We consider four dimensional CHL models with sixteen spacetime
supersymmetries obtained from orbifolds of type IIA superstring on
K3
Article funded by SCOAP3
0}\prod_{t=0}^{M-1}(1-e^{\frac{2\pi it}{M}}q^{\frac{m}{N}}y^\ell p^d)^{\hat c_{g,h}(d,m,\ell ,t)}, \end{align} where the first product runs over \be d,m\in\ZZ_{\ge 0}\quad \text{and} \quad \begin{cases} \ell\in\ZZ,\ \ell<0, & \text{if }m=0=d\,,\\ \ell\in\ZZ, & \text{otherwise.} \end{cases} \ee We proved that the functions $\Phi_{g,h}(\sigma,\tau,z)$ are Siegel modular forms with respect to certain discrete subgroups $\Gamma_{g,h}^{(2)}\subset \Sp(4;\mathbb{R})$. The automorphic properties of $\Phi_{g,h}(\sigma,\tau,z)$ imply, in particular, that they satisfy the relation: \be\label{Sduality} \Phi_{g,h}(\sigma,\tau,z)=\Phi_{g,h'}\bigg(\frac{\tau}{N},N\sigma,z\bigg)\,, \ee where we note that $h$ and $h'$ may not be in the same conjugacy class of $M_{24}$. For the case $(g,h)=(e,e)$ one obtains the famous Igusa cusp form $\Phi_{10}$, whose inverse is the generating function of 1/4 BPS-states in $\mathcal{N}=4$ string theory. In this case,~(\ref{Sduality}) reduces to ordinary electric-magnetic duality. It was conjectured in~\cite{Persson-ml-2013xpa} that $\Phi_{g,h}^{-1}$ similarly count twisted dyons in the CHL model obtained through orbifolding by $g$. We therefore conclude that whenever these functions have a physical interpretation, the associated CHL model should exhibit a generalization of electric-magnetic dual which acts on the vector of electric-magnetic charges $(Q,P)$ in the following way: \be \left(\begin{array}{c} Q \\ P \end{array}\right) \longmapsto \left(\begin{array}{c} -\frac{1}{\sqrt{N}} P \\ \phantom{-}\sqrt{N} Q \end{array}\right). \ee This electric-magnetic duality precisely corresponds to the $N$-modularity of the associated charge lattices. Thus, the results of the present paper show that this modular property is a consequence of the fact that CHL models satisfy Fricke S-duality. \label{sec:moonshine} ]]>
1$ is left to appendix~\ref{s:STdualGen}. Then, $\delta$ is simply a shift of $2\pi R/N$ along the circle $S^1$. Let us first take the weak coupling limit of heterotic string theory, with all the other moduli fixed. In the type IIA picture this corresponds to the limit of large volume of $S^1\times \tilde S^1$, by the usual exchange of the axio-dilaton heterotic modulus with the type IIA K\"ahler modulus. For very large $R$, $\tilde R$, a local observer in the type IIA picture sees a six dimensional space-time. The orbifold construction implies that the winding $w$ and momentum $m$ around $S^1$ are quantized as \be w=\frac{nR}{N},\qquad \qquad m=\frac{k}{R},\qquad \qquad\qquad n,k\in\ZZ\ . \ee We label the states in this six-dimensional orbifold theory by \be |w=\frac{nR}{N},m=\frac{k}{R},\chi_{n,k}\rangle\,, \ee \looseness=-1 where $\chi_{n,k}$ denotes a state in the $g^n$-twisted sector of the internal K3 model $\C$, with $g$-eigenvalue $e^{\frac{2\pi i k}{N}}$. As $R\to \infty$, local excitations carry winding number $w=0$ and the spectrum of momenta $m$ approaches a continuum. Such excitations are associated with states $\chi_{0,k}$ of the untwisted sector of the internal K3 model with any possible $g$-eigenvalue. Therefore, in the large $R$ limit, the local physics is simply described by type IIA string theory compactified on the internal K3 model $\C$. By string-string duality, the same physics can be equivalently described by heterotic string compactified on $T^4$. The orbifold construction has a `global' effect: as a local observer moves by $2\pi R/N$ around the $S^1$ direction, the states get mapped into themselves up to a $g$ transformation acting on the internal K3 model. More precisely, a shift by $2\pi R/N$ multiplies the eigenstates of momentum $m=k/R$ by $e^{\frac{2\pi ik}{N}}$. This is the same as the action of $g$ on the internal state $\chi_{0,k}$, since this is a $g$-eigenstate with the same eigenvalue. The same `global' effect occurs in the heterotic string compactified on $\frac{T^4\times S^1}{\ZZ_{\hat N}}$, and one can then argue that the two orbifold models are equivalent~\cite{VafaWitten95}. Now, let us consider the strongly coupled limit of the heterotic string on $\frac{T^4\times S^1}{\ZZ_{\hat N}}\times \tilde S^1$, with all the other moduli being fixed. In the type IIA picture, this means that both radii $R$ and $\tilde R$ of $S^1$ and $\tilde S^1$ become infinitesimal, with their ratio being fixed. It is useful to consider T-duality both in the $S^1$ and $\tilde S^1$ directions, leading again to a compactification of a six dimensional type IIA theory along two circles $ {S'}^1$ and $\tilde {S'}^1$ of very large radii\footnote{Here and in the following, we set the type IIA string length to $1$.} \be\label{Tdual} R'=\frac{N}{R}\qquad \tilde R'=\frac{1}{\tilde R}\ . \ee Notice the non-standard factor $N$ in~\eqref{Tdual}: one can check that upon setting $n'=k$ and $k'=n$ the action on the Narain left- and right-moving momenta \be (p_L,p_R)\equiv \frac{1}{\sqrt2} (m+w,m-w)=\frac{1}{\sqrt2}\biggl(\frac{k}{R}+\frac{nR}{N},\frac{k}{R}-\frac{nR}{N}\biggr) \qquad\qquad n,k\in\ZZ\,, \ee is the expected one \be (p_L,p_R)\to (p'_L,p'_R)=(p_L,-p_R)\,, \ee thus assuring that this is indeed a symmetry of the string theory. As in the previous case, in the limit $R'\to \infty$ a local observer will see local excitations with vanishing winding $w'=0$ and spectrum of momenta $m'=k'/{R'}$, $k'\in \ZZ$ approaching a continuum. However, the $\hat g$ symmetry acts differently in this T-dual picture. Namely, the states with $w'=0$ and $k'$ quanta of momentum are tensored with $g$-invariant states $\chi_{k',0}$ in the $g^{k'}$-twisted sector of the internal K3 model $\C$. Thus, in the large $R'$ limit, the local physics appears as type IIA compactified on an internal CFT whose spectrum consists of $g$-invariant twisted and untwisted states. But this is simply the orbifold $\C'$ of the internal K3 model by $g$. Since we assumed that $\lambda=1$, $\C'$ is again a consistent $\N=(4,4)$ superconformal theory with central charge $c=6$, and can be either a non-linear sigma model on K3 or on a torus $T^4$. As we move around the circle $S^1$ by a shift $2\pi R'/N$, a state $|w'=0,m',\chi_{k',0}\rangle$ of momentum $m'=k'/{R'}$ picks up a phase $e^{\frac{2\pi ik'}{N}}$. One can think of this transformation as the action on $\chi_{k',0}$ of the quantum symmetry $\Q$: by