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Topological K-theory of quasi-BPS categories of symmetric quivers with potential

Tudor Pădurariu, Yukinobu Toda

TL;DR

This work develops a topological K-theory framework for quasi-BPS categories attached to symmetric quivers with potential, establishing filtrations whose graded pieces recover monodromy-invariant BPS cohomologies. It constructs Chern character and cycle maps from the topological K-theory of matrix-factorization categories to monodromy-invariant vanishing-cycle cohomology, and proves a Grothendieck–Riemann–Roch theorem in this setting. A key achievement is linking the K-theory of quasi-BPS categories to BPS invariants via a decomposition-theorem, and showing compatibility with Koszul equivalence and dimensional reduction, allowing computations through preprojective algebras and reduced moduli. The results yield exact dimension matches between topological K-theory of quasi-BPS categories and monodromy-invariant BPS data, and they enable explicit identifications in cases like preprojective algebras and K3-local geometries, with applications to categorical DT theory and local Calabi–Yau geometries.

Abstract

In previous works, we introduced and studied certain categories called quasi-BPS categories associated to symmetric quivers with potential, preprojective algebras, and local surfaces. They have properties reminiscent of BPS invariants/ cohomologies in enumerative geometry, for example they play important roles in categorical wall-crossing formulas. In this paper, we make the connections between quasi-BPS categories and BPS cohomologies more precise via the cycle map for topological K-theory. We show the existence of filtrations on topological K-theory of quasi-BPS categories whose associated graded are isomorphic to the monodromy invariant BPS cohomologies. Along the way, we also compute the topological K-theory of categories of matrix factorizations in terms of the monodromy invariant vanishing cycles (a version of this comparison was already known by work of Blanc-Robalo-Toën-Vezzosi), prove a Grothendieck-Riemann-Roch theorem for matrix factorizations, and prove the compatibility between the Koszul equivalence in K-theory and dimensional reduction in cohomology. In a separate paper, we use the results from this paper to show that the quasi-BPS categories of K3 surfaces recover the BPS invariants of the corresponding local surface, which are Euler characteristics of Hilbert schemes of points on K3 surfaces.

Topological K-theory of quasi-BPS categories of symmetric quivers with potential

TL;DR

This work develops a topological K-theory framework for quasi-BPS categories attached to symmetric quivers with potential, establishing filtrations whose graded pieces recover monodromy-invariant BPS cohomologies. It constructs Chern character and cycle maps from the topological K-theory of matrix-factorization categories to monodromy-invariant vanishing-cycle cohomology, and proves a Grothendieck–Riemann–Roch theorem in this setting. A key achievement is linking the K-theory of quasi-BPS categories to BPS invariants via a decomposition-theorem, and showing compatibility with Koszul equivalence and dimensional reduction, allowing computations through preprojective algebras and reduced moduli. The results yield exact dimension matches between topological K-theory of quasi-BPS categories and monodromy-invariant BPS data, and they enable explicit identifications in cases like preprojective algebras and K3-local geometries, with applications to categorical DT theory and local Calabi–Yau geometries.

Abstract

In previous works, we introduced and studied certain categories called quasi-BPS categories associated to symmetric quivers with potential, preprojective algebras, and local surfaces. They have properties reminiscent of BPS invariants/ cohomologies in enumerative geometry, for example they play important roles in categorical wall-crossing formulas. In this paper, we make the connections between quasi-BPS categories and BPS cohomologies more precise via the cycle map for topological K-theory. We show the existence of filtrations on topological K-theory of quasi-BPS categories whose associated graded are isomorphic to the monodromy invariant BPS cohomologies. Along the way, we also compute the topological K-theory of categories of matrix factorizations in terms of the monodromy invariant vanishing cycles (a version of this comparison was already known by work of Blanc-Robalo-Toën-Vezzosi), prove a Grothendieck-Riemann-Roch theorem for matrix factorizations, and prove the compatibility between the Koszul equivalence in K-theory and dimensional reduction in cohomology. In a separate paper, we use the results from this paper to show that the quasi-BPS categories of K3 surfaces recover the BPS invariants of the corresponding local surface, which are Euler characteristics of Hilbert schemes of points on K3 surfaces.
Paper Structure (78 sections, 91 theorems, 471 equations, 1 figure)

This paper contains 78 sections, 91 theorems, 471 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. The main result
  4. Cohomological DT theory
  5. Categorical DT theory
  6. Matrix factorizations and vanishing cycles
  7. Quasi-BPS categories for symmetric quivers with potential
  8. Main ingredients used in the proof
  9. Quasi-BPS categories for preprojective algebras
  10. Weight-independence
  11. Plan of the paper
  12. Acknowledgements
  13. Preliminaries
  14. Some notations and conventions
  15. Stacks and semiorthogonal decompositions
  16. ...and 63 more sections

Key Result

Theorem 1.1

(Corollary corollarytheorem61) Let $(Q, W)$ be a symmetric quiver with potential. Then, for each dimension vector $d$, there exists a real cocharacter $\delta$ such that Here, the right hand side is the monodromy invariant part of the BPS cohomology.

Figures (1)

  • Figure 1: Notation introduced in the paper

Theorems & Definitions (187)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Theorem 2.1
  • Theorem 2.2
  • ...and 177 more