A Rational Logarithmic Conformal Field Theory

TL;DR

This work demonstrates that the triplet algebra at $c=-2$ admits a finite set of representations closed under fusion, forming a rational logarithmic CFT. It identifies four irreducible highest-weight representations ${\cal V}_0$, ${\cal V}_{-1/8}$, ${\cal V}_1$, ${\cal V}_{3/8}$ and two indecomposables ${\cal R}_0$, ${\cal R}_1$, with explicit fusion rules producing indecomposable structures ${\cal R}_0$ and ${\cal R}_1$. The authors construct the corresponding characters, analyze their modular properties, and show that a straightforward Verlinde analysis fails, requiring a modified approach to recover fusion data from modular data. These results suggest that the $c=-2$ triplet theory is a genuine rational logarithmic CFT and motivate conjectures for other $(1,q)$ triplet models. The work deepens understanding of how logarithmic features arise from indecomposable representations within a finite fusion-closed spectrum.

Abstract

We analyse the fusion of representations of the triplet algebra, the maximally extended symmetry algebra of the Virasoro algebra at c=-2. It is shown that there exists a finite number of representations which are closed under fusion. These include all irreducible representations, but also some reducible representations which appear as indecomposable components in fusion products.