Parafermionic character formulae

Abstract

We study various aspects of parafermionic theories such as the precise field content, a description of a basis of states (that is, the counting of independent states in a freely generated highest-weight module) and the explicit expression of the parafermionic singular vectors in completely irreducible modules. This analysis culminates in the presentation of new character formulae for the $Z_N$ parafermionic primary fields. These characters provide novel field theoretical expressions for $\su(2)$ string functions.Comment: Harvmac (b mode : 37 p

Quantum groups and non-Abelian braiding in quantum Hall systems

“…[14,15] Before we write down any explicit expression for the RR-states, we recall some well known facts about this CFT. The Z k parafermionic CFT may be described completely in terms of a chiral algebra generated by the modes of k parafermionic currents (see [14,15] and also [16] for some more recent work in this vein), but it also has two different coset descriptions. The cosets involved are sl(2) k / U(1) k and sl(k) 1 × sl(k) 1 / sl(k) 2 .…”

“…Also, let us define D(Λ, n) = 0 if there is no point with coordinates (Λ, n). The number of independent n-quasihole states encoded by (16) is then D(0, n) in case 2N + n = 0 ( mod 2k), D(k, n) in case 2N + n = k ( mod 2k), and zero otherwise. It should be obvious from looking at the Bratteli diagram that the D(Λ, n) satisfy the following recursion relation:…”

“…First of all, we have to restrict to the states in a truncated tensor product with a given bracketing, as explained in sections 3.2.4 and 3.2.8. Second, there is a restriction that comes from the fact that the conformal block in (16) vanishes unless all the fields that appear in it fuse into the vacuum sector. This now means that the system as a whole is in one of the A q 1 ,q 2 -representations π 0 0 , π k k or in the B q 3 ,q 4 -representation π 0 .…”

“…The braiding representations associated to this U q (sl(2))-representation were described in detail in section 3.2.12 (and they were also included in the material of section 4.3). The only difference between the braid representations described there and the braiding for the RR-quasiholes lie in a scalar factor for every exchange, which comes from the CZ 4k,q 2 part of A q 1 ,q 2 and from the explicit factors in the wave function (16). Thus, a basis for the space of states with n quasiholes in fixed positions is labelled by the paths on the fusion diagram of figure 1 which start at the point (0, 0) and which end at the point (n, Λ), where Λ = −N mod k, so that the A q 1 ,q 2 -charge of the whole system corresponds to the vacuum sector of the CFT.…”

Higher String Functions, Higher-Level Appell Functions, and the Logarithmic $${{{\widehat{s\ell}2_k/u(1)}}}$$ CFT Model

“…Instead of working out the details of the resulting "logarithmic parafermion" model starting from representation theory, which seems to be quite a laborious task (cf. [27] in the nonlogarithmic case), we work at the level of characters, and this is how the A and B functions appear. The logarithmically extended parafermion model is, strictly speaking, presently nonexistent beyond as much as can be deduced from its proposed characters and the modular group representation generated from them, derived in what follows.…”