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

Abstract

This work develops new ideas and tools to establish equivariant wall-crossing in Calabi-Yau four categories. In the process, I introduce several necessary formalisms, including an equivariant deformation of Joyce's vertex algebras built on a novel and explicitly computable definition of equivariant homology for stacks. The proof of the wall-crossing formula is then given for Calabi-Yau four 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 follows from the wall-crossing formula for Joyce-Song stable pairs. In joint work with Kuhn-Liu-Thimm, these frameworks together with functoriality established here for Park's virtual pullback diagrams are used to prove the general CY4 wall-crossing conjecture.

Added properness condition for local DT/PT wall-crossing and further minor corrections. Referenced the now finished joined work with Kuhn-Liu-Thimm