Quasi-BPS categories for symmetric quivers with potential

TL;DR

This paper introduces and analyzes quasi-BPS categories associated with symmetric quivers with potential, establishing their structural properties, decompositions, and connections to noncommutative hyperk"ahler varieties and crepant resolutions.

Contribution

It constructs semiorthogonal decompositions for categories of matrix factorizations of symmetric quivers with potential, generalizing previous work and introducing reduced quasi-BPS categories related to preprojective algebras.

Findings

Proved wall-crossing equivalence of quasi-BPS categories

Established strong generation and support properties

Identified reduced quasi-BPS categories as noncommutative hyperk"ahler varieties

Abstract

We study certain categories associated to symmetric quivers with potential, called quasi-BPS categories. We construct semiorthogonal decompositions of the categories of matrix factorizations for moduli stacks of representations of (framed or unframed) symmetric quivers with potential, where the summands are categorical Hall products of quasi-BPS categories. These results generalize our previous results about the three loop quiver. We prove several properties of quasi-BPS categories: wall-crossing equivalence, strong generation, and categorical support lemma in the case of tripled quivers with potential. We also introduce reduced quasi-BPS categories for preprojective algebras, which have trivial relative Serre functor and are indecomposable when the weight is coprime with the total dimension. In this case, we regard the reduced quasi-BPS categories as noncommutative local hyperk\"ahler varieties, and as (twisted) categorical versions of crepant resolutions of singularities of good moduli spaces of representations of preprojective algebras. The studied categories include the local models of quasi-BPS categories of K3 surfaces. In a follow-up paper, we establish analogous properties for quasi-BPS categories of K3 surfaces.