Quasi-BPS categories for K3 surfaces

TL;DR

This paper introduces quasi-BPS categories for K3 surfaces, establishing their properties, wall-crossing equivalences, and connections to BPS invariants, thus providing a categorical framework for understanding moduli spaces and invariants of K3 surfaces.

Contribution

It constructs and studies quasi-BPS categories for K3 surfaces, including their semiorthogonal decompositions, wall-crossing equivalences, and properties of reduced categories, linking them to BPS invariants.

Findings

Reduced quasi-BPS categories are proper, smooth, and have trivial Serre functor étale locally.

Topological K-theory of these categories recovers BPS invariants of K3 surfaces.

The categories serve as noncommutative hyperkähler varieties and categorical crepant resolutions.

Abstract

We introduce and begin the study of quasi-BPS categories for K3 surfaces, which are a categorical version of the BPS cohomologies for K3 surfaces. We construct semiorthogonal decompositions of derived categories of coherent sheaves on moduli stacks of semistable objects on K3 surfaces, where each summand is a categorical Hall product of quasi-BPS categories. We also prove the wall-crossing equivalence of quasi-BPS categories, which generalizes Halpern-Leistner's wall-crossing equivalence of moduli spaces of stable objects for primitive Mukai vectors on K3 surfaces. We also introduce and study a reduced quasi-BPS category. When the weight is coprime to the Mukai vector, the reduced quasi-BPS category is proper, smooth, and its Serre functor is trivial \'{e}tale locally on the good moduli space. Moreover we prove that its topological K-theory recovers the BPS invariants of K3 surfaces, which are known to be equal to the Euler characteristics of Hilbert schemes of points on K3 surfaces. We regard reduced quasi-BPS categories as noncommutative hyperk\"ahler varieties which are categorical versions of crepant resolutions of singular symplectic moduli spaces of semistable objects on K3 surfaces.