K-Theory Equivalence of Hall and Quasi-BPS Categories

Definitions and mathematical statement

The supplied record concerns coherent $D$-module categories and quasi-BPS categories, denoted respectively by \[ \mathcal A_{d,w}:=D\text{-}\operatorname{mod}_{\mathrm{coh}}(\mathcal X(d))_{w/d}, \qquad \mathcal B_{d,w}:=\mathbb T(d)_w. \] These are names for the categories in the source, not complete definitions: the record does not define $\mathcal X(d)$, the quasi-BPS construction $\mathbb T(d)_w$, or the subscript $w/d$. The admissible parameters, geometric hypotheses, and categorical enhancements must therefore be supplied before a fully specified conjecture can be stated.

The requested $K$-theory assertion has the following schematic form. For the admissible parameters $(d,w)$ intended by the source, the specialization comparison is to induce an isomorphism \[ K^{\varepsilon}(\mathcal A_{d,w}) \cong K^{\varepsilon}(\mathcal B_{d,w}), \qquad \varepsilon\in\{\mathrm{alg},\mathrm{top}\}. \] Here $K^{\mathrm{alg}}$ and $K^{\mathrm{top}}$ stand for the intended algebraic and topological $K$-theory invariants. The record does not specify whether this means spectra, graded groups, or individual groups, nor does it construct or orient the specialization map. Its wording "algebraic or topological" also leaves unresolved whether both versions are asserted independently or a choice of version is intended. The display records the proposed comparison without resolving these qualifications.

The source additionally requests a corresponding equivalence involving a slope-zero monoidal Hall category, written $D\text{-}\mathrm{HA}(Q)_0$ in a damaged display. That display contains categories $D\text{-}\operatorname{mod}_{\mathrm{coh}}(\mathcal X(d))$ indexed by $d\in{\mathbb{N}}^I$, but does not reliably specify their assembly, the slope-zero restriction, or the monoidal product. Neither $Q$ nor $I$ is defined. In particular, the relationship between this vector-valued index $d$ and the quotient $w/d$ is not supplied. The excerpt ends with a reference to "Conjecture 1" without its mathematical content. Consequently, the target category and functor for the requested Hall-category equivalence cannot be stated from this record. This missing assertion is retained as an unresolved part of the problem.

Short English statement

The problem asks whether specialization identifies the algebraic or topological $K$-theory of certain coherent $D$-module categories with that of quasi-BPS categories, and asks for a related slope-zero monoidal Hall-category equivalence. Essential definitions, parameter restrictions, and the target of the categorical equivalence are missing from the supplied excerpt, so only a qualified schematic statement is possible.

Sources

• Supplied local problem registry record, permanent UnsolvedMath ID $30006271$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied title is "K-Theory Equivalence of Hall and Quasi-BPS Categories."
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2025), Oberwolfach Reports, pp. 1271--1332, DOI 10.4171/owr/2025/25. https://doi.org/10.4171/owr/2025/25. The record attributes the conjectural discussion to this report; no individual proposer is supplied.
• Supplied literature source: "OWR 2025 report." https://ems.press/content/serial-article-files/51851.
• Supplied literature source: Quasi-BPS categories. https://arxiv.org/abs/2309.08425.
• Supplied literature source: Topological K-theory of quasi-BPS categories. https://arxiv.org/abs/2309.08432.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that the $K$-theory comparisons and slope-zero Hall-category identification are conjectures in the report, and reports that no general proof was located. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that the official report was checked and notation repaired. These are supplied source-verification claims; the present statement-only draft does not independently verify them.