definition, $\Q$ is the symmetry of the internal orbifold CFT $\C'$ that multiplies each state in the $g^{k'}$-twisted sector by $e^{\frac{2\pi ik'}{N}}$. Therefore, this model is equivalently described by a type IIA theory compactified on $\frac{\C'\times {S'}^1}{\ZZ_{N}}\times \tilde {S'}^1$, where $\C'$ is the $g$-orbifold of the original K3 model and the $\ZZ_{N}$-symmetry $\hat{g}'=(\Q,\delta')$ acts by the quantum symmetry $\Q$ on $\C'$ and by a shift by $2\pi R'/N$ along ${S'}^1$. To summarize, we have shown that type IIA on $\frac{\C\times T^2}{\ZZ_N}$, where $\C$ is a non-linear sigma model on K3 and $\ZZ_N$ is generated by $(\delta,g)$, is dual to type IIA on $\frac{\C'\times T^2}{\ZZ_N}$, where $\C'$ is the orbifold of $\C$ by $g$ and $\ZZ_N$ is generated by $(\delta',\Q)$. The duality acts on the type IIA axio-dilaton and on the K\"ahler and complex structure moduli of $T^2$ by a \emph{Fricke involution} \be\label{FrickeTdual} S_{\text{IIA}}\to S_{\text{IIA}}\qquad T_{\text{IIA}}\to -\frac{1}{NT_{\text{IIA}}}\qquad U_{\text{IIA}}\to -\frac{1}{NU_{\text{IIA}}}\ . \ee At the special points in the moduli space where \be T_{\text{IIA}}=i\frac{R_{\text{IIA}}\tilde R_{\text{IIA}}}{N}\qquad U_{\text{IIA}}=i\frac{R_{\text{IIA}}}{N\tilde R_{\text{IIA}}}\,, \ee eq.\eqref{FrickeTdual} reduces to~\eqref{Tdual}. Furthermore, the moduli determining the non-linear sigma model $\C$ on K3 are mapped to the moduli determining the orbifold model $\C'$. A similar T-duality was noticed before in~\cite{Vafa-ml-1997mh}, where ALE spaces were considered, rather than K3 surfaces. If the orbifold theory $\C'$ can be interpreted again as a non-linear sigma model on K3, then this is dual to a weakly-coupled heterotic string compactified on ${T^6}/{\ZZ_{N}}$. Furthermore, if $\hat g'$ is in the same $O(6,22,\ZZ)$ conjugacy class as $\hat g$, then this duality relates two points in the moduli space of the same heterotic CHL model. Using the standard correspondence between type IIA and heterotic moduli, we obtain the following transformations \be\label{hetmodSdual} T_{\text{het}}\to T_{\text{het}}\qquad S_{\text{het}}\to -\frac{1}{NS_{\text{het}}}\qquad U_{\text{het}}\to -\frac{1}{NU_{\text{het}}}\,, \ee We refer to this transformation as (heterotic) \emph{Fricke S-duality}. When $\C'$ is not a non-linear sigma model on K3, then the strong coupling dual of the heterotic CHL model is a genuinely different theory. More precisely, since, in this case, $\C'$ is necessarily a non-linear sigma model on a torus $T^4$, the S-dual theory is type IIA on $\frac{T^4\times {S'}^1}{\ZZ_{N}}\times \tilde {S'}^1$. Notice that type IIA on $\frac{T^4\times {S'}^1}{\langle\hat g' \rangle}\times \tilde {S'}^1$ has no dual heterotic description. It is natural to ask whether there exists any string theory related to this model by an exchange of the dilaton and the K\"ahler moduli $S\leftrightarrow T$. In fact, this is the case, as shown in~\cite{Sen-ml-1995ff}: the dual theory is again a compactification of type IIA on $\frac{T^4\times {S'}^1}{\langle\hat g'' \rangle}\times \tilde {S'}^1$, where $\hat g''$ acts by a shift along ${S'}^1$ as well as by a symmetry of the non-linear sigma model on $T^4$ acting only on the right-moving oscillators. The symmetry $\hat g''$ breaks all spacetime supersymmetries coming from right-moving sector of type IIA superstring, but preserves all the left-moving ones, thus giving rise to a four dimensional $\N=4$ theory. Following~\cite{Sen-ml-1995ff}, we call such a theory of $(4,0)$ type. This should be contrasted with the properties of the $\hat g'$ symmetry, which breaks half of the spacetime supersymmetries in both the left- and the right-moving sector; therefore, type IIA on $\frac{T^4\times {S'}^1}{\langle\hat g' \rangle}\times \tilde {S'}^1$ is of $(2,2)$ type. Invariance of the four-dimensional low energy equations of motion implies that the electric and magnetic charges of the two S-dual models are related by \be\label{emcharges} \begin{pmatrix} Q'\\ P' \end{pmatrix}= \begin{pmatrix} 0 & -\frac{1}{\sqrt{N}}\\ \sqrt{N} & 0 \end{pmatrix}\begin{pmatrix} Q\\ P \end{pmatrix}=\begin{pmatrix} -\frac{1}{\sqrt{N}}P\\ \sqrt{N} Q \end{pmatrix} \ .\ee Let $\Lambda_e'$ and $\Lambda_m'$ denote the electric and magnetic charges in the dual theory. By~\eqref{emcharges}, \be\label{duallatte} \Lambda_e'=\Lambda_m\bigg(\frac{1}{N}\bigg)\,, \ee and dually \be\label{duallattm} \Lambda_m'\cong \Lambda_e(N) \ee where $L(n)$ denotes a lattices $L$ with the quadratic form rescaled by $n$ (i.e., each vector is rescaled by $\sqrt{n}$). If a theory is self-dual under the Fricke S-duality, using the relation $\Lambda_e\cong \Lambda_m^*$, we obtain the \be \Lambda_e^*\cong \Lambda_e(N) \ee or, equivalently, \be \Lambda_m\cong \Lambda_m^*(N)\ . \ee A lattice $L$ satisfying the condition $L\cong L^*(N)$ is called $N$-modular. Although all integral lattices of dimension $2$ are $N$-modular, for higher dimensional lattices $N$-modularity is a very special property. Thus, the fact that the lattice of magnetic charges $\Lambda_m$ of a $\ZZ_N$ self-dual CHL model is an integral $N$-modular lattice is a very non-trivial prediction of S-duality. We will verify this prediction by explicitly computing the lattices $\Lambda_e$ and $\Lambda_m$ for several CHL models. When a CHL model is not self-dual, eqs.\eqref{duallatte} and~\eqref{duallattm} imply \be \Lambda_e'= \Lambda_m\bigg(\frac{1}{N}\bigg)\qquad \Lambda_m'= \Lambda_e(N)\,, \ee but we cannot argue that $\Lambda_e'\cong \Lambda_e$ and $\Lambda_m'\cong \Lambda_e$, and hence $\Lambda_e$ and $\Lambda_m$ need not be $N$-modular in this case. We will verify that, in all cases where $\C'$ is a torus model, $N$-modularity does not hold. ]]>
