In order to compactify ${\cal M}_{c\leq c_{0}}$ we consider degenerations of CFTs as $E_{min}\to 0$. A degeneration is given by a one-parameter (discrete or continuous) family $H_{\varepsilon},\varepsilon\to 0$ of bi-graded spaces as above, where $(p,q)=(p(\varepsilon),q(\varepsilon))$. These spaces are equipped with OPEs. The subspace of fields with conformal dimensions vanishing as $\varepsilon\to 0$ gives rise to a commutative algebra $H^{small}=\oplus_{p(\varepsilon)\ll 1}H_{\varepsilon}^{p(\varepsilon),p(\varepsilon)}$ (the algebra structure is given by the leading terms in OPEs). The spectrum $X$ of $H^{small}$ is expected to be a compact space (“manifold with singularities”) such that $dim\,X\leq c_{0}$. It follows from the conformal invariance and the OPE, that the grading of $H^{small}$ (rescaled as $\varepsilon\to 0$) is given by the eigenvalues of a second order differential operator defined on the smooth part of $X$. The operator has positive eigenvalues and is determined up to multiplication by a scalar. This implies that the smooth part of $X$ carries a metric $g_{X}$, which is also defined up to multiplication by a scalar. Other terms in OPEs give rise to additional differential-geometric structures on $X$.

Thus, as a first approximation to the real picture, we assume the following description of a “simple collapse” of a family of CFTs. The degeneration of the family is described by the point of the boundary of $\overline{{\cal M}}_{c\leq c_{0}}$ which is a triple $(X,{{\bf R}}_{+}^{\ast}\cdot g_{X},\phi_{X})$, where the metric $g_{X}$ is defined up to a positive scalar factor, and $\phi_{X}:X\to{\cal M}_{c\leq c_{0}-dim\,X}$ is a map. One can have some extra conditions on the data. For example, the metric $g_{X}$ can satisfy the Einstein equation.

Although the scalar factor for the metric is arbitrary, one should imagine that the curvature of $g_{X}$ is “small”, and the injectivity radius of $g_{X}$ is “large”. The map $\phi_{X}$ appears naturally from the point of view of the simple collapse of CFTs described above. Indeed, in the limit $\varepsilon\to 0$, the space $H_{\varepsilon}$ becomes an $H^{small}$-module. It can be thought of as a space of sections of an infinite-dimensional vector bundle $W\to X$. One can argue that fibers of $W$ generically are spaces of states of CFTs with central charges less or equal than $c_{0}-dim\,X$. This is encoded in the map $\phi_{X}$. In the case when CFTs from $\phi_{X}(X)$ have non-trivial symmetry groups, one expects a kind of a gauge theory on $X$ as well.

Purely bosonic sigma-models correspond the case when $c_{0}=c(\varepsilon)=dim\,X$ and the residual theories (CFTs in the image of $\phi_{X}$) are all trivial. The target space $X$ in this case should carry a Ricci flat metric. In the supersymmetric case the target space $X$ is a Calabi-Yau manifold, and the residual bundle of CFTs is a bundle of free fermion theories.