Strict enlargement from finite-phase extended critical loop spectra

Determine whether replacing the critical loop-model spectrum \mathcal{B}^1 by the finite-phase spectrum \mathcal{B}^P strictly enlarges the space of conformal bootstrap solutions for P>1, thereby producing additional correlators with nonabelian monodromies.

Background

The extended critical loop models enlarge the spectrum from fields with integer spin to fields whose phases are P-th roots of unity. Since \mathcal{B}^1 is contained in \mathcal{B}^P, the enlarged spectrum cannot reduce the solution space of the conformal bootstrap equations.

The unresolved issue is whether the inclusion is strict. The paper notes that P=2 produces additional fermionic torus one-point solutions, but asks whether higher finite-phase extensions generally yield more solutions and correlators with nonabelian monodromies.