Let X be a compact manifold, G a Lie group, $P\to X$ a principal G-bundle, and ${\mathcal{B}}_P$ the infinite-dimensional moduli space of connections on P modulo gauge, as a topological stack. For a real elliptic operator $E_{•}$ we previously studied orientations on the real determinant line bundle over ${\mathcal{B}}_P$, twisting $E_{•}$ by connections ${\mathrm{\nabla }}_{\mathrm{Ad}(P)}$. These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson [15], [16], [17].

Here we consider complex elliptic operators $F_{•}$ and introduce the idea of spin structures, square roots of the complex determinant line bundle of $F_{•}$ twisted over ${\mathcal{B}}_P$. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures.

Our main result identifies spin structures on X with orientations on $X\times {\mathcal{S}}^1$. Thus, if $P\to X$ and $Q\to X\times {\mathcal{S}}^1$ are principal G-bundles with $Q{|}_{X\times \{1\}}\cong P$, we relate spin structures on $({\mathcal{B}}_P,F_{•})$ to orientations on $({\mathcal{B}}_Q,E_{•})$ for a certain class of operators $F_{•}$ on X and $E_{•}$ on $X\times {\mathcal{S}}^1$.

Combined with [25], we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups $G=\mathrm{U}(m),\mathrm{SU}(m)$. In a sequel [26] we will apply this to define canonical orientation data for all Calabi–Yau 3-folds X over $\mathbb{C}$, as in Kontsevich and Soibelman [28, §5.2], solving a long-standing problem in Donaldson–Thomas theory.

令X为紧流形，G为李群，$P\to X$本金G束${\mathcal{乙}}_P$P模量规上的连接的无穷维模空间，作为拓扑堆栈。对于真正的椭圆算子${Ë}_{•}$ 我们之前在实行列式线束上研究了方向 ${\mathcal{乙}}_P$，扭曲 ${Ë}_{•}$ 通过连接 ${\mathrm{\nabla }}_{\mathrm{广告}（P）}$。这些用于构造通常意义上的光滑规范理论模空间上的取向，自唐纳森[15]，[16]，[17]的工作以来，已经进行了广泛的研究。

这里我们考虑复数椭圆算子$F_{•}$并介绍了自旋结构的概念，复行列式束的平方根。$F_{•}$ 扭曲 ${\mathcal{乙}}_P$。这些可以用来在光滑复数规范理论模空间上按一般意义构造自旋结构。我们研究了这种自旋结构的存在和分类。

我们的主要结果确定了X上具有方向为的自旋结构$X\times {\mathcal{小号}}^{\mathrm{1个}}$。因此，如果$P\to X$ 和 $问\to X\times {\mathcal{小号}}^{\mathrm{1个}}$是主要的G束$问{|}_{X\times \{\mathrm{1个}\}}\cong P$，我们将自旋结构与 $（{\mathcal{乙}}_P，F_{•}）$ 转向 $（{\mathcal{乙}}_{问}，{Ë}_{•}）$ 对于特定类别的运营商 $F_{•}$在X和${Ë}_{•}$ 在 $X\times {\mathcal{小号}}^{\mathrm{1个}}$。

结合[25]，我们获得了自旋6流形和规范组上正狄拉克数的规范自旋结构 $G=\mathrm{ü}（米），\mathrm{苏}（米）$。在续集[26]我们会将此定义规范的方向数据对所有卡拉比-丘3折痕X在$\mathbb{C}$就像Kontsevich和Soibelman [28，§5.2]中一样，解决了唐纳森-托马斯理论中一个长期存在的问题。