Graded parafermions: standard and quasiparticle bases

Abstract

Two bases of states are presented for modules of the graded parafermionic conformal field theory associated to the coset osp(1, 2) k / u(1). The first one is formulated in terms of the two fundamental (i.e., lowest dimensional) parafermionic modes. In that basis, one can identify the completely reducible representations, i.e., those whose modules contain an infinite number of singular vectors; the explicit form of these vectors is also given. The second basis is a quasi-particle basis, determined in terms of a modified version of the Z 2k exclusion principle. A novel feature of this model is that none of its bases are fully ordered and this reflects a hidden structural Z 3 exclusion principle. 09/01This is the core restriction; it characterizes the combinatorics of the Andrews-Gordon identities that generalize those of Rogers-Ramanujan-Schur. 2 The above restriction is rooted in the existence of one relation at the level of fields, a relation that is model dependent. The original derivation of Lepowsky-Primc applies to the coset su(2) k / u(1) and the relation that leads to the restriction rule is (cf. [4], prop. 5.5)More precisely, e(z) k+1 or f (z) k+1 , applied on any field associated to a state in an integrable su(2) k module, vanishes. This null-field condition leads to a linear relation among the spanning states at a given grade, whose effect is captured by the restriction rule (1.2). 3Now these characters, in a slightly modified form, also turn out to provide a fermionicsum representation for the Virasoro minimal models M(2, 2k + 1) [6,7]. In this context, the restriction rule is rooted in the null field condition T (z) k + · · · (the power of T being suitably normal ordered and the dots stand for a differential polynomial in T of degree 2k). This is the field version of the non-trivial vacuum singular vector, the one at level 2k [7,8].Although the mathematical structure of the irreducible modules of the su(2) k / u(1) and the M(2, 2k + 1) models are similar -i.e., there exists a linear relation that depends upon k), their physical excitations are quite different. 4 In fact, the Virasoro modes (in terms of which the M(2, 2k + 1) space of states are described) can even not be regarded as physical 2 Note that an equivalent way of formulating this basis is by writing the states (1.1) under thewhere the arrow indicates that the index i increases toward the left; the constraint (1.2) is equivalent to imposing a i + a i+1 < k.3 Here e and f are the 'raising' and 'lowering' su(2) currents in the Chevalley basis. This relation is easily shown for k = 1 and the result is lifted to general k by tensor product. The same relation also appears in the su(2) k fermionic bases constructed in [5]. 4 Another example exhibiting a similar mathematical structure but with a quite different source for the constraining linear relation, is provided by the spinon description of the su(k) 1 WZW models [9,10,11]. In that case, the restrictions on the ordered spanning states originates from the Yangian symmetry inherited from the fini...

A new path description for the {\cal M} (k+1,2k+3) models and the dual {\cal Z}_k graded parafermions

“…The dual models are actually the graded Z k parafermions [12,24]. Note that our result goes beyond establishing the duality at the level of the characters since the dual M(k + 1, 2k + 3) paths have been shown to be in a one-to-one correspondence with the graded parfermionic states.…”

Characters of graded parafermion conformal field theory

“…When the model is viewed directly from the point of view of the parafermionic algebra, the modes of the parafermion ψ 1/2 are the natural choice for these quasi-particles. A descendant can thus be represented by the ordered sequence of its modes, and up to an overall sign this sequence forms a jagged partition [20]. The generalized exclusion principle takes the form of certain (k-dependent) difference conditions on parts of the jagged partitions.…”

K-matrices for 2D conformal field theories

“…An interesting open question is whether similar K-matrices can be used for more general CFTs, such as the twisted affine Lie algebras (and their parafermions), which were studied in [16] and [17]. Another interesting class of theories which might be addressed in a similar fashion are the affine Lie superalgebras and the related parafermions (see, for instance, [14] and [32] for the case osp(1|2)).…”