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

This lecture is the second part of a series by Dominic Joyce from the University of Oxford, presented at the M-Seminar at Kansas State University on May 1, 2025. Explore the complex mathematical challenge of defining DT4 invariants on compact Calabi-Yau 4-folds through orientations on moduli spaces of coherent sheaves. Learn about the mistake discovered in the Cao-Gross-Joyce 2020 proof regarding the orientability of moduli stack M of perfect complexes, and understand the proposed solution involving additional hypotheses on H^3(X,Z) cohomology. Discover how "flag structures" on H^4(X,Z) can determine canonical orientations on M. Delve into the broader project connecting orientability in gauge theory and special submanifolds through "bordism categories" Bord_n(BG), where orientations are encoded as functors to Z_2-torsors. Understand how these questions relate to spin bordism groups and challenging calculations in Algebraic Topology, based on joint work with Markus Upmeier.