Curiosities at c=-2

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.