On the Uniqueness of the Twisted Representation in the Z_2 Orbifold Construction of a Conformal Field Theory from a Lattice

TL;DR

This paper investigates the uniqueness of twisted representations in Z_2 orbifold conformal field theories, providing a general framework and explicit proof of the conjectured uniqueness, with applications to lattice-based models.

Contribution

It introduces a general solution for the state defining representations and proves the uniqueness of twisted representations in Z_2 orbifold CFTs, extending to lattice constructions.

Findings

Explicit proof of the uniqueness of twisted representations.

General framework for states defining representations.

Application to lattice-based conformal field theories.

Abstract

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.