W-symmetries on the Homogeneous Space G/U(1)^r

TL;DR

This paper constructs W-symmetries on the homogeneous space G/U(1)^r using parafermionic fields, extending the stress tensor and analyzing operator product expansions for simple-laced Lie algebras.

Contribution

It provides a nonlocal field construction of W-symmetries on G/U(1)^r and explores the conditions for consistent extensions of the stress tensor.

Findings

Explicit construction of W-symmetries using parafermions.

Calculation of OPEs for spin three primary fields.

Emergence of a spin four primary field.

Abstract

A construction of $W$-symmetries is given only in terms of the nonlocal fields (parafermions ${\ps}_{\al}$), which take values on the homogeneous space $G/U(1)^r$, where $G$ is a simply connected compact Lie group manifold (its accompanying Lie algebra ${\cal G}$ is a simple one of rank $r$). Only certain restriction of the root set of Lie algebra on which the parafermionic fields take values are satisfied, then a consistent and non-trivial extension of the stress momentum tensor may exist. For arbitrary simple-laced algebras, i.e. the $A-D-E$ cases, a more detailed discussion is given. The OPE of spin three primary field are calculated, in which a primary field with spin four is emerging.