Curiosities at c=-2

TL;DR

This paper explores the $c=-2$ conformal field theory, revealing the necessity of indecomposable representations and constructing logarithmic operators with symplectic fermions, highlighting unique features compared to minimal models.

Contribution

It demonstrates that at $c=-2$, conformal theories involve reducible but indecomposable representations and introduces a construction of logarithmic operators using symplectic fermions.

Findings

Logarithmic operators require indecomposable Virasoro representations.

The $(\xi,\xi)$ ghost system at $c=-2$ cannot be described by primary fields alone.

Orbifolds with $SL(2)$ symmetry resemble $ADE$ classification but are isolated models.

Abstract

Conformal field theory at $c=-2$ provides the simplest example of a theory with ``logarithmic'' operators. We examine in detail the $(\xi,\eta)$ 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.