Path representation of su(2)_k states II: Operator construction of the fermionic character and spin-1/2--RSOS factorization

TL;DR

This paper constructs fermionic characters for su(2)_k WZW models using path representations, revealing a factorization into spinon and RSOS parts and connecting to minimal models and spectral decompositions.

Contribution

It provides a new operator-based derivation of fermionic characters at level k, demonstrating a path interpretation and factorization into simpler components.

Findings

Constructed level-k fermionic characters from level-1 operators.

Unveiled a direct factorization into spinon and RSOS paths.

Connected the path construction to minimal models and spectral decomposition.

Abstract

This is the second of two articles (independent of each other) devoted to the analysis of the path description of the states in su(2)_k WZW models. Here we present a constructive derivation of the fermionic character at level k based on these paths. The starting point is the expression of a path in terms of a sequence of nonlocal (formal) operators acting on the vacuum ground-state path. Within this framework, the key step is the construction of the level-k operator sequences out of those at level-1 by the action of a new type of operators. These actions of operators on operators turn out to have a path interpretation: these paths are precisely the finitized RSOS paths related to the unitary minimal models M(k+1,k+2). We thus unravel -- at the level of the path representation of the states --, a direct factorization into a k=1 spinon part times a RSOS factor. It is also pointed out that since there are two fermionic forms describing these finite RSOS paths, the resulting fermionic su(2)_k characters arise in two versions. Finally, the relation between the present construction and the Nagoya spectral decomposition of the path space is sketched.