $(\tilde{q}_{1},q_{2})$ given by (3.53), we obtain the
four copies of massless scalars transforming in the representation which
consists of a set of elements satisfying the conditions on short root and
eight neutral elements.

The result of the solutions is summarized in table
LABEL:table:_C9_x_C1_weights. In the vector conjugacy class, the solutions
correspond to the nonzero roots of $C_{1}\times C_{9}$. There is no long
weight element in the spinor conjugacy class, and combining the short weight
solutions with eight neutral elements, we obtain the massless spinors
transforming in $\bm{152}\oplus\bm{1}$ of the $C_{9}$. We also find the co-
spinors in $\left(\bm{2},\bm{18}\right)$ of the $C_{1}\times C_{9}$ in the
twisted sector. It is natural for the co-spinors to transform in the bi-
fundamental representation because $G_{\text{CHL}}=C_{10}$,
$G_{\mathcal{N}=0}=C_{9}\times C_{1}$ and the residual $\Delta_{C_{9}\times
C_{1}}\backslash\Delta_{C_{10}}$ gives a set of weight vectors associated to
the co-spinors, as mentioned in subsection 3.3. As we have seen above, there
are the four copies of massless states transforming in $\bm{152}\oplus\bm{1}$
of the $C_{9}$ in the scalar conjugacy class.

####  3.5.2 Example: $A_{4}\times C_{2}\times C_{4}$ gauge symmetry

As a second example, let us consider the moduli

$\displaystyle E=\left(\begin{array}[]{cc}2&0\\\
0&1\end{array}\right),~{}~{}~{}a_{1}=\frac{\omega_{6}}{2},~{}~{}~{}a_{2}=\frac{\omega_{3}}{5},$
(3.61)

where

$\displaystyle\omega_{6}=\left(0,0,0,0,0,0,-1,1\right),~{}~{}~{}~{}\omega_{3}=\left(0,0,0,-1,-1,-1,-1,4\right).$
(3.62)

In the CHL model with this background, the gauge symmetry is enhanced to
$A_{4}\times C_{6}$ [51]. Inserting the above $E$ into eq.(2.40), we get

$\displaystyle n_{1}$ $\displaystyle=-\pi_{+}\cdot a_{1}+2w^{1}\in\mathbb{Z},$
(3.63a) $\displaystyle n_{2}$ $\displaystyle=-\pi_{+}\cdot
a_{2}+w^{2}\in\mathbb{Z}.$ (3.63b)

Obviously $B=0$, and the metric is

$\displaystyle G=E-a\cdot
a=\left(\begin{array}[]{cc}\frac{3}{2}&-\frac{1}{2}\\\
-\frac{1}{2}&\frac{1}{5}\end{array}\right).$ (3.66)

Using this metric $G$, the condition on short root elements is expressed as

$\displaystyle\left(\pi_{+}-2w^{i}a_{i}\right)^{2}+\left(w^{1}\right)^{2}+\frac{1}{5}\left(5w^{1}-2w^{2}\right)^{2}=2.$
(3.67)

As for long roots, using $\pi^{\prime}_{+}=\pi_{+}/2\in\Gamma_{E_{8}}$, we
obtain

$\displaystyle\left(\pi^{\prime}_{+}-w^{i}a_{i}\right)^{2}+\frac{1}{4}\left(w^{1}\right)^{2}+\frac{1}{20}\left(5w^{1}-2w^{2}\right)^{2}=1.$
(3.68)

The condition (3.16) in the scalar conjugacy class is

$\displaystyle\left(2q_{1}+5q_{2}\right)^{2}+5q_{2}^{2}=\left(2q_{1}+5q_{2}\right)^{2}+\frac{1}{5}\left(5q_{2}\right)^{2}=2,$
(3.69)

where $q_{1}=\pi_{+}\cdot a_{1}-2w^{1}+n_{1}$ and $q_{2}=\pi_{+}\cdot
a_{2}-w^{2}+n_{2}$. Since $2q_{1}$ and $5q_{2}$ are integers (note
$2a_{1}\in\Gamma_{E_{8}}$ and $5a_{2}\in\Gamma_{E_{8}}$), there is no a pair
$\left(2q_{1},5q_{2}\right)$ of integers satisfying eq.(3.69), i.e. no scalars
in the massless spectrum.

All the solutions with the moduli (3.61) are given in table
LABEL:table:_A4_x_C4_x_C2_weights. We identify the solutions in the vector
conjugacy class as the nonzero roots of $A_{4}\times C_{2}\times C_{4}$. As
expected by (3.27) with $G_{\text{CHL}}=A_{4}\times C_{6}$ and
$G_{\mathcal{N}=0}=A_{4}\times C_{2}\times C_{4}$, the massless co-spinors
transform in $\left(\bm{4},\bm{8}\right)$ of the $C_{2}\times C_{4}$. In this
example, unlike the previous one, there are solutions for long weights with
$n_{1}$ odd, and we obtain the spinors in
$\left(\bm{24},\bm{1},\bm{1}\right)\oplus\left(\bm{1},\bm{5}\oplus\bm{1},\bm{1}\right)\oplus\left(\bm{1},\bm{1},\bm{42}\oplus\bm{1}\oplus\bm{1}\right)$
of the $A_{4}\times C_{2}\times C_{4}$.

##  4 Cosmological constant

In this section, we evaluate the leading behavior of the cosmological constant
in the region where supersymmetry is asymptotically restored. We will see that
as in (1.1), the leading contribution is proportional to
$\left(n_{F}-n_{B}\right)$ under a constraint on the Wilson line $a_{1}$. The
derivation is almost the same as in other interpolating models (e.g., see
section IV in ref. [25] or appendix in ref. [34]).

The one-loop cosmological constant is expressed as the integral of the torus
partition function over the fundamental domain $\mathcal{F}$ of the modular
group:

$\displaystyle\Lambda^{(10-d)}=-\frac{1}{2}\left(4\pi^{2}\right)^{-\frac{10-d}{2}}\int_{\mathcal{F}}\frac{d^{2}\tau}{\tau_{2}^{2}}Z_{\mathcal{N}=0}^{(10-d)},$
(4.1)

In order to make the behavior of the supersymmetry breaking scale $m_{3/2}$
clear, we restrict our attention to the following background:

$\displaystyle
e^{1}_{m}=R\delta^{1}_{m},~{}~{}~{}e^{a}_{1}=R\delta^{m}_{1},~{}~{}~{}B_{m1}=0.$
(4.2)

These choices of the veirbeil $e^{a}_{m}$ implies that $T^{d}$ is decomposed
into $S^{1}\times{T^{d-1}}^{\prime}$, i.e. $G_{m1}=R^{2}\delta^{1}_{m}$. For
states with $w^{1}=0$, which are dominant in the region we are focusing on,
the internal momenta can be written as

