Characters of graded parafermion conformal field theory

Abstract

The graded parafermion conformal field theory at level k is a close cousin of the much-studied Z_k parafermion model. Three character formulas for the graded parafermion theory are presented, one bosonic, one fermionic (both previously known) and one of spinon type (which is new). The main result of this paper is a proof of the equivalence of these three forms using q-series methods combined with the combinatorics of lattice paths. The pivotal step in our approach is the observation that the graded parafermion theory -- which is equivalent to the coset osp(1,2)_k/ u(1) -- can be factored as (osp(1,2)_k/ su(2)_k) x (su(2)_k/ u(1)), with the two cosets on the right equivalent to the minimal model M(k+2,2k+3) and the Z_k parafermion model, respectively. This factorisation allows for a new combinatorial description of the graded parafermion characters in terms of the one-dimensional configuration sums of the (k+1)-state Andrews--Baxter--Forrester model.