${\mathcal{F}}_{2}(q,\bar{\rho})=\frac{1}{2}\sum_{L\in\Gamma_{8}}q^{(L+\frac{\rho}{2})^{2}}\,.$
(A.44)

The functions ${\mathcal{F}}_{1}$, ${\mathcal{F}}_{2}$, and
${\mathcal{F}}_{3}$ below, match the Mikhailov’s generating functions $F_{1}$,
$F_{2}$ and $F_{3}$ given in equations (3.3)-(3.5) in [11] up to an overall
factor $q/\eta^{24}$. The meaning of these functions will be explained
shortly.

Twisted sector

The twisted sector is much simpler. The two terms in
${\mathcal{Z}}_{\mathrm{II}}(g)$ are given in equations (A.24) and (A.26).
Combining them gives

${\mathcal{Z}}_{\mathrm{II}}(g)=\frac{1}{\bar{\eta}\eta^{17}}\sum_{n\in{\mathbb{Z}},\,\ell\in
2{\mathbb{Z}}+1}\bar{q}^{\frac{1}{2}p_{R}^{2}}\,q^{\frac{1}{2}p_{L}^{2}}\,\sum_{\rho\hskip
1.0pt\in\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}\,{\mathcal{F}}_{3}\\!\left(q,(\tfrac{\rho^{2}}{2}+n)\\!\\!\\!\mod\\!2\right)\,.$
(A.45)

The momenta $p_{R}$ and $p_{L}$ are again given in (A.31). The function
${\mathcal{F}}_{3}$ reads

${\mathcal{F}}_{3}\\!\left(q,(\tfrac{\rho^{2}}{2}+n)\\!\\!\\!\\!\mod\\!2\right)=\frac{1}{2}\left[\left(\frac{\eta^{3}}{\vartheta_{4}}\right)^{\\!\\!4}-e^{i\pi(n+\frac{\rho^{2}}{2})}\left(\frac{\eta^{3}}{\vartheta_{3}}\right)^{\\!\\!4}\right]\,.$
(A.46)

The twisted states constitute class C characterized by $\rho\in\Gamma_{8}$ and
$\ell\in 2{\mathbb{Z}}+1$.

Full orbifold

Adding ${\mathcal{Z}}_{\mathrm{II}}({\mathbb{1}})$ and
${\mathcal{Z}}_{\mathrm{II}}(g)$ yields

${\mathcal{Z}}_{{\mathrm{II}}_{\text{o}}}=\frac{1}{\bar{\eta}\eta^{17}}\left\\{\sum_{\begin{subarray}{c}n\in{\mathbb{Z}}\\\
\ell\in
2{\mathbb{Z}}\end{subarray}}\bar{q}^{\frac{1}{2}p_{R}^{2}}\,q^{\frac{1}{2}p_{L}^{2}}\left[\sum_{\rho\hskip
1.0pt\in 2\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}\,{\mathcal{F}}_{1}\
+\sum_{\rho\hskip
1.0pt\in\Gamma_{8}/2\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}{\mathcal{F}}_{2}\right]+\sum_{\begin{subarray}{c}n\in{\mathbb{Z}}\\\
\ell\in
2{\mathbb{Z}}+1\end{subarray}}\bar{q}^{\frac{1}{2}p_{R}^{2}}\,q^{\frac{1}{2}p_{L}^{2}}\,\sum_{\rho\hskip
1.0pt\in\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}\,{\mathcal{F}}_{3}\right\\}\,.$
(A.47)

The arguments of ${\mathcal{F}}_{1}$, ${\mathcal{F}}_{2}$ and
${\mathcal{F}}_{3}$ are omitted to simplify the expression. From this result
we can read off the content of orbifold states classified according to the
possible domains of $\ell$, $n$ and $\rho$. This information is summarised in
Table 8.

class |  $\ell$ |  $n$ |  $\rho$ | sector | generating function
---|---|---|---|---|---
A |  $2{\mathbb{Z}}$ |  ${\mathbb{Z}}$ |  $2\Gamma_{8}$ | untwisted | ${\mathcal{F}}_{1}(q,n\\!\\!\\!\mod\\!2)$
B |  $2{\mathbb{Z}}$ |  ${\mathbb{Z}}$ |  $\Gamma_{8}/2\Gamma_{8}$ | untwisted | ${\mathcal{F}}_{2}(q,\bar{\rho})$
C |  $2{\mathbb{Z}}+1$ |  ${\mathbb{Z}}$ |  $\Gamma_{8}$ | twisted | ${\mathcal{F}}_{3}\\!\left(q,(\tfrac{\rho^{2}}{2}+n)\\!\\!\\!\mod\\!2\right)$
Table 8: Classes of orbifold states