$\displaystyle p_{L,1}$ $\displaystyle=\frac{1}{\sqrt{2}R}\left(\pi_{+}\cdot
a_{1}-w^{\prime}a^{\prime}\cdot a_{1}+n_{1}\right),$ (4.3a) $\displaystyle
p_{R,1}$ $\displaystyle=\frac{1}{\sqrt{2}R}\left(\pi_{+}\cdot
a_{1}-w^{\prime}a^{\prime}\cdot a_{1}+n_{1}\right),$ (4.3b)

and

$\displaystyle\left(P^{\prime}_{L};p^{\prime}_{R}\right)=\left(\pi_{+},w^{\prime},n^{\prime}\right)\mathcal{E}^{(d-1)}\left(e^{\prime},B^{\prime},a^{\prime}\right),$
(4.4)

where $e^{\prime}=e^{a^{\prime}}_{m^{\prime}}$,
$B^{\prime}=B_{m^{\prime}n^{\prime}}$, $a^{\prime}=a_{m^{\prime}}^{I}$ and
$\left(w^{\prime},n^{\prime}\right)=(w^{m^{\prime}},n_{m^{\prime}})$ with
$m^{\prime},n^{\prime},a^{\prime}=2,\cdots,d$. Note that states with $w^{1}=0$
only exist in the untwisted sector, and the massless conditions for those
states are given by (2.46) and (2.40) with $(w,n,E,a)$ replaced by
$(w^{\prime},n^{\prime},E^{\prime},a^{\prime})$, in addition to $p_{R,1}=0$:

$\displaystyle n_{1}=-\left(\pi_{+}-w^{\prime}a^{\prime}\right)\cdot
a_{1}\in\mathbb{Z}.$ (4.5)

For long roots, in particular, recall that whether the state is bosonic or
fermionic depends on whether $n_{1}$ is even or odd.

Let us compute $\Lambda^{(10-d)}$ in the background (4.2) with $R$ large. In
this regime, the contributions with $w^{1}\neq 0$ are suppressed at least by
the factor of $e^{-R^{2}}$. Inserting the partition function (3.1) into the
cosmological constant (4.1) and using the Jacobi abstruse identity
$V_{8}-S_{8}=0$, we find

$\displaystyle\Lambda^{(10-d)}$
$\displaystyle\sim-\frac{1}{2}\left(4\pi^{2}\right)^{-\frac{10-d}{2}}\int_{\mathcal{F}}\frac{d^{2}\tau}{\tau_{2}^{2}}Z_{B}^{(8-d)}\bar{V}_{8}\left.Z(1,g)\right|_{w^{1}=0}$
$\displaystyle\sim-\frac{1}{2}\left(4\pi^{2}\right)^{-\frac{10-d}{2}}\int_{\mathcal{F}}\frac{d^{2}\tau}{\tau_{2}^{\frac{12-d}{2}}}\bar{V}_{8}\frac{1}{\bar{\eta}^{8}\eta^{24}}\left(\frac{2\eta^{3}}{\vartheta_{2}}\right)^{4}\left.\sum_{\left(P_{L};p_{R}\right)\in\mathbb{I}}q^{\frac{1}{2}P_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}e^{\pi
in_{1}}\right|_{w^{1}=0}.$ (4.6)

The sum over the invariant lattice $\mathbb{I}$ is expressed in terms of the
theta functions:

$\displaystyle\left.\sum_{\left(P_{L};p_{R}\right)\in
I}q^{\frac{1}{2}P_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}e^{\pi
in_{1}}\right|_{w^{1}=0}$ $\displaystyle=\sum_{\begin{subarray}{c}\pi_{+}\in
2\Gamma_{E_{8}}\\\
w^{\prime},n^{\prime}\in\mathbb{Z}\end{subarray}}q^{\frac{1}{2}P^{\prime
2}_{L}}\bar{q}^{\frac{1}{2}p^{\prime
2}_{R}}\sum_{n_{1}\in\mathbb{Z}}\left\\{e^{\frac{4\pi\tau_{2}}{R^{2}}\left(n_{1}+\frac{\rho\cdot
a_{1}}{2}\right)^{2}}-e^{\frac{4\pi\tau_{2}}{R^{2}}\left(n_{1}+\frac{\rho\cdot
a_{1}+1}{2}\right)^{2}}\right\\}$
$\displaystyle=\sum_{\begin{subarray}{c}\pi_{+}\in 2\Gamma_{E_{8}}\\\
w^{\prime},n^{\prime}\in\mathbb{Z}\end{subarray}}q^{\frac{1}{2}P^{\prime
2}_{L}}\bar{q}^{\frac{1}{2}p^{\prime
2}_{R}}\left\\{\vartheta\begin{bmatrix}\frac{\rho\cdot a_{1}}{2}\\\ 0\\\
\end{bmatrix}\left(\frac{4i\tau_{2}}{R^{2}}\right)-\vartheta\begin{bmatrix}\frac{\rho\cdot
a_{1}+1}{2}\\\ 0\\\
\end{bmatrix}\left(\frac{4i\tau_{2}}{R^{2}}\right)\right\\},$ (4.7)

where we define $\rho:=\pi_{+}-w^{\prime}a^{\prime}$. By using the
$S$-transformation law (A.15) of the theta functions, eq.(4) can be rewritten
as

$\displaystyle\frac{2R}{\sqrt{\tau_{2}}}\sum_{\pi_{+}\in
2\Gamma_{E_{8}}}\sum_{w^{\prime},n^{\prime}\in\mathbb{Z}^{d-1}}q^{\frac{1}{2}P^{\prime
2}_{L}}\bar{q}^{\frac{1}{2}p^{\prime
2}_{R}}\left\\{\vartheta\begin{bmatrix}0\\\ \frac{\rho\cdot a_{1}}{2}\\\
\end{bmatrix}\left(\frac{iR^{2}}{4\tau_{2}}\right)-\vartheta\begin{bmatrix}0\\\
\frac{\rho\cdot a_{1}+1}{2}\\\
\end{bmatrix}\left(\frac{iR^{2}}{4\tau_{2}}\right)\right\\}$
$\displaystyle=\frac{2R}{\sqrt{\tau_{2}}}\sum_{\pi_{+}\in
2\Gamma_{E_{8}}}\sum_{w^{\prime},n^{\prime}\in\mathbb{Z}^{d-1}}q^{\frac{1}{2}P^{\prime
2}_{L}}\bar{q}^{\frac{1}{2}p^{\prime 2}_{R}}\sum_{n\geq
1}\cos\left[\pi(2n-1)\rho\cdot
a_{1}\right]e^{-\frac{\pi(2n-1)^{2}R^{2}}{4\tau_{2}}}.$ (4.8)

