Orientation data for moduli spaces of coherent sheaves over Calabi–Yau 3-folds - Joyce, D; Upmeier, M (Elsevier, 2021)
Let X be a compact Calabi–Yau 3-fold, and write M,M for the moduli stacks of objects in coh(X), Dbcoh(X). There are natural line bundles KM → M, KM → M, analogues of canonical bundles. Orientation data on M,M is an isomorphism class of square root line bundles K1/2 M , K1/2 M , satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman [35, §5] in their theory of motivic Donaldson–Thomas invariants, and is also important in categorifying Donaldson– Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi–Yau 3-folds X, and also for compactlysupported coherent sheaves and perfect complexes on noncompact Calabi–Yau 3-folds X that admit a spin smooth projective compactification X → Y . This proves a longstanding conjecture in Donaldson–Thomas theory. These are special cases of a more general result. Let X be a spin smooth projective 3-fold. Using the spin structure we construct line bundles KM → M, KM → M. We define spin structures on M,M to be isomorphism classes of square roots K1/2 M , K1/2 M . We prove that natural spin structures exist on M,M. They are equivalent to orientation data when X is a Calabi–Yau 3-fold with the trivial spin structure.