Quasi-BPS Categories in Moduli Theory

• Quasi-BPS categories are defined as distinguished subcategories that capture the primitive BPS sector in diverse moduli problems including symmetric quivers, K3 surfaces, and Higgs bundles.
• They are constructed using matrix factorizations, window subcategories, and Hall products to yield semiorthogonal decompositions that reflect categorical DT theory.
• In reduced or primitive regimes, these categories recover BPS cohomology and exhibit strong geometric properties, acting as noncommutative analogues of crepant resolutions and hyperkähler varieties.

Quasi-BPS categories are dg or triangulated categories introduced as categorical versions of BPS invariants or BPS cohomologies in Donaldson–Thomas-type moduli problems. In existing work they arise for symmetric quivers with potential, for moduli stacks of semistable objects on K3 surfaces, and for moduli stacks of semistable Higgs bundles. In all of these settings they are extracted as distinguished subcategories of ambient derived or matrix-factorization categories, and they appear as the basic summands in semiorthogonal decompositions governed by categorical Hall products. In reduced or primitive regimes they are further interpreted as noncommutative analogues of crepant, hyperkähler, or Hitchin-type geometries (Pădurariu et al., 2023, Pădurariu et al., 2023, Pădurariu et al., 2024, Pădurariu et al., 27 Aug 2025).

1. Origins and scope

The quiver-theoretic origin of the subject is the construction of quasi-BPS categories for symmetric quivers with potential as full subcategories of categories of matrix factorizations. Those categories were introduced as categorical avatars of BPS invariants and BPS cohomologies in DT theory, and they were shown to control semiorthogonal decompositions of framed and unframed DT categories (Pădurariu et al., 2023). The same period saw a parallel development for K3 surfaces, where quasi-BPS categories were introduced as a categorical version of the BPS cohomologies for K3 surfaces (Pădurariu et al., 2023).

The notion was then extended to Higgs-bundle geometry. For twisted Higgs bundles on a smooth projective curve, quasi-BPS categories were introduced as canonical triangulated or dg subcategories inside the derived categories of moduli stacks of semistable twisted Higgs bundles, and they were described as the building blocks of those derived categories. Under a primitivity condition, the resulting BPS categories were identified as noncommutative analogues of Hitchin integrable systems (Pădurariu et al., 2024). In a further reformulation, quasi-BPS categories for semistable Higgs bundles were recast as special cases of limit categories for cotangent stacks, placing them in a proposed Dolbeault geometric Langlands framework (Pădurariu et al., 27 Aug 2025).

Taken together, these constructions exhibit a common pattern. Quasi-BPS categories are not defined by a single universal formula across all moduli problems; rather, they form a family of closely related categorical constructions whose role is stable: they isolate the primitive or BPS-like sector inside larger Hall-theoretic categories. This suggests that the term names a program as much as a definition.

2. Quiver and preprojective constructions

For a symmetric quiver $Q=(I,E)$ with potential $W$, dimension vector $d$, representation space $R(d)$, and gauge group $G(d)$, the basic quotient stack is

$X(d)=R(d)/G(d).$

The construction begins with a “magic” window subcategory

$\mathbb M(d;\delta_d)\subset D^b(X(d)),$

defined by the condition that the dominant weights $\chi$ of the generators satisfy

$\chi+\rho-\delta_d\in \mathbf W(d),$

where $\mathbf W(d)$ is the polytope built from the $W$0-weights of $W$1. The quasi-BPS category is then

$W$2

with shorthand $W$3 when $W$4. For tripled quivers there is also a graded version $W$5, and via Koszul equivalence this has a preprojective incarnation

$W$6

inside the derived category of the corresponding preprojective stack (Pădurariu et al., 2023).

These categories are the Hall atoms of the theory. The unframed matrix-factorization category admits a semiorthogonal decomposition

$W$7

and the framed category admits the analogous decomposition with slope inequalities shifted by the framing parameter. The embedding functors are the corresponding categorical Hall products. The same framework yields wall-crossing equivalences $W$8 for generic stability conditions, and the quasi-BPS categories are proved to be strongly generated (Pădurariu et al., 2023).

A major refinement is the reduced theory for preprojective algebras. Writing $W$9 for the reduced derived stack cut out by the traceless moment map, the reduced quasi-BPS category

$d$0

is defined by the same window condition $d$1. When the weight is coprime with the total dimension, these reduced categories have trivial relative Serre functor and no nontrivial semiorthogonal decomposition, and they are regarded as noncommutative local hyperkähler varieties and as twisted categorical versions of crepant resolutions of the good moduli spaces of preprojective representations (Pădurariu et al., 2023).

