Automorphisms of W-Algebras and Extended Rational Conformal Field Theories

TL;DR

This paper investigates the automorphisms of W-algebras and their impact on extended rational conformal field theories, providing formulas for models with two generators and exploring applications in statistical mechanics.

Contribution

It introduces formulas for extended rational conformal field theories with W-algebras, including conjectures for minimal series and analysis of twisted sectors.

Findings

Outer automorphisms enable twisted boundary conditions.

Formulas for W-algebras with two generators are provided.

Applications to $Z_n$-quantum spin chains are demonstrated.

Abstract

Many extended conformal algebras with one generator in addition to the Virasoro field as well as Casimir algebras have non-trivial outer automorphisms which enables one to impose `twisted' boundary conditions on the chiral fields. We study their effect on the highest weight representations. We give formulae for the enlarged rational conformal field theories in both series of W-algebras with two generators and conjecture a general formula for the additional models in the minimal series of Casimir algebras. A third series of W-algebras with two generators which includes the spin three algebra at $c=-2$ also has finitely many additional fields in the twisted sector although the model itself is apparently not rational. The additional fields in the twisted sector have applications in statistical mechanics as we demonstrate for $Z_n$-quantum spin chains with a particular type of boundary conditions.