Determine whether more general graded orbifold constructions are possible

Determine whether the proposed generalization of the Gepner orbifold projection, using distinct subgroups for different spin-structure sectors, can be made consistent by allowing more general discrete-torsion data.

Background

The mirror construction considered in the paper uses the same orbifold group in all spin-structure sectors. The authors note a possible broader formulation in which the NS and R sectors are associated with different components of a Z/2-graded group.

For the discrete-torsion class developed in the paper, modular T-invariance fails when the sector-dependent groups differ. The unresolved issue is whether a more general discrete-torsion formalism could restore consistency and permit this broader class of orbifolds.

References

Such a generalization may still be possible by allowing discrete torsion to be more general (cf \,), but we do not explore the possibility in this article.

— Notes on Gepner Construction  (2608.26017 - Nishikawa et al., 26 Aug 2026) in Section 4, Mirror Orbifold Group, footnote following the necessary condition ${}^\circ\Gamma^{NS}={}^\circ\Gamma^R$