####  A.4.3 Reading ${\mathrm{II}}_{(1)}$ from the partition function

It turns out that some identities relating the functions ${\mathcal{F}}_{1}$,
${\mathcal{F}}_{2}$ and ${\mathcal{F}}_{3}$ are needed to show that the
orbifold states lie in the lattice ${\mathrm{II}}_{(1)}$. In [11] the
relations are proven analytically for the generating functions $F_{c}$,
$c=1,2,3$, connected to the ${\mathcal{F}}_{c}$ by

$F_{c}=\frac{q}{\eta^{24}}{\mathcal{F}}_{c}\,.$ (A.48)

The identities can be verified by comparing the $q$-expansions of the
${\mathcal{F}}_{c}$. For ${\mathcal{F}}_{1}(q,n\\!\\!\\!\mod\\!2)$ and
${\mathcal{F}}_{3}\\!\left(q,(\tfrac{\rho^{2}}{2}+n)\\!\\!\\!\mod\\!2\right)$
these expansions are easily found from their definitions. The expansion of
${\mathcal{F}}_{2}(q,\bar{\rho})$ is less direct because it depends on
$\bar{\rho}$, which is the conjugacy class of $\rho$ in
$\Gamma_{8}/2\Gamma_{8}$. As explained in [11], if $\frac{1}{2}\rho^{2}$ is
odd, then $\rho$ equals a root modulo $2\Gamma_{8}$. We denote the conjugacy
class by $\Delta_{2}$. If instead $\frac{1}{2}\rho^{2}$ is even, then $\rho$
modulo $2\Gamma_{8}$ is equal to a vector $\varrho\in\Gamma_{8}$ with
$\varrho^{2}=4$. We denote the conjugacy class by $\Delta_{4}$. From the
expansions we can check the identities

$\displaystyle{\mathcal{F}}_{1}(q,1)$
$\displaystyle={\mathcal{F}}_{2}(q,\Delta_{4})={\mathcal{F}}_{3}(q,0)=8q+64q^{2}+224q^{3}+512q^{4}+\cdots,$
(A.49a) $\displaystyle{\mathcal{F}}_{2}(q,\Delta_{2})$
$\displaystyle={\mathcal{F}}_{3}(q,1)=q^{\frac{1}{2}}(1+28q+126q^{2}+344q^{3}+\cdots)\,.$
(A.49b)

The Mikhailov’s generating functions $F_{c}$ verify the same identities
because they are equal to the ${\mathcal{F}}_{c}$ up to an overall factor.

The meaning of the generating functions can be understood by looking at the
simple lattice partition function of the 10-dimensional heterotic string in
(A.2). In this case there is a generating function $1/\eta^{16}$. Now, we know
that for each vector in the lattice there is a tower of excited states created
by acting with the oscillators of the $Y^{\hat{I}}$, $\hat{I}=1,\cdots,16$.
Moreover, the coefficients in the $q$-expansion of $1/\eta^{16}$ precisely
count the number of states at each excited level. The meaning of the
Mikhailov’s generating functions $F_{c}$ is completely analogous. The
prefactor $q/\eta^{24}$ in the relation with the ${\mathcal{F}}_{c}$ is well
justified. The power of $\eta$ corresponds to the 24 left-moving coordinates,
and the power of $q$ just offsets the normal ordering constant. In this way
their $q$-expansion will be of the form

$F(q)=\sum_{N^{\prime}}d(N^{\prime},(\ell,n,\rho))q^{N^{\prime}}\,,$ (A.50)

where now $N^{\prime}$ corresponds to the full left-moving oscillator number.
For each state $(\ell,n,\rho)$, the coefficients $d(N^{\prime},(\ell,n,\rho))$
count the number of states with oscillators acting on it and given oscillator
number $N^{\prime}$. The dependence on $(\ell,n,\rho)$ is necessary because,
as seen in Table 8, for each type of state there is an associated generating
function.

