Wall-crossing for invariants of equivariant 3CY categories

TL;DR

This paper develops a wall-crossing framework for invariants of equivariant 3-Calabi--Yau categories, enabling explicit comparisons and formulas for DT, PT, BS, and Vafa--Witten invariants.

Contribution

It introduces a novel wall-crossing approach preserving symmetry in obstruction theories and applies it to derive explicit invariants relations and formulas.

Findings

Established a wall-crossing framework for equivariant 3CY categories

Derived explicit DT/PT descendent vertex correspondence in the Calabi--Yau limit

Proved wall-crossing formulas for refined Vafa--Witten invariants

Abstract

We provide a wall-crossing framework for operational enumerative invariants of equivariant 3-Calabi--Yau categories arising from virtual cycles. The strategy follows ideas of Joyce's ``universal'' wall-crossing framework arXiv:2111.04694, using the authors' symmetrized pullback technique to preserve the symmetry of the (almost-perfect) obstruction theories throughout. As an application, we define and study wall-crossings of simple type between operational equivariant Donaldson--Thomas (DT), Pandharipande--Thomas (PT), and Bryan--Steinberg (BS) vertices. In particular, we give an explicit DT/PT descendent vertex correspondence in the Calabi--Yau limit. As another application, we construct and prove wall-crossing formulas for operational refined semistable Vafa--Witten invariants.