arXiv subjects
Publications and source records attributed to P. Montague.
The radical $J(V)$ of a vertex operator algebra $V$ is defined to be the subspace of $V$ consisting of vectors $v$ such that the zero mode $o(v)=0$ on $V$ where $o(v)=v_{wt v-1}$ if $v$ is homogeneous. We establish various facts about $o(v),$ including the determination of $J(V)$ which is shown to be essentially equal to $(L(0)+L(-1))V.$
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, $H(Λ)$ and $\tilde H(Λ)$ respectively, which may be constructed from a suitable even Euclidean lattice $Λ$. Similarly, one may construct lattices $Λ_C$ and $\tildeΛ_C$ 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 $\tilde H(Λ)$ are self-dual if and only if $Λ$ is. We show that $H(Λ_C)$ has a natural ``triality'' structure, which induces an isomorphism $H(\tildeΛ_C)\equiv\tilde H(Λ_C)$ and also a triality structure on $\tilde H(\tildeΛ_C)$. For $C$ the Golay code, $\tildeΛ_C$ is the Leech lattice, and the triality on $\tilde H(\tildeΛ_C)$ 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 $\tilde H(Λ)$ 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.
We discuss questions arising from the work of Schellekens. After introducing the concept of complementary representations, we examine $Z_2$-orbifold constructions in general, and propose a technique for identifying the orbifold theory without knowledge of its explicit construction. This technique is then generalised to twists of order 3, 5 and 7, and we proceed to apply our considerations to the FKS constructions $H(Λ)$ ($Λ$ an even self-dual lattice) and the reflection-twisted orbifold theories $\widetilde H(Λ)$, which together remain the only $c=24$ theories which have so far been proven to exist. We also make, in the course of our arguments, some comments on the automorphism groups of the theories $H(Λ)$ and $\widetilde H(Λ)$, and of meromorphic theories in general, introducing the concept of deterministic theories.
We describe a pair of constructions of Eisenstein lattices from ternary codes, and a corresponding pair of constructions of conformal field theories from lattices which turn out to have a string theoretic interpretation. These are found to interconnect in a similar way to results for binary codes, which led to a generalisation of the triality structure relevant in the construction of the Monster module. We therefore make some comments regarding a series of constructions of $V^\natural$. In addition, we present a complete construction of the Niemeier lattices from ternary codes, which in view of the above analogies should prove to be of great importance in the problem of the classification of self-dual $c=24$ conformal field theories. Other progress towards this problem is summarised, and some comments arise from this discussion regarding the uniqueness of the Monster conformal field theory. (Talk presented at the "Monster Bash", Ohio State University, May 1993.)
We discuss questions arising from the recent work of Schellekens, and also from an earlier paper by Schellekens and Yankielowicz. We summarise Schellekens' results, and proceed to discuss the uniqueness of the c=24 self-dual conformal field theory with no weight one states, i.e. the Monster module $V^\natural$. After introducing the concept of complementary representations, we examine $Z_2$-orbifold constructions in general, and then proceed to apply our considerations firstly to the specific case of the FKS constructions $H(Λ)$ and then to the reflection twisted theories $\widetilde H(Λ)$. Our techniques provide evidence for the existence of several new theories beyond those proven to exist in previous work and conjectured to exist in Schellekens and Yankielowicz.