Quasi-BPS categories for symmetric quivers with potential

Abstract

We study certain categories associated to symmetric quivers with potential, called quasi-Bogomol’nyi–Prasad–Sommerfield (BPS) categories. We construct semiorthogonal decompositions of the categories of matrix factorizations for moduli stacks of representations of (framed or unframed) symmetric quivers with potential, where the summands are categorical Hall products of quasi-BPS categories. These results generalize our previous results about the three-loop quiver. We prove several properties of quasi-BPS categories: wall-crossing equivalence, strong generation, and a categorical support lemma in the case of tripled quivers with potential. We also introduce reduced quasi-BPS categories for preprojective algebras, which have trivial relative Serre functor and are indecomposable when the weight is coprime with the total dimension. In this case, we regard the reduced quasi-BPS categories as noncommutative local hyperkähler varieties and as (twisted) categorical versions of crepant resolutions of singularities of good moduli spaces of representations of preprojective algebras. The studied categories include the local models of quasi-BPS categories of K3 surfaces. In a follow-up paper, we establish analogous properties for quasi-BPS categories of K3 surfaces.

1. Introduction

1.1 Motivation

The BPS (named after Bogomol’nyi-Prasad–Sommerfield states) invariants [PT14, § 2 and a half] and BPS cohomologies [DM20] are central objects in the study of Donaldson–Thomas (DT) theory and of (Kontsevich–Soibelman [KS11]) cohomological Hall algebras of a Calabi–Yau $3$ -fold (CY3) or a quiver with potential. In this paper, we study certain subcategories of matrix factorizations associated with symmetric quivers with potential, called quasi-BPS categories. They were introduced by the first named author in [Păd21] to prove a categorical version of the PBW theorem for cohomological Hall algebras [DM20]. As proved by the second named author [Tod18], quivers with potential describe the local structure of moduli of sheaves on a CY3. Thus, the study of quasi-BPS categories for quivers with potential is expected to help in understanding the (yet to be defined) Donaldson–Thomas (DT) categories or quasi-BPS categories for global CY3 geometries.

A particular case of interest is that of tripled quivers with potential. A subclass of tripled quivers with potential gives a local model of the moduli stack of (Bridgeland semistable and compactly supported) sheaves on the local K3 surface

(1.1) \begin{equation} X=S\times \mathbb {C}, \end{equation}

where $S$ is a K3 surface. This local description was used by Halpern-Leistner [HL20] to prove the D-equivalence conjecture for moduli spaces of stable sheaves on K3 surfaces; see [Tod19, Corollary 5.4.8] for its generalization. Tripled quivers with potential are also of interest in representation theory: the Hall algebras of a tripled quiver with potential are Koszul equivalent to the preprojective Hall algebras introduced by Schiffmann and Vasserot [SV13], Yang and Zhao [YZ16], and Varagnolo and Vasserot [VV22], which are categorifications of positive halves of quantum affine algebras [NSS21].

The tripled quiver with potential for the Jordan quiver is the quiver with one vertex and three loops $\{X,Y,Z\}$ and with potential $X[Y,Z]$ . In our previous papers [PT22a, PT22b], motivated by the search for a categorical analogue of the MacMahon formula, we studied quasi-BPS categories for the three-loop quiver. In particular, we constructed semiorthogonal decompositions for the framed and unframed stacks of representations of the tripled quiver, proved a categorical support lemma, and so on. The purpose of this paper is to generalize the results in [PT22a, PT22b] to more general symmetric quivers with potential, with special attention to tripled quivers with potential. We also prove new results on quasi-BPS categories: first, we show that quasi-BPS categories are equivalent under wall-crossing; next, we introduce reduced quasi-BPS categories and show that they are indecomposable when the weight is coprime with the total dimension.

In [PT23a], we use the results of this paper to introduce and study quasi-BPS categories for (local) K3 surfaces, and we discuss their relationship with (twisted) categorical crepant resolutions of singular symplectic moduli spaces of semistable sheaves on K3 surfaces.

1.2 Quasi-BPS categories

For a symmetric quiver $Q=(I,E)$ and a dimension vector $d\in \mathbb {N}^I$ , consider

\begin{align*} {\textrm{Tr}} W : \mathscr{X}(d):=R(d)/G(d) \to \mathbb{C}, \end{align*}

the moduli stack of representations of $Q$ of dimension $d$ , together with the regular function determined by the potential $W$ . Let $M(d)_{\mathbb {R}}^{W_d}$ be the set of Weyl-invariant real weights of the maximal torus $T(d) \subset G(d)$ . For $\delta \in M(d)_{\mathbb {R}}^{W_d}$ , consider the (ungraded or graded) quasi-BPS category

(1.2) \begin{equation} \mathbb {S}^\bullet (d; \delta )\subset \mathrm {MF}^\bullet (\mathscr {X}(d), \mathrm {Tr}\,W)\quad\text {for }\bullet \in \{\emptyset , \mathrm {gr}\}, \end{equation}

which is the category of matrix factorizations with factors in

(1.3) \begin{align} \mathbb {M}(d; \delta )\subset D^b(\mathscr {X}(d)), \end{align}

a noncommutative resolution of the coarse space of $\mathscr {X}(d)$ constructed by Špenko and Van den Bergh [ŠVdB17].

When $d$ is primitive and $\delta , \ell \in M(d)^{W_d}_{\mathbb {R}}$ are generic of weight zero with respect to the diagonal torus $\mathbb {C}^{\ast } \subset G(d)$ , Halpern-Leistner and Sam’s magic window theorem [HLS20] says that there is an equivalence

(1.4) \begin{align} \mathbb {M}(d; \delta ) \stackrel {\sim }{\to } D^b(X(d)^{\ell \text {-ss}}). \end{align}

Here, $\mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ is the GIT quotient of the $\ell$ -semistable locus, which is a $\mathbb {C}^{\ast }$ -gerbe, and $X(d)^{\ell \text {-ss}}$ is a smooth quasi-projective variety. However, there is no equivalence (1.4) for non-primitive $d$ . In this case, the stack $\mathscr {X}(d)^{\ell \text {-ss}}$ contains strictly semistable representations, the morphism $\mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ is more complicated, and $X(d)^{\ell \text {-ss}}$ is usually singular. Nevertheless, under some conditions on $\delta$ , we expect $\mathbb {M}(d; \delta )$ to behave as the derived category of a smooth quasi-projective variety. Thus it is interesting to investigate the structure of $\mathbb {M}(d; \delta )$ or $\mathbb {S}(d; \delta )$ , especially when $d$ is non-primitive.

As its name suggests, the category $\mathbb {S}(d; \delta )$ was introduced in [Păd21] as a categorical version of BPS invariants. The BPS invariants for CY3s are fundamental enumerative invariants which determine other enumerative invariants of interest, such as Donaldson–Thomas (DT) and Gromov–Witten invariants [PT14, § 2 and a half]. There are BPS cohomologies whose Euler characteristics equal the BPS invariants, defined by Davison and Meinhardt [DM20] in the case of symmetric quivers with potential and by Davison, Hennecart, Schlegel and Mejia [DHSM] in the case of local K3 surfaces. For a general CY3, up to the existence of certain orientation data, the BPS cohomologies are defined in [Tod23, Definition 2.11].

In [PT23b], we make the relation between quasi-BPS categories and BPS cohomologies more precise: we describe the topological K-theory of (1.2) in terms of BPS cohomologies and show that, under some extra conditions, they are isomorphic.

1.3 Semiorthogonal decompositions

In [Păd21], the first named author constructed semiorthogonal decompositions of the categorical Hall algebra of $(Q, W)$ in Hall products of quasi-BPS categories for all symmetric quivers $Q$ and all potentials $W$ . However, the (combinatorial) data which parametrize the summands are not easy to determine and are not very convenient for studying explicit wall-crossing geometries.

In this paper, we construct, for certain symmetric quivers, a different semiorthogonal decomposition which is more amenable to wall-crossing. We state the result in a particular case; see Theorem 4.2 for a more general statement which applies to all tripled quivers with potential.

Before stating Theorem 1.1, we introduce some notation. For a dimension vector $d=(d^i)_{i\in I}$ , let $\underline {d}=\sum _{i\in I}d^i$ be its total length. We set $\tau _d= ({1}/{\underline {d}})(\sum _{i\in I}\sum _{j=1}^{d^i}\beta _i^j)$ , where $\beta _i^j$ are weights of the standard representation of $G(d)$ . We consider the following particular examples of quasi-BPS categories (1.2):

\begin{equation*}\mathbb {S}^\bullet (d)_v:=\mathbb {S}^\bullet (d; v\tau _d).\end{equation*}

Theorem 1.1 (Theorem 4.21). Let $(Q, W)$ be a symmetric quiver with potential such that the number of loops at each vertex is odd and the number of edges between any two different vertices is even. Let $\bullet \in \{\emptyset , \mathrm {gr}\}$ . There is a semiorthogonal decomposition

(1.5) \begin{equation} \mathrm {MF}^\bullet (\mathscr {X}(d), \mathop {\textrm {Tr}} W)= \biggl\langle \bigotimes _{i=1}^k \mathbb {S}^\bullet (d_i)_{v_i}: \frac {v_1}{\underline {d}_1} \lt \cdots \lt \frac {v_k}{\underline {d}_k} \biggr \rangle , \end{equation}

where $(d_i)_{i=1}^k$ is a partition of $d$ and $(v_i)_{i=1}^k\in \mathbb {Z}^k$ .

As in [Păd21], we regard the above semiorthogonal decomposition as a categorical version of the PBW theorem for cohomological Hall algebras of Davison and Meinhardt [DM20].

We also study semiorthogonal decompositions for the moduli spaces $\mathscr {X}^f(d)^{\mathrm {ss}}$ of semistable framed representations of $Q$ , consisting of framed $Q$ -representations generated by the image of the maps from the framed vertex. We state a particular case here; see Theorem 4.1 for a general statement which includes all tripled quivers with potential.

Theorem 1.2 (Theorem 4.20). In the setting of Theorem 1.1, we further take $\mu \in \mathbb {R}\setminus \mathbb {Q}$ . Then there is a semiorthogonal decomposition

(1.6) \begin{equation} \mathrm {MF}^\bullet (\mathscr {X}^f(d)^{\text {ss}}, \mathop {\textrm {Tr}} W)=\biggl \langle \bigotimes _{i=1}^k \mathbb {S}^\bullet (d_i)_{v_i}: \mu \leqslant \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}\lt 1+\mu \biggr \rangle , \end{equation}

where $(d_i)_{i=1}^k$ is a partition of $d$ and $(v_i)_{i=1}^k\in \mathbb {Z}^k$ .

We proved Theorem 1.2 for the three-loop quiver in [PT22a, Theorem 1.1] in order to give a categorical analogue of the MacMahon formula for Hilbert schemes of points on $\mathbb {C}^3$ . Theorem 1.7 gives a generalization of [PT22a, Theorem 1.1]. As explained in [PT22a], the semiorthogonal decomposition (1.6) is regarded as a categorical analogue of the DT/BPS wall-crossing formula [Bri11, Tod10, Tod12], whose motivic/cohomological version is due to Meinhardt and Reineke [MR19].

Theorems 1.1 and 1.2 are two of the main tools we use to further investigate quasi-BPS categories. For example, they are central in the proof of Theorem 1.4 and in the results of [PT23b].

1.4 Categorical wall-crossing of quasi-BPS categories

Let $d$ be a primitive dimension vector and let $\ell , \ell '\in M(d)^{W_d}_{\mathbb {R}}$ be generic stability conditions. Then there is a birational map between two crepant resolutions of $X(d)$ .

(1.7)

As a corollary of the magic window theorem (1.4) of Halpern-Leistner and Sam, there is a derived equivalence

\begin{equation*}D^b(X(d)^{\ell \text {-ss}})\simeq D^b(X(d)^{\ell '\text {-ss}}),\end{equation*}

which proves the D/K equivalence conjecture of Bondal and Orlov [BO95] and Kawamata [Kaw02] for the resolutions (1.9).

We prove an analogous result when $d$ is not necessary primitive and hence there is no stability condition such that the $\mathbb {C}^{\ast }$ -rigidified moduli stack of semistable representations is a Deligne–Mumford stack. For a stability condition $\ell$ on $Q$ , we define a quasi-BPS category

\begin{equation*}\mathbb {S}^\ell (d; \delta )\subset \mathrm {MF} (\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W )\end{equation*}

which is, locally on the good moduli space $\mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ , modeled by a category (1.2).

Theorem 1.3 (Theorem 3.13). Let $(Q, W)$ be a symmetric quiver with potential, and let $\ell \mbox{ and } \ell '$ be stability conditions. Then there is a dense open subset $U \subset M(d)_{\mathbb {R}}^{W_d}$ such that for $\delta \in U$ there is an equivalence

(1.8) \begin{equation} \mathbb {S}^\ell (d; \delta ) \simeq \mathbb {S}^{\ell '}(d; \delta ). \end{equation}

Note that BPS invariants are preserved under wall-crossing [Tod23, Lemma 4.7]. We regard Theorem 1.3 as the categorical analogue of this property.

1.5 Categorical support lemma for quasi-BPS categories of tripled quivers

For a quiver $Q^{\circ }=(I, E^{\circ })$ , let $(Q^{\circ , d}, \mathscr {I})$ be its doubled quiver with relation $\mathscr {I}$ , and let $(Q, W)$ be its tripled quiver with potential; see § 2.2.6. Tripled quivers with potential form an important class of symmetric quivers with potential. Hall algebras of tripled quivers with potential are isomorphic to preprojective Hall algebras, which are themselves positive parts of quantum affine algebras [NSS21]. An important ingredient in the study of these Hall algebras is Davison’s support lemma [Dav16, Lemma 4.1] for BPS sheaves of tripled quivers with potential, which is used to prove purity of various cohomologies [Dav16, Dav21].

Inspired by Davison’s support lemma, we studied in [PT22b, Theorem 1.1] the support of objects in quasi-BPS categories for the tripled quiver with potential of the Jordan quiver, and we used it to obtain generators for the integral equivariant K-theory of certain quasi-BPS categories [PT22b, Theorem 1.2].

We prove an analogous result to [PT22b, Theorem 1.1] for (certain) tripled quivers with potential; see Theorem 1.4. The examples we study include all Ext-quivers of polystable sheaves (for a Bridgeland stability condition) on a local K3 surface as in (1.1). We use Theorem 1.4 to show relative properness of reduced quasi-BPS categories in Theorem 1.5.

Let $d=(d^i)_{i\in I}\in \mathbb {N}^I$ be a dimension vector and let $\mathfrak {g}(d)$ be the Lie algebra of $G(d)$ . There is a projection map which remembers the linear map on the added loops (i.e. edges of the tripled quiver which are added to the doubled quiver):

\begin{equation*}\mathscr {X}(d)\to \mathfrak {g}(d)/G(d).\end{equation*}

It induces a map

\begin{equation*}\pi \colon \mathrm {Crit}(\mathrm {Tr}\,W)\hookrightarrow \mathscr {X}(d)\to \mathfrak {g}(d)/G(d)\to \mathfrak {g}(d)/\!\!/ G(d)=\prod _{i \in I} \mathrm {Sym}^{d^i}(\mathbb {C}).\end{equation*}

Consider the diagonal map

\begin{equation*}\Delta \colon \mathbb {C} \hookrightarrow \prod _{i\in I} \mathrm {Sym}^{d^i}(\mathbb {C}).\end{equation*}

For two vertices $a, b \in I$ of the quiver $Q^{\circ }$ , let $\delta _{ab}=1$ if $a=b$ and $\delta _{ab}=0$ otherwise, and define

(1.9) \begin{align} \alpha _{a, b} &:=\sharp (a \to b\text { in }E^\circ )+\sharp (b \to a\text { in }E^\circ )-2\delta _{ab}. \end{align}

Theorem 1.4 (Theorem 5.1). Let $Q^{\circ }=(I, E^{\circ })$ be a quiver such that $\alpha _{a, b}$ is even for any $a, b \in I$ . Let $(Q,W)$ be the tripled quiver with potential of $Q^{\circ }$ . If $\gcd (v, \underline {d})=1$ , then any object of $\mathbb {S}(d)_v$ is supported over $\pi ^{-1}(\Delta )$ .

1.6 Quasi-BPS categories for reduced stacks

We now explain a modification of the categories (1.2) with better geometric properties in the case of tripled quivers with potential. We first introduce notation related to stacks of representations of doubled quivers.

Let $Q^\circ =(I, E^{\circ })$ be a quiver with stack of representations $\mathscr {X}^{\circ }(d)=R^{\circ }(d)/G(d)$ . Let $ \mathscr {P}(d):=\mu ^{-1}(0)/G(d)$ be the derived moduli stack of dimension- $d$ representations of the preprojective algebra of $Q^\circ$ (equivalently, of $(Q^{\circ , d}, \mathscr {I})$ -representations), where $\mu ^{-1}(0)$ is the derived zero locus of the moment map

(1.10) \begin{equation} \mu \colon T^{\ast }R^{\circ }(d) \to \mathfrak {g}(d). \end{equation}

Consider the good moduli space

\begin{align*} \mathscr {P}(d)^{\textrm {cl}} \to P(d)=\mu ^{-1}(0)/\!\!/ G(d). \end{align*}

In many cases, $P(d)$ is a singular symplectic variety and the study of its (geometric or noncommutative or categorical) resolutions is related to the study of hyperkähler varieties. Note that the variety $P(d)$ may not have geometric crepant resolutions of singularities, for example if $Q^\circ$ is the quiver with one loop and $g\geqslant 2$ loops, $d\geqslant 2$ , and $(g, d)\neq (2,2)$ ; see [KLS06, Proposition 3.5, Theorem 6.2].

Under the Koszul equivalence, the graded quasi-BPS category $\mathbb {S}^{\mathrm {gr}}(d)_v$ for the tripled quiver with potential of the quiver $Q^\circ$ is equivalent to the preprojective quasi-BPS category

\begin{equation*}\mathbb {T}(d)_v\subset D^b(\mathscr {P}(d)).\end{equation*}

The stack $\mathscr {P}(d)$ is never classical because the image of the moment map $\mu$ lies in the Lie subalgebra $\mathfrak {g}(d)_0\subset \mathfrak {g}(d)$ of traceless elements. We consider the reduced stack

\begin{equation*}\mathscr {P}(d)^{\mathrm {red}}:=\mu _0^{-1}(0)/G(d),\end{equation*}

where $\mu _0\colon T^*R^\circ (d)\to \mathfrak {g}(d)_0$ . We study the reduced quasi-BPS category

(1.11) \begin{equation} \mathbb {T}:=\mathbb {T}(d)_v^{\mathrm {red}}\subset D^b (\mathscr {P}(d)^{\mathrm {red}} ). \end{equation}

Recall $\alpha _{a, b}$ from (1.12) and define $\alpha _{Q^{\circ }}:=\mathrm {min}\{\alpha _{a, b} \mid a, b \in I\}$ . We use Theorem 1.4 to prove the following result.

Theorem 1.5 (Propositions 4.24 and 5.9, Theorem 5.10, Corollary 5.14). In the setting of Theorem 1.4, suppose that $\gcd (v, \underline {d})=1$ . Then the following hold.

1. (i) If $\alpha _{Q^{\circ }} \geqslant 2$ , the category $\mathbb {T}$ is regular, and it is proper over $P(d)$ .

2. (ii) Suppose furthermore that $P(d)$ is Gorenstein, e.g. $\alpha _{Q^{\circ }} \geqslant 3$ . Then there exists a relative Serre functor $\mathbb {S}_{\mathbb {T}/P(d)}$ of $\mathbb {T}$ over $P(d)$ , and it satisfies $\mathbb {S}_{\mathbb {T}/P(d)}\cong \mathrm {id}_{\mathbb {T}}$ .

3. (iii) In the situation of (ii), $\mathbb {T}$ does not admit any non-trivial semiorthogonal decomposition.

Inspired by the above theorem, we regard (1.11) as a noncommutative local hyperkähler variety, which is a (twisted) categorical version of a crepant resolution of singularities of $P(d)$ . It is an interesting question to examine the relation with categorical crepant resolutions in the sense of Kuznetsov [Kuz14] or noncommutative crepant resolutions in the sense of Van den Bergh [VdB22]. We plan to investigate this relation in future work.

In [PT23a], we use Theorem 1.5 to study reduced quasi-BPS categories for a K3 surface $S$ . In particular, we show that these categories are a (twisted) categorical version of a crepant resolution of the moduli space

\begin{equation*}M^{H}_S(v)\end{equation*}

of $H$ -Gieseker semistable sheaves on $S$ , where $H$ is a generic stability condition and $v$ is a non-primitive Mukai vector such that $\langle v, v\rangle \geqslant 2$ ; compare with [KLS06] in the geometric case.

1.7 Notation

We list the main notation used in the paper in Table 1.

Table 1. Notation used in the paper.

All the spaces $\mathscr {X}=X/G$ considered are quasi-smooth (derived) quotient stacks over $\mathbb {C}$ , where $G$ is an algebraic group. The classical truncation of $\mathscr {X}$ is denoted by $\mathscr {X}^{\textrm {cl}}=X^{\mathrm {cl}}/G$ . We assume that $X^{\mathrm {cl}}$ is a quasi-projective scheme. We denote by $\mathbb {L}_{\mathscr {X}}$ the cotangent complex of $\mathscr {X}$ . Any dg-category considered is a $\mathbb {C}$ -linear pre-triangulated dg-category; in particular, its homotopy category is a triangulated category. We denote by $D_{\textrm {qc}}(\mathscr {X})$ the unbounded derived category of quasi-coherent sheaves, by $D^b(\mathscr {X})$ the bounded derived category of coherent sheaves, and by $\mathrm {Perf}(\mathscr {X})$ its subcategory of perfect complexes.

Let $R$ be a set. Consider a set $O\subset R\times R$ such that for any $i, j\in R$ we have $(i,j)\in O$ or $(j,i)\in O$ , or both $(i,j)\in O$ and $(j,i)\in O$ . Let $\mathbb {T}$ be a pre-triangulated dg-category. We will construct semiorthogonal decompositions

(1.12) \begin{equation} \mathbb {T}=\langle \mathbb {A}_i \mid i \in R \rangle \end{equation}

with summands being pre-triangulated subcategories $\mathbb {A}_i$ indexed by $i\in R$ such that for any $i,j\in R$ with $(i, j)\in O$ and for any objects $\mathscr {A}_i\in \mathbb {A}_i$ and $\mathscr {A}_j\in \mathbb {A}_j$ , we have $\textrm {Hom}_{\mathbb {T}}(\mathscr {A}_i,\mathscr {A}_j)=0$ .

Consider a morphism $\pi \colon \mathscr {X}\to S$ . We say that the semiorthogonal decomposition (1.12) is $S$ -linear if $\mathbb {A}_i\otimes \pi ^*\mathrm {Perf}(S)\subset \mathbb {A}_i$ for all $i\in R$ .

We use the terminology of good moduli spaces of Alper; see [Alp13, § 8] for examples of stacks with good moduli spaces.

2. Preliminaries

2.1 Matrix factorizations

We briefly review the definition of categories of matrix factorizations. For more details, see [PT22a, § 2.6].

Consider a smooth quotient stack $\mathscr {X}=X/G$ , where $G$ is an algebraic group acting on a smooth affine scheme $X$ , with a regular function $f\colon \mathscr {X}\to \mathbb {C}$ . Consider the category of matrix factorizations by

\begin{equation*}\mathrm {MF}(\mathscr {X}, f),\end{equation*}

whose objects are tuples

(2.1) \begin{align} (\alpha \colon A \leftrightarrows B \colon \beta ), \quad \alpha \circ \beta =\cdot f, \quad \beta \circ \alpha =\cdot f, \end{align}

where $A, B \in \textrm {Coh}(\mathscr {X})$ . If $\mathbb {M}\subset D^b(\mathscr {X})$ is a subcategory, let

(2.2) \begin{align} \mathrm {MF}(\mathbb {M}, f)\subset \mathrm {MF}(\mathscr {X}, f) \end{align}

be the subcategory consisting of totalizations of tuples (2.1) with $A, B \in \mathbb {M}$ ; see [PT22a, § 2.6] for the precise definition. If $\mathbb {M}$ is generated by a set of vector bundles $\{\mathscr {V}_i\}_{i\in I}$ on $\mathscr {X}$ , then (2.2) is generated by matrix factorizations whose factors are direct sums of vector bundles from $\{\mathscr {V}_i\}_{i\in I}$ ; see [PT22a, Lemma 2.3].

Given an action of $\mathbb {C}^*$ on $\mathscr {X}$ for which $f$ is of weight $2$ , we also consider the category of graded matrix factorizations $\mathrm {MF}^{\mathrm {gr}}(\mathscr {X}, f)$ . Its objects consist of tuples (2.1) where $A\mbox{ and } B$ are $\mathbb {C}^{\ast }$ -equivariant and $\alpha \mbox{ and } \beta$ are of $\mathbb {C}^{\ast }$ -weight one. For a subcategory $\mathbb {M}\subset D^b(\mathscr {X})$ , we define $\mathrm {MF}^{\mathrm {gr}}(\mathbb {M}, f)\subset \mathrm {MF}^{\mathrm {gr}}(\mathscr {X}, f)$ similarly to (2.2).

Let $\mathscr {Z}\subset \mathscr {X}$ be a closed substack. A matrix factorization $F$ in $\mathrm {MF}(\mathscr {X}, f)$ has support in $\mathscr {Z}$ if its restriction to $\mathrm {MF}(\mathscr {X}\setminus \mathscr {Z}, f)$ is zero. Every matrix factorization $F$ has support included in $\mathrm {Crit}(f)\subset \mathscr {X}$ , so for any open substack $\mathscr {U} \subset \mathscr {X}$ which contains $\mathrm {Crit}(f)$ , the following restriction functor is an equivalence:

(2.3) \begin{align} \mathrm {MF}(\mathscr {X}, f) \stackrel {\sim }{\to } \mathrm {MF}(\mathscr {U}, f). \end{align}

We deduce semiorthogonal decompositions for a quiver with potential $(Q,W)$ from the case of zero potential; see for example [PT22a, Proposition 2.5] or [Păd22, Proposition 2.1]. We extensively use the Koszul equivalence; see Theorem 2.5.

We consider either ungraded categories of matrix factorizations or graded categories which are Koszul equivalent to derived categories of bounded complexes of coherent sheaf on a quasi-smooth stack. When considering the product of two categories of matrix factorizations, which is in the context of the Thom–Sebastiani theorem, we consider the product of dg-categories over $\mathbb {C}(\!(\beta )\!)$ for $\beta$ of homological degree $-2$ in the ungraded case (see [Pre11, Theorem 4.1.3]) and the product of dg-categories over $\mathbb {C}$ in the graded case (see [BFK14, Corollary 5.18]; alternatively, in the graded case, one can use the Koszul equivalence).

