text:
\section{Introduction}

String theories on the asymmetric orbifolds, which should correspond to some non-geometric backgrounds,
have  interesting aspects.
Among others
it is  remarkable that the string vacua with  the vanishing cosmological constant
can be realized without the help of
unbroken SUSY.
Such studies
have been initiated
in type II string theories
by \cite{Kachru1,Kachru2,Kachru3}
based on some non-abelian orbifolds,
followed by studies {\em e.g.} in
\cite{Harvey,Shiu-Tye,Blumenhagen:1998uf,Angelantonj:1999gm,Antoniadis,Aoki:2003sy}.
More recently, several non-SUSY vacua with this property have been constructed
as asymmetric orbifolds \cite{Narain:1986qm} by simpler cyclic groups in \cite{SSW,SWada}.
In heterotic string theories, on the other hand, there have been many studies on the string vacua with
the cosmological constant exponentially suppressed with respect to some moduli
(for instance, the radii of tori of compactifications as given in \cite{SS})
in  \cite{IT,Harvey}, and more recently, {\em e.g.} in \cite{Blaszczyk:2014qoa,Angelantonj:2014dia,Faraggi:2014eoa,Abel:2015oxa,
Kounnas:2015yrc,Abel:2017rch,ItoyamaN} closely related with the model buildings in string phenomenology.

However, in all of these models, the vanishing (or exponentially suppressed) cosmological constant
is eventually achieved by the bose-fermi cancellation,
even though the space-time SUSY is broken.
Toward realistic particle content,
 it would be more preferable to realize the vanishing (or very small) cosmological constant without the bose-fermi cancellation.
In string theory this means that
\begin{equation}
Z(\tau) \not\equiv 0, ~~ \mbox{but,} ~~  \Lambda \equiv \int_{{\cal F}} \frac{d^2 \tau}{\tau_2^2}\, Z(\tau) =0
\hspace{1cm} \left(\tau \equiv \tau_1 + i\tau_2 \in \Bbb H \right) ,
\label{vanising cc string}
\end{equation}
at the one-loop level (at least).
Here,
$Z(\tau)$ is the torus partition function, and
${\cal F}$ denotes the familiar fundamental region of
the full modular group $\Gamma(1) \equiv SL(2, \Bbb Z)$, in other words, the moduli space of the world-sheet torus,
\begin{equation}
{\cal F}:= \left\{ \tau \in \Bbb H^+ ~:~ -\frac{1}{2} \leq \tau_1 < \frac{1}{2} , ~ \left|\tau \right| \geq 1 \right\}.
\label{cF}
\end{equation}
Unfortunately it has been known to be very difficult to
find the string vacua with this property.
An elaborated mechanism has been proposed
in \cite{Moore}
 to achieve them based on some modular symmetry argument
by the Atkin-Lehner involution.
However, concrete constructions of such string vacua that are physically consistent
along this line are still very difficult
\cite{Taylor,Balog:1988dt,Dienes}.

In this paper, instead of searching the string vacua, we discuss how one can systematically construct
the {\em particle theories}
with the preferable property mentioned above, that is, the vanishing one-loop cosmological constant without the bose-fermi cancellation,
\begin{equation}
Z_{\msc{particle}}(\ell) \not\equiv 0,
~~ \mbox{but,} ~~  \Lambda \equiv \lim_{\epsilon\, \rightarrow\, +0} \int_{\epsilon}^{\infty} \frac{d \ell}{\ell}\, Z_{\msc{particle}}(\ell) =0,
\label{vanishing cc particle}
\end{equation}
where $\ell $ denotes the Schwinger parameter (modulus of the world-line circle)
and $\epsilon$ is the UV cut-off,
based on some non-geometric orbifolds of superstring vacua.
Indeed,
as will be demonstrated,
 $Z_{\msc{particle}}(\ell) $ given here originates from
the partition sum only of the {\em untwisted sector} of
the relevant orbifold models
rather than the total one,
which is
only invariant under the modular T-transformations $\tau \, \rightarrow \, \tau + n$ (${}^{\forall} n \in \Bbb Z$).
Consequently, the natural integration region of modulus $\tau$ should be
the `strip region'
\begin{equation}
{\cal S}:= \left\{ \tau \in \Bbb H^+ ~:~ -\frac{1}{2} \leq \tau_1 < \frac{1}{2} \right\},
\label{cS}
\end{equation}
in place of ${\cal F}$. We then naturally obtain $Z_{\msc{particle}}(\ell)$ by identifying $\tau_2 $ with $ \ell $ and integrating $\tau_1$ out, which just amounts to
imposing the level-matching condition.
In the main part of this article,
we will clarify how to construct the appropriate non-geometric orbifold models
whose untwisted sector
yields the particle theories with the desired property
\eqn{vanishing cc particle}.

~

This paper is organized as follows:
In section 2, we discuss how one can obtain
the particle theories with the property \eqn{vanishing cc particle} from rather general setups of orbifolds.
We propose the conditions that should be satisfied by the wanted models,
and prove that  they are indeed  sufficient to  realize \eqn{vanishing cc particle}.
In section 3, we demonstrate  a systematic construction of the concrete models satisfying these conditions based on the type II and heterotic
string vacua compactified on toroidal asymmetric orbifolds.
Especially, the models we focus on are the ones
given in \cite{SSLie}, with some extensions and refinements included.
The main part of section 3 would look a little  technical, but,
we also present simple examples in order to clarify the general features of our models.
In section 4,
we conclude with several comments and discussion.

~

\section{How to Achieve Vanishing Cosmological Constant}
\label{sec 2}

In this section we discuss how to obtain the particle theories from general setups of orbifolds which possess the property \eqn{vanishing cc particle}.
Among others, we clarify the conditions that should be satisfied by the relevant orbifold models.

We start with a generic  superstring vacuum with unbroken space-time SUSY, which we
tentatively denote as `${\cal M}_0$'.
More precisely, we assume that ${\cal M}_0$ is defined as the following background,
$$
{\cal M}_0 \cong \Bbb R^{D-1,1} \times {\cal K} \times S^1_R,
$$
where
$S^1_R$ denotes the circle with radius $R \sqrt{\alpha'}$.
The `internal sector', which we also denote by ${\cal K}$,
only has a discrete spectrum.

Let $g$ be an order $N_0$ automorphism that acts on
${\cal K}$ as well as the world-sheet fermions
and does not commute
with any space-time supercharges.
We further assume the existence of the modular invariant partition function for
the $\Bbb Z_N$-orbifold of $\Bbb R^{D-1,1} \times {\cal K} \left( \otimes [\mbox{fermions}]\right) $ defined by the $g$-action,
where $N$ is a certain multiple of $N_0$ \footnote{We allow the cases of $N \neq N_0$, which typically happens for the {\em asymmetric\/} orbifolds due to
the existence of non-trivial phase factors in the twisted sectors
 (see e.g.\cite{Aoki:2004sm,SSW,Harvey:2017rko}).
We will actually work with the orbifolds of this type in section \ref{sec:simple models}.
}.
This is written in the form,
\begin{equation}
Z_{\msc{orb}}(\tau) \equiv \frac{1}{N} \sum_{a, b \in \Bbb Z_N}\, Z_{(a,b)}(\tau),
\label{ZN-orb}
\end{equation}
where
$a,b \in \Bbb Z_N$ label the twistings
along the spatial and the temporal direction respectively
with respect to the orbifold action $g$ in the world-sheet torus.
In other words, one may write
\begin{equation}
Z_{(a,b)}(\tau) \equiv \mbox{Tr}_{{\cal H}_{a}
} \left[g^{b}\,
q^{L_0-\frac{c_L}{24}} \overline{q^{\tilde{L}_0-\frac{c_R}{24}}}  \right],
\label{Z al beta def}
\end{equation}
where ${\cal H}_{a}$ denotes the Hilbert space of the  twisted sector
associated to the $g^{a}$ action. $Z_{(a,b)}(\tau)$ should possess the periodicity
\begin{equation}
Z_{(a+N,b)}(\tau) = Z_{(a,b+N)}(\tau) = Z_{(a,b)}(\tau),
\end{equation}
for consistency.
Moreover, one should require
the `modular covariance';
\begin{align}
&
\left. Z_{(a,b)}(\tau) \right|_S \left( \equiv Z_{(a,b)} \left(-\frac{1}{\tau} \right) \right)
= Z_{(b,-a)}(\tau),
\label{mod cov S}
\\
& \left. Z_{(a,b)}(\tau)\right|_T \left(\equiv Z_{(a,b)}\left(\tau+1 \right) \right)
= Z_{(a, a+b)}(\tau),
\label{mod cov T}
\end{align}
in the standard fashion.

Now,
setting $R = N \epsilon $ with a small positive number $\epsilon$, let us consider the orbifold of ${\cal M}_0\equiv \Bbb R^{D-1,1} \times {\cal K} \times S^1_{N\epsilon}$ defined by the operator
\begin{equation}
\mbox{{\bf m}}{g} := g \otimes {\cal T}_{2\pi \epsilon},
\label{bm g}
\end{equation}
where ${\cal T}_{2\pi \epsilon}$ denotes the translation
$
X \, \rightarrow \, X + 2\pi \epsilon \sqrt{\alpha'} \,
$
along $S^1_{N\epsilon}$.
One may regard this as a generalization of the Scherk-Schwarz type compactification \cite{SS}.
We shall later take the $\epsilon \, \rightarrow \, +0$ limit.
The torus partition function of the $\mbox{{\bf m}}{g}$-orbifold is written as
\begin{align}
\mbox{{\bf m}}{Z}_{\msc{orb}}(\tau)
& =
\frac{1}{N} \sum_{a, b \in \Bbb Z_N}\,
Z_{(a,b)}(\tau) \, \frac{N \epsilon}{\sqrt{\tau_2} \left|\eta(\tau)\right|^2}
\sum_{w,m\in \Bbb Z}\, e^{- \frac{\pi \left(N \epsilon\right)^2}{\tau_2}\left| \left(w + \frac{a}{N}\right) \tau+ \left(m + \frac{b}{N}\right)\right|^2}
\nonumber\\
& \equiv  \frac{ \epsilon}{\sqrt{\tau_2} \left|\eta(\tau)\right|^2} \sum_{w,m \in \Bbb Z}\, Z_{(w,m)}(\tau) \, e^{- \frac{\pi \epsilon^2}{\tau_2} \left|w \tau + m \right|^2} ,
\label{total ZN-orb}
\end{align}
which is manifestly modular invariant.

Then,
we define the `theory ${\cal M} [\epsilon]$' as that contains
the mass spectrum
read off from the level-matched
sector of the following  partition function
\begin{equation}
\mbox{{\bf m}}{Z}_{{\cal M}[\epsilon]} (\tau) := \frac{ \epsilon}{\sqrt{\tau_2} \left|\eta(\tau)\right|^2}  \sum_{m\in \Bbb Z}\, Z_{(0,m)}(\tau) \, e^{-\frac{\pi}{\tau_2} \epsilon^2 m^2}.
\label{ZcM}
\end{equation}
This represents  the contributions to $ \mbox{{\bf m}}{Z}_{\msc{orb}}(\tau) $
for the $\mbox{{\bf m}}{g}$-orbifold
from the untwisted sector
with  {\em no spatial winding number\/} along $S^1_{N\epsilon}$.
The exponential factor $e^{-\frac{\pi}{\tau_2} \epsilon^2 m^2}$
makes
the infinite summation
over $m\in \Bbb Z$ converge absolutely as long as $\epsilon>0$.
One can  read off the mass spectrum from the partition function \eqn{ZcM} by making the Poisson resummation for the temporal winding $m \in \Bbb Z$.
The overall factor $\epsilon$ is absorbed after this resummation.
Since we have
$
{\cal T}_{2\pi \epsilon} \equiv \exp \left[- i 2\pi \epsilon \sqrt{\alpha'} P \right],
$
where $P$ denotes the KK momentum operator along $S^1_{N\epsilon}$,
the states with the $g$-eigenvalue $e^{2\pi i \frac{r}{N}}$ have to possess the KK momenta $n \equiv r ~ (\mod\, N)$, implying that these excitations
at least have masses of order
$\displaystyle \sim  \frac{1}{\epsilon} M_s \left(\equiv \frac{1}{\epsilon \sqrt{\alpha'}} \right)$
except for the $r=0$ case.
In particular all the massless states in ${\cal M}[\epsilon]$ are lying in $\displaystyle Z_0(\tau)\equiv \frac{1}{N} \sum_{b \in \Bbb Z_N}\, Z_{(0,b)}(\tau)$.