We are finally ready to state Mikhailov’s proof that the spectrum of orbifold
states can be put into correspondence with the points in the lattice
${\mathrm{II}}_{(1)}$ displayed in (A.35). This correspondence is summarised
in Table 9. For example, the points of type 1 where $(\ell,n,\rho)\in
2{\mathrm{II}}_{(1)}$ can only correlate with points of orbifold class A which
have $\rho\in 2\Gamma_{8}$, and $\ell\in 2{\mathbb{Z}}$, provided that also
$n\in 2{\mathbb{Z}}$.

${\mathrm{II}}_{(1)}$ type |  $(l,n,\rho)$ |  $\frac{1}{2}\rho^{2}+\ell n$ | orbifold class
---|---|---|---
1 |  $2{\mathrm{II}}_{(1)}$ |  $2{\mathbb{Z}}$ | $\left[{\bf A},n\in 2{\mathbb{Z}}\right]$
2 |  ${\mathrm{II}}_{(1)}/2{\mathrm{II}}_{(1)}$ |  $2{\mathbb{Z}}$ |  $\left[{\bf A},n\in 2{\mathbb{Z}}+1\right]$, $\left[{\bf B},\frac{1}{2}\rho^{2}\in 2{\mathbb{Z}}\right]$, $\left[{\bf C},\left(\frac{1}{2}\rho^{2}+n\right)\in 2{\mathbb{Z}}\right]$
3 |  ${\mathrm{II}}_{(1)}$ |  $2{\mathbb{Z}}+1$ |  $\left[{\bf B},\frac{1}{2}\rho^{2}\in 2{\mathbb{Z}}+1\right]$, $\left[{\bf C},\left(\frac{1}{2}\rho^{2}+n\right)\in 2{\mathbb{Z}}+1\right]$
Table 9: Points in ${\mathrm{II}}_{(1)}$ vs orbifold classes

For points of type 2 and 3 there can be more than one orbifold class as can be
understood by looking at Table 8. For these points consistency requires
precise identities among the generating functions. For example, the points of
type 3 must appear with the same generating function whether they arise in
class B with $\frac{1}{2}\rho^{2}\in 2{\mathbb{Z}}+1$, or in class C with
$(\frac{1}{2}\rho^{2}+n)\in 2{\mathbb{Z}}+1$. This means that
${\mathcal{F}}_{2}(q,\Delta_{2})$ must be equal to ${\mathcal{F}}_{3}(q,1)$,
which is precisely the identity in (A.49b). Similarly for the points of type 2
the functions ${\mathcal{F}}_{1}(q,1)$, ${\mathcal{F}}_{2}(q,\Delta_{4})$ and
${\mathcal{F}}_{3}(q,0)$ must be the same, which is true by virtue of of the
identity (A.49a). For points of type 1 they just fall in class A and occur
with generating function ${\mathcal{F}}_{1}(q,0)$.

####  A.4.4 T-duality

The previous results can be used to show that the partition function of the
$S^{1}/{\mathbb{Z}}_{2}$ orbifold is invariant under T-duality. We consider
the simpler situation with Wilson line $a=0$ in which T-duality is the action
$n\leftrightarrow\ell$, $R_{M}\to 1/R_{M}$. The relevant piece of the
partition function is the lattice contribution
${\mathcal{Z}}_{{\mathrm{II}}_{\text{o}}}$ displayed in (A.47). From the
corresponding spectrum of states summarised in Table 8 it is evident that
T-duality mixes twisted and untwisted states as remarked in [11].

To establish T-duality it is enough to show that the quantity between brackets
in (A.47) is invariant. The parts with both $n$ and $\ell$ even (odd), arising
in the untwisted (twisted) sector, are clearly invariant by themselves. The
remaining question is whether the untwisted sector terms with $n$ odd and
$\ell$ even do match the twisted sector terms with $n$ even and $\ell$ odd.
The answer is yes as follows from the equality

$\sum_{\begin{subarray}{c}n\in 2{\mathbb{Z}}+1\\\ \ell\in
2{\mathbb{Z}}\end{subarray}}\bar{q}^{\frac{1}{2}p_{R}^{2}}\,q^{\frac{1}{2}p_{L}^{2}}\left[\sum_{\rho\hskip
1.0pt\in 2\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}\,{\mathcal{F}}_{1}(q,1)\
+\sum_{\rho\hskip
1.0pt\in\Gamma_{8}/2\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}{\mathcal{F}}_{2}(q,\bar{\rho})\right]\\!=\\!\\!\sum_{\begin{subarray}{c}n\in
2{\mathbb{Z}}\\\ \ell\in
2{\mathbb{Z}}+1\end{subarray}}\bar{q}^{\frac{1}{2}p_{R}^{2}}\,q^{\frac{1}{2}p_{L}^{2}}\,\sum_{\rho\hskip
1.0pt\in\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}\,{\mathcal{F}}_{3}\\!\left(q,\tfrac{\rho^{2}}{2}\\!\\!\\!\\!\mod\\!2\right)\,.$
(A.51)

