Cosets, characters and fusion for admissible-level osp(1|2) minimal models

Abstract

A. We study the minimal models associated to osp(1 |2), otherwise known as the fractional-level Wess-Zumino-Witten models of osp(1 |2). Since these minimal models are extensions of the tensor product of certain Virasoro and sl 2 minimal models, we can induce the known structures of the representations of the latter models to get a rather complete understanding of the minimal models of osp(1 |2). In particular, we classify the irreducible relaxed highest-weight modules, determine their characters and compute their Grothendieck fusion rules. We also discuss conjectures for their (genuine) fusion products and the projective covers of the irreducibles. IThis project is part of a programme to understand the admissible-level Wess-Zumino-Witten (WZW) models for a Lie algebra or superalgebra g. While the theories with non-negative integer levels and simple Lie algebras lead to rational conformal field theories and, as such, are very well understood, the situation is much more complicated and rich for other levels or when superalgebras are involved. Indeed, the non-rational admissible-level WZW models are expected to be prime examples of logarithmic conformal field theories, these being models that admit representations on which the hamiltonian acts non-diagonalisably, leading to correlation functions with logarithmic singularities.Another interesting feature of these models is that they have a continuous spectrum of modules.We view our programme as complementary to older approaches. In particular, Quella, Saleur, Schomerus et al.[1-9] approached supergroup WZW theories via free field realisations and semiclassical limits (the minisuperspace analysis), the interest being rather in features of the WZW theory of the supergroup at integer levels. Another approach employed was to learn more about the conformal field theory using the mock modular behaviour of certain irreducible characters [10][11][12]. The relatively accessible case of g = gl(1|1) has also been studied from a more algebraic perspective by two of us [13,14].Presently, we have a very good picture in the case of g = sl 2 [15][16][17][18][19][20][21][22][23]. In order to extend our understanding to more sophisticated theories, one has to develop some basic strategies. First, one has to study the general theory of relaxed highest-weight modules. These natural generalisations of the usual highest-weight modules were introduced in the conformal field theory literature in [24] for g = sl 2 , though they had already appeared in mathematics classifications such as [15], but have only recently been formalised in a general setting [25]. Since then, the role played by irreducible relaxed highest-weight modules in facilitating the study of general admissible-level WZW models has been widely appreciated and the field has been rapidly developing, see [26][27][28][29] for example.Second, one should develop techniques to reconstruct, at least in favourable circumstances, the representation theory of the algebra of interest in terms of those of subalgebras. We call this techniq...

Unitary and non-unitary N = 2 minimal models

“…Based on these results, the actual fusion rules have also been conjectured [54] and those involving just the L r, 0 have recently been proven [66]. These are all listed in Appendix A.…”

“…We have already applied this powerful realisation to the example of non-unitary (logarithmic) parafermions in [52]. A similar application involving a non-Heisenberg coset (and the vice versa direction) has also recently appeared [53,54]. The example that concerns us here has V as the tensor product of the simple affine vertex operator algebra of sl (2), at admissible level k = −2 + u , and the fermionic ghost vertex operator superalgebra (of central charge 1).…”

“…Using the same methodology as detailed in Section 6.2, conjectures for the fusion rules of A 1 (u, ) yield (conjectural) M(u, ) fusion rules. For example, the conjectural fusion rule (A.5) (originally made in [54]) gives providing that 2 s − 2. When s = 1 or − 1, we must remove those [i ′′ ] C E p ′′ ;r,s ′ with s ′ = 0 or from the right-hand side.…”

“…It was recently conjectured [54] that these these staggered modules are projective (in an appropriate category).…”

“…A.4] for an elementary introduction to this concept. The structural conjecture of [54] is then that the Loewy diagram of S ± r,s is…”