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.

arXiv commentUpdated to address the additional assumption in Joyce-Upmeier