In turn this identity can be shown using the properties
${\mathcal{F}}_{1}(q,1)={\mathcal{F}}_{3}(q,0)$,
${\mathcal{F}}_{2}(q,\Delta_{4})={\mathcal{F}}_{3}(q,0)$ and
${\mathcal{F}}_{2}(q,\Delta_{2})={\mathcal{F}}_{3}(q,1)$, given in (A.49).

##  Appendix B World-sheet realisation of gauge symmetries

In this appendix we briefly discuss the Kac-Moody algebras that realize the
space-time gauge symmetries of the CHL theory in 9 dimensions and its toroidal
compactifications.

The space-time ${\mathrm{E}}_{8}\times{{\mathrm{E}}}^{\prime}_{8}$ gauge
symmetry is realized on the world-sheet by dimension (1,0) currents
$J_{1}^{a}\otimes 1$ and $1\otimes J^{b}_{2}$, $a,b=1,...,248$, that obey the
OPE

${J}_{i}^{a}(z){J}_{i}^{b}(0)\sim\frac{\tilde{k}_{i}\delta^{ab}}{z^{2}}+\frac{i}{z}f^{ab}{}_{c}{J}_{i}^{c}(0)\,,\quad
i=1,2$ (B.1)

at level $k_{i}=\frac{2\tilde{k}_{i}}{\psi_{i}^{2}}=1$, where
$\tilde{k}_{i}=1$, $\psi_{i}^{2}=2$ is the norm of the highest root and
$f^{ab}{}_{c}$ are the structure constants of the simply-laced Lie algebra of
${\mathrm{E}}_{8}$.

The Sugawara construction induces a representation of the Virasoro algebra
with central charge

$c=\sum_{i=1}^{2}c_{i}=\sum_{i=1}^{2}\frac{k_{i}\,{\rm dim}\ G_{i}}{k_{i}+{\rm
g}_{i}}\,,$ (B.2)

where ${\rm g}_{i}$ is the dual Coxeter number of the group $G_{i}$ (see Table
10). These formulae hold in general for arbitrary products of groups [35, 36].
For simply laced algebras at level one it follows that $c_{i}={\rm
rank}\,G_{i}$. In the ten dimensional heterotic string with
$G_{i}={\mathrm{E}}_{8}$, clearly $c_{i}=8$ and $c=16$.

$G$ | ${\mathrm{A}}_{n}$ | ${\mathrm{D}}_{n}$ | ${\mathrm{E}}_{6}$ | ${\mathrm{E}}_{7}$ | ${\mathrm{E}}_{8}$ | ${\mathrm{B}}_{n}$ | ${\mathrm{C}}_{n}$ | ${\mathrm{F}}_{4}$ | ${\mathrm{G}}_{2}$
---|---|---|---|---|---|---|---|---|---
${\rm g}$ | $n+1$ | $2n-2$ | 12 | 18 | 30 | $2n-1$ | $n+1$ | 9 | 4
dim $G$ | $n(n+2)$ | $n(2n-1)$ | 78 | 133 | 248 | $n(2n+1)$ | $n(2n+1)$ | 52 | 14
Table 10: Dual Coxeter number ${\rm g}$ and dimension of the gauge group $G$

The currents of the level $k=1$ untwisted affine Kac Moody algebras associated
with simple Lie algebras which are simply-laced were constructed using the
vertex operators of the massless gauge bosons of the string spectrum in [37,
38]. In the ten dimensional theory, the 248 gauge bosons of each
${\mathrm{E}}_{8}$ comprise the 8 Cartan $\alpha^{I}_{-1}|0,0\rangle$ or
$\alpha^{I+8}_{-1}|0,0\rangle,I=1,...,8$ and the 240 roots
$|p^{I},p^{I+8}\rangle=|r_{1}^{I},0\rangle$ or $|0,r_{2}^{I}\rangle$, with
$r_{1}^{I},r_{2}^{I}\in\Gamma_{8}$. Their vertex operators can be written in
terms of the free bosons $Y^{I}(z)$ and ${Y^{\prime}}^{I}(z)$, and the
corresponding currents $J_{1,2}^{a}$ have the following realisation in the
Cartan basis