We emphasize that $\mbox{{\bf m}}{Z}_{{\cal M}[\epsilon]}(\tau)$
is {\em not\/}  invariant under the full modular group.
This is invariant only under the modular T-transformations.
Thus, we cannot adopt the usual fundamental region ${\cal F}$
of $\Bbb H^+/\Gamma(1)$ given in \eqn{cF}
for the integration of the modulus $\tau$.
In other words, the theory ${\cal M}[\epsilon]$ is {\em not\/} identified with
some string vacuum, although we have an infinite number of mass spectrum.
It is rather natural to take the `strip region' ${\cal S}$ defined in \eqn{cS}
as the appropriate integration region,
which is identified as the fundamental region for
$\Bbb H^+/\Gamma_0(\infty) \equiv \Bbb H^+/\left\langle T \right\rangle$.
We note that the factor $e^{-\frac{\pi}{\tau_2} \epsilon^2 m^2}$  for $m\neq 0$ effectively truncates the UV region $\tau_2\, \sim \, +0$,  which makes the moduli
integral well-defined, even if taking ${\cal S}$ instead of ${\cal F}$. In other words the parameter $\epsilon$ plays the role of  the UV cut-off.

Let us recall the familiar relationship between the one-loop cosmological constants of closed string theory
and of point particle theory (see {\em e.g.}
section
7.3 of \cite{Pol book 1}).
We assume the bosonic (fermionic) mass spectrum $\left\{ m_i \right\}$, $i\in {\cal H}_B$ ($i \in {\cal H}_F$).
The one-loop cosmological constant of the particle theory on the space-time with $D$-dimensional non-compact directions
is schematically expressed in terms of
the summation of path-integrals over a world-line circle
(with modulus $\ell$);
\begin{align}
\Lambda_{\msc{particle}} & = \frac{1}{V_{D}}
\left[ \sum_{i\in {\cal H}_{\msc{B}}}\, - \sum_{i\in {\cal H}_{\msc{F}}}\,\right] \, Z_{S^1}(m_i^2)
\nonumber\\
& \equiv  \left[ \sum_{i\in {\cal H}_{\msc{B}}}\, - \sum_{i\in {\cal H}_{\msc{F}}}\,\right]
\, \int_0^{\infty}\frac{d\ell}{\ell}\, \int \frac{d^D p_i}{(2\pi)^D}\, e^{-\ell \frac{\alpha'}{2}   \left[  p_i^2 + m_i^2 \right]},
\label{La general particle}
\end{align}
where $p_i$ are the zero-mode momenta along $\Bbb R^{D-1,1}$ and $V_D$ denotes the volume factor.
This is naturally compared with that of string theory.
For instance,  consider the type II string on
the $\Bbb R^{D-1,1} \times
[\mbox{internal sector}]
$,
where the `internal sector' only includes the discrete spectrum.
The 1-loop cosmological constant is written schematically as
\begin{align}
\Lambda_{\msc{string}} & = \frac{1}{V_{D}}  \int_{{\cal F}} \frac{d^2\tau}{\tau^2_2}\, Z(\tau)  ,
\label{La general string}
\end{align}
where
\begin{equation}
Z(\tau) \equiv \frac{V_D}{\tau_2^{\frac{D-2}{2}}}\,
 \sum_{(i, \tilde{i}) \in {\cal H}_{\perp}} \, D(h_i, \tilde{h}_{\tilde{i}} ) \,
q^{h_i-\frac{1}{2}} \overline{q^{\tilde{h}_{\tilde{i}}-\frac{1}{2}}}
\hspace{1cm}\left(q\equiv e^{2\pi i \tau}\right) ,
\end{equation}
denotes the modular invariant partition function on the world-sheet torus\footnote
  {`$(i,\tilde{i}) \in {\cal H}_{\perp}$' indicates the transverse degrees of freedom
in
$ \Bbb R^{D-2} \times [\mbox{internal sector}]$,
and the summation is only taken over the discrete
spectrum of conformal weights.  The Gaussian integral of zero-mode momenta along the transverse direction $\Bbb R^{D-2}$
just yields the factor $\tau_2^{- \frac{D-2}{2}}$, while one more factor $\tau_2^{-1}$ due to the  temporal and longitudinal
zero-mode integrals are
incorporated into the modular invariant measure $d^2\tau/\tau_2^2$.
}.
The coefficients of `degeneracy'
$D(h_i, \tilde{h}_{\tilde{i}})$ are positive (negative) integers for the bosonic (fermionic) states.

The correspondence between \eqn{La general particle} and \eqn{La general string} is  clear,
if we rewrite
$\displaystyle \tau \equiv \frac{\th+ i\ell}{2\pi}$ and identify the imaginary part
$\ell$ with the circle modulus (Schwinger parameter).
Indeed, we obtain
{\em by formally replacing the integration region ${\cal F}$ with ${\cal S}$};
\begin{align}
\hspace{-5mm}
\Lambda_{\msc{string}} \left[\mbox{${\cal F}$ replaced with ${\cal S}$}\right]
& \equiv  \int_{{\cal S}} \frac{d^2\tau}{\tau^2_2}\, \frac{1}{\tau_2^{\frac{D-2}{2}}}
\, \sum_{i,\tilde{i}}
\, D(h_i, \tilde{h}_{\tilde{i}} ) \, q^{h_i-\frac{1}{2}}
\overline{q^{\tilde{h}_{\tilde{i}}-\frac{1}{2}}},
\nonumber\\
& = \int_0^{\infty}\frac{d\ell}{\ell}\, \int_{-\pi}^{\pi} \frac{d\th}{2\pi} \, \left( 2\pi \alpha' \ell \right)^{-\frac{D}{2}}
\nonumber\\
& \hspace{1cm} \times  \sum_{i, \tilde{i}}
\, D(h_i, \tilde{h}_{\tilde{i}}) \,
\exp \left[-\left(h_i+ \tilde{h}_{\tilde{i}} -1 \right)\ell + i \theta \left(h_i-\tilde{h}_{\tilde{i}} \right)\right],
\label{La string particle}
\end{align}
which easily reduces to  the  expression
in the form as  \eqn{La general particle}.
In fact, the $\th$-integral just amounts to imposing the level-matching condition
$h_i= \tilde{h}_{\tilde{i}}$, and
the mass spectrum is identified  as
\begin{equation}
\frac{\alpha'}{4}m_i^2 = h_i-\frac{1}{2}.
\end{equation}

Based on these simple arguments, we propose that ${\cal M}[\epsilon]$ defined above  represents a theory
with an infinite number of {\em particle spectrum,}
in which the one-loop cosmological constant is given as
\begin{align}
\Lambda_{\msc{${\cal M}[\epsilon]$}} & \equiv \frac{1}{V_{D}}  \int_{{\cal S}} \frac{d^2\tau}{\tau^2_2}\, \mbox{{\bf m}}{Z}_{{\cal M}[\epsilon]}(\tau)
\nonumber\\
& = \int_0^{\infty} \frac{d\ell}{\ell} \, (2\pi \alpha' \ell)^{-\frac{D}{2}}\, \sum_{i \in {\cal H}[{\cal M}[\epsilon]]} \, D(m_i) \, e^{- 2\pi \ell \frac{\alpha'}{2} m_i^2} ,
\label{La cM}
\end{align}
where the mass spectrum, which is read off from the partition function
\eqn{ZcM} and by imposing the level-matching condition, is expressed as ${\cal H}[{\cal M}[\epsilon]]$.
Again $D(m_i)$ denotes the degeneracy of the mass spectrum including the minus sign for fermions.
Recall that the  integration region suitable for $\mbox{{\bf m}}{Z}_{{\cal M}[\epsilon]}(\tau)$ should be ${\cal S}$ rather than ${\cal F}$, as mentioned above.
We can thus naturally identify the partition function $\mbox{{\bf m}}{Z}_{{\cal M}[\epsilon]}(\tau)$ as describing a spectrum of infinite particles, rather than a string spectrum,
 without making any formal replacement of
the integration region of modulus.

~

Under these preparations  let us exhibit the fundamental requirements to be satisfied by the models we want (some  are already mentioned):

\begin{description}
\item[(1)] The unorbifolded string vacuum ${\cal M}_0$ has the vanishing partition function,
\begin{equation}
Z_{{\cal M}_0}(\tau) \equiv \frac{N\epsilon}{\sqrt{\tau_2} \left|\eta(\tau)\right|^2}\, \sum_{w,m\in \Bbb Z}\, e^{- \frac{\pi }{\tau_2} N^2 \epsilon^2\left|w \tau + m\right|^2} \, Z_{(0,0)}(\tau)
= 0.
\end{equation}
Needless to say, it is enough to start with any supersymmetric vacuum ${\cal M}_0 \equiv \Bbb R^{d-1, 1} \times {\cal K} \times S^1_{N\epsilon}$.

\item[(2)] The partition function $\mbox{{\bf m}}{Z}_{{\cal M}[\epsilon]}(\tau)$ defined in \eqn{ZcM} does not vanish.
This may imply
\begin{equation}
Z_0(\tau) \equiv \frac{1}{N} \sum_{b\in \Bbb Z_N}\, Z_{(0,b)}(\tau) \neq 0.
\nonumber
\end{equation}
Namely, all the supercharges existing originally in ${\cal M}_0$ are removed by the  $g$-projection.

\item[(3)] The spectrum of the level-matched states appearing in $\mbox{{\bf m}}{Z}_{{\cal M}[\epsilon]}(\tau)$ is consistent with unitarity.
Alternatively, we require that  $Z_0(\tau)$ possess  the same property.

\item[(4)] The spectra of the level-matched
states in the sectors $\displaystyle Z_a(\tau) \equiv \frac{1}{N} \sum_{b\in \Bbb Z_N} Z_{(a,b)}(\tau)$ for ${}^{\forall} a \in \Bbb Z_N$ do not contain
tachyons. In other words, we require\footnote
   {The factor $e^{4\pi \tau_2 \frac{1}{24}}$ originates from the factor $\left|\eta\right|^{-2}$ appearing in the $S^1_{N\epsilon}$-sector.
   We note that the non-tachyonic behavior for $a \neq 0$ is necessary in order to prove the statement \eqn{La cM 0} (see below (\ref{eval La ep 3})),
even though only the $a=0$ sector contributes to the spectrum of the theory ${\cal M}[\epsilon]$.}
\begin{equation}
\lim_{\tau_2 \rightarrow +\infty} \, e^{4\pi \tau_2 \frac{1}{24}} \left| \int_{-1/2}^{1/2} d\tau_1\, Z_{a}(\tau) \right|
< + \infty
\hspace{1cm} ({}^{\forall} a \in \Bbb Z_N) .
\label{cond no tachyon}
\end{equation}
(We allow the existence of  level-mismatched tachyons.)

\item[(5)] The orbifold partition function  \eqn{ZN-orb} vanishes.

\end{description}

Now, we  state our main claim:
under the requirements {\bf (1)} $\sim$ {\bf (5)} given above\footnote
   {In fact, only the requirements {\bf (1)}, {\bf (4)}, {\bf (5)} are sufficient to show the claim \eqn{La cM 0}.},
the theory ${\cal M}[\epsilon]$ ($\epsilon>0$)
yields the finite cosmological constant $\Lambda_{{\cal M}[\epsilon]}$,
and we further obtain
\begin{equation}
\lim_{\epsilon\, \rightarrow \, +0}\,  \Lambda_{{\cal M}[\epsilon]} =0, ~~~ \mbox{for}~ {}^{\forall} D \geq 1.
\label{La cM 0}
\end{equation}

~

\noindent
{\bf [proof of the claim]}

We first note that $\Lambda_{{\cal M}[\epsilon]}$ defined in \eqn{La cM} is explicitly written as
\begin{align}
\Lambda_{\msc{${\cal M}[\epsilon]$}}
= \frac{1}{V_D} \int_{{\cal S}} \frac{d^2\tau}{\tau^2_2}\, \frac{\epsilon}{\sqrt{\tau_2} \left|\eta(\tau) \right|^2}\,
\sum_{m\in \Bbb Z}\, Z_{(0,m)}(\tau)  \, e^{-\frac{\pi}{\tau_2} \epsilon^2 m^2}.
\label{La cM explicit}
\end{align}
Then, $\Lambda_{{\cal M}[\epsilon]}$ should be
finite for ${}^{\forall} \epsilon >0$.
Indeed, the $m=0$ term is absent due to the condition {\bf (1)}, and
the potential UV-divergence around $\tau_2 \sim 0$ is removed
by the damping factor $e^{-\frac{\pi}{\tau_2} \epsilon^2 m^2 }$  for ${}^{\forall} m \neq 0$,
as mentioned above.
Moreover,
the convergence in the IR-region $\tau_2 \rightarrow +\infty$ is ensured by
the condition {\bf (4)}.

