  We discuss the structure of 2D conformal field theories (CFT) at central
charge c=0 describing critical disordered systems, polymers and percolation. We
construct a novel extension of the c=0 Virasoro algebra, characterized by a
number b measuring the effective number of massless degrees of freedom, and by
a logarithmic partner of the stress tensor. It is argued to be present at a
generic random critical point, lacking super Kac-Moody, or other higher
symmetries, and is a tool to describe and classify such theories.
Interestingly, this algebra is not only consistent with, but indeed naturally
accommodates in general an underlying global supersymmetry. Polymers and
percolation realize this algebra. Unexpectedly, we find that the c=0 Kac table
of the degenerate fields contains two distinct theories with b=5/6 and b=-5/8
which we conjecture to correspond to percolation and polymers respectively. A
given Kac-table field can be degenerate only in one of them. Remarkably, we
also find this algebra, and thereby an ensuing hidden supersymmetry, realized
at general replica-averaged critical points, for which we derive an explicit
formula for b.
