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The Z_k^(su(2),3/2) Parafermions

We introduce a novel parafermionic theory for which the conformal dimension of the basic parafermion is 3(1-1/k)/2, with k even. The structure constants and the central charges are obtained from mode-type 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+2 beta) models to be identified with generalized parafermionic models whose lowest dimensional parafermion has dimension beta(1-1/k).


Introduction
The Fateev-Zamolodchikov Z k parafermionic conformal field theory, originally introduced in [1], can be generalized along two quite distinct lines. The first one relies on the observation that the standard parafermionic model is equivalent to the coset su(2) k / u (1). In that vein, a natural generalization amounts to consider the cosets g/ u(1) r where r is the rank of the affine Lie algebra g [2]. Originally formulated in terms of untwisted affine simple Lie algebras, this approach has been further extended to the cases where g is either twisted [3] or contains fermionic generators [4].
For the second line of generalization, one preserves the structure of the standard theory, i.e., the cyclic structure of the OPEs (namely ψ n × ψ m ∼ ψ n+m with ψ k ∼ I), but attributes different conformal dimension to the parafermionic fields. We recall in that regard that for a candidate chiral algebra whose OPE has a Z k invariance, the dimension of the basic parafermionic fields, h ψn , have to be introduced as an input. The Z k model of [1] amounts to choose the dimension of the parafermions ψ n to be h ψn = n(k − n)/k. This is the simplest choice compatible with the monodromy constraints. A general solution to these constraints has also been displayed in [1]. It reads where β is an integer and a n are integers that satisfy a n = a k−n . The second type of generalization is thus defined in terms of such a generalized form of the dimension of the parafermionic fields.
Actually, both types of generalization (with a n = 0 and β positive) can be combined, leading to a much larger set of parafermionic theories, denoted as Z (g,β) k , classified in terms of both their underlying structural algebra 1 g and the parameter β. In this notation, the original parafermionic theories of [1] would correspond to the Z (su(2),1) k models.
In this work we explore a generalization of the second type. The corresponding models will be referred to as the Z (su(2),β) k models, written Z (β) k for short. Some Z (β) k models have been considered previously. The unitary sequence of the Z (2) 3 models, first introduced in appendix A of [1], have been studied in more detail in [5]. Further developments are presented in [6]. More recently, the unitary sequence of the Z (2) k models for arbitrary k have been analyzed in depth in [7]. The only known results for higher values of β seems to be those presented in [8] pertaining to β = 4 and k = 3.
Here we explore a novel possibility, which is to consider β to be half-integer. This is allowed whenever k is even. The possibility of having β non-integer seems to have first been mentioned in [9], although it has not been studied there. This enhancement in the range of the possible values of β in relation with the parity of k is a direct consequence of the requirement kβ ∈ Z. This, in turn, ensures that moving the parafermionic field ψ 1 (z) k times around ψ 1 (w) should not produce any phase.
The special case β = 3/2 is analyzed here in some detail. 2 Since the dimension of the basic parafermion reads this provides a sort of parafermionic deformation of supersymmetry. 3 Like for the Z k models, an associativity analysis is enough to fix the central charge of the Z (3/2) k 1 More precisely, the algebra g is the finite form of the affine algebra involved in the coset g/ u(1) r pertaining to the β = 1 version, the algebra that governs the form of the parafermionic OPE within this class. 2 This turns out to be the simplest possibility when β is positive since the associativity conditions are not satisfied for β = 1/2. 3 Exotic supersymmetry has been explored in [9] also from a parafermionic point of view. However, the models considered there are such that the basic parafermionic field has dimension 1 + 1/k, which requires β = −1 and a 1 = 2. This choice of dimension is motivated by the aim of reproducing the relation Q k ∼ P between the exotic supercharge Q and the translation operator P . models to For k = 2, the central charge is −3/5, which is precisely that of the Virasoro minimal model M (3,5). The Z (3/2) 2 model is indeed equivalent to M (3,5). 4 This relation between the Z (3/2) 2 model and a particular Virasoro minimal model is the simplest example of a general relationship between Z (3/2) k and W k models, whose precise phrasing is Note that for k even (and positive), k + 1 and k + 3 are ensured to be relatively prime.
2 Structure of the Z Consider the set of k parafermionic fields ψ n , n = 0, · · · , k − 1, of conformal dimensions As usual with parafermions, the mode decomposition of ψ 1 depends upon the sector on which it acts. These sectors are labeled by integers t = 0, · · · , k − 1. In the sector t, we have i.e., the sector determines the fractional power of z. In the second relation, we have traded t for its rescaled version q, and wrote 3/2k as λ: A similar expression holds for the decomposition of ψ † 1 , with q→ − q. The inverted versions of (6) and its dagger version are: Interpreting q as the charge that characterizes the sector t amounts to assign the charge q = 2 to A (actually this is essentially the reference charge, the charge concept being relative) while that of A † is −2. It is convenient to drop the fractional part (more precisely, the part proportional to λ) since it is easily reconstructed from the charge of the state on which the mode acts), signaling this omission in the mode writing by replacing A by A. Thus, when acting on an arbitrary state of charge q, we write The defining (holomorphic) OPEs of the Z k parafermionic conformal algebra [1] are where ψ 0 = I and ψ † n = ψ k−n . The remaining OPEs are obtained by conjugation, with c n,n ′ = c k−n,k−n ′ .
Following [1], the commutation relations are found to be where (with c given by (3)) In the same way, we easily obtain the following commutation relations involving only A modes: There is an identical expression with A replaced by A † 5 .
The highest-weight states |hws are naturally defined from being annihilated by the action of the positive parafermionic modes: The vacuum |0 is certainly a particular example of a highest-weight state. The parafermionic field ψ n itself is associated to the following vacuum descendant state:

