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Quasi-BPS categories for K3 surfaces

Tudor Pădurariu, Yukinobu Toda

TL;DR

The paper defines and studies reduced quasi-BPS categories $\mathbb{T}_S(v)_w^{\rm{red}}$ for K3 surfaces, showing primitive vectors recover twisted derived moduli categories while non-primitive cases with $\gcd(v,w)=1$ yield smooth, proper, and étale-locally Serre-trivial categories that serve as categorical crepant resolutions of singular moduli. It constructs semiorthogonal decompositions of derived moduli stacks into categorical Hall products of quasi-BPS blocks, and proves wall-crossing equivalences for these categories, both globally and on formal fibers. Topological K-theory of the reduced categories is shown to recover BPS invariants, linking the categorification to Behrend-function-weighted DT theory, with a DT-theoretic interpretation in the local K3 setting. The work connects DT theory, hyperkähler geometry, and noncommutative crepant resolutions by presenting a categorical framework that mirrors PBW-type decompositions, establishes (étale-local) Serre properties, and proposes a conjectural HK-equivalence class: the reduced quasi-BPS categories are deformation-equivalent to derived categories of Hilbert schemes of points on $S$, at least in low-genus/genera cases. Overall, the paper provides a robust categorical apparatus for BPS/Dt invariants on K3s, including explicit local-quiver models, wall-crossing machinery, and a rich interplay between geometry, representation theory, and noncommutative geometry.

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 étale 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 hyperkähler varieties which are categorical versions of crepant resolutions of singular symplectic moduli spaces of semistable objects on K3 surfaces.

Quasi-BPS categories for K3 surfaces

TL;DR

The paper defines and studies reduced quasi-BPS categories for K3 surfaces, showing primitive vectors recover twisted derived moduli categories while non-primitive cases with yield smooth, proper, and étale-locally Serre-trivial categories that serve as categorical crepant resolutions of singular moduli. It constructs semiorthogonal decompositions of derived moduli stacks into categorical Hall products of quasi-BPS blocks, and proves wall-crossing equivalences for these categories, both globally and on formal fibers. Topological K-theory of the reduced categories is shown to recover BPS invariants, linking the categorification to Behrend-function-weighted DT theory, with a DT-theoretic interpretation in the local K3 setting. The work connects DT theory, hyperkähler geometry, and noncommutative crepant resolutions by presenting a categorical framework that mirrors PBW-type decompositions, establishes (étale-local) Serre properties, and proposes a conjectural HK-equivalence class: the reduced quasi-BPS categories are deformation-equivalent to derived categories of Hilbert schemes of points on , at least in low-genus/genera cases. Overall, the paper provides a robust categorical apparatus for BPS/Dt invariants on K3s, including explicit local-quiver models, wall-crossing machinery, and a rich interplay between geometry, representation theory, and noncommutative geometry.

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 étale 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 hyperkähler varieties which are categorical versions of crepant resolutions of singular symplectic moduli spaces of semistable objects on K3 surfaces.
Paper Structure (61 sections, 59 theorems, 397 equations, 1 figure)

This paper contains 61 sections, 59 theorems, 397 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Semiorthogonal decompositions into quasi-BPS categories
  3. Quasi-BPS categories for reduced stacks
  4. Topological K-theory of quasi-BPS categories
  5. Motivation from Donaldson-Thomas theory
  6. Motivation from hyperkähler geometry
  7. Acknowledgements
  8. Preliminaries
  9. Notations for (derived) stacks
  10. DG-categories
  11. Generation of dg-categories
  12. Semiorthogonal decompositions
  13. Graded matrix factorizations
  14. The Koszul equivalence
  15. Window categories
  16. ...and 46 more sections

Key Result

Theorem 1.1

(Theorem thm:sodK3) Let $\sigma \in \mathrm{Stab}(S)$ be a generic stability condition. Then there exists a subcategory (called quasi-BPS category) such that there is a semiorthogonal decomposition The right hand side is after all partitions $(d_i)_{i=1}^k$ of $d$ and all weights $(w_i)_{i=1}^k\in\mathbb{Z}^k$ such that and each fully-faithful functor in (intro:sod) is given by the categorical

Figures (1)

  • Figure 1: Notation used in the paper

Theorems & Definitions (124)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 114 more