2.2 Quivers, weights, and partitions

2.2.1 Basic notions

Let $Q=(I,E)$ be a quiver, i.e. a directed graph with set of vertices $I$ and set of edges $E$ . Let $d=(d^a)_{a\in I}\in \mathbb {N}^I$ be a dimension vector. Denote by

\begin{equation*}\mathscr {X}(d)=R(d)/G(d)\end{equation*}

the stack of representations of $Q$ of dimension $d$ . Here $R(d)$ and $G(d)$ are given by

\begin{align*} R(d)=\bigoplus _{(a\to b) \in E}\textrm {Hom}(V^a, V^b), \quad G(d)=\prod _{a\in I}GL(V^a). \end{align*}

We say that $Q$ is symmetric if for any $a, b \in I$ , the number of arrows from $a$ to $b$ is the same as the number from $b$ to $a$ . In this case, $R(d)$ is a self-dual $G(d)$ -representation. We have the good moduli space morphism (or GIT quotient)

\begin{align*} \pi _{X,d}=\pi _X \colon \mathscr {X}(d) \to X(d):=R(d)/\!\!/ G(d). \end{align*}

For a quiver $Q$ , let $\mathbb {C}[Q]$ be its path algebra. A potential $W$ of a quiver $Q$ is an element

\begin{align*} W \in \mathbb {C}[Q]/[\mathbb {C}[Q], \mathbb {C}[Q]]. \end{align*}

A pair $(Q, W)$ is called a quiver with potential. Given a potential $W$ , there is a regular function

(2.4) \begin{align} \mathop {\textrm {Tr}} W \colon \mathscr {X}(d) \to \mathbb {C}. \end{align}

By the property of the good moduli space, the function $\mathop {\textrm {Tr}} W$ factors through the good moduli space, $\mathop {\textrm {Tr}} W \colon \mathscr {X}(d) \xrightarrow {\pi _{X,d}} X(d) \to \mathbb {C}$ .

We will consider the derived category $D^b(\mathscr {X}(d))$ of coherent sheaves on $\mathscr {X}(d)$ and the category of matrix factorizations $\mathrm {MF}(\mathscr {X}(d), \mathop {\textrm {Tr}} W)$ . Since the diagonal torus $\mathbb {C}^{\ast } \subset T(d)$ acts on $R(d)$ trivially, there are orthogonal decompositions

(2.5) \begin{align} D^b(\mathscr {X}(d))=\bigoplus _{w\in \mathbb {Z}} D^b(\mathscr {X}(d))_w, \quad \mathrm {MF}(\mathscr {X}(d), \mathop {\textrm {Tr}} W)=\bigoplus _{w\in \mathbb {Z}} \mathrm {MF}(\mathscr {X}(d), \mathop {\textrm {Tr}} W)_w, \end{align}

where each summand corresponds to the diagonal $\mathbb {C}^{\ast }$ -weight $w$ -part.

2.2.2 The weight lattice

We fix a maximal torus $T(d)$ of $G(d)$ . Let $M(d)$ be the weight lattice of $T(d)$ . For $a\in I$ and $d^a\in \mathbb {N}$ , denote by $\beta ^a_i$ for $1\leqslant i\leqslant d^a$ the weights of the standard representation of $T(d^a)$ . We have

\begin{align*} M(d)=\bigoplus _{a \in I} \bigoplus _{1\leqslant i\leqslant d^a} \mathbb {Z} \beta ^a_i. \end{align*}

With an abuse of notation, for a $T(d)$ -representation $U$ we also denote by $U \in M(d)$ the sum of weights in $U$ or, equivalently, the class of the character $\det U$ . A weight

\begin{equation*}\chi =\sum _{a\in I}\sum _{1\leqslant i\leqslant d^a} x_i^a \beta ^a_i\end{equation*}

is dominant (or antidominant) if $x_i^a \leqslant x_{i+1}^a$ (or $x_i^a \geqslant x_{i+1}^a$ ) for all $a\in I$ and $1\leqslant i\leqslant d^a$ . For a dominant weight $\chi$ , we denote by $\Gamma _{G(d)}(\chi )$ the irreducible representation of $G(d)$ with highest weight $\chi$ . The dominant or antidominant cocharacters are also defined for elements of the cocharacter lattice $N(d)=\textrm {Hom}(M(d), \mathbb {Z})$ . We denote by $1_d \in N(d)$ the diagonal cocharacter.

We denote by $M(d)_0\subset M(d)$ the hyperplane of weights with sum of coefficients equal to zero, and set

\begin{align*} M(d)_{\mathbb {R}}:=M(d)\otimes _{\mathbb {Z}}\mathbb {R}, \quad M(d)_{0,\mathbb {R}}:=M(d)_0\otimes _{\mathbb {Z}}\mathbb {R}. \end{align*}

Note that $M(d)_0$ is the weight lattice of the subtorus $ST(d) \subset T(d)$ defined by

\begin{align*} ST(d):=\text {ker}(\det \colon T(d)\to \mathbb {C}^*), \quad (g^a)_{a \in I} \stackrel {\det }{\mapsto } \prod _{a \in I} \det (g^a). \end{align*}

We denote by $\langle \,,\,\rangle \colon N(d)\times M(d)\to \mathbb {Z}$ the natural pairing, and we use the same notation for its real version. If $\lambda$ is a cocharacter of $T(d)$ and $V$ is a $T(d)$ -representation, we may abuse notation and write

\begin{equation*}\langle \lambda , V\rangle =\langle \lambda ,\quad \det (V)\rangle \end{equation*}

to ease the notation.

We denote by $W_d$ the Weyl group of $G(d)$ and by $M(d)^{W_d} \subset M(d)$ the Weyl-invariant subset. For $d=(d^a)_{a\in I}$ , let $\underline {d}=\sum _{a\in I}d^a$ be its total length. Define the Weyl-invariant weights in $M(d)_{\mathbb {R}}$ :

\begin{align*} \sigma _{d}:=\sum _{a \in I} \sum _{1\leqslant i \leqslant d^a}\beta ^a_i, \quad \tau _d:=\frac {\sigma _d}{\underline {d}}. \end{align*}

We denote by $\mathfrak {g}(d)$ the Lie algebra of $G(d)$ and by $\rho$ half the sum of the positive roots of $\mathfrak {g}(d)$ :

\begin{align*} \rho =\frac {1}{2} \sum _{a \in I}\sum _{1\leqslant i\lt j \leqslant d^a} (\beta _j^a -\beta _i^b). \end{align*}

2.2.3 Partitions

Let $(d_i)_{i=1}^k$ be a partition of $d$ . There is an identification

\begin{equation*}\bigoplus _{i=1}^k M(d_i)\cong M(d),\end{equation*}

where $\beta ^a_1, \ldots , \beta ^a_{d_1}$ correspond to the weights of the standard representation of $GL(d^a_1)$ in $M(d^a_1)$ for $a\in I$ , etc.

Definition 2.1. Let $\underline {e}=(e_i)_{i=1}^l$ and $\underline {d}=(d_i)_{i=1}^k$ be two partitions of $d\in \mathbb {N}^I$ . We write $\underline {e}\geqslant \underline {d}$ if there exist integers

\begin{equation*}a_0=0\lt a_1\lt \cdots \lt a_{k-1}\leqslant a_k=l\end{equation*}

such that for any $0\leqslant j\leqslant k-1$ , we have

\begin{equation*}\sum _{i=a_{j}+1}^{a_{j+1}} e_i=d_{j+1}.\end{equation*}

We next define a tree which is useful in decomposing dominant weights of $M(d)_{\mathbb {R}}$ .

Definition 2.2. We define $\mathcal {T}$ to be the unique (oriented) tree such that:

1. (1) each vertex is indexed by a partition $(d_1, \ldots , d_k)$ of some $d \in \mathbb {N}^I$ ;

2. (2) for each $d \in \mathbb {N}^I$ , there is a unique vertex indexed by the partition $(d)$ of size one;

3. (3) if $\bullet$ is a vertex indexed by $(d_1, \ldots , d_k)$ and $d_m=(e_1, \ldots , e_s)$ is a partition of $d_m$ for some $1\leqslant m \leqslant k$ , then there is a unique vertex $\bullet '$ indexed by $(d_1, \ldots , d_{m-1}, e_1, \ldots , e_s, d_{m+1}, \ldots , d_k)$ and with an edge from $\bullet$ to $\bullet '$ ;

4. (4) all edges in $\mathcal {T}$ are as in (3).

Note that each partition $(d_1, \ldots , d_k)$ of some $d \in \mathbb {N}^I$ gives an index of some (not necessary unique) vertex. A subtree $T \subset \mathcal {T}$ is called a path of partitions if it is connected, contains a vertex indexed by $(d)$ for some $d \in \mathbb {N}^I$ , and has a unique end vertex $\bullet$ . The partition $(d_1, \ldots , d_k)$ at the end vertex $\bullet$ is called the associated partition of $T$ . We define the Levi group associated to $T$ to be

\begin{align*} L(T):=\times _{i=1}^k G(d_i). \end{align*}

2.2.4 Framed quivers

Consider a quiver $Q=(I, E)$ . Define the framed quiver

\begin{align*} Q^f=(I^f, E^f) \end{align*}

with set of vertices $I^f=I\sqcup \{\infty \}$ and set of edges $E^f=E\sqcup \{e_a\mid a\in I\}$ , where $e_a$ is an edge from $\infty$ to $a\in I$ . Let $V(d)=\bigoplus _{a\in I}V^a$ , where $V^a$ is a $\mathbb {C}$ -vector space of dimension $d^a$ . Denote by

\begin{equation*}R^f(d)=R(d)\oplus V(d)\end{equation*}

the affine space of representations of $Q^f$ of dimension $d$ and consider the moduli stack of framed representations

\begin{equation*}\mathscr {X}^f(d):=R^f(d)/G(d).\end{equation*}

We consider GIT stability on $Q^f$ given by the character $\sigma _{\underline {d}}$ . It coincides with the King stability condition on $Q^f$ such that the (semi)stable representations of dimension $(1,d)$ are the representations of $Q^f$ with no subrepresentations of dimension $(1,d')$ for $d'$ different from $d$ ; see [Tod19, Lemma 5.1.9]. Consider the smooth variety obtained as a GIT quotient:

\begin{equation*}\mathscr {X}^f(d)^{\text {ss}}:=R^f(d)^{\text {ss}}/G(d).\end{equation*}

2.2.5 The categorical Hall product

For a cocharacter $\lambda \colon \mathbb {C}^{\ast } \to T(d)$ and a $T(d)$ -representation $V$ , let $V^{\lambda \geqslant 0}\subset V$ be the subspace generated by weights $\beta$ such that $\langle \lambda , \beta \rangle \geqslant 0$ , and let $V^\lambda \subset V$ be the subspace generated by weights $\beta$ such that $\langle \lambda , \beta \rangle =0$ . We denote by $G(d)^{\lambda } \subset G(d)^{\lambda \geqslant 0}$ the associated Levi and parabolic subgroup of $G(d)$ . If $V$ is a $G(d)$ -representation, consider the quotient stack $\mathscr {X}=V/G(d)$ and set

\begin{align*} \mathscr {X}^{\lambda \geqslant 0}=V^{\lambda \geqslant 0}/G(d)^{\lambda \geqslant 0}, \quad \mathscr {X}^{\lambda }=V^{\lambda }/G(d)^{\lambda }. \end{align*}

The projection $V^{\lambda \geqslant 0} \twoheadrightarrow V^{\lambda }$ and the inclusion $V^{\lambda \geqslant 0} \hookrightarrow V$ induce maps

(2.6) \begin{align} \mathscr {X}^{\lambda } \leftarrow \mathscr {X}^{\lambda \geqslant 0} \to \mathscr {X}. \end{align}

We apply the above construction for $V=R(d)$ to obtain maps

\begin{equation*}\mathscr {X}(d)^\lambda =\times _{i=1}^k\mathscr {X}(d_i)\xleftarrow {q_\lambda }\mathscr {X}(d)^{\lambda \geqslant 0}\xrightarrow {p_\lambda }\mathscr {X}(d).\end{equation*}

Suppose that $\lambda$ is antidominant with associated partition $(d_i)_{i=1}^k$ of $d\in \mathbb {N}^I$ , meaning that

\begin{equation*}\mathscr {X}(d)^\lambda =\times _{i=1}^k\mathscr {X}(d_i).\end{equation*}

The multiplication for the categorical Hall algebra of $(Q, 0)$ (or of $(Q,W)$ for a potential $W$ of $Q$ and possibly a grading) is defined by the following functors [Păd22], where $\bullet \in \{\emptyset , \mathrm {gr}\}$ :

(2.7) \begin{align} &m_\lambda :=p_{\lambda *}q_\lambda ^{\ast } \colon \boxtimes _{i=1}^k D^b(\mathscr {X}(d_i)) \to D^b(\mathscr {X}(d)),\nonumber \\ &m_\lambda :=p_{\lambda *}q_\lambda ^{\ast } \colon \boxtimes _{i=1}^k \mathrm {MF}^\bullet (\mathscr {X}(d_i), \mathop {\textrm {Tr}} W) \to \mathrm {MF}^\bullet (\mathscr {X}(d), \mathop {\textrm {Tr}} W). \end{align}

2.2.6 Doubled quiver

Let $Q^\circ =(I, E^\circ )$ be a quiver. Its doubled quiver is the quiver

\begin{align*} Q^{\circ , d}=(I, E^{\circ , d}) \end{align*}

with the set of edges $E^{\circ , d}=\{e, \overline {e} \mid e \in E^{\circ }\}$ , where $\overline {e}$ is the edge with the opposite orientation to $e\in E^\circ$ . Consider the following relation $\mathscr {I}$ of $\mathbb {C}[Q^{\circ , d}]$ :

(2.8) \begin{align} \mathscr {I}:=\sum _{e \in E^{\circ }}[e, \overline {e}] \in \mathbb {C}[Q^{\circ , d}]. \end{align}

For $d\in \mathbb {N}^I$ , consider the stack of representations of the quiver $Q^{\circ , d}$ of dimension $d$ ,

\begin{equation*}\mathscr {Y}(d):=\overline {R}(d)/G(d):=T^*R^\circ (d)/G(d),\end{equation*}

with good moduli space map

\begin{equation*}\pi _{Y,d}=\pi _Y\colon \mathscr {Y}(d)\to Y(d):=\overline {R}(d)/\!\!/ G(d).\end{equation*}

Note that

\begin{align*} \overline {R}(d):=T^*R^\circ (d)=\bigoplus _{(a\to b) \in E^{\circ }}\textrm {Hom}(V^a, V^b) \oplus \textrm {Hom}(V^b, V^a). \end{align*}

For $x \in T^*R^\circ (d)$ and an edge $(a \to b) \in E^{\circ }$ , consider the components $x(e) \in \textrm {Hom}(V^a, V^b)$ and $x(\overline {e}) \in \textrm {Hom}(V^b, V^a)$ . The relation (2.8) determines the moment map

(2.9) \begin{align} \mu \colon T^*R^\circ (d) \to \mathfrak {g}(d), \quad x \mapsto \sum _{e \in E^{\circ }}[x(e), x(\overline {e})]. \end{align}

Let $\mu ^{-1}(0)$ be the derived zero locus of $\mu$ . Define the stack of representations of $Q^{\circ , d}$ with relation $\mathscr {I}$ or, alternatively, of the preprojective algebra $\Pi _{Q^\circ }:=\mathbb {C}[Q^{\circ , d}]/(\mathscr {I})$ :

(2.10) \begin{align} j \colon \mathscr {P}(d):=\mu ^{-1}(0)/G(d) \hookrightarrow \mathscr {Y}(d). \end{align}

The stack $\mathscr {P}(d)^{\mathrm {cl}}$ has a good moduli space map

(2.11) \begin{equation} \pi _{P,d}=\pi _P\colon \mathscr {P}(d)^{\mathrm {cl}}\to P(d):=\mu ^{-1}(0)^{\textrm {cl}}/\!\!/ G(d). \end{equation}

Let $\lambda$ be an antidominant cocharacter of $T(d)$ corresponding to the decomposition $(d_i)_{i=1}^k$ of $d$ . Similarly to (2.7), there is a categorical Hall product [PS23, VV22]; see also (2.19):

(2.12) \begin{align} m_{\lambda } \colon \boxtimes _{i=1}^k D^b(\mathscr {P}(d_i)) \to D^b(\mathscr {P}). \end{align}

Remark 2.3. As mentioned in the introduction, moduli stacks of representations of preprojective algebras are interesting for at least two reasons: they describe locally the moduli stack of (Bridgeland semistable) sheaves on a Calabi–Yau surface, and their K-theory (or Borel–Moore homology) can be used to construct positive halves of quantum affine algebras (or of Yangians).

2.2.7 Tripled quivers with potential

Consider a quiver $Q^{\circ }=(I, E^{\circ })$ . Its tripled quiver

\begin{align*} Q=(I, E) \end{align*}

has set of edges $E=E^{\circ , d}\sqcup \{\omega _a \mid a\in I\}$ , where $\omega _a$ is a loop at the vertex $a \in I$ . The tripled potential $W$ (which is a potential of $Q$ ) is defined as follows:

(2.13) \begin{equation} W:=\biggl(\sum _{a\in I}\omega _a \biggr) \biggl (\sum _{e\in E^\circ }[e, \overline {e}]\biggr )\in \mathbb {C}[Q]. \end{equation}

The quiver with potential $(Q, W)$ constructed above is called the tripled quiver with potential associated to $Q^{\circ }$ .

For $d \in \mathbb {N}^I$ , let $\mathscr {X}(d)$ be the stack of representations of $Q$ . It is given by

(2.14) \begin{align} \mathscr {X}(d)=(T^*R^\circ (d) \oplus \mathfrak {g}(d))/G(d)= (\overline {R}(d) \oplus \mathfrak {g}(d))/G(d). \end{align}

Recall the function $\mathop {\textrm {Tr}} W$ on $\mathscr {X}(d)$ from (2.4). The critical locus

\begin{align*} \mathrm {Crit}(\mathop {\textrm {Tr}} W) \subset \mathscr {X}(d) \end{align*}

is the moduli stack of $(Q, W)$ -representations or, alternatively, of the Jacobi algebra $\mathbb {C}[Q]/\mathscr {J}$ , where $\mathscr {J}$ is the two-sided ideal generated by the partial derivatives $\partial W/\partial e$ for all $e\in E$ .

Consider the grading on $\mathscr {X}(d)$ which is of weight $0$ on $\overline {R}(d)$ and of weight $2$ on $\mathfrak {g}(d)$ . The Koszul equivalence (which we recall later in Theorem 2.5) says that there is an equivalence

(2.15) \begin{equation} \Theta \colon D^b(\mathscr {P}(d))\stackrel {\sim }{\to } \mathrm {MF}^{\mathrm {gr}}(\mathscr {X}(d), \mathrm {Tr}\,W). \end{equation}

Remark 2.4. Tripled quivers with potential are interesting for at least two reasons: they model the local structure of moduli stacks of semistable sheaves on an arbitrary CY3, and they can be used, in conjunction with dimensional reduction techniques (such as the Koszul equivalence (2.15)), to study (moduli of representations of) preprojective algebras.

2.3 The Koszul equivalence

Let $Y$ be an affine smooth scheme with an action of a reductive group $G$ . Consider the quotient stack $\mathscr {Y}=Y/G$ . Let $\mathscr {V} \to \mathscr {Y}$ be a vector bundle and let $s\in \Gamma (\mathscr {Y}, \mathscr {V})$ . Let $\mathscr {P}=s^{-1}(0)$ be the derived zero locus of $s$ , so its structure complex is the Koszul complex

\begin{equation*}\mathcal {O}_{\mathscr {P}}:= (\mathrm {Sym}_{\mathcal {O}_{\mathscr {Y}}}(\mathscr {V}^{\vee }[1]), d_{\mathscr {P}} )\end{equation*}

with differential $d_{\mathscr {P}}$ induced by $s$ . Let $\mathscr {X}=\mathrm {Tot}_{\mathscr {X}} (\mathscr {V}^{\vee })$ and define the function $f$ by

\begin{align*} f \colon \mathscr {X} \to \mathbb {C}, \; f(y, v)=\langle s(y), v \rangle \end{align*}

for $y \in \mathscr {Y}$ and $v \in \mathscr {V}^{\vee }|_{y}$ . There are maps

(2.16) \begin{align} \mathscr {P} \stackrel {j}{\hookrightarrow } \mathscr {Y} \stackrel {\eta }{\leftarrow } \mathscr {X} \stackrel {f}{\to } \mathbb {C}, \end{align}

where $j$ is the natural inclusion and $\eta$ is the natural projection. The following is the Koszul equivalence.

Theorem 2.5 [Isi13, Hir17, Tod19]. There are equivalences

\begin{align*} \Theta \colon D^b(\mathscr {P}) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}(\mathscr {X}, f), \quad \Theta \colon \textrm {Ind}\, D^b(\mathscr {P}) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}_{\textrm {qc}}(\mathscr {X}, f). \end{align*}

The grading on the right-hand side has weight zero on $\mathscr {Y}$ and weight $2$ on the fibers of $\eta \colon \mathscr {X} \to \mathscr {Y}$ .

The equivalence $\Theta$ is given by the functor

(2.17) \begin{align} \Theta (-)=(-) \otimes _{\mathcal {O}_{\mathscr {P}}}\mathcal {K}, \end{align}

where $\mathcal {K}$ is the Koszul factorization $\mathcal {K}:= (\mathcal {O}_{\mathscr {P}} \otimes _{\mathcal {O}_{\mathscr {Y}}}\mathcal {O}_{\mathscr {X}}, d_{\mathcal {K}} )$ with $d_{\mathcal {K}}=d_{\mathscr {P}} \otimes 1 +\kappa$ , where $\kappa \in \mathcal {V}^{\vee } \otimes \mathcal {V}$ corresponds to $\textrm {id} \in \textrm {Hom}(\mathcal {V}, \mathcal {V})$ ; see [Tod19, Theorem 2.3.3]. The following lemma is easily proved from the above description of $\Theta$ .

Lemma 2.6. Let $\{V_j\}_{j\in J}$ be a set of $G$ -representations and let $\mathbb {S} \subset \mathrm {MF}^{\textrm {gr}}(\mathscr {X}, f)$ be the subcategory generated by matrix factorizations whose factors are direct sums of vector bundles $\mathcal {O}_{\mathscr {X}} \otimes V_j$ . Then an object $\mathcal {E} \in D^b(\mathscr {P})$ satisfies $\Theta (\mathcal {E}) \in \mathbb {S}$ if and only if $j_{\ast }\mathcal {E} \in D^b(\mathscr {Y})$ is generated by $\mathcal {O}_{\mathscr {Y}} \otimes V_j$ for $j\in J$ .

Proof. The same argument used to prove [PT22a, Lemma 4.5] applies here.

For the later use, we will compare internal homomorphisms under Koszul equivalence. For $\mathcal {E}_1, \mathcal {E}_2 \in D^b(\mathscr {P})$ , there exists an internal homomorphism (see [DG13, Remark 3.4])

\begin{align*} \mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2) \in D_{\textrm {qc}}(\mathscr {P}). \end{align*}

It satisfies the following tensor-Hom adjunction: for any $A \in D_{\textrm {qc}}(\mathscr {P})$ , there are natural isomorphisms

\begin{align*} \textrm {Hom}_{D_{\textrm {qc}}(\mathscr {P})}(A, \mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2)) \cong \textrm {Hom}_{\textrm {Ind} D^b(\mathscr {P})}(A \otimes \mathcal {E}_1, \mathcal {E}_2). \end{align*}

On the other side of the Koszul equivalence, consider the internal Hom of $\mathcal {F}_1, \mathcal {F}_2 \in \mathrm {MF}^{\textrm {gr}}(\mathscr {X}, f)$ ,

\begin{align*} \mathcal {H}om(\mathcal {F}_1, \mathcal {F}_2) \in \mathrm {MF}^{\textrm {gr}}(\mathscr {X}, 0). \end{align*}

Lemma 2.7. For $\mathcal {E}_1, \mathcal {E}_2 \in D^b(\mathscr {P})$ , the equivalence $\Theta$ induces an isomorphism

(2.18) \begin{align} j_{\ast }\mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2) \stackrel {\cong }{\to } \eta _{\ast } \mathcal {H}om(\Theta (\mathcal {E}_1), \Theta (\mathcal {E}_2)) \end{align}

in $D_{\textrm {qc}}(\mathscr {Y})=\mathrm {MF}^{\textrm {gr}}_{\mathrm {qc}}(\mathscr {Y}, 0)$ .

Proof. For $A \in D_{\textrm {qc}}(\mathscr {Y})$ , we have

\begin{align*} \textrm {Hom}(A, j_{\ast }\mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2)) &\cong \textrm {Hom}(j^{\ast }A, \mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2)) \\ &\cong \textrm {Hom}(j^{\ast }A \otimes \mathcal {E}_1, \mathcal {E}_2). \end{align*}

We also have

\begin{align*} \textrm {Hom}(A, \eta _{\ast }\mathcal {H}om(\Theta (\mathcal {E}_1), \Theta (\mathcal {E}_2)) &\cong \textrm {Hom}(\eta ^{\ast }A, \mathcal {H}om(\Theta (\mathcal {E}_1), \Theta (\mathcal {E}_2))) \\ &\cong \textrm {Hom}(\eta ^{\ast }A \otimes \Theta (\mathcal {E}_1), \Theta (\mathcal {E}_2)) \\ &\cong \textrm {Hom}(\Theta (j^{\ast }A \otimes \mathcal {E}_1), \Theta (\mathcal {E}_2)), \end{align*}

where the last isomorphism follows from the explicit form of $\Theta$ in (2.17). Therefore $\Theta$ induces an isomorphism

\begin{align*} \textrm {Hom}(A, j_{\ast }\mathcal {H}om(\mathcal {E}_1, \mathcal {E}_2)) \stackrel {\sim }{\to } \textrm {Hom}(A, \eta _{\ast }\mathcal {H}om(\Theta (\mathcal {E}_1), \Theta (\mathcal {E}_2))), \end{align*}

which implies the isomorphism (2.18).

Let $\lambda$ be a cocharacter of $G$ . Consider the sections induced by $s$ ,

\begin{align*} s^{\lambda \geqslant 0}\in \Gamma (\mathscr {Y}^{\lambda \geqslant 0}, \mathscr {V}^{\lambda \geqslant 0}), \quad s^\lambda \in \Gamma (\mathscr {Y}^{\lambda }, \mathscr {V}^{\lambda }), \end{align*}

and their derived zero loci

\begin{equation*}\mathscr {P}^{\lambda \geqslant 0}:=(s^{\lambda \geqslant 0})^{-1}(0),\quad \mathscr {P}^{\lambda }:=(s^\lambda)^{-1}(0).\end{equation*}

Similarly to (2.6), consider the maps

\begin{align*} \mathscr {P}^{\lambda } \stackrel {q}{\leftarrow } \mathscr {P}^{\lambda \geqslant 0} \stackrel {p}{\to } \mathscr {P}, \end{align*}

where $q$ is quasi-smooth and $p$ is proper. Consider the functor, which is a generalization of the categorical Hall product for preprojective algebras [PS23, VV22]:

(2.19) \begin{align} m_{\lambda }= p_{\ast }q^{\ast } \colon D^b(\mathscr {P}^{\lambda }) \to D^b(\mathscr {P}). \end{align}

We have the following compatibility of categorical Hall products under Koszul equivalence. For the proof, see [Păd23, Proposition 3.1] or [Tod19, Lemmas 2.4.4 and 2.4.7].

