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Wall-crossing for invariants of equivariant 3CY categories

Nikolas Kuhn, Henry Liu, Felix Thimm

TL;DR

The article develops a comprehensive wall-crossing framework for operational K-homology invariants of equivariant 3-Calabi–Yau categories arising from virtual cycles, generalizing Joyce’s universal wall-crossing; central to the construction is a symmetrized pullback that preserves symmetric obstruction theories and yields a vertex-algebraic, Lie-algebraic structure on framed auxiliary moduli. It defines semistable invariants z_α(τ) and an overarching universal coefficient system, proving a dominant wall-crossing formula that expresses changes across stability conditions as nested Lie brackets, and demonstrating framing-independence via master-space arguments. The framework is applied to DT/PT/BS vertices, providing explicit DT/PT descendent and PT/BS vertex correspondences in Calabi–Yau limits, and to refined Vafa–Witten invariants, illustrating the power of operational K-homology in capturing wall-crossing phenomena. The results connect to cohomological analogues, offer potential modularity insights for VW invariants, and set up a flexible, renormalizable approach to wall-crossing across a broad class of equivariant 3CY categories. Overall, the work advances a robust geometric and algebraic toolkit for tracking how enumerative invariants transform under stability variations in a highly structured, symmetry-preserving setting.

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.

Wall-crossing for invariants of equivariant 3CY categories

TL;DR

The article develops a comprehensive wall-crossing framework for operational K-homology invariants of equivariant 3-Calabi–Yau categories arising from virtual cycles, generalizing Joyce’s universal wall-crossing; central to the construction is a symmetrized pullback that preserves symmetric obstruction theories and yields a vertex-algebraic, Lie-algebraic structure on framed auxiliary moduli. It defines semistable invariants z_α(τ) and an overarching universal coefficient system, proving a dominant wall-crossing formula that expresses changes across stability conditions as nested Lie brackets, and demonstrating framing-independence via master-space arguments. The framework is applied to DT/PT/BS vertices, providing explicit DT/PT descendent and PT/BS vertex correspondences in Calabi–Yau limits, and to refined Vafa–Witten invariants, illustrating the power of operational K-homology in capturing wall-crossing phenomena. The results connect to cohomological analogues, offer potential modularity insights for VW invariants, and set up a flexible, renormalizable approach to wall-crossing across a broad class of equivariant 3CY categories. Overall, the work advances a robust geometric and algebraic toolkit for tracking how enumerative invariants transform under stability variations in a highly structured, symmetry-preserving setting.

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.
Paper Structure (42 sections, 102 theorems, 516 equations, 3 figures)

This paper contains 42 sections, 102 theorems, 516 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General motivation
  3. The geometric setting
  4. Summary of main results
  5. Summary of applications
  6. Notation, conventions, acknowledgements
  7. Background and setup
  8. Equivariant K-theory
  9. Equivariant operational K-homology
  10. Monoidal stacks and vertex algebras
  11. Auxiliary framed stacks
  12. Obstruction theories
  13. Symmetrized enumerative invariants
  14. Semistable invariants
  15. Construction and main result
  16. ...and 27 more sections

Key Result

Theorem 1

Suppose $\tau$ is a weak stability condition on $\mathscr{A}$ for which Assumption assump:semistable-invariants holds. Then there exists a unique collection of operational K-homology classes satisfying the following properties:

Figures (3)

  • Figure 1: A one-parameter family of weak stability conditions defining a wall-crossing problem of simple type. Central charges of all classes not of type $A$ and $B$ stay constant and far away from central charges of classes of type $A$ and $B$.
  • Figure 2: Proof strategy for dominant wall-crossing formula.
  • Figure 3: Proof strategy for \ref{['wc:eq:aux-wc-formula']} for $(\beta,\bm{{e}})=(\alpha,\bm{{0}})$ and $(s,x)=(1,-1)$.

Theorems & Definitions (261)

  • Example
  • Theorem : Operational semistable invariants
  • Remark
  • Definition : Universal coefficients
  • Lemma : joyce-config-iv
  • Definition : Dominance conditions
  • Theorem : Dominant wall-crossing formula
  • Remark
  • Remark
  • Proposition
  • ...and 251 more