Substituting eq.(4) into eq.(4), we find

$\displaystyle\Lambda^{(10-d)}\sim-\frac{R}{\left(4\pi^{2}\right)^{\frac{10-d}{2}}}\sum_{N_{+},N_{-}}a_{N_{+},N_{-}}$
$\displaystyle\sum_{\pi_{+}\in
2\Gamma_{E_{8}}}\sum_{w^{\prime},n^{\prime}\in\mathbb{Z}^{d-1}}\sum_{n\geq
1}\cos\left[\pi(2n-1)\rho\cdot a_{1}\right]$
$\displaystyle\times\int_{\mathcal{F}}\frac{d^{2}\tau}{\tau_{2}^{\frac{13-d}{2}}}e^{2\pi
i\tau_{1}M^{\prime 2}_{-}}e^{-\pi\left(2\tau_{2}M^{\prime
2}_{+}+\frac{(2n-1)^{2}R^{2}}{4\tau_{2}}\right)},$ (4.9)

where we expand the contribution from the oscillators as

$\displaystyle\bar{V}_{8}\frac{1}{\bar{\eta}^{8}\eta^{24}}\left(\frac{2\eta^{3}}{\vartheta_{2}}\right)^{4}=\sum_{N,\tilde{N}}\hat{a}_{N,\tilde{N}}q^{N}\bar{q}^{\tilde{N}}=\sum_{N_{+},N_{-}}a_{N_{+},N_{-}}e^{2\pi
i\tau_{1}N_{-}}e^{-2\pi\tau_{2}N_{+}},$ (4.10)

and define as

$\displaystyle M^{\prime 2}_{+}=N_{+}+\frac{1}{2}\left(P^{\prime
2}_{L}+p^{\prime 2}_{R}\right),~{}~{}~{}~{}M^{\prime
2}_{-}=N_{-}+\frac{1}{2}\left(P^{\prime 2}_{L}-p^{\prime
2}_{R}\right)=N_{-}+\frac{1}{4}\pi_{+}^{2}+w^{\prime}n^{\prime}.$ (4.11)

In order to evaluate the integration, let us decompose the fundamental domain
$\mathcal{F}$ into $\mathcal{F}_{\geq
1}=\left.\mathcal{F}\right|_{\tau_{2}\geq 1}$ and
$\mathcal{F}_{<1}=\left.\mathcal{F}\right|_{\tau_{2}<1}$. For the integration
over $\mathcal{F}_{<1}$, the domain itself is finite and the integrand itself
is non-singular. Then, we can bound the integration as

$\displaystyle\int_{\mathcal{F}_{<1}}\frac{d^{2}\tau}{\tau_{2}^{\frac{13-d}{2}}}e^{2\pi
i\tau_{1}M^{\prime 2}_{-}}e^{-\pi\left(2\tau_{2}M^{\prime
2}_{+}+\frac{(2n-1)^{2}R^{2}}{4\tau_{2}}\right)}<e^{-\frac{\pi(2n-1)^{2}R^{2}}{4}}\int_{\mathcal{F}_{<1}}\frac{d^{2}\tau}{\tau_{2}^{\frac{13-d}{2}}}e^{2\pi
i\tau_{1}M^{\prime 2}_{-}}e^{-2\pi\tau_{2}M^{\prime 2}_{+}}.$ (4.12)

So, this contribution is suppressed at least by the factor $e^{-\pi R^{2}/4}$.
As for $\mathcal{F}_{\geq 1}$, the integration over $\tau_{1}$ gives the
level-matching(-like) condition $M^{\prime 2}_{-}=0$. By the arithmetic-
geometric mean, one can find the bound of the $\tau_{2}$-dependent factor:

$\displaystyle e^{-\pi\left(2\tau_{2}M^{\prime
2}_{+}+\frac{(2n-1)^{2}R^{2}}{4\tau_{2}}\right)}\leq
e^{-\sqrt{2}\pi(2n-1)M^{\prime}_{+}R}.$ (4.13)

The contribution from $\mathcal{F}_{\geq 1}$ is suppressed at least by the
factor $e^{-\sqrt{2}\pi M^{\prime}_{+}R}$ unless $M^{\prime 2}_{+}=0$. Up to
the exponentially suppressed terms, thus, the leading contribution is

$\displaystyle\Lambda^{(10-d)}$
$\displaystyle\sim-\frac{R}{\left(4\pi^{2}\right)^{\frac{10-d}{2}}}\sum_{n\geq
1}\int_{1}^{\infty}\frac{d\tau_{2}}{\tau_{2}^{\frac{13-d}{2}}}e^{-\frac{\pi(2n-1)^{2}}{4\tau_{2}}R^{2}}\sum_{\left\\{N_{\pm},P^{\prime}_{L},p^{\prime}_{R}\right\\}\in\mathbb{L}_{0}}a_{N_{+},N_{-}}\cos\left[\pi(2n-1)\rho\cdot
a_{1}\right]$
$\displaystyle\sim-\frac{2\Gamma\left(\frac{11-d}{2}\right)}{\sqrt{\pi}\left(\pi^{\frac{3}{2}}R\right)^{10-d}}\sum_{n\geq
1}(2n-1)^{d-11}\sum_{\left\\{N_{\pm},P^{\prime}_{L},p^{\prime}_{R}\right\\}\in\mathbb{L}_{0}}a_{N_{+},N_{-}}\cos\left[\pi(2n-1)\rho\cdot
a_{1}\right],$ (4.14)

where we define

$\displaystyle\mathbb{L}_{0}=\left\\{N_{\pm},P^{\prime}_{L},p^{\prime}_{R}\left|~{}M^{\prime
2}_{+}=M^{\prime 2}_{-}=0\right.\right\\}.$ (4.15)

Note that the constraint in $\mathbb{L}_{0}$ can be rewritten as

$\displaystyle N+\frac{1}{2}P^{\prime 2}_{L}=\tilde{N}+\frac{1}{2}p^{\prime
2}_{R}=0,$ (4.16)

and $N$, $\tilde{N}$ and $\tilde{a}_{N,\tilde{N}}$ can be identified by
expanding eqs.(4.10) as

$\displaystyle\frac{1}{\eta^{24}}\left(2\frac{\eta^{3}}{\vartheta_{2}}\right)^{4}$
$\displaystyle=q^{-1}+8+\mathcal{O}(q),~{}~{}~{}~{}\frac{\bar{V}_{8}}{\bar{\eta}^{8}}=8+\mathcal{O}(\bar{q}).$
(4.17)

In order to satisfy (4.16), $\tilde{N}=p^{\prime}_{R}=0$ and either
$N=P^{\prime}_{L}=0$ or $N=-1$, $P^{\prime 2}_{L}=2$, and we find

$\displaystyle\Lambda^{(10-d)}$
$\displaystyle\sim-\frac{2\Gamma\left(\frac{11-d}{2}\right)}{\sqrt{\pi}\left(\pi^{\frac{3}{2}}R\right)^{10-d}}\sum_{n\geq
1}(2n-1)^{d-11}\left(64+8\sum_{\rho\in\mathbb{L}^{\prime}_{0}}\cos\left[\pi(2n-1)\rho\cdot
a_{1}\right]\right),$ (4.18)

where the first term $64$ in the parentheses comes from $\hat{a}_{0,0}$, and
we define

$\displaystyle\mathbb{L}^{\prime}_{0}=\left\\{\rho=\pi_{+}-w^{\prime}a^{\prime}\left|~{}P^{\prime
2}_{L}=2,~{}p^{\prime}_{R}=0\right.\right\\}.$ (4.19)

