Orientations on Moduli Spaces of Coherent Sheaves on Calabi-Yau 4-folds - Part I

This lecture by Dominic Joyce from the University of Oxford, presented at the M-Seminar at Kansas State University on April 24, 2025, explores the challenges in defining DT4 invariants on compact Calabi-Yau 4-folds. Discover how orientations on moduli spaces of semistable coherent sheaves are essential for this process, and learn about the mistake found in the Cao-Gross-Joyce 2020 proof claiming orientability of the moduli stack M of perfect complexes. Follow the explanation of how this mistake can be fixed under additional hypotheses on the cohomology H^3(X,Z), and understand how "flag structures" on H^4(X,Z) can determine canonical orientations. The lecture also covers the broader project on orientability in gauge theory, introducing bordism categories Bord_n(BG) and their connection to spin bordism groups, ultimately reducing orientability questions to calculations in Algebraic Topology. This is part of joint work with Markus Upmeier and represents the first part of a series on this advanced mathematical topic.