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

This paper establishes a precise connection between quasi-BPS categories of symmetric quivers with potential and BPS cohomologies by constructing filtrations on their topological K-theory whose associated graded pieces are isomorphic to monodromy-invariant BPS cohomologies, while also proving a Grothendieck-Riemann-Roch theorem for matrix factorizations and demonstrating the compatibility between Koszul equivalence and dimensional reduction.

Original authors: Tudor Pădurariu, Yukinobu Toda

Published 2026-03-27

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CC BY 4.0

Original authors: Tudor Pădurariu, Yukinobu Toda

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to count the number of unique, stable structures that can be built out of a specific set of Lego bricks. In the world of advanced mathematics (specifically algebraic geometry), these "structures" are called sheaves, and the "bricks" are complex shapes defined by equations.

For a long time, mathematicians have had a way to count these structures using numbers called invariants. One famous type of invariant is the Donaldson-Thomas (DT) invariant. Think of this as a simple headcount: "There are 5 stable structures here."

However, sometimes the structures are messy. They might be slightly unstable or have "ghost" versions that make a simple headcount inaccurate. To fix this, mathematicians invented BPS invariants. Instead of just a number, BPS invariants are like a detailed blueprint or a fingerprint of the collection of structures. They tell you not just how many there are, but what kind they are, and they remain stable even when the rules of the game change (a phenomenon called "wall-crossing").

The Problem: The Blueprint is Missing

The problem is that while we know these "BPS fingerprints" exist and have powerful properties, we haven't been able to build a concrete category (a mathematical toolbox containing all the rules and relationships) that perfectly matches them. It's like knowing a perfect recipe exists for a cake, but not having the actual kitchen or the ingredients to bake it.

The Solution: Quasi-BPS Categories

In previous work, the authors (Tudor Pădurariu and Yukinobu Toda) built a new kind of toolbox called Quasi-BPS categories. Think of these as "practice kitchens" or "simulated environments" designed to mimic the behavior of the real BPS fingerprints. They are built using symmetric quivers with potential.

Quivers: Imagine a map with dots (vertices) and arrows (edges).
Potential: A set of rules or a "recipe" that tells you how the arrows interact.
Symmetric: The map looks the same if you flip it or rotate it (it has balance).

These categories are mathematical objects that behave almost exactly like the elusive BPS invariants. But the big question remained: Do they actually match the real thing?

The Breakthrough: The "Cycle Map"

This paper answers "Yes" by building a bridge between two different languages:

Topological K-Theory: A way of measuring the "shape" and "size" of these mathematical toolboxes (the Quasi-BPS categories).
BPS Cohomology: The actual, physical "fingerprint" or blueprint of the structures we want to count.

The authors constructed a Cycle Map. Imagine this as a universal translator.

On one side, you have the complex, abstract Lego set (the Quasi-BPS category).
On the other side, you have the final, perfect blueprint (BPS cohomology).
The Cycle Map translates the abstract Lego instructions into the blueprint.

The Main Result: A Perfect Match

The paper proves that if you take the "Topological K-Theory" of their Quasi-BPS categories and apply this translator, you get an exact match with the monodromy-invariant BPS cohomology.

In simple terms:

The authors built a mathematical machine (the Quasi-BPS category). They proved that if you measure the "size" of this machine using a specific ruler (Topological K-Theory), the result is exactly the same as the "weight" of the real-world object (BPS cohomology) it was designed to mimic.

Why This Matters (The "So What?")

Categorification: This is a huge step in "categorification." Instead of just having a number (like "5"), we now have a rich, structured object (a category) that contains that number and much more information. It's like upgrading from a receipt (just the total price) to the full itemized invoice with all the details.
New Tools: By proving these categories work, mathematicians now have a new, powerful toolkit to study complex shapes in geometry, specifically those related to Calabi-Yau 3-folds (shapes that are crucial in string theory and mirror symmetry).
Matrix Factorizations: The paper also solves a side puzzle involving "Matrix Factorizations" (a way of breaking down complex equations). They showed how to count these using the same "fingerprint" method, proving a famous theorem (Grothendieck-Riemann-Roch) in this new context.

The Analogy of the "Ghost"

Imagine you are trying to count the number of ghosts in a haunted house.

Old Method: You just count the shadows. Sometimes a shadow looks like two ghosts, or one ghost splits into two. It's messy.
BPS Invariants: You realize there is a "true count" that doesn't change even if the ghosts move around.
Quasi-BPS Categories: You build a virtual reality simulation of the house.
This Paper: You prove that if you run a specific diagnostic test on your virtual simulation, the data it spits out is identical to the "true count" of the ghosts.

