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Register a new userConjectured Z_2-Orbifold Constructions of Self-Dual Conformal Field Theories at Central Charge 24 - the Neighborhood Graph
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P.S. Montague
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By considering constraints on the dimensions of the Lie algebra corresponding to the weight one states of Z_2 and Z_3 orbifold models arising from imposing the appropriate modular properties on the graded characters of the automorphisms on the underlying conformal field theory, we propose a set of constructions of all but one of the 71 self-dual meromorphic bosonic conformal field theories at central charge 24. In the Z_2 case, this leads to an extension of the neighborhood graph of the even self-dual lattices in 24 dimensions to conformal field theories, and we demonstrate that the graph becomes disconnected.
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Following on from earlier work relating modules of meromorphic bosonic conformal field theories to states representing solutions of certain simple equations inside the theories, we show, in the context of orbifold theories, that the intertwiners between twisted sectors are unique and described explicitly in terms of the states corresponding to the relevant modules. No explicit knowledge of the structure of the twisted sectors is required. Further, we propose a general set of sufficiency conditions, illustrated in the context of a third order no-fixed-point twist of a lattice theory, for verifying consistency of arbitrary orbifold models in terms of the states representing the twisted sectors.
An account is given of the structure and representations of chiral bosonic meromorphic conformal field theories (CFTs), 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 $tilde H(Lambda)$ respectively, which may be constructed from a suitable even Euclidean lattice $Lambda$. Similarly, one may construct lattices $Lambda_C$ and $tildeLambda_C$ 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 $tilde H(Lambda)$ are self-dual if and only if $Lambda$ is. We show that $H(Lambda_C)$ has a natural ``triality structure, which induces an isomorphism $H(tildeLambda_C)equivtilde H(Lambda_C)$ and also a triality structure on $tilde H(tildeLambda_C)$. For $C$ the Golay code, $tildeLambda_C$ is the Leech lattice, and the triality on $tilde H(tildeLambda_C)$ is the symmetry which extends the natural action of (an extension of) Conways group on this theory to the Monster, so setting triality and Frenkel, Lepowsky and Meurmans 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 CFTs. 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 isomorphism detailed above, induced by the existence of a doubly-even self-dual binary code.
Following on from a general observation in an earlier paper, we consider the continuous symmetries of a certain class of conformal field theories constructed from lattices and their reflection-twisted orbifolds. It is shown that the naive expectation that the only such (inner) symmetries are generated by the modes of the vertex operators corresponding to the states of unit conformal weight obtains, and a criterion for this expectation to hold in general is proposed.
Following on from recent work describing the representation content of a meromorphic bosonic conformal field theory in terms of a certain state inside the theory corresponding to a fixed state in the representation, and using work of Zhu on a correspondence between the representations of the conformal field theory and representations of a particular associative algebra constructed from it, we construct a general solution for the state defining the representation and identify the further restrictions on it necessary for it to correspond to a ground state in the representation space. We then use this general theory to analyze the representations of the Heisenberg algebra and its $Z_2$-projection. The conjectured uniqueness of the twisted representation is shown explicitly, and we extend our considerations to the reflection-twisted FKS construction of a conformal field theory from a lattice.
We investigate orbifold constructions of conformal field theories from lattices by no-fixed-point automorphisms (NFPAs) $Z_p$ for $p$ prime, $p>2$, concentrating on the case $p=3$. Explicit expressions are given for most of the relevant vertex operators, and we consider the locality relations necessary for these to define a consistent conformal field theory. A relation to constructions of lattices from codes, analogous to that found in earlier work in the $p=2$ case which led to a generalisation of the triality structure of the Monster module, is also demonstrated.
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