PATH SPACES AND $\mathcal W$ FUSION IN MINIMAL MODELS

Abstract

Product forms of characters of Virasoro minimal models are obtained which factorize into (2, odd) × (3, even) characters. These are related by generalized Rogers-Ramanujan identities to sum forms allowing for a quasiparticle interpretation. The corresponding dilogarithm identities are given and the factorization is used to analyse the related path space structure as well as the fusion of the maximally extended chiral algebra.

Factorized Combinations of Virasoro Characters

“…The factorization of some Virasoro characters in the M(3, 4) and M(4, 5) models was already observed in [1], whereas the factorization of all characters of type χ 2n,t n,m (q) and χ 3n,t n,m (q) was discovered in [2]. It was already discussed in [3,2,4,5,6] that the factorization of characters in these series may be obtained by exploiting the Gauß-Jacobi and Watson identities. Nevertheless, we wish to present here a systematic derivation of these results based on alternative arguments which will also be applicable in a more general situation.…”

“…. , (C. 4) which means that after expanding we will generate a term q nm+∆h . Since nm + ∆h < ∆h + m(s − n), we have to include a factor (1 − q nm+∆h ) on the r. h. s. of (C.4) in order to cancel this term.…”

Dilogarithm Identities in Conformal Field Theory

“…Of course, this method to extract dilogarithm identities from the asymptotics of the character functions can also be applied to other cases -provided a sum form of the characters similar to (2.5) is known [4,12]. Furthermore, in view of the next-to-leading term in the expansion (2.6), it should be possible to obtain dilogarithm expressions for c − 24h i as well, namely from the asymptotic behaviour of suitable linear combinations of characters.…”

“…First, it would be nice if we could write at least one character of any CFT in a sum-form as in the Andrews-Gordon identities (see [19] for an interpretation of this in the language of affine Lie algebras) in order to generate dilogarithm expressions for the central charge and possibly for the conformal dimensions in any RCFT. Ways to achieve this might be to study the annihilating ideals [6] of the models and perhaps to find path representations of the highest weight modules [11,12], or to introduce suitable filtrations on the space of fields. However, as these sum-formulas need not be at all unique, this could lead to further relations for the dilogarithm function.…”

From Path Representations to Global Morphisms for a Class of Minimal Models

“…One can compare these formulas to the well-known Rocha-Caridi expressions for the characters of Virasoro minimal models, which follow directly from the Feigin-Fuchs results on the chain of singular vectors. If one applies the Jacobi triple product identity to the Rocha-Caridi characters, see [41], one obtains the product form…”