Conformal field theories, representations and lattice constructions

Abstract

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 andZ ₂-twisted theories, ℋ(Λ) and$\tilde H(\Lambda )$ respectively, which may be constructed from a suitable even Euclidean lattice Λ. Similarly, one may construct lattices$\Lambda _C$ and$\tilde \Lambda _C$ 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(Λ) and$\tilde H(\Lambda )$ are self-dual if and only if Λ is. We show that$H(\Lambda _C )$ has a natural “triality” structure, which induces an isomorphism$H(\tilde \Lambda _C )$ ≡$\tilde H(\Lambda _C )$ and also a triality structure on$\tilde H(\tilde \Lambda _C )$. For$C$ the Golay code,$\tilde \Lambda _C$ is the Leech lattice, and the triality on$\tilde H(\tilde \Lambda _C )$ 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(Λ) 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 isomorphism detailed above, induced by the existence of a doubly-even self-dual binary code.