[Paper Review] Paths for Z_k parafermionic models

P. Jacob, Pierre Mathieu|CERN Bulletin|Mar 1, 2007

Algebraic structures and combinatorial models21 references3 citations

TL;DR

This paper establishes a weight-preserving bijection between restricted Bressoud lattice paths and RSOS paths in regime II of the Andrews-Baxter-Forrester model, demonstrating a direct correspondence between RSOS paths and parafermionic states in a quasi-particle basis. The key result is a unified fermionic character expression for ${\cal Z}_k$ parafermionic modules, reconciling two distinct path descriptions through a duality transformation and finite-length path construction.

ABSTRACT

We present a simple bijection between restricted (Bressoud) lattice paths and RSOS paths in regime II. Both types of paths describe states in Z_k parafermionic irreducible modules. The bijection implies a direct correspondence between a RSOS path and a parafermionic state in a quasi-particle basis.

Motivation & Objective

• To establish a direct correspondence between two distinct path descriptions of states in ${\cal Z}_k$ parafermionic irreducible modules: RSOS paths in regime II and restricted Bressoud lattice paths.
• To demonstrate that these path descriptions yield equivalent fermionic character expressions, reconciling two seemingly different combinatorial formulations of the same conformal field theory.
• To provide a finite-length path construction that yields the finitized parafermionic vacuum character, offering a self-contained derivation of the fermionic form.
• To extend the correspondence to generic highest-weight modules by modifying initial path conditions and weight functions to account for non-vacuum states.

Proposed method

• Construct a weight-preserving bijection between RSOS paths in regime II and Bressoud lattice paths, using the combinatorial constraints of the quasi-particle basis.
• Define the weight of an RSOS path via a novel formula derived from the quasi-particle basis, with correction terms accounting for fractional conformal dimensions of parafermionic modes.
• Apply a duality transformation between regime II and III of the RSOS model, relating $q \to q^{-1}$ and mapping minimal-weight configurations to maximal-weight ones.
• Derive the finitized parafermionic vacuum character by constructing minimal-weight configurations for a given charge content $m_j$, adjusting for initial height $\ell$.
• Modify the path constraints and weight functions to describe generic highest-weight modules, introducing conditions on peak positions and height offsets.
• Use the generating function of Bressoud paths with adjusted weights to recover the fermionic character expression in the infinite-length limit.

Experimental results

Research questions

• RQ1 Can a direct, weight-preserving bijection be established between RSOS paths in regime II and Bressoud lattice paths for ${\cal Z}_k$ parafermionic models?
• RQ2 How does the duality transformation between regime II and III of the RSOS model relate the fermionic character expressions of the ${\cal Z}_k$ and ${\cal M}(k+1,k+2)$ models?
• RQ3 What is the correct weight assignment for an RSOS path that corresponds to a state in the quasi-particle basis of the parafermionic module?
• RQ4 How can the finitized parafermionic vacuum character be derived directly from path configurations without relying on duality?
• RQ5 How are the path constraints and weight functions modified to describe generic highest-weight modules beyond the vacuum?

Key findings

• A weight-preserving bijection is established between RSOS paths in regime II and Bressoud lattice paths, proving their equivalence in describing parafermionic states.
• The fermionic character expression for the ${\cal Z}_k$ parafermionic modules is unified across both path descriptions, with the same form derived from both constructions.
• The weight of an RSOS path is given by $w_{{\rm mwc}(\ell)} = w_{{\rm mwc}(0)} + \sum_{j=1}^{\ell} j m_{k-j+1}$, incorporating corrections from initial height $\ell$.
• The finitized vacuum character is derived as $\chi^{(m')}_{\ell}(q) = \sum_{\mathbf{m}} q^{h_{{\rm mwc}(\ell)}} \prod_{j=1}^{k-1} \begin{bmatrix} p_j + m_j \\ m_j \end{bmatrix}$, with $h_{{\rm mwc}(\ell)} = w_{{\rm mwc}(\ell)} - \frac{m'(m'+\ell)}{k}$.
• The path constraints for generic modules are modified to include $\gamma^{(j)}_{m_j} \geq j + \max(j+\ell-k,0) + 2j(m_{j+1} + \cdots + m_k)$ and $\gamma^{(j)}_1 \leq -j + \max(j+\ell-k,0) + 2[ jm_j + \cdots + km_k ]$.
• The RSOS formulation provides a natural finitization of the fermionic character, unlike the Bressoud path description, which lacks an intrinsic notion of path length.