Let us next prove the more non-trivial statement \eqn{La cM 0}.
Set
\begin{equation}
\mbox{{\bf m}}{Z}_{(w,m)}(\tau; \epsilon) := Z_{(w,m)}(\tau) \, \frac{\epsilon}{\sqrt{\tau_2} \left|\eta(\tau)\right|^2}\, e^{- \frac{\pi}{\tau_2}\epsilon^2 \left|w\tau +m \right|^2} ,
\hspace{1cm} ({}^{\forall} w,m \in \Bbb Z).
\end{equation}
Because of the  modular covariance
$$
\left. \mbox{{\bf m}}{Z}_{(w,m)}(\tau; \epsilon)\right|_S = \mbox{{\bf m}}{Z}_{(m,-w)}(\tau;\epsilon), \hspace{1cm}
\left. \mbox{{\bf m}}{Z}_{(w,m)}(\tau; \epsilon)\right|_T = \mbox{{\bf m}}{Z}_{(w,w+m)}(\tau;\epsilon),
$$
together with the condition {\bf (1)}, that is,  $\mbox{{\bf m}}{Z}_{(0,0)}(\tau; \epsilon)\equiv 0$,
we can rewrite  \eqn{La cM explicit} as
\begin{align}
\Lambda_{{\cal M}[\epsilon]}
& \equiv \frac{1}{V_D} \int_{{\cal S}} \frac{d^2\tau}{\tau^2_2}\,  \sum_{m\in\Bbb Z}\, \mbox{{\bf m}}{Z}_{(0,m)}(\tau;\epsilon)
= \frac{1}{V_D} \int_{{\cal F}} \frac{d^2\tau}{\tau^2_2}\,  \sum_{w, m\in\Bbb Z}\, \mbox{{\bf m}}{Z}_{(w,m)}(\tau;\epsilon) ,
\label{eval La ep 1}
\end{align}
due to the arguments given in
\cite{Polchinski,OBrien,McClain},
which are commonly used in thermal string theory.
Introducing the `Fourier transform',
\begin{align}
\widehat{Z}_{(\alpha,\beta)}(\tau) := \frac{1}{N}
\sum_{a,b\in \Bbb Z_N}\, Z_{(a,b)}(\tau) e^{2\pi i \frac{1}{N} \left(\alpha b - \beta a\right)},
\label{hZ}
\end{align}
and making use of the Poisson resummation,
we can further rewrite the R.H.S of \eqn{eval La ep 1} as follows,
\begin{align}
\Lambda_{{\cal M}[\epsilon]}
& = \frac{1}{V_D} \int_{{\cal F}} \frac{d^2\tau}{\tau^2_2}\, \frac{1}{N \epsilon} \frac{1}{\sqrt{\tau_2} \left|\eta(\tau)\right|^2} \sum_{w, m\in\Bbb Z}\,
\widehat{Z}_{(w,m)}(\tau) e^{-\frac{\pi}{\tau_2} \frac{1}{ N^2 \epsilon^2} \left| w\tau+ m\right|^2}
\nonumber\\
& = \frac{1}{V_D} \int_{{\cal F}} \frac{d^2\tau}{\tau^2_2}\, \frac{1}{N \epsilon} \frac{1}{\sqrt{\tau_2} \left|\eta(\tau)\right|^2}
 \sum_{w, m\in\Bbb Z, \atop  (w,m)\neq (0,0)}\,

\widehat{Z}_{(w,m)}(\tau) e^{-\frac{\pi}{\tau_2} \frac{1}{ N^2 \epsilon^2} \left| w\tau+ m\right|^2}.
\label{eval La ep 2}
\end{align}
In the second line we made use of the fact that
$$
\widehat{Z}_{(0,0)}(\tau) \equiv \frac{1}{N}\sum_{a,b\in \Bbb Z_N}\, Z_{(a,b)}(\tau) = 0,
$$
due to the condition {\bf (5)}.

We first estimate the non-zero winding sectors
with
$w\neq 0$ in \eqn{eval La ep 2}.
For each fixed $w \neq 0$, the summation over $m\in \Bbb Z$ is evaluated by using the Poisson resummation, giving the inequality,
\begin{align}
& \frac{1}{V_D N \epsilon} \frac{1}{\sqrt{\tau_2} \left|\eta(\tau)\right|^2}\,
 \sum_{m\in\Bbb Z}\, \left|\widehat{Z}_{(w,m)}(\tau) \right| \, e^{-\frac{\pi}{\tau_2} \frac{1}{ N^2 \epsilon^2} \left| w\tau+ m\right|^2}
\leq \frac{{\cal C}}{\epsilon} \, \frac{1}{\tau_2^{\frac{D-2}{2}}} \, e^{-4 \pi \tau_2 \left(h_0 - \frac{1}{2}\right)}\, e^{-\pi\frac{1}{ N^2 \epsilon^2} w^2 \tau_2}
\nonumber\\
&
\hspace{10cm} ({}^{\forall} w\neq 0, ~ {}^{\forall} \tau \in {\cal F}) ,
\nonumber
\end{align}
with some finite constant ${\cal C}$, $h_0$.
Here, $h_0$ is the lowest conformal weight in the discrete part of the spectrum and {\em allowed to be tachyonic.}

Therefore,
replacing the integration region ${\cal F}$ with the slightly larger one,
\begin{equation}
\left\{ \tau \in \Bbb H^+\, :\, -\frac{1}{2} \leq \tau_1 < \frac{1}{2}, ~ \tau_2 >  \frac{\sqrt{3}}{2} \right\},
\label{int region}
\end{equation}
we find
\begin{align}
\left| \Lambda_{\epsilon, w} \right|
& \leq \frac{{\cal C}}{\epsilon} \int_{\sqrt{3}/2}^{\infty} \frac{dt}{t^2} \frac{1}{t^{\frac{D-2}{2}}}\,
 e^{-4 \pi t \left(h_0 - \frac{1}{2}\right)}\, e^{-\pi\frac{1}{ N^2 \epsilon^2} w^2 t}
\hspace{1cm} ({}^{\forall} w\neq 0) .
\nonumber
\end{align}
This integral
is obviously finite
for sufficiently small $\epsilon >0$, and we readily obtain
\begin{equation}
\lim_{\epsilon\rightarrow +0} \, \sum_{w\neq 0}\, \Lambda_{\epsilon, w} =0.
\label{limit La w}
\end{equation}

We next focus on the $w=0$ contribution.
Replacing again the integration region with \eqn{int region},
we obtain the following evaluation
\begin{align}
\left| \Lambda_{\epsilon, w=0} \right| & \equiv \frac{1}{N \epsilon V_D} \left| \int_{{\cal F}} \frac{d^2\tau}{\tau^2_2}\,
\frac{1}{\sqrt{\tau_2} \left|\eta(\tau)\right|^2}\,
\sum_{m\neq 0} \, \widehat{Z}_{(0,m)}(\tau)
e^{-\frac{\pi}{\tau_2} \frac{m^2}{N^2 \epsilon^2}} \right|
\nonumber\\
& \leq \frac{{\cal C}'}{\epsilon}
\int_{\sqrt{3}/2}^{\infty} \frac{dt}{t^{\frac{5}{2}}} \,  \frac{1}{\left|\eta(it)\right|^2}\,
\left| \sum_{m\neq 0} \,  \left.
\widehat{Z}_{(0,m)}(it)\right|_{\msc{level-matched}}\right|  \,
e^{-\frac{\pi}{t} \frac{m^2}{N^2 \epsilon^2}},
\label{eval La ep 3}
\end{align}
with some finite constant ${\cal C}'$.
Moreover, we note that
$$
\left. e^{4\pi \frac{1}{24} \tau_2} \, \widehat{Z}_{(0,m)}(\tau)\right|_{\msc{level-matched}}
\equiv e^{4\pi \frac{1}{24} \tau_2} \,  \frac{1}{N} \sum_{a,b\in \Bbb Z_N}\, \left.
Z_{(a,b)}(\tau) \right|_{\msc{level-matched}}
 e^{-2\pi i \frac{m}{N}a},
$$
is non-tachyonic due to the condition {\bf (4)}.
Thus,
we obtain the following evaluation with a finite constant ${\cal C}''$;
\begin{align}
\left| \Lambda_{\epsilon, w=0} \right| & \leq  \frac{{\cal C}''}{\epsilon} \int_{\sqrt{3}/2}^{\infty} \frac{dt}{t^{\frac{5}{2}}} \, \frac{1}{t^{\frac{D-2}{2}}}\,
  \sum_{m=1}^{\infty}\,
e^{-\frac{\pi}{t} \frac{m^2}{N^2 \epsilon^2}}
\nonumber\\
& < \frac{{\cal C}''}{\epsilon} \pi^{- \frac{D+1}{2}} \left(N\epsilon \right)^{D+1}\,
\int_0^{\infty} \frac{ds}{s} \, s^{\frac{D+1}{2}} \, \sum_{m=1}^{\infty} \, e^{-m^2 s}\,
\nonumber\\
&
= {\cal C}'' N^{D+1} \epsilon^{D} \, \widehat{\zeta}(D+1),
\end{align}
where $\widehat{\zeta} (s) \equiv \pi^{-\frac{s}{2}} \Gamma\left(\frac{s}{2}\right) \zeta(s)$ is the completed zeta function.

In this way, as long as $D \geq 1$, we obtain the desired result;
$$
\lim_{\epsilon \, \rightarrow\, +0} \, \Lambda_{{\cal M}[\epsilon]} \equiv  \lim_{\epsilon \, \rightarrow\, +0} \, \left[\Lambda_{\epsilon, w=0} + \sum_{w\neq 0} \Lambda_{\epsilon, w} \right] =0.
\hspace{2cm}
\mbox{\bf (Q.E.D)}
$$

~

A few comments are in order:
\begin{description}
\item[(i)]
Under the requirement {\bf (2)}
the bosonic and fermionic mass spectra in the point particle theory ${\cal M}[\epsilon]$
are obviously mismatched.
Nevertheless, the 1-loop cosmological constant $\Lambda_{{\cal M}[\epsilon]}$ vanishes
under the $\epsilon\, \rightarrow \, +0$ limit in the cases of $D \geq 1$.

\item[(ii)]
As $\epsilon$ plays the role of the UV cut-off,
the integration region is truncated roughly as follows,
$$
\int_{{\cal S}} \frac{d^2\tau}{\tau_2^2} ~ \longrightarrow ~ \int_{\epsilon^2}^{\infty} \frac{d\tau_2}{\tau_2^2} \, \int_{-1/2}^{1/2} d\tau_1.
$$
In other words, we have a {\em substringy\/} cut-off mass scale
$
\displaystyle
M_{\epsilon} \sim \frac{1}{\epsilon \sqrt{\alpha'}} \left( \equiv \frac{1}{\epsilon} M_s \right),
$
and the low energy mass spectrum $M \ll M_{\epsilon}$ is
nearly equal to that of $\displaystyle Z_0(\tau) \equiv \frac{1}{N} \sum_{m\in \Bbb Z_N}\, Z_{(0,m)}(\tau) $,
which only contains the $g$-invariant particle spectrum.
On the other hand,
the `super-partners'  that are not $g$-invariant acquire
the masses greater than $M_{\epsilon}$.
$M_{\epsilon}$ is  thus interpreted as the SUSY breaking scale in our theory ${\cal M}[\epsilon]$.
It is remarkable that,
{\em while the naive dimensional estimation implies
$$
\Lambda_{{\cal M}[\epsilon]} \sim M_{\epsilon}^D \equiv \epsilon^{-D} M_s^D,
$$
we indeed obtain
$\Lambda_{{\cal M}[\epsilon]} \sim \epsilon^D M_s^D$, }
as was shown above.

