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) over cap (1, 2)(k)/(u) over cap (1) - can be factored as ((osp) over cap (1, 2)(k)/(su) over cap (2)(k)) x ((su) over cap (2)(k)/(u) over cap (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 factorization allows for a new combinatorial description of the graded parafermion characters in terms of the one-dimensional con. guration sums of the (k + 1)-state Andrews-Baxter-Forrester model.