Let X be a compact Calabi–Yau 3-fold, and write $\mathcal{M},\bar{\mathcal{M}}$ for the moduli stacks of objects in $\mathrm{coh}(X),D^b\mathrm{coh}(X)$. 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 [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 $X\hookrightarrow Y$. 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 [33], which constructs ‘spin structures’ (square roots of a certain complex line bundle $K_P^{E_{•}}\to {\mathcal{B}}_P$) on differential-geometric moduli stacks ${\mathcal{B}}_P$ of connections on a principal $\mathrm{U}(m)$-bundle $P\to X$ over a compact spin 6-manifold X.

令X为紧凑的Calabi–Yau的3倍，然后写$\mathcal{中号}，\stackrel{〜}{\mathcal{中号}}$ 用于对象的模堆栈 $\mathrm{h}（X），d^b\mathrm{h}（X）$。有自然线束${ķ}_{\mathcal{中号}}\to \mathcal{中号}$， ${ķ}_{\stackrel{〜}{\mathcal{中号}}}\to \stackrel{〜}{\mathcal{中号}}$，规范束的类似物。定位数据上$\mathcal{中号}，\stackrel{〜}{\mathcal{中号}}$ 是平方根线束的同构类 ${ķ}_{\mathcal{中号}}^{\mathrm{1个}/2}，{ķ}_{\stackrel{〜}{\mathcal{中号}}}^{\mathrm{1个}/2}$，满足短精确序列堆栈上的兼容性条件。它是由Kontsevich和Soibelman [35，§5]在他们的唐纳森-托马斯不变量原动力理论中引入的，并且在使用反向滑轮对唐纳森-托马斯理论进行分类中也很重要。

我们表明，可以为所有紧凑的Calabi–Yau 3折X构造自然取向数据，还可以为紧凑支撑的连贯滑轮和非紧凑的Calabi–Yau 3折X的完美配合物构造，从而允许自旋光滑射影压实$X\hookrightarrow ÿ$。这证明了唐纳森-托马斯理论中的一个长期猜想。

这些是更普遍结果的特殊情况。令X为自旋光滑投影3倍。使用旋转结构，我们构造线束${ķ}_{\mathcal{中号}}\to \mathcal{中号}$， ${ķ}_{\stackrel{〜}{\mathcal{中号}}}\to \stackrel{〜}{\mathcal{中号}}$。我们定义的自旋结构上$\mathcal{中号}，\stackrel{〜}{\mathcal{中号}}$ 成为平方根的同构类 ${ķ}_{\mathcal{中号}}^{\mathrm{1个}/2}，{ķ}_{\stackrel{〜}{\mathcal{中号}}}^{\mathrm{1个}/2}$。我们证明自然自旋结构存在于$\mathcal{中号}，\stackrel{〜}{\mathcal{中号}}$。当X是具有微不足道的自旋结构的Calabi–Yau的3倍时，它们等效于方向数据。

我们使用先前的论文[33]证明了这一点，该论文构建了“旋转结构”（某些复杂线束的平方根） ${ķ}_P^{{Ë}_{•}}\to {\mathcal{乙}}_P$）在微分几何模量堆栈上 ${\mathcal{乙}}_P$ 主体上的连接数 $\mathrm{ü}（米）$-捆 $P\to X$在紧凑自旋6歧管X上。