Path representation of states II: Operator construction of the fermionic character and spin-–RSOS factorization

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.

Path representation of states I: Operators and particles for

“…This will be quite useful in the following construction of the generating functions for S paths. But the real power of this observation will show up in the analysis of the higher level cases, for which it will not even be necessary to define the analogous P paths: it suffices to guess the expression for w(γ) and demonstrate it [31]. In brief, the immediate off-shoot of the present section is the gain of a new and simpler weighting of the S paths.…”

“…Our first objective is to obtain a bijective correspondence between these two types of paths. This bijection has an immediate spin-off: it provides an alternative and much simpler weighting of the su(2) 1 Kyoto paths, an expression that turns out to remain valid to all levels [31]. On the other hand, we develop a formal operator representation of the paths that is instrumental in our constructive approach to the generating function, both at level 1 and for the general case worked out in the sequel article.…”

“…The main goal of the present article and its sequel [31] is to work out the fermionic su(2) k characters directly from the (Kyoto) path representation [13,28] by means of a variant of the constructive method pioneered in [39] in the context of unitary minimal models (and rooted in the work [9]). The motivation for focussing on the constructive method is that it seems to be best suited to eventually unravel the physics behind the fermionic forms.…”

“…Using j = j L+1 , we recover the value (26). Similarly, for the sector s, starting from ( 9), we make the substitution (31), which transforms the sum into telescopic type, thereby readily evaluated:…”

“…We now perform a similar analysis for S paths to show that the resulting weight modification ∆ ŵ(β l ) is equal to the above expression ∆w(β l ). The action of β l on a S path is defined via the bijection (31): it consists in replacing a NE edge between l and l + 1 by a SE one as:…”