On duality and extended chiral symmetry in the SL(2,R) WZW model

TL;DR

This paper explores duality and extended chiral symmetry in the SL(2,R) WZW model, clarifying primary field conjugation, addressing fusion rule contradictions, and revealing spectral flow automorphisms related to extended chiral algebras.

Contribution

It introduces a natural definition of primary field conjugation, investigates fusion rule issues, and identifies spectral flow automorphisms linked to extended chiral symmetry.

Findings

Conjugation of primary fields aligns with two-point functions.

Fusion rules may be affected by nonsemisimplicity of the category.

Spectral flow automorphisms suggest an extended chiral algebra.

Abstract

Two chiral aspects of the SL(2,R) WZW model in an operator formalism are investigated. First, the meaning of duality, or conjugation, of primary fields is clarified. On a class of modules obtained from the discrete series it is shown, by looking at spaces of two-point conformal blocks, that a natural definition of contragredient module provides a suitable notion of conjugation of primary fields, consistent with known two-point functions. We find strong indications that an apparent contradiction with the Clebsch-Gordan series of SL(2,R), and proposed fusion rules, is explained by nonsemisimplicity of a certain category. Second, results indicating an infinite cyclic simple current group, corresponding to spectral flow automorphisms, are presented. In particular, the subgroup corresponding to even spectral flow provides part of a hypothetical extended chiral algebra resulting in proposed modular invariant bulk spectra.