Non-supersymmetric toroidal model is described by the so-called Narain lattice, endowed with some additional data. More precisely, let us fix the central charge $c=n$ which is a positive integer number. What physicists call the Narain lattice $\Gamma^{n,n}$ is a unique unimodular lattice of rank $2n$ and the signature $(n,n)$. It can be described as ${{\bf Z}}^{2n}$ equipped with the quadratic form $Q(x_{1},...,x_{n},y_{1},...,y_{n})=\sum_{i}x_{i}y_{i}$. The moduli space of toroidal CFTs is

$${\cal M}_{c=n}^{tor}=O(n,n,{{\bf Z}})\backslash O(n,n,{{\bf R}})/O(n,{{\bf R}})\times O(n,{{\bf R}}).$$

Equivalently, it is a quotient of the open part of the Grassmannian $\{V_{+}\subset{{\bf R}}^{n,n}|\,dim\,V_{+}=n,Q_{|V}>0\}$ by the action of $O(n,n,{{\bf Z}})=Aut(\Gamma^{n,n},Q)$. Let $V_{-}$ be the orthogonal complement to $V_{+}$. Then every vector of $\Gamma^{n,n}$ can be uniquely written as $\gamma=\gamma_{+}+\gamma_{-}$, where $\gamma_{\pm}\in V_{\pm}$. For the corresponding CFT one has

$$\sum_{p,q}dim(\,H^{p,q})z^{p}\bar{z}^{q}=\bigl|\prod_{k\geq 1}(1-z^{k})\bigr|^{-2n}\sum_{\gamma\in\Gamma^{n,n}}z^{Q(\gamma_{+})}\bar{z}^{\,-Q(\gamma_{-})}$$

Let us try to compactify the moduli space ${\cal M}_{c=n}^{tor}$. Suppose that we have a one-parameter family of toroidal theories such that $E_{min}(\varepsilon)$ approaches zero. Then for corresponding vectors in $H_{\varepsilon}$ one gets $p(\varepsilon)=q(\varepsilon)\to 0$. It implies that $Q(\gamma(\varepsilon))=0,Q(\gamma_{+}(\varepsilon))\ll 1$. It is easy to see that one can add vectors $\gamma(\varepsilon)$ satisfying these conditions. Thus one gets a (part of) lattice of the rank less or equal than $n$. In the case of “maximal” simple collapse the rank will be equal to $n$. One can see that the corresponding points of the boundary give rise to the following data: $(X,{{\bf R}}_{+}^{\ast}\cdot g_{X},\phi_{X}^{triv};B)$, where $(X,g_{X})$ is a flat $n$-dimensional torus, $B\in H^{2}(X,{{\bf R}}/{{\bf Z}})$ and $\phi_{X}^{triv}$ is the constant map form $X$ to the trivial theory point in the moduli space of CFTs. These data in turn give rise to a toroidal CFT, which can be realized as a sigma model with the target space $(X,g_{X})$ and given B-field $B$. The residual bundle of CFTs on $X$ is trivial.

Let us consider a $1$-parameter family of CFTs defined by the family $(X,\lambda g_{X},\phi_{X}^{triv};B=0)$, where $\lambda\in(0,+\infty)$. There are two degenerations of this family, which define two points of the boundary $\partial\overline{\cal M}_{c=n}^{tor}$. As $\lambda\to+\infty$, we get a toroidal CFT defined by $(X,{{\bf R}}_{+}^{\ast}\cdot g_{X},\phi_{X}^{triv};B=0)$. As $\lambda\to 0$ we get $(X^{\vee},{{\bf R}}_{+}^{\ast}\cdot g_{X^{\vee}},\phi_{X}^{triv};B=0)$, where $(X^{\vee},g_{X^{\vee}})$ is the dual flat torus.

There might be further degenerations of the lattice. Thus one obtains a stratification of the compactified moduli space of lattices (and hence CFTs). Points of the compactification are described by flags of vector spaces $0=V_{0}\subset V_{1}\subset V_{2}\subset...\subset V_{k}\subset{{\bf R}}^{n}$. In addition one has a lattice $\Gamma_{i+1}\subset V_{i+1}/V_{i}$, considered up to a scalar factor. These data give rise to a tower of torus bundles $X_{k}\to X_{k-1}\to...\to X_{1}\to\{pt\}$ over tori with fibers $(V_{i+1}/V_{i})/\Gamma_{i+1}$. If $V_{k}\simeq{{\bf R}}^{n-l},l\geq 1$, then one has also a map from the total space $X_{k}$ of the last torus bundle to the point $[H_{k}]$ in the moduli space of toroidal theories of smaller central charge: $\phi_{n}:X_{k}\to{\cal M}_{c=l}^{tor}$, $\phi_{k}(X_{k})=[H_{k}]$.