Proposition 2.8. The following diagram commutes.

The horizontal arrows are categorical Hall products for $\mathscr {P}$ and $\mathscr {X}$ . We denote by $\Theta$ the Koszul equivalence for both $\mathscr {X}$ and $\mathscr {X}^\lambda$ , and the left vertical map is the functor $\Theta '(-):=\Theta (-)\otimes \det (\mathcal {V}^{\lambda \gt 0})^{\vee }[\mathrm {rank}\, \mathcal {V}^{\lambda \gt 0}]$ .

2.4 Polytopes and categories

Let $Q=(I,E)$ be a symmetric quiver and let $d=(d^a)_{a\in I}\in \mathbb {N}^I$ be a dimension vector. Consider the multisets of $T(d)$ -weights

(2.20) \begin{align} \mathscr {A}&:=\{\beta ^a_i-\beta ^b_j \mid a,b\in I,\, (a \to b) \in E, \,1\leqslant i\leqslant d^a,\, 1\leqslant j\leqslant d^b\},\nonumber \\ \mathscr {B}&:=\{\beta ^a_i\mid a\in I,\, 1\leqslant i\leqslant d^a\}, \\ \mathscr {C}&:=\mathscr {A}\sqcup \mathscr {B}. \nonumber\end{align}

Here, $\mathscr {A}$ is the set of $T(d)$ -weights of $R(d)$ and $\mathscr {C}$ is the set of $T(d)$ -weights of $R^f(d)$ . Define the polytopes

(2.21) \begin{align} \mathbf {W}(d)&:=\tfrac {1}{2}\mathrm {sum}_{\beta \in \mathscr {A}}[0, \beta ]\subset M(d)_{0,\mathbb {R}}\subset M(d)_{\mathbb {R}},\nonumber \\ \mathbf {V}(d)&:=\tfrac {1}{2}\mathrm {sum}_{\beta \in \mathscr {C}}[0,\beta ]\subset M(d)_{\mathbb {R}}, \end{align}

where the sums above are Minkowski sums in the space of weights $M(d)_{\mathbb {R}}$ .

Definition 2.9. For a weight $\delta _d\in M(d)_{\mathbb {R}}^{W_d}$ , define

(2.22) \begin{equation} \mathbb {M}(d; \delta _d)\subset D^b(\mathscr {X}(d)) \end{equation}

to be the full subcategory of $D^b(\mathscr {X}(d))$ generated by vector bundles $\mathcal {O}_{\mathscr {X}(d)}\otimes \Gamma _{G(d)}(\chi )$ , where $\chi \in M(d)$ is a dominant weight such that

(2.23) \begin{equation} \chi +\rho -\delta _d\in \mathbf {W}(d). \end{equation}

The category (2.22) is a particular case of the noncommutative resolutions of quotient singularities introduced by Špenko and Van den Bergh [ŠVdB17]. We may call (2.22) a `magic category’ following [HLS20]. For a cocharacter $\lambda$ of $T(d)$ , define

(2.24) \begin{align} n_{\lambda }=\langle \lambda , \det (\mathbb {L}_{\mathscr {X}(d)}|_{0}^{\lambda \gt 0} ) \rangle = \langle \lambda , \det (R(d)^{\vee })^{\lambda \geqslant 0} \rangle - \langle \lambda , \det (\mathfrak {g}(d)^{\vee } )^{\lambda \gt 0}\rangle . \end{align}

The category (2.22) also has the following alternative description.

Lemma 2.10 [HLS20, Lemma 2.9]. The subcategory $\mathbb{M}(d; \delta_d)$ of $D^b(\mathscr{X}(d))$ is generated by vector bundles $\mathcal{O}_{\mathscr{X}(d)} \otimes \Gamma$ for a $G(d)$ -representation $\Gamma$ such that for any $T(d)$ -weight $\chi$ of $\Gamma$ and any cocharacter $\lambda$ of $T(d)$ , we have that

(2.25) \begin{align} \langle \lambda , \chi -\delta _d\rangle \in [{-}\tfrac {1}{2}n_{\lambda }, \tfrac {1}{2}n_{\lambda } ]. \end{align}

Remark 2.11. The subcategory (2.22) is contained in $D^b(\mathscr {X}(d))_w$ for $w=\langle 1_d, \delta _d \rangle$ . In particular, if (2.22) is non-zero, then $\langle 1_d, \delta _d \rangle \in \mathbb {Z}$ .

We also define a larger subcategory corresponding to the polytope $\mathbf {V}(d)$ . Let

(2.26) \begin{align} \mathbb {D}(d; \delta _d) \subset D^b(\mathscr {X}(d)) \end{align}

be generated by vector bundles $\mathcal {O}_{\mathscr {X}(d)}\otimes \Gamma _{G(d)}(\chi )$ , where $\chi$ is a dominant weight of $G(d)$ such that

\begin{equation*}\chi +\rho -\delta _d\in \mathbf {V}(d).\end{equation*}

The following definition will be used later in decomposing $\mathbb {D}(d; \delta _d)$ into categorical Hall products of $\mathbb {M}(d; \delta _d)$ .

Definition 2.12. A weight $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ is said to be good if for all dominant cocharacters $\lambda$ such that $\langle \lambda , \beta ^i_a\rangle \in \{-1, 0\}$ for all $i\in I$ and $1\leqslant a\leqslant d^i$ , one has that $\langle \lambda , 2\delta _d\rangle \notin \mathbb {Z}$ .

Next, for a quiver with potential $(Q, W)$ , we define quasi-BPS categories as follows.

Definition 2.13. Define the quasi-BPS category to be

(2.27) \begin{equation} \mathbb {S}(d; \delta _d):=\mathrm {MF}(\mathbb {M}(d; \delta _d), \mathrm {Tr}\,W)\subset \mathrm {MF}(\mathscr {X}(d), \mathop {\textrm {Tr}} W). \end{equation}

Suppose that $(Q, W)$ is a tripled quiver of a quiver $Q^{\circ }=(I, E^{\circ })$ . In this case, the graded version is similarly defined as

(2.28) \begin{equation} \mathbb {S}^{\mathrm {gr}}(d; \delta _d):=\mathrm {MF}^{\mathrm {gr}}(\mathbb {M}(d; \delta _d), \mathop {\textrm {Tr}} W) \subset \mathrm {MF}^{\textrm {gr}}(\mathscr {X}(d), \mathop {\textrm {Tr}} W). \end{equation}

Next, we define quasi-BPS categories for preprojective algebras. Let $Q^\circ$ be a quiver, consider the moduli stack $\mathscr {P}(d)$ of dimension- $d$ representations of the preprojective algebra of $Q^\circ$ , and recall the closed immersion $j \colon \mathscr {P}(d) \hookrightarrow \mathscr {Y}(d)$ . The quasi-BPS category for $\mathscr {P}(d)$ is defined as follows.

Definition 2.14. Let $\widetilde {\mathbb {T}}(d; \delta _d)\subset D^b(\mathscr {Y}(d))$ be the subcategory generated by vector bundles $\mathcal {O}_{\mathscr {Y}(d)} \otimes \Gamma _{G(d)}(\chi )$ , where $\chi$ is a dominant weight of $G(d)$ satisfying

\begin{equation*}\chi +\rho -\delta _d\in \tfrac {1}{2}\text {sum}_{\beta \in \mathscr {A}}\,[0,\beta ],\end{equation*}

where $\mathscr {A}$ is the set of $T(d)$ -weights of $\overline {R}(d)\oplus \mathfrak {g}(d)$ . Define the preprojective quasi-BPS category (also called the quasi-BPS category of preprojective algebra)

\begin{align*} \mathbb {T}(d; \delta _d) \subset D^b(\mathscr {P}(d)) \end{align*}

as the full subcategory of $D^b(\mathscr {P}(d))$ with objects $\mathcal {E}$ such that $j_{\ast }\mathcal {E}\in \widetilde {\mathbb {T}}(d; \delta _d)$ .

By Lemma 2.6, the Koszul equivalence (2.15) restricts to the equivalence

(2.29) \begin{align} \Theta \colon \mathbb {T}(d; \delta _d) \stackrel {\sim }{\to } \mathbb {S}^{\textrm {gr}}(d; \delta _d). \end{align}

For $v\in \mathbb {Z}$ and $\bullet \in \{\emptyset , \mathrm {gr}\}$ , we will use the following shorthand notation:

\begin{equation*}\mathbb {M}(d)_v:=\mathbb {M}(d; \delta _d),\quad \mathbb {S}^\bullet (d)_v:=\mathbb {S}^\bullet (d; v\tau _d),\quad \mathbb {T}(d)_v:=\mathbb {T}(d; v\tau _d).\end{equation*}

3. The categorical wall-crossing equivalence

In this section, we prove a wall-crossing equivalence for quasi-BPS categories for symmetric quivers. We thus generalize (the particular case of moduli stacks of representations of quivers of) the theorem of Halpern-Leistner and Sam [HLS20, Theorem 3.2] to the situation where there is no stability condition such that the $\mathbb {C}^{\ast }$ -rigidified stack of semistable representations is a Deligne–Mumford stack.

3.1 Preliminaries

Let $Q=(I, E)$ be a symmetric quiver. For $d \in \mathbb {N}^I$ , recall from § 2.4 that $\mathbf {W}(d) \subset M(d)_{0, \mathbb {R}}$ is the polytope given by the Minkowski sum

\begin{align*} \mathbf {W}(d)=\tfrac {1}{2}\mathrm {sum}_{\beta \in \mathscr {A}}\,[0, \beta ] \subset M(d)_{0, \mathbb {R}}, \end{align*}

where

\begin{equation*}\mathscr {A}:=\{\beta ^a_i-\beta ^b_j \mid a,b\in I, \,(a \to b) \in E,\, 1\leqslant i\leqslant d^a,\, 1\leqslant j\leqslant d^b\}.\end{equation*}

We denote by

\begin{equation*}H_1, \ldots , H_m \subset \mathbf {W}(d)\end{equation*}

the codimension-one faces of $\mathbf {W}(d)$ and by

(3.1) \begin{equation} \lambda , \ldots , \lambda _m\colon \mathbb {C}^*\to ST(d) \end{equation}

the cocharacters of $ST(d)$ such that $\lambda _i^{\perp } \subset M(d)_{0, \mathbb {R}}$ is parallel to $H_i$ for all $1\leqslant i\leqslant m$ .

Note that there is a natural pairing (we abuse notation and use the same notation as for the pairing in § 2.2.2)

(3.2) \begin{align} \langle -, -\rangle \colon \mathbb {R}^I\times M(d)_{\mathbb {R}}^{W_d} \to \mathbb {R} \end{align}

defined by

\begin{equation*}\langle e^b, \det V^a\rangle =\delta ^{ab},\end{equation*}

where $e^b$ is the basis element of $\mathbb {R}^I$ corresponding to $b \in I$ . Note that an element $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ is written as

(3.3) \begin{align} \ell =\sum _{a}\ell ^{a}\det V^{a}, \quad \langle d, \ell \rangle =\sum _{a}d^a \ell ^a=0, \end{align}

i.e. $\ell$ is a $\mathbb {R}$ -character of $G(d)$ which is trivial on the diagonal torus $\mathbb {C}^{\ast } \subset G(d)$ . Note the following lemma.

Lemma 3.1. Let $\lambda \in \{\lambda _1,\ldots ,\lambda _m\}$ be an antidominant cocharacter such that $\mathscr {X}(d)^\lambda =\times _{j=1}^k \mathscr {X}(d_j)$ . Let $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ satisfy $\langle \lambda , \ell \rangle =0$ . Then we have $\langle d_j, \ell \rangle =0$ for all $1 \leqslant j \leqslant k$ . In particular, if $M(d)_{0, \mathbb {R}}^{W_d} \subset \lambda ^{\perp }$ , then $d_j$ is proportional to $d$ for all $1\leqslant j\leqslant k$ .

Proof. For simplicity, suppose that $\lambda$ corresponds to a decomposition $d=d_1+d_2$ . Since $\lambda ^{\perp }$ is spanned by a subset of weights in $R(d)$ , for fixed vertices $a, b \in I$ we can write

(3.4) \begin{align} \ell =\sum _{i, j, p, q}{c_{ij}^{pq}}(\beta _i^{p}-\beta _j^{q}) \end{align}

for some $c_{ij}^{pq}\in \mathbb {R}$ , where the sum on the right-hand side is over all $p, q \in I$ and all $1 \leqslant i \leqslant d_1^{p}\mbox{ and } 1\leqslant j \leqslant d_1^{q}$ or $d_1^{p}\lt i \leqslant d^{p}\mbox{ and } d_1^{q}\lt j \leqslant d^{q}$ . On the other hand, as $l$ is Weyl-invariant, we can write

(3.5) \begin{align} \ell =\sum _{a \in I}\ell ^a(\beta _1^a+\cdots +\beta _{d^a}^a) \end{align}

for some $\ell ^a \in \mathbb {R}$ ; see (3.3).

In the right-hand side of (3.4), the sum of the coefficients of $\beta _i^{p}$ for all $p \in I$ and $1\leqslant i \leqslant d_1^{p}$ is zero. Therefore, by taking the sum of coefficients of such $\beta _i^p$ in the right-hand side of (3.5), we conclude that

\begin{align*} \sum _{a \in I}\ell ^a d_1^a=\langle d_1, \ell \rangle =0. \end{align*}

If $M(d)_{0, \mathbb {R}}^{W_d} \subset \lambda ^{\perp }$ , we have $\langle d_j, \ell \rangle =0$ for all $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ , and hence $d_j$ is proportional to $d$ .

For $\ell \in M(d)^{W_d}_{0, \mathbb {R}}$ , consider the open substack $\mathscr {X}(d)^{\ell \text {-ss}} \subset \mathscr {X}(d)$ of $\ell$ -semistable points. By [Kin94], the $\ell$ -semistable locus consists of $Q$ -representations $R$ such that for any subrepresentation $R' \subset R$ of dimension vector $d'$ , we have that $\langle d', \ell \rangle \geqslant 0$ . Consider the good moduli space morphism

\begin{align*} \mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}. \end{align*}

A closed point of $X(d)^{\ell \text {-ss}}$ corresponds to a direct sum

\begin{align*} \bigoplus _{i=1}^k V^{(i)} \otimes R^{(i)}, \end{align*}

where $V^{(i)}$ is a finite-dimensional $\mathbb {C}$ -vector space and $R^{(i)}$ is an $\ell$ -stable $Q$ -representation whose dimension vector $d^{(i)}$ satisfies $\langle d^{(i)}, \ell \rangle =0$ for each $1\leqslant i \leqslant k$ .

3.2 Quasi-BPS categories for semistable stacks

For each $\delta _d \in M(d)_{\mathbb {R}}^{W_d}$ , recall from Definition (2.9) the quasi-BPS category

(3.6) \begin{align} \mathbb {M}(d; \delta _d) \subset D^b(\mathscr {X}(d)). \end{align}

By Lemma 2.10, it is the subcategory of objects $P$ such that for any cocharacter $\lambda \colon \mathbb {C}^{\ast } \to T(d)$ , we have that

(3.7) \begin{align} \mathrm {wt}_{\lambda }(P) \subset [{-}\tfrac {1}{2}n_{\lambda }, \tfrac {1}{2}n_{\lambda } ]+\langle \lambda , \delta \rangle , \end{align}

where $n_{\lambda }:=\langle \lambda , \det (\mathbb {L}_{\mathscr {X}(d)}^{\lambda \gt 0}|_{0}) \rangle$ ; see (2.24).

Consider a complex $A\in D^b(B\mathbb {C}^*)$ . Write $A=\bigoplus _{w\in \mathbb {Z}}A_w$ , where $\mathbb {C}^*$ acts with weight $w$ on $A_w$ . Define the set of weights

(3.8) \begin{equation} \mathrm {wt}(A):=\{w\mid A_w\neq 0\}\subset \mathbb {Z}. \end{equation}

Define also the integers

(3.9) \begin{equation} \mathrm {wt}^{\mathrm {max}}(A):=\mathrm {max}(\mathrm {wt}(A)),\quad \mathrm {wt}^{\mathrm {min}}(A):=\mathrm {min}(\mathrm {wt}(A)). \end{equation}

We define a version of quasi-BPS categories for semistable loci. Let

(3.10) \begin{align} \mathbb {M}^{\ell }(d; \delta _d) \subset D^b(\mathscr {X}(d)^{\ell \text {-ss}}) \end{align}

be the subcategory of objects $P$ such that for any map $\nu \colon B\mathbb {C}^{\ast } \to \mathscr {X}(d)^{\ell \text {-ss}}$ , we have

(3.11) \begin{align} \mathrm {wt}(\nu ^{\ast }P) \subset [{-}\tfrac {1}{2}n_{\nu }, \tfrac {1}{2}n_{\nu } ]+\mathrm {wt}(\nu ^{\ast }\delta _d), \end{align}

where $n_{\nu }:=\mathrm {wt}(\det ((\nu ^{\ast }\mathbb {L}_{\mathscr {X}(d)})^{\gt 0})) \in \mathbb {Z}$ . In Corollary 3.10, we show that for $\ell =0$ , the category (3.10) is $\mathbb {M}(d; \delta _d)$ .

Consider the restriction functor

(3.12) \begin{align} \mathrm {res} \colon D^b(\mathscr {X}(d)) \twoheadrightarrow D^b(\mathscr {X}(d)^{\ell \text {-ss}}). \end{align}

Lemma 3.2. The functor (3.12) restricts to the functor

(3.13) \begin{align} \mathrm {res} \colon \mathbb {M}(d; \delta _d) \to \mathbb {M}^{\ell }(d; \delta _d). \end{align}

Proof. A map $\nu \colon \mathbb {C}^{\ast } \to \mathscr {X}(d)$ corresponds (possibly after conjugation) to a point $x \in R(d)$ together with a cocharacter $\lambda \colon \mathbb {C}^{\ast } \to T(d)$ which fixes $x$ . Then $n_{\nu }=n_{\lambda }$ , so the functor (3.12) restricts to the functor (3.13).

There is an orthogonal decomposition

\begin{equation*}D^b(\mathscr {X}(d)^{\ell \text {-ss}})=\bigoplus _{w\in \mathbb {Z}}D^b(\mathscr {X}(d)^{\ell \text {-ss}})_w\end{equation*}

as in (2.5). There are analogous such decompositions for $D^b(\mathscr {X}(d)^\lambda )$ . The following lemma is a generalization of [HLS20, Proposition 3.11], where a similar result was obtained when the $\mathbb {C}^{\ast }$ -rigidification of $\mathscr {X}(d)^{\ell \text {-ss}}$ is a Deligne–Mumford stack.

Lemma 3.3. Let $w\in \mathbb {Z}$ . For $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ and $\delta _d \in M(d)_{\mathbb {R}}^{W_d}$ with $\langle 1_d, \delta _d\rangle =w$ , the category $D^b(\mathscr {X}(d)^{\ell \text {-ss}})_w$ is generated by $\mathrm {res}(\mathbb {M}(d; \delta _d) )$ and objects of the form $m_{\lambda }(P)$ , where $P \in D^b(\mathscr {X}(d)^{\lambda })_w$ is generated by $\Gamma _{G^{\lambda }(d)}(\chi '') \otimes \mathcal {O}_{\mathscr {X}^{\lambda }}(d)$ such that

(3.14) \begin{align} \langle \lambda , \chi '' \rangle \lt -\tfrac {1}{2}n_{\lambda }+\langle \lambda , \delta _d \rangle , \end{align}

and $\lambda \in \{\lambda _1, \ldots , \lambda _m\}$ is an antidominant cocharacter of $T(d)$ such that $\langle \lambda , \ell \rangle =0$ . The functor $m_{\lambda }$ is the categorical Hall product for the cocharacter $\lambda$ ; see (2.7).

Proof. We explain how to modify the proof of [HLS20, Proposition 3.11] to obtain the desired conclusion. The vector bundles

(3.15) \begin{equation} \Gamma _{G(d)}(\chi ) \otimes \mathcal {O}_{\mathscr {X}(d)^{\ell \text {-ss}}}, \end{equation}

for $\chi$ a dominant weight of $G(d)$ such that $\langle 1_d, \chi \rangle =w$ , generate the category $D^b(\mathscr {X}(d)^{\ell \text {-ss}})_w$ . We show that (3.15) is generated by the objects in the statement using induction on the pair $(r_{\chi }, p_{\chi })$ (with respect to the lexicographic order), where

\begin{align*} r_{\chi }:=\mathrm {min} \{r \geqslant 0 \mid \chi +\rho -\delta _d \in r\mathbf {W}(d) \} \end{align*}

and $p_{\chi }$ is the smallest possible number of $a_\beta$ equal to $-r_{\chi }$ among all ways of writing

\begin{equation*}\chi +\rho -\delta _d=\sum _{\beta \in \mathscr {A}}a_\beta \beta\end{equation*}

with $a_\beta \in [-r_{\chi }, 0]$ . If $r_{\chi } \le 1/2$ , then $\Gamma _{G(d)}(\chi ) \otimes \mathcal {O}_{\mathscr {X}(d)^{\ell \text {-ss}}}$ is an object in $\mathrm {res}(\mathbb {M}(d; \delta _d) )$ , so we may assume that $r_{\chi }\gt 1/2$ .

As in the argument of [HLS20, Proposition 3.11], there is an antidominant cocharacter $\lambda$ of $ST(d)$ such that the following hold:

1. – $\langle \lambda , \chi \rangle \leqslant \langle \lambda , \mu \rangle$ for any $\mu \in -\rho +r\mathbf {W}(d)+\delta _d$ ;

2. – $\lambda ^{\perp }$ is parallel to a face in $\mathbf {W}(d)$ .

Suppose first that $\langle \lambda , \ell \rangle \neq 0$ . Then as in the proof of [HLS20, Proposition 3.11], there is a complex of vector bundles consisting of $\Gamma _{G(d)}(\chi ) \otimes \mathcal {O}_{\mathscr {X}(d)}$ (which appears once) and $\Gamma _{G(d)}(\chi ') \otimes \mathcal {O}_{\mathscr {X}(d)}$ such that $(r_{\chi '}, p_{\chi '})$ is smaller than $(r_{\chi }, p_{\chi })$ and supported on the $\ell$ -unstable locus. The conclusion then follows by induction.

Suppose next that $\langle \lambda , \ell \rangle =0$ (observe that this case did not occur in [HLS20, Proposition 3.11]). The object

\begin{equation*}m_{\lambda }(\Gamma _{G(d)^{\lambda }}(\chi ) \otimes \mathcal {O}_{\mathscr {X}^{\lambda }(d)})\end{equation*}

is quasi-isomorphic to a complex of vector bundles (see [PT22a, Proposition 2.1]) consisting of $\Gamma _{G(d)}(\chi ) \otimes \mathcal {O}_{\mathscr {X}(d)}$ (which appears once) and $\Gamma _{G(d)}(\chi ') \otimes \mathcal {O}_{\mathscr {X}(d)}$ such that $(r_{\chi '}, p_{\chi '})$ is smaller than $(r_{\chi }, p_{\chi })$ ; see the last part of the proof of [Păd21, Proposition 4.1]. Since we have

\begin{align*} r_{\chi }=-\frac {\langle \lambda , \chi +\rho -\delta _d\rangle }{\langle \lambda , R^{\lambda \gt 0}(d) \rangle }\gt \tfrac {1}{2} \end{align*}

and $\lambda$ is antidominant, the inequality (3.14) holds for $\chi ''=\chi$ . The conclusion then follows using induction on $(r_{\chi }, p_{\chi })$ .

3.3 The categorical wall-crossing equivalence for symmetric quivers

In this subsection, we give some sufficient conditions for the functor (3.13) to be an equivalence. As a corollary, we obtain the categorical wall-crossing equivalence for quasi-BPS categories of symmetric quivers (and potential zero). The argument is similar to that for [HLS20, Theorem 3.2], with a modification due to the existence of faces of $\mathbf {W}(d)$ parallel to hyperplanes which contain $M(d)_{0, \mathbb {R}}^{W_d}$ .

For a stability condition $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ , let

(3.16) \begin{align} \mathscr {X}(d)=\mathscr {S}_1 \sqcup \cdots \sqcup \mathscr {S}_N \sqcup \mathscr {X}(d)^{\ell \text {-ss}} \end{align}

be the Kempf–Ness stratification of $\mathscr {X}(d)$ with center $\mathscr {Z}_i \subset \mathscr {S}_i$ for $1\leqslant i\leqslant N$ . Consider the associated one-parameter subgroups

(3.17) \begin{equation} \mu _1,\ldots ,\mu _N\colon \mathbb {C}^{\ast } \to T(d); \end{equation}

see [HL15, § 2] for a review of Kempf–Ness stratification.

Proposition 3.4. Suppose that $2\langle \mu _i, \delta _d \rangle \notin \mathbb {Z}$ for all $1\leqslant i\leqslant N$ . Then the functor (3.13) is fully faithful:

\begin{equation*}\mathrm {res} \colon \mathbb {M}(d; \delta _d) \hookrightarrow \mathbb {M}^{\ell }(d; \delta _d).\end{equation*}

Proof. For a choice $k_\bullet =(k_i)_{i=1}^N \in \mathbb {R}^N$ , there is a `window category’ $\mathbb {W}_{k_{\bullet }}^{\ell } \subset D^b(\mathscr {X}(d))$ such that the functor (3.12) restricts to the equivalence (see [HL15, BFK19]):

\begin{align*} \mathrm {res} \colon \mathbb {W}_{k_{\bullet }}^{\ell } \stackrel {\sim }{\to } D^b(\mathscr {X}(d)^{\ell \text {-ss}}). \end{align*}

The subcategory $\mathbb {W}_{k_{\bullet }}^{\ell }$ consists of objects $P$ such that $P|_{\mathscr {Z}_i}$ has $\mu _i$ -weights contained in the interval $[k_i, k_i+n_{\mu _i})$ , where the width $n_{\mu _i}$ is defined by (2.24) for $\lambda =\mu _i$ . For $1\leqslant i\leqslant N$ , let

\begin{equation*}k_i:=-n_{\mu _i}/2+\langle \mu _i, \delta _d \rangle .\end{equation*}

By the assumption of $\delta _d$ , we have that $\mathbb {M}(d; \delta _d) \subset \mathbb {W}_{k_{\bullet }}^{\ell }$ . It thus follows that the functor (3.13) is fully faithful.