Summary

This paper is a bridge. It connects a newly invented, abstract mathematical structure (Quasi-BPS categories) with the established, physical reality of counting geometric objects (BPS cohomology). By proving they are isomorphic (mathematically identical in their measurable properties), the authors have validated their new tool and opened the door to using it to solve even harder problems in geometry and physics.

1. Problem Statement and Context

Background:
In enumerative geometry, particularly for Calabi-Yau (CY) 3-folds, BPS invariants are fundamental integer-valued invariants that count semistable coherent sheaves. They generalize Donaldson-Thomas (DT) invariants and play a crucial role in wall-crossing formulas. While DT invariants are defined via integration over moduli spaces, BPS invariants are conjectured to be the Euler characteristics of BPS cohomology spaces (specifically, the hypercohomology of a perverse sheaf called the BPS sheaf).

The Gap:
A major open problem is the categorification of these invariants. The authors aim to construct a dg-category, denoted $\mathcal{BPS}$ , whose periodic cyclic homology or topological K-theory recovers the BPS invariants. Previous work by the authors introduced quasi-BPS categories ( $S(d; \delta)$ ) for symmetric quivers with potential, which serve as local models for these global BPS categories. However, the precise relationship between the topological K-theory of these quasi-BPS categories and the actual BPS cohomology (specifically the monodromy-invariant part) remained to be established rigorously.

Objective:
The paper aims to bridge the gap between the categorical world (quasi-BPS categories) and the cohomological world (BPS cohomology) by constructing a cycle map from the topological K-theory of quasi-BPS categories to the monodromy-invariant BPS cohomology, proving it is an isomorphism (over $\mathbb{Q}$ ).

2. Methodology

The authors employ a sophisticated blend of derived algebraic geometry, representation theory, and homotopy theory. The methodology can be broken down into the following key components:

A. Quasi-BPS Categories and Matrix Factorizations

Setup: For a symmetric quiver $Q$ with potential $W$ , let $X(d)$ be the stack of representations of dimension $d$ . The potential induces a function $\text{Tr } W: X(d) \to \mathbb{C}$ .
Construction: The quasi-BPS category $S(d; \delta)$ is defined as a subcategory of the category of matrix factorizations $\text{MF}(X(d), \text{Tr } W)$ . It is constructed using a "window" subcategory $M(d; \delta) \subset \text{Db}(X(d))$ (based on the work of Špenko–Van den Bergh) determined by a weight $\delta$ and a polytope in the weight lattice.
Goal: Compute $K^{\text{top}}_i(S(d; \delta))$ , the topological K-theory spectrum of this category.

B. Topological K-Theory and Chern Characters

Blanc's Theory: The authors utilize the topological K-theory of dg-categories defined by Blanc, which generalizes topological K-theory of spaces.
Relative K-Theory: They employ Moulinos' theory of relative topological K-theory ( $K^{\text{top}}_M(\mathcal{D})$ ), which associates a sheaf of spectra to a dg-category over a base scheme. This is crucial for relating the K-theory of the stack to the cohomology of the underlying space.
Chern Character Maps: They construct Chern character maps from topological K-theory to cohomology. For matrix factorizations, the target is the cohomology of monodromy-invariant vanishing cycles ( $\phi^{\text{inv}}_f \mathbb{Q}_X$ ).

C. Dimensional Reduction and Koszul Equivalence

Koszul Equivalence: For preprojective algebras (which arise from doubled quivers), the authors use the Koszul equivalence (Isik, Hirano, Toda) to relate the derived category of the preprojective stack to the category of graded matrix factorizations.
Dimensional Reduction: They establish compatibility between the Koszul equivalence in K-theory and the dimensional reduction theorem in cohomology (Davison, Meinhardt). This allows them to translate results from the "tripled" quiver setting (with potential) to the preprojective setting (without potential, but with a specific stack structure).

D. The Cycle Map and Decomposition Theorems

Coproduct-like Maps: A central technical innovation is the construction of "coproduct-like" maps in both K-theory and cohomology associated with partitions of the dimension vector $d$ .
Restriction to BPS Summands: By analyzing the weights of objects in the quasi-BPS category, they show that the image of the cycle map is restricted to specific summands in the cohomology decomposition (BBDG decomposition theorem).
Injectivity and Surjectivity:
- Injectivity: Proved using the admissibility of the quasi-BPS category and the properties of the Chern character for matrix factorizations.
- Surjectivity (Dimension Counting): They use relative topological K-theory to show that the BPS sheaf appears as a direct summand in the relative K-theory of the quasi-BPS category. This ensures the dimension of the K-theory is at least that of the BPS cohomology. Combined with injectivity, this proves the isomorphism.

