Resolving Spin(7) Orbifolds: From Local ALE Geometry to Global Exceptional Holonomy

Resolving singular Spin(7) spaces links exceptional geometry with gauge-theory vacuum structure. This talk builds torsion-free structures on compact orbifold resolutions from local ALE data encoded by harmonic self-dual forms. It explains the relation to supersymmetric gauge equations and asks when local solutions extend globally. Gluing methods and Dirac operators identify analytic obstructions, with explicit examples attaining full Spin(7) holonomy.