Fusion of irreducible modules in

Abstract

Based on symmetry principles, we derive a fusion algebra generated from repeated fusions of the irreducible modules appearing in the -extended logarithmic minimal model . 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 and . 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 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 Δkp − 1, 1, k = 1, 2, 3. We also identify polynomial fusion rings associated with our fusion algebras.