Proposition 3.5. For any $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ , the functor (3.13) is essentially surjective:

\begin{equation*}\mathrm {res} \colon \mathbb {M}(d; \delta _d) \twoheadrightarrow \mathbb {M}^{\ell }(d; \delta _d).\end{equation*}

Proof. We will use Lemma 3.3. We first explain that it suffices to show that

(3.18) \begin{align} \textrm {Hom}(\mathbb {M}^{\ell }(d; \delta _d), \mathrm {res}(m_{\lambda }(P)))=0, \end{align}

where $m_\lambda$ is the categorical Hall product and $\lambda$ and $P$ are given as in Lemma 3.3. If the above vanishing holds, then

\begin{align*} \textrm {Hom}(\mathrm {res}(\mathbb {M}(d; \delta _d)), \mathrm {res}(m_{\lambda }(P)))=0, \end{align*}

and hence by Lemma 3.3 there is a semiorthogonal decomposition

\begin{align*} D^b(\mathscr {X}(d)^{\ell \text {-ss}}) = \langle \mathcal {C}, \mathrm {res}(\mathbb {M}(d; \delta _d)) \rangle , \end{align*}

where $\mathcal {C}$ is the subcategory of $D^b(\mathscr {X}(d)^{\ell \text {-ss}})$ generated by objects $\mathrm {res}(m_{\lambda }(P))$ as above (or as in Lemma 3.3). Observe that $\mathbb {M}^{\ell }(d; \delta _d)$ is in the left complement of $\mathcal {C}$ in $D^b(\mathscr {X}(d)^{\ell \text {-ss}})$ . Thus the vanishing (3.18) implies that indeed (3.13) is essentially surjective.

Below we show the vanishing (3.13). The stack $\mathscr {X}^{\lambda \geqslant 0}(d)$ is the stack of filtrations

(3.19) \begin{align} 0=R_0 \subset R_1 \subset R_2 \subset \cdots \subset R_k=R \end{align}

of $Q$ -representations such that $R_i/R_{i-1}$ has dimension vector $d_i$ . By Lemma 3.1, we have that $\langle \ell , d_i\rangle =0$ for $1\leqslant i\leqslant k$ . It follows that $R$ is $\ell$ -semistable if and only if every $R_i/R_{i-1}$ is $\ell$ -semistable. Therefore there are Cartesian squares

(3.20)

where each vertical arrow is an open immersion. The stack $\mathcal {N}$ is the moduli stack of filtrations (3.19) such that each $R_i/R_{i-1}$ is $\ell$ -semistable with dimension vector $d_i$ . Using proper base change and adjunction, the vanishing (3.18) is equivalent to the vanishing of

(3.21) \begin{align} \textrm {Hom}(E, p_{\lambda \ast }^{\ell }q_{\lambda }^{\ell \ast } j^{\ast }P) =\textrm {Hom}(p_{\lambda }^{\ell \ast }E, q_{\lambda }^{\ell \ast }j^{\ast }P) \end{align}

for any $E \in \mathbb {M}^{\ell }(d; \delta _d)$ . Let $\pi$ be the following composition:

\begin{align*} \pi \colon \mathcal {N} \to \mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}. \end{align*}

By the local-to-global Hom spectral sequence, it is enough to show the vanishing of

(3.22) \begin{align} \pi _{\ast }\mathcal {H}om (p_{\lambda }^{\ell \ast }E, q_{\lambda }^{\ell \ast }j^{\ast }P)=0. \end{align}

Furthermore, it suffices to show the vanishing (3.22) formally locally at each point $p \in X(d)^{\ell \text {-ss}}$ . We abuse notation and also denote by $p$ the unique closed point in the fiber of $\mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ , and we may assume that $\lambda$ is a cocharacter of $G_p:=\mathrm {Aut}(p)$ . By Lemma 3.6 below, the top diagram of (3.20) is, formally locally over $p\in X(d)^{\ell \text {-ss}}$ , of the form

(3.23) \begin{align} A^{\lambda }/G_p^{\lambda } \leftarrow A^{\lambda \geqslant 0}/G_p^{\lambda \geqslant 0} \to A/G_p, \end{align}

where $A$ is a smooth affine scheme (of finite type over a complete local ring) with a $G_p$ -action and good moduli space map $\pi _A\colon A/G_p\to A/\!\!/ G_p$ . Then the vanishing (3.22) at $p$ holds since the $\lambda$ -weights of $p_{\lambda }^{\ell \ast }E$ are strictly larger than those of $q_{\lambda }^{\ell \ast }j^{\ast }P$ by the definition of $\mathbb {M}^{\ell }(d; \delta _d)$ and the inequality (3.14); see [HL15, Corollary 3.17 and Amplification 3.18] and [Păd21, Proposition 4.2].

We have used the following lemma.

Lemma 3.6. Let $X(d)_p^{\ell \text {-ss}}=\textrm {Spec} \widehat {\mathcal {O}}_{X(d)^{\ell \text {-ss}}, p}$ . The top diagram of (3.20) pulled back via $X(d)_p^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ is of the form (3.23).

Proof. Consider the stack $\Theta =\mathbb {A}^1/\mathbb {C}^{\ast }$ . Since $\mathcal {N}$ is the moduli stack of filtrations of $\ell$ -semistable objects (3.19), the top diagram of (3.20) is a component of the diagram

(3.24) \begin{align} \mathrm {Map}(B\mathbb {C}^{\ast }, \mathscr {X}(d)^{\ell \text {-ss}}) \leftarrow \mathrm {Map}(\Theta , \mathscr {X}(d)^{\ell \text {-ss}}) \to \mathscr {X}(d)^{\ell \text {-ss}}, \end{align}

where the horizontal arrows are evaluation maps for mapping stacks from $B\mathbb {C}^{\ast }$ or $\Theta$ ; see [HL18]. Specifically, $\mathcal {N}$ is an open and closed substack of $\mathrm {Map}(\Theta , \mathscr {X}(d)^{\ell \text {-ss}})$ , and the top diagram in (3.20) is the restriction of (3.24) to $\mathcal {N}$ . By the Luna étale slice theorem, the pull-back of $\mathscr {X}(d)^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ via $X(d)_p^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ is of the form $A/G_p$ , where $A$ is a smooth affine scheme of finite type over $X(d)_p^{\ell \text {-ss}}$ with an action of $G_p$ and good moduli space map $\pi _A\colon A/G_p\to A/\!\!/ G_p$ . Since the mapping stacks from $B\mathbb {C}^{\ast }$ or $\Theta$ commute with pull-backs of maps to good moduli spaces (see [HL18, Corollary 1.30.1]), the pull-back of the top diagram in (3.20) via $X(d)_p^{\ell \text {-ss}} \to X(d)^{\ell \text {-ss}}$ consists of compatible connected components of the stacks:

(3.25) \begin{align} \mathrm {Map}(B\mathbb {C}^{\ast }, A/G_p) \leftarrow \mathrm {Map}(\Theta , A/G_p) \to A/G_p. \end{align}

Such connected components are of the form (3.23) for some cocharacter $\lambda$ of $G_p$ (see [HL18, Theorem 1.37]), and thus the conclusion follows.

Recall the cocharacters $\{\lambda _1,\ldots ,\lambda _m\}$ from (3.1) and the cocharacters $\{\mu _1,\ldots ,\mu _N\}$ from (3.17).

Definition 3.7. For $\ell \in M(d)_{0,\mathbb {R}}^{W_d}$ , let $U_\ell \subset M(d)^{W_d}_{\mathbb {R}}$ be the dense open subset of weights $\delta _d$ such that $2\langle \mu _i, \delta _d \rangle \notin \mathbb {Z}$ .

Propositions 3.4 and 3.5 imply the following result.

Theorem 3.8. Suppose that the pair $(\ell , \delta _d)\in M(d)_{0,\mathbb {R}}^{W_d}\times M(d)^{W_d}_{\mathbb {R}}$ satisfies $\delta _d \in U_{\ell }$ . Then the functor (3.13) is an equivalence

\begin{equation*}\mathrm {res} \colon \mathbb {M}(d; \delta _d) \stackrel {\sim }{\to } \mathbb {M}^{\ell }(d; \delta _d).\end{equation*}

We mention two corollaries of Theorem 3.8.

Corollary 3.9. For weights $\ell , \ell ' \in M(d)_{0, \mathbb {R}}^{W_d}$ and $\delta _d \in U_{\ell } \cap U_{\ell '}$ , there is an equivalence

\begin{align*} \mathbb {M}^{\ell }(d; \delta _d) \simeq \mathbb {M}^{\ell '}(d; \delta _d). \end{align*}

Corollary 3.10. For any $\delta _d \in M(d)_{\mathbb {R}}^{W_d}$ , we have $\mathbb {M}(d; \delta _d)=\mathbb {M}^{\ell =0}(d; \delta _d)$ .

Proof. For $\ell =0$ , there are no Kempf–Ness loci, so the condition in Theorem 3.8 is automatic.

Remark 3.11. The condition in Theorem 3.8 is satisfied for $\delta _d=\varepsilon \cdot \ell$ for $0\lt \varepsilon \ll 1$ since $\langle \ell , \mu _i \rangle \in \mathbb {Z}\setminus \{0\}$ . Similarly, in Corollary 3.9, the weight $\delta _d=\varepsilon \cdot \ell +\varepsilon ' \cdot \ell '$ satisfies $\delta _d \in U_{\ell } \cap U_{\ell '}$ if $0\lt \varepsilon , \varepsilon \ll 1$ and $(\varepsilon , \varepsilon ')$ are linearly independent over $\mathbb {Q}$ .

3.4 The categorical wall-crossing equivalence for symmetric quivers with potential

Let $(Q, W)$ be a symmetric quiver with potential. Similarly to (3.10), we define a category

(3.26) \begin{align} \mathbb {S}^{\ell }(d; \delta _d) \subset \mathrm {MF}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W). \end{align}

First, for an object $A\in \mathrm {MF}(B\mathbb {C}^*, 0)$ , write $A=\bigoplus _{w\in \mathbb {Z}} A_w$ with $A_w\in \mathrm {MF}(B\mathbb {C}^*, 0)_w$ . Consider the set of weights

\begin{equation*}\mathrm {wt}(A):=\{w\mid A_w\neq 0\}\subset \mathbb {Z}.\end{equation*}

Then (3.26) is the subcategory of $\mathrm {MF}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W)$ containing objects $P$ such that for any map $\nu \colon B\mathbb {C}^{\ast } \to \mathscr {X}(d)^{\ell \text {-ss}}$ with $\nu ^{\ast } \mathop {\textrm {Tr}} W=0$ , the set $\mathrm {wt} (\nu ^{\ast }P)$ satisfies the condition (3.11). Note that if $\nu ^{\ast } \mathop {\textrm {Tr}} W \neq 0$ , then $\mathrm {MF}(B\mathbb {C}^{\ast }, \nu ^{\ast } \mathop {\textrm {Tr}} W)=0$ so that the condition on weights in $\nu ^{\ast }P$ is vacuous in this case.

In the graded case (of a tripled quiver), the subcategory

\begin{align*} \mathbb {S}^{\mathrm {gr}, \ell }(d; \delta _d) \subset \mathrm {MF}^{\mathrm {gr}}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W) \end{align*}

is defined to be the pull-back of $\mathbb {S}^{\ell }(d; \delta _d)$ by the forget-the-grading functor

(3.27) \begin{align} \mathrm {forg} \colon \mathrm {MF}^{\textrm {gr}}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W) \to \mathrm {MF}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W). \end{align}

Let $\bullet \in \{\emptyset , \mathrm {gr}\}$ . The following results are proved as Theorem 3.8, Corollary 3.9, and Corollary 3.10.

Theorem 3.12. Suppose that the pair $(\ell , \delta _d)\in M(d)_{0,\mathbb {R}}^{W_d}\times M(d)^{W_d}_{\mathbb {R}}$ satisfies $\delta _d \in U_{\ell }$ . Then the restriction functor induces an equivalence

\begin{equation*}\mathrm {res} \colon \mathbb {S}^{\bullet }(d; \delta _d) \stackrel {\sim }{\to } \mathbb {S}^{\bullet , \ell }(d; \delta _d).\end{equation*}

Corollary 3.13. For weights $\ell , \ell ' \in M(d)_{0, \mathbb {R}}^{W_d}$ , and $\delta _d \in U_{\ell } \cap U_{\ell '}$ , there is an equivalence

\begin{align*} \mathbb {S}^{\bullet , \ell }(d; \delta _d) \simeq \mathbb {S}^{\bullet , \ell '}(d; \delta _d). \end{align*}

Corollary 3.14. For any $\delta _d \in M(d)_{\mathbb {R}}^{W_d}$ , we have $\mathbb {S}^{\bullet }(d; \delta _d)=\mathbb {S}^{\bullet , \ell =0}(d; \delta _d)$ .

3.5 The categorical wall-crossing for preprojective algebras

In this subsection we will use the notation from §§ 2.2.6 and 2.2.7. Consider a quiver $Q^{\circ }=(I, E^{\circ })$ . Let $Q^{\circ , d}=(I, E^{\circ , d})$ be its doubled quiver. For $\ell \in M(d)_{0, \mathbb {R}}^{W_d}$ , the moment map $\mu \colon T^*R^\circ (d)\to \mathfrak {g}(d)$ induces a map $\mu ^{\ell \text {-ss}}\colon (T^*R^\circ (d))^{\ell \text {-ss}}\to \mathfrak {g}(d)$ . Let

\begin{equation*}\mathscr {P}(d)^{\ell \text {-ss}}:=(\mu ^{\ell \text {-ss}} )^{-1}(0) /G(d) \subset \mathscr {P}(d)\end{equation*}

be the derived open substack of $\ell$ -semistable representations of the preprojective algebra of $Q^\circ$ . Consider the restriction functor

(3.28) \begin{align} \mathrm {res} \colon D^b(\mathscr {P}(d)) \twoheadrightarrow D^b(\mathscr {P}(d)^{\ell \text {-ss}}). \end{align}

The closed immersion (2.10) restricts to the closed immersion

(3.29) \begin{align} j \colon \mathscr {P}(d)^{\ell \text {-ss}} \hookrightarrow \mathscr {Y}(d)^{\ell \text {-ss}}. \end{align}

For $\delta _d \in M(d)_{\mathbb {R}}^{W_d}$ , define the subcategory

(3.30) \begin{align} \mathbb {T}^{\ell }(d; \delta _d) \subset D^b(\mathscr {P}(d)^{\ell \text {-ss}}) \end{align}

with objects $\mathcal {E}$ such that for any map $\nu \colon B\mathbb {C}^{\ast } \to \mathscr {P}(d)^{\ell \text {-ss}}$ , we have

(3.31) \begin{align} \mathrm {wt}(\nu ^{\ast }j^{\ast }j_{\ast }\mathcal {E}) \subset [{-}\tfrac {1}{2}n_{\nu }, \tfrac {1}{2}n_{\nu } ] +\mathrm {wt}(\nu ^{\ast }\delta _d). \end{align}

Here, we let $n_{\nu }:=\mathrm {wt}(\det (\nu ^{\ast }\mathbb {L}_{\mathscr {X}(d)}|_{\mathscr {P}(d)})^{\nu \gt 0})$ , where $\mathscr {X}(d)$ is the moduli stack of representations (2.14) of the tripled quiver $Q$ of $Q^{\circ }$ .

Remark 3.15. The subcategory (3.30) is the intrinsic window subcategory for the quasi-smooth stack $\mathscr {P}(d)^{\ell \text {-ss}}$ defined in [Tod19, Definition 5.2.13].

Proposition 3.16. The Koszul equivalence (2.15) descends to an equivalence

(3.32) \begin{align} \Theta \colon D^b(\mathscr {P}(d)^{\ell \text {-ss}}) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W), \end{align}

which restricts to an equivalence

(3.33) \begin{align} \Theta \colon \mathbb {T}^{\ell }(d; \delta _d) \stackrel {\sim }{\to } \mathbb {S}^{\mathrm {gr}, \ell }(d; \delta _d). \end{align}

Proof. Let $\eta \colon \mathscr {X}(d) \to \mathscr {Y}(d)$ be the projection. Then we have

(3.34) \begin{align} \mathrm {Crit}(\mathop {\textrm {Tr}} W) \cap \mathscr {X}(d)^{\ell \text {-ss}} \subset \eta ^{-1}(\mathscr {Y}(d)^{\ell \text {-ss}}) \subset \mathscr {X}(d)^{\ell \text {-ss}}, \end{align}

where the first inclusion is proved in [HL20, Lemma 4.3.22] and the second inclusion is immediate from the definition of $\ell$ -stability. We obtain equivalences

\begin{align*} D^b(\mathscr {P}(d)^{\ell \text {-ss}}) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}(\eta ^{-1}(\mathscr {Y}(d)^{\ell \text {-ss}}), \mathop {\textrm {Tr}} W) \stackrel {\sim }{\leftarrow } \mathrm {MF}^{\textrm {gr}}(\mathscr {X}(d)^{\ell \text {-ss}}, \mathop {\textrm {Tr}} W), \end{align*}

where the first equivalence is the Koszul equivalence in Theorem 2.5 and the second equivalence follows from (3.34) together with the fact that matrix factorizations are supported on the critical locus; see (2.3). Therefore we obtain the equivalence (3.32).

For an object $\mathcal {E} \in D^b(\mathscr {P}(d)^{\ell \text {-ss}})$ , the object $P=\Theta (\mathcal {E})$ is in $\mathbb {S}^{\textrm {gr}, \ell }(d; \delta _d)$ if and only if for any map $\nu \colon B\mathbb {C}^{\ast } \to \mathscr {X}(d)^{\ell \text {-ss}}$ with $\nu ^{\ast }\mathop {\textrm {Tr}} W=0$ , the set $\mathrm {wt}(\nu ^{\ast }\mathrm {forg}(P))$ satisfies the weight condition (3.11). As $P$ is supported on $\mathrm {Crit}(\mathop {\textrm {Tr}} W)$ , we may assume that the image of $\nu$ is contained in $\mathrm {Crit}(\mathop {\textrm {Tr}} W) \cap \mathscr {X}(d)^{\ell \text {-ss}}$ . By the $\mathbb {C}^{\ast }$ -equivariance of $P$ for the fiberwise weight $2$ -action on $\eta \colon \mathscr {X}(d)\to \mathscr {Y}(d)$ and upper semicontinuity, we have that

\begin{equation*}\mathrm {wt}(\nu ^{\ast }\mathrm {forg}(P) )\subset \mathrm {wt}(\nu '^{\ast }\mathrm {forg}(P) ),\end{equation*}

where $\nu '$ is the composition

\begin{align*} \nu ' \colon B\mathbb {C}^{\ast } \stackrel {\nu }{\to } \mathscr {X}(d) \stackrel {\eta }{\to } \mathscr {Y}(d) \stackrel {0}{\hookrightarrow } \mathscr {X}(d). \end{align*}

The image of $\nu '$ lies in $\mathscr {P}(d)^{\ell \text {-ss}} \hookrightarrow \mathscr {Y}(d)^{\ell \text {-ss}}\stackrel {0}{\hookrightarrow } \mathscr {X}(d)^{\ell \text {-ss}}$ . Therefore we may assume that the image of $\nu$ is contained in $\mathscr {P}(d)^{\ell \text {-ss}}$ . Since the object $P$ is represented by

\begin{align*} P=\Theta (\mathcal {E})= (\mathcal {E} \otimes _{\mathcal {O}_{\mathscr {P}(d)}}\mathcal {O}_{\mathscr {X}(d)})|_{\mathscr {X}(d)^{\ell \text {-ss}}} =(j_{\ast }\mathcal {E} \otimes _{\mathcal {O}_{\mathscr {Y}(d)}} \mathcal {O}_{\mathscr {X}(d)})|_{\mathscr {X}(d)^{\ell \text {-ss}}}, \end{align*}

it follows that $\mathrm {wt}(\nu ^{\ast }\mathrm {forg}(P))$ satisfies the condition (3.11) if and only if $\mathrm {wt}(\nu ^{\ast }(j_{\ast }\mathcal {E}) )$ satisfies the condition (3.31). Therefore $\Theta (\mathcal {E})$ is in $\mathbb {S}^{\textrm {gr}, \ell }(d; \delta _d)$ if and only if $\mathcal {E}$ is in $\mathbb {T}^{\ell }(d; \delta _d)$ .

By combining Proposition 3.16 with Theorem 3.12, Corollary 3.13, and Corollary 3.14, we obtain the following.

Theorem 3.17. Suppose that the pair $(\ell , \delta _d)\in M(d)_{0,\mathbb {R}}^{W_d}\times M(d)^{W_d}_{\mathbb {R}}$ satisfies $\delta _d \in U_{\ell }$ . Then the restriction functor (3.28) induces an equivalence

\begin{equation*}\mathrm {res} \colon \mathbb {T}(d; \delta _d) \stackrel {\sim }{\to } \mathbb {T}^{\ell }(d; \delta _d).\end{equation*}

Corollary 3.18. For $\ell , \ell ' \in M(d)_{0, \mathbb {R}}^{W_d}$ and $\delta _d \in U_{\ell } \cap U_{\ell '}$ , there is an equivalence

\begin{align*} \mathbb {T}^{\ell }(d; \delta _d) \simeq \mathbb {T}^{\ell '}(d; \delta _d). \end{align*}

Corollary 3.19. For any $\delta _d \in M(d)_{\mathbb {R}}^{W_d}$ , we have $\mathbb {T}(d; \delta _d)=\mathbb {T}^{\ell =0}(d; \delta _d)$ .

3.6 Quasi-BPS categories under Knörrer periodicity

In this subsection, we apply Corollaries 3.10 and 3.14 to obtain an equivalence of quasi-BPS categories under Knörrer periodicity, which is a particular case of the Koszul equivalence. Other than the use of Corollaries 3.10 and 3.14, the current subsection is independent of the other results and constructions discussed in § 3. We will use the results of this subsection in § 4 and in [PT23b].

We will use the notation from § 2.2. For a symmetric quiver $Q$ and $d \in \mathbb {N}^I$ , let $U$ be a $G(d)$ -representation. Consider the closed immersion

\begin{align*} j \colon \mathscr {X}(d):=R(d)/G(d) \hookrightarrow \mathscr {Y}=(R(d) \oplus U)/G(d) \end{align*}

into $(R(d)\oplus \{0\})/G(d)$ . We consider the quotient stack $\mathscr {X}^\gimel$ and regular function $f^\gimel$ :

\begin{align*} \mathscr {X}^\gimel =(R(d) \oplus U \oplus U^{\vee })/G(d) \stackrel {f}{\to } \mathbb {C}, \quad f(x, u, u')=\langle u, u'\rangle . \end{align*}

We consider a Cartesian diagram

where $\eta$ is the natural projection. Let $\mathbb {C}^{\ast }$ act with weight $0$ on $R(d)\oplus U$ and with weight $2$ on $U^\vee$ . In this case, the Koszul equivalence in Theorem 2.5 is given by (see [Tod19, Remark 2.3.5])

(3.35) \begin{align} \Theta =s_{\ast }v^{\ast } \colon D^b(\mathscr {X}(d)) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}(\mathscr {X}^\gimel , f). \end{align}

Such an equivalence is also called Knörrer periodicity [Orl06, Hir17]. For a weight $\delta ^\gimel _d \in M(d)_{\mathbb {R}}^{W_d}$ , define the subcategory

(3.36) \begin{align} \mathbb {S}^{\textrm {gr}}(d; \delta ^\gimel _d) \subset \mathrm {MF}^{\textrm {gr}}(\mathscr {X}^\gimel , f) \end{align}

in a way similar to (2.28). By Lemma 2.10, it consists of matrix factorizations whose factors are direct sums of vector bundles $\mathcal {O}_{\mathscr {X}^\gimel }\otimes \Gamma$ , where $\Gamma$ is a $G(d)$ -representations such that for any weight $\chi '$ of $\Gamma$ and any cocharacter $\lambda$ of $T(d)$ , we have

(3.37) \begin{align} \langle \lambda , \chi '-\delta ^\gimel _d \rangle \in [{-}\tfrac {1}{2}n^\gimel _{\lambda }, \tfrac {1}{2}n^\gimel _{\lambda } ]. \end{align}

Here, we define $n^\gimel _{\lambda }$ by

(3.38) \begin{align} n^\gimel _{\lambda }=\langle \lambda , \mathbb {L}_{\mathscr {X}^\gimel }^{\lambda \gt 0} \rangle = n_{\lambda }+\langle \lambda , U^{\lambda \gt 0} \rangle + \langle \lambda , (U^{\vee })^{\lambda \gt 0} \rangle , \end{align}

where we recall the definition of $n_{\lambda }$ from (2.24). The following is the main result we prove in this subsection.

Proposition 3.20. Let $\delta ^\gimel _d=\delta _d-\frac {1}{2}\det U\in M(d)^{W_d}_{\mathbb {R}}$ . The equivalence (3.35) restricts to the equivalence

(3.39) \begin{align} \Theta \colon \mathbb {M}(d; \delta _d) \stackrel {\sim }{\to } \mathbb {S}^{\textrm {gr}}(d; \delta ^\gimel _d). \end{align}

Proof. We first note that, by Lemma 2.6, an object $\mathcal {E} \in D^b(\mathscr {X}(d))$ satisfies $\Theta (\mathcal {E}) \in \mathbf {S}^{\textrm {gr}}(d; \delta ^\gimel _d)$ if and only if $j_{\ast }\mathcal {E}$ is generated by vector bundles $\mathcal {O}_{\mathscr {Y}} \otimes \Gamma '$ , where any weight $\chi '$ of $\Gamma '$ satisfies (3.37).

By Lemma 2.10, the category $\mathbb {M}(d; \delta _d)$ is generated by vector bundles $\mathcal {O}_{\mathscr {X}(d)}\otimes \Gamma$ such that any weight $\chi$ of $\Gamma$ satisfies (2.25) for any $\lambda$ . Consider the Koszul resolution

(3.40) \begin{align} j_{\ast }(\Gamma \otimes \mathcal {O}_{\mathscr {X}(d)}) =\Gamma \otimes \mathrm {Sym}_{\mathscr {Y}} (\mathcal {U}^{\vee }[1]), \end{align}

where $\mathcal {U} \to \mathscr {Y}$ is the vector bundle associated with the $G(d)$ -representation $U$ . Therefore, the category $j_{\ast }\mathbb {M}(d; \delta )$ is generated by vector bundles $\mathcal {O}_{\mathscr {Y}} \otimes \Gamma '$ such that any weight $\chi '$ of $\Gamma '$ satisfies

(3.41) \begin{align} \langle \lambda , \chi '-\delta _d \rangle \in \left [{-}\frac {n_{\lambda }}{2}+ \langle \lambda , (U^{\vee })^{\lambda \lt 0}), \frac {n_{\lambda }}{2}+ \langle \lambda , (U^{\vee })^{\lambda \gt 0}) \right ] \end{align}

for any $\lambda$ . By (3.38), we have

(3.42) \begin{align} \frac {n^\gimel _{\lambda }}{2} =\frac {n_{\lambda }}{2}+\langle \lambda , (U^{\vee })^{\lambda \gt 0}\rangle +\frac {1}{2}\langle \lambda , U \rangle =\frac {n_{\lambda }}{2}- \langle \lambda , (U^{\vee })^{\lambda \lt 0}\rangle -\frac {1}{2}\langle \lambda , U \rangle . \end{align}

Therefore (3.41) implies (3.37) for $\delta ^\gimel _d=\delta _d-\frac {1}{2} \det U$ , and hence the functor (3.35) sends $\mathbb {M}(d; \delta )$ to $\mathbb {S}^{\textrm {gr}}(d; \delta ^\gimel _d)$ , which shows the fully faithfulness of (3.39).