1$ is treated in appendix~\ref{a:quantsymm}). It follows that the orbifold CFT $\C'$ is either a non-linear sigma model on K3 or on a four dimensional torus $T^4$. Furthermore, $\C'$ has a symmetry $\Q$ of order $N$ (the quantum symmetry) that acts by multiplication by $e^{\frac{2\pi i r}{N}}$ on the $g^r$-twisted states. Let $\prod_{a|N} a^{m(a)}$, $m(a)\in \ZZ$, be the cycle shape associated to $g$. By~\eqref{Elevel}, the condition that the orbifold is consistent, i.e $\lambda=1$, is equivalent to \be\label{consorbif} \sum_{a|N}m(N/a)a\in 24\ZZ\ .\ee The cycle shape determines the eigenvalues of $g$ on the $24$ dimensional space Ramond-Ramond ground states of $\C$. In this section, we will prove that the cycle shape describing the action of $\Q$ on the RR ground states of the orbifold theory $\C'$ is $\prod_{a|N} a^{m(N/a)}$. This statement needs a refinement when $\C'$ is a torus model; we will be more precise below. In general, the topology of the target space of a $\N=(4,4)$ superconformal field theory $\C$ with central charge $c=6$ can be distinguished by the Witten index\footnote{Of course, the Witten index can be defined even for theories with a smaller superconformal algebra, in particular for $\N=(2,2)$ theories.} \be \I^\C(\tau.\bar\tau)=\Tr_{RR}((-1)^{F+\tilde F}q^{L_0-\frac{c}{24}}\bar q^{\bar L_0-\frac{c}{24}})=\begin{cases}24 & \text{if $\C$ is a K3 model}\\ 0 & \text{if $\C$ is a torus model}\,,\end{cases} \ee \looseness=-1 where $q=e^{2\pi i\tau}$. The Witten index is the specialization to $z=0$ of the elliptic genus $\phi(\tau,z)$. By standard arguments, the only non-vanishing contribution to this trace comes from the RR ground states, i.e.\ states with conformal weights $(h,\bar h)=(\frac{c}{24},\frac{\bar c}{24})=(\frac{1}{4},\frac{1}{4})$, so that the Witten index is actually a constant and it equals the Euler characteristic of the target space. If the theory has a symmetry $g$ one can consider a $g$-twined version of the Witten index (the specialization to $z=0$ of the twining elliptic genus $\phi_g(\tau,z)$) \be \I^\C_g:=\Tr_{RR}(g(-1)^{F+\tilde F}q^{L_0-\frac{c}{24}}\bar q^{\bar L_0-\frac{c}{24}})\,, \ee which is again a constant. In the case we are considering, where $\C$ is a K3 model and $g$ has cycle shape $\prod_{a|N} a^{m(a)}$, the twining index $\I^\C_g$ is simply the trace of $g$ on its $24$-dimensional representation (the RR ground states in K3 models have all positive fermion number $(-1)^{F+\tilde F}$), so that \be\label{firstIg} \I^\C_{g}=\sum_{a|N} m(a) \sum_{k=1}^a e^{\frac{2\pi i k}{a}}=m(1)\,,\ee which is just the dimension of the fixed subspace of $g$. More generally, for any power $g^j$ we have \be\label{secondIg} \I^\C_{g^j}= \sum_{u|(j,N)} m(u)u\,, \ee where the right hand side is the dimension of the fixed subspace of $g^j$. Notice that, once the twined indices $\I^\C_{g^j}$ are known for all $j=1,\ldots,N$, using~\eqref{secondIg} one can reconstruct the cycle shape of $g$. Similarly, one can consider the twined Witten indices $\I^{\C'}_{\Q^k}$, $k=1,\ldots,N$, associated with the action of the quantum symmetry $\Q$ on the RR ground states of the orbifold theory $\C'$. Using standard orbifold techniques, this index can be computed in terms of the Frame shape of $g$ (see appendix~\ref{a:quantsymm}) \be\label{Qindex} \I^{\C'}_{\Q^k}=\sum_{n|(k,N)}m(N/n)n\ . \ee By specializing to the case $k=N$, we obtain the Witten index of the orbifold theory $\C'$ \be \I^{\C'}=\sum_{n|N}m(N/n)n\ . \ee Thus, the condition~\eqref{consorbif} of consistency of the orbifold $\C'$ implies that the elliptic genus of the orbifold theory is a multiple of $24$. Now, we see that, if~\eqref{consorbif} is satisfied, then the only two possibilities are $\I^{\C'}=0$ for a torus model and $\I^{\C'}=24$ for a K3 model. This can be also verified a posteriori for all possible cycle shapes of Conway elements. \looseness=-1 If $\I^{\C'}=24$, then $\C'$ is a K3 model, and by comparing this formula with~\eqref{secondIg}, we conclude that $\Q$ must correspond to a cycle shape $\prod_{a|N}a^{m(N/a)}$. In particular, $\Q$ has the same cycle shape as $g$ if and only if the cycle shape is `balanced' with balancing number $N$, i.e. \be\label{balance} m(a)=m(N/a)\ . \ee If~\eqref{balance} holds, then~\eqref{consorbif} is automatically satisfied and $\C$ is a K3 model. When $\C'$ is a torus model, i.e.\ $\I^{\C'}=0$, the cycle shape $\prod_{a|N}a^{m(N/a)}$ determines the eigenvalues of $\Q(-1)^{F+\tilde F}$ on the RR ground states. Notice that, in this case, the fermion number has to be taken into account, as it acts non-trivially on the RR ground states. To summarise, the above analysis shows that there are three possible cases to take into account: \paragraph{Case 1:} if the Frame shape is \emph{balanced} with balancing number $N$, i.e.\ if \be m(N/a)=m(a)\,, \ee for all $a|N$, then \be \sum_{a|N}m(N/a)a=24\,,\ee the orbifold $\C'$ is again a K3 model and the quantum symmetry $\Q$ has the same Frame shape as $g$. Thus, in this case, the CHL model is self-dual and the lattice $\Lambda_m$ is $N$-modular. \paragraph{Case 2:} if the Frame shape is not balanced, but \be \sum_{a|N}m(N/a)a=24\,, \ee then the orbifold $\C'$ is again a K3 model, but Frame shape $\prod_{a|n}a^{m(N/a)}$ of the quantum symmetry $\Q$ is different from $g$. Thus, in this case, S-duality relates two different CHL models. In particular, if $\Lambda_m$ and $\Lambda_m'$ are the lattices of magnetic charges in these two models, we have the relation \be \Lambda_m^*\cong\Lambda'_m(N)\ . \ee \paragraph{Case 3:} if \be \sum_{a|N}m(N/a)a=0\,, \ee then the orbifold $\C'$ is a torus model. Thus, the CHL model is dual to type IIA on $(T^4\times S')/\ZZ_N\times \tilde S'$. ]]>
1$, i.e.\ when the rotation $g$ within the symmetry $\hat g=(\delta,g)$ does not satisfy the level-matching condition by itself, as will be shown in appendix~\ref{s:genlatticeem}. In this case the natural choice of electric and magnetic charge lattices does not yield a Lagrangian decomposition, as in~(\ref{emsplit}). Although such a decomposition always exists --- indeed for any lattice with bilinear form~\eqref{DiracQuant} there is a choice of maximal isotropic sublattices $\Lambda_e$ and $\Lambda_m$ for which~\eqref{emsplit} holds --- this is not necessarily the natural choice in the heterotic string frame and, most importantly, it might not be the choice where the S-duality group and space parity transformation have the standard action. We deal with this situation in appendix~\ref{s:genlatticeem}. ]]>
