Conformal field theory, triality and the Monster group

From an even self-dual N-dimensional lattice, Λ, it is always possible to construct two (chiral) conformal field theories, an untwisted theory H (Λ), and a Z2-twisted theory H (Λ), constructed using the reflection twist. (N must be a multiple of 8 and the theories are modular invariant if it is a multiple of 24.) Similarly, from a doubly-even self-dual binary code C, it is possible to construct two even self-dual lattices, an untwisted one ΛC and a twisted one anti ΛC. It is shown that H(ΛC) always has a triality structure, and that this triality induces first an isomorphism H(anti ΛC)≅H(ΛC) and, through this, a triality of H(anti ΛC). In the case where C is the Golay code, anti ΛC is the Leech lattice and the induced triality is the extra symmetry necessary to generate the Monster group from (an extension of) Conway's group. Thus it is demonstrated that triality is a generic symmetry. The induced isomorphism accounts for all 9 of the coincidences between the 48 conformal field theories H(Λ) and H(Λ) with N=24. (orig.)