\item[(iii)]
Intuitively, the counterpart of the bose-fermi cancellation in the original
orbifold
model is disguised by the modular transformation
as other particles
in the resultant
particle theory ${\cal M}[\epsilon]$, which contribute to $\Lambda_{{\cal M}[\epsilon]} $ in the integration
region ${\cal S}$ outside ${\cal F}$.

\end{description}

~

\section{Simple Models from String Vacua}
\label{sec:simple models}

In this section we exhibit  simple examples
satisfying the requirements {\bf (1)} $\sim$ {\bf (5)} given in the previous section
based on the toroidal asymmetric orbifolds of type II or heterotic superstring vacua.
To this end we shall make use of the modular invariants, or modular covariant functions more generally,
constructed in \cite{SSLie}.
They are associated to  Lie algebra lattices satisfying the `self-duality condition'  under the T-duality twist.
We start with reviewing  briefly the results  given in \cite{SSLie}
for our construction of the relevant models.
Since we only need appropriate modular forms,
slight extensions and refinements are also included below.

~

\subsection{Asymmetric Orbifolds based on Lie Algebra Lattices}
\label{review asymm}

Let us consider the $r$-dimensional torus $T^r[X_r]$
associated to the Englert-Neveu
lattice \cite{EN} (see also \cite{Lerche lattice} for a review)
for  the semi-simple Lie algebra
$X_r$ (rank $r$), for which we have the symmetry enhancement to the affine $X_r$-symmetry with level 1 on the string world-sheet.
We simply call it the `Lie algebra lattice for $X_r$' in this section.
We assume that the simple parts of $X_r$ are composed only of
\begin{equation}
A_1, ~ E_7, ~ D_n~ ({}^{\forall} n \in \Bbb Z_{>0}) .
\label{AED}
\end{equation}
Here, we have slightly extended the argument  in \cite{SSLie}
by  including $D_n$
with odd $n$.
The number of `$D_{\msc{odd}}$-components'
is supposed to be even to preserve the unitarity, as explained below.
Then the asymmetric orbifold model of $T^r[X_r]$ for the chiral reflection (or the `T-duality transformation')
$$
(\-_R)^{\otimes r} ~ : ~ \left(X^i_L, X^i_R\right) ~ \longrightarrow ~  \left(X^i_L, - X^i_R\right)
$$
is described by  the following partition function
\cite{SSLie},
\begin{align}
& Z^{T^r[X_r]}_{\msc{orb}} (\tau) \equiv \frac{1}{16} \sum_{a,b \in \Bbb Z_{16}} \, Z^{T^r[X_r]}_{(a,b)} (\tau),
\label{Z TXr orb}
\end{align}
where
\begin{align}
 Z^{T^r[X_r]}_{(a,b)} (\tau) &  \equiv
\left\{
\begin{array}{ll}
\displaystyle
 \epsilon^{[r]}_{(a,b)} \, \overline{\left( \tilde{\chi}^{A_1}_{(a,b)} (\tau) \right)^r} \,
\chi^{X_r}_{(a,b)} (\tau)
& ~~
\left(a \in 2\Bbb Z+1 ~ \mbox{or} ~ b \in 2\Bbb Z+1 \right) ,
\\
\displaystyle
Z^{T^r[X_r]} (\tau)
& ~~ \left(a, b \in 2\Bbb Z\right) ,
\end{array}
\right.
\label{Z TXr ab}
\end{align}
and we introduced the notation
\begin{align}
\tilde{\chi}^{A_1}_{(a,b)}(\tau) & :=
\left\{
\begin{array}{ll}
\sqrt{
\frac{\th_3 \th_4}{\eta^2}
}
& ~~ (a\in 2\Bbb Z, ~ b\in 2\Bbb Z+1) , \\
\sqrt{
\frac{\th_3 \th_2}{\eta^2}
}
& ~~ (a\in 2\Bbb Z+1, ~ b\in 2\Bbb Z) , \\
\sqrt{
\frac{\th_4 \th_2}{\eta^2}
}
& ~~ (a\in 2\Bbb Z+1, ~ b\in 2\Bbb Z+1) , \\
\end{array}
\right.
\label{tchi A1ab}
\\
\epsilon^{[r]}_{(a,b)} & := \left\{
\begin{array}{ll}
e^{\frac{i \pi r}{8}ab}
& ~~ \left(a \in 2\Bbb Z, ~ b \in 2\Bbb Z+1\right) ,
\\
e^{-\frac{i \pi r}{8}ab}
& ~~ \left(a \in 2\Bbb Z+1, ~ b \in 2\Bbb Z\right) ,
\\
\left(\kappa_a\right)^r
e^{-\frac{i \pi r}{8}ab}
& ~~ \left(a , ~ b \in 2\Bbb Z+1\right) ,
\end{array}
\right.
\label{ep r ab}
\\
\kappa_a & :=
e^{-\frac{i\pi }{8}(a^2-1 )}
\equiv
 \left\{
\begin{array}{ll}
+1 & ~~ (a \equiv 1,7 ~ \mod \, 8) ,
\\
-1 & ~~ (a \equiv 3,5 ~ \mod \,  8) .
\end{array}
\right.
\end{align}
Moreover, $\chi^{X_r}_{(a,b)}(\tau)$ is defined as the simple product
$$
\chi^{X_r}_{(a,b)}(\tau) := \prod_{i} \, \chi^{X_{r_i}^{(i)}}_{(a,b)} (\tau) ,
\hspace{1cm} \sum_i \, r_i = r,
$$
and each component is
associated to the simple Lie algebra $X_{r_i}^{(i)}$ appearing in \eqn{AED}.
The explicit forms of  $Z^{T^r[X_r]} (\tau)$
and  $\chi^{X_r}_{(a,b)}(\tau)$ are summarized in appendix A.

One can derive \eqn{Z TXr ab} by first evaluating the trace,
$$
Z^{T^r[X_r]}_{(0,1)}(\tau) \equiv \mbox{Tr} \left[\, \left(\-_R\right)^{\otimes r} q^{L_0-\frac{c}{24}} \overline{q^{\tilde{L}_0-\frac{c}{24}}}\, \right] ,
$$
with $c=r$,
and by requiring the modular covariance,
\begin{equation}
\left. Z^{T^r[X_r]}_{(a,b)}(\tau)\right|_T = Z^{T^r[X_r]}_{(a,a+b)}(\tau),
\hspace{1cm}
\left. Z^{T^r[X_r]}_{(a,b)}(\tau)\right|_S = Z^{T^r[X_r]}_{(b,-a)}(\tau),
\label{mod cov ZTXr}
\end{equation}
which ensures the modular invariance of \eqn{Z TXr orb}.
See \cite{SSLie} for details.
We emphasize that  the non-trivial phase factor $\epsilon^{[r]}_{(a,b)}$  is necessary to achieve
the  modular covariance \eqn{mod cov ZTXr},
and  thus, the building blocks  \eqn{Z TXr ab} generically possess an order 16 periodicity,
rather than the naive expectation of
order 2, for the twisted sectors with $a\neq 0$.
These phases appear because  the action of the
chiral
reflection in the target space is uplifted
 on the world-sheet
(see e.g.\cite{Aoki:2004sm,SSW,Harvey:2017rko}),
and will play a crucial role in our argument given below.

We also note another interpretation of the asymmetric orbifolds given above.
The Narain lattice defining the torus $T^r[X_r]$ can be generically decomposed as
\begin{align}
\Gamma^{r,r}[X_r] & = \sum_{\alpha_{L}, \, \alpha^{(i)}_{R}}
\Lambda_{(\alpha_L)}^{X_r} \bigoplus \sum_{i=1}^r \, \Lambda^{A_1}_{(\alpha_R^{(i)})}
\equiv \Lambda_{(0)}^{X_r} \bigoplus \left[ \Lambda^{A_1}_{(0)} \oplus \cdots \oplus \Lambda^{A_1}_{(0)} \right] + \cdots ,
\label{lattice decomp 1}
\end{align}
where `$\Lambda^{X_r}_{(\alpha)}$' denotes the Lie algebra lattice of $X_r$ associated to the conjugacy class labeled by
$\alpha$ (corresponding to a certain integrable representation
 of affine $X_r$ with level 1).
Especially, $\Lambda^{X_r}_{(0)}$ is nothing but the root lattice of $X_r$ associated to the basic representation.
Only  the term including  $\left[ \Lambda^{A_1}_{(0)} \oplus \cdots \oplus \Lambda^{A_1}_{(0)}\right]$
can contribute to the trace $\displaystyle \mbox{Tr} \left[\left(\-_R\right)^{\otimes r} \, \cdots \right]$ for the untwisted sector
in this decomposition.
If focusing on each of the right-moving momentum lattice $\Lambda^{A_1}_{(\alpha)}$, the chiral reflection
$\-_R$ is naturally described by the $SU(2)$-current algebra $\{ J_R^a \}$ of level 1.
Namely, one can simply identify
$\displaystyle \-_R \equiv e^{i\pi J^1_{R,0}}$
in the basic
representation (see appendix B).
One can also adopt the `chiral half-shift' $s_R \equiv e^{i\pi J^3_{R,0}}$ as the involutive operator
in the untwisted sector
to define the relevant
asymmetric orbifolds,
and obtains the same  building blocks $Z^{T^r[X_r]}_{(a,b)}(\tau)$
because of the obvious reason of the $SU(2)$ invariance\footnote
   {More precisely, the chiral half-shift operator $s_R$ (as well as the chiral reflection) has to contain the phase factor
to achieve the involutive property $s_R^2 = \mbox{{\bf m}}{1}$.
Namely, we should adopt the definition
$
s_R \equiv e^{-\frac{i\pi}{2} \ell}\, e^{i\pi J^3_{R, 0}},
$
for the spin $\ell/2$ representation of $\widehat{A_1}$ with level 1 ($\ell=0,1$).
}.

In the  following,  we will mainly focus on the chiral half-shifts rather than the chiral reflections
to obtain desired models, which modifies the construction in \cite{SSLie}.
We will also maintain the world-sheet ${\cal N}=1$ superconformal symmetry,
since we expect that the consistency of
the original superstring theories is important to ensure
that of the resultant particle theories after interactions are turned o
\footnote
{However, it may not be necessary to assume the world-sheet superconformal symmetry, since our purpose in this paper
    is to construct the non-SUSY {\em particle theories\/}  with vanishing cosmological constant, rather than the non-SUSY string vacua.
   In other words one may be able to relax the requirements of the consistency as string vacua.
 We will again discuss this point in section 4.}.
The chiral half-shift preserves the superconformal symmetry
even if trivially acting on the world-sheet fermions
$\psi^i_R$,
whereas we need to require
$$
\left(\-_R\right)^{\otimes r} ~: ~ (\psi_L^i, \psi_R^i) ~ \longrightarrow ~ (\psi_L^i, - \psi_R^i),
$$
for the chiral reflection.

~

\subsection{Type II Models}

Let us start with the type II string on $\Bbb R^{3,1} \times T^5 \times S^1_{N \epsilon}$, where $N$ is the order of the $g$-orbifold introduced below.
In other words we focus on the cases with ${\cal K} = T^5$ in the notations given in section \ref{sec 2}.
As discussed in section \ref{sec 2},
we consider the orbifold  by $\mbox{{\bf m}}{g} \equiv g \otimes {\cal T}_{2\pi \epsilon}$ with
appropriate choices of $g$ acting on the $T^5$-sector as well as the world-sheet fermions.
The modular invariant partition function of the type II string on this orbifold is written in the form as
\eqn{total ZN-orb}, and we shall only focus  on the building block `$Z_{(a,b)}(\tau)$' in \eqn{total ZN-orb},
which includes the contributions from  $\Bbb R^{3,1} \times T^5$
with the  $g^a$($g^b$)-twist along the spatial(temporal) direction.