3. K3-surface quasi-BPS categories

For a K3 surface $d$2, a generic Bridgeland stability condition $d$3, a Mukai vector $d$4, and an integer weight $d$5, the relevant geometry is the derived moduli stack

$d$6

of $d$7-semistable objects, together with its classical truncation $d$8 and good moduli space $d$9. Scalar automorphisms induce an orthogonal decomposition

$R(d)$0

Choosing a determinant-type real line bundle $R(d)$1, the quasi-BPS category is defined by the intrinsic window construction

$R(d)$2

and for generic $R(d)$3 it depends only on $R(d)$4, yielding

$R(d)$5

There is likewise a reduced version

$R(d)$6

on the reduced moduli stack (Pădurariu et al., 2023).

The local mechanism is quiver-theoretic. Near a polystable object $R(d)$7, formality of $R(d)$8 identifies the formal neighborhood of the K3 moduli stack with the formal neighborhood of a preprojective-algebra representation stack attached to the Ext-quiver. In the deepest stratum, for $R(d)$9 with $G(d)$0, the local model is the $G(d)$1-loop preprojective stack. This imports the quasi-BPS categories of doubled quivers and preprojective algebras directly into the K3 setting (Pădurariu et al., 2023).

The main structural theorem is a categorical PBW-type semiorthogonal decomposition of $G(d)$2 and of its reduced counterpart into Hall products of quasi-BPS categories indexed by partitions of the multiplicity $G(d)$3 and by strict inequalities among the normalized weights. The paper also proves the wall-crossing equivalence

$G(d)$4

for any generic $G(d)$5 (Pădurariu et al., 2023).

The reduced categories carry the strongest geometric interpretation. If $G(d)$6 is primitive, then

$G(d)$7

More generally, when $G(d)$8 and $G(d)$9, the reduced quasi-BPS category is proper and smooth, and its Serre functor is étale locally trivial on the good moduli space. The authors accordingly regard reduced quasi-BPS categories as noncommutative hyperkähler varieties and as categorical versions of crepant resolutions of singular symplectic moduli spaces (Pădurariu et al., 2023).

4. Higgs bundles, BPS conditions, and limit categories

For a smooth projective curve $X(d)=R(d)/G(d).$0, a line bundle $X(d)=R(d)/G(d).$1 with $X(d)=R(d)/G(d).$2 or $X(d)=R(d)/G(d).$3, rank $X(d)=R(d)/G(d).$4, Euler characteristic $X(d)=R(d)/G(d).$5, and weight $X(d)=R(d)/G(d).$6, let

$X(d)=R(d)/G(d).$7

be the derived moduli stack of semistable $X(d)=R(d)/G(d).$8-twisted Higgs bundles. Scalar automorphisms give the orthogonal decomposition

$X(d)=R(d)/G(d).$9

For $\mathbb M(d;\delta_d)\subset D^b(X(d)),$0, the quasi-BPS category

$\mathbb M(d;\delta_d)\subset D^b(X(d)),$1

is defined by a window condition on the allowed weights of $\mathbb M(d;\delta_d)\subset D^b(X(d)),$2 for every map $\mathbb M(d;\delta_d)\subset D^b(X(d)),$3. For $\mathbb M(d;\delta_d)\subset D^b(X(d)),$4, the moduli stack is quasi-smooth, and the definition is modified by embedding it as a derived zero locus inside $\mathbb M(d;\delta_d)\subset D^b(X(d)),$5; this also yields the reduced category

$\mathbb M(d;\delta_d)\subset D^b(X(d)),$6

(Pădurariu et al., 2024).

These categories again appear as Hall-theoretic building blocks. The paper proves a semiorthogonal decomposition of $\mathbb M(d;\delta_d)\subset D^b(X(d)),$7 into Hall products of quasi-BPS categories indexed by partitions of $\mathbb M(d;\delta_d)\subset D^b(X(d)),$8 with equal slope $\mathbb M(d;\delta_d)\subset D^b(X(d)),$9 and strict inequalities in normalized shifted weights. Under the BPS condition

$\chi$0

the category $\chi$1 is called a BPS category, and for $\chi$2 the corresponding reduced primitive category is a reduced BPS category (Pădurariu et al., 2024).

The BPS condition is exactly the regime in which the categories acquire the geometry of a noncommutative Hitchin system. For $\chi$3, $\chi$4 is smooth over $\chi$5, proper over the Hitchin base, and Calabi–Yau over the Hitchin base. For $\chi$6, the reduced BPS category $\chi$7 has the analogous smooth, proper, and Calabi–Yau properties over the Hitchin base. Over the smooth spectral locus it restricts to the usual derived category of twisted coherent sheaves on a relative Picard variety. This is the explicit basis for describing BPS categories as noncommutative analogues of Hitchin integrable systems (Pădurariu et al., 2024).

A further development is the symmetry exchanging Euler characteristic and weight. For ordinary Higgs bundles, the main conjecture is

$\chi$8

and the corresponding topological $\chi$9-theory statement is proved rationally (Pădurariu et al., 2024). In the limit-category approach, quasi-BPS categories are defined by

$\chi+\rho-\delta_d\in \mathbf W(d),$0

and the full limit category $\chi+\rho-\delta_d\in \mathbf W(d),$1 admits a semiorthogonal decomposition into quasi-BPS categories indexed by Levi subgroups. This formulation is used to propose a precise Dolbeault geometric Langlands conjecture and the expected equivalence

$\chi+\rho-\delta_d\in \mathbf W(d),$2

as a categorical form of topological mirror symmetry for Higgs bundles (Pădurariu et al., 27 Aug 2025).

