Conformal Field Theory at central charge c=0 and Two-Dimensional Critical Systems with Quenched Disorder

TL;DR

This paper explores the structure of two-dimensional conformal field theories at central charge c=0, revealing an extra logarithmic field t(z) with unique properties that distinguish different theories and impact their mathematical structure.

Contribution

It identifies a new anomaly parameter b in c=0 CFTs, characterizes the extra field t(z), and discusses implications for the algebraic and differential structures of these theories.

Findings

Introduction of a new anomaly parameter b in c=0 CFTs.

Existence of a non-primary, logarithmic field t(z) with conformal weight two.

Potential supersymmetry in the OPEs of conformal weight-two fields.

Abstract

We examine two-dimensional conformal field 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 field 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 fixed up to a single number b, defining a new `anomaly' which is a characteristic of any c=0 CFT, and which may be used to distinguish between different such CFTs. The extra field t(z) is not primary (unless b=0), and is a so-called `logarithmic operator' except in special cases which include affine (Kac-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 Kac table may be set to zero or not, in these nonunitary theories. This has, in the familiar manner, implications on the existence of differential equations satisfied by conformal blocks involving primary operators with Kac-table dimensions. It is shown that c=0 theories where t(z) is logarithmic, contain, besides T and t, additional fields 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(1|1) supersymmetry. A full extension of the Virasoro algebra by the Laurent modes of these extra conformal weight-two fields, including t(z), remains an interesting question for future work.