We consider the orbifolding by the following involutive operator
with $N_0=2$ in the notation of section \ref{sec 2},
\begin{equation}
g:=  (-1)^{F_L+F_R}  \otimes s_R[k+r] \otimes s_L[k+\ell],
\label{g type II}
\end{equation}
where {\em e.g.} `$s_R[m]$' denotes the chiral half-shift along the $m$ directions in $T^5$ and we set
$d+k+\ell+r =5$ $(d,k,\ell,r \geq 0)$.
We  consider the case where
the Narain lattice of $T^5$ can be decomposed as
\begin{align}
\Gamma^{5,5} & = \Gamma^{d,d} \bigoplus
\left[\left(\Lambda^{A_1}_{(0)}\right)^{k+\ell} \oplus \Lambda^{X_{r}}_{(0)} \right] \bigoplus
\left[\left(\Lambda^{A_1}_{(0)}\right)^{k+r} \oplus \Lambda^{X'_{\ell}}_{(0)} \right]
+ \cdots ,
\label{decomp Gamma 66}
\end{align}
with some Lie algebra lattices for $X_r$, $X'_{\ell}$
with rank $r$, $\ell$
in order to make the operator $g$ well-defined.
The standard symbol $(-1)^{F_L}$ ($(-1)^{F_R}$) acts on the left(right)-moving  Ramond sector as  the sign-flip  (`space-time fermion number mod 2').
We assume that $X_r$, $X'_{\ell}$ are composed only of $A_1$,
$D_n$, and
 the total number of the $D_n$  components with $n \in 2\Bbb Z+1$
 in  $X_r$, $X'_{\ell}$
 should be even to maintain unitarity condition {\bf (2)}, as explained shortly.

The building block $Z_{(a,b)}(\tau)$ is explicitly written in terms of the twisted characters given in
section \ref{review asymm} (see also appendix A).
For the  `even sectors', {\em i.e.} sectors with $a,b\in 2\Bbb Z$,
the building blocks
are equal to the partition function of
the  unorbifolded model.
On the other hand,
those for
the `odd sectors',  {\em i.e.} sectors with $a\in 2\Bbb Z+1$ or $b \in 2\Bbb Z+1$,
are non-trivial. They are explicitly written as
\begin{align}
Z_{(a,b)}(\tau) & =
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{T^d}(\tau) \cdot \left|h_{(a,b)}\right|^2 \cdot
\left|\tilde{\chi}^{A_1}_{(a,b)}\right|^{2k}
\nonumber\\
& \hspace{1cm}
\times  \epsilon^{[r]}_{(a,b)} \,
\overline{\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{r}} \, \chi^{X_r}_{(a,b)} \cdot
\epsilon^{[-\ell]}_{(a,b)} \,
\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{\ell} \, \overline{\chi^{X'_{\ell}}_{(a,b)}}
\nonumber\\
& = \frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\, Z^{T^d}(\tau)
\cdot
\left|h_{(a,b)}\right|^2
\nonumber\\
& \hspace{1cm} \times
\epsilon^{[r-\ell]}_{(a,b)} \,
\overline{\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{r+k}} \,
\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{\ell+k} \,
\chi^{X_r}_{(a,b)} \, \overline{\chi^{X'_{\ell}}_{(a,b)}} .
\label{type II ab general}
\end{align}
Here we introduced the free fermion chiral blocks twisted by $(-1)^{F_L}$,
\begin{align}
h_{(a,b)} & := \left\{
\begin{array}{ll}
\left(\frac{\th_3}{\eta}\right)^4 -  \left(\frac{\th_4}{\eta}\right)^4 + \left(\frac{\th_2}{\eta}\right)^4
& ~~ \left(a \in 2\Bbb Z, ~ b \in 2\Bbb Z+1\right) ,
\\
 \left(\frac{\th_3}{\eta}\right)^4 +  \left(\frac{\th_4}{\eta}\right)^4 - \left(\frac{\th_2}{\eta}\right)^4
& ~~ \left(a \in 2\Bbb Z+1, ~ b \in 2\Bbb Z\right) ,
\\
 - \left[ \left(\frac{\th_3}{\eta}\right)^4 +  \left(\frac{\th_4}{\eta}\right)^4 + \left(\frac{\th_2}{\eta}\right)^4 \right]
& ~~ \left(a , ~ b \in 2\Bbb Z+1\right) .
\end{array}
\right.
\end{align}

One can readily confirm the modular covariance,
\begin{align}
\left. Z_{(a,b)}(\tau)\right|_{T} = Z_{(a,a+b)}(\tau),
\hspace{1cm}
\left. Z_{(a,b)}(\tau)\right|_{S} = Z_{(b,-a)}(\tau),
\label{mod cov type II}
\end{align}
which ensures the modular invariance of the total partition function.
In fact, the first line on
the R.H.S of \eqn{type II ab general} is written in the form that is manifestly modular covariant.
We again emphasize the importance of
the phase factor $\epsilon^{[*]}_{(a,b)}$ to achieve this relation.

~

Now, we discuss whether
the model obtained from this setup
satisfies
the requirements {\bf (1)} $\sim$ {\bf (5)} given in the previous section.
First of all, it is easy to check that {\bf (1)} $\sim$ {\bf (3)} are satisfied.
We note that $g^2 = {\bf 1}$ in the untwisted sector, and the operator $(-1)^{F_L+F_R}$
removes, say, all the gravitinos in the spectrum read off from $\displaystyle Z_0(\tau) \equiv \frac{1}{2} \sum_{b\in \Bbb Z_2}\,
Z_{(0,b)}(\tau)$. We also note that $\epsilon^{[*]}_{(0,b)}=1$ \footnote
   {The assumption that `the number of $D_n$-components with $n\in 2\Bbb Z+1$ should be even' was necessary for this statement.}, thus $Z_0(\tau)$ clearly yields a unitary
spectrum.

We next focus on the more non-trivial requirement {\bf (5)}.
For the `even sectors' with $a,b \in 2\Bbb Z$,
the building blocks $Z_{(a,b)}(\tau)$
vanish as they
are the same as the partition function of the original supersymmetric
model.
We thus need to show that the summation over the `odd sectors'
with $a \in 2\Bbb Z+1$ or $b \in 2\Bbb Z+1$ vanishes;
\begin{equation}
\sum_{\msc{$a \in 2\Bbb Z+1$ or $b\in 2\Bbb Z+1$}}\,  Z_{(a,b)} (\tau) =0.
\label{sum odd sec}
\end{equation}
To do so it is sufficient to confirm the  following `criterion'
\begin{equation}
\sum_{b\in 2\Bbb Z \cap \Bbb Z_{16}}\, Z_{(a,b)} (\tau) =0
\hspace{1cm}
\left({}^{\forall} a \in 2\Bbb Z+1\right) .
\label{criterion}
\end{equation}
Indeed, if this is the case, we readily obtain \eqn{sum odd sec}
due to the modular covariance \eqn{mod cov type II}
\footnote{
Almost the same argument has been  used in \cite{AS,AS2} in a different context.
}.

Therefore, the question is what type of the
Lie algebra lattices for $X_r$, $X'_{\ell}$
in \eqn{type II ab general}
can satisfy the criterion \eqn{criterion}.
We note that
each term of the theta functions
appearing in
$Z_{(a,b)} (\tau)$
contains the various phase factors of the form
$
e^{\frac{i\pi}{8} K a b}
$
with $K \equiv r- \ell ~ (\mod \, 2)$.
(Recall the definitions of phase $\epsilon^{[*]}_{(a,b)}$ and the twisted characters
$\tilde{\chi}^{A_1}_{(a,b)}$, $\chi^{X_r}_{(a,b)}$ given in appendix A.
)
Thus,
it is enough that all the theta function terms contain a {\em non-vanishing\/} phase factor in order
to satisfy \eqn{criterion}.
Although it seems hard to fully classify the general choices of $X_r$, $X'_{\ell}$ satisfying \eqn{criterion},
the following two simple cases are obviously sufficient\footnote
   {However, we would like to emphasize that these two are sufficient but not necessary.
Namely, there are many examples
 that satisfy \eqn{criterion} apart from these cases {\bf (i)} and {\bf (ii)}.
};
\begin{description}
\item[(i)]
$\ell=0$, $r \not\in 8\Bbb Z $ ~($r=0$, $\ell \not\in 8\Bbb Z)$, and
$X_r$ ($X'_{\ell}$) is simple ({\em i.e.} made up only of
a single piece of $A_1$ (when $r=1$) or $D_r$).

\item[(ii)]
$r-\ell \in 2\Bbb Z+1$.

\end{description}

Finally, let us examine the requirement {\bf (4)}.
At first glance, it seems that the tachyonic behavior would appear due to the fermion chiral block $h_{(a,b)}$ with $a \in 2\Bbb Z+1$.
However,
once \eqn{criterion} is satisfied,
by using its T-transformation
 we eventually obtain
$$
\sum_{b\in \Bbb Z_{16}}\, Z_{(a,b)} (\tau) = 0.
$$
Moreover, it is easy to show that we do not have any tachyonic modes in all the sectors of $a\in 2\Bbb Z$
from \eqn{criterion}.

~

\subsection{Heterotic Models}

We next try to construct the models
based on the heterotic string
compactified on $T^5 \times S^1_{N\epsilon}$, where the left-mover is given by the 26 dimensional bosonic
theory.
We shall only consider the $E_8\times E_8$-cases, and the $SO(32)$-cases can be similarly treated.

We take the orbifold action
\begin{align}
g & :=  (-1)^{F_R}  \otimes s_R[k+r] \otimes
s_L[k+\ell_1+\ell_2+\ell_3]
\nonumber\\
& \equiv (-1)^{F_R}  \otimes s_R[k+r] \otimes s_L[k+\ell_1] \otimes s_L[\ell_2] \otimes s_L[\ell_3] ,
\label{g hetero}
\end{align}
where all the  notation is  defined as in \eqn{g type II}.
Here, we assume
$d+k+\ell_1+r =5$, $\ell_2, \ell_3 \leq 8$.
$s_R[k+r]$, $s_L[k+\ell_1]$ act along the $T^5$-directions, while
$s_L[\ell_2]$, $s_L[\ell_3]$ are assigned to the two $E_8$-factors.
We consider the case where the total Narain lattice is
decomposed as
\begin{align}
\Gamma^{21,5}  = & \Gamma^{d,d} \bigoplus
\left[\left(\Lambda^{A_1}_{(0)}\right)^{k+\ell_1+\ell_2+\ell_3} \oplus \Lambda^{X_{r}}_{(0)} \oplus
\Lambda_{(0)}^{X_{8-\ell_2}} \oplus
\Lambda_{(0)}^{X_{8-\ell_3}}\right]
\nonumber\\
&
 \bigoplus
\left[\left(\Lambda^{A_1}_{(0)}\right)^{k+r} \oplus \Lambda^{X'_{\ell_1}}_{(0)} \right]
+ \cdots .
\label{decomp Gamma 22 6}
\end{align}
Here,
$X_r$, $X'_{\ell_1}$ are composed only of $A_1$, $D_n$, and
$X_{8-\ell_2}$, $X_{8-\ell_3}$ can contain the pieces of
$A_1$, $D_n$, $E_7$, or $E_8$.
The chiral half-shifts $s_L[*]$, $s_R[*]$ are associated to the $A_1$-pieces $\left(\Lambda_{(0)}^{A_1}\right)^*$.
We again require that
the number of total $D_n$ components with $n \in 2\Bbb Z+1$
in the left- and right-movers
should be even
in order to satisfy the `unitarity condition' {\bf (3)}.

