On Representations of Conformal Field Theories and the Construction of Orbifolds

TL;DR

This paper studies representations of conformal field theories, showing they are fully determined by specific states, and extends these ideas to orbifolds, with applications to lattice theories and their self-duality properties.

Contribution

It provides a complete characterization of representations via states and extends these results to orbifolds, including new proofs of self-duality for lattice conformal field theories.

Findings

Representation of a conformal field theory is determined by a state.

Conditions for extending representations to larger theories are derived.

Reflection-twisted lattice theories are shown to be self-dual.

Abstract

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 Dong's 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.