5. Recurring structural features

A striking feature of the theory is the recurrence of the same formal architecture in very different moduli problems. For symmetric quivers with potential, the ambient matrix-factorization category decomposes into Hall products of quasi-BPS categories ordered by slope (Pădurariu et al., 2023). For K3 surfaces, the derived category of the moduli stack of semistable objects decomposes into Hall products of quasi-BPS categories indexed by partitions and weight inequalities (Pădurariu et al., 2023). For twisted Higgs bundles, the same Hall-product mechanism produces semiorthogonal decompositions of the weight pieces of the derived category of the Higgs moduli stack (Pădurariu et al., 2024). In all three settings, quasi-BPS categories play the role of categorical PBW factors.

Wall-crossing is another common theme, but its form depends on the geometry. In the quiver case, quasi-BPS categories are invariant under changing generic stability conditions, through equivalences of the form

$\chi+\rho-\delta_d\in \mathbf W(d),$3

(Pădurariu et al., 2023). In the K3 case, the corresponding statement is the equivalence

$\chi+\rho-\delta_d\in \mathbf W(d),$4

and its reduced analogue (Pădurariu et al., 2023). In Higgs-bundle theory, the central symmetry is instead the exchange of Euler characteristic and weight, extending the Fourier–Mukai transform on the smooth spectral locus and proved in topological $\chi+\rho-\delta_d\in \mathbf W(d),$5-theory (Pădurariu et al., 2024, Pădurariu et al., 2024).

The strongest geometric properties consistently appear in reduced or primitive regimes. Reduced preprojective quasi-BPS categories have trivial relative Serre functor and, under coprimality, no nontrivial semiorthogonal decomposition (Pădurariu et al., 2023). Reduced quasi-BPS categories for K3 surfaces are proper and smooth with étale locally trivial Serre functor when $\chi+\rho-\delta_d\in \mathbf W(d),$6 (Pădurariu et al., 2023). Reduced BPS categories for Higgs bundles are smooth, proper, and Calabi–Yau over the Hitchin base under the BPS condition (Pădurariu et al., 2024). In the three-dimensional affine case, the categorical analogue of Davison’s support lemma states that if $\chi+\rho-\delta_d\in \mathbf W(d),$7, then every object of $\chi+\rho-\delta_d\in \mathbf W(d),$8 is supported on $\chi+\rho-\delta_d\in \mathbf W(d),$9, the preimage of the small diagonal in $\mathbf W(d)$0 (Pădurariu et al., 2022).

6. Topological $\mathbf W(d)$1-theory, BPS cohomology, and geometric meaning

One of the main achievements of the subject is the precise relation between quasi-BPS categories and BPS sheaves or BPS cohomology. For symmetric quivers with potential, the topological $\mathbf W(d)$2-theory of the quasi-BPS category $\mathbf W(d)$3 carries a filtration whose associated graded is monodromy-invariant BPS cohomology: $\mathbf W(d)$4 and the relative topological $\mathbf W(d)$5-theory sheaf is the periodicized monodromy-invariant BPS sheaf. The same paper proves the analogous statement for the preprojective categories $\mathbf W(d)$6 (Pădurariu et al., 2023).

For K3 surfaces, the topological $\mathbf W(d)$7-theory theorem is

$\mathbf W(d)$8

Its Euler characteristic recovers the BPS invariant, which in the local K3 case is known to equal the Euler characteristic of the corresponding Hilbert scheme of points on the K3 surface (Pădurariu et al., 2023). For Higgs bundles, the topological $\mathbf W(d)$9-theory of BPS categories is likewise identified with BPS-theoretic objects: in the twisted case with the intersection complex of the Higgs moduli space, and in the untwisted reduced case with the BPS sheaf

$W$00

(Pădurariu et al., 2024).

The affine threefold case exhibits the same principle in a Hall-algebraic form. Fixing a primitive slope $W$01, the torsion-free localized equivariant $W$02-theory

$W$03

becomes a commutative and cocommutative $W$04-bialgebra isomorphic to the ring of symmetric functions, and the localized equivariant $W$05-theoretic BPS spaces, defined as primitive elements for the coproduct, are one-dimensional in every degree (Pădurariu et al., 2022).

This recurring passage from quasi-BPS categories to BPS sheaves, BPS cohomology, primitive Hall data, and DT product formulas suggests that quasi-BPS categories function as categorical BPS pieces. Their ambient role is categorical and derived, but their additive invariants recover the structures traditionally associated with BPS theory. In that sense, quasi-BPS categories form a bridge between noncommutative geometry, Hall-algebraic factorization, and the cohomological or $W$06-theoretic realization of BPS phenomena.