To show essential surjectivity of (3.39), let $\mathcal {E} \in D^b(\mathscr {X}(d))$ be such that $j_{\ast }\mathcal {E}$ is generated by the vector bundles $\mathcal {O}_{\mathscr {Y}} \otimes \Gamma '$ , where any weight $\chi '$ of $\Gamma '$ satisfies (3.37). We will show that $\mathcal {E}\in \mathbb {M}^{\ell =0}(d; \delta _d)$ and thus that $\mathcal {E}\in \mathbb {M}(d; \delta _d)$ by Corollary 3.10.

Let $\nu \colon B\mathbb {C}^{\ast } \to \mathscr {X}(d)$ be a map, which corresponds to a point $x \in R(d)$ and a cocharacter $\lambda \colon \mathbb {C}^{\ast } \to T(d)$ which fixes $x$ . By the condition (3.37) for weights of $\Gamma '$ , we have

\begin{align*} \mathrm {wt}^{\mathrm {max}}(\nu ^{\ast }j^{\ast }j_{\ast }\mathcal {E}) \leqslant \tfrac {1}{2} n_{\lambda }' +\langle \lambda , \delta _d' \rangle ; \end{align*}

see (3.9) for the definition of $\mathrm {wt}^{\mathrm {max}}$ . On the other hand, by the Koszul resolution (3.40), we have

\begin{align*} \mathrm {wt}^{\mathrm {max}}(\nu ^{\ast }j^{\ast }j_{\ast }\mathcal {E}) = \mathrm {wt}^{\mathrm {max}}(\nu ^{\ast }\mathcal {E}) -\langle \lambda , (U^{\vee })^{\lambda \gt 0} \rangle . \end{align*}

Therefore we have

\begin{align*} \mathrm {wt}^{\mathrm {max}}(\nu ^{\ast }\mathcal {E}) \leqslant \frac {n_{\lambda }'}{2}+\langle \lambda , \delta _d' \rangle -\langle \lambda , (U^{\vee })^{\lambda \gt 0} \rangle =\frac {n_{\lambda }}{2}+\langle \lambda , \delta _d\rangle , \end{align*}

where the last equality follows from (3.42). The lower bound

\begin{equation*}\mathrm {wt}^{\mathrm {min}}(\nu ^{\ast }\mathcal {E}) \geqslant -\frac {n_{\lambda }}{2}+\langle \lambda , \delta _d\rangle \end{equation*}

is proved similarly. We then have that

\begin{equation*}\mathrm {wt}(\nu ^{\ast }\mathcal {E})\subset \left [{-}\frac {n_{\lambda }}{2}+\langle \lambda , \delta _d\rangle , \frac {n_{\lambda }}{2}+\langle \lambda , \delta _d\rangle \right ].\end{equation*}

Thus $\mathcal {E} \in \mathbb {M}(d; \delta _d)^{\ell =0}$ , and then $\mathcal {E} \in \mathbb {M}(d; \delta _d)$ by Corollary 3.10.

Let $W$ be a potential of $Q$ . With an abuse of notation, we denote by $\mathop {\textrm {Tr}} W \colon \mathscr {X}^\gimel \to \mathbb {C}$ the pull-back of $\mathop {\textrm {Tr}} W \colon \mathscr {X}(d) \to \mathbb {C}$ by the natural projection $\mathscr {X}^\gimel \to \mathscr {X}(d)$ . There is an equivalence similar to (3.35), also called Knörrer periodicity (see [Hir17, Theorem 4.2] and [Orl06]):

(3.43) \begin{align} \Theta =s_{\ast }v^{\ast } \colon \mathrm {MF}(\mathscr {X}(d), \mathop {\textrm {Tr}} W) \stackrel {\sim }{\to } \mathrm {MF}(\mathscr {X}^\gimel , \mathop {\textrm {Tr}} W+f). \end{align}

The subcategory

\begin{align*} \mathbb {S}(d; \delta _d) \subset \mathrm {MF}(\mathscr {X}^\gimel , \mathop {\textrm {Tr}} W+f) \end{align*}

is defined similarly to (3.36). The following proposition is proved as Proposition 3.20, using Corollary 3.14 instead of Corollary 3.10.

Proposition 3.21. Let $\delta ^\gimel _d=\delta _d-\frac {1}{2}\det U\in M(d)^{W_d}_{\mathbb {R}}$ . The equivalence (3.43) restricts to the equivalence

\begin{align*} \Theta \colon \mathbb {S}(d; \delta _d) \stackrel {\sim }{\to } \mathbb {S}(d; \delta ^\gimel _d). \end{align*}

4. The semiorthogonal decompositions of DT categories

In this section, we construct semiorthogonal decompositions for the moduli of (framed or unframed) representations of certain symmetric quivers (see § 4.8) in terms of quasi-BPS categories; see Theorems 4.1, 4.2, and 4.21 and Corollary 4.18. The results generalize the decomposition of DT categories of points on $\mathbb {C}^3$ from [PT22a, Theorem 1.1] and the decomposition of the Hall algebra of $\mathbb {C}^3$ (or, equivalently, of the Porta–Sala Hall algebra of $\mathbb {C}^2$ ) from [Păd21, Theorem 1.1] and [Păd23, Theorem 1.1].

4.1 Semiorthogonal decompositions

The following is the main result in this section, which provides a semiorthogonal decomposition of $D^b(\mathscr {X}^f(d)^{\text {ss}})$ in products of quasi-BPS categories of $Q$ . Recall the definitions of a good weight from Definition 2.12, the category $\mathbb {M}(d;\delta )$ from Definition (2.9), and the Weyl-invariant real weights $\tau _d\mbox{ and } \sigma _d$ from § 2.2.2. Recall also the convention about the product of categories of matrix factorizations from § 2.1.

Theorem 4.1. Let $Q$ be a symmetric quiver such that the number of loops at each vertex $i\in I$ has the same parity. Let $d\in \mathbb {N}^I$ , let $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ , and let $\mu \in \mathbb {R}$ be such that $\delta _d+\mu \sigma _d$ is a good weight. For a partition $(d_i)_{i=1}^k$ of $d$ , let $\lambda$ be an associated antidominant cocharacter and define the weights $\delta _{d_i}\in M(d_i)_{\mathbb {R}}^{W_{d_i}}$ and $\theta _i \in \frac {1}{2} M(d_i)^{W_{d_i}}$ for $1\leqslant i\leqslant k$ by

(4.1) \begin{equation} \sum _{i=1}^k \delta _{d_i}=\delta _d, \quad \sum _{i=1}^k \theta _i=-\frac {1}{2}R(d)^{\lambda \gt 0}+\frac {1}{2}\mathfrak {g}(d)^{\lambda \gt 0}. \end{equation}

There is a semiorthogonal decomposition

(4.2) \begin{equation} D^b(\mathscr {X}^f(d)^{\text {ss}})=\bigg \langle \bigotimes _{i=1}^k \mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i}): \mu \leqslant \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}\lt 1+\mu \bigg \rangle , \end{equation}

where the right-hand side in (4.2) is over all partitions $(d_i)_{i=1}^k$ of $d$ and real numbers $(v_i)_{i=1}^k\in \mathbb {R}^k$ such that the sum of coefficients of $\theta _{i}+\delta _{d_i}+v_i\tau _{d_i}$ is an integer for all $1\leqslant i\leqslant k$ . The order on the summands is as in § 4.6 .

The functor from a summand on the right-hand side to $D^b(\mathscr {X}^f(d)^{\text {ss}})$ is the composition of the Hall product with the pull-back along the forget-the-framing map $\mathscr {X}^f(d)^{\mathrm {ss}}\to \mathscr {X}(d)$ . Further, the decomposition (4.2) is $X(d)$ -linear for the map $\pi _{f,d}\colon \mathscr {X}^f(d)^{\mathrm {ss}}\to \mathscr {X}(d)\xrightarrow {\pi _{X,d}} X(d)$ .

The same argument also applies to obtain the following similar semiorthogonal decomposition using Theorem 4.15 for unframed moduli stacks. Note that this decomposition is different from the one discussed in [Păd21, Theorem 1.1], which we could not use to obtain a decomposition of $D^b(\mathscr {X}^f(d)^{\mathrm {ss}} )$ as in Theorem 4.1.

Theorem 4.2. Let $Q$ be a symmetric quiver such that the number of loops at each vertex $i\in I$ has the same parity. Let $d\in \mathbb {N}^I$ and let $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ . For a partition $(d_i)_{i=1}^k$ of $d$ , define the weights $\delta _{d_i}\in M(d_i)_{\mathbb {R}}^{W_{d_i}}$ and $\theta _i \in \frac {1}{2} M(d_i)^{W_{d_i}}$ for $1\leqslant i\leqslant k$ as in (4.1). There is a semiorthogonal decomposition

(4.3) \begin{equation} D^b(\mathscr {X}(d))=\bigg \langle \bigotimes _{i=1}^k \mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i}) : \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k} \bigg \rangle , \end{equation}

where the right-hand side in (4.3) is over all partitions $(d_i)_{i=1}^k$ of $d$ and real numbers $(v_i)_{i=1}^k\in \mathbb {R}^k$ such that the sum of coefficients of $\theta _{i}+\delta _{d_i}+v_i\tau _{d_i}$ is an integer for all $1\leqslant i\leqslant k$ . The order on the summands is as in § 4.6 .

The functor from a summand on the right-hand side to $D^b(\mathscr {X}^f(d)^{\text {ss}} )$ is given by the Hall product. The decomposition (4.3) is $X(d)$ -linear.

The plan of proof for both of Theorems 4.1 and 4.2 is as follows: we first prove them in a particular case, that of very symmetric quivers; then we use Knörrer periodicity to obtain the more general statements stated above.

Using [PT22a, Proposition 2.3] and [Păd22, Proposition 2.1], one obtains versions for quivers with potentials of Theorems 4.1 and 4.2; see § 4.12. Note that both theorems apply to tripled quivers and thus to tripled quivers with potential.

4.2 Very symmetric quivers

We introduce a class of quivers for which we can prove Theorems 4.1 and 4.4 using the arguments employed for the quiver with one vertex and three loops in [PT22a, Păd23].

Definition 4.3. A quiver $Q=(I,E)$ is a very symmetric quiver if there exists an integer $A \in \mathbb {Z}_{\geqslant 1}$ such that for any vertices $a, b\in I$ , the number of edges from $a$ to $b$ is $A$ .

Remark 4.4. We will use the fact that the categories of matrix factorizations for a symmetric quiver as in Theorem 4.1 (i.e. a symmetric quiver with the number of loops at each vertex having the same parity) are equivalent, under Knörrer periodicity, to the categories of matrix factorizations for a very symmetric quiver. Thus we will study in detail the categories of matrix factorizations for very symmetric quivers, for which one can use the same combinatorial techniques as in [Păd23, PT22a]; see Remark 4.7.

The first step in proving Theorem 4.1 is the following.

Theorem 4.5. Let $Q$ be a very symmetric quiver. Then Theorem 4.1 holds for $Q$ .

Until § 4.8, we assume that the quiver $Q$ is very symmetric.

The proof of Theorem 4.5 follows closely the proof of [PT22a, Theorem 3.2]. As in loc. cit., the claim follows from the following semiorthogonal decomposition for subcategories of $D^b(\mathscr {X}(d))$ using `window categories’. Recall the definition of the categories $\mathbb {D}(d; \delta )$ from (2.26).

Theorem 4.6. Let $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ and let $\mu \in \mathbb {R}$ be such that $\delta _d+\mu \sigma _d$ is a good weight. Then there is a semiorthogonal decomposition

(4.4) \begin{equation} \mathbb {D}(d; \delta _d+\mu \sigma _d) =\bigg \langle \bigotimes _{i=1}^k \mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i}) \bigg \rangle , \end{equation}

where the right-hand side is as in Theorem 4.1 .

Proof of Theorem 4.5 assuming Theorem 4.6. We briefly explain why Theorem 4.6 implies Theorem 4.5; for full details see [PT22a, proof of Theorem 3.2]. Consider the morphisms

(4.5) \begin{equation} \mathscr {X}^f(d)^{\text {ss}} \stackrel {j}{\hookrightarrow } \mathscr {X}^f(d)\stackrel {\pi }{\twoheadrightarrow } \mathscr {X}(d), \end{equation}

where $j$ is an open immersion and $\pi$ is the natural projection. Let

\begin{align*} \mathbb {E}(d; \delta _d+\mu \sigma _d) \subset D^b(\mathscr {X}^f(d)) \end{align*}

be the subcategory generated by the complexes $\pi ^*(D)$ for $D\in \mathbb {D}(d; \delta _d+\mu \sigma _d)$ . If $\delta _d+\mu \sigma _d$ is a good weight, then there is an equivalence of categories

\begin{equation*}j^*\colon \mathbb {E}(d; \delta _d+\mu \sigma _d)\xrightarrow {\sim } D^b(\mathscr {X}^f(d)^{\text {ss}});\end{equation*}

see [PT22a, proof of Proposition 3.16]. The equivalence follows from the theory of `window categories’ of Halpern-Leistner [HL15] and of Ballard, Favero, and Katzarkov [BFK19], and from the description of window categories via explicit generators (due to Halpern-Leistner and Sam [HLS20]) for the self-dual representation of $G(d)$ :

\begin{align*} R^f(d) \oplus V(d)^{\vee }=R(d)\oplus V(d)\oplus V(d)^{\vee }. \end{align*}

Furthermore, there is a semiorthogonal decomposition

\begin{align*} D^b(\mathscr {X}^f(d))= \langle \pi ^{\ast } D^b(\mathscr {X}(d))_w : w \in \mathbb {Z} \rangle \end{align*}

and equivalences $\pi ^{\ast } \colon D^b(\mathscr {X}(d))_w \stackrel {\sim }{\to } \pi ^{\ast }D^b(\mathscr {X}(d))_w$ for each $w\in \mathbb {Z}$ . Therefore, by Theorem 4.6, there is a semiorthogonal decomposition

\begin{align*} \mathbb {E}(d; \delta _d+\mu \sigma _d)= \bigg \langle \bigotimes _{i=1}^k \mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i})\bigg \rangle , \end{align*}

where the right-hand side is as in Theorem 4.5. Therefore the theorem holds.

4.3 Decompositions of weights

The proof of Theorem 4.6 follows closely the proof of [PT22a, Proposition 3.12].

Remark 4.7. The assumption that the quiver $Q$ is very symmetric is used only to obtain a (convenient) decomposition (4.9) of the weight $\chi$ , which allows us to perform a similar combinatorial analysis as in [Păd23, PT22a]. Note that the decomposition (4.9) is a corollary of Proposition 4.9, where it is essential that $Q$ is very symmetric.

The proof of Theorem 4.6 uses the decomposition of categorical Hall algebras for quivers with potential in quasi-BPS categories from [Păd21]. The summands in this semiorthogonal decomposition are indexed by decompositions of weights of $T(d)$ , which we now briefly review.

Before stating the results, we introduce some notation. For a summand $d_a$ of a partition of $d$ , denote by $M(d_a)\subset M(d)$ the subspace as in the decomposition from § 2.2.3. Assume that $\ell$ is a partition of a dimension $d_a\in \mathbb {N}^I$ or, alternatively, that $\ell$ is an edge of the tree $\mathcal {T}$ introduced in § 2.2. Let $\lambda _{\ell }$ be the corresponding antidominant cocharacter of $T(d_a)$ . Recall the set

\begin{equation*}\mathscr {A}=\{(\beta ^a_i-\beta ^b_j)^{\times A}\mid a,b\in I,\: 1\leqslant i\leqslant i\leqslant d^a,\: 1\leqslant j\leqslant d^b\}\end{equation*}

from (2.20). Let

\begin{equation*}\mathscr {A}_{\ell }\subset M(d_a)\cap \mathscr {A}\end{equation*}

be the multiset of weights in $M(d_a)\cap \mathscr {A}$ such that $\langle \lambda _{\ell }, \beta \rangle \gt 0$ . Define $N_{\ell }$ by

\begin{align*} N_\ell :=\sum _{\beta \in \mathscr {A}_\ell }\beta . \end{align*}

Proposition 4.8. Let $\chi$ be a dominant weight in $M(d)_{\mathbb {R}}$ , let $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ , and let $w=\langle 1_{d}, \chi -\delta _d\rangle \in \mathbb {R}$ . Then the following exist:

1. (1) a path of partitions $T$ (see § 2.2) with decomposition $(d_i)_{i=1}^k$ at the end vertex;

2. (2) coefficients $r_\ell$ for $\ell \in T$ such that $r_{\ell }\gt 1/2$ if $\ell$ corresponds to a partition with length greater than $1$ and $r_{\ell }=0$ otherwise and such that, furthermore, if $\ell , \ell '\in T$ are vertices corresponding to partitions with length greater than 1 and with a path from $\ell$ to $\ell '$ , then $r_{\ell }\gt r_{\ell '}\gt {1}/{2}$ ;

3. (3) dominant weights $\psi _i\in \mathbf {W}(d_i)$ for $1\leqslant i\leqslant k$ such that

(4.6) \begin{equation} \chi +\rho -\delta _d=-\sum _{\ell \in T}r_\ell N_\ell +\sum _{i=1}^k\psi _i+w\tau _d. \end{equation}

Proof. The above is proved in [Păd22, § 3.1.2]; see also [Păd23, § 3.2.8].

We briefly explain the process of obtaining the decomposition (4.6). Choose $r$ such that $\chi +\rho -\delta _d-w\tau _d$ is on the boundary of $2r\mathbf {W}(d)$ (that is, let $r$ be the $r$ -invariant of $\chi +\rho -w\tau _d$ ). The first partition $\ell _1$ in $T$ corresponds to the face of $2r\mathbf {W}(d)$ which contains $\chi +\rho -\delta _d-w\tau _d$ in its interior. Assume that $\ell _1$ corresponds to a partition $(e_i)_{i=1}^s$ . Then there exist weights $\chi '_i\in M(e_i)_0$ for $1\leqslant i\leqslant s$ such that

\begin{equation*}\chi +\rho -\delta _d-w\tau _d+r_{\ell _1}N_{\ell _1}=\sum _{i=1}^s \chi '_i.\end{equation*}

By the choice of $r$ and $\ell _1$ , the weights $\chi '_i$ are inside the polytopes $2r\mathbf {W}(e_i)$ . One repeats the process above to further decompose the weights $\chi '_i$ until the decomposition (4.6) is obtained.

Let $w\in \mathbb {Z}$ and assume that $\langle 1_d, \delta _d\rangle =w$ . Let $M(d)^+_w$ be the subset of $M(d)$ of integral dominant weights $\chi$ with $\langle 1_d, \chi \rangle =w$ . We denote by $L^d_{\delta _d}$ the set of all paths of partitions $T$ with coefficients $r_\ell$ for $\ell \in T$ satisfying (2) from the statement of Proposition 4.9 for a dominant integral weight $\chi \in M(d)^+_w$ . By Proposition 4.8, there is a map

\begin{equation*}\Upsilon \colon M(d)^+_w\to L^d_{\delta _d}, \,\, \Upsilon (\chi )=(T, r_\ell ).\end{equation*}

The set $L^d_{\delta _d}$ was used in [Păd21] to index summands in semiorthogonal decompositions of $D^b(\mathscr {X}(d))_w$ . We will show in the next subsection that $L^d_{\delta _d}$ has an explicit description.

4.4 Partitions associated to dominant weights

We continue with the notation from § 4.3. Using the following proposition, the weight $-\sum _{\ell \in T}r_\ell N_\ell$ from (4.6) is a linear combinations of $\tau _{d_i}$ for $1\leqslant i\leqslant k$ .

Proposition 4.9. Let $\lambda$ be an antidominant cocharacter associated to the partition $(d_i)_{i=1}^k$ of $d$ . Then $R(d)^{\lambda \gt 0}$ is a linear combination of the weights $\tau _{d_i}$ (or, alternatively, of $\sigma _{d_i}$ ) for $1 \leqslant i\leqslant k$ .

Proof. This follows from a direct computation; for example when $k=2$ , one computes directly that

\begin{equation*}R(d)^{\lambda \gt 0}=A \sum _{a, b \in I}\sum _{d_1^a\lt i \leqslant d^a, 1 \leqslant j\leqslant d_1^b}(\beta ^b_j-\beta ^a_i)=A(\underline {d}_2\sigma _{d_1}-\underline {d}_1\sigma _{d_2}).\end{equation*}

Remark 4.10. The conclusion of Proposition 4.9 is not true for a general symmetric quiver.

Consider the decomposition (4.6) and let $\lambda$ be the antidominant cocharacter corresponding to $(d_i)_{i=1}^k$ . We define $v_i \in \mathbb {R}$ for $1\leqslant i \leqslant k$ by

(4.7) \begin{align} \sum _{i=1}^k v_i \tau _{d_i}=-\sum _{\ell \in T}\Big(r_{\ell }-\frac {1}{2}\Big)N_{\ell }+w\tau _d=-\sum _{\ell \in T} r_{\ell } N_{\ell } +\frac {1}{2}R(d)^{\lambda \gt 0}+w\tau _d. \end{align}

Here the right-hand side is a linear combination of $\tau _{d_i}$ by Proposition 4.9, so $v_i \in \mathbb {R}$ is well-defined. We define the weights $\theta _i \in M(d_i)^{W_{d_i}}_{\mathbb {R}}$ by

(4.8) \begin{align} \sum _{i=1}^k \theta _i=-\frac {1}{2}R(d)^{\lambda \gt 0}+\frac {1}{2}\mathfrak {g}(d)^{\lambda \gt 0}. \end{align}

Then we rewrite (4.4) as

(4.9) \begin{align} \chi =\sum _{i=1}^k \theta _i+\sum _{i=1}^k v_i\tau _{d_i}+\sum _{i=1}^k(\psi _i-\rho _i+\delta _{d_i}), \end{align}

where $\rho _i$ is half the sum of the positive roots of $\mathfrak {g}(d_i)$ . The next proposition follows as in [PT22a, Proposition 3.5].

Proposition 4.11. Let $\chi$ be a dominant weight in $M(d)$ and consider the weights $(v_i)_{i=1}^k$ from (4.7). Then

(4.10) \begin{equation} \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}. \end{equation}

Let $T^d_w$ be the set of tuples $A=(d_i, v_i)_{i=1}^k$ of $(d,w)$ such that $(d_i)_{i=1}^k$ is a partition of $d$ and the real numbers $(v_i)_{i=1}^k\in \mathbb {R}^k$ are as follows:

1. – $\sum _{i=1}^k v_i=w$ ;

2. – the inequality (4.10) holds;

3. – for each $1\leqslant i\leqslant k$ , the sum of coefficients of $\theta _i+v_i\tau _{d_i}+\delta _{d_i}$ is an integer.

By Proposition 4.11, there is a map

(4.11) \begin{equation} \varphi \colon L^d_{\delta _d}\to T^d_w. \end{equation}

Proposition 4.12. The map (4.11) is a bijection.

The above follows similarly to [PT22a, Proposition 3.8]. The proof proceeds by constructing an inverse $\varphi '$ . To construct $\varphi '$ , one chooses $\psi _i\in \textbf {W}(d_i)$ such that $\chi$ defined in (4.9) is an integral weight.

Thus, the summands appearing in the semiorthogonal decomposition of $D^b(\mathscr {X}(d))_w$ from [Păd21, Theorem 1.1] are labeled by elements of $T^d_w$ .

4.5 Partitions for framed quivers

The following proposition is the analogue of [PT22a, Proposition 3.7].

Proposition 4.13. Let $\delta _d\in M(d)^{W_d}_{\mathbb {R}}$ and let $\chi \in M(d)$ be a dominant weight. Consider the decomposition (4.9) with associated partition $(d_i)_{i=1}^k$ and weights $(v_i)_{i=1}^k\in \mathbb {R}^k$ . Let $\mu \in \mathbb {R}$ and assume that

(4.12) \begin{align} \chi +\rho -\delta _d-\mu \sigma _d\in \mathbf {V}(d). \end{align}

Then

(4.13) \begin{equation} \mu \leqslant \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}\leqslant 1+\mu . \end{equation}

Proof. Using the decomposition (4.9), we have that

(4.14) \begin{align} \chi +\rho -\delta _d-\mu \sigma _d =-\frac {1}{2}R(d)^{\lambda \gt 0}+\sum _{i=1}^k (v_i-\mu \underline {d}_i)\tau _{d_i}+\sum _{i=1}^k\psi _i. \end{align}

Let $\alpha _k$ be the (dominant) cocharacter of $T(d)$ which acts with weight $1$ on $\beta ^a_i$ for $a\in I$ and $d^a-d^a_k\lt i\leqslant d^a$ and with weight $0$ on $\beta ^a_i$ for $a\in I$ and $d^a-d^a_k\geqslant i$ . By (4.14), we have that

(4.15) \begin{align} \langle \alpha _k, \chi +\rho -\delta _d-\mu \sigma _d\rangle = \langle \alpha _k, -\tfrac {1}{2}R(d)^{\lambda \gt 0} \rangle +v_k-\mu \underline {d}_k. \end{align}

On the other hand, we have

\begin{align*} \mathbf {V}(d)=\tfrac {1}{2} \mathrm {sum}_{\beta \in \mathscr {C}}[0, \beta ], \end{align*}

where we recall that

\begin{equation*}\mathscr {C}=\{(\beta ^a_i-\beta ^b_j)^{\times A}, \beta ^a_i\mid a,b\in I,\: 1\leqslant i\leqslant d^a,\: 1\leqslant j\leqslant d^b\}.\end{equation*}

Then $-R(d)^{\lambda \gt 0}/2+\underline {d}_k \tau _{d_k}$ has maximum $\alpha _k$ -weight among weights in $\mathbf {V}(d)$ . Therefore, from (4.12) we obtain that

(4.16) \begin{align} \langle \alpha _k, \chi +\rho -\delta _d-\mu \sigma _d\rangle \leqslant \langle \alpha _k, -\tfrac {1}{2}R(d)^{\lambda \gt 0} \rangle +\underline {d}_k. \end{align}

By comparing (4.15) and (4.16), we conclude that $v_k-\mu \underline {d}_k\leqslant \underline {d}_k$ , so $v_k/\underline {d}_k \leqslant 1+\mu$ . A similar argument also shows the lower bound.

Corollary 4.14. In the setting of Proposition 4.13, consider the decomposition (4.14). Recall the sets $\mathscr {A}$ and $\mathscr {B}$ from (2.20) and define $\mathscr {A}_{\lambda }:=\{\beta \in \mathscr {A} \mid \langle \lambda , \beta \rangle \gt 0\}$ . Then there are $\psi _i\in \mathbf {W}(d_i)$ for $1\leqslant i\leqslant k$ such that

\begin{align*} -\frac {1}{2}R(d)^{\lambda \gt 0}&\in \frac {1}{2}\mathrm {sum}_{\beta \in \mathscr {A}_\lambda }[0, -\beta ],\quad \sum _{i=1}^k (v_i-\mu \underline {d}_i)\tau _{d_i}\in \mathrm {sum}_{\beta \in \mathscr {B}}[0,\beta ]. \end{align*}

Proof. The inclusion

\begin{align*} \sum _{i=1}^k (v_i-\mu \underline {d}_i)\tau _{d_i}\in \mathrm {sum}_{\beta \in \mathscr {B}}[0,\beta ] \end{align*}

follows from Proposition 4.13; see [PT22a, Proposition 3.8]. The rest of the decomposition is immediate.