1$ is considered in appendix~\ref{s:pureelectgen}. We can write a basis of perturbative states in the parent (unorbifolded) heterotic string picture as a tensor product \be |\N;(\tilde p,p,v)\rangle=\N |(\tilde p,p,v)\rangle \ . \ee where $|(\tilde p,p,v)\rangle$ is a lowest level state with Narain winding-momentum \be (\tilde p,p,v)\in \Gamma_{\tilde S^1}\oplus\Gamma_{S^1}\oplus\Gamma_{K3}\,, \ee and $\N$ is a combination of left- and right-moving oscillators. The orbifold projection excludes $20-d$ right-moving bosonic oscillators, so that the gauge group of the low energy effective field theory is reduced from $\U(1)^{28}$ to $\U(1)^{8+d}$ and the state $|\N;(\tilde p,p,v)\rangle$ has electric charge \be \tilde p+p+\Pi_g(v)\in \RR^{4,4+d}\,, \ee with respect to this reduced gauge group. Here, \be \Pi_g(v):=\frac{1}{N}\sum_{i=1}^N g^i(v)\in \Gamma_{K3}^g\otimes \RR\,,\qquad v\in \Gamma_{K3}\,,\ee is the projection of $v\in \Gamma_{K3}$ onto the $g$-invariant subspace $\Gamma_{K3}^g\otimes \RR$. Since $\Gamma_{K3}$ is self-dual, it follows that \be \Pi_g(\Gamma_{K3})=(\Gamma_{K3}^g)^*\ .\ee For each electric charge vector $\tilde p+p+\Pi_g(v)$, one can find a configuration of oscillators such that the corresponding state is $\hat g$-invariant. We conclude that the lattice of electric charges for the untwisted sector is \be\label{untwproj} \Lambda_e^{U}=(\Gamma_{K3}^g)^*\oplus \Gamma_{S^1} \oplus \Gamma_{\tilde S^1}\ . \ee \looseness=-1 In the $\hat g$-twisted sector, \emph{before} the projection over the $\hat g$-invariant states, the possible $\U(1)^{8+d}$ charges form a translate of the lattice~\eqref{untwproj} by $\delta=u/ N$. After projecting onto the $\hat g$ invariant subspace, we expect only a subset of these charges to survive, and this subset should form a translate of the lattice~\eqref{untwproj}, in order to have a consistent OPE between untwisted and twisted states. More generally, since the spectrum of the orbifold is generated by the OPE of $\hat g$-invariant $\hat g$-twisted fields, we obtain that the lattice of electric charges is \be\label{electLattsimpl} \Lambda_e=(\Gamma_{K3}^g)^*\oplus \langle \frac{u}{N}, u^*\rangle \oplus \Gamma_{\tilde S^1}\,,\qquad \text{for }\lambda=1\ . \ee We stress that this result is only valid when $\lambda=1$, i.e.\ when the level matching condition is satisfied. ]]>
1$, the argument showing that $\Gamma_{K3}^g\subset \Lambda_m$ goes through without essential modifications. However, as explained in appendix~\ref{s:genlatticeem}, the analysis of the charges carried by NS5-branes and KK monopoles is more subtle in this case. The outcome is that the inclusion $\Lambda_m\subset \Lambda_e^*$ is strict for $\lambda>1$ and, as a consequence, $\Lambda_{em}$ is not simply the direct sum of the purely electric and purely magnetic sublattices. ]]>
1$, as described in section~\ref{s:genlatticeem}. A highly non-trivial example is provided by the Frame shape $1^82^8$, which corresponds on the type IIA side to a symplectic automorphism of order $2$ of the K3 manifold (Nikulin involution), and on the $E_8\times E_8$ heterotic side to the involution exchanging the two $E_8$ factors. In this case, one obtains \be \Gamma_{K3}^g= \Gamma^{4,4}\oplus E_8(-2)\,, \ee and \be (\Gamma_{K3}^g)^*= \Gamma^{4,4}\oplus E_8(-1/2)\ . \ee S-duality, in this case, predicts the existence of an isomorphism \be \Gamma^{4,4}(2)\oplus E_8\cong \Gamma^{4,4}\oplus E_8(-2)\ . \ee This isomorphism can be verified by first checking that the lattices on both sides are in the same genus, and then noting that the genus has only one isomorphism class (as follows from theorem~1.14.2 of~\cite{Nikulin}). In case 2, S-duality relates pairs of distinct CHL models, related to symmetries $g, g'\in O(\Gamma_{K3})$ of the same order but different Frame shape, and we have (table~\ref{t:case3}) \be (\Gamma_{K3}^g)^*(N)\cong \Gamma_{K3}^{g'}\ . \ee Finally, for case 3, i.e.\ when the orbifold of the type IIA K3 model by $g$ is a sigma model on $T^4$, one has \be (\Gamma_{K3}^g)^*(N)\cong (\Gamma^{4,4})^{g'}\,, \ee where $\Gamma^{4,4}$ is the lattice of even D-brane charges on a non-linear sigma model on $T^4$ and $g'\in O(\Gamma^{4,4})$ is a symmetry of this model preserving the (small) $\N=(4,4)$ superconformal algebra~\cite{Volpato-ml-2014zla} (see table~\ref{t:case2} in appendix~\ref{s:lattices}). ]]>
$ of the monster CFT $V^{\natural}$ give back the same theory, while orbifolding by non-Fricke elements instead yields the Leech lattice CFT. Tuite was able to demonstrate this for a large class of elements of the monster. This observation is very similar to what we find in the present paper. Namely, it is precisely when the orbifold $\mathcal{C}'=\mathcal{C}/\left
1$. The inclusion of the summand $\langle N\lambda u^*,\frac{u}{\lambda}+N\E_gu^* \rangle \oplus \Gamma_{\tilde S^1}$, however, is more subtle. String-string duality exchanges NS5-branes wrapping $S^1\times K3$ (or its orbifold) with fundamental strings winding $S^1$. Via this duality, eqs.\eqref{electLatt} and~\eqref{Lambdamnaive} provide the lattice of winding and momenta around $S^1$ and $\tilde S^1$ for the type IIA fundamental string in the orbifold. One can check that this lattice is compatible with an orbifold by a symmetry $\hat g=(\delta,g)$ in the type IIA frame only if the $g$-twisted sector of the K3 sigma model satisfies the level-matching condition. However, the only case where the level-mismatch of an element with $\lambda>1$ has been computed in the type IIA picture ($g$ of Frame shape $2^{12}$) seems to contradict this conclusion. This lattice of magnetic charges in this CHL, therefore, cannot be described by~\eqref{Lambdamnaive}. Furthermore, the non-perturbative type IIA charges seem to break the perturbative T-duality. These issues suggest that the naive guess~\eqref{Lambdamnaive} might be wrong and that the lattice of electric magnetic charges is more complicated in this case. The adiabatic argument and the analysis of perturbative states in type IIA model seem to suggest that string-string duality and type IIA S-duality should hold also in the case $\lambda>1$. In section~\ref{s:genlatticeem}, we will therefore \emph{assume} that these dualities hold and derive the most general form of the lattice $\Lambda_{em}$ compatible with these assumptions. ]]>
