The Zk(su(2),3/2) parafermions

Abstract

We introduce a novel parafermionic theory for which the conformal dimension of the basic parafermion is 3 2 (1 − 1/k), with k even. The structure constants and the central charges are obtained from modetype associativity calculations. The spectrum of the completely reducible representations is also determined. The primary fields turns out to be labeled by two positive integers instead of a single one for the usual parafermionic models. The simplest singular vectors are also displayed. It is argued that these models are equivalent to the non-unitary minimal W k (k + 1, k + 3) models. More generally, we expect all W k (k + 1, k + 2β) models to be identified with generalized parafermionic models whose lowest dimensional parafermion has dimension β(1 − 1/k). *

Relating Jack wavefunctions to \textrm{WA}_{k-1} theories

“…As it has been observed in [23,24], the Z 2 parafermionic operator Ψ 1 (18) coincides with the Φ (1|2) operator , Ψ = Ψ 1 = Φ (1|2) . The operator Ψ has conformal dimension ∆ = ∆ (1|2) = r/4 and the ΨΨ fusion realizes the Z …”

“…(36) is a more general form of the Virasoro Ward identity (24). Suming over the singular vector equation (35) resulting from each field Ψ and using the Eq.…”

“…These fusion rules are only valid when p = k + 1, because in that case the fields Ψ i belong to the boundary of the Kac table, namely: (19) and the usual fusion rules are truncated accordingly. The set of operators Ψ i , which are degenerate representations of the WA k−1 algebras, form then an associative Z (r) k parafermionic algebras [22,23,24,25,26] with a fixed central charge given by c(k + 1, k + r), see Eq.(10). In particular the Ψ i operators can be identified with the parafermionic chiral currents with Z k charge equal to i.…”

Clustering properties, Jack polynomials and unitary conformal field theories

“…For r = 2 (∆ = 1/2), Ψ(z) is a free-fermion field and the associated function P (2,2) n ({z i }) describes are Moore-Read states, see section (5). For r = 3, 4 · · · , it has been observed in [25,26,27] that the non-unitary minimal models M (3, 2 + r) (= W A 1 (3, 2 + r)) present an operator in their Kac table (more specifically the φ 1,2 operator in the standard notation) whose fusion realizes the Z (r) 2 algebra with central charge c = c W (2, r) = r(5 − 2r)/(2 + r). In particular, one can show that for c = c W (2, r) the P (2,6) n ({z i }) satisfy Eq.…”

“…Now, consider the case r = 3 and k taking arbitrary integer values. It was proved in [26] that there is an associative algebra Z (3) k where the central charge c is fixed to the value c = c W (k = 3, r). The corresponding model is shown to be equivalent to the…”