For a given CFT one can consider its group of symmetries (i.e. automorphisms of the space $H=\oplus_{p,q}H^{p,q}$ preserving all the structures). It is expected that the group of symmetries is a compact Lie group of dimension less or equal than $dim\,H^{1,0}$.

Let us fix $c_{0}\geq 0$ and $E_{min}>0$, and consider the moduli space ${\cal M}_{c\leq c_{0}}^{E_{min}}$ of all irreducible CFTs with the central charge $c\leq c_{0}$ and

$$min\{p+q>0|H^{p,q}\neq 0\}\geq E_{min}$$

It is expected that ${\cal M}_{c\leq c_{0}}^{E_{min}}$ is a compact real analytic stack of finite local dimension. The dimension of the base of the minimal versal deformation of a given CFT is less or equal than $dim\,H^{1,1}$. We define ${\cal M}_{c\leq c_{0}}=\cup_{E_{min}>0}{\cal M}_{c\leq c_{0}}^{E_{min}}$. We would like to compactify this stack by adding boundary components corresponding to certain asymptotic descriptions of the theories with $E_{min}\to 0$. The compactified space is expected to be a compact stack $\overline{{\cal M}}_{c\leq c_{0}}$. In what follows we will loosely use the word “space” instead of the word “stack”.