Fixing Z/N charge to make the twist-defect topological spin θ_N well-defined

Ascertain whether a canonical choice of Z/N charge for twist defects in the stack of N copies of a two-dimensional quantum spin system can be specified so that the topological spin θ_N itself (rather than only its Nth power) is well-defined, thereby yielding a more refined invariant of invertible phases.

Background

In the construction of twist defects for N copies with cyclic permutation symmetry, there is a local ambiguity near the insertion point that affects the Z/N charge assignment of the defect. This ambiguity leads to an Nth-root indeterminacy in the topological spin θ_N, making only (θ_N)^N an invariant.

Resolving whether the Z/N charge can be canonically fixed would strengthen the index from (θ_N)^N to θ_N itself, potentially enabling a more refined invariant and clarifying connections to the chiral central charge in conformal field theory contexts.