The building blocks for the even sectors
are again equal to the partition function of the unorbifolded model,
and those of the odd sectors are evaluated as
\begin{align}
Z_{(a,b)}(\tau) & =
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{T^d}(\tau)
\cdot \overline{h_{(a,b)}} \cdot
\left|\tilde{\chi}^{A_1}_{(a,b)}\right|^{2k}
\nonumber\\
& \hspace{1cm}
\times  \epsilon^{[r]}_{(a,b)} \,
\overline{\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{r}} \, \chi^{X_r}_{(a,b)} \cdot
\epsilon^{[-\ell_1]}_{(a,b)} \,
\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{\ell_1} \, \overline{\chi^{X'_{\ell_1}}_{(a,b)}}
\nonumber\\
& \hspace{1cm}
\times \prod_{i=2,3}\, \left[\epsilon^{[8-\ell_i]}_{(a,b)} \, (-1)^{ab}\, \left(\tilde{\chi}_{(a,b)}^{A_1} \right)^{\ell_i} \, \chi^{X_{8-\ell_i}}_{(a,b)} \right]
\nonumber\\
& = \frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\, Z^{T^d}(\tau)
\cdot
 \overline{h_{(a,b)}}
\nonumber\\
& \hspace{1cm} \times
\epsilon^{[r-\sum_{i} \ell_i]}_{(a,b)} \,
\overline{\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{k+r}} \,
\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{k+\sum_i \ell_i} \,
\chi^{X_r}_{(a,b)} \, \overline{\chi^{X'_{\ell_1}}_{(a,b)}} \,
\prod_{i=2,3}\, \chi^{X_{8-\ell_i}}_{(a,b)} .
\label{hetero ab general}
\end{align}
The first line of \eqn{hetero ab general} shows its modular covariance manifestly\footnote
   {Note that $(-1)^{ab} \left[\tilde{\chi}^{A_1}_{(a,b)}(\tau)\right]^8$ behaves  modular covariantly upto the phase factor
   arising from the $T$-transformation of $\eta(\tau)^{-8}$. }.
One can likewise show that the total partition function of this model generally satisfies {\bf (1)} $\sim$ {\bf (3)},
and the requirements {\bf (4)}, {\bf (5)}
are achieved if the condition \eqn{criterion} is satisfied.
As in the type II models, we can exhibit the two simple cases satisfying \eqn{criterion}, though the general classification
would be hard to describe;
\begin{description}

\item[(i)]
Only one of $\{ r, \ell_1, \ell_2, \ell_3\}$ is non-zero (mod 8), and the Lie algebra lattice corresponding to the non-vanishing integer
({\em e.g.}  $X_{8-\ell_2}$ when $\ell_2 \not\equiv 0 ~ (\mod \, 8)$)  is simple ({\em i.e.} made up only of
a single piece of $A_1$, $D_r$ or $E_7$).

\item[(ii)]
$\displaystyle r-\sum_{i=1}^3 \ell_i \in 2\Bbb Z+1$.

\end{description}

~

\subsection{Simple Examples}
\label{subsec: ex}

Here we present simple examples to demonstrate the general features given in the previous subsections.

~

\noindent
{\bf 1. Example from type II string on
$T^4[D_4] \times S^1 \times S^1_{N\epsilon}$ : }

~

The first example we consider is
a particle model from
the type II string on
$
\Bbb R^{3,1} \times   T^4[D_4] \times S^1 \times S^1_{N\epsilon},
$
where
$T^4[D_4]$ is the 4-dim. torus for
the Englert-Neveu lattice of $D_4$, in other words,
at the $SO(8)$-symmetry enhancement point.
$S^1$ is a circle with an arbitrary radius,
which is not important below.
We simply choose $k=\ell=0$, $r=4$, that is, the orbifold action is defined as
\begin{equation}
g= (-1)^{F_L+F_R} \otimes s_R[4],
\label{g ex1}
\end{equation}
where $s_R[4]$ acts on $T^4[D_4]$.

The building blocks \eqn{type II ab general} for the odd sectors
with  $a \in 2\Bbb Z+1$ or $b\in 2\Bbb Z+1$ are written in this case as
\begin{align}
Z_{(a,b)}(\tau) & =
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot \left|h_{(a,b)}\right|^2
\cdot
  \epsilon^{[4]}_{(a,b)} \,
\overline{\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{4}} \, \chi^{D_4}_{(a,b)} .
\label{ex1 ab general}
\end{align}
Especially, for the sectors with $a \in 2\Bbb Z+1$, $b\in 2\Bbb Z$,
we can explicitly write it down as
\begin{align}
Z_{(a,b)}(\tau) & =
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot
\left|
\left(\frac{\th_3}{\eta}\right)^4 + \left(\frac{\th_4}{\eta}\right)^4 -\left(\frac{\th_2}{\eta}\right)^4
\right|^2
\nonumber\\
& \hspace{1cm} \times  (-1)^{\frac{b}{2}}\,
\overline{\left(\frac{\th_3 \th_2}{\eta^2}\right)^2} \,
\frac{1}{2} \left[\left(\frac{\th_3}{\eta}\right)^4 + \left(\frac{\th_2}{\eta}\right)^4\right] .
\label{ex1 ab odd}
\end{align}
Due to the existence of
the phase factor $(-1)^\frac{b}{2}$ the criterion \eqn{criterion} is obviously satisfied;
$$
\sum_{b\,:\,\msc{even}}\, Z_{(a,b)}(\tau) =0.
$$

On the other hand, the partition function of the untwisted sector ($a=0$) is evaluated as
\begin{align}
\hspace{-0.5cm}
Z_0(\tau)
& = \frac{1}{2}
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot
\left|
\left(\frac{\th_3}{\eta}\right)^4 - \left(\frac{\th_4}{\eta}\right)^4 - \left(\frac{\th_2}{\eta}\right)^4
\right|^2
\frac{1}{2} \left[ \left|\frac{\th_3}{\eta}\right|^8
+ \left|\frac{\th_4}{\eta}\right|^8
+ \left|\frac{\th_2}{\eta}\right|^8
\right]
\nonumber\\
& + \frac{1}{2}
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot
\left|
\left(\frac{\th_3}{\eta}\right)^4 - \left(\frac{\th_4}{\eta}\right)^4 +\left(\frac{\th_2}{\eta}\right)^4
\right|^2
\overline{\left(\frac{\th_3 \th_4}{\eta^2}\right)^2} \,
\frac{1}{2} \left[\left(\frac{\th_3}{\eta}\right)^4 + \left(\frac{\th_4}{\eta}\right)^4\right] ,
\nonumber\\
&
\label{ex1 a=0 b}
\end{align}
which does not identically vanish.
In fact,
this partition function describes
128 massless space-time bosons in the standard fashion, while all the space-time fermions get massive.
Especially, all of the space-time fermions belonging to the massless super-multiplets
in the unorbifolded theory are removed by the $g$-projection, which means that they
acquire the masses of order $\displaystyle M_{\epsilon} \sim
\frac{1}{\epsilon \sqrt{\alpha'}} $ in the theory ${\cal M}[\epsilon]$.
(See the comment {\bf (ii)} in section 2.)
Thus, the bose-fermi cancellation is `maximally' broken in
${\cal M}[\epsilon]$.
Nevertheless, the cosmological constant $\Lambda_{{\cal M}[\epsilon]}$ vanishes under the $\epsilon\, \rightarrow\, +0$ limit, as we have proven in section 2.

~

\noindent
{\bf 2. Counter example from type II string on
$ T^4[D_2 \oplus D_2] \times S^1 \times S^1_{N\epsilon}  $}

~

As a digression, here we give an example which does not
satisfy our criterion.
We adopt almost the same background as the first one, but with the $T^4[D_4]$ replaced with
\begin{equation}
T^4[D_2 \oplus D_2] \left( \cong T^4 [A_1 \oplus A_1 \oplus A_1 \oplus A_1] \right),
\end{equation}
and the orbifold action is again given by \eqn{g ex1}.

The total building blocks \eqn{type II ab general} now become
\begin{align}
Z_{(a,b)}(\tau) & =
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot \left|h_{(a,b)}\right|^2
\cdot
  \epsilon^{[4]}_{(a,b)} \,
\overline{\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{4}} \, \left[\chi^{D_2}_{(a,b)} \right]^2,
\label{ex2 ab general}
\end{align}
and  we  obtain for the sectors with $a \in 2\Bbb Z+1$, $b\in 2\Bbb Z$,
\begin{align}
Z_{(a,b)}(\tau) & =
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot
\left|
\left(\frac{\th_3}{\eta}\right)^4 + \left(\frac{\th_4}{\eta}\right)^4 -\left(\frac{\th_2}{\eta}\right)^4
\right|^2
\nonumber\\
& \hspace{1cm} \times  (-1)^{\frac{b}{2}}\,
\overline{\left(\frac{\th_3 \th_2}{\eta^2}\right)^2} \,
\frac{1}{4} \left[\left(\frac{\th_3}{\eta}\right)^4 + \left(\frac{\th_{2}}{\eta}\right)^4
+ 2 (-1)^{\frac{b}{2}} \left(\frac{\th_3 \th_2}{\eta^2}\right)^2\right] ,
\label{ex2 ab odd}
\end{align}
in place of \eqn{ex1 ab odd}.
We thus find that
$$
\sum_{b\,:\,\msc{even}}\, Z_{(a,b)}(\tau)
\propto
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot
\left|
\left(\frac{\th_3}{\eta}\right)^4 + \left(\frac{\th_4}{\eta}\right)^4 -\left(\frac{\th_2}{\eta}\right)^4
\right|^2
\cdot \left| \frac{\th_3 \th_2}{\eta^2}\right|^4 \neq 0,
$$
because of the cancellation of the phase factors in the coefficient of the term $\displaystyle \left| \frac{\th_3 \th_2}{\eta^2}\right|^4$
appearing in \eqn{ex2 ab odd}.
In this way we conclude that  the criterion \eqn{criterion} is not satisfied in this case.

~

\noindent
{\bf 3.
Example from heterotic string on
$ T^4[D_4] \times S^1 \times S^1_{N\epsilon}$ :  }

The third example is a model from the heterotic string on
$
\Bbb R^{3,1} \times T^4[D_4] \times S^1 \times S^1_{N\epsilon},
$
and
\begin{equation}
g= (-1)^{F_R} \otimes s_R[4],
\label{g ex 3}
\end{equation}
where $s_R[4]$ acts on $T^4[D_4]$.
The total building blocks \eqn{hetero ab general} are written as
\begin{align}
Z_{(a,b)}(\tau) & =
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot \overline{h_{(a,b)}}
\cdot
  \epsilon^{[4]}_{(a,b)} \,
\overline{\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{4}} \, \chi^{D_4}_{(a,b)} \cdot \left[\chi^{E_8}_0\right]^2 .
\label{ex3 ab general}
\end{align}
Here,
$\chi^{E_8}_0$ denotes the character of the basic representation of affine $E_8$ with level 1 given in \eqn{ch E8}.
We find that the sectors with $a\in 2\Bbb Z+1$, $b\in 2\Bbb Z$ satisfy the criterion \eqn{criterion} due to the phase factor $\epsilon^{[4]}_{(a,b)}$.
We thus obtain the vanishing cosmological constant $\displaystyle \lim_{\epsilon\, \rightarrow \, +0} \Lambda_{{\cal M}[\epsilon]} =0$,  as in the first example.

Moreover, we obtain for the untwisted sector
\begin{align}
\hspace{-0.5cm}
Z_0(\tau)
& = \frac{1}{2}
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot
\overline{
\left[
\left(\frac{\th_3}{\eta}\right)^4 - \left(\frac{\th_4}{\eta}\right)^4 - \left(\frac{\th_2}{\eta}\right)^4
\right]
}
\nonumber\\
& \hspace{2cm} \times
\frac{1}{2} \left[ \left|\frac{\th_3}{\eta}\right|^8
+ \left|\frac{\th_4}{\eta}\right|^8
+ \left|\frac{\th_2}{\eta}\right|^8
\right]
\cdot\left[\chi^{E_8}_0 \right]^2
\nonumber\\
& + \frac{1}{2}
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot
\overline{
\left[
\left(\frac{\th_3}{\eta}\right)^4 - \left(\frac{\th_4}{\eta}\right)^4 +\left(\frac{\th_2}{\eta}\right)^4
\right]
}\,
\nonumber\\
& \hspace{2cm} \times
\overline{\left(\frac{\th_3 \th_4}{\eta^2}\right)^2} \,
\frac{1}{2} \left[\left(\frac{\th_3}{\eta}\right)^4 + \left(\frac{\th_4}{\eta}\right)^4\right]
\cdot\left[\chi^{E_8}_0 \right]^2 .
\label{ex3 a=0 b}
\end{align}
This does not vanish, meaning that we do not have the bose-fermi cancellation in the theory ${\cal M}[\epsilon]$ again.

As is familiar, there appear various non-abelian gauge symmetries in the heterotic string models, in contrast with the type II cases,
which originate from the left-moving vertex operators with
conformal weight $h=1$.
In the present case the relevant orbifold action \eqn{g ex 3} does not affect such bosonic massless spectrum lying in
$Z_0(\tau) $ and also the $S^1_{N\epsilon}$-sector omitted here.
We then find the non-abelian gauge symmetry
$U(1) \times U(1) \times SO(8)\times E_8 \times E_8$
(at generic points of the moduli space of the $S^1$-sector), while we always have an abelian gauge group $U(1)^6$ that originates from the right-mover.
On the other hand,  no massless space-time fermions appear
in the manner similar to  the first example.