Let us henceforth assume that $a_{1}$ is chosen such that $\rho\cdot a_{1}$ is
an integer for all solutions of $p^{\prime}_{R}=0,~{}P^{\prime 2}_{L}=2$ for
given $E^{\prime}$ and $a^{\prime}$. Since we focus on non-winding states
along $x^{1}$, it turns out from (4.5) that the integer value $\rho\cdot
a_{1}$ is nothing but $n_{1}$. The cosine factor in r.h.s. of (4.18) gives
$+1$ ($-1$) with $n_{1}$ even (odd), and the second term in the parentheses is
indeed the net number of massless bosonic and fermionic states with long
roots. As mentioned in section 3, the net number of massless bosons and
fermions with short roots is always zero, and the factor $64$ in the
parentheses in (4.18) agrees with the degrees of freedom in the gravitational
sector. Therefore, the cosmological constant (4.18) can be interpreted as

$\displaystyle\Lambda^{(10-d)}\sim\frac{\Gamma\left(\frac{11-d}{2}\right)}{\sqrt{\pi}\left(\pi^{\frac{3}{2}}R\right)^{10-d}}2^{d-10}\zeta\left(11-d,\frac{1}{2}\right)\left(n_{F}-n_{B}\right),$
(4.20)

where $n_{F}$ and $n_{B}$ are the degrees of freedom of massless fermionic
states and bosonic states respectively. Note again that we restrict our
attention to the region where $R$ is sufficiently larger than the string
length scale.

One can easily see that the cosmological constant can not be suppressed with
$d=1$. In the case $d=1$, there is no solutions of $P^{\prime 2}_{L}=2$ since
$P^{\prime}_{L}=\frac{1}{\sqrt{2}}\pi_{+}=\frac{1}{\sqrt{2}}\rho\in\sqrt{2}\Gamma_{E_{8}}$.
Then, the leading term is independent of the Wilson line $a_{1}$ and always
negative:

$\displaystyle\Lambda^{(9)}\sim-\frac{24}{\pi^{14}R^{9}}2^{-9}\zeta\left(10,\frac{1}{2}\right)64,$
(4.21)

where the Hurwitz zeta function
$\zeta(s,a):=\sum_{n=0}^{\infty}\left(n+a\right)^{-s}$.

In the following subsections, we will give explicit examples of the points at
which the cosmological constant is exponentially suppressed in the cases $d=2$
and $d=3$.

###  4.1 Suppression of $\Lambda^{(8)}$

In the case $d=2$, the background is $S^{1}\times{S^{1}}^{\prime}$ where the
Scherk-Schwarz mechanism is implemented on $S^{1}$ and the radius $R$ of
$S^{1}$ is supposed to be large. Let us consider the following special radius
and Wilson line for ${S^{1}}^{\prime}$:

$\displaystyle
R^{\prime}=\frac{1}{2},~{}~{}~{}~{}a^{\prime}=\left(0,0,0,0,-\frac{1}{4},-\frac{1}{4},-\frac{1}{4},\frac{3}{4}\right),~{}~{}~{}~{}E^{\prime}=R^{\prime
2}+a^{\prime}\cdot a^{\prime}=1.$ (4.22)

Inserting the moduli (4.22) into $P^{\prime 2}_{L}=2$ and $p^{\prime}_{R}=0$,
we get

$\displaystyle\left(\pi^{\prime}_{+}-w^{\prime}a^{\prime}\right)^{2}+\frac{1}{4}w^{\prime
2}=1,~{}~{}~{}n^{\prime}=-2\pi^{\prime}_{+}\cdot
a^{\prime}+w^{\prime}\in\mathbb{Z},$ (4.23)

where $\pi^{\prime}_{+}=\pi_{+}/2\in\Gamma_{E_{8}}$. There are eight solutions
to the above equations:

$\displaystyle\left(w^{\prime};n^{\prime};\pi_{+}\right)=\pm\left(1;1;0,0,0,0,0,0,0,0\right),\pm\left(1;-1;0,0,0,0,\underline{-2,0,0},2\right).$
(4.24)

In order to realize $n_{F}-n_{B}=0$, $a_{1}$ must be chosen such that
$\rho\cdot a_{1}=(\pi_{+}-w^{\prime}a^{\prime})\cdot a_{1}$ is odd for all of
the solutions (4.24). Assuming that $a_{1}$ is restricted to be an element in
$\Gamma_{E_{8}}$ (i.e. $\rho\cdot a_{1}=-w^{\prime}a^{\prime}\cdot a_{1}$ mod
$2$), one can easily find $a_{1}$ satisfying the requirement, e.g,
$a_{1}=(0,0,0,0,0,0,-1,1)$. With those moduli, by following the procedure in
subsection 3.2, one can find that the enhanced non-Abelian gauge symmetry is
$D_{4}\times D_{5}$ and massless chiral fermions transform in
$(\bm{1},\bm{45})\oplus(\bm{35}\oplus\bm{1},\bm{1})$ of the $D_{4}\times
D_{5}$.

###  4.2 Suppression of $\Lambda^{(7)}$

Let us turn to the case $d=3$, i.e., the background
$S^{1}\times{T^{2}}^{\prime}$. We suppose the moduli for ${T^{2}}^{\prime}$ to
be chosen as

$\displaystyle E^{\prime}=\left(\begin{array}[]{cc}2&0\\\
0&1\end{array}\right),~{}~{}~{}a_{2}=0,~{}~{}~{}a_{3}=\frac{1}{8}\left(1,1,1,1,1,1,1,-7\right).$
(4.27)

In the CHL model on $T^{2}$ with the above moduli, the gauge group is enhanced
to $A_{1}\times A_{1}\times C_{8}$ [51]. The conditions $P^{\prime 2}_{L}=2$
and $p^{\prime}_{R}=0$ with the moduli (4.27) give

$\displaystyle\left(\pi^{\prime}_{+}-w^{3}a_{3}\right)^{2}+2\left(w^{2}\right)^{2}+\frac{1}{8}\left(w^{3}\right)^{2}=1,$
(4.28)

and

$\displaystyle
n_{2}=2w^{2}\in\mathbb{Z},~{}~{}~{}n_{3}=-2\pi^{\prime}_{+}\cdot
a_{3}+w^{3}\in\mathbb{Z}.$ (4.29)

Solving the above equations, we get the sixteen solutions
$(w^{\prime};\pi_{+})$101010We omit $n^{\prime}$ which does not matter to
check whether the suppression is realized or not.:

$\displaystyle\pm\left(0,1;0,0,0,0,0,0,0\right),~{}\pm\left(0,1;\underline{2,0,0,0,0,0,0},-2\right).$
(4.30)

One can check that $n_{F}-n_{B}=0$ is achieved if $a_{1}$ is chosen as111111Of
course, this $a_{1}$ is not the unique choice to realize $n_{F}-n_{B}=0$. For
instance, if one permutates the first seven components of (4.30),
$n_{F}-n_{B}$ remains vanishing.

