Quasi-BPS categories for K3 surfaces

BPS invariants and cohomology are central objects in the study of (Kontsevich-Soibelman) Hall algebras or in enumerative geometry of Calabi-Yau 3-folds. In joint work with Yukinobu Toda, we introduce and study a categorical version of BPS cohomology for local K3 surfaces, called quasi-BPS categories. For a generic stability condition, we construct semiorthogonal decompositions of (Porta-Sala) Hall algebra of a K3 surface in products of quasi-BPS categories. When the weight and the Mukai vector are coprime, the quasi-BPS category is smooth, proper, and with trivial Serre functor etale locally on the good moduli space. Thus quasi-BPS categories provide (twisted) categorical (etale locally) crepant resolutions of the moduli space of semistable sheaves on a K3 surface for a generic stability condition and a general Mukai vector. We also discuss a categorical version of the \chi-independence phenomenon for BPS invariants/ cohomology.