4.6 Comparison of partitions

In this subsection, we explain the order used in the semiorthogonal decomposition from Theorem 4.5; see also the discussion in § 1.8.

Fix $d\in \mathbb {N}^I$ and a weight $\delta ^\circ \in M(d)^{W_d}_{0,\mathbb {R}}$ . Define $L^d_{\delta ^\circ , w}:=L^{d}_{\delta ^\circ +w\tau _d}$ and $L^d_{\delta ^\circ }:=\bigcup _{w\in \mathbb {Z}}L^d_{\delta ^\circ , w}$ . We define a set

\begin{equation*}O\subset L^d_{\delta ^\circ }\times L^d_{\delta ^\circ }\end{equation*}

which is used to compare summands of semiorthogonal decompositions; see § 1.8.

For $w\gt w'$ , let $O_{w, w'}:=L^d_{\delta ^\circ , w}\times L^d_{\delta ^\circ , w'}$ . For $w\lt w'$ , let $O_{w, w'}$ be the empty set.

We now define $O_{w,w}\subset L^d_{\delta ^\circ , w}\times L^d_{\delta ^\circ , w}$ . The general procedure for defining such a set, or equivalently for comparing two partitions for an arbitrary symmetric quiver, is described in [Păd21, § 3.3.4]. Consider the path of partitions $T_A$ with coefficients $r_{\ell , A}$ as in (4.6) corresponding to $A\in L^d_{\delta ^\circ , w}$ . Order the coefficients $r_{\ell , A}$ in decreasing order, $r'_{1,A}\gt r'_{2,A}\gt \cdots \gt r'_{f(A),A}.$ Each $r'_{i, A}$ for $1\leqslant i\leqslant f(A)$ corresponds to a partition $\pi _{i,A}$ . Similarly, consider the path of partitions $T_B$ with coefficients $r_{\ell , B}$ corresponding to $B\in L^d_{\delta ^\circ , w}$ . Define similarly $r'_{1,B}\gt \cdots \gt r'_{f(B),B}$ and $\pi _{i,B}$ for $1\leqslant i\leqslant f(B)$ .

Define the set $R\subset L^d_{\delta ^\circ , w}\times L^d_{\delta ^\circ , w}$ which contains pairs $(A,B)$ such that one of the following holds:

1. – there exists $n\geqslant 1$ such that $r'_{n, A}\gt r'_{n, B}$ and $r'_{i, A}=r'_{i, B}$ for $i\lt n$ ;

2. – there exists $n\geqslant 1$ such that $r'_{i, A}=r'_{i, B}$ for $i\leqslant n$ , $\pi _{i, A}=\pi _{i, B}$ for $i\lt n$ , and $\pi _{n, B}\geqslant \pi _{n, A}$ (see § 2.1);

3. – they are of the form $(A, A)$ .

We then let $O_{w,w}:=L^d_{\delta ^\circ , w}\times L^d_{\delta ^\circ , w}\setminus R$ and

(4.17) \begin{equation} O:=\bigcup _{w,w'\in \mathbb {Z}}O_{w, w'}. \end{equation}

We will only use that such an order exists in the current paper.

In order to make the above process more accessible, we explain how to compute $r'_{1,A}$ and $\pi _{1,A}$ . From Proposition 4.12, there is an isomorphism of sets $L^d_{\delta ^\circ , w}\cong T^d_w$ .

For a dominant weight $\theta \in M(d)^+_{\mathbb {R}}$ with $\langle 1_d, \theta \rangle =w$ , its $r$ -invariant is the smallest $r$ such that $ \theta -w\tau _d\in 2r\textbf {W}$ . Equivalently, the $r$ -invariant of a dominant weight $\theta \in M(d)^+_{\mathbb {R}}$ is the maximum over all dominant cocharacters $\mu$ of $ST(d)$ :

(4.18) \begin{equation} r(\theta )=\text {max}_\mu \frac {\langle \mu , \theta \rangle }{ \langle \mu , R(d)^{\mu \gt 0} \rangle }; \end{equation}

see [Păd21, § 3.1.1]. Assume that $A=(d_i, v_i)_{i=1}^k\in T^d_w$ ; we also denote by $A$ the corresponding element of $T^d_{w}$ . Then one can show that

\begin{equation*}r'_{1,A}=r(\chi _A+\rho ),\end{equation*}

where $\chi _A:=\sum _{i=1}^k v_i\tau _{d_i}+\sum _{i=1}^k\theta _i\in M(d)_{\mathbb {R}}$ ; see (4.8) for the definition of $\theta _i$ . We let $\lambda$ be the antidominant cocharacter corresponding to the partition $(d_i)_{i=1}^k$ . By (4.7) and Proposition 4.9, write

\begin{equation*}\chi _A+\frac {1}{2}\mathfrak {g}(d)^{\lambda \lt 0}=\sum _{i=1}^k w_i\tau _{d_i}.\end{equation*}

There is a transformation

\begin{equation*}(d_i, v_i)_{i=1}^k\mapsto (d_i, w_i)_{i=1}^k.\end{equation*}

We will compute $r'_{1,A}$ in terms of $(w_i)_{i=1}^k$ . Let $\mu$ be a dominant cocharacter attaining the maximum above and assume that the associated partition of $\mu$ is $(e_i)_{i=1}^s$ . Then the maximum in (4.18) is also attained for the cocharacter $\mu$ with associated partition $\big(\sum _{i\leqslant b} e_i, \sum _{i\gt b} e_i\big)$ ; see [Păd21, Proposition 3.2]. We have that $(d_i)_{i=1}^k\geqslant (e_i)_{i=1}^s$ ; see § 2.1 for the notation and Proposition 4.8.

Let $\mu _a$ be a cocharacter which acts with weight $\sum _{i\leqslant a} d_i$ on $\beta _j$ for $j\gt a$ and weight $-\sum _{i\gt a} d_i$ on $\beta _j$ for $j\leqslant a$ . We have that

\begin{equation*}r(\chi _A+\rho )=\text {max}_a \frac {\langle \mu _a, \chi _A+\rho \rangle }{ \langle \mu _a, R(d)^{\mu _a\gt 0} \rangle },\end{equation*}

where the maximum is taken over all $1\leqslant a\lt k$ . We compute

\begin{equation*} \langle \mu _a, R(d)^{\mu _a\gt 0} \rangle = A\underline {d}\bigg(\sum _{i\gt a} \underline {d}_i\bigg)\bigg(\sum _{i\leqslant a}\underline {d}_i\bigg). \end{equation*}

Then

\begin{equation*}r(\chi _A+\rho )=r'_{1,A}=\frac {1}{A}\text {max}_a\biggl (\frac {\sum _{i\gt a} w_i}{\sum _{i\gt a} \underline {d}_i}-\frac {\sum _{i\leqslant a} w_i}{\sum _{i\leqslant a} \underline {d}_i}\biggr ),\end{equation*}

where the maximum is taken over $1\leqslant a\leqslant k$ . The partition $\pi _{1,A}$ can be reconstructed from all $1\leqslant a\lt k$ for which $\mu _a$ attains the maximum of (4.18). Assume that the set of all such $1\leqslant a \lt k$ is $1\leqslant a_2\lt \cdots \lt a_s\lt k$ . Then $\pi _{1,A}=(e_i)_{i=1}^s$ is the partition of $d$ with terms

\begin{equation*}(d_1+\cdots +d_{a_2},\cdots , d_{a_s+1}+\cdots +d_k).\end{equation*}

4.7 Semiorthogonal decompositions for very symmetric quivers

We now prove Theorem 4.6 and thus Theorem 4.5.

Proof of Theorem 4.6. The same argument used to prove [PT22a, Proposition 3.9] applies here. The argument in loc. cit. was organized in three steps:

1. (1) for a partition $(d_i)_{i=1}^k$ of $d$ and $(v_i)_{i=1}^k\in \mathbb {R}^k$ as in the statement of Theorem 4.6, the categorical Hall product

   \begin{equation*}\boxtimes _{i=1}^k \mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i})\to D^b(\mathscr {X}(d))\end{equation*}
   has image in $\mathbb {D}(d; \delta _d+\mu \sigma _d)$ ;

2. (2) the categories on the left-hand side of (4.4) are semiorthogonal for the ordering introduced in § 4.6 (see [Păd21, § 3.3] and [PT22a, § 3.4]);

3. (3) the categories on the left-hand side of (4.4) generate $\mathbb {D}(d; \delta _d+\mu \sigma _d)$ .

The proofs of (2) and (3) are exactly as in loc. cit. and follow from the semiorthogonal decomposition of the categorical Hall algebra for a quiver from [Păd21].

We explain the shifts used in defining the categories on the right-hand side of (4.4). Consider the weights $\chi _i\in M(d_i)$ such that $\sum _{i=1}^k\chi _i=\chi$ . The decomposition (4.9) can be rewritten as

\begin{equation*}\sum _{i=1}^k (\chi _i+\rho _i-\delta _{d_i})=\sum _{i=1}^k (\theta _i+v_i\tau _{d_i}+\psi _i ),\end{equation*}

and so

\begin{equation*}\chi _i+\rho _i-(\theta _i+\delta _{d_i}+v_i\tau _{d_i})=\psi _i\in \mathbf {W}(d_i).\end{equation*}

The proof of (1) follows from Corollary (4.14) and an explicit resolution by vector bundles of the Hall product of generators of categories $\mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i})$ for $1\leqslant i\leqslant k$ ; see Step 1 in the proof of [PT22a, Proposition 3.9] and [PT22a, Proposition 2.1].

In Theorem 4.6, we obtained a semiorthogonal decomposition after decomposing the polytope $\mathbf {V}(d)$ into translates of direct sums of the polytopes $\mathbf {W}(e)$ for dimension vectors $e$ which are parts of partitions of $d$ . We can also decompose the full weight lattice $M(d)$ to prove Theorem 4.15, which follows [Păd21, Theorem 1.1] for the quiver $Q$ and Proposition 4.11; see also the argument in [Păd23, Corollary 3.3].

Theorem 4.15. Let $Q$ be a very symmetric quiver. Then Theorem 4.2 holds for $Q$ .

Note that Theorem 4.15 follows from the analogues (for the full weight lattice) of Steps (2) and (3) from the proof of Theorem 4.6.

4.8 A class of symmetric quivers

In this subsection, we discuss some preliminaries before proving Theorems 4.1 and 4.2. We stop assuming that $Q$ is a very symmetric quiver.

Let $Q=(I, E)$ be a symmetric quiver such that the number of loops at each vertex $a\in I$ has the same parity $\varepsilon \in \mathbb {Z}/2\mathbb {Z}$ . We construct a very symmetric quiver $Q^\gimel =(I, E^\gimel )$ with potential as follows. For $a,b\in I$ , let $e_{ab}$ be the number of edges from $a$ to $b$ in $Q$ . Choose $A \in \mathbb {Z}_{\geqslant 1}$ such that

\begin{equation*}A\geqslant \text {max}\{e_{ab}\mid a,b\in I\}\quad\text {and}\quad A\equiv \varepsilon \,(\text {mod }2).\end{equation*}

For $a\in I$ , let $c_a\in \mathbb {N}$ be defined by

\begin{equation*}c_{a}:=\frac {A-e_{aa}}{2}.\end{equation*}

Add loops $\{\omega _k\mid 1\leqslant k\leqslant c_a\}$ and their opposites $\{\overline {\omega }_k\mid 1\leqslant k\leqslant c_a\}$ at $a$ and define the potential

\begin{equation*}W_a:=\sum _{k=1}^{c_a}\omega _k \overline {\omega }_k.\end{equation*}

Fix a total ordering on $I$ . For two different vertices $a\lt b$ in $I$ , let $c_{ab}:=A-e_{ab}$ . Add edges $\{e_k\mid 1\leqslant k\leqslant c_{ab}\}$ from $a$ to $b$ and their opposites $\{\overline {e}_k\mid 1\leqslant k\leqslant c_{ab}\}$ from $b$ to $a$ . Let

\begin{equation*}W_{ab}:=\sum _{k=1}^{c_{ab}}e_k\overline {e}_k.\end{equation*}

Consider the potential $W^\gimel$ of $Q^\gimel$ defined by

\begin{equation*}W^\gimel :=\sum _{a\in I}W_a+\sum _{a, b \in I, a\lt b}W_{ab}\end{equation*}

of the quiver $Q^\gimel$ . For $d\in \mathbb {N}^I$ , let $U(d)$ be the affine space of linear maps corresponding to the edges

\begin{equation*}\bigsqcup _{a\in I}\{\omega _k\mid 1\leqslant k\leqslant c_a\}\sqcup \bigsqcup _{a\lt b}\{e_k\mid 1\leqslant k\leqslant c_{ab}\}.\end{equation*}

The stack of representations of dimension $d$ of $Q^\gimel$ is

\begin{equation*}\mathscr {X}^\gimel (d):=R^\gimel (d)/G(d):= (R(d)\oplus U(d)\oplus U(d)^{\vee })/G(d).\end{equation*}

Consider the action of $\mathbb {C}^*$ on

\begin{align*} R^\gimel (d):=R(d)\oplus U(d)\oplus U(d)^{\vee } \end{align*}

of weight $(0, 0, 2)$ . Let $s\mbox{ and }v$ be the maps

\begin{equation*}\mathscr {X}(d)\xleftarrow {v}(R(d)\oplus U(d)^{\vee } )/G(d)\xrightarrow {s}\mathscr {X}'(d),\end{equation*}

where $s$ is the inclusion and $v$ is the projection. The Koszul equivalence (3.35) gives an equivalence (Knörrer periodicity)

(4.19) \begin{equation} \Theta _d=s_{\ast }v^{\ast }\colon D^b(\mathscr {X}(d))\xrightarrow {\sim }\mathrm {MF}^{\mathrm {gr}}(\mathscr {X}^\gimel (d), \mathrm {Tr}\,W^\gimel ). \end{equation}

We may write $\Theta$ instead of $\Theta _d$ when the dimension vector $d$ is clear from the context.

Denote by $\mathscr {X}^f(d)^{\text {ss}}$ and $\mathscr {X}^{\gimel f}f(d)^{\text {ss}}$ the varieties of stable framed representations of $Q$ and $Q^\gimel$ , respectively.

We will prove Theorem 4.1 in the next section. The same argument also applies to obtain Theorem 4.2 using Theorem 4.15.

4.9 Proof of Theorem 4.1

The order of summands in the semiorthogonal decomposition is induced from the order in § 4.6 for the quiver $Q^\gimel$ . Note that $Q^\gimel$ depends on the choice of a certain integer $A$ , but we do not discuss the dependence of the order on this choice and only claim that such an order exists.

For a partition $(d_i)_{i=1}^k$ of $d$ and an associated antidominant cocharacter $\lambda$ , we define the weights $\theta ^\gimel _i\in M(d_i)_{\mathbb {R}}$ such that

\begin{equation*}\sum _{i=1}^k \theta ^\gimel _i=-\frac {1}{2}R^\gimel (d)^{\lambda \gt 0}+\frac {1}{2}\mathfrak {g}(d)^{\lambda \gt 0}.\end{equation*}

For $\delta _d\in M(d)_{\mathbb {R}}^{W_d}$ , denote by $\mathbb {M}^\gimel (d; \delta _d) \subset D^b(\mathscr {X}^\gimel (d))$ the magic categories (2.22) for the quiver $Q^\gimel$ . Consider the quasi-BPS categories

\begin{align*} \mathbb {S}^{\gimel \textrm {gr}}(d; \delta _d) :=\mathrm {MF}^{\textrm {gr}}(\mathbb {M}^\gimel (d; \delta ), \mathop {\textrm {Tr}} W') \subset \mathrm {MF}^{\textrm {gr}}(\mathscr {X}^\gimel (d), \mathop {\textrm {Tr}} W^\gimel ). \end{align*}

Define

\begin{equation*}\delta ^\circ _d:=-\tfrac {1}{2}\det U(d)=-\tfrac {1}{2} U(d),\end{equation*}

where above, as elsewhere, we abuse notation and denote by $U(d)$ the sum of weights of $U(d)$ . Further, define

\begin{equation*}\delta ^\gimel _d=\delta _d+\delta ^\circ _d.\end{equation*}

We will use the notation $\delta ^\gimel _{d_i}\mbox{ and } \delta _{d_i}$ from (4.1). Note that $\delta ^\gimel _d+\mu \sigma _d$ is a good weight if and only if $\delta _d+\mu \sigma _d$ is a good weight because $\langle \lambda , \delta ^\circ _d\rangle \in \frac {1}{2}\mathbb {Z}$ for all $\lambda$ as in Definition 2.12.

Step 1. There is a semiorthogonal decomposition

\begin{equation*}\mathrm {MF}^{\mathrm {gr}}(\mathscr {X}^{\gimel f}(d)^{\text {ss}}, \mathrm {Tr}\,W^\gimel )=\bigg \langle \bigotimes _{i=1}^k \mathbb {S}^{\gimel \mathrm {gr}}(d_i; \theta ^\gimel _i+\delta ^\gimel _{d_i}+v_i \tau _{d_i}) : \mu \leqslant \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}\lt 1+\mu \bigg \rangle ,\end{equation*}

where the right-hand side is as in Theorem 4.5 .

Proof. The claim follows by applying matrix factorizations [PT22a, Proposition 2.5] and [Păd22, Proposition 2.1] for the potential $W^\gimel$ to the semiorthogonal decomposition of Theorem 4.5 for the quiver $Q^\gimel$ .

Step 2. There is an equivalence

\begin{equation*}\Theta _d^f\colon D^b(\mathscr {X}^f(d)^{\text {ss}})\xrightarrow {\sim }\mathrm {MF}^{\mathrm {gr}}(\mathscr {X}^{\gimel f}(d)^{\text {ss}}, \mathrm {Tr}\,W^\gimel ).\end{equation*}

Proof. Consider the natural projection map $\pi ^\gimel \colon \mathscr {X}^{\gimel f}(d)\to \mathscr {X}^f(d).$ Then

\begin{equation*}(\pi ^{\gimel })^{-1}(\mathscr {X}^f(d)^{\text {ss}})\subset \mathscr {X}^{f \gimel }(d)^{\text {ss}}\end{equation*}

is an inclusion of open sets and

(4.20) \begin{align} (\pi ^{\gimel })^{-1}(\mathscr {X}^f(d)^{\text {ss}})\cap \mathrm {Crit}(\mathrm {Tr}\,W^\gimel )=\mathscr {X}^{\gimel f}(d)^{\text {ss}}\cap \mathrm {Crit}(\mathrm {Tr}\,W^\gimel ). \end{align}

We have equivalences

\begin{equation*}\mathrm {MF}^{\mathrm {gr}}(\mathscr {X}^{\gimel f}(d)^{\text {ss}}, \mathrm {Tr}\,W^\gimel )\stackrel {\sim }{\to } \mathrm {MF}^{\mathrm {gr}}(\pi ^{\gimel -1}(\mathscr {X}^f(d)^{\text {ss}}), \mathrm {Tr}\,W^\gimel )\stackrel {\sim }{\leftarrow }D^b(\mathscr {X}^f(d)^{\text {ss}}).\end{equation*}

Here the first equivalence follows from (4.20) and (2.3), and the second equivalence is an instance of the Koszul equivalence from Theorem 2.5.

The claim of Theorem 4.1 then follows from this next step.

Step 3. The equivalence (4.19) restricts to the equivalence

\begin{align*} \Theta _d\colon \bigotimes _{i=1}^k\mathbb {M}(d_i; \theta _i+\delta _{d_i}+v_i\tau _{d_i})\stackrel {\sim }{\to } \bigotimes _{i=1}^k\mathbb {S}^{\gimel \mathrm {gr}}(d_i; \theta ^\gimel _i+\delta ^\gimel _{d_i}+v_i\tau _{d_i}), \end{align*}

where the tensor products on the left- and right-hand sides are embedded by categorical Hall products into $D^b(\mathscr {X}(d))$ and $\mathrm {MF}^{\textrm {gr}}(\mathscr {X}^\gimel (d), \mathop {\textrm {Tr}} W^\gimel )$ , respectively.

Proof. By the compatibility of the Koszul equivalence with categorical Hall products in Proposition 2.8 and with quasi-BPS categories in Proposition 3.20, it is enough to check that

\begin{equation*}\sum _{i=1}^k (\theta _i+v_i\tau _{d_i}+\delta _{d_i})-U(d)^{\lambda \gt 0}=\sum _{i=1}^k\Bigl (\theta ^\gimel _i+\delta ^\gimel _{d_i}+v_i\tau _{d_i}+\frac {1}{2}U(d_i)\Bigr ).\end{equation*}

Recall that

\begin{align*} &\sum _{i=1}^k U(d_i)=U(d)^\lambda ,\quad \sum _{i=1}^k \delta ^\circ _{d_i}=-\frac {1}{2}U(d),\\ & \sum _{i=1}^k (\theta ^\gimel _i-\theta _i)=-\frac {1}{2}(R^\gimel (d)^{\lambda \gt 0}-R(d)^{\lambda \gt 0}). \end{align*}

It thus suffices to show that

\begin{equation*}-\tfrac {1}{2}(R^\gimel (d)^{\lambda \gt 0}-R(d)^{\lambda \gt 0})-\tfrac {1}{2}U(d)+\tfrac {1}{2}U(d)^\lambda +U(d)^{\lambda \gt 0}=0,\end{equation*}

which can be verified by a direct computation.

4.10 More classes of quivers

Note that Theorem 4.1 applies to any tripled quiver. The semiorthogonal decomposition in Theorem 4.1 is particularly simple for quivers satisfying the following assumption.

Assumption 4.16. The quiver $Q=(I,E)$ is symmetric and has the following properties:

1. – for all $a, b \in I$ different, the number of edges from $a$ to $b$ is even;

2. – for all $a\in I$ , the number of loops at $a$ is odd.

Examples of quivers satisfying Assumption 4.16 are tripled quivers $Q$ of quivers $Q^\circ =(I, E^\circ )$ satisfying the following assumption, where we recall $\alpha _{a, b}$ from (1.9).

Assumption 4.17. For all $a, b \in I$ , we have $\alpha _{a, b} \in 2\mathbb {Z}$ .

For example, Assumption 4.17 is satisfied if $Q^{\circ }$ is symmetric. Further, the moduli stack of semistable sheaves on a K3 surface is locally described by the stack of representations of a preprojective algebra of a quiver satisfying Assumption 4.17; see [PT23a].

We discuss the particular case of Theorem 4.1 for quivers satisfying Assumption 4.16.

Corollary 4.18. Let $Q$ be a quiver satisfying Assumption 4.16. Let $\mu \in \mathbb {R}$ be such that $\mu \sigma _d$ is a good weight. Then there is a $X(d)$ -linear semiorthogonal decomposition

(4.21) \begin{equation} D^b(\mathscr {X}^f(d)^{\text {ss}})=\biggl \langle \bigotimes _{i=1}^k \mathbb {M}(d_i)_{v_i} : \mu \leqslant \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}\lt 1+\mu \biggr \rangle , \end{equation}

where the right-hand side is over all partitions $(d_i)_{i=1}^k$ of $d$ and integers $(v_i)_{i=1}^k\in \mathbb {Z}^k$ satisfying the above inequality.

Proof. We set $\delta _d=0$ in Theorem 4.1. Fix $d\in \mathbb {N}^I$ . For $a\in I$ , let $V^a$ be a $\mathbb {C}$ -vector space of dimension $d^a$ . For each $a, b\in I$ , let $V^{ab}:=\text {Hom} (V^a, V^b )$ , and let $e^{ab}$ denote the number of edges from $a$ to $b$ . Then

\begin{equation*}\frac {1}{2}R(d)^{\lambda \gt 0}-\frac {1}{2}\mathfrak {g}(d)^{\lambda \gt 0}=\sum _{a\in I}\frac {e^{aa}-1}{2}\mathfrak {g}(d)^{\lambda \gt 0}+\sum _{a\neq b\in I}\frac {e^{ab}}{2}(V^{ab})^{\lambda \gt 0}\in M(d).\end{equation*}

Thus $\theta _i\in M(d_i)^{W_{d_i}}$ , so the weights $v_i$ are integers for $1\leqslant i\leqslant k$ . Moreover, there is an equivalence $\mathbb {M}(d_i)_{v_i}=\mathbb {M}(d_i; v_i \tau _{d_i})\stackrel {\sim }{\to } \mathbb {M}(d_i; \theta _i+v_i \tau _{d_i})$ by taking the tensor product with $\theta _i$ . The claim then follows from Theorem 4.1.

4.11 More framed quivers

The semiorthogonal decompositions in Theorem 4.5 and Corollary 4.18 also hold for spaces of semistable representations of the quivers $Q^{\alpha f}$ , where $Q=(I,E)$ is as in the statements of these theorems, $\alpha \in \mathbb {Z}_{\geqslant 1}$ , and $Q^{\alpha f}$ has set of vertices $I\sqcup \{\infty \}$ and set of edges $E$ together with $\alpha$ edges from $\infty$ to any vertex of $I$ . For future reference, we state the version of Corollary 4.18 for the space of semistable representations $\mathscr {X}^{\alpha f}(d)^{\text {ss}}$ of the quiver $Q^{\alpha f}$ .

Corollary 4.19. Let $Q$ be a quiver satisfying Assumption 4.17. Let $\mu \in \mathbb {R}$ be such that $\mu \sigma _d$ is a good weight and let $\alpha \in \mathbb {N}$ . Then there is a $X(d)$ -linear semiorthogonal decomposition

\begin{equation*}D^b(\mathscr {X}^{\alpha f}(d)^{\text {ss}} )=\biggl \langle \bigotimes _{i=1}^k \mathbb {M}(d_i)_{v_i} : \mu \leqslant \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}\lt \alpha +\mu \biggr \rangle ,\end{equation*}

where the right-hand side is over all partitions $(d_i)_{i=1}^k$ of $d$ and integers $(v_i)_{i=1}^k\in \mathbb {Z}^k$ satisfying the above inequality.

4.12 Semiorthogonal decompositions for general potentials

Let $W$ be a potential of $Q$ . By [PT22a, Proposition 2.5], there are analogous semiorthogonal decompositions to those in Theorems 4.1 and 4.2 (and also to those in Corollaries 4.18 and 4.19) for categories of matrix factorizations. Recall the definition of (graded or not) quasi-BPS categories from (2.27) and (2.28). We first state the version for Corollary 4.18, which we use in [PT23b].

Theorem 4.20. Let $Q$ be a quiver satisfying Assumption 4.17 and let $W$ be a potential of $Q$ (and possibly a grading as in § 2.4). Let $\mu \in \mathbb {R}$ be such that $\mu \sigma _d$ is a good weight and let $\alpha \in \mathbb {Z}_{\geqslant 1}$ . There is a semiorthogonal decomposition

\begin{equation*} \mathrm {MF}^{\bullet }(\mathscr {X}^{\alpha f}(d)^{\text {ss}}, \mathrm {Tr}\,W)=\biggl \langle \bigotimes _{i=1}^k \mathbb {S}^{\bullet }(d_i)_{v_i}: \mu \leqslant \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}\lt \mu +\alpha \biggr \rangle ,\end{equation*}

where the right-hand side is over all partitions $(d_i)_{i=1}^k$ of $d$ and integers $(v_i)_{i=1}^k \in \mathbb {Z}^k$ satisfying the above inequality and where $\bullet \in \{\emptyset , \mathrm {gr}\}$ .

