ar X iv : 0 90 6 . 54 14 v 2 [ he p - th ] 1 2 Ju l 2 00 9 Fusion of irreducible modules in WLM ( p , p ′ )

Based on symmetry principles, we derive a fusion algebra generated from repeated fusions of the irreducible modules appearing in the W-extended logarithmic minimal model WLM(p, p′). In addition to the irreducible modules themselves, closure of the commutative and associative fusion algebra requires the participation of a variety of reducible yet indecomposable modules. We conjecture that this fusion algebra is the same as the one obtained by application of the Nahm-Gaberdiel-Kausch algorithm and find that it reproduces the known such results for WLM(1, p′) and WLM(2, 3). For p > 1, this fusion algebra does not contain a unit. Requiring that the spectrum of modules is invariant under a natural notion of conjugation, however, introduces an additional (p− 1)(p′ − 1) reducible yet indecomposable rank-1 modules, among which the identity is found, still yielding a well-defined fusion algebra. In this greater fusion algebra, the aforementioned symmetries are generated by fusions with the three irreducible modules of conformal weights ∆kp−1,1, k = 1, 2, 3. We also identify polynomial fusion rings associated with our fusion algebras.