Conformal field theory at central charge c=0 and two-dimensional critical systems with quenched disorder

We examine two-dimensional conformal eld theories (CFTs) at central charge c = 0. These arise typically in the description of critical systems with quenched disorder, but also in other contexts including dilute self-avoiding polymers and percolation. We show that such CFTs must in general possess, in addition to their stress energy tensor T (z), an extra eld whose holomorphic part, t(z), has conformal weight two. The singular part of the Operator Product Expansion (OPE) between T (z) and t(z) is uniquely xed up to a single number b, dening a new ‘anomaly’ which is a characteristic of any c = 0 CFT, and which may be used to distinguish between dieren t such CFTs. The extra eld t(z) is not primary (unless b = 0), and is a so-called ‘logarithmic operator’ except in special cases which include ane (Ka c{Moody) Lie-super current algebras. The number b controls the question of whether Virasoro null-vectors arising at certain conformal weights contained in the c = 0 Ka c table may be set to zero or not, in these nonunitary theories. This has, in the familiar manner, implications on the existence of dieren tial equations satised by conformal blocks involving primary operators with Ka c-table dimensions. It is shown that c = 0 theories where t(z) is logarithmic, contain, besides T and t, additional elds with conformal weight two. If the latter are a fermionic pair, the OPEs between the holomorphic parts of all these conformal weight-two operators are automatically covariant under a global U(1j1) supersymmetry. A full extension of the Virasoro algebra by the Laurent modes of these extra conformal weight-two elds, including t(z), remains an interesting question for future work.