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An account is given of the structure and representations of chiral bosonicmeromorphic conformal field theories (CFT's), and, in particular, theconditions under which such a CFT may be extended by a representation to form anew theory. This general approach is illustrated by considering the untwistedand $Z_2$-twisted theories, $H(\Lambda)$ and $\tilde H(\Lambda)$ respectively,which may be constructed from a suitable even Euclidean lattice $\Lambda$.Similarly, one may construct lattices $\Lambda_C$ and $\tilde\Lambda_C$ byanalogous constructions from a doubly-even binary code $C$. In the case when$C$ is self-dual, the corresponding lattices are also. Similarly, $H(\Lambda)$and $\tilde H(\Lambda)$ are self-dual if and only if $\Lambda$ is. We show that$H(\Lambda_C)$ has a natural ``triality'' structure, which induces anisomorphism $H(\tilde\Lambda_C)\equiv\tilde H(\Lambda_C)$ and also a trialitystructure on $\tilde H(\tilde\Lambda_C)$. For $C$ the Golay code,$\tilde\Lambda_C$ is the Leech lattice, and the triality on $\tildeH(\tilde\Lambda_C)$ is the symmetry which extends the natural action of (anextension of) Conway's group on this theory to the Monster, so setting trialityand Frenkel, Lepowsky and Meurman's construction of the natural Monster modulein a more general context. The results also serve to shed some light on theclassification of self-dual CFT's. We find that of the 48 theories $H(\Lambda)$and $\tilde H(\Lambda)$ with central charge 24 that there are 39 distinct ones,and further that all 9 coincidences are accounted for by the isomorphismdetailed above, induced by the existence of a doubly-even self-dual binarycode.Comment: 65 page

Conformal field theories, representations and lattice constructions