On the uniqueness of the twisted representation in the Z$_{2}$ orbifold construction of a conformal field theory from a lattice

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