On the Quantum Symmetry of the Chiral Ising Model

TL;DR

This paper introduces rational Hopf algebras as a framework to describe superselection symmetries in two-dimensional rational quantum field theories, exemplified by the chiral Ising model.

Contribution

It demonstrates that a six-dimensional rational Hopf algebra can reproduce key features of the chiral Ising model, linking algebraic structures to physical symmetries.

Findings

Reproduces fusion rules, conformal weights, and quantum dimensions of the Ising model

Shows the Hopf algebra acts as a global symmetry algebra

Establishes the existence of covariant primary fields with algebraic properties

Abstract

We introduce the notion of rational Hopf algebras that we think are able to describe the superselection symmetries of two dimensional rational quantum field theories. As an example we show that a six dimensional rational Hopf algebra $H$ can reproduce the fusion rules, the conformal weights, the quantum dimensions and the representation of the modular group of the chiral Ising model. $H$ plays the role of the global symmetry algebra of the chiral Ising model in the following sense: 1) a simple field algebra $\FA$ and a representation $\pi$ on $\HS_\pi$ of it is given, which contains the $c=1/2$ unitary representations of the Virasoro algebra as subrepresentations; 2) the embedding $U\colon H\to\BOH$ is such that the observable algebra $\pi(\OA)^-$ is the invariant subalgebra of $\BOH$ with respect to the left adjoint action of $H$ and $U(H)$ is the commutant of $\pi(\OA)$; 3) there exist $H$-covariant primary fields in $\BOH$, which obey generalized Cuntz algebra properties and intertwine between the inequivalent sectors of the observables.