$\displaystyle H^{I}_{1}(z)=i\partial Y^{I}(z)\,,\qquad\qquad E_{1}^{\pm
r_{1}}(z)=c_{r_{1}}:e^{\pm ir_{1}\cdot Y(z)}:\,,$ (B.3) $\displaystyle
H^{I}_{2}(z)=i\partial{Y^{\prime}}^{I}(z)\,,\qquad\qquad E_{2}^{\pm
r_{2}}(z)=c_{r_{2}}:e^{\pm ir_{2}\cdot Y^{\prime}(z)}:\,,$ (B.4)

where $c_{r_{1}},c_{r_{2}}$ are cocycle factors. Using the OPEs

$\partial Y^{I}(z)\partial
Y^{J}(0)=-\frac{\delta^{IJ}}{z^{2}}\,,\qquad\partial{Y^{\prime}}^{I}(z)\partial{Y^{\prime}}^{J}(0)=-\frac{\delta^{IJ}}{z^{2}}\,,$
(B.5)

the current algebra of
$\widehat{{\mathrm{E}}_{8}\times{\mathrm{E}}^{\prime}_{8}}$ is realized at
level $k_{1}=k_{2}=\frac{2\tilde{k}_{i}}{|\psi_{i}|^{2}}=1$, as can be read
from

$\displaystyle H^{I}_{i}(z)H^{J}_{j}(0)$
$\displaystyle\sim\frac{\delta_{ij}\delta^{IJ}}{z^{2}}\,,$ (B.6a)
$\displaystyle H_{i}^{I}(z)E_{j}^{\pm r_{j}}(0)$ $\displaystyle\sim\frac{\pm
r_{j}^{I}E_{j}^{\pm r_{j}}(0)\delta_{ij}}{z}\,,$ (B.6b) $\displaystyle
E_{i}^{r_{i}}(z)E_{i}^{-r_{i}}(0)$
$\displaystyle\sim\frac{1}{z^{2}}+\frac{r_{i}\cdot H_{i}(0)}{z}\,,$ (B.6c)

As we have seen, the CHL string in 9 and lower dimensions can be constructed
as a ${\mathbb{Z}}_{2}$ orbifold involving the outer automorphism that
exchanges ${\mathrm{E}}_{8}$ and ${{\mathrm{E}}}^{\prime}_{8}$. In 10
dimensions the orbifold by this exchange simply reproduces the original
theory. This can be verified computing the partition function as discussed in
Appendix A, with $g$ corresponding to the action
${\mathrm{E}}_{8}\leftrightarrow{\mathrm{E}}^{\prime}_{8}$, and using the
identities (A.49). However, if the exchange of ${\mathrm{E}}_{8}$ and
${\mathrm{E}}^{\prime}_{8}$ is accompanied by an additional $2\pi$ rotation of
the ten dimensional space-time one gets the non-supersymmetric
${\mathrm{E}}_{8}$ string [39, 40] in which some sectors of the Hilbert space
are projected out. As explained in [40], only the products ${\cal H}_{\rm
spF}\otimes{\cal H}_{\rm s}$ and ${\cal H}_{\rm spB}\otimes{\cal H}_{\rm as}$
survive, where ${\cal H}_{\rm spB}({\cal H}_{\rm spF})$ denotes the Hilbert
subspace of space-time bosons (fermions) which is symmetric (antisymmetric)
under the $2\pi$ rotation. Since the internal Hilbert space ${\cal H}_{\rm
int}$ is an irreducible representation of
$\widehat{{\mathrm{E}}_{8}\times{\mathrm{E}}^{\prime}_{8}}$, its symmetric and
antisymmetric subspaces ${\cal H}_{\rm s}$ and ${\cal H}_{\rm as}$ are not
invariant under the full ${\mathrm{E}}_{8}\times{\mathrm{E}}^{\prime}_{8}$
current algebra, but are invariant under the algebra of the diagonal currents