A version of Theorem 4.2 (for quivers satisfying Assumption 4.17) is the following.

Theorem 4.21. Let $Q$ be a quiver satisfying Assumption 4.17 and let $W$ be a potential of $Q$ (and possibly consider a grading as in § 2.4). There is a semiorthogonal decomposition

\begin{equation*} \mathrm {MF}^{\bullet }(\mathscr {X}(d), \mathrm {Tr}\,W)=\biggl \langle \bigotimes _{i=1}^k \mathbb {S}^{\bullet }(d_i; v_i\tau _{d_i}): \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}\biggr \rangle ,\end{equation*}

where the right-hand side is over all partitions $(d_i)_{i=1}^k$ of $d$ and integers $(v_i)_{i=1}^k \in \mathbb {Z}^k$ satisfying the above inequality and where $\bullet \in \{\emptyset , \mathrm {gr}\}$ .

In the case of a doubled quiver, by combining Theorems 4.2 and 4.21 with the Koszul equivalence in Theorem 2.5 and the compatibility of Koszul equivalence with the categorical Hall product in Proposition 2.8, we obtain the following.

Theorem 4.22. Let $Q^{\circ }$ be a quiver and let $(Q^{\circ , d}, \mathscr {I})$ be its doubled quiver relation. For a partition $(d_i)_{i=1}^k$ of $d$ , let $\lambda$ be an associated antidominant cocharacter, and define $\theta _i \in \frac {1}{2}M(d_i)^{W_{d_i}}$ by

\begin{equation*} \sum _{i=1}^k \theta _i=-\frac {1}{2}\overline {R}(d)^{\lambda \gt 0}+\mathfrak {g}(d)^{\lambda \gt 0}. \end{equation*}

There is a semiorthogonal decomposition

(4.22) \begin{align} D^b(\mathscr {P}(d))=\biggl \langle \bigotimes _{i=1}^k \mathbb {T}(d_i, \theta _i+ v_i \tau _{d_i})\biggr \rangle, \end{align}

where the right-hand side is over all partitions $(d_i)_{i=1}^k$ of $d$ and tuples $(v_i)_{i=1}^k\in \mathbb {R}^k$ such that the sum of coefficients of $\theta _i+v_i\tau _{d_i}$ is an integer for all $1\leqslant i\leqslant k$ and such that

(4.23) \begin{equation} \frac {v_1}{\underline {d}_1}\lt \cdots \lt \frac {v_k}{\underline {d}_k}. \end{equation}

Moreover, each summand is given by the image of the categorical Hall product (2.12).

If, furthermore, $Q^{\circ }$ satisfies Assumption 4.17, then $v_i \in \mathbb {Z}$ and $\theta _i \in M(d_i)^{W_{d_i}}$ , so the right-hand side of (4.22) is over all partitions $(d_i)_{i=1}^k$ of $d$ and all tuples $(v_i)_{i=1}^k\in \mathbb {Z}^k$ satisfying (4.23).

The following example will be used in [PT23a].

Example 4.23. Let $Q^{\circ }$ be the quiver with one vertex and $g\geqslant 1$ loops. Then $d \in \mathbb {N}$ and the semiorthogonal decomposition (4.22) is

\begin{align*} D^b(\mathscr {P}(d))=\biggl \langle \bigotimes _{i=1}^k \mathbb {T}(d_i)_{w_i} :\frac {v_1}{d_1}\lt \cdots \lt \frac {v_k}{d_k} \biggr \rangle . \end{align*}

Here, $w_i\in \mathbb {Z}$ for $1\leqslant i\leqslant k$ is given by

\begin{align*} w_i=v_i+(g-1)d_i\biggl(\sum _{i\gt j}d_j-\sum _{i\lt j}d_j \biggr). \end{align*}

Note that $\mathbb {T}(d_i)_{w_i}\cong \mathbb {T}(d_i)_{v_i}$ for all $1\leqslant i\leqslant k$ .

4.13 Strong generation of quasi-BPS categories

We use Theorem 4.1 to prove the strong generation of the (graded or not) quasi-BPS categories

\begin{equation*}\mathbb {S}^\bullet (d; \delta _d)\quad \text {for }\bullet \in \{\emptyset , \mathrm {gr}\},\end{equation*}

where the grading is as in § 1.8. We first recall some terminology.

Let $\mathcal {D}$ be a pre-triangulated dg-category. For a set of objects $\mathcal {S} \subset \mathcal {D}$ , we denote by $\langle \mathcal {S} \rangle$ the smallest subcategory which contains $S$ and is closed under shifts, finite direct sums, and direct summands. For subcategories $\mathcal {C}_1, \mathcal {C}_2 \subset \mathcal {D}$ , we denote by $\mathcal {C}_1 \star \mathcal {C}_2 \subset \mathcal {D}$ the smallest subcategory which contains objects $E$ that fit into distinguished triangles $A_1 \to E \to A_2\to A_1[1]$ for $A_i \in \mathcal {C}_i$ and is closed under shifts, finite direct sums, and direct summands.

We say that $\mathcal {D}$ is strongly generated by $C \in \mathcal {D}$ if $\mathcal {D}=\langle C \rangle ^{\star n}$ for some $n\geqslant 1$ . A dg-category $\mathcal {D}$ is said to be regular if it has a strong generator. It is said to be smooth if the diagonal dg-module of $\mathcal {D}$ is perfect. It is proved in [Lun10, Lemmas 3.5 and 3.6] that if $\mathcal {D}$ is smooth, then $\mathcal {D}$ is regular.

Proposition 4.24. Let $Q$ be a symmetric quiver such that the number of loops at each vertex $i\in I$ has the same parity, let $W$ be any potential of $Q$ , let $d\in \mathbb {N}^I$ , and let $\delta _d\in M(d)_{\mathbb {R}}^{W_d}$ . The category $\mathbb {S}^\bullet (d; \delta _d)$ has a strong generator, and thus it is regular.

Proof. The category $\mathbb {S}^\bullet (d; \delta _d)$ is admissible in $\mathrm {MF}^\bullet (\mathscr {X}^f(d),\mathrm {Tr}\,W )$ by the variant for an arbitrary potential (see [PT22a, Proposition 2.5]) of Theorem 4.1. Let

\begin{equation*}\Phi \colon \mathrm {MF}^\bullet (\mathscr {X}^f(d),\mathrm {Tr}\,W)\to \mathbb {S}^\bullet (d; \delta _d)\end{equation*}

be the adjoint of the inclusion. The category $\mathrm {MF}^\bullet (\mathscr {X}^f(d),\mathrm {Tr}\,W)$ is smooth; see [FK18, Lemma 2.11]. Therefore it is regular, so it has a strong generator $C$ . Then $\mathbb {S}^\bullet (d; \delta _d)$ has the strong generator $\Phi (C)$ .

Remark 4.25. We do not discuss smoothness of the category $\mathbb {S}^\bullet (d; \delta _d)$ in this paper. Note that smoothness is equivalent to regularity if the category is proper [Orl16, Theorem 3.18], but this is not the case for the categories $\mathbb {S}^\bullet (d; \delta _d)$ . However, it should be possible to prove the smoothness of $\mathbb {S}^{\bullet }(d; \delta _d)$ using noncommutative matrix factorizations as in [PT24, Proposition 3.29]. The details may appear elsewhere.

5. Quasi-BPS categories for tripled quivers

In this section, we prove a categorical analogue of Davison’s support lemma [Dav16, Lemma 4.1] for tripled quivers with potential of quivers $Q^\circ$ satisfying Assumption 4.17; see Theorem 5.1. We then use Theorem 5.1 to construct reduced quasi-BPS categories $\mathbb {T}$ for preprojective algebras, which are proper over the good moduli space $P$ of representations of the preprojective algebra and are regular. When the reduced stack of representations of the preprojective algebra is classical, we show that the relative Serre functor of $\mathbb {T}$ over $P$ is trivial and, further, that the category $\mathbb {T}$ is indecomposable.

Throughout this section, we consider tripled quivers with potential or preprojective algebra for quivers $Q^\circ$ satisfying Assumption 4.17.

5.1 The categorical support lemma

Let $Q^{\circ }=(I, E^{\circ })$ be a quiver satisfying Assumption 4.17, and consider its tripled quiver $Q=(I, E)$ with potential $W$ ; see § 2.2.6. We will use the notation from § 2.2.6. Recall that

\begin{align*} \mathscr {X}(d)=R(d)/G(d),\quad R(d)=\overline {R}(d) \oplus \mathfrak {g}(d), \end{align*}

where $\overline {R}(d)$ is the representation space of the doubled quiver of $Q^{\circ }$ . There is thus a projection map onto the second summand (which records the linear maps corresponding to the loops in the tripled quiver not in the doubled quiver):

\begin{equation*}\tau \colon \mathscr {X}(d)\to \mathfrak {g}(d)/G(d).\end{equation*}

We consider the good moduli space morphism

(5.1) \begin{align} \pi _{\mathfrak {g}}\colon \mathfrak {g}(d)/G(d) \to \mathfrak {g}(d)/\!\!/ G(d)=\prod _{a \in I}\mathrm {Sym}^{d^a}(\mathbb {C}). \end{align}

The above map sends $z \in \mathfrak {g}(d)=\bigoplus _{a\in I}\mathrm {End}(V^a)$ to its generalized eigenvalues. Let $\Delta$ be the diagonal

\begin{align*} \Delta \colon \mathbb {C} \hookrightarrow \prod _{a\in I}\mathrm {Sym}^{d^a}(\mathbb {C}),\quad x\mapsto \prod _{a\in I} (\overbrace {x,\ldots , x}^{d^a})=(\overbrace {x,\ldots , x}^{\underline {d}}). \end{align*}

Consider the composition

(5.2) \begin{align} \pi \colon \mathrm {Crit}(\mathop {\textrm {Tr}} W) \hookrightarrow \mathscr {X}(d) \xrightarrow {\tau } \mathfrak {g}(d)/G(d) \xrightarrow {\pi _{\mathfrak {g}}} \mathfrak {g}(d)/\!\!/ G(d). \end{align}

The following is the main result of this section and is a generalization of [PT22b, Theorem 3.1] which discusses the case of the tripled quiver with potential of the Jordan quiver.

Theorem 5.1. Let $v \in \mathbb {Z}$ be such that $\gcd (v, \underline {d})=1$ . Then any object in $\mathbb {S}(d)_v$ is supported on $\pi ^{-1}(\Delta )$ .

Before the proof of Theorem 5.1, we introduce some notation related to formal completions of fibers over $\mathfrak {g}(d)/\!\!/ G(d)$ . For $p \in \mathfrak {g}(d)/\!\!/ G(d)$ , we denote by $\mathscr {X}_p(d)$ the pull-back of the morphism

\begin{equation*}\pi _{\mathfrak {g}} \circ \tau \colon \mathscr {X}(d) \to \mathfrak {g}(d)/\!\!/ G(d)\end{equation*}

by $\textrm {Spec} \widehat {\mathcal {O}}_{\mathfrak {g}(d)/\!\!/ G(d), p} \to \mathfrak {g}(d)/\!\!/ G(d)$ . We write an element $p \in \prod _{a\in I}\mathrm {Sym}^{d^a}(\mathbb {C})$ as

(5.3) \begin{align} p=\biggl (\sum _{j=1}^{l^a} d^{a, (j)} x^{a, (j)}\biggr )_{a\in I} \end{align}

where $x^{a, (j)} \in \mathbb {C}$ with $x^{a, (j)} \neq x^{a, (j')}$ for $1\leqslant j\neq j'\leqslant l^a$ and $d^{a, (j)} \in \mathbb {Z}_{\geqslant 1}$ for $1\leqslant j\leqslant l^a$ are such that $\sum _{j=1}^{l^a}d^{a,(j)}=d$ . There is an isomorphism

(5.4) \begin{align} \mathscr {X}_p(d)\cong \biggl (\overline {R}(d) \times \prod _{a, j} \widehat {\mathfrak {g}}^{a, (j)}\biggr )/G_p \end{align}

where $V^a=\bigoplus _{j} V^{a, (j)}$ is the decomposition into generalized eigenspaces corresponding to $p$ , $G_p:=\prod _{a, j} GL(V^{a, (j)})$ , and $\mathfrak {g}^{a, (j)}:=\mathrm {End}(V^{a, (j)})$ .

Remark 5.2. For a point $p$ as in (5.3), let $J$ be the set of pairs $(a, j)$ such that $a \in I$ and $1\leqslant j \leqslant l^a$ . The support of $p$ is defined to be

\begin{equation*} \mathrm{supp}(p):=\{x^{a,(j)} \mid (a,j)\in J \}\subset \mathbb{C}.\end{equation*}

Let $Q_p^{\circ }$ be a quiver with vertex set $J$ and such that the number of arrows from $(a, j)$ to $(b, j')$ is the number of arrows from $a$ to $b$ in $Q^{\circ }$ . Since we have

(5.5) \begin{align} &\overline {R}(d) \oplus \bigoplus _{(a, j)\in J}\mathfrak {g}^{a, (j)} \\ \notag &\quad =\bigoplus _{(a\to b) \in E^{\circ }, j, j'} \textrm {Hom}(V^{a, (j)}, V^{b, (j')}) \oplus \textrm {Hom}(V^{b, (j')}, V^{a, (j)}) \oplus \bigoplus _{(a, j)\in J}\mathrm {End}(V^{a, (j)}), \end{align}

the space (5.5) is the representation space of the tripled quiver $Q_p$ of $Q_p^{\circ }$ with dimension vector $d=(d^{a, (j)})_{(a, j)\in J}$ . Note that $Q_p^{\circ }$ satisfies Assumption 4.17 since $Q^{\circ }$ satisfies Assumption 4.17. There is a correspondence from dimension vectors of $Q_p$ to dimension vectors of $Q$ :

\begin{align*} (d^{a, (j)})_{(a, j)\in J} \mapsto \biggl(d^{a}=\sum _{j}d^{a, (j)} \biggr)_{a\in I}. \end{align*}

Proof of Theorem 5.1. The proof is similar to the proof of [PT22b, Theorem 3.1] but simpler by the use of Proposition 3.21. Consider an object $\mathcal {F} \in \mathbb {S}(d)_v$ . Let $p\in \prod _{a\in I}\mathrm {Sym}^{d^a}(\mathbb {C})$ be a point which decomposes as $p=p'+p''$ such that the supports of $p'$ and $p''$ are disjoint. Write $p$ as in (5.3). Assume that the support of $\mathcal {F}$ intersects $\pi ^{-1}(p)$ . We will reach a contradiction with the assumption $\gcd (v, \underline {d})=1$ . Define

(5.6) \begin{align} \mathbb {S}_p(d)_v \subset \mathrm {MF}(\mathscr {X}_p(d), \mathop {\textrm {Tr}} W ) \end{align}

to be the full subcategory generated by matrix factorizations whose factors are direct summands of the vector bundles $\mathcal {O}_{\mathscr {X}_p(d)} \otimes \Gamma _{G_p}(\chi )$ , where $\chi$ is a $G_p$ -dominant $T(d)$ -weight satisfying

(5.7) \begin{align} \chi +\rho _p -v\tau _d\in \mathbf {W}_p(d):=\tfrac {1}{2}\mathrm {sum}[0, \beta ]. \end{align}

Here, $\rho _p$ is half the sum of the positive roots of $G_p$ , and the Minkowski sum for $\mathbf {W}_p(d)$ is over all weights $\beta$ of the $T(d)$ -representation (5.5). Define $n_{\lambda , p}$ by

(5.8) \begin{align} n_{\lambda , p}:=\big \langle \lambda , \det \big (\mathbb {L}_{\mathscr {X}_p(d)}^{\lambda \gt 0}|_{0}\big )\big \rangle =\big \langle \lambda , \det \big (\mathbb {L}_{\mathscr {X}(d)}^{\lambda \gt 0}|_{0}\big )\big \rangle . \end{align}

As in Lemma 2.10, the subcategory (5.6) is generated by matrix factorizations whose factors are of the form $\Gamma \otimes \mathcal {O}_{\mathscr {X}_p(d)}$ , where $\Gamma$ is a $G_p$ -representation such that any $T(d)$ -weight $\chi$ of $\Gamma$ satisfies

\begin{align*} \langle \lambda , \chi -v\tau _d \rangle \in \left [{-}\tfrac {1}{2} n_{\lambda , p}, \tfrac {1}{2} n_{\lambda , p} \right ]. \end{align*}

In particular, by the identity (5.8), the restriction along with the natural morphism $\iota _p \colon \mathscr {X}_p(d) \to \mathscr {X}(d)$ restricts to the functor

\begin{align*} \iota _p^{\ast } \colon \mathbb {S}(d)_v \to \mathbb {S}_p(d)_v. \end{align*}

Therefore, by the assumption that the support of $\mathcal {F}$ intersects $\pi ^{-1}(p)$ , we have $0\neq \iota _p^{\ast }\mathcal {F} \in \mathbb {S}_p(d)_v$ , and in particular $\mathbb {S}_p(d)_v \neq 0$ . We show that, in this case, $v$ is not coprime with $\underline {d}$ .

The decomposition $p=p'+p''$ corresponds to decomposition $V^{a}=V'^{a} \oplus V''^{a}$ , where

\begin{align*} {V'}^{a}=\bigoplus _{x^{a, (j)} \in \mathrm {supp}(p')} V^{a, (j)}, \quad {V''}^{a}=\bigoplus _{x^{a, (j)} \in \mathrm {supp}(p'')} V^{a, (j)} \end{align*}

for all $a\in I$ . Let $d'^{a}=\dim V'^{a}$ , $d''^{a}=\dim V''^{a}$ , $d'=(d'^{a})_{a\in I}$ , and $d''=(d''^{a})_{a\in I}$ , so $d=d'+d''$ . By Lemma 5.3, after possibly replacing the isomorphism (5.4), the regular function $\mathop {\textrm {Tr}} W$ restricted to $\mathscr {X}_p(d)$ is written as

(5.9) \begin{align} \mathop {\textrm {Tr}} W|_{\mathscr {X}_p(d)}= \mathop {\textrm {Tr}} W'\boxplus \mathop {\textrm {Tr}} W'\boxplus f. \end{align}

Here, $\mathop {\textrm {Tr}} W'$ and $\mathop {\textrm {Tr}} W''$ are the regular functions given by $\mathop {\textrm {Tr}} W$ on $\mathscr {X}(d')$ and $\mathscr {X}(d'')$ , respectively, restricted to $\mathscr {X}_{p'}(d')$ and $\mathscr {X}_{p''}(d'')$ , and $f$ is a nondegenerate $G_p$ -invariant quadratic form on $U\oplus U^{\vee }$ given by $f(u, v)=\langle u, v \rangle$ , where $U$ is the $G_p$ -representation

\begin{align*} U:=\bigoplus _{(a\to b) \in E^{\circ }}\textrm {Hom}({V'}^{a}, {V''}^{b}) \oplus \bigoplus _{(b\to a) \in E^{\circ }}\textrm {Hom}({V'}^{a}, {V''}^{b}). \end{align*}

Note that we have the decomposition as $G_p$ -representations

(5.10) \begin{align} \overline {R}(d)=\overline {R}(d') \oplus \overline {R}(d'') \oplus U \oplus U^{\vee }. \end{align}

We have a diagram

where $\mathcal {U}$ is the vector bundle on $\mathscr {X}_{p'}(d') \times \mathscr {X}_{p''}(d'')$ determined by the $G_p$ -representation $U$ , $i$ is the closed immersion $x \mapsto (x, 0)$ , and $j$ is the natural morphism induced by the formal completion which induces the isomorphism on critical loci of $\mathop {\textrm {Tr}} W$ . Consider the functor

(5.11) \begin{align} \Psi := & j^{\ast }i_{\ast }q^{\ast } \colon \mathrm {MF}(\mathscr {X}_{p'}(d'), \mathop {\textrm {Tr}} W') &\boxtimes \mathrm {MF}(\mathscr {X}_{p''}(d''), \mathop {\textrm {Tr}} W'') \\ \notag &\stackrel {\sim }{\to }\mathrm {MF}(\mathcal {U} \oplus \mathcal {U}^{\vee }, \mathop {\textrm {Tr}} W) \stackrel {j^{\ast }}{\hookrightarrow } \mathrm {MF}(\mathscr {X}_p(d), \mathop {\textrm {Tr}} W), \end{align}

where the first arrow is an equivalence by Knörrer periodicity (3.43) and the second arrow is fully faithful with dense image; see [Tod21, Lemma 6.4]. Let $v', v''\in \mathbb {Q}$ be such that

(5.12) \begin{equation} v\tau _d=v'\tau _{d'}+v''\tau _{d''}, \end{equation}

and let $\delta '\in M(d')_{\mathbb {R}}$ and $\delta ''\in M(d'')_{\mathbb {R}}$ be such that

(5.13) \begin{align} \delta ' +\delta ''=\tfrac {1}{2}U. \end{align}

The quiver $Q^\circ$ satisfies Assumption 4.17, and thus $\delta ' \in M(d')^{W_{d'}}$ and $\delta '' \in M(d'')^{W_{d''}}$ . By Proposition 3.21, the functor (5.11) restricts to the fully faithful functor with dense image:

(5.14) \begin{align} \mathbb {S}_{p'}(d'; \delta '+v'\tau _{d'})\boxtimes \mathbb {S}_{p''}(d''; \delta ''+v''\tau _{d''}) \to \mathbb {S}_p(d)_v. \end{align}

Then we have

\begin{align*} \mathbb {S}_{p'}(d')_{v'} & \simeq \mathbb {S}_{p'}(d'; \delta '+v' \tau _{d'}) \neq 0, \\ \mathbb {S}_{p''}(d'')_{v''} & \simeq \mathbb {S}_{p''}(d''; \delta '' +v'' \tau _{d''}) \neq 0. \end{align*}

In particular, we have $v' \in \mathbb {Z}$ and $v'' \in \mathbb {Z}$ by Remark 2.11. By (5.12), we further have that

\begin{align*} \frac {v'}{\underline {d}'} =\frac {v''}{\underline {d}''}=\frac {v}{\underline {d}}, \end{align*}

which contradicts the assumption that $\gcd (v, \underline {d})=1$ .

We have postponed the proof of the following.

Lemma 5.3. By replacing the isomorphism (5.4) if necessary, the identity (5.9) holds.

Proof. (See [PT22b, Lemma 3.3].) For $p \in \prod _{a \in I}\mathrm {Sym}^{d^a}(\mathbb {C})$ as in (5.3), let $u \in \mathfrak {g}(d)/G(d)$ be the unique closed point over $p$ . Note that $u$ is represented by a diagonal matrix with eigenvalues $x^{a, (j)}$ . In particular, we can assume that

\begin{align*} u \in \bigoplus _{(a, j)\in J}\mathfrak {g}^{a, (j)} \subset \mathfrak {g}(d). \end{align*}

We construct a map

\begin{align*} \nu \colon \biggl (\overline {R}(d) \oplus \bigoplus _{(a, j)\in J}\mathfrak {g}^{a, (j)}\biggr )/G_p \to \mathscr {X}(d) \end{align*}

given by $(\alpha , \beta =\beta ^{a, (j)}) \mapsto (\alpha , \beta +u)$ . The morphism $\nu$ is étale at $\nu (0)$ . Indeed, the tangent complex of $\mathscr {X}(d)$ at $\nu (0)$ is

\begin{align*} \mathbb {T}_{\mathscr {X}(d)}|_{\nu (0)}= (\mathfrak {g}(d) \to \overline {R}(d) \oplus \mathfrak {g}(d) ), \quad \gamma \mapsto (0, [\gamma , u]). \end{align*}

The kernel of the above map is $\bigoplus _{(a, j)\in J}\mathfrak {g}^{a, (j)}$ and the cokernel is $\overline {R}(d) \oplus \bigoplus _{(a, j)\in J}\mathfrak {g}^{a, (j)}$ , so the morphism $\nu$ induces a quasi-isomorphism on tangent complexes at $\nu (0)$ .

For $x \in \overline {R}(d)$ , a vertex $a \in I$ , and an edge $e=(a \to b)$ in $E^{\circ }$ , write the corresponding maps as $x(e) \colon V^a \to V^b$ and $x(\overline {e}) \colon V^b \to V^a$ . For $\theta \in \mathfrak {g}(d)$ , write its corresponding map as $\theta (a) \colon V^a \to V^a$ . The function $\mathop {\textrm {Tr}} W$ is given by

\begin{align*} \mathop {\textrm {Tr}} W(x, \theta )=\mathop {\textrm {Tr}} \biggl (\sum _{e \in E^{\circ }}[x(e), x(\overline {e})] \biggr )\biggl (\sum _{a\in I} \theta (a)\biggr ). \end{align*}

We set $\mathfrak {g}'$ and $\mathfrak {g}''$ to be

\begin{align*} \mathfrak {g}'=\bigoplus _{x^{a, (j)} \in \mathrm {supp}(p')} \mathfrak {g}^{a, (j)}, \quad \mathfrak {g}''=\bigoplus _{x^{a, (j)} \in \mathrm {supp}(p'')} \mathfrak {g}^{a, (j)} \end{align*}

and write an element $\gamma \in \bigoplus _{(a, j)\in J}\mathfrak {g}^{a, (j)}$ as $\gamma =\gamma '+\gamma ''$ for $\gamma ' \in \mathfrak {g}'$ and $\gamma '' \in \mathfrak {g}''$ . Note that there are isomorphisms

\begin{align*} \mathscr {X}_{p'}(d')\cong (\overline {R}(d') \times \widehat {\mathfrak {g}}')/G_{p'}, \quad \mathscr {X}_{p''}(d'')\cong (\overline {R}(d'') \times \widehat {\mathfrak {g}}'')/G_{p''}. \end{align*}

For $x \in \overline {R}(d)$ , we write

\begin{align*} x=x'+x''+x_U+x_{U^{\vee }} \end{align*}

for $x' \in \overline {R}(d')$ , $x'' \in \overline {R}(d'')$ , $x_U \in U$ , and $x_{U^{\vee }} \in U^{\vee }$ . Then $\nu ^{\ast } \mathop {\textrm {Tr}} W$ is calculated as

\begin{align*} \nu ^{\ast }\mathop {\textrm {Tr}} W(x, \gamma )&= \mathop {\textrm {Tr}} \biggl (\sum _{e \in E^{\circ }}[x'(e), x'(\overline {e})] \biggr )\biggl (\sum _{a\in I} \gamma '(a)\biggr ) \\ &\quad +\mathop {\textrm {Tr}} \biggl (\sum _{e \in E^{\circ }}[x''(e), x''(\overline {e})] \biggr )\biggl (\sum _{a\in I} \gamma ''(a)\biggr ) \\ &\quad+\mathop {\textrm {Tr}} \biggl (\sum _{e\in E^{\circ }} x_{U^{\vee }}(e)(x_{U}(\overline {e})\gamma '+x_U(\overline {e})u'-\gamma ''x_U(\overline {e})-u''x_{U}(\overline {e})) \biggr ) \\ &\quad+\mathop {\textrm {Tr}} \biggl (\sum _{e\in E^{\circ }} x_{U}(e) (x_{U^{\vee }}(\overline {e})\gamma ''+x_{U^{\vee }}(\overline {e})u''- \gamma 'x_{U^{\vee }}(\overline {e})-u'x_{U^\vee }(\overline {e})) \biggr ). \end{align*}

We take the following $G_p$ -equivariant variable change:

\begin{align*} x_U(\overline {e}) & \mapsto x_{U}(\overline {e})\gamma '+x_U(\overline {e})u'-\gamma ''x_U(\overline {e})-u''x_{U}(\overline {e}), \\ x_{U^{\vee }}(\overline {e}) & \mapsto x_{U^{\vee }}(\overline {e})\gamma ''+x_{U^{\vee }}(\overline {e})u''- \gamma 'x_{U^{\vee }}(\overline {e})-u'x_{U^\vee }(\overline {e}). \end{align*}

Since $u'$ and $u''$ are diagonal matrices with different eigenvalues, the above variable change determines an automorphism of $\mathscr {X}_p(d)$ . Therefore we obtain the desired identity (5.9).