$\displaystyle a_{1}=\frac{1}{2}\left(0,0,0,0,0,0,7,1\right).$ (4.31)

Note that $a_{3}\cdot a_{1}=0$ and then $\rho\cdot a_{1}=\pi_{+}\cdot a_{1}$.
So, we can divide the sixteen solutions into four and twelve depending on
$n_{1}$ being even or odd:

$\displaystyle n_{1}\text{
even}:~{}\pm\left(0,1;0,0,0,0,0,0,0\right),~{}\pm\left(0,1;0,0,0,0,0,0,2,-2\right),$
(4.32a) $\displaystyle n_{1}\text{
odd}:~{}\pm\left(0,1;\underline{2,0,0,0,0,0},0,-2\right).$ (4.32b)

Thus, the second term in the parentheses in eq.(4.18) is $-64$ and the leading
contribution of $\Lambda^{(7)}$ is canceled out. By following the procedure in
subsection 3.2, we find that at the point (4.27), (4.31) and with large $R$,
the enhanced non-Abelian gauge group is $A_{1}\times C_{2}\times D_{6}$ and
there exist chiral fermions transforming in
$(\bm{3},\bm{1},\bm{1})\oplus(\bm{1},\bm{5}\oplus\bm{1},\bm{1})\oplus(\bm{1},\bm{1},\bm{77}\oplus\bm{1})$
of the $A_{1}\times C_{2}\times D_{6}$.

##  5 Summary and discussion

In this paper, we have studied the reduced rank non-supersymmetric heterotic
string model constructed by the asymmetric orbifold with the
$\mathbb{Z}_{2}$-twist $\bm{\mathrm{R}}\bm{\mathrm{T}}(-1)^{F}$. We first
reviewed the CHL string in which all sixteen supercharges are preserved and
the rank of gauge groups is reduced to $8+d$. We then considered a natural
extension of the CHL model to a non-supersymmetric one by combining the
orbifold twist with a $2\pi$ spatial rotation. The conditions on the internal
momenta of massless states have been figured out. We gave concrete examples of
the symmetry enhancements and the matter spectra in the cases $d=1,2$. In
$d=2$, in particular, the enhanced groups are non-simply-laced and the
fermions (and scalars) that non-trivially transform under the gauge group are
found. We also evaluated the cosmological constant in the regime with $R$
large and argued that its exponential suppression is possible if $d\geq 2$. It
turned out that in order to realize the suppression, the existence of massless
fermions with long weight elements of gauge groups is crucial.

Although the conditions that internal momenta of massless states should
satisfy are given in this paper, it is not shown how to solve them in an
efficient way. In refs. [57, 51], the algorithms to systematically explore the
maximal enhancement of gauge groups have been proposed in heterotic strings
with sixteen supercharges. By using the relations (3.26) and (3.27), it is
expected that the methods in refs. [57, 51] can be applied to the non-
supersymmetric case, and patterns of the maximal enhancements and the
corresponding representations of co-spinors can be figured out. However, it is
unclear whether it can be also done in the spinor and scalar conjugacy classes
because the orbifold projection in these classes is opposite to the CHL
strings and the momentum lattices do not have simple relevance, such as (3.26)
and (3.27).

###  5.1 Moduli stabilization

In this paper, we did not take the moduli stabilization into consideration. In
general, some moduli in non-supersymmetric string theory are frozen because
they acquire potentials at the quantum level. For instance, we have already
obtained the one-loop effective potential of the Wilson line moduli by
eq.(4.18), and we can perform the stability analysis of the Wilson lines. Let
us again assume that the Wilson line $a_{1}$ satisfy $\rho\cdot
a_{1}=n_{1}\in\mathbb{Z}$ for all solutions of $p^{\prime}_{R}=0$ and
$P^{\prime 2}_{R}=2$. Note that such points are critical points for the
$a_{1}$-directions:

$\displaystyle\frac{\partial\Lambda^{(10-d)}}{\partial
a_{1}^{I}}\sim\sum_{\rho}\rho^{I}\sin\left[\pi(2n-1)\rho\cdot a_{1}\right]=0.$
(5.1)

Let $\Delta_{+}$ and $\Delta_{-}$ be sets of $\rho$ satisfying
$p^{\prime}_{R}=0$ and $P^{\prime 2}_{R}=2$ with $\rho\cdot a_{1}$ even and
odd respectively:

$\displaystyle\Delta_{+}=\left\\{\rho\left|~{}p^{\prime}_{R}=0,~{}P^{\prime
2}_{R}=2,~{}\rho\cdot a_{1}\in
2\mathbb{Z}\right.\right\\},~{}~{}~{}~{}\Delta_{-}=\left\\{\rho\left|~{}p^{\prime}_{R}=0,~{}P^{\prime
2}_{R}=2,~{}\rho\cdot a_{1}\in 2\mathbb{Z}+1\right.\right\\}.$ (5.2)

In order for the cosmological constant (4.18) to vanish, the difference
between the numbers of elements in $\Delta_{+}$ and $\Delta_{-}$ must be $-8$:

$\displaystyle|\Delta_{+}|-|\Delta_{-}|=-8.$ (5.3)

The Hessian $H_{IJ}$ for $a_{1}^{I}$ is

$\displaystyle H_{IJ}:=\frac{\partial^{2}\Lambda^{(10-d)}}{\partial
a_{1}^{I}\partial
a_{1}^{J}}\sim\sum_{\rho}\rho^{I}\rho^{J}\cos\left[\pi(2n-1)\rho\cdot
a_{1}\right]=\sum_{\rho}\rho^{I}\rho^{J}(-1)^{\rho\cdot a_{1}}.$ (5.4)

The necessary condition for the critical point to be a minimum requires that
all diagonal components of the Hessian be positive, i.e.,

$\displaystyle\sum_{\rho\in\Delta_{+}}\left(\rho^{I}\right)^{2}-\sum_{\rho\in\Delta_{-}}\left(\rho^{I}\right)^{2}>0~{}~{}~{}\text{for
all $I=1,\cdots,8$}.$ (5.5)

The question is whether a point $(E^{\prime},a^{\prime},a_{1})$ satisfying
both eq.(5.3) and eq.(5.5) exists or not. Actually, one can check that the
examples given in the previous section that satisfy eq.(5.3) do not satisfy
eq.(5.5). For $d=2$, since $|\Delta_{+}|=0$ and $|\Delta_{-}|=8$, this point
corresponds to a maximum of the potential. For $d=3$, by eqs. (4.32), we find

$\displaystyle H_{11}>0,~{}~{}~{}H_{II}<0~{}~{}\text{for $I\neq 1$},$ (5.6)

and the $a_{1}^{I\neq 1}$-directions are tachyonic. Thus, the point that we
found in subsection 4.2 realizes the exponential suppression of the
cosmological constant but corresponds to an unstable saddle point.

Also, in this paper, we just assume that the radion $R$, which has the runaway
behavior, is sufficiently larger than the string length scale. It is an open
question whether (meta)stable de Sitter vacua exist or not. In order to
realize that, some extra ingredients (e.g. fluxes) are expected to be needed
such as in refs. [58, 59, 60].

## Acknowledgments

The author would like to thank Yuta Hamada and Shun’ya Mizoguchi for valuable
discussions, and Hiroshi Itoyama and Yuichi Koga for collaborations in related
subjects. This work is supported in part by JSPS KAKENHI Grant Number
21J15497.

