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

Abstract

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 KM1/2,KM‾1/2, 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 compactly-supported coherent sheaves and perfect complexes on noncompact Calabi–Yau 3-folds X that admit a spin smooth projective compactification X↪Y. This proves a long-standing 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 KM1/2,KM‾1/2. 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.We prove this using our previous paper [33], which constructs ‘spin structures’ (square roots of a certain complex line bundle KPE•→BP) on differential-geometric moduli stacks BP of connections on a principal U(m)-bundle P→X over a compact spin 6-manifold X.