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Generalized K-theoretic invariants and wall-crossing via non-abelian localization

Ivan Karpov, Miguel Moreira

TL;DR

This work develops a non-abelian localization–based framework to define and study generalized $K$-theoretic wall-crossing invariants for abelian categories under stability conditions. By constructing a new $K$-Hall algebra on the $K$-homology of the rigidified moduli stack, the authors define $\delta$-invariants and their logarithms, the $\varepsilon$-invariants, and prove wall-crossing formulas that are valid without framing functors. The theory unifies and extends Joyce–Liu invariants, connects to motivic wall-crossing, and relates $K$-theory to cohomology through a virtual Riemann–Roch framework, including homological lifts and a comparison with Joyce’s vertex algebra. The results apply to non-standard hearts of $D^b(X)$ and tilt stability, offering a conceptual and technically streamlined approach to wall-crossing with broad potential applications in DT-type theories. The paper also outlines conjectures about no-pole phenomena and homological lifts in greater generality, signaling directions for future work in Calabi–Yau fourfolds and beyond.

Abstract

Given an abelian category and a stability condition satisfying appropriate conditions, we define generalized $K$-theoretic invariants and prove that they satisfy wall-crossing formulas. For this, we introduce a new associative algebra structure on the $K$-homology of the stack of objects of an abelian category, which we call the $K$-Hall algebra. We first define $δ$-invariants directly coming from the stack of semistable objects and use the $K$-Hall algebra to take a formal logarithm and construct $\varepsilon$-invariants. We prove that these satisfy appropriate wall-crossing formulas using the non-abelian localization theorem. Based on work of Joyce in the cohomological setting, Liu had previously defined similar invariants assuming the existence of a framing functor; we show that when their definition of invariants makes sense it agrees with ours. Our results extend Joyce--Liu wall-crossing to non-standard hearts of $D^b(X)$, for which framing functors are not known to exist.

Generalized K-theoretic invariants and wall-crossing via non-abelian localization

TL;DR

This work develops a non-abelian localization–based framework to define and study generalized -theoretic wall-crossing invariants for abelian categories under stability conditions. By constructing a new -Hall algebra on the -homology of the rigidified moduli stack, the authors define -invariants and their logarithms, the -invariants, and prove wall-crossing formulas that are valid without framing functors. The theory unifies and extends Joyce–Liu invariants, connects to motivic wall-crossing, and relates -theory to cohomology through a virtual Riemann–Roch framework, including homological lifts and a comparison with Joyce’s vertex algebra. The results apply to non-standard hearts of and tilt stability, offering a conceptual and technically streamlined approach to wall-crossing with broad potential applications in DT-type theories. The paper also outlines conjectures about no-pole phenomena and homological lifts in greater generality, signaling directions for future work in Calabi–Yau fourfolds and beyond.

Abstract

Given an abelian category and a stability condition satisfying appropriate conditions, we define generalized -theoretic invariants and prove that they satisfy wall-crossing formulas. For this, we introduce a new associative algebra structure on the -homology of the stack of objects of an abelian category, which we call the -Hall algebra. We first define -invariants directly coming from the stack of semistable objects and use the -Hall algebra to take a formal logarithm and construct -invariants. We prove that these satisfy appropriate wall-crossing formulas using the non-abelian localization theorem. Based on work of Joyce in the cohomological setting, Liu had previously defined similar invariants assuming the existence of a framing functor; we show that when their definition of invariants makes sense it agrees with ours. Our results extend Joyce--Liu wall-crossing to non-standard hearts of , for which framing functors are not known to exist.
Paper Structure (49 sections, 43 theorems, 285 equations)

This paper contains 49 sections, 43 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. $\delta$-invariants
  4. $K$-Hall algebra and $\varepsilon$-invariants
  5. Wall-crossing formula
  6. Comparison with Joyce--Liu invariants
  7. Analogy with motivic wall-crossing
  8. Conjectures and future directions
  9. Notation and conventions
  10. Acknowledgments
  11. Preliminaries: abelian categories and stability conditions
  12. Abelian categories and moduli stacks
  13. Stability conditions, $\Theta$-stratifications and good moduli spaces
  14. Hearts of $D^b(X)$ and tilt stability
  15. Preliminaries: $K$-theory and $K$-homology
  16. ...and 34 more sections

Key Result

Theorem A

Definition def: KHall makes an associative algebra.

Theorems & Definitions (121)

  • Theorem A: =Theorem \ref{['thm: associativity']}
  • Example 1.1
  • Theorem B: =Theorem \ref{['thm: generalwc']}
  • Corollary 1.2
  • Theorem C: =Theorem \ref{['thm: comparisonJL']}
  • Theorem D: =Theorem \ref{['thm: liealgebrahom']}, Proposition \ref{['prop: nopolesknown']}(3)
  • Theorem E: =Theorem \ref{['thm: homliftsjoyce']}
  • Example 1.3
  • Remark 2.4
  • Definition 2.5
  • ...and 111 more