##  Appendix A Lattices and characters

Irreducible representations of $SO(2n)$ ($D_{n}$) can be classified into the
four conjugacy classes:

  * •

The trivial conjugacy class (the root lattice):

$\displaystyle\Gamma^{(n)}_{g}=\left\\{\left(n_{1},\cdots,n_{n}\right)\left|~{}n_{i}\in\mathbb{Z},~{}\sum_{i=1}^{n}n_{i}\in
2\mathbb{Z}\right.\right\\}.$ (A.1)

  * •

The vector conjugacy class:

$\displaystyle\Gamma^{(n)}_{v}=\left\\{\left(n_{1},\cdots,n_{n}\right)\left|~{}n_{i}\in\mathbb{Z},~{}\sum_{i=1}^{n}n_{i}\in
2\mathbb{Z}+1\right.\right\\}.$ (A.2)

  * •

The spinor conjugacy class:

$\displaystyle\Gamma^{(n)}_{s}=\left\\{\left(n_{1}+\frac{1}{2},\cdots,n_{n}+\frac{1}{2}\right)\left|~{}n_{i}\in\mathbb{Z},~{}\sum_{i=1}^{n}n_{i}\in
2\mathbb{Z}\right.\right\\}.$ (A.3)

  * •

The conjugate spinor conjugacy class:

$\displaystyle\Gamma^{(n)}_{c}=\left\\{\left(n_{1}+\frac{1}{2},\cdots,n_{n}+\frac{1}{2}\right)\left|~{}n_{i}\in\mathbb{Z},~{}\sum_{i=1}^{n}n_{i}\in
2\mathbb{Z}+1\right.\right\\}.$ (A.4)

The $E_{8}$ lattice is expressed in terms of the conjugacy classes of
$SO(16)$:

$\displaystyle\Gamma_{E_{8}}=\Gamma_{g}^{(8)}+\Gamma_{s}^{(8)}.$ (A.5)

The $SO(2n)$ characters of the corresponding conjugacy classes are defined as

$\displaystyle O_{2n}$
$\displaystyle=\frac{1}{\eta^{n}}\sum_{\pi\in\Gamma^{(n)}_{g}}q^{\frac{1}{2}|\pi|^{2}}=\frac{1}{2\eta^{n}}\left(\vartheta_{3}^{n}(\tau)+\vartheta_{4}^{n}(\tau)\right),$
(A.6) $\displaystyle V_{2n}$
$\displaystyle=\frac{1}{\eta^{n}}\sum_{\pi\in\Gamma^{(n)}_{v}}q^{\frac{1}{2}|\pi|^{2}}=\frac{1}{2\eta^{n}}\left(\vartheta_{3}^{n}(\tau)-\vartheta_{4}^{n}(\tau)\right),$
(A.7) $\displaystyle S_{2n}$
$\displaystyle=\frac{1}{\eta^{n}}\sum_{\pi\in\Gamma^{(n)}_{s}}q^{\frac{1}{2}|\pi|^{2}}=\frac{1}{2\eta^{n}}\left(\vartheta_{2}^{n}(\tau)+\vartheta_{1}^{n}(\tau)\right),$
(A.8) $\displaystyle C_{2n}$
$\displaystyle=\frac{1}{\eta^{n}}\sum_{\pi\in\Gamma^{(n)}_{c}}q^{\frac{1}{2}|\pi|^{2}}=\frac{1}{2\eta^{n}}\left(\vartheta_{2}^{n}(\tau)-\vartheta_{1}^{n}(\tau)\right),$
(A.9)

where the Dedekind eta function and the theta functions with characteristics
are defined as

$\displaystyle\eta(\tau)$
$\displaystyle=q^{\frac{1}{24}}\prod_{n=1}^{\infty}\left(1-q^{n}\right),$
(A.10) $\displaystyle\vartheta\begin{bmatrix}\alpha\\\ \beta\\\
\end{bmatrix}(\tau)$ $\displaystyle=\sum_{n=-\infty}^{\infty}\exp\left(\pi
i(n+\alpha)^{2}\tau+2\pi i(n+\alpha)\beta\right),$ (A.11)

and

$\displaystyle\vartheta_{3}=\vartheta\begin{bmatrix}0\\\ 0\\\
\end{bmatrix},~{}~{}~{}\vartheta_{4}=\vartheta\begin{bmatrix}0\\\
\frac{1}{2}\\\
\end{bmatrix},~{}~{}~{}\vartheta_{2}=\vartheta\begin{bmatrix}\frac{1}{2}\\\
0\\\
\end{bmatrix},~{}~{}~{}\vartheta_{1}=\vartheta\begin{bmatrix}\frac{1}{2}\\\
\frac{1}{2}\\\ \end{bmatrix}.$ (A.12)

The $SO(8)$ characters satisfy the Jacobi’s abstruse identity

$\displaystyle V_{8}-S_{8}=0.$ (A.13)

In order to check the modular invariance of the partition functions, it is
useful to reveal how the $SO(2n)$ characters transform under $T:\tau\to\tau+1$
and $S:\tau\to-1/\tau$. The eta function and the theta function satisfy the
following identities:

$\displaystyle\eta(\tau+1)$ $\displaystyle=e^{\frac{\pi
i}{12}}\eta(\tau),~{}~{}~{}~{}\vartheta\begin{bmatrix}\alpha\\\ \beta\\\
\end{bmatrix}(\tau+1)=e^{-\pi
i\alpha(\alpha-1)}\vartheta\begin{bmatrix}\alpha\\\
\alpha+\beta-\frac{1}{2}\\\ \end{bmatrix}(\tau),$ (A.14)
$\displaystyle\eta\left(-\frac{1}{\tau}\right)$
$\displaystyle=(-i\tau)^{\frac{1}{2}}\eta(\tau),~{}~{}~{}~{}\vartheta\begin{bmatrix}\alpha\\\
\beta\\\
\end{bmatrix}\left(-\frac{1}{\tau}\right)=(-i\tau)^{\frac{1}{2}}e^{2\pi
i\alpha\beta}\vartheta\begin{bmatrix}\beta\\\ -\alpha\\\ \end{bmatrix}(\tau).$
(A.15)

Then we find the transformation laws of the $SO(2n)$ characters:

$\displaystyle
T:\left(O_{2n},V_{2n},S_{2n},C_{2n}\right)\to\left(O_{2n},V_{2n},S_{2n},C_{2n}\right)\mathcal{T}_{2n},$
(A.16) $\displaystyle
S:\left(O_{2n},V_{2n},S_{2n},C_{2n}\right)\to\left(O_{2n},V_{2n},S_{2n},C_{2n}\right)\mathcal{S}_{2n},$
(A.17)

where $\mathcal{T}_{2n}$ and $\mathcal{S}_{2n}$ are $4\times 4$ matrices
defined as follows:

