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

TL;DR

This paper establishes a connection between topological K-theory and BPS cohomologies for quasi-BPS categories of symmetric quivers with potential, revealing filtrations and computing K-theory in relation to vanishing cycles.

Contribution

It introduces filtrations on topological K-theory of quasi-BPS categories and relates them to BPS cohomologies, also computing K-theory of matrix factorizations and proving a Grothendieck-Riemann-Roch theorem.

Findings

Filtrations on K-theory with graded pieces isomorphic to BPS cohomologies.

Computed topological K-theory of matrix factorizations via vanishing cycles.

Proved compatibility between Koszul equivalence in K-theory and dimensional reduction in cohomology.

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-To\"en-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.