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 $\mathcal{M},\bar{\mathcal{M}}$ for the moduli stacks of objects in coh$(X),D^b$coh$(X)$. There are natural line bundles $K_{\mathcal{M}}\to \mathcal{M}$, $K_{\bar{\mathcal{M}}}\to \bar{\mathcal{M}}$, analogues of canonical bundles. Orientation data on $\mathcal{M},\bar{\mathcal{M}}$ is an isomorphism class of square root line bundles $K_{\mathcal{M}}^{1/2},K_{\bar{\mathcal{M}}}^{1/2}$, satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman arXiv:1006.270 in their theory of motivic Donaldson-Thomas invariants, and is 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, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds $X$ with a spin smooth projective compactification $X\hookrightarrow Y$. 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 $K_{\mathcal{M}}\to \mathcal{M}$, $K_{\bar{\mathcal{M}}}\to \bar{\mathcal{M}}$. We define spin structures on $\mathcal{M},\bar{\mathcal{M}}$ to be isomorphism classes of square roots $K_{\mathcal{M}}^{1/2},K_{\bar{\mathcal{M}}}^{1/2}$. We prove that natural spin structures exist on $\mathcal{M},\bar{\mathcal{M}}$. 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 arXiv:1908.03524, which constructs 'spin structures' (square roots of a certain complex line bundle $K_P\to {\mathcal{B}}_P$) on differential-geometric moduli stacks ${\mathcal{B}}_P$ of connections on a principal U$(m)$-bundle $P\to X$ over a compact spin 6-manifold $X$.

46 pages. (v2) final version, to appear in Advances in Mathematics