Equivalence of the oriented and unorientable eight-supercharge compactifications

Determine whether M-theory compactification on the unorientable manifold $(M\times T^3)/\langle\sigma\rho_1\rho_2\rho_3\rangle$ and compactification on the oriented manifold $Y\times S^1$, where $Y$ is the Enriques Calabi–Yau three-fold, define equivalent theories or distinct ultraviolet completions with identical massless spectra.

Background

The paper compares two seven-dimensional compactification geometries that preserve eight supercharges: the oriented product $Y\times S^1$, with holonomy contained in $SO(7)$, and the unorientable quotient $$(M\times T^3)/\langle\sigma\rho_1\rho_2\rho_3\rangle$$, whose holonomy is instead contained in $O(7)$ but not in $SO(7)$. Despite this geometric distinction, both compactifications produce a four-dimensional theory with eleven vector multiplets and twelve hypermultiplets.

The unresolved issue is whether the identical massless spectra imply full equivalence or whether the different global geometry and holonomy encode distinct ultraviolet completions, analogous to global variants known in lower-dimensional string-theory compactifications.