3. Key Contributions and Results

Main Theorem (Theorem 1.2 / Theorem 8.14)

For a symmetric quiver with potential $(Q, W)$ and a dimension vector $d$ , there exists a Weyl-invariant real weight $\delta$ such that the cycle map induces an isomorphism of $\mathbb{Q}$ -vector spaces:
$c: \text{gr}^\ell K^{\text{top}}_i(S(d; \delta))_{\mathbb{Q}} \xrightarrow{\sim} H^{\dim X(d) - 2\ell - i + 1}(X(d), \text{BPS}^{\text{inv}}_{d, \delta})$
Here, $\text{BPS}^{\text{inv}}_{d, \delta}$ is the monodromy-invariant part of the direct sum of BPS sheaves associated with partitions in a specific set $S^d_\delta$ .

Corollary (Theorem 1.1):
The total dimension of the topological K-theory of the quasi-BPS category recovers the total dimension of the monodromy-invariant BPS cohomology:
$\dim_{\mathbb{Q}} K^{\text{top}}_i(S(d; \delta))_{\mathbb{Q}} = \sum_{j \in \mathbb{Z}} \dim_{\mathbb{Q}} H^j(X(d), \text{BPS}_d)^{\text{inv}}$

Results for Preprojective Algebras (Theorem 1.6)

Using dimensional reduction, the authors compute the topological K-theory of quasi-BPS categories for preprojective algebras ( $T(d; \delta)$ ). They prove:
$\text{gr}^\ell K^{\text{top}}_0(T(d; \delta))_{\mathbb{Q}} \cong H^{-2\ell}(P(d), \text{BPS}^p_{d, \delta})$
where $\text{BPS}^p_{d, \delta}$ are the preprojective BPS sheaves.

Weight Independence (Corollary 1.8)

The authors prove a $\chi$ -independence phenomenon for the dimensions of the K-theory. If $\gcd(v, d) = \gcd(v', d)$ , then:
$\dim_{\mathbb{Q}} K^{\text{top}}_i(S(d)_v)_{\mathbb{Q}} = \dim_{\mathbb{Q}} K^{\text{top}}_i(S(d)_{v'})_{\mathbb{Q}}$
This mirrors known phenomena in DT theory and suggests a deeper categorical independence of the weight parameter.

Sheaf-Theoretic Version (Theorem 1.5)

The paper establishes a stronger, sheaf-theoretic isomorphism:
$K^{\text{top}}_{X(d)}(S(d; \delta))_{\mathbb{Q}} \cong \text{BPS}^{\text{inv}}_{d, \delta}[\beta^{\pm 1}][-\dim X(d)]$
This identifies the relative topological K-theory sheaf directly with the BPS sheaf (shifted and periodic).

4. Significance and Impact

Categorification of BPS Invariants: This work provides the first rigorous proof that the topological K-theory of the quasi-BPS categories (constructed purely from quiver representation theory) recovers the dimensions of the BPS cohomology. This validates the quasi-BPS categories as the correct categorical objects for BPS invariants.
Bridge between K-Theory and Cohomology: The paper successfully constructs a "cycle map" that translates between the non-commutative world of dg-categories and the commutative world of constructible sheaves, generalizing the classical Chern character.
New Tools for Enumerative Geometry: The results provide a new method to compute BPS invariants (and intersection cohomology of moduli spaces) by computing the K-theory of matrix factorizations, which can be more tractable.
Applications to K3 Surfaces: As noted in the abstract and introduction, these local results are the foundation for a separate paper where the authors compute BPS invariants for local K3 surfaces, recovering the Euler characteristics of Hilbert schemes of points on K3 surfaces.
Technical Advances: The development of the relative topological K-theory for semiorthogonal summands and the explicit construction of the cycle map for matrix factorizations with monodromy invariance are significant technical contributions to the field of non-commutative algebraic geometry and motivic homotopy theory.

In summary, the paper resolves a key step in the program of categorifying Donaldson-Thomas theory, proving that the proposed categorical structures (quasi-BPS categories) faithfully encode the enumerative invariants (BPS cohomology) via topological K-theory.