$\displaystyle\mathcal{T}_{2n}=\begin{pmatrix}e^{-\frac{n\pi i}{12}}&0&0&0\\\
0&-e^{-\frac{n\pi i}{12}}&0&0\\\ 0&0&e^{\frac{n\pi i}{6}}&0\\\
0&0&0&e^{\frac{n\pi i}{6}}\\\
\end{pmatrix},~{}~{}~{}\mathcal{S}_{2n}=\frac{1}{2}\begin{pmatrix}1&1&1&1\\\
1&1&-1&-1\\\ 1&-1&i^{n}&-i^{n}\\\ 1&-1&-i^{n}&i^{n}\\\ \end{pmatrix}.$ (A.18)

##  Appendix B Non-supersymmetric $E_{8}$ theory

In this appendix, we briefly review the non-supersymmetric $E_{8}$ theory
constructed by the orbifold with $\bm{\mathrm{R}}(-1)^{F}$. The starting point
is the $10$D $E_{8}\times E^{\prime}_{8}$ theory of which the partition
function is given by

$\displaystyle Z_{E_{8}\times
E^{\prime}_{8}}^{(10)}=Z_{B}^{(8)}\left(\bar{V}_{8}-\bar{S}_{8}\right)\sum_{\pi\in\Gamma_{E_{8}}}\sum_{\pi^{\prime}\in\Gamma_{E^{\prime}_{8}}}q^{\frac{1}{2}\left(\pi^{2}+\pi^{\prime
2}\right)}.$ (B.1)

As in the CHL model, the partition function of the $E_{8}$ model is expressed
as the sum of the building blocks:

$\displaystyle
Z_{\mathcal{N}=0}^{(10)}=\frac{1}{2}Z_{B}^{(8)}\sum_{a,b=0,1}Z^{(10)}(g^{a},g^{b}),$
(B.2)

where

$\displaystyle Z^{(10)}(1,1)$
$\displaystyle=\left(\bar{V}_{8}-\bar{S}_{8}\right)\frac{1}{\eta^{16}}\sum_{\pi\in\Gamma_{E_{8}}}\sum_{\pi^{\prime}\in\Gamma_{E^{\prime}_{8}}}q^{\frac{1}{2}\left(\pi^{2}+\pi^{\prime
2}\right)},$ (B.3) $\displaystyle Z^{(10)}(1,g)$
$\displaystyle=\left(\bar{V}_{8}+\bar{S}_{8}\right)\frac{1}{\eta^{16}}\left(\frac{2\eta^{3}}{\vartheta_{2}}\right)^{4}\sum_{\pi\in\sqrt{2}\Gamma_{E_{8}}}q^{\frac{1}{2}\pi^{2}},$
(B.4) $\displaystyle Z^{(10)}(g,1)$
$\displaystyle=\left(\bar{O}_{8}-\bar{C}_{8}\right)\frac{1}{\eta^{16}}\left(\frac{\eta^{3}}{\vartheta_{4}}\right)^{4}\sum_{\pi\in\frac{1}{\sqrt{2}}\Gamma_{E_{8}}}q^{\frac{1}{2}\pi^{2}},$
(B.5) $\displaystyle Z^{(10)}(g,g)$
$\displaystyle=\left(\bar{O}_{8}+\bar{C}_{8}\right)\frac{1}{\eta^{16}}\left(\frac{\eta^{3}}{\vartheta_{3}}\right)^{4}\sum_{\pi\in\frac{1}{\sqrt{2}}\Gamma_{E_{8}}}q^{\frac{1}{2}\pi^{2}}e^{\pi
i\pi^{2}}.$ (B.6)

The total partition function is then written as

$\displaystyle
Z_{\mathcal{N}=0}^{(10)}=\frac{1}{2}Z_{B}^{(8)}\frac{1}{\eta^{16}}$
$\displaystyle\left\\{\bar{V}_{8}\left(\sum_{\pi\in\Gamma_{E_{8}}}\sum_{\pi^{\prime}\in\Gamma_{E^{\prime}_{8}}}q^{\frac{1}{2}\left(\pi^{2}+\pi^{\prime
2}\right)}+\left(\frac{2\eta^{3}}{\vartheta_{2}}\right)^{4}\sum_{\pi\in\sqrt{2}\Gamma_{E_{8}}}q^{\frac{1}{2}\pi^{2}}\right)\right.$
$\displaystyle\left.-\bar{S}_{8}\left(\sum_{\pi\in\Gamma_{E_{8}}}\sum_{\pi^{\prime}\in\Gamma_{E^{\prime}_{8}}}q^{\frac{1}{2}\left(\pi^{2}+\pi^{\prime
2}\right)}-\left(\frac{2\eta^{3}}{\vartheta_{2}}\right)^{4}\sum_{\pi\in\sqrt{2}\Gamma_{E_{8}}}q^{\frac{1}{2}\pi^{2}}\right)\right.$
$\displaystyle\left.+\bar{O}_{8}\sum_{\pi\in\frac{1}{\sqrt{2}}\Gamma_{E_{8}}}q^{\frac{1}{2}\pi^{2}}\left(\left(\frac{\eta^{3}}{\vartheta_{4}}\right)^{4}+\left(\frac{\eta^{3}}{\vartheta_{3}}\right)^{4}e^{\pi
i\pi^{2}}\right)\right.$
$\displaystyle\left.-\bar{C}_{8}\sum_{\pi\in\frac{1}{\sqrt{2}}\Gamma_{E_{8}}}q^{\frac{1}{2}\pi^{2}}\left(\left(\frac{\eta^{3}}{\vartheta_{4}}\right)^{4}-\left(\frac{\eta^{3}}{\vartheta_{3}}\right)^{4}e^{\pi
i\pi^{2}}\right)\right\\}.$ (B.7)

The spectrum can be identified in the same way as in the CHL model. In the
massless level, there are gauge bosons of $E_{8}$ and non-chiral fermions
transforming in the adjoint representation of the $E_{8}$. The tachyonic
state, which is a singlet of the $E_{8}$, also exists. The spectrum of the
non-supersymmetric $E_{8}$ string is summarized in table 2.

spacetime $SO(8)$ rep | $\bm{8}_{V}$ | $\bm{8}_{S}$ | $\bm{8}_{C}$ | $\bm{1}$
---|---|---|---|---
$E_{8}$ rep. | $\bm{248}$ | $\bm{248}$ | $\bm{248}$ | $\bm{1}$
$M_{R}^{2}=M_{L}^{2}$ | $0$ | $0$ | $0$ | $-1$
Table 2: The spectrum of the $E_{8}$ theory. The vector, spinor and co-spinor
are massless while the scalar is tachyonic.

##  Appendix C Solutions in $d=2$

In this appendix, we present the tables which summarize the solutions of the
massless conditions in the case $d=2$ studied in subsection 3.5.

