Comments on nonunitary conformal field theories

Abstract

As is well-known, nonunitary RCFTs are distinguished from unitary ones in a number of ways, two of which are that the vacuum 0 doesn't have minimal conformal weight, and that the vacuum column of the modular S matrix isn't positive. However there is another primary field, call it o, which has minimal weight and has positive S column. We find that often there is a precise and useful relationship, which we call the Galois shuffle, between primary o and the vacuum; among other things this can explain why (like the vacuum) its multiplicity in the full RCFT should be 1. As examples we consider the minimal WSU(N) models. We conclude with some comments on fractional level admissible representations of affine algebras. As an immediate consequence of our analysis, we get the classification of an infinite family of nonunitary W 3 minimal models in the bulk.

The W N Minimal Model Classification

“…This lemma is crucial to the 'modern' approach (see Steps 1-3 below) to modular invariant classifications, but fails in general for non-unitary modular data. Because of this, non-unitary modular invariant classifications can look very different (see [18,27] for dramatic examples) and will in general require new arguments.…”

“…Fortunately, in many non-unitary RCFTs, the minimal primary and the vacuum are related in a definite way called the Galois shuffle [18]. When this holds, M oo = 1 and Lemma 2.1 remains valid.…”

“…the matrices S and T defining the modular group representation) looks like that of SU(N ) × SU(N ) WZW modular data at fractional heights p/q and q/ p (height = N + level). Applying to this modular data the Galois shuffle of [18], we associate to each W N ( p, q) minimal model a unique SU(N ) × SU(N ) modular invariant at integral heights p, q. For N = 2 and N = 3, the SU(N ) × SU(N ) modular invariants corresponding to W N minimal models can always be expressed in terms of those of SU(N ) together with simple-current modular invariants (though the proof for N = 3 is difficult, as we see in this paper).…”

“…[11], and for necessary aspects of modular data and modular invariants, see [21]. The subtleties occurring in non-unitary theories are discussed in [18]. The standard references on W -algebras are the review [6] and the reprint volume [7].…”

Notes on the Verlinde formula in nonrational conformal field theories

“…An interesting set of generic comments was made in [16]. Most striking is the fact that one can argue for the identification of a special primary that not necessarily corresponds to the vacuum (but that can be seen as a unity).…”