Critical RSOS and Minimal Models I: Paths, Fermionic Algebras and Virasoro Modules

TL;DR

This paper connects lattice models of minimal conformal field theories with Virasoro modules, using paths and fermionic algebras to generate states and relate them to transfer matrix spectra.

Contribution

It introduces a lattice perspective linking RSOS paths, fermionic algebras, and Virasoro modules, extending to parafermion models and establishing a bijection with transfer matrix eigenstates.

Findings

Finite Virasoro modules generated by path-based states.

Bijection between configuration paths and transfer matrix eigenstates.

Extension of results to Z_{L-1} parafermion models.

Abstract

We consider sl(2) minimal conformal field theories on a cylinder from a lattice perspective. To each allowed one-dimensional configuration path of the A_L Restricted Solid-on-Solid (RSOS) models we associate a physical state |h> and a monomial in a finite fermionic algebra. The orthonormal states produced by the action of these monomials on the primary states generate finite Virasoro modules with dimensions given by the finitized Virasoro characters $\chi^{(N)}_h(q)$. These finitized characters are the generating functions for the double row transfer matrix spectra of the critical RSOS models. We argue that a general energy-preserving bijection exists between the one-dimensional configuration paths and the eigenstates of these transfer matrices and exhibit this bijection for the critical and tricritical Ising models in the vacuum sector. Our results extend to Z_{L-1} parafermion models by duality.