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Register a new userContinuous Symmetries of Lattice Conformal Field Theories and their $Z_2$-Orbifolds
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Authors
P.S. Montague
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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.
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We consider representations of meromorphic bosonic chiral conformal field theories, and demonstrate that such a representation is completely specified by a state within the theory. The necessary and sufficient conditions upon this state are derived, and, because of their form, we show that we may extend the representation to a representation of a suitable larger conformal field theory. In particular, we apply this procedure to the lattice (FKS) conformal field theories, and deduce that Dongs proof of the uniqueness of the twisted representation for the reflection-twisted projection of the Leech lattice conformal field theory generalises to an arbitrary even (self-dual) lattice. As a consequence, we see that the reflection-twisted lattice theories of Dolan et al are truly self-dual, extending the analogies with the theories of lattices and codes which were being pursued. Some comments are also made on the general concept of the definition of an orbifold of a conformal field theory in relation to this point of view.
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.
For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the corresponding twisted partition functions
Supersymmetric theories with the same bosonic content but different fermions, aka emph{twins}, were thought to exist only for supergravity. Here we show that pairs of super conformal field theories, for example exotic $mathcal{N}=3$ and $mathcal{N}=1$ theories in $D=4$ spacetime dimensions, can also be twin. We provide evidence from three different perspectives: (i) a twin S-fold construction, (ii) a double-copy argument and (iii) by identifying candidate twin holographically dual gauged supergravity theories. Furthermore, twin W-supergravity theories then follow by applying the double-copy prescription to exotic super conformal field theories.
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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