~

\noindent
{\bf 4. Example from heterotic string on
$ T^4[D_4] \times S^1 \times S^1_{N\epsilon}$ :  }

The fourth example is defined  for
the same background of example 3,
$
\Bbb R^{3,1} \times T^4[D_4] \times S^1 \times S^1_{N\epsilon},
$
but we here take
\begin{equation}
g= (-1)^{F_R} \otimes s_L[4],
\label{g ex 4}
\end{equation}
instead of \eqn{g ex 3} as the orbifold action.
Then,
the total building blocks \eqn{hetero ab general} become
\begin{align}
Z_{(a,b)}(\tau) & =
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot \overline{h_{(a,b)}}
\cdot
  \epsilon^{[-4]}_{(a,b)} \,
\left(\tilde{\chi}^{A_1}_{(a,b)}\right)^{4} \, \overline{\chi^{D_4}_{(a,b)}} \cdot \left[\chi^{E_8}_0\right]^2 ,
\label{ex4 ab general}
\end{align}
which again satisfy the criterion \eqn{criterion}.

The untwisted partition function becomes
\begin{align}
\hspace{-0.5cm}
Z_0(\tau)
& = \frac{1}{2}
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot
\overline{
\left[
\left(\frac{\th_3}{\eta}\right)^4 - \left(\frac{\th_4}{\eta}\right)^4 - \left(\frac{\th_2}{\eta}\right)^4
\right]
}
\nonumber\\
& \hspace{2cm} \times
\frac{1}{2} \left[ \left|\frac{\th_3}{\eta}\right|^8
+ \left|\frac{\th_4}{\eta}\right|^8
+ \left|\frac{\th_2}{\eta}\right|^8
\right]
\cdot\left[\chi^{E_8}_0 \right]^2
\nonumber\\
& + \frac{1}{2}
\frac{1}{\tau_2 \left|\eta(\tau)\right|^4}\,
Z^{S^1}(\tau) \cdot
\overline{
\left[
\left(\frac{\th_3}{\eta}\right)^4 - \left(\frac{\th_4}{\eta}\right)^4 +\left(\frac{\th_2}{\eta}\right)^4
\right]}\,
\nonumber\\
& \hspace{2cm} \times
\left(\frac{\th_3 \th_4}{\eta^2}\right)^2 \,
\frac{1}{2} \overline{ \left[\left(\frac{\th_3}{\eta}\right)^4 + \left(\frac{\th_4}{\eta}\right)^4\right] }
\cdot\left[\chi^{E_8}_0 \right]^2 ,
\label{ex4 a=0 b}
\end{align}
which again does not vanish.

This time, the unbroken gauge symmetry originating from the left-mover is given as
$U(1) \times U(1) \times SO(4)\times SO(4) \times E_8 \times E_8$,
that is, the $SO(8)$ gauge symmetry is broken to $SO(4) \times SO(4)$ by the relevant orbifolding \eqn{g ex 4}.
On the other hand, as opposed to the third example, we have massless space-time fermions which belong to the vector representation for
both factors of the $SO(4)$ gauge groups\footnote
  {The simplest way to observe these aspects is as follows;
The $T^4[D_4]$ sector is fermionized in the standard manner, that is, described by the 8 pairs of chiral fermions $(\lambda_L^i, \, \lambda_R^i)$ ($i=1,\ldots,8$),
and $s_L[4]$ is just regarded  as the sign-flip of the 4 left-moving fermions, say, $\lambda_L^i$, $i=1,\ldots, 4$.
Then, one can easily understand the massless spectrum mentioned here.
}.

~

\section{Discussions and Comments}

In this paper we have demonstrated how one can systematically construct the {\em point particle theories}
that realize the vanishing one-loop cosmological constant without the bose-fermi cancellation, namely,
the theories with the property \eqn{vanishing cc particle}.
The main idea to reach the desired theories is to utilize the building blocks of
certain asymmetric orbifolds of supersymmetric string vacua, in which all the space-time supercharges are removed  at least in the untwisted sectors.
We have interpreted
the partition function for
these untwisted sectors,
which is not modular invariant,
as  that  of the particle theories satisfying \eqn{vanishing cc particle}.
We have clarified the several conditions to be satisfied by the relevant orbifold models,
and assumed the consistency of the  superstring theories on these orbifolds.
Especially, we required the world-sheet superconformal symmetry in the RNS formalism.

However,
if one focuses only on the spectrum,
it is possible to relax the consistency
conditions of the original superstring theories, or even not to
start from superstring theories.
In such cases, the possibility of the construction
is largely enhanced, since appropriate modular forms
can be used as building blocks,
regardlessly of the consistency at intermediate stages.

An important issue here is the consistency {\em after interactions are turned on.}
If the resultant particle theories are still consistent,
we may adopt them,
as our purpose is to construct the non-SUSY {\em particle theories\/}
rather than the non-SUSY string vacua.
We have, however,  considered the particle theories
which descend from consistent superstring theories
in this paper,
since  the consistency of the latter
may be inherited by the former.

At one loop, for example, the multi-particle scattering amplitudes must be compatible with the requirement of unitarity.
It would be possible to reduce this issue to that in superstring theory.
Indeed, the superstring multi-particle amplitudes
are given by the integral of
the contributions from each spin structure over the torus modulus
 (see e.g. \cite{Green:1987mn,DHoker:1988pdl}).
 Each contribution is
 invariant under the modular subgroup $\Gamma(2)$,
which preserves the spin structures. Furthermore,
it is decomposed into
the factors coming from the partition functions and from the
vertex operators, and those factors are separately modular
covariant.
One may thus expect that the consistency
of the relevant amplitudes in the particle theories
is deduced from
that of superstring
by an argument similar to the one given in section \ref{sec 2},
with the help of the modular property of the amplitudes.
It would be also possible to confirm the UV-finiteness of the multi-particle amplitudes
in our particle theories,
even though the integration region of the modulus (or Schwinger parameter) is
${\cal S}$, rather than ${\cal F}$, as for the cosmological constant.
We would like to discuss these issues in more detail elsewhere.

~

Let us further add several comments:
\begin{description}
\item[(i)]
We have demonstrated the two types of constructions based on type II and heterotic strings by utilizing the building blocks
studied in \cite{SSLie}. They might look quite similar. However, we have a crucial difference between them.

In the type II cases, the spectra read off from $Z_0(\tau)$ (and thus, the theory ${\cal M}[\epsilon]$) do not include
massless space-time fermions,
while the heterotic cases can do. This aspect originates from the fact  that
the chiral half-shift only acts on the bosonic coordinates.
In fact, the possible massless fermions
for the type II cases
should be of the form
$$
\ket{\mbox{R-vacuum}}_L \otimes \psi^i_{R,\,-1/2} \ket{0}_R, ~~ \mbox{or} ~~
 \psi^i_{L,\,-1/2} \ket{0}_L \otimes \ket{\mbox{R-vacuum}}_R .
$$
However, both of them are projected out by the orbifold action \eqn{g type II}, since
the chiral half-shifts $s_L[*]$, $s_R[*]$ assign +1 to any NS massless states
in addition to $ (-1)^{F_L+F_R} = -1$.

On the other hand, in the heterotic models, one can obtain
 the massless fermionic states of the form
$$
\ket{p_L}_L \otimes \ket{\mbox{R-vacuum}}_R,
$$
where $\ket{p_L}_L$ is any massless Fock vacuum ({\em i.e.} $h=1$) with $s_L[*]=-1$.
(See example 4
presented in subsection \ref{subsec: ex}.)

~

\item[(ii)]
As mentioned above, the models we constructed satisfy the unitarity condition {\bf (3)},
that is, the unitarity
in the untwisted sector
with $a=0$ in $Z_{(a,b)}$, which is easily confirmed.
We need not impose the same condition
in the twisted sectors with $a\neq 0$
at least at the level of the spectrum, though the condition for $a\neq 0$ is also
satisfied in our examples.

~

\item[(iii)]
We here only treated
the twist for the world-sheet fermions
by $(-1)^{F_L+F_R}$ in the type II cases and
by $(-1)^{F_R}$ in  the heterotic cases
in order to break the space-time SUSY,
in other words, to achieve the condition {\bf (2)},
$Z_0(\tau) \neq 0$.
As another possibility to break SUSY, one can also make use of the chiral reflections
$$
(X_L^i, \, X_R^i) ~ \longrightarrow (-X^i_L, \, X^i_R),
\hspace{1cm}
(\psi_L^i, \, \psi_R^i) ~ \longrightarrow (-\psi^i_L, \, \psi^i_R),
$$
or those for the right-mover.
The fermionic building blocks for the chiral reflection acting on the
$2p$ components $(\-_L)^{\otimes 2p}$ are explicitly given in
appendix A and denoted as $g^{[p]}_{(a,b)}(\tau)$. They do not identically vanish for $p=1, 3$, which means the absence of  the bose-fermi cancellation.

One can construct more elaborated
models satisfying the requirements {\bf (1)} $\sim$ {\bf (5)}
by combining the chiral half-shifts and the chiral reflections.
The crucial point is that the total building blocks
$Z_{(a,b)}(\tau)$ for the  odd sectors
should include the non-trivial phase factors
such as $e^{\frac{i \pi}{8} K ab}$,
especially to satisfy the condition {\bf (5)}.
To this aim the inclusions of the chiral half-shifts are useful, since the phases that originate from the chiral reflections
tend to be canceled out after combining the bosonic and fermionic building blocks. (Compare \eqn{ep r ab} and \eqn{gp ab}.)
We also note that the massless fermions can appear in these models
{\em even in the type II cases}. This is in a sharp contrast with those constructed only with the chiral half-shifts
and $(-1)^{F_L+F_R}$ mentioned in the comment {\bf (i)}.

\end{description}

~

\subsection*{Acknowledgments}
We would like to thank S. Iso for useful discussion.
This work is supported in part by
JSPS KAKENHI Grant Number JP17K05406.

\newpage

redpajama_set_name: RedPajamaArXiv


This week I received my new wrist watch case from Zeugma Straps. This is definitely an exceptional piece of craftsmanship, most likely rivaled by none. Nuri (Zeugma Straps) tells me that his company made 120 of these cases and that they worked 10 months to complete the task.
The finest thick and sturdy leather was used to create a unique product of the highest quality. Every case is completely hand crafted and uniquely numbered on the bottom side of the case (see photo below). No more than the current 120 cases will be produced.
The case is closed by two large strap loops as shown below.
The case safely holds up to eight large wristwatches. The inside of the lid is made from suede and is heavily padded. The board that holds the watch cushions can be taken out of the case eg to store the board in your safe or you could even use the case as a regular briefcase. If you're interested in the watches shown - from left to right Panerai PAM233, PAM360, PAM2A, PAM243.
A few more photos to show the excellent quality and fine craftsmanship. You can click the photos for a larger version.
If you are looking for a high quality hand made watch case then stop looking. You won't find anything better than this. If you want to buy one of these cases you can contact Nuri by sending an e-mail to info@zeugmastraps.com. Mind you, quality comes at a price - the cases sell for 590 USD (paypal) or 540 USD if you pay with Western Union. Considering that these cases are hand made, limited and that the finest quality leather has been used - it's a bargain. The prices include worldwide shipping with DHL.
All photos © 2011 M.Wilmsen - do not use my photos without permission.
Beauty! I'm crazy about leather anything, especially with such good make! Do they ship internationally??
Hi, those beautiful pictures do look as though they do go some way toward doing that beautiful case justice!
Quick question, are only the watch cushions removable or is the whole board displaying the watches removable as well (i.e. so that the watch case can also be used like a regular briefcase as well)?
Actually, you can take the board out of the case. I just edited the text of my post to reflect this, because it was a good question! Thanks.
Just saw one of those beauties at P.day and it sold for 1600$ at the auction!
Really? Wow! Great for the auction!
Can someone please list the outside dimensions of the case?


Sunday Morning, April 9 (Palm Sunday): We will be continuing with our series on Paul's letter to the Galatians, and considering Paul's declaration, "cursed is everyone who hangs on a tree" from Galatians 3:1-14). Our worship service begins at 10:30 a.m.
Sunday Afternoon: We are in that section of the Heidelberg Catechism dealing with the Lord's Supper (Lord's Day 28, Q & A 75-77). Our catechism service begins @ 1:15 p.m.
Wednesday Night Bible Study, April 5, (7:30 p.m.): We continue our series on personal evangelism entitled, "Telling the Truth in Love." We will be considering the spirit of the age and the difficultly this presents in evangelism.
Academy, Friday, April 7, (7:30 p.m.): We will be viewing and discussing Allen Guelzo's Teaching Company Course, The American Mind. Our text for this series will be Hollinger and Capper, The American Intellectual Tradition. Be sure to get a used copy! They are much cheaper!