1$. The heterotic and type IIA frame are very similar and can be treated together. Let us generically denote by $\C$ the internal CFT in a six dimensional compactification of string theory. For heterotic strings compactified on $T^4$, $\C$ is a $\N=(4,0)$ superconformal field theory with central charges $(c,\tilde c)=(6,16)$. In type IIA compactified on K3, $\C$ denotes a $\N=(4,4)$ SCFT with central charges $(c,\tilde c)=(6,6)$. Let us consider a symmetry $g\in O(\Gamma_{K3})$ of $\C$, preserving the superconformal algebra, such that the level-matching condition for the $g$-twisted sector is not satisfied. We define the four dimensional CHL model as the orbifold of our string theory (heterotic or type IIA) on $\C\times S^1\times \tilde S^1$ by a symmetry $\hat g=(g,\delta)$, where $\delta$ has order $\hat N=N\lambda$. Once again, $R$ and $\tilde R$ are the radii of $S^1$ and $\tilde S^1$ and, for simplicity, we consider a point in the moduli space where the K\"ahler and the complex moduli of the torus $S^1\times\tilde S^1$ are \be T=i\frac{R\tilde R}{N\lambda}\qquad U=i\frac{R}{N\lambda\tilde R} \ee and there are no Wilson line along $S^1\times \tilde S^1$. The left- and right-moving momenta along $S^1$ are quantized as (see appendix~\ref{s:pureelectgen}) \be (p_L,p_R)=\frac{1}{\sqrt2}\biggl(\frac{k\lambda-n\E_g}{R}+\frac{nR}{N\lambda },\frac{k\lambda-n\E_g}{R}-\frac{nR}{N\lambda }\biggr)\,,\qquad n,k\in\ZZ \ee and are tensored with states in the $\hat g^n$-twisted sector of $\C$ with $\hat g$-eigenvalue $e^{2\pi i\frac{k\lambda-n\E_g}{N\lambda}}$. In the limit $R,\tilde R\to \infty$ we recover the original six dimensional theory with internal CFT $\C$. The T-dual picture is obtained by \be R\to R'=\frac{N\lambda }{R}\qquad \begin{pmatrix} k\\ n \end{pmatrix} \to \begin{pmatrix} k'\\ n' \end{pmatrix}=\begin{pmatrix} \E_g & \frac{1-\E_g^2}{\lambda}\\ \lambda & -\E_g \end{pmatrix}\begin{pmatrix} k\\ n \end{pmatrix} \ee (together with the analogous transformation along $\tilde S^1$), so that \be (p_L,p_R)\to (p'_L,p'_R)=(p_l,-p_R)\,, \ee as appropriate for a T-duality. Since $\lambda|24$ and $(\E_g,\lambda)=1$, it follows that $\E_g^2\equiv 1\mod \lambda$ for all $g$, so that $k',n'$ are integers, as expected. In the limit $R'\to\infty$, local excitations satisfy \be\label{orbiproj} n'=k\lambda-n\E_g=0\,, \ee so that the limit enforces a projection onto the $\hat g$-invariant states. In fact, since $\E_g$ and $\lambda$ are coprime, the condition~\eqref{orbiproj} can only be satisfied if $n$ is a multiple of $\lambda$. Define $a=0,\ldots,N/\lambda-1$ and $b=0,\ldots, \lambda-1$, such that \be n\equiv a\lambda+bN\mod N\lambda\,,\ee and denote by $\Hh_n$ the $g^n$-twisted sector of the internal CFT $\C$. Since $g$ has order $N$ on the original untwisted theory $\C$, the $\hat g^{a\lambda+bN}$-twisted sectors, for $a$ fixed and $b=0,\ldots, \lambda-1$, are all isomorphic as vector spaces \be\label{twistisom} \Hh_{a\lambda}\stackrel{\cong}{\longrightarrow} \Hh_{a\lambda+N}\stackrel{\cong}{\longrightarrow} \ldots\stackrel{\cong}{\longrightarrow} \Hh_{a\lambda+bN}\stackrel{\cong}{\longrightarrow} \ldots\stackrel{\cong}{\longrightarrow}\Hh_{a\lambda+N(\lambda-1)}\ . \ee One can think of this isomorphism as being generated by the OPE with the vertex operator $\V_N$ associated with the unique $g^N$-twisted ground state, which has conformal dimension $0$. However, the action of $g$ on the twisted sectors $\Hh_{a\lambda+bN}$ depends on \emph{both} $a$ and $b$. To see this, first notice that by definition $g^{N+1}$ acts by $e^{2\pi i(N+1)(L-\bar L_0)}$ on the $g$-twisted sector $\Hh_1$, while it acts by $e^{2\pi i(L-\bar L_0)}$ on $\Hh_{N+1}$. Since on $\Hh_1$ and $\Hh_{N+1}$ the eigenvalues of $L_0-\bar L_0$ take value in $\frac{\E_g}{N\lambda}+\frac{1}{N}\ZZ$, and since the OPE by $\V_N$ maps states of $\Hh_1$ to states of the same spin on $\Hh_{N+1}$, we obtain that $g^{N+1}(\V_N)=e^{-\frac{2\pi i\E_g}{\lambda}}\V_N$. Since $g$ has order $N\lambda$ on the twisted sectors, it follows that $g^\lambda(\V_N)=g^{(N+1)\lambda}(\V_N)=\V_N$ and since $\lambda$ divides $N$ we finally get \be g(\V_N)=e^{-\frac{2\pi i\E_g}{\lambda}}\V_N\ . \ee Thus, if we denote by $\Hh_n(g=\zeta)$ the $g$-eigenspace of $\Hh_n$ with $g$-eigenvalue $\zeta$, we have \be \Hh_{a\lambda}(g\!=\!\zeta)\stackrel{\cong}{\longrightarrow} \ldots\stackrel{\cong}{\longrightarrow} \Hh_{a\lambda+bN}(g\!=\!e^{-\frac{2\pi i b\E_g}{\lambda}}\zeta)\stackrel{\cong}{\longrightarrow} \ldots\stackrel{\cong}{\longrightarrow}\Hh_{a\lambda+N(\lambda-1)}(g\!=\!e^{-\frac{2\pi i (\lambda-1)\E_g}{\lambda}}\zeta)\ . \ee Therefore, the direct sum of the $g$-invariant subspaces of the $g^n$-twisted sectors is \be \bigoplus_{n=0}^{\hat N-1} \Hh_n(g=1)\cong \bigoplus_{a=0}^{N/\lambda-1}\biggl(\bigoplus_{b=0}^{\lambda-1} \Hh_{a\lambda}(g=e^{\frac{2\pi i \E_g b}{\lambda}})\biggr)\cong \bigoplus_{a=0}^{N/\lambda-1} \Hh_{a\lambda}(g^\lambda=1)\ . \ee \looseness=-1 The latter space is the spectrum of the orbifold $\C'$ of $\C$ by the symmetry $g^\lambda$. This is a consistent SCFT with the same world-sheet supersymmetry as $\C$; in particular, the level-matching is satisfied (while the orbifold by $g$ is inconsistent). Thus, the local physics in this case is described by type IIA (or heterotic) superstring theory compactified on the internal CFT $\C'$. As we move along $S'^1$ by a shift of $2\pi R'/(N\lambda)$, a state with zero winding $n'=0$ and momentum $k'=n/\lambda$ picks up a phase $e^{\frac{2\pi i k'\lambda}{N\lambda}}=e^{\frac{2\pi i n/\lambda}{N}}$. This is equivalent to a symmetry $g'$ of order $N$ in the orbifold CFT $\C'$, acting by $e^{\frac{2\pi i a}{N}}g^{\E_g^{-1}}$ on the $g^{a\lambda}$-twisted sector ($\E_g^{-1}\in \ZZ/\lambda\ZZ$ is determined by $\E_g\E_g^{-1}\equiv 1\mod \lambda$). Therefore, the model is actually a type IIA (or heterotic) string theory compactified on $\frac{\C'\times {S'}^1}{\langle g'\rangle}\times \tilde {S'}^1$. For type IIA T-duality (i.e., heterotic S-duality), if $\C'$ is a K3 model and $g'$ is in the same duality class as the original $g$, the corresponding CHL is self-dual. In fact, one can show (see appendix~\ref{a:quantsymm}) that whenever $\lambda>1$, the Frame shape of $g$ is balanced with balancing number $\hat N$ and the model is self-dual. In the heterotic case, as explained in appendix~\ref{a:hetTdual}, the T-dual theory is the same CHL model, possibly at a different point in the moduli space. The effect of these dualities on non-perturbative states and the lattice of electric-magnetic charges is considerably more complicated for $\lambda>1$ and is considered in appendix~\ref{s:genlatticeem}. ]]>
1$. In this case, the orbifold of $\C$ by $g$ is not consistent and $\lambda$ is the smallest power such that $g^\lambda$ gives rise to a consistent orbifold. Failure of the level matching condition implies that the twining genus $\phi_g(\tau,z)$ is modular up to a phase. This implies that the twining Witten index $\I^\C_g=\phi(\tau,0)$ must vanish. The same argument applies to $g^k$, for any $k$ with $(\lambda,k)\neq \lambda$
\be\label{Ikvanish} \I^\C_{g^k}=0\qquad \text{if }(\lambda, k)\neq \lambda\ .
