Wall-crossing for Calabi-Yau fourfolds: framework, tools, and applications

TL;DR

This paper develops new tools and frameworks to establish wall-crossing phenomena in Calabi-Yau fourfolds, including a refined vertex algebra and a stable $mbda$-categorical approach, with applications to invariants and stability conditions.

Contribution

It introduces a refined vertex algebra, a stable $mbda$-categorical framework, and proves wall-crossing for Calabi-Yau four categories, advancing the understanding of invariants and stability.

Findings

Wall-crossing formulas are established for Calabi-Yau four dg-quivers.

A new language refining Joyce's vertex algebras to equivariant homology is developed.

The functoriality of Park's virtual pullback diagrams is proved using stable $mbda$-categories.

Abstract

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 $\infty$-categorical formulation of Park's virtual pullback diagrams in the appendix. This implies their functoriality, which is used repeatedly throughout this work.