Orientations for DT invariants on quasi-projective Calabi-Yau 4-folds

Abstract

For a Calabi-Yau 4-fold $(X,\omega)$, where $X$ is quasi-projective and $\omega$ is a nowhere vanishing section of its canonical bundle $K_X$, the (derived) moduli stack of compactly supported perfect complexes $\mathcal{M}_X$ is $-2$-shifted symplectic and thus has an orientation bundle $O^\omega\to \mathcal{M}_X$ in the sense of Borisov-Joyce, necessary for defining Donaldson-Thomas type invariants of $X$. We first extend the orientability result of Cao-Gross-Joyce/Joyce-Upmeier to projective spin 4-folds. Then for any smooth projective compactification $Y$, such that $D=Y\backslash X$ is strictly normal crossing, we define orientation bundles on the stack $\mathcal{M}_{Y}\times_{\mathcal{M}_D}\mathcal{M}_{Y}$ and express these as pullbacks of orientation bundles from gauge theory, constructed using positive Dirac operators on the double of $X$. As a result, we relate the orientation bundle $O^\omega\to \mathcal{M}_X$ to a gauge-theoretic orientation on the classifying space of compactly supported K-theory. Using orientability of the latter under the compactly supported version of the assumption of Joyce-Upmeier, we obtain orientability of $\mathcal{M}_X$. We also prove orientability of moduli spaces of stable pairs and Hilbert schemes of proper subschemes. Finally, we consider the compatibility of orientations under direct sums.