2025/07/08 by Arkadij Bojko, Bojko, Arkadij
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2507.05922
openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
This work develops new ideas and tools to establish wall-crossing in Calabi-Yau four categories as originally conjectured by Gross-Joyce-Tanaka. In the process, I set up some necessary new language, including a natural refinement of Joyce's vertex algebras to equivariant homology. The proof is then given for Calabi-Yau four dg-quivers and local CY fourfolds. A crucial part of the problem is showing that the generalized invariants counting stable objects are well-defined. Using a conceptual argument akin to the quantum Lefschetz principle, I show that for torsion-free sheaves, this is already implied by the wall-crossing formula for Joyce-Song stable pairs. Lastly, I introduce an important framework in the form of a stable ∞-categorical formulation of Park's virtual pullback diagrams in the appendix. This implies their functoriality, which is used repeatedly throughout this work.