Conformal field theory, triality and the monster group

Abstract

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 Z 2-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 Λ C and a twisted one Λ̃ C . It is shown that H ̃ (Λ̃ C ) always has a triality structure, and that this triality induces first an isomorphism H Λ ̃ C )≌ H ̃ (Λ C and, through this, a triality of H Λ̃(Λ̃ C ). In the case where C is the Golay code, Λ̃ C 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.