Orientation data for moduli spaces of coherent sheaves over Calabi-Yau 3-folds

TL;DR

This paper proves the existence of natural orientation data for moduli stacks of sheaves on Calabi-Yau 3-folds, confirming a long-standing conjecture and extending to noncompact cases using spin structures.

Contribution

It constructs natural orientation data for all compact Calabi-Yau 3-folds and certain noncompact cases, resolving a key conjecture in Donaldson-Thomas theory.

Findings

Constructed orientation data for all compact Calabi-Yau 3-folds.

Extended the construction to noncompact Calabi-Yau 3-folds with spin compactifications.

Proved the existence of natural spin structures on moduli stacks of sheaves.

Abstract

Let $X$ be a compact Calabi-Yau 3-fold, and write $\mathcal M,\bar{\mathcal M}$ for the moduli stacks of objects in coh$(X),D^b$coh$(X)$. There are natural line bundles $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal M}$, analogues of canonical bundles. Orientation data on $\mathcal M,\bar{\mathcal M}$ is an isomorphism class of square root line bundles $K_{\mathcal M}^{1/2},K_{\bar{\mathcal M}}^{1/2}$, satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman arXiv:1006.270 in their theory of motivic Donaldson-Thomas invariants, and is important in categorifying Donaldson-Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi-Yau 3-folds, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds $X$ with a spin smooth projective compactification $X\hookrightarrow Y$. This proves a long-standing conjecture in Donaldson-Thomas theory. These are special cases of a more general result. Let $X$ be a spin smooth projective 3-fold. Using the spin structure we construct line bundles $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal M}$. We define spin structures on $\mathcal M,\bar{\mathcal M}$ to be isomorphism classes of square roots $K_{\mathcal M}^{1/2},K_{\bar{\mathcal M}}^{1/2}$. We prove that natural spin structures exist on $\mathcal M,\bar{\mathcal M}$. They are equivalent to orientation data when $X$ is a Calabi-Yau 3-fold with the trivial spin structure. We prove this using our previous paper arXiv:1908.03524, which constructs 'spin structures' (square roots of a certain complex line bundle $K_P\to\mathcal B_P$) on differential-geometric moduli stacks $\mathcal B_P$ of connections on a principal U$(m)$-bundle $P\to X$ over a compact spin 6-manifold $X$.