Structure constants and central charge
In this section, we will fix the value of the central charge and the structure constants for the Z (3/2) k parafermionic theory. This will be done by a rather simple method using mode computations (cf. [13]).
Let us start by fixing the central charge. The trick is to start with a simple string of three modes acting on the vacuum. For the first mode A u |0 we chose the highest value of u that makes the state non-vanishing. This is u = −3/2. The second mode is chosen to be such that the resulting state is proportional to L −1 |0 . This second mode is thus A † −1−u = A † 1/2 . Finally, we choose the third mode such that the resulting state is proportional to A u |0 . We thus consider By commuting the two rightmost term (which is indicated below by the underbrace) with (11), we have 5 The relative sign in (11) indicates that ψ 1 and ψ † 1 are mutually fermionic. The one in (13) shows that ψ 1 is not fermionic with respect to itself. These signs have been obtained by associativity.
This, of course, is compatible with the fact that A † 1/2 A −3/2 |0 is proportional to L −1 |0 = 0. On the other hand, by commuting the two left-most terms still using (11), we obtain In establishing this result, we used the case n = 1 of the following relations: which are easily proved by means of the commutation relations (11) and (13) supplemented by an inductive argument. Coming back the computation of the central charge, equating the two expressions obtained for A −1/2 A † 1/2 A −3/2 |0 yields the announced result (3). This value can be recalculated in several different ways, confirming thereby the associativity of the theory.
Let us now turn to the computation of the structure constants. We first show that the constants c n,n ′ are fixed by c 1,n . Introducing the compact notation (c) n = c 1,1 · · · c 1,n−1 , we have, in a schematic notation (i.e., by dropping the z dependence and regarding these equalities in terms of leading terms): From this, we conclude that the sought for relationship. Note also that We are now in position to calculate c 2 1,n . From the previous relation, we get We next commute A † 3/2 with the first A −3/2 factor using (11): where ∆ n stands for and used In the intermediate steps, the conditions (19) have been taken into account. Now by iterating this result, we get, with c fixed as above, Comparing the two distinct expressions we have derived for this state, we end up with Note that the generic structure constant have to be imaginary. In the simplest case k = 2, we see that c 2 1,1 = c 2 1,k−1 = 1 as it should. Finally, using (22), we have