###  C.1 Enhancement to $C_{9}\times C_{1}$

The solutions in the scalar conjugacy class are those of eq.(3.44) and
eq.(3.50) with $w^{1,2}\in\mathbb{Z}$ and $\pi_{+}\in\Gamma_{E_{8}}$ or
$(w^{1},w^{2};\pi_{+})=(0,0;0)$ shifted by the following four vectors $v_{a}$:

$\displaystyle v_{1}$
$\displaystyle=-v_{2}=\left(\frac{3}{2},1;-3,2;\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{5}{2}\right),$
(C.1) $\displaystyle v_{3}$
$\displaystyle=-v_{4}=\left(\frac{3}{2},2;-3,1;\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{5}{2}\right).$
(C.2)

Table 3: The solutions of the massless conditions with the moduli (3.42).  $\bm{SO(8)}$ rep. | $\bm{P_{L}^{2}}$ | $\bm{\left(w;n;\pi_{+}\right)}$
---|---|---
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(0,0;0,0;1,\underline{1,0,0,0,0,0},0\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(0,0;-1,0;1,0,0,0,0,0,0,1\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\left(0,0;0,0;0,\underline{1,-1,0,0,0,0},0\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\left(0,0;1,0;0,\underline{1,0,0,0,0,0},-1\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(0,0;0,0;-\frac{1}{2},\underline{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}},\frac{1}{2}\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(0,0;1,0;-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(0,0;0,0;\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(1,1;-2,1;0,\underline{-1,-1,0,0,0,0},2\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(1,1;-2,1;1,\underline{-1,0,0,0,0,0},2\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(1,1;-1,1;0,\underline{-1,0,0,0,0,0},1\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(1,1;-1,1;1,0,0,0,0,0,0,1\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(1,1;-1,1;\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(1,1;-2,1;\frac{3}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{3}{2}\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(1,1;-2,1;\frac{1}{2},\underline{-\frac{3}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}},\frac{3}{2}\right)$
$\bm{8}_{V}$ | $2$ | $\pm\left(0,1;-2,1;0,0,0,0,0,0,0,0\right)$
$\bm{8}_{V}$ | $2$ | $\pm\left(1,1;0,1;0,0,0,0,0,0,0,0\right)$
$\bm{8}_{V}$ | $2$ | $\pm\left(1,1;-2,1;0,\underline{-2,0,0,0,0,0},2\right)$
$\bm{8}_{V}$ | $2$ | $\pm\left(1,1;-2,1;2,0,0,0,0,0,0,2\right)$
$\bm{8}_{V}$ | $2$ | $\pm\left(1,1;-2,1;1,-1,-1,-1,-1,-1,-1,1\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},0;1,0;0,0,0,0,0,0,0,0\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},0;0,0;0,\underline{-1,0,0,0,0,0},1\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},0;0,0;1,0,0,0,0,0,0,1\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},0;0,0;\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},1;-1,1;0,0,0,0,0,0,0,0\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},1;-2,1;0,\underline{-1,0,0,0,0,0},1\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},1;-2,1;1,0,0,0,0,0,0,1\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},1;-2,1;\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right)$
$\bm{1}$ | $0$ | $v_{1,2}=\pm\left(\frac{3}{2},1;-3,2;\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{5}{2}\right)$
$\bm{1}$ | $0$ | $v_{3,2}=\pm\left(\frac{3}{2},2;-3,1;\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{5}{2}\right)$
$\bm{1}$ | $1$ | $\pm\left(0,0;0,0;1,\underline{1,0,0,0,0,0},0\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(0,0;-1,0;1,0,0,0,0,0,0,1\right)+v_{a}$
$\bm{1}$ | $1$ | $\left(0,0;0,0;0,\underline{1,-1,0,0,0,0},0\right)+v_{a}$
$\bm{1}$ | $1$ | $\left(0,0;1,0;0,\underline{1,0,0,0,0,0},-1\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(0,0;0,0;-\frac{1}{2},\underline{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}},\frac{1}{2}\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(0,0;1,0;-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(0,0;0,0;\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(1,1;-2,1;0,\underline{-1,-1,0,0,0,0},2\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(1,1;-2,1;1,\underline{-1,0,0,0,0,0},2\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(1,1;-1,1;0,\underline{-1,0,0,0,0,0},1\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(1,1;-1,1;1,0,0,0,0,0,0,1\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(1,1;-1,1;\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(1,1;-2,1;\frac{3}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{3}{2}\right)+v_{a}$
$\bm{1}$ | $1$ | $\pm\left(1,1;-2,1;\frac{1}{2},\underline{-\frac{3}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}},\frac{3}{2}\right)+v_{a}$

###  C.2 Enhancement to $A_{4}\times C_{4}\times C_{2}$

In this example, there are no solutions in the scalar conjugacy class.

Table 4: The solutions of the massless conditions with the moduli (3.61).  $\bm{SO(8)}$ rep. | $\bm{P_{L}^{2}}$ | $\bm{\left(w;n;\pi_{+}\right)}$
---|---|---
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\left(0,0;0,0;\underline{\pm 1,\pm 1,0},0,0,0,0,0\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(0,0;0,0;\underline{\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2}}_{+},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\left(0,0;0,0;0,0,0,\underline{+1,-1,0},0,0\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(0,1;-1,-1;0,0,0,\underline{-1,-1,0},0,2\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(1,2;-1,-2;0,0,0,\underline{-1,-1,0},-2,4\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(1,3;-2,-3;0,0,0,\underline{-1,-1,-2},-2,6\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(0,0;1,1;0,0,0,0,0,0,1,-1\right)$
$\bm{8}_{V}$, $\bm{8}_{S}$ | $1$ | $\pm\left(0,1;-1,0;0,0,0,0,0,0,-1,1\right)$
$\bm{8}_{V}$ | $2$ | $\pm\left(0,1;0,1;0,0,0,0,0,0,0,0\right)$
$\bm{8}_{V}$ | $2$ | $\pm\left(0,1;-2,-1;0,0,0,0,0,0,-2,2\right)$
$\bm{8}_{V}$ | $2$ | $\pm\left(1,1;0,-1;0,0,0,0,0,0,-2,2\right)$
$\bm{8}_{V}$ | $2$ | $\pm\left(1,3;-2,-3;0,0,0,\underline{-2,-2,0},-2,6\right)$
$\bm{8}_{S}$ | $2$ | $\pm\left(0,1;-1,-1;0,0,0,\underline{-2,0,0},0,2\right)$
$\bm{8}_{S}$ | $2$ | $\pm\left(1,2;-1,-2;0,0,0,\underline{-2,0,0},-2,4\right)$
$\bm{8}_{S}$ | $2$ | $\pm\left(1,4;-3,-4;0,0,0,-2,-2,-2,-2,8\right)$
$\bm{8}_{S}$ | $2$ | $\pm\left(2,5;-3,-5;0,0,0,-2,-2,-2,-4,10\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},1;-1,-2;0,0,0,\underline{-1,-1,0},-1,3\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},1;0,-1;0,0,0,\underline{-1,-1,0},0,2\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},1;0,0;0,0,0,0,0,0,-1,1\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},1;-1,-1;0,0,0,0,0,0,-2,2\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},2;-2,-2;0,0,0,\underline{-1,-1,0},-2,4\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},2;-1,-1;0,0,0,\underline{-1,-1,0},-1,3\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},0;1,0;0,0,0,0,0,0,0,0\right)$
$\bm{8}_{C}$ | $1$ | $\pm\left(\frac{1}{2},0;1,0;0,0,0,0,0,0,-1,1\right)$

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