$\displaystyle T^{a}(z)=J_{1}^{a}(z)\otimes 1+1\otimes J^{a}_{2}(z)\,,$ (B.7)

since $T^{a}$ is invariant under the exchange of the two ${\mathrm{E}}_{8}$’s.
The diagonal $\hat{\mathrm{E}}_{8}$ is a subalgebra of
$\widehat{{\mathrm{E}}_{8}\times{\mathrm{E}}^{\prime}_{8}}$ and clearly, the
current algebra is realized at level $k_{1}+k_{2}=2$. In this case, the
central charge obtained from (B.2) is $\frac{31}{2}$, and the missing
$\frac{1}{2}$ is provided by the coset theory
$\frac{({\mathrm{E}}_{8}\times{\mathrm{E}}_{8})_{k=1}}{({\mathrm{E}}_{8})_{k=2}}$,
which is equivalent to the Ising model [40].

Let us now turn to CHL strings. As we have reviewed in the main text, the
9-dimensional theory can be described by a $S^{1}/{\mathbb{Z}}_{2}$ orbifold,
with ${\mathbb{Z}}_{2}$ action given by the exchange
${\mathrm{E}}_{8}\leftrightarrow{\mathrm{E}}^{\prime}_{8}$ together with a
translation in the compactified direction $x^{9}\rightarrow x^{9}+\pi R$. For
arbitrary values of the compactification radius $R$ and Wilson lines $a$, only
the 8 diagonal Cartan gauge bosons in the untwisted massless sector survive
the orbifold projection, and together with the KK gauge boson of the
compactified $x^{9}$, they account for the ${\mathrm{U}(1)}^{9}$ abelian
symmetry of the theory, with generators

$T^{I}_{+}=H^{I}_{1}+H^{I}_{2}=i\left(\partial
Y^{I}(z)+\partial{Y}^{{}^{\prime}I}(z)\right)\qquad{\rm and}\qquad
H^{9}=i\partial X^{9}(z)\,,$ (B.8)

where $X^{9}$ is a free boson.

For arbitrary $R$ and $a=0$ the untwisted states
$\frac{1}{\sqrt{2}}\left(|r^{I},0\rangle+|0,r^{I}\rangle\right)$ are also
massless when $r^{I}$ is a root of ${\mathrm{E}}_{8}$ and $m=n=0$. Together
with the nine Cartan above, they give rise to the rank nine gauge symmetry
${\mathrm{E}}_{8}\times{\mathrm{U}(1)}$, with ${\mathrm{E}}_{8}$ the diagonal
subgroup of the original ${\mathrm{E}}_{8}\times{\mathrm{E}}_{8}$. The
corresponding raising and lowering currents are

$T_{+}^{\pm r^{I}}=E_{1}^{\pm r^{I}}+E_{2}^{\pm r^{I}}\,.$ (B.9)

The ${\mathrm{E}}_{8}$ current algebra is realized at level 2, just as in the
10-dimensional ${\mathrm{E}}_{8}$ string theory.

Lower rank groups and higher level algebras are a hallmark of CHL strings. The
total central charge $c$ of the Kac-Moody algebra associated to the gauge
symmetry also gives useful information. In general, in $(10-d)$ dimensions
there is a bound $c\leq c_{L}^{int}$, where the internal piece is
$c_{L}^{int}=16+d$. This follows because keeping transverse degrees of freedom
the total left-moving central charge is $c_{L}=24$ and the world-sheet bosons
corresponding to the space-time coordinates contribute $(8-d)$. It is
convenient to write the bound on $c$ as

$\Delta=16+d-c\geq 0\,.$ (B.10)

We will refer to $\Delta$ as the missing central charge. A consistency
condition is that when $\Delta<1$ it must be equal to the central charge of a
unitary minimal model given by

$c_{j}=1-\frac{6}{j(j+1)},\quad j=3,4,\ldots.$ (B.11)

For instance, in the above ${\mathrm{E}}_{8}\times{\mathrm{U}(1)}$ example the
Kac-Moody central charge is $c=\frac{31}{2}+1$ and the missing
$\Delta=\frac{1}{2}$ is provided by the $j=3$ minimal model, i.e. the Ising
model which is furthermore equivalent to the coset theory
$\frac{({\mathrm{E}}_{8}\times{\mathrm{E}}_{8})_{k=1}}{({\mathrm{E}}_{8})_{k=2}}$.

