Invariance of orientation data for ind-constructible Calabi-Yau $A_{\infty}$ categories under derived equivalence

Abstract

We study orientation data, as introduced by Kontsevich and Soibelman in order to define well-behaved integration maps from the motivic Hall algebra of 3-dimensional Calabi-Yau categories to rings of motives. We start with an example that demonstrates the role of orientation data in this story, before working through the technical details. We give an account of orientation data in the case of categories of compactly supported sheaves on noncompact Calabi-Yau three-folds. We finally study how this structure behaves under pullbacks along quasi-equivalences of categories, prove Kontsevich and Soibelman's conjecture regarding this behaviour, and also some stronger theorems regarding flops and more general tilts.