Conformal field theories, representations and lattice constructions

An account is given of the structure and representations of chiral bosonic meromorphic conformal field theories (CFT's), and, in particular, the conditions under which such a CFT may be extended by a representation to form a new theory. This general approach is illustrated by considering the untwisted and Z(2)-twisted theories, H(Lambda) and (H) over tilde(Lambda) respectively, which may be constructed from a suitable even Euclidean lattice Lambda. Similarly, one may construct lattices Lambda(C) and <(Lambda)over tilde>C by analogous 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 (H) over tilde(Lambda) are self-dual if and only if Lambda is. We show that H(Lambda(C)) has a natural ''triality'' structure, which induces an isomorphism H(<(Lambda)over tilde>(C)) = <(Lambda)over tilde>(Lambda(C)) and also a triality structure on (H) over tilde(<(Lambda)over tilde>(C)). For C the Golay code, <(Lambda)over tilde>(C) is the Leech lattice, and the triality on (H) over tilde(<(Lambda)over tilde>(C)) is the symmetry which extends the natural action of (an extension of) Conway's group on this theory to the Monster, so setting triality and Frenkel, Lepowsky and Meurman's construction of the natural Monster module in a more general context. The results also serve to shed some light on the classification of self-dual CFT's. We find that of the 48 theories H(Lambda) and (H) over tilde(Lambda) with central charge 24 that there are 39 distinct ones, and further that all 9 coincidences are accounted for by the isomorphism detailed above, induced by the existence of a doubly-even self-dual binary code.