Continuing with the 9-dimensional CHL string, at the particular radius
$R=\sqrt{2}$ and $a=0$, the states with $\ell=\pm 1,n=\pm 1,\rho=0$ in the
twisted sector become massless and enhance the ${\mathrm{U}(1)}$ of the KK
vector to ${\mathrm{SU}}(2)$. The vertex operators that create these states
involve the left moving currents

$\qquad H^{1}(z)=i\partial X^{1}(z)\,,\qquad E^{\pm}(z)=c_{\pm}\Lambda e^{\pm
ip_{L}\cdot X(z)}\,,$ (B.12)

where the fields $X^{\rm a}(z)=e^{\rm a}_{i}X^{i}(z)$ with ${\rm a},i=1,...,d$
have tangent space indices ${\rm a}$ and standard propagator

$\langle X^{\rm a}(z)X^{\rm b}(w)\rangle=-\delta^{\rm{ab}}ln(z-w)\,.$ (B.13)

The momentum in the tangent space is $p_{L{\rm
a}}=\hat{e}_{\rm{a}}{}^{i}p_{Li}=1$ and $\Lambda$ is a twist field with
conformal dimension $h=\frac{1}{2}$ and OPE

$\Lambda(z)\Lambda(0)=\frac{1}{z}+{\rm reg}\,,$ (B.14)

which is necessary to build spin 1 currents [41, 42]. From the OPEs

$\displaystyle H(z)H(0)$ $\displaystyle\sim$ $\displaystyle\frac{1}{z^{2}}\,,$
(B.15) $\displaystyle H(z)E^{\pm}(0)$ $\displaystyle\sim$
$\displaystyle\pm\frac{p_{L{\rm a}}E^{\pm}(0)}{z}\,,$ (B.16) $\displaystyle
E^{+}(z)E^{-}(0)$ $\displaystyle\sim$
$\displaystyle\frac{1}{z^{2}}+\frac{p_{L{\rm a}}H^{\rm a}(0)}{z}\,,$ (B.17)

we see that the affine ${\mathrm{SU}}(2)$ algebra is realized at level
$k=\frac{2\tilde{k}}{p_{L}^{2}}=2$. The central charge of the
${\mathrm{E}}_{8}\times{\mathrm{A}}_{1}$ model at level $k=2$ saturates
$c_{L}^{int}=17$, as may be verified using the data in Table 10.

The central charges of all the maximal enhancements listed in Table 3 can be
readily computed. Except for the ${\mathrm{D}}_{9}$ and
${\mathrm{E}}_{8}\times{\mathrm{A}}_{1}$ models, the internal Kac-Moody
algebras do not saturate $c_{L}^{int}=17$. In some cases, the missing central
charge $\Delta$ is provided by unitary minimal models, cf. (B.11). For
instance the ${\mathrm{E}}_{6}\times{\mathrm{A}}_{3}$ at level 2 requires
$\Delta=6/7=c_{6}$. On the other hand, the
${\mathrm{A}}_{1}\times{\mathrm{A}}_{2}\times{\mathrm{A}}_{6}$ current algebra
at level 2 leads to $\Delta=\frac{49}{30}$ which could arise combining two
minimal models with $j=4$ and $j=9$. However in the case
${\mathrm{D}}_{5}\times{\mathrm{A}}_{4}$, with $\Delta=8/7$, a candidate
world-sheet CFT is not obvious. It would be interesting to understand if there
is a realisation of the missing CFTs in terms of coset models involving the
original and the enhanced gauge groups.

In compactifications of the CHL string to 8 dimensions, the gauge group is
${\mathrm{U}(1)}^{10}$ for generic values of the background fields. To analyze
maximal enhancement at a special point in moduli space let us choose
$E_{11}=2,E_{22}=1,E_{12}=E_{21}=0$ and $a_{1}=a_{2}=0$. These moduli actually
correspond to starting with the 9-dimensional CHL model with group
${\mathrm{E}}_{8}\times{\mathrm{A}}_{1}$ at level $k=2$ discussed above, and
further compactifying on a circle of radius $R_{2}=1$. Following the analysis
in section 3, and using equations (3.7) and (3.8), we see that there are
additional untwisted states with $Z^{2}=4$, having $\rho=0$ and
$(\ell^{1},\ell^{2},n_{1},n_{2})=\pm(0,2,0,1)$. These states enhance the
$({\mathrm{E}}_{8}\times{\mathrm{A}}_{1})_{2}\times{\mathrm{U}(1)}$ gauge
symmetry to
$({\mathrm{E}}_{8}\times{\mathrm{A}}_{1})_{2}\times({\mathrm{C}}_{1})_{1}$.
Using that $p_{L}^{\rm a}=\frac{1}{\sqrt{2}}l^{i}e_{i}{}^{\rm
a}=(0,\sqrt{2})$, the vertex operators contain the currents

$H^{2}=i\partial X^{2}\,,\qquad E^{\pm}=c^{\pm}e^{\pm i\sqrt{2}X^{2}}\,,$
(B.18)

which realize the current algebra of ${\mathrm{C}}_{1}$ at level $k=1$.

