GW/PT-correspondence for CY 4-folds-I: Canonical Orientations

Abstract

We build a cohesive topological theory of orientations on moduli stacks for non-compact complex-analytic Calabi-Yau $4$-folds by extending and generalising the work of Joyce and Upmeier, and the work of Bojko. As a result, we get, in particular, canonical family orientations for all moduli spaces of Pandharipande-Thomas pairs. Our proofs use the algebraic topology of mapping spaces representing $KU$- and $KO$- theory, as well as some properties of the Lie groups $E_8$ and, surprisingly, $\operatorname{HSpin}(16)$. This is part of a joint project with Felix Thimm dedicated to the proof of the $GW/PT$-correspondence for Calabi-Yau $4$-folds by implementing Pardon's programme in dimension $4$.