Smooth G2 resolutions of non-free quotients of Calabi–Yau×S1

Determine whether non-free quotients of the form (X6 × S1)/Z2, obtained from a global involution that acts anti-holomorphically on the six-dimensional Calabi–Yau space X6 and reflects the circle S1, admit smooth G2 resolutions in general.

Background

In the massless type IIA dual frame, the seven-dimensional geometry is globally described by a Z2 quotient of X6 × S1. When the involution acts freely, one obtains smooth barely G2 manifolds. In the constructions considered here, the global action is not free, leading to singular quotients.

The authors note that while non-free quotients of this type can be defined, it is not established in general whether these singular spaces admit smooth G2 resolutions. Resolving this would clarify the geometric status of the backgrounds and potential uplifts to M-theory.

References

More generally, non-free quotients of this type can still be considered, as also discussed in the references above, but the existence of a smooth G$_2$ resolution is not known in full generality, see .

— T-dualities and scale-separated AdS$_3$ in massless IIA on $(X_6 \times S^1)/\mathbb{Z}_2$  (2603.26615 - Tringas, 27 Mar 2026) in Subsection 2.2, Orbifold for new massive and massless IIA

Moreover, the most challenging open problem is to show that the topological $G_2$-manifolds we have constructed are actually (singular degenerations of) $G_2$-manifolds. To do so, one needs to construct a torsion-free $G_2$-holonomy metric on these spaces.

— $G_2$-Manifolds from 4d $\mathcal{N}=1$ Quivers  (2608.21238 - Braun et al., 21 Aug 2026) in Section 'Discussion and Outlook', paragraph 'Future questions'

As the field theory predicts 4d $N=1$ supersymmetry, we conjecture that the resulting geometry is a (topological) $G_2$ holonomy manifold.

— $G_2$-Manifolds from 4d $\mathcal{N}=1$ Quivers  (2608.21238 - Braun et al., 21 Aug 2026) in Section 1, Introduction; Section 'Discussion and Outlook', paragraph 'Concatenating Building Blocks'