Determine whether smooth Spin(7) manifolds admit suitable quantised self-dual flux vacua

Determine whether a smooth compact Spin(7) manifold admits an integral four-form flux class that becomes self-dual at a point in its torsion-free Spin(7)-structure moduli space and satisfies the remaining singlet supersymmetry condition, thereby establishing the existence of candidate supersymmetric Minkowski vacua.

Background

For general compact Spin(7) manifolds, the paper derives geometric supersymmetry conditions for type IIA flux compactifications involving $F_0$, $F_4$, and $F_8$. A supersymmetric Minkowski vacuum requires the anti-self-dual component of $F_4$ to vanish, a relation between the Romans mass and the eight-form flux, and an additional condition involving the integral of $F_4$ against the Cayley form.

The authors explain that counting self-duality equations does not determine the flux cost of stabilising all shape moduli: for a fixed quantised cohomology class, the self-duality locus may be empty or non-isolated, and the norm of the class is constrained by the tadpole. Consequently, the existence of candidate vacua on smooth compact Spin(7) manifolds remains unresolved and requires control of both the integral flux lattice and the relevant self-duality locus.