Ising anyons in the $SU(2)_2$ Chern--Simons theory

TL;DR

This work explores the relationship between the Ising minimal model and the $SU(2)_2$ Chern--Simons theory, focusing on their observable equivalence despite differences in representation structures.

Contribution

It analyzes the discrepancies in representation structures between the two theories and demonstrates that these do not impact their observable aspects relevant to topological quantum computation.

Findings

Representation structures differ but do not affect observables.

Discrepancies are examined for tensor products of low degree.

Observable equivalence holds despite representation differences.

Abstract

The present work is motivated by the statement that the Ising minimal model $\mathcal{M}(4,3)$ is equivalent, at the level of observables, to the $SU(2)_2$ Chern--Simons theory. At first glance, however, these two theories appear to differ substantially. For instance, the number of irreducible highest-weight representations does not match the number of Ising anyons. For tensor products of low degree, these discrepancies are examined in this work. While representation structure differs, it does not affect the observables underlying topological quantum computation algorithms.