In just a few more months, rock fans are in for the show of a lifetime when Motley Crue and Kiss hit the road together on their co-headlining trek, The Tour. The 40-date tour will kick off on July 20 in Bristow, Va. and is promised to be packed full of the same edge that both bands have always had.
And speaking of Aerosmith, they will also be hitting the road this summer on their own co-headlining venture with Cheap Trick, which kicks off in June.


text:
De Blasio defends bike lanes at Queens town hall, but says he's open to looking at problems
By Suzanne Monteverdi Posted on November 3, 2017
Photos by Suzanne Monteverdi/QNS
Ahead of next week's election, Mayor Bill de Blasio headed to Queens Village on Thursday night for a three-hour town hall meeting with residents.
In front of a full audience in the gymnasium of Martin Van Buren High School, the mayor and commissioners of various city agencies fielded individual questions on a number of topics, including immigration, over-development and bike lanes.
The city's Department of Transportation is currently working on two protected bike lane projects in Douglaston and Bayside: along Northern Boulevard between 223rd Street to Douglaston Parkway and along the western edge of Alley Pond Park, as well as along Oceania Street and 210th Street.
Attendees on both sides of the issue spoke up at the Nov. 2 event. The first person to breach the subject told the mayor how much she appreciates both bike lane projects — which was immediately met with both boos and cheers from some in the crowd.
The resident told the mayor the bike lane creates a safer condition for cyclists using the busy roadway and said she looks forward to more bike lane projects in the future.
After assuring those making heckling comments "everyone's gonna get heard," de Blasio said he believes bike lanes "done right" offer important transit alternatives, protect the environment and reduce congestion.
"I also want to say, we look at [bike lanes] constantly to see if they're working," he continued. "If something's not working, we also have to have the decency to say, 'Hey, that didn't work,' and we'll make the change."
A representative from Bayside's Windsor Park co-op followed up, stating the community was not given adequate notice about the project, which "displaced" hundreds of residents' cars.
De Blasio promised the resident that DOT Commissioner Polly Trottenburg would come to the site to "see it with her own eyes." Trottenburg, who was in attendance, agreed.
About a half hour later, the Northern Boulevard bike lane was brought up again by another resident. The bike lane there has been under scrutiny by certain Community Board 11 members, residents and state Senator Tony Avella, who has been particularly vocal in his opposition to the current DOT plan, since June.
Earlier in the week, the lawmaker commented on an accident at the site involving a school bus carrying special needs adults. While there were no reported injuries, the lawmaker said the accident has left the community "worried that the next incident will not be so lucky."
The next speaker at the town hall said while he "philosophically" agreed with the bike lane and the protection it provides cyclists, he believes it was "not well thought out." Since the installation of the bike lane — which eliminated one lane of westbound traffic — the resident's commute has increased from two minutes to 20 minutes, he told the mayor.
"[Northern Boulevard] is a major artery throughout northeastern Queens," he said.
De Blasio and Trottenburg again assured attendees that the city would "continue to monitor [the lane]." The DOT commissioner also reminded the crowd that a fatality had occurred at each of the sites, triggering the city to study the area and make the decision to install the protected lanes.
"If it really isn't working, we'll continue to make adjustments," Trottenburg said.
On the topic of the property tax system, which de Blasio called both "a mess" and "inconsistent," the mayor promised attendees reform efforts would be a top priority should he be re-elected. He also reminded attendees multiple times that property assessments, not rates, have increased under his administration.
De Blasio started the evening touting his approach to affordable housing, a property tax break for seniors and the city's low crime numbers. He also held a moment of silence for the victims of the recent terror attack in Manhattan, which left eight people dead.
Prior to the town hall event, City Council candidate Joseph Concannon and a group of protesters assembled outside of the venue. Upon the mayor's arrival, the outspoken group began chants of "lock him up" and "dump the mayor."
Free Fun Day with Soccer Shots at Hunter's Point – LIC
Hunter's Point South Park

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to access an Array containing hash-like objects I received  an array from an API that appears to contain, for lack of a better phrase, object-style notation.
Since it is an array, I cannot obviously access it by a key or value.  I don't see a method in this particular object type that allows me to convert it to a hash, or JSON and then to a hash.
What is the most appropriate way of converting this to a hash where I can access the data by key and value?
Output:
 [#<ObjectType::ObjectRef:0x30f2862
 @_ref="record:full/data/location/setting/partition",
 @configured=false,
 @name="John Doe",
 @addr="10.10.10.10">]

Alternatively, if this can be converted to an array with multiple elements (instead of one big chunked together element), I could parse it to CSV and filter outside of Ruby.  The elements in the new array would contain the "values" (e.g. false, "John Doe", "10.10.10.10).

A: Try this:
array # =>  [#<ObjectType::ObjectRef:0x30f2862
             @_ref="record:full/data/location/setting/partition",
             @configured=false,
             @name="John Doe",
             @addr="10.10.10.10">]
array.map { |a| {configured: a.configured, name: a.name, addr: a.addr} }

# or if you can't access the instance variables
array.map do |a|
  {
    configured: a.instance_variable_get(:@configured),
    name: a.instance_variable_get(:@name),
    addr: a.instance_variable_get(:@addr)
  }
end # => [{configured: false, name: "John Doe", addr: "10.10.10.10"}]

# and if you want an array
array.map do |a|
  [
    a.instance_variable_get(:@configured),
    a.instance_variable_get(:@name),
    a.instance_variable_get(:@addr)
  ]
end # => [[false, "John Doe", "10.10.10.10"]]

redpajama_set_name: RedPajamaStackExchange


text:
allmediascotland...Your key to the media > Media News > Prince Giveaway Adds 45,000 to Record Sales
Prince Giveaway Adds 45,000 to Record Sales
By Administrator in Media News, Press · July 13, 2010 · No comments
Sales of the Daily Record were boosted by 45,000, following a giveaway on Saturday of a new album by pop star, Prince.
The promotion of 20TEN also saw UK sister title, the Daily Mirror, enjoy a 334,000 sales boost.
The 10 track CD was given away free to readers in the Daily Mirror and Daily Record after a week-long marketing campaign, including Prince's first British interview for more than a decade.
The reclusive singer gave the Mirror's associate editor. Peter Willis, a tour of his Paisley Park studios, claiming that the "internet is completely over" and defended his decision to boycott websites including iTunes and YouTube. His controversial comments sparked debate around the world – including on websites such as CNN and ABC – and the interview has become one of the most read articles of the year on the Mirror website mirror.co.uk.
The Daily Record's average sale in May – according to latest Audit Bureau of Circulation figures – was 300,800.
In a statement, Mark Hollinshead, managing director of Trinity Mirror Nationals, said: "We are delighted with the huge response to the Prince promotion which has proved to be a successful sampling opportunity for both titles.
"This was also a great example of our editorial and commercial teams working closely together which resulted in the most effective way for Prince to reach his legion of UK fans."
« Herald Diarist Enjoys 'Grave' Error
Scheme Set to Return to TV Screens »


Data often have structure that needs to be modeled explicitly. For example, when investigating students' outcomes we need to account for the fact that students are nested inside classes that are in turn nested inside schools. If we are watching students develop over time, we need to account for the dependence of measurements across time. If we do not, our inferences will tend to be overly optimistic and wrong. The course provides an overall framework, the multilevel and generalized multilevel (hierarchical) model, for thinking about and analyzing these forms of data. We will focus on specific versions of these tools for the most common forms of longitudinal and clustered data. This course will focus on applied work, using real data sets and the statistical software R. R will be specifically taught and supported. While the primary focus will be on the linear model with continuous outcomes (i.e., the classic regression framework) we will also discuss binary, categorical, and ordinal outcomes. We will emphasize how to think about the applicability of these methods, how they might fail, and what one might do to protect oneself in such circumstances. Applications of hierarchical (multi-level) models will include the canonical specific cases of random-slope, random-intercept, mixed effect, crossed effect, marginal, and growth-curve models.


SPRINGFIELD, Mo. — After meeting with Florida school shooting survivors this week, President Trump said he wants "highly trained" teachers to carry guns in an effort to deter the next mass casualty event.
When Aaron Sydow heard about the president's position, he was encouraged.
After all, Sydow's school district has been doing it for years.
More than a dozen other Missouri schools have followed Fairview's lead.
The American Federation of Teachers this week came out against arming teachers. And the three biggest school districts in the Springfield area told the News-Leader they have no plans to adopt the president's suggestion.
"It's sad that it's come to this," Sydow said.
"If there was an event out here at this particular school and if a sheriff's deputy was sitting in his car at the sheriff's office waiting for us to call, it would be a minimum of 8 minutes before arrival," Sydow said.
Greg Martin is the CEO of Shield Solutions, the company that trains Fairview's teachers. Martin said his company also provides the same services to more than 20 other schools in Missouri.
"Why not give them the tools they need to stop something like this?" Martin asked.


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I am a printmaker, drawer, and human being. This body of work intends to illustrate a narrative, story and composition that could resinate with you all. I want to tell a story. it might be my story, their story, or your own story, so feel free to deepen into my work and take it all personal.


Knees down. Hands together. Finally we saw Sung Ji Hyun give the signature celebration when she wins a tournament, as she took her second career Superseries victory, with both titles coming at the VICTOR Korea Open. It was almost three years of wait.
And it was much tougher this time. She had to fight past Taiwan's top seed Tai Tzu Ying in the quarterfinals, before knocking out last week's Japan Open runner-up Akane Yamaguchi and China's fourth seed Wang Yihan, both with a 21-18 decider, to finally capture her second Superseries title. Teary-eyed, Sung was handed the winner's check worth S45,000 from VICTOR president Bill Chen.
This is Sung's second BWF Tour title in the 2015 season, after posting a win over Spaniard Carolina Marin at the German Open Grand Prix Gold tournament in February which was her sixth G.P. Gold level success.
Korea National Team head coach Lee Deuk Choon (L) cheering for Sung.
Top seed Tai Tzu Ying fell to Sung in the quarterfinals.


Barium nitrite represented by the chemical formula Ba(NO2)2 or BaN2O4  that bears the IUPAC name barium dinitride  is a white crystalline powder that is soluble in water and alcohol . It is a barium salt and an ionic compound .
Barium nitrite can be prepared by decomposing silver nitrite by barium chloride .
It can also be made by pouring a warm solution of barium nitrate over lead and boiling the mixture for 1.5 to 2 hours .
In research work in laboratories, in diazotization [1, 6].
As a corrosion inhibitor .
The compound can be highly toxic if swallowed or inhaled .


text:
ANZAC LEGEND STRONG IN LISMORE ELECTORATE
THE SPIRIT of ANZAC and its deep meaning to Australians has been highlighted in an informative booklet produced by Nationals Member for Lismore Thomas George.
The eight-page booklet titled �The Spirit of ANZAC� gives an interesting insight into Australia�s military history and the role personnel from within the Electorate played in various wars and conflicts.
The booklet also carries a handy reference guide to many of the ANZAC Day dawn services and marches across the Electorate.
Mr George said he hoped the booklet would also serve as an educational resource for young Australians who are keeping the ANZAC Spirit alive through their interest and attendance at the national day of remembrance on April 25.
�In recent years there have been record attendances at ANZAC Day marches and services across Australia and that enthusiasm is also reflected in the Lismore Electorate,� Mr George said.
�The young people lining the streets and participating in marches on this very important day is a fantastic sight � the Spirit of ANZAC is alive.�
ANZAC Day commemorates the landing of Australian soldiers at Gallipoli on April 25, 1915 � this year is the 97th anniversary of the start of the Gallipoli campaign.
�On ANZAC Day, a national ceremony, we honour those who answered the call to defend this great nation in conflicts which have threatened our safety,� Mr George said.
�Their strength, service and sacrifice is something we must never forget. We must also keep in our thoughts those men and women who are currently on overseas duty on peacekeeping missions and those serving in Afghanistan.�
You can download or view the booklet by clicking here.