Path representation of su(2)_k states I: Operators and particles for k=1,2

TL;DR

This paper analyzes path representations of su(2)_k states for levels 1 and 2, introducing operator and particle descriptions, and deriving new fermionic character formulas for these models.

Contribution

It provides explicit bijections between path representations at levels 1 and 2, introduces new operator constructions, and derives novel fermionic character formulas.

Findings

Explicit bijection between two path representations for su(2)_1

New simple weighting for path representations that generalizes to level k

Derivation of new fermionic character formulas for su(2)_2

Abstract

This is the first of two articles devoted to the analysis of the path description of the states in su(2)_k WZW models, a representation well suited for constructive derivations of the fermionic characters. In this first article, the cases k=1,2 are treated in detail, emphasizing a different description in each case (operators vs particles). For k=1, we first prove, as a side result, the equivalence of two known path representations for the finitized su(2)_1 states by displaying an explicit bijection. An immediate offshoot is the gain of a new and simple weighting for the (Kyoto) path representation that generalizes to level k. The bijection also suggests two operator constructions for the su(2)_1 paths, a local and a nonlocal one, both interrelated. These are formal operators that map a path to another path, so that any path can be obtained by successive applications of these operators on a simple reference (ground-state) path. The nonlocal operator description is the starting point for a direct and elementary derivation of the su(2)_1 spinon character. The second part presents an extensive study of the su(2)_2 paths from their particle point of view, where the particles are defined as the path building blocks. The resulting generating functions appear to provide new (at least superficially) fermionic forms of the characters. In particular, a nice relationship between the sum of the j=0,1 characters at k=2 and the two ones at k=1 is unravelled.