On Fusion Rules in Logarithmic Conformal Field Theories

TL;DR

The article extends rationality concepts to logarithmic CFTs by analyzing the $c_{p,1}$ series, revealing how exceptional representations and logarithmic character pairs modify modular data and fusion rules. Through a detailed construction of modular-invariant partition functions and several strategies for applying the Verlinde formula, it shows that fusion coefficients can be negative but that a sign-free, effective fusion structure emerges that mirrors the BPZ fusion rules of a virtual minimal model with central charge $c_{3p,3}$. The work identifies distinct schemes (cases I–III) for obtaining consistent fusion data, demonstrates explicit examples (e.g., $c=-2$ with $p=2$), and connects these LCFT fusion structures to a broader view of the 2d field theory landscape, including limits of non-unitary minimal models and a conjectured broader class of rational LCFTs. It concludes with speculative links to supersymmetry and a proposed hierarchical view ${\rm LCFTs} = \partial\overline{{\rm CFTs}}$, positioning rational LCFTs as a bridge between RCFTs and non-conformal quantum field theories."

Abstract

We find the fusion rules for the c_{p,1} series of logarithmic conformal field theories. This completes our attempts to generalize the concept of rationality for conformal field theories to the logarithmic case. A novelty is the appearance of negative fusion coefficients which can be understood in terms of exceptional quantum group representations. The effective fusion rules (i.e. without signs and multiplicities) resemble the BPZ fusion rules for the virtual minimal models with conformal grid given via c = c_{3p,3}. This leads to the conjecture that (almost) all minimal models with c = c_{p,q}, gcd(p,q) > 1, belong to the class of rational logarithmic conformal field theories.