We next consider an example with short and long roots. Taking $E_{11}=2$,
$E_{12}=-2$, $E_{21}=0$, $E_{22}=1$ and $a_{1}=a_{2}=0$, gives gauge symmetry
${\mathrm{E}}_{8}\times{\mathrm{C}}_{2}$. The quantum numbers
$(\ell^{1},\ell^{2},n_{1},n_{2})$ of the massless states that enhance the
${\mathrm{U}(1)}^{2}$ to ${\mathrm{C}}_{2}$ are $\pm(0,2,-2,1)$,
$\pm(2,2,0,1)$, $\pm(1,0,1,0)$ and $\pm(1,2,-1,1)$, and they all have
$\rho=0$. The vertex operators contain the currents

$\displaystyle E_{1}^{\pm}(z)$ $\displaystyle=c_{1}^{\pm}\Lambda(z)e^{\pm
iX^{1}(z)},\hskip 56.9055ptp^{\rm a}_{1L}=\alpha_{1}=(1,0)\,,$ (B.19a)
$\displaystyle E_{2}^{\pm}(z)$ $\displaystyle=c_{2}^{\pm}e^{\mp
i(X^{1}(z)-X^{2}(z))},\qquad\qquad p^{\rm a}_{2L}=\alpha_{2}=(-1,1)\,,$
(B.19b) $\displaystyle E_{3}^{\pm}(z)$
$\displaystyle=c_{3}^{\pm}\Lambda(z)e^{\pm iX^{2}(z)},\hskip 48.36958pt\quad
p^{\rm a}_{3L}=\alpha_{3}=(0,1)\,,$ (B.19c) $\displaystyle E_{4}^{\pm}(z)$
$\displaystyle=c_{4}^{\pm}e^{\pm i(X^{1}(z)+X^{2}(z))},\qquad\qquad p^{\rm
a}_{4L}=\alpha_{4}=(1,1)\,.$ (B.19d)

Together with the Cartan operators $H^{1}=i\partial X^{1}$ and
$H^{2}=i\partial X^{2}$, the current algebra of $\hat{\mathrm{C}}_{2}$ is
realized at level $k=1$ since, $\tilde{k}=1$ and the square of the highest
root $\alpha_{4}$ is 2.

It is straightforward to calculate the central charge of the Kac-Moody
algebras of the eight dimensional models listed in Table 7. As in the nine-
dimensional case, they do not saturate $\Delta=0$ in general, but again in
most cases one can find combinations of minimal models that account for the
missing contribution. A consistency check is that when $\Delta<1$ it is always
equal to the central charge of a unitary minimal model.

Finally, let us remark that vertex operators for the twisted states in the
examples (B.18) and (B.19) discussed above do not involve the fields
$Y^{I},Y^{\prime I}$ explicitly. However, when $\rho$ is non-vanishing, they
are expected to be part of the exponentials in the currents. For instance,
with Wilson line $a=\frac{1}{2}w_{6}$ and $R^{2}=\frac{3}{2}$ in nine
dimensions, the states with quantum numbers $(\ell,n,\rho)$ given by
$\pm(1,0,-w_{6})$ and $\pm(1,1,0)$ become massless and enhance the gauge group
to ${\mathrm{E}}_{7}\times{\mathrm{A}}_{2}$, with current algebra realized at
level 2. Free field representations of the affine ${\mathrm{SU}}(3)$ current
algebra at level 2 are known [35] as well as of the level 1 non-simply laced
algebras [43, 44, 45] involved in the enhanced gauge groups of the eight
dimensional theory (see also [46] for constructions of Kac Moody algebras in
terms of free fields). But not all of them can be directly related to the
vertex operators of the twisted states of the CHL theory. We postpone a
detailed analysis of the twisted vertex operators that realize these current
algebras to a future publication.

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