Exhaustiveness of the c=1 unitary 2d CFT catalog

Establish whether the currently known unitary two-dimensional conformal field theories with central charge c=1—comprising the moduli space of compactified free boson theories, their orbifolds, and the discrete exceptional cases—exhaust all unitary c=1 CFTs, i.e., prove or disprove that no additional unitary c=1 CFTs exist beyond these classes.

Background

At c=1, there is a widely studied set of unitary 2d CFTs that includes a continuous moduli space of compactified bosons, their orbifolds, and a finite list of exceptional theories. This set is commonly referred to as an exhaustive catalog in the physics literature.

However, the authors explicitly note that a rigorous proof of exhaustiveness is lacking. Resolving this would close a longstanding gap between common belief and formal classification, and would clarify whether any additional unitary c=1 CFTs exist outside the known families.