On Fusion Rules in Logarithmic Conformal Field Theories

TL;DR

This paper determines the fusion rules for a series of logarithmic conformal field theories, revealing novel negative coefficients and suggesting a broader class of rational models beyond traditional minimal models.

Contribution

It completes the derivation of fusion rules for the c_{p,1} series and introduces the concept of rationality in logarithmic conformal field theories.

Findings

Negative fusion coefficients linked to quantum group representations

Effective fusion rules resemble BPZ rules for virtual minimal models

Conjecture that many minimal models are rational logarithmic CFTs

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.