On spin structures and orientations for gauge-theoretic moduli spaces

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. For a real elliptic operator $E_\bullet$ we previously studied orientations on the real determinant line bundle over $\mathcal{B}_P$. 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. Here we consider complex elliptic operators $F_\bullet$ and introduce the idea of spin structures, square roots of the complex determinant line bundle of $F_\bullet$. 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 $X \times S^1$. Thus, if $P \to X$ and $Q \to X \times S^1$ are principal $G$-bundles with $Q|_{X\times\{1\}} \cong P$, we relate spin structures on $(\mathcal{B}_P,F_\bullet)$ to orientations on $(\mathcal{B}_Q,E_\bullet)$ for a certain class of operators $F_\bullet$ on $X$ and $E_\bullet$ on $X\times S^1$. Combined with arXiv:1811.02405, we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups $G=U(m), SU(m)$. In a sequel arXiv:2001.00113 we apply this to define canonical orientation data for all Calabi-Yau 3-folds $X$ over the complex numbers, as in Kontsevich-Soibelman arXiv:0811.2435, solving a long-standing problem in Donaldson-Thomas theory.

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