Minimal model fusion rules from 2-groups

TL;DR

This paper reveals that the fusion rules of (p,q)-minimal models in conformal field theory can be derived from a simple algebraic structure involving the group Z_2^{p+q-5} and a specific partitioning scheme.

Contribution

It introduces a minimal model fusion rule framework based on 2-group structures, simplifying the understanding of fusion rules in minimal models.

Findings

Fusion rules correspond to a partitioned group structure.

A bijection links group partitions to model sectors.

Fusion operations are represented through set addition within partitions.

Abstract

The fusion rules for the $(p,q)$-minimal model representations of the Virasoro algebra are shown to come from the group $G = \boZ_2^{p+q-5}$ in the following manner. There is a partition $G = P_1 \cup ...\cup P_N$ into disjoint subsets and a bijection between $\{P_1,...,P_N\}$ and the sectors $\{S_1,...,S_N\}$ of the $(p,q)$-minimal model such that the fusion rules $S_i * S_j = \sum_k D(S_i,S_j,S_k) S_k$ correspond to $P_i * P_j = \sum_{k\in T(i,j)} P_k$ where $T(i,j) = \{k|\exists a\in P_i,\exists b\in P_j, a+b\in P_k\}$.