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

TL;DR

This paper establishes orientability results for moduli stacks of perfect complexes on quasi-projective Calabi-Yau 4-folds, extending previous results and connecting algebraic geometry with gauge theory to define DT invariants.

Contribution

It extends orientability results to quasi-projective Calabi-Yau 4-folds and relates algebraic orientations to gauge-theoretic constructions, enabling the definition of DT invariants.

Findings

Extended orientability to projective spin 4-folds.

Defined orientation bundles on moduli stacks with normal crossing divisors.

Related algebraic orientations to gauge theory and established orientability of moduli spaces.

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 arXiv:1504.00690 necessary for defining Donaldson-Thomas type invariants of $X$. We extend first the orientability result of Cao-Gross-Joyce arXiv:1811.09658 to projective spin 4-folds. Then for any smooth projective compactification $\bar{X}$, such that $D=\bar{X}\backslash X$ is strictly normal crossing, we define orientation bundles on the stack $\mathcal{M}_{\bar{X}}\times_{\mathcal{M}_D}\mathcal{M}_{\bar{X}}$ and express these as pullbacks of $\mathbb{Z}_2$-bundles in 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, 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.