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arXiv · 2512.22360

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

Ivan Karpov·Miguel Moreira
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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.

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Ivan Karpov, Miguel Moreira. 2026-04-07. Generalized K-theoretic invariants and wall-crossing via non-abelian localization. https://arxiv.org/abs/2512.22360

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