Orientation data for moduli spaces of coherent sheaves over Calabi–Yau 3-folds

Let X be a compact Calabi–Yau 3-fold, and write M , ? for the moduli stacks of objects in coh ( ) D b . There are natural line bundles K ? analogues canonical bundles. Orientation data on is an isomorphism class square root 1 / 2 satisfying compatibility condition stack short exact sequences. It was introduced by Kontsevich Soibelman [35, §5] their theory motivic Donaldson–Thomas invariants, also important categorifying using perverse sheaves. We show that orientation can constructed all 3-folds compactly-supported coherent sheaves perfect complexes noncompact admit spin smooth projective compactification ? Y This proves long-standing conjecture theory. These special cases more general result. 3-fold. Using structure we construct define structures to classes roots prove exist They equivalent when 3-fold with trivial structure. this our previous paper [33] which constructs ‘spin structures’ (square certain complex bundle P E • B differential-geometric connections principal U m -bundle over 6-manifold