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

TL;DR

This paper introduces generalized K-theoretic invariants for abelian categories, establishes wall-crossing formulas using a new K-Hall algebra structure, and extends Joyce--Liu results to broader contexts.

Contribution

It defines new K-theoretic invariants, constructs a K-Hall algebra, and proves wall-crossing formulas without requiring framing functors, extending previous work.

Findings

Defined generalized K-theoretic invariants from abelian categories.

Proved wall-crossing formulas using non-abelian localization.

Extended Joyce--Liu wall-crossing results to non-standard hearts.

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 $\delta$-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.