Curiosities at c=-2

TL;DR

This work analyzes conformal field theories at central charge $c=-2$ as canonical examples of logarithmic CFTs, focusing on the $(\xi,\eta)$ ghost system and its Coulomb gas realization. It demonstrates that the theory contains reducible but indecomposable Virasoro representations and logarithmic operators, which arise from twist-field fusion and require extending the operator content beyond primaries; a non-chiral realization via symplectic fermions with an $SL(2)$ symmetry provides a concrete framework. The paper then constructs orbifolds by finite subgroups of $SL(2)$, linking their spectra to Coulomb gas data and showing these theories are isolated under marginal deformations. These insights illuminate the role of logarithmic representations in $c=-2$ and hint at broader implications for polymers, 2D gravity, and related non-minimal CFTs near the $c=1$ line.

Abstract

Conformal field theory at $c=-2$ provides the simplest example of a theory with ``logarithmic'' operators. We examine in detail the $(ξ,η)$ ghost system and Coulomb gas construction at $c=-2$ and show that, in contradistinction to minimal models, they can not be described in terms of conformal families of {\em primary\/} fields alone but necessarily contain reducible but indecomposable representations of the Virasoro algebra. We then present a construction of ``logarithmic'' operators in terms of ``symplectic'' fermions displaying a global $SL(2)$ symmetry. Orbifolds with respect to finite subgroups of $SL(2)$ are reminiscent of the $ADE$ classification of $c=1$ modular invariant partition functions, but are isolated models and not linked by massless flows.