The boundary of the compactified moduli space $\overline{{\cal M}}^{N=2}$ of $N=2$ SCFTs with a given central charge contains an open stratum given by sigma models with Calabi-Yau targets. Each stratum is parametrized by the classes of equivalence of quadruples $(X,J_{X},{{\bf R}}_{+}^{\ast}\cdot g_{X},B)$ where $X$ is a compact real manifold, $J_{X}$ a complex structure, $g_{X}$ is a Calabi-Yau metric, and $B\in H^{2}(X,i{\bf R}/{\bf Z})$ is a $B$-field. The residual bundle of CFTs is a bundle of free fermion theories.

As a consequence of supersymmetry, the moduli space ${\cal M}^{N=2}$ of superconformal field theories is a complex manifold which is locally isomorphic to the product of two complex manifolds. ¹¹ 1 Strictly speaking, one should exclude models with chiral fields of conformal dimension $(2,0)$, e.g. sigma models on hyperkähler manifolds, see [AM]. It is believed that this decomposition (up to certain corrections) is global. Also, there are two types of sigma models with Calabi-Yau targets: $A$-models and $B$-models. Hence, the traditional picture of the compactified moduli space looks as follows:

Here the boundary consists of two open strata (A-stratum and B-stratum) and a mysterious meeting point. This point corresponds, in general, to a submanifold of codimension one in the closure of A-stratum and of B-stratum.

We argue that this picture should be modified. There is another open stratum of $\partial\overline{{\cal M}}^{N=2}$ (we call it T-stratum). It consists of toroidal models (i.e. CFTs associated with Narain lattices), parametrized by a manifold $Y$ with a Riemannian metric defined up to a scalar factor. This subvariety meets both $A$ and $B$ strata along the codimension one stratum corresponding to the double collapse. Therefore the “true” picture is obtained from the traditional one by the real blow-up at the corner: