ar X iv : h ep - t h / 92 05 06 7 v 2 2 5 M ay 1 99 2 Discussion of Self - Dual c = 24 Conformal Field Theories

We discuss questions arising from the recent work of Schellekens[17], and also from an earlier paper by Schellekens and Yankielowicz[19]. We summarise Schellekens' results, and proceed to discuss the uniqueness of the c = 24 self-dual conformal field theory with no weight one states, i.e. the Monster module V ♮ [8]. After introducing the concept of complementary representations, we examine Z 2-orbifold constructions in general, and then proceed to apply our considerations firstly to the specific case of the FKS constructions H(Λ) and then to the reflection twisted theories H(Λ)[5]. Our techniques provide evidence for the existence of several new theories beyond those proven to exist in [5] and conjectured to exist in [19].