\ee As a consequence, a cycle of length $a$ in the cycle shape of $g$ can have non-zero multiplicity $m(a)\neq 0$ only if $a$ is a multiple of $\lambda$. To show this, consider by absurdity that there is some cycle of length $b$ such that $m(b)\neq 0$ and $(b,\lambda)<\lambda$. Without loss of generality, we can assume that $b$ is the smallest such cycle. Then, we have the contradiction
\be 0=\I^\C_{g^b}=\sum_{u|(b,N)} u m(u)=bm(b)\,,
\ee where the last equality follows because all $u
1$ under the assumption that string-string duality and heterotic S-duality (or, equivalently, type IIA T-duality along $S^1\times \tilde S^1$) hold. As stressed in section~\eqref{s:pureelectgen}, one expects the lattice of purely magnetic charges $\Lambda_m$ to contain a summand of the form $\Gamma_{K3}^g$, corresponding to the $\hat g$-invariant states of the original unorbifolded theory. By~\eqref{electLatt} and by the Dirac quantization condition, this is the finest possible lattice of magnetic charges with respect to the $\U(1)^{4+d}$ gauge group associated with the gauge fields descending from the ${4+d}$ six dimensional vector multiplets. Let us focus on the lattice of electric and magnetic charges with respect to the remaining $\U(1)^4$ gauge fields, associated with the metric and B-field along $S^1\times \tilde S^1$. As explained in section~\ref{s:pureelectgen}, the analysis of the spectrum of perturbative heterotic string leads to a lattice of purely electric charges of the form \be\label{hetlattice} \begin{pmatrix} m \\ \tilde m\\ w\\ \tilde w\end{pmatrix}\in\{\begin{pmatrix} \lambda & -\E_g & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 1/N\lambda & 0 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} a_1 \\ a_2\\ a_3\\ a_4\end{pmatrix},\ a_1,a_2,a_3,a_4\in\ZZ\}\,, \ee with standard quadratic form~\eqref{quadform}. Here, $\E_g/N\lambda\mod \frac{1}{N}\ZZ$ is the level-mismatch~\eqref{Elevel} for a heterotic string in the $g$-twisted sector, where $g\in O(\Gamma_{K3})$ is the pure rotational part of $\hat g=(\delta,g)$. Without loss of generality, we can assume that $\E_g\in \ZZ/\lambda \ZZ$ and $\lambda$ be coprime; since $\lambda$ is always a divisor of $24$, this is equivalent to \be\label{coprime2} \gcd(\E_g,\lambda)=1\qquad \Leftrightarrow \qquad \E_g^2\equiv 1\mod \lambda\ . \ee Similarly, in type IIA, the momenta and winding along the torus $S^1$ and $\tilde S^1$ are quantized as follows \be\label{IIAlattice} \begin{pmatrix} m' \\ \tilde m'\\ w'\\ \tilde w'\end{pmatrix}\in\{\begin{pmatrix} \lambda & -\E'_g & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 1/N\lambda & 0 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} a_1' \\ a_2'\\ a_3'\\ a_4'\end{pmatrix},\ a_1',a_2',a_3',a_4'\in\ZZ\} \ee where, $\E'_g/N\lambda\mod \frac{1}{N}\ZZ$ is the level-mismatch for a type IIA $g$-twisted sector in a non-linear sigma model on K3. We do \emph{not} assume, a priori, that $\E_g'$ and $\lambda$ are coprime --- we will derive this property as a consequence of string-string duality, T-duality and S-duality. Let $m$, $\tilde m$, $w$, $\tilde w$, $M$, $\tilde M$, $W$, $\tilde W$ denote, respectively, momentum around $S^1$, and $\tilde S^1$, winding around $S^1$ and $\tilde S^1$, NS5 brane wrapping $\tilde S^1$ and $S^1$ and KK monopole with asymptotic circle $S^1$ and $\tilde S^1$ in the heterotic string frame. Let us impose the Dirac quantization condition and assume that these charges be compatible with string-string duality and heterotic T-duality along $S^1\times \tilde S^1$ (corresponding to S-duality in the type IIA frame). We find that the most general lattice of electric-magnetic charges satisfying these constraints and correctly reproducing the perturbative spectra~\eqref{hetlattice} and~\eqref{IIAlattice} has the following structure \be\label{hetlattice2} {}^t\begin{pmatrix} m & \tilde m& w& \tilde w& \vline & M& \tilde M& W& \tilde W \end{pmatrix}\in \Bigl\{\M_{N,\lambda,\E_g,\E_g',y}\, v,\ v\in\ZZ^8\Bigr\} \ee where $\M_{N,\lambda,\E_g,\E_g',y}$ is the $8\times 8$ matrix \be\label{genemlattice} \M_{N,\lambda,\E_g,\E_g',y}:=\begin{pmatrix} \lambda & -\E_g & 0 & 0 &\vline & 0 & -\E_g' & y & 0\\ 0 & 0 & 1 & 0 &\vline & 0 & 0 &-\E_g\E_g'/\lambda & 0\\ 0 & 1/N\lambda & 0 & 0 &\vline & 0 & 0& 0& -\E_g'/N\lambda\\ 0 & 0 & 0 & 1&\vline & 0 & 0 & -\E_g'/\lambda & 0\\ \hline 0 & 0 & 0 & 0 &\vline & N\lambda & 0 & \E_g N & 0\\ 0 & 0 & 0 & 0 &\vline & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 &\vline & 0 & 0 & 1/\lambda & 0\\ 0 & 0 & 0 & 0 &\vline & 0 & 0 & 0 & 1 \end{pmatrix}\ . \ee The constraints that we are imposing are not sufficient to fix the parameter $y\in\ZZ/\lambda\ZZ$. In order to determine its precise value a more detailed analysis of non-perturbative string states is necessary. The analogous type IIA lattice of winding, momenta, NS5 branes and KK monopoles around $S^1$ and $\tilde S^1$ has exactly the same form with $\E_g$ and $\E_g'$ exchanged: \be\label{IIAlattice2} {}^t\begin{pmatrix} m' & \tilde m'& w'& \tilde w'& \vline & M'& \tilde M'& W'& \tilde W' \end{pmatrix}\in \Bigl\{\M_{N,\lambda,\E_g',\E_g,y}\, v',\ v'\in\ZZ^8\Bigr\}\ . \ee The derivation of the matrix~\eqref{genemlattice} is an easy but tedious exercise. We will skip the details, but just show that, a posteriori, the lattice above is compatible with all expected dualities. This can be checked by comparing the BPS mass formulas \be\label{BPSmass} M^2=\frac{|\tilde mU+\frac{m}{N\lambda}+T(N\lambda U w-\tilde w)+S(\tilde MU+\frac{M}{N\lambda})+ST(N\lambda UW-\tilde W) |^2}{4U_2T_2S_2} \ee for the heterotic and the type IIA frame. \looseness=-1 The BPS mass formula is invariant under string-string duality acting on the moduli by \be T'= S\qquad S'=T\qquad U'=U\,, \ee provided that the lattices~\eqref{hetlattice2} and~\eqref{IIAlattice2} are identified via the $GL(8,\ZZ)$ transformation \be v\mapsto v'=\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & \E_g \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & -\E_g' \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}v\ . \ee The mass formula is also compatible with heterotic T-duality (type IIA S-duality). In order to preserve the spectrum of perturbative heterotic states, the action of T-duality on the K\"ahler and complex moduli of $S^1\times \tilde S^1$ must be \be T\to -\frac{1}{N\lambda T}\qquad U\to -\frac{1}{N\lambda U}\qquad S\to S\ . \ee This transformation must be accompanied by an automorphism of the electric magnetic lattice such that the mass formula is invariant: \be v\rightarrow \begin{pmatrix} \E_g & \frac{1-\E_g ^2}{\lambda} & 0 & 0 & -y & 0 & 0 & 0 \\ \lambda & -\E_g & 0 & 0 & 0 & 0 & y & 0 \\ 0 & 0 & 0 & 1 & \E_g {\E_g'} & 0 & \frac{{-\E_g'} (1-\E_g ^2) }{\lambda} & 0 \\ 0 & 0 & 1 & 0 & {\E_g'} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -\E_g & 0 & \frac{1-\E_g ^2}{\lambda} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & \lambda & 0 & \E_g & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{pmatrix}v\ . \ee This is a $GL(8,\ZZ)$ transformation, thanks to~\eqref{coprime2}. The assumptions of string-string duality and heterotic T-duality are sufficient to fix the form of the electric-magnetic lattice as a functions of the three parameters $\E_g,\E_g',y$, with the constraint $\E_g^2\equiv 1\mod \lambda$. Let us now impose heterotic S-duality (or, equivalently, type IIA T-duality). One can check that, in order to preserve the BPS mass formula, one needs to act on the moduli via \be T\to T\qquad U\to -\frac{1}{N\lambda U}\qquad S\to -\frac{1}{N\lambda S}\ . \ee and on the charges by \be v\rightarrow \begin{pmatrix} {\E_g'} & 0 & 0 & y & 0 & \frac{1-{\E_g'}^2}{\lambda} & 0 & \frac{{-\E_g} ({\E_g'}^2-1)}{\lambda} \\ 0 & {\E_g'} & 0 & 0 & 0 & 0 & 0 & 1-{\E_g'}^2 \\ 0 & 0 & 0 & -{\E_g} {\E_g'} & -1 & 0 & \frac{-{\E_g}(1- {\E_g'}^2)}{\lambda} & 0 \\ 0 & 0 & 0 & {-\E_g'} & 0 & 0 & \frac{{\E_g'}^2-1}{\lambda} & 0 \\ 0 & 0 & -1 & {\E_g} & 0 & 0 & 0 & 0 \\ \lambda & -{\E_g} & 0 & 0 & 0 & -{\E_g'} & y & 0 \\ 0 & 0 & 0 & -\lambda & 0 & 0 & {\E_g'} & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & -{\E_g'} \end{pmatrix}v\ . \ee This is a $GL(8,\ZZ)$ transformation if and only if \be\label{coprime} {\E'_g}^2\equiv 1\mod \lambda\ . \ee We conclude that, upon imposing string-string duality, T-duality and S-duality, the level mismatch in the $g$-twisted sector in the heterotic and type IIA frames must be related \be \gcd(\E_g,\lambda)=1\qquad \Leftrightarrow \qquad \gcd(\E_g',\lambda)=1\ . \ee The physical interpretation of the lattice~\eqref{genemlattice} is not obvious. The form of the upper right block in~\eqref{genemlattice} suggests that (in the heterotic frame) each unit of NS5 brane wrapping $S^1$ (respectively, $\tilde S^1$), must carry $ \E_g'$ units of momentum along $S^1$ (respectively, $\E_g'/N\lambda$ units of momentum along $\tilde S^1$). Furthermore (and compatibly with T-duality), each unit of KK monopole with asymptotic circle $S^1$ (respectively, $\tilde S^1$), carries also $\E_g'$ units of winding along $\tilde S^1$ (respectively, $ \E_g'/N\lambda$ units of winding along $S^1$). A similar observation holds in the type IIA picture, with $ \E_g'$ replaced by $\E_g$. In general, it is known that, in type IIA, a NS5-brane wrapping a non-trivial 4-manifold (such as K3) times a circles $S^1$, must carry also a `geometrically induced' momentum charge along $S^1$ (by a chain of dualities, this is related to the D1-brane charge geometrically induced on a type IIB D5 brane wrapping K3). In this case, the four manifold is an asymmetric orbifold of a K3 sigma model (or, in the heterotic picture, an asymmetric orbifold of a compactification on $T^4$), so that the precise computation of the induced charges is more complicated. This interpretation suggests that the parameter $y$ should vanish. Let us now consider the electric and magnetic charges under the full $\U(1)^{8+d}$ gauge group. From~\eqref{genemlattice}, it follows that the sublattice $\Lambda_m$ of purely magnetic charges is \be \Lambda_m= \Gamma_{K3}^g\oplus \langle N\lambda u^*,u+N\lambda\E_gu^* \rangle \oplus \langle \lambda \tilde u^*,\tilde u+\E_g\tilde u^*\rangle\,, \ee \sloppy{where each vector $u,u^*,\tilde u,\tilde u^*$ represent, respectively, one unit of magnetic charges $W,M,\tilde W,\tilde M$. This is a proper sublattice of the lattice $\Lambda_e^*$ thus showing that, as anticipated in section~\ref{s:pureelectgen}, the naive guess~\eqref{Lambdamnaive} is incompatible with all expected dualities. As a consequence, the full lattice $\Lambda_{em}$ is finer than the direct sum $\Lambda_e\oplus\Lambda_m$, i.e.\ that there are dyonic charges that cannot be written as sums of purely electric and purely magnetic. The last three columns of~\eqref{genemlattice} are examples of such charges. The full lattice of electric magnetic charges is given by} \be \Lambda_{em}=\M_{N,\lambda,\E_g,\E_g',y}\ZZ^8\oplus (\Gamma_{K3}^g)^*\oplus \Gamma_{K3}^g\subset \Lambda_e^*\oplus\Lambda_m^*\,, \ee where $\Gamma_{K3}^g\subset \Lambda_m$ is purely magnetic and its dual $(\Gamma_{K3}^g)^*\subset \Lambda_e$ is purely electric. The summand $\M_{N,\lambda,\E_g,\E_g',y}\ZZ^8\subset \Lambda_{em}$ is invariant under heterotic S-duality by construction. Notice that all $g$ such that $\lambda>1$ have balanced Frame shape, with balancing number $\hat N=N\lambda$. Therefore, the corresponding CHL models are self-dual under S-duality and the lattice $\Gamma_{K3}^g$ must be $\hat N$-modular \be \Gamma_{K3}^g\cong (\Gamma_{K3}^g)^*(\hat N)\ . \ee As announced in section~\ref{s:Nmodular}, this prediction of S-duality can be verified for all possible Frame shapes with $\lambda>1$ (see table~\ref{t:case1} in appendix~\ref{s:lattices}). ]]>