Primary singular vectors and spectrum
Verma modules are obtained by acting on the highest-weight states with the different lowering operators. A spanning set of states is given by: with m 1 , m 2 and m 3 running over all integers. Note that the above states have relative charge 2(m 2 − m 3 ). We are interested in finding the characteristics of those highest-weight states that make the modules completely reducible, that is, for which there is an infinite number of the above states that are not independent.
Like for any parafermionic theory, if a string made of A −1/2 or A † −1/2 modes, acting on a highestweight state, is allowed to run freely, it will eventually reach a conformal dimension lower than that of the highest-weight state. (We stress that in order to see this, we need to take into account the fractional part of the modes.) For that reason, it is natural to look for singular vectors in the form of (A −1/2 ) r+1 |hws and (A † −1/2 ) r ′ +1 |hws for some integers r and r ′ . Yet, we have no characterization of the highest-weight states. For sure, it must belong to a definite sector t (and recall for the usual parafermions, the sector label characterizes the highest-weight state uniquely).
So we first require the singular vectors to obey the highest-weight conditions (14). This fixes the conformal dimensions of the highest-weight states to be We next look for singular vectors in the form (A † −1/2 ) r ′ +1 |hws leading the to same conformal dimension for the highest-weight states. We thereby obtain It is clear at this point that we cannot get rid of this second parameter r. Consequently, it must become a second quantum number characterizing our highest-weight states: At this stage, the absence of any dependence upon k, meaning that the different models would have the same primary singular vectors, indicate us that we certainly do not have obtained the full set of primary singular vectors that would characterize the completely reducible representations. In order to unravel further constraints, observe that we have yet no way of removing states of the form (A −3/2 ) ℓ (A −1/2 ) r |t, r , which, for ℓ sufficiently large, will once again have negative relative conformal dimension. It is thus unsurprising to discover another set of null states that does depend upon k: and Note however that these null states |χ have been obtained as the solutions of the conditions instead of (14). These conditions are enough to ensure their decoupling from the whole module, but they also suggest that these states do arise as descendants of genuine singular vectors. We found the simplest of such singular vectors in the module of relative charge 0 to be: This is indeed a primary singular vector, obeying the highest-weight conditions (14). By acting on this vector (39) with (A −1/2 ) r+1 and taking (32) into account, we get which is identical to (36) for k = 2r +t. We can also generate (36) in a similrar way. We thus naturally expect a whole sequence of similar singular vectors to be at the source of (36) and (37). 6 By definition, r has to be a non-negative integer. From (36) or (37), we deduce that 0 ≤ r ≤ (k − t)/2. This bound, together with 0 ≤ t ≤ k − 1, allow us to fix the spectrum of the theory.

Conclusion
We have introduced new Z k parafermionic models that belongs to the general class of models introduced in [1]: the structure of the algebra is still a cyclic Z k -one but the dimension of the basic parafermion is modified as compared with those of the usual su(2) k / u(1) model, by the multiplicative factor 3/2. In this letter, we have confined ourself to the study of the basic properties of these models, such as the determination of the essential parameters of the theory (structure constants and central charge) as well as a determination of the spectrum of the reducible modules. Much remains to be done and/or clarified: the determination of the field identifications, the complete analysis of the singular vectors, the derivation of character formulae, the unravelling of a quasi-particle basis and the corresponding fermionic form of the character, etc. These topics will be considered elsewhere. Our main aim here was to establish the well-definiteness of these models. Further support for this comes from their proposed identification with the W k (k + 1, k + 3) models, which has been explicitly checked for k = 2, 4, 6, 8. 7 With regard to the last point, let us indicate that the identification (4) can be generalized in a very natural way, for all integer or half-integer values of β ≥ 1 as follows: (41) 6 The complete singular-vector structure and the resulting character formula will be presented elsewhere. 7 Note that the central charge of the W