1$ denotes a failure of the level matching condition in the $g$-twisted sector), the Frame shape of $g$, and the fixed lattice $\Gamma_{K3}^g$. Our notation is as follows: $L(n)$ denotes a lattice $L$ with the quadratic form rescaled by $n$; $A_m$, $D_m$, $E_8$ denote the root lattices of the corresponding Lie algebras and $A_m^*,D_m^*$ denote their duals (the weight lattices); $\Gamma^{n,m}$ is an indefinite even unimodular lattice of signature $n,m$. For some entries, we simply provide the quadratic form of the lattice $\Gamma_{K3}^g$. For certain Frame shapes, there are several isomorphism classes of lattices $\Gamma_{K3}^g$; we list all of them. This phenomenon is related to the fact that the class of $g$ in the Conway group does not always determine the $O(\Gamma_{K3})$ class uniquely --- in fact, even when the lattice $\Gamma_{K3}^g$ is unique, there might be different $O(\Gamma_{K3})$ classes with the same Frame shape, corresponding to the inequivalent ways of `gluing' $\Gamma_{K3}^g$ and its orthogonal complement $\Gamma_{K3,g}$ to obtain $\Gamma_{K3}\cong \Gamma^{4,20}$. Note however that, by the results of section~\ref{s:CHLclass}, the full lattice of electric charges~\eqref{electLatt} (and in particular $\Gamma^{1,1}\oplus \Gamma_{K3}^g$) only depends on the Frame shape of $g$, up to isomorphisms. Fricke S-duality implies that the lattices $\Gamma_{K3}^g$ in table~\ref{t:case1} must be $N\lambda$-modular $(\Gamma_{K3}^g)^*(N\lambda)\cong \Gamma_{K3}^g$. The entries of table~\ref{t:case2} are grouped into S-dual pairs, whose corresponding lattices satisfy $(\Gamma_{K3}^g)^*(N)\cong \Gamma_{K3}^{g'}$. In most cases, these isomorphisms can be verified easily by noticing that $A_2$ is $3$-modular and $D_4$ is $2$-modular. The only non-trivial case, which is described in section~\ref{s:emlattices}, corresponds to the Frame shape $1^82^8$ in table~\ref{t:case1}. \begin{table}\centering \newcolumntype{C}{>{$}c<{$}} \begin{tabular}{|CCC|} \hline N|\lambda & g & \Gamma_{K3}^g\cong (\Gamma_{K3}^g)^*(N\lambda)\\ \hline 1|1& 1^{24} & \Gamma^{4,20} \\ 2|1& 1^82^8 & \Gamma^{4,4}\oplus E_8(-2) \\ 3|1 & 1^63^6 & \Gamma^{2,2}\oplus \Gamma^{2,2}(3)\oplus (A_2(-1))^{\oplus 2} \\ 4|1 & 1^42^24^4 & \Gamma^{2,2}\oplus\Gamma^{2,2}(4)\oplus \ZZ(-2)^{\oplus 2} \\ 5|1 & 1^45^4 & \Gamma^{2,2}\oplus \Gamma^{2,2}(5) \\ 6|1 & 1^22^23^26^2 & \Gamma^{2,2}\oplus \Gamma^{2,2}(6) \\ 7|1 & 1^37^3 & \Gamma^{1,1}\oplus \Gamma^{1,1}(7)\oplus \left[\begin{smallmatrix} 4 & 1\\ 1& 2 \end{smallmatrix}\right] \\ 8|1 & 1^22^14^18^2 & \Gamma^{1,1}\oplus \Gamma^{1,1}(8)\oplus \left[\begin{smallmatrix} 2 & 0\\ 0& 4 \end{smallmatrix}\right] \\ 11|1 & 1^211^2 & \left[\begin{smallmatrix} 4 & 2 & 1 & 1 \\ 2 & 4 & 0 & 1 \\ 1 & 0 & 4 & 2 \\ 1 & 1 & 2 & 4 \end{smallmatrix}\right],\left[\begin{smallmatrix} 2 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ 1 & 0 & 6 & 0 \\ 0 & 1 & 0 & 6 \end{smallmatrix}\right],\left[\begin{smallmatrix} 2 & 1 & 1 & 1 \\ 1 & 2 & 0 & 1 \\ 1 & 0 & 8 & 4 \\ 1 & 1 & 4 & 8 \end{smallmatrix}\right] \\[10pt] 14|1 & 1^12^17^114^1 & \left[\begin{smallmatrix} 4 & 1 & 1 & 0 \\ 1 & 4 & 0 & 1 \\ 1 & 0 & 4 & -1 \\ 0 & 1 & -1 & 4\end{smallmatrix}\right],\left[\begin{smallmatrix} 2 & 0 & 1 & 1 \\ 0 & 2 & 1 & 1 \\ 1 & 1 & 8 & 1 \\ 1 & 1 & 1 & 8 \end{smallmatrix}\right],\left[\begin{smallmatrix} 2 & 1 & 0 & 0 \\ 1 & 4 & 0 & 0 \\ 0 & 0 & 4 & 2 \\ 0 & 0 & 2 & 8 \end{smallmatrix}\right] \\[10pt] 15|1 & 1^13^15^115^1 & \left[\begin{smallmatrix} 4 & 2 & 1 & 1 \\ 2 & 4 & -1 & 2 \\ 1 & -1 & 6 & 2 \\ 1 & 2 & 2 & 6 \end{smallmatrix}\right], \left[\begin{smallmatrix} 2 & 1 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 0 & 0 & 10 & 5 \\ 0 & 0 & 5 & 10 \end{smallmatrix}\right], \left[\begin{smallmatrix} 2 & 0 & 0 & 1 \\ 0 & 4 & 1 & 0 \\ 0 & 1 & 4 & 0 \\ 1 & 0 & 0 & 8 \end{smallmatrix}\right] \\[10pt] 2|2 & 2^{12} & \ZZ(2)^4\oplus \ZZ(-2)^{\oplus 8} \\ 3|3 & 3^{8} & \Gamma^{4,4}(3) \\ 4|4 & 4^6 & \ZZ(4)^{\oplus 4}\oplus \ZZ(-4)^{\oplus 2} \\ 4|2 & 2^{4}4^4 & D_4(2)\oplus D_4(-2) \\ 6|6 & 6^{4} & \ZZ(6)^{\oplus 4} \\ 10|2 & 2^210^2 & \left[\begin{smallmatrix} 6 & 4\\ 4 & 6 \end{smallmatrix}\right]^{\oplus 2} \\ 12|2 & 2^14^16^112^1 & \left[\begin{smallmatrix} 4 & 2 & 0 & 0\\ 2 & 4 & 0 & 0\\ 0 & 0 & 8 & 4\\ 0& 0& 4& 8 \end{smallmatrix}\right] \\[10pt] \hline && \\[-10pt] 6|2 & 2^36^3 & A_2(2)^{\oplus 2}\oplus A_2(-2) \\ 8|4 & 4^28^2 & \left[\begin{smallmatrix} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 8 & 0 \\ 0 & 0 & 0 & 8 \end{smallmatrix}\right] \\[10pt] \hline &&\\[-10pt] 4|1 & 1^82^{-8}4^8 & \Gamma^{4,4}(2) \\ 8|1 & 1^42^{-2}4^{-2}8^4 & D_4(2) \\ 8|2 & 2^44^{-4}8^4 & \ZZ(4)^{\oplus 4} \\ 9|1 & 1^33^{-2}9^3 & A_2\oplus A_2(3)\\ 12|1 & 1^{2}2^{-2}3^24^{2}6^{-2}12^2 & A_2(2)\oplus A_2(2) \\ \hline ]]>
{$}c<{$}} \begin{tabular}{|CCC|} \hline N|\lambda & g & \Gamma_{K3}^g \\ \hline 6|1 & 1^{4}2^13^{-4}6^5 & \Gamma^{2,2}(2)\oplus A_2(2) \\ 6|1 & 1^{5}2^{-4}3^16^4 & \Gamma^{2,2}(3)\oplus A_2 \\ \hline 10|1 & 1^{2}2^15^{-2}10^3 & A_4(2) \\ 10|1 & 1^{3}2^{-2}5^110^2 & A_4^*(5) \\ \hline 12|1 & 1^{1}2^23^14^{-2}12^2 & A_2\oplus \ZZ(6)^{\oplus 2} \\ 12|1 & 1^{2}3^{-2}4^16^212^1 & A_2(4)\oplus \ZZ(2)^{\oplus 2} \\ \hline \end{tabular} ]]>
{$}c<{$}} \begin{tabular}{|CCCC|} \hline N|\lambda & g & \Gamma_{K3}^g & (\Gamma_{K3}^g)^*(N\lambda)\\ \hline 2|1 & 1^{-8}2^{16} & \Gamma^{4,4}(2) & \Gamma^{4,4} \\ 3|1 & 1^{-3}3^{9} & \Gamma^{2,2}(3)\oplus A_2 & \Gamma^{2,2}\oplus A_2 \\ 4|1 & 1^{-4}2^64^4 & \Gamma^{2,2}(4)\oplus\ZZ(2)^{\oplus 2} & \Gamma^{2,2}\oplus \ZZ(2)^{\oplus 2} \\ 4|1 & 4^82^{-4} & D_4(2) & D_4 \\ 5|1 & 1^{-1}5^5 & A_4^*(5) & A_4 \\ 6|1 & 1^{-4}2^53^46^1 & \Gamma^{2,2}(6)\oplus A_2(2) & \Gamma^{2,2}\oplus A_2 \\ 6|1 & 1^{-2}2^43^{-2}6^4 & A_2(2)^{\oplus 2} & A_2^{\oplus 2} \\ 6|1 & 1^{-1}2^{-1}3^36^3 & D_4(3) & D_4 \\ 8|1 & 1^{-2}2^34^18^2 & \ZZ(4)\oplus A_3^*(8) & \ZZ(2)\oplus A_3 \\ 10|1 & 1^{-2}2^35^210^1 & A_4^*(10) & A_4 \\ 12|1 & 1^{-2}2^23^24^112^1 & \ZZ(6)^{\oplus 2}\oplus A_2(4) & \ZZ(2)^{\oplus 2}\oplus A_2 \\ \hline ]]>