5.2 Reduced stacks

We use the notations introduced in §§ 2.2.6 and 2.2.7. Let $\mathfrak {g}(d)_0 \subset \mathfrak {g}(d)$ be the traceless subalgebra, i.e. the kernel of the map

\begin{align*} \mathfrak {g}(d)=\bigoplus _{a\in I} \textrm {Hom}(V^a, V^a) \to \mathbb {C}, \quad (g^a)_{a \in I} \mapsto \sum _{a\in I} \mathop {\textrm {Tr}}(g^a). \end{align*}

The moment map $\mu$ in (2.9) factors through the map

\begin{align*} \mu _0 \colon \overline {R}(d) \to \mathfrak {g}(d)_0. \end{align*}

We define the following reduced derived stack:

(5.15) \begin{align} \mathscr {P}(d)^{\textrm {red}}:=\mu _0^{-1}(d)/G(d). \end{align}

Note that we have a commutative diagram

where the horizontal arrows are closed immersions and $\pi _{P}=\pi _{P,d}$ and $\pi _{Y}=\pi _{Y,d}$ are the good moduli space morphisms.

Furthermore, consider the stack

\begin{align*} \mathscr {X}_0(d):=(\overline {R}(d) \oplus \mathfrak {g}(d)_0 )/G(d) \end{align*}

and the regular function

\begin{align*} \mathop {\textrm {Tr}} W_0 =\mathop {\textrm {Tr}} W|_{\mathscr {X}_0(d)}\colon \mathscr {X}_0(d) \to \mathbb {C}. \end{align*}

Let $(Q,W)$ be the tripled quiver with potential associated to $Q^\circ$ ; see § 2.2.6. Denote by $(\partial W)$ the two-sided ideal of $\mathbb {C}[Q]$ generated by $\partial W/\partial e$ for $e\in E$ . Consider the Jacobi algebra $J(Q, \partial W):=\mathbb {C}[Q]/(\partial W)$ . Let $w_a\in J(Q, \partial W)$ be the image of the element corresponding to the loop $\omega _a$ . The critical locus

\begin{align*} \mathrm {Crit}(\mathop {\textrm {Tr}} W_0) \subset \mathscr {X}_0(d) \end{align*}

is the moduli stack of $(Q, W)$ -representations such that the action of $\theta$ has trace zero, where $\theta$ is the element

(5.16) \begin{align} \theta :=\sum _{a\in I} w_a \in J(Q, \partial W). \end{align}

The element $\theta$ is a central element in $J(Q, \partial W)$ from the definition of the potential $W$ ; see (2.13). We have the following diagram.

(5.17)

Here, recall the good moduli space map $\pi _X=\pi _{X,d}$ , $p$ and $\eta$ are projections, the morphism $0 \colon \mathscr {Y}(d) \to \mathscr {X}_0(d)$ is the zero-section of $\eta \colon \mathscr {X}_0(d) \to \mathscr {Y}(d)$ , the middle vertical arrows are the induced maps on good moduli spaces, and $\left (\prod _{a\in I}\mathrm {Sym}^{d^a}(\mathbb {C})\right )_0$ is the fiber at $0\in \mathbb {C}$ of the addition map

\begin{align*} \prod _{a\in I} \mathrm {Sym}^{d^a}(\mathbb {C}) \to \mathbb {C}. \end{align*}

Let $\overline {\mathrm {Crit}}(\mathop {\textrm {Tr}} W_0)\hookrightarrow X(d)$ be the good moduli space of $\mathrm {Crit}(\mathop {\textrm {Tr}} W_0)$ . The above diagram restricts to a diagram

(5.18)

where the arrows $\pi _C$ and $\pi _P$ are good moduli space morphisms. We abuse notation and write

\begin{equation*}0:=(\overbrace {0,\ldots , 0}^{\underline {d}})\in \biggl (\prod _{a\in I} \mathrm {Sym}^{d^a}(\mathbb {C})\biggr )_0.\end{equation*}

Define

(5.19) \begin{equation} \mathscr {N}_{\textrm {nil}}:=\pi _C^{-1}\overline {p}_C^{-1}(0) \subset \mathrm {Crit}(\mathop {\textrm {Tr}} W_0). \end{equation}

The substack $\mathscr {N}_{\textrm {nil}} \subset \mathrm {Crit}(\mathop {\textrm {Tr}} W_0)$ corresponds to $(Q, W)$ -representations such that the action of $\theta$ is nilpotent. Alternatively, it can be described as follows.

Lemma 5.4. We have $\overline {p}_{C}^{-1}(0)=\mathrm {Im}(\overline {0}_C)$ in the diagram (5.18). Hence $\mathscr {N}_{\textrm {nil}}=\pi _{C}^{-1}(\mathrm {Im}(\overline {0}_C))$ .

Proof. The inclusion $\mathrm {Im}(\overline {0}_C) \subset \overline {p}_C^{-1}(0)$ is obvious. Below we show that $\overline {p}_C^{-1}(0) \subset \mathrm {Im}(\overline {0}_C)$ .

Let $Q^{\circ , d}$ be the doubled quiver of $Q^{\circ }=(I, E^{\circ })$ and let $(\mathscr {I}) \subset \mathbb {C}[Q^{\circ , d}]$ be the two-sided ideal generated by the relation $\mathscr {I}:=\sum _{e\in E^{\circ }}[e, \overline {e}]$ . Since $\sum _{e\in E^{\circ }}[e, \overline {e}] \in (\partial W)$ , we have the functor

\begin{align*} \eta _{\ast } \colon J(Q, \partial W)\text {-mod} \to \mathbb {C}[Q^{\circ , d}]/(\mathscr {I})\text {-mod} \end{align*}

which forgets the action of $\theta$ , where $\theta \in J(Q, \partial W)$ is defined in (5.16).

For a simple $J(Q, \partial W)$ -module $R$ , we show that $\eta _{\ast }R$ is a simple module over $\mathbb {C}[Q^{\circ , d}]/(\mathscr {I})$ . We first note that the action of $\theta$ on $R$ has equal generalized eigenvalues. Indeed, $\theta \in J(Q, \partial W)$ is central, so the generalized eigenspaces for different eigenvalues would give a direct sum decomposition of $R$ , which contradicts that $R$ is simple. Moreover, if $R' \subset R$ is an eigenspace for $\theta$ , then $R'$ is preserved by the $J(Q, \partial W)$ -action, so $R'=R$ as $R$ is simple. It follows that the action of $\theta$ on $R$ is multiplication by $\lambda$ for some $\lambda \in \mathbb {C}$ . It follows that any submodule of $\eta _{\ast }R$ is preserved by the action of $\theta$ , so $\eta _{\ast }R$ is also simple.

Let $\bigoplus _{i=1}^k R_i^{\oplus n_i}$ be a semisimple $J(Q, \partial W)$ -module. Then the morphism $\overline {\eta }_C$ in the diagram (5.18) sends it to the semisimple $\mathbb {C}[Q^{\circ , d}]/(\mathscr {I})$ -module $\bigoplus _{i=1}^k \eta _{\ast }R_i^{\oplus n_i}$ , as $\eta _{\ast }R_i$ is simple. Thus, given a semisimple $\mathbb {C}[Q^{\circ , d}]/(\mathscr {I})$ -module $T=\bigoplus _{i=1}^k T_i^{\oplus n_i}$ corresponding to a point $r\in P(d)$ , the set of points of the fiber of $\overline {\eta }_C$ at $r$ consists of choices of $\lambda _{ij}\in \mathbb {C}$ for $1\leqslant i\leqslant k$ and $1\leqslant j\leqslant n_i$ , such that $\theta$ acts on the $j$ th copy of $T_i$ in $T$ by multiplication by $\lambda _{ij}$ . If it lies in $\overline {p}_C^{-1}(0)$ , we must have $\lambda _{ij}=0$ for all $1\leqslant i\leqslant k$ and $1\leqslant j\leqslant n_i$ . Therefore $\overline {p}_C^{-1}(0) \subset \mathrm {Im}(\overline {0}_C)$ holds.

5.3 Quasi-BPS categories for reduced stacks

We abuse notation and also denote by $j$ the closed immersion

\begin{align*} j^r \colon \mathscr {P}(d)^{\textrm {red}} \hookrightarrow \mathscr {Y}(d):=\overline {R}(d)/G(d). \end{align*}

We define the subcategory

(5.20) \begin{align} \mathbb {T}(d)_v^{\textrm {red}} \subset D^b(\mathscr {P}(d)^{\textrm {red}}) \end{align}

to consist of objects $\mathcal {E}$ such that $j^r_{\ast }\mathcal {E}$ is generated by $\mathcal {O}_{\mathscr {Y}(d)} \otimes \Gamma _{G(d)}(\chi )$ for a dominant weight $\chi$ satisfying (2.23) for $\delta _d=v\tau _d$ , i.e.

(5.21) \begin{align} \chi +\rho -v\tau _d \in \mathbf {W}(d), \end{align}

where $\mathbf {W}(d)$ is the polytope defined by (2.21) for the tripled quiver $Q$ of $Q^{\circ }$ .

The Koszul equivalence in Theorem 2.5 gives an equivalence

(5.22) \begin{align} \Theta _0 \colon D^b(\mathscr {P}(d)^{\textrm {red}}) \stackrel {\sim }{\to } \mathrm {MF}^{\textrm {gr}}(\mathscr {X}_0(d), \mathop {\textrm {Tr}} W_0). \end{align}

Define the reduced quasi-BPS category

\begin{equation*}\mathbb {S}^{\mathrm {gr}}(d)_v^{\mathrm {red}}\subset \mathrm {MF}^{\textrm {gr}}(\mathscr {X}_0(d), \mathop {\textrm {Tr}} W_0)\end{equation*}

as in (2.28), that is,

(5.23) \begin{equation} \mathbb {S}^{\mathrm {gr}}(d)_v^{\mathrm {red}}:= \mathrm {MF}^{\textrm {gr}}(\mathbb {M}(d)_v^{\mathrm {red}}, \mathop {\textrm {Tr}} W_0), \end{equation}

where $\mathbb {M}(d)_v^{\mathrm {red}}$ is the full subcategory of $D^b(\mathscr {X}_0(d))$ generated by the vector bundles $\mathcal {O}_{\mathscr {X}_0(d)}\otimes \Gamma _{G(d)}(\chi )$ for a dominant weight $\chi$ satisfying (5.21). The Koszul equivalence (5.22) restricts to the equivalence (see Lemma 2.26)

\begin{align*} \Theta _0 \colon \mathbb {T}(d)_v^{\textrm {red}} \stackrel {\sim }{\to } \mathbb {S}^{\textrm {gr}}(d)_v^{\textrm {red}}. \end{align*}

We use Theorem 5.1 to study the singular support of sheaves [AG15] in reduced quasi-BPS categories.

Corollary 5.5. Let $v\in \mathbb {Z}$ be such that $\gcd (\underline {d}, v)=1$ . Then any object $\mathcal {E} \in \mathbb {T}(d)_v^{\textrm {red}}$ has singular support,

(5.24) \begin{align} \mathrm {Supp}^{\textrm {sg}}(\mathcal {E}) \subset \mathscr {N}_{\textrm {nil}}. \end{align}

Proof. The singular support of $\mathcal {E}$ equals the support of $\Theta _0(\mathcal {E})$ ; see [Tod19, Proposition 2.3.9]. Then the corollary follows from Theorem 5.1 together with

\begin{align*} \Delta \cap \biggl(\prod _{a\in I} \mathrm {Sym}^{d^a}(\mathbb {C}) \biggr)_{\!0}=\{0\}. \end{align*}

Remark 5.6. The closed substack $\mathrm {Im}(0_C) \subset \mathscr {N}_{\textrm {nil}}$ is given by the equation $\theta =0$ , and the condition $\mathrm {Supp}^{\textrm {sg}}(\mathcal {E}) \subset \mathrm {Im}(0_C)$ is equivalent to $\mathcal {E}$ being perfect; see [AG15, Theorem 4.2.6]. The condition (5.24) is weaker than $\mathcal {E}$ being perfect, and indeed there may exist objects in $\mathbb {T}(d)_v^{\textrm {red}}$ which are not perfect.

5.4 Relative properness of quasi-BPS categories

We continue to assume that the quiver $Q^\circ =(I, E^\circ )$ satisfies Assumption 4.17. We will make a further assumption on the quiver $Q^{\circ }$ that guarantees, for example, that the reduced stack $\mathscr {P}(d)^{\textrm {red}}$ is classical. Recall $\alpha _{a, b}$ defined in (1.9) and let

\begin{equation*}\alpha _{Q^{\circ }}:=\mathrm {min}\{\alpha _{a, b} \mid a, b \in I\}.\end{equation*}

Recall the good moduli spaces $X(d)$ , $Y(d)$ , and $P(d)$ from §§ 2.2.1 and 2.2.6.

Lemma 5.7.

1. (i) If $\alpha _{Q^{\circ }} \geqslant 1$ , then $X(d)$ and $Y(d)$ are Gorenstein with trivial canonical module.

2. (ii) If $\alpha _{Q^{\circ }} \geqslant 2$ , then $\mathscr {P}(d)^{\textrm {red}}$ is a classical stack and $P(d)$ is a normal irreducible variety with

   (5.25) \begin{align} \dim P(d)=2+\sum _{a, b \in I}d^a d^b \alpha _{a, b}. \end{align}
   If, furthermore, $P(d)$ is Gorenstein, then its canonical module is trivial.

3. (iii) If $\alpha _{Q^{\circ }} \geqslant 3$ , or if $\alpha _{Q^{\circ }}=2$ and $\underline {d} \geqslant 3$ , then $P(d)$ is Gorenstein.

Proof. (i) We only prove the case of $Y(d)$ , as the case of $X(d)$ is similar. Let $\mathscr {Y}(d)^{\textrm {s}} \subset \mathscr {Y}(d)$ be the open substack corresponding to simple representations. By [Kno89, corollary 2], it is enough to show that the codimension of $\mathscr {Y}(d) \setminus \mathscr {Y}(d)^{\textrm {s}}$ is at least two. Let $\lambda$ be a cocharacter corresponding to $d=d_1+d_2$ such that $d_1$ and $d_2$ are non-zero. A simple calculation shows that

\begin{align*} \dim \mathscr {Y}(d) -\dim \mathscr {Y}(d)^{\lambda \geqslant 0} =\sum _{a, b \in I}d_1^a d_2^b \alpha _{a, b} +\sum _{a \in I} d_1^a d_2^a \geqslant 2 \underline {d}_1 \underline {d}_2 \alpha _{Q^{\circ }}\geqslant 2. \end{align*}

Therefore $\mathscr {Y}(d) \setminus \mathscr {Y}(d)^{\textrm {s}}$ is of codimension at least two in $\mathscr {Y}(d)$ .

(ii) If $\alpha _{Q^{\circ }} \geqslant 2$ , then $\mu _0^{-1}(0) \subset \overline {R}(d)$ is an irreducible variety of dimension $\dim \overline {R}(d)-\dim \mathfrak {g}(d)_0$ by [KLS06, Proposition 3.6]. In particular, $\mathscr {P}(d)^{\textrm {red}}$ is classical. Moreover, in the proof of [KLS06, Proposition 3.6], it is also proved that $\mu _0^{-1}(0)$ contains a dense open subset of points with trivial stabilizer groups in $G(d)/\mathbb {C}^{\ast }$ . Therefore the dimension (5.25) is easily calculated from

\begin{align*} \dim P(d)=1+\dim \mathscr {P}(d)^{\textrm {red}}=2+\dim \overline {R}(d)-2\dim \mathfrak {g}(d). \end{align*}

The last statement holds since the canonical module of $\mu _0^{-1}(0)$ is a $G(d)$ -equivariantly trivial line bundle.

(iii) By [Fle88] and [Nam01], it is enough to show that the codimension of the singular locus of $P(d)$ is at least $4$ ; see [Fu06, Proposition 1.1 and Theorem 1.2]. The singular locus of $P(d)$ is contained in the union of images of

\begin{align*} \oplus \colon P(d_1) \times P(d_2) \to P(d) \end{align*}

for $d=d_1+d_2$ with $d_i \neq 0$ . The codimension of the image of the above map is at least

\begin{align*} \dim P(d)-\dim P(d_1)-\dim P(d_2) &=-2+\sum _{a, b \in I}(d_1^a d_2^b +d_1^b d_2^a)\alpha _{a, b} \\ &\geqslant 2 \underline {d}_1 \underline {d}_2 \alpha _{Q^{\circ }}-2 \geqslant 4. \end{align*}

Here the identity follows from (5.25), the first inequality follows from the definition of $\alpha _{Q^{\circ }}$ , and the last inequality follows from the assumption on $\alpha _{Q^{\circ }}$ and $d$ . Therefore (iii) holds.

Example 5.8. As in Example 4.23, let $Q^{\circ }$ be a quiver with one vertex and $g$ loops. Then $\alpha _{Q^{\circ }}=2g-2$ . By Lemma 5.7, the stack $\mathscr {P}(d)^{\textrm {red}}$ is classical if $g\geqslant 2$ , and $P(d)$ is Gorenstein if $g\geqslant 3$ or if $g=2$ and $d\geqslant 3$ . When $g=d=2$ , the singular locus of $P(d)$ is of codimension two, but nevertheless it is also Gorenstein because it admits a symplectic resolution of singularities [O’G99].

Below we assume that $\mathscr {P}(d)^{\textrm {red}}$ is classical, e.g. $\alpha _{Q^{\circ }} \geqslant 2$ . Then we have the good moduli space morphism

\begin{align*} \pi _P=\pi _{P,d} \colon \mathscr {P}(d)^{\textrm {red}} \to P(d). \end{align*}

In particular, for $\mathcal {E}_1, \mathcal {E}_2 \in D^b(\mathscr {P}(d)^{\textrm {red}})$ , the Hom space $\textrm {Hom}(\mathcal {E}_1, \mathcal {E}_2)$ is a module over $\mathcal {O}_{P(d)}$ .

In general, for a dg-category $\mathcal {C}$ that is linear over a commutative algebra $R$ , we say that it is proper over $\textrm {Spec} R$ if $\textrm {Hom}^{\ast }(E, F)$ is a finitely generated $R$ -module. The categorical support condition in Corollary 5.5 implies the relative properness of quasi-BPS categories, which is a non-trivial statement as objects in $\mathbb {T}(d)_v^{\textrm {red}}$ may not be perfect; see Remark 5.6.

Proposition 5.9. Suppose that the stack $\mathscr {P}(d)^{\textrm {red}}$ is classical, e.g. $\alpha _{Q^{\circ }} \geqslant 2$ . For $v\in \mathbb {Z}$ such that $\gcd (\underline {d}, v)=1$ and objects $\mathcal {E}_i \in \mathbb {T}(d)_v^{\textrm {red}}$ for $i=1, 2$ , the $\mathcal {O}_{P(d)}$ -module

\begin{align*} \bigoplus _{i\in \mathbb {Z}} \textrm {Hom}^i(\mathcal {E}_1, \mathcal {E}_2) \end{align*}

is finitely generated, i.e. the category $\mathbb {T}(d)_v^{\textrm {red}}$ is proper over $P(d)$ . In particular, we have $\textrm {Hom}^i(\mathcal {E}_1, \mathcal {E}_2)=0$ for $\lvert i \rvert \gg 0$ .

Proof. Let $F_i$ be defined by

\begin{align*} F_i=\mathrm {forg}\circ \Theta _0(\mathcal {E}_i) \in \mathrm {MF}(\mathscr {X}_0(d), \mathop {\textrm {Tr}} W_0) \end{align*}

where $\Theta _0$ is the equivalence (5.22) and $\mathrm {forg}$ is the forget-the-grading functor. Consider its internal Hom,

\begin{align*} \mathcal {H}om(F_1, F_2) \in \mathrm {MF}(\mathscr {X}_0(d), 0)=D^{\mathbb {Z}/2}(\mathscr {X}_0(d)). \end{align*}

Here $D^{\mathbb {Z}/2}(-)$ is the $\mathbb {Z}/2$ -graded derived category of coherent sheaves. The above object is supported over $\pi _C^{-1}(\mathrm {Im}(\overline {0}_C))$ by Corollary 5.5 and Lemma 5.4. As $\pi _X$ is a good moduli space morphism, $\pi _{X\ast }$ sends a coherent sheaf to a bounded complex of coherent sheaves. Therefore we have

(5.26) \begin{align} \pi _{X \ast }\mathcal {H}om(F_1, F_2) \in D^{\mathbb {Z}/2}(X(d)) \end{align}

and, furthermore, that $\pi _{X\ast }\mathcal {H}om(F_1, F_2)$ is supported over $\mathrm {Im}(\overline {0}_C) \subset \mathrm {Im}(\overline {0})$ ; see the diagrams (5.17) and (5.18). Since $\overline {0}$ is a section of $\overline {p}$ , the restriction of $\overline {\eta }$ to $\mathrm {Im}(\overline {0})$ is an isomorphism, $\overline {\eta } \colon \mathrm {Im}(\overline {0}) \stackrel {\cong }{\to } Y(d)$ . In particular, (5.26) has proper support over $Y(d)$ . Therefore, we have that

\begin{align*} \overline {\eta }_{\ast }\pi _{X\ast }\mathcal {H}om(F_1, F_2) \in D^{\mathbb {Z}/2}(Y(d)), \end{align*}

so $\textrm {Hom}^{\ast }(F_1, F_2)$ is finitely generated over $Y(d)$ and hence over $P(d)$ as $P(d) \hookrightarrow Y(d)$ is a closed subscheme. Now, for $i \in \mathbb {Z}/2$ , we have

\begin{align*} \textrm {Hom}^{i}(F_1, F_2)=\bigoplus _{k\in \mathbb {Z}}\textrm {Hom}^{2k+i}(\mathcal {E}_1, \mathcal {E}_2), \end{align*}

and hence the finite generation over $P(d)$ of the left-hand side implies the proposition.

5.5 Relative Serre functor on quasi-BPS categories

We keep the situation of Proposition 5.9. The category $\mathbb {T}=\mathbb {T}(d)_v$ is a module over $\mathrm {Perf}(P(d))$ . We have the associated internal homomorphism (see § 2.3)

\begin{align*} \mathcal {H}om_{\mathbb {T}}(\mathcal {E}_1, \mathcal {E}_2) \in D_{\textrm {qc}}(\mathscr {P}(d)^{\textrm {red}}), \quad \mathcal {E}_i \in \mathbb {T}(d)_v^{\textrm {red}}. \end{align*}

Then Proposition 5.9 implies that

\begin{align*} \pi _{\ast } \mathcal {H}om_{\mathbb {T}}(\mathcal {E}_1, \mathcal {E}_2) \in D^b(P(d)). \end{align*}

A functor $S_{\mathbb {T}/P} \colon \mathbb {T} \to \mathbb {T}$ is called a relative Serre functor if it satisfies the isomorphism

\begin{align*} \textrm {Hom}_{P(d)}(\pi _{\ast }\mathcal {H}om_{\mathbb {T}}(\mathcal {E}_1, \mathcal {E}_2), \mathcal {O}_{P(d)}) \cong \textrm {Hom}_{\mathbb {T}}(\mathcal {E}_2, S_{\mathbb {T}/P}(\mathcal {E}_1)). \end{align*}

The following result shows that $\mathbb {T}$ is strongly crepant in the sense of [VdB22, § 2.2].

Theorem 5.10. Suppose that $\alpha _{Q^{\circ }} \geqslant 2$ and $P(d)$ is Gorenstein, e.g. $\alpha _{Q^{\circ }} \geqslant 3$ . Let $v\in \mathbb {Z}$ be such that $\gcd (\underline {d}, v)=1$ . Then the relative Serre functor $S_{\mathbb {T}/P}$ exists and satisfies $S_{\mathbb {T}/P} \cong \textrm {id}_{\mathbb {T}}$ .

Proof. We have the following commutative diagram.

(5.27)

Let $\Theta _0$ be the Koszul duality equivalence in (5.22). Then, by Lemma 2.7, we have

\begin{align*} j^r_{\ast }\mathcal {H}om_{\mathbb {T}}(\mathcal {E}_1, \mathcal {E}_2)=\eta _{\ast }\mathcal {Q}, \end{align*}

where $\mathcal {Q}$ is the internal homomorphism of matrix factorizations,

\begin{align*} \mathcal {Q}=\mathcal {H}om_{\mathrm {MF}}(\Theta _0(\mathcal {E}_1), \Theta _0(\mathcal {E}_2)) \in \mathrm {MF}^{\textrm {gr}}(\mathscr {X}_0(d), 0). \end{align*}

Since the left vertical arrows $\pi _P$ and $\pi _Y$ have relative dimension $-1$ and the codimension of the closed substack $\mathscr {P}(d)^{\textrm {red}}$ in $\mathscr {Y}(d)$ is $\dim \mathfrak {g}(d)_0$ , the codimension of $P(d)$ in $Y(d)$ is $\dim \mathfrak {g}(d)_0$ . By Lemma 5.7, we have the following descriptions of dualizing complexes:

(5.28) \begin{align} \omega _{Y(d)}=\mathcal {O}_{Y(d)}[\dim Y(d)], \quad \omega _{P(d)}=\mathcal {O}_{P(d)}[\dim Y(d)-\dim \mathfrak {g}(d)_0]. \end{align}

As $\overline {j}^{!}\omega _{Y(d)}=\omega _{P(d)}$ , we have

\begin{align*} \overline {j}^{!}\mathcal {O}_{Y(d)}=\mathcal {O}_{P(d)}[{-}\dim \mathfrak {g}(d)_0]. \end{align*}

Then we have the isomorphisms

\begin{align*} &\textrm {Hom}_{P(d)}(\pi _{P\ast }\mathcal {H}om_{\mathbb {T}}(\mathcal {E}_1, \mathcal {E}_2), \mathcal {O}_{P(d)}) \\[5pt] &\quad\cong \textrm {Hom}_{Y(d)}(\overline {j}_{\ast }\pi _{P\ast }\mathcal {H}om_{\mathbb {T}}(\mathcal {E}_1, \mathcal {E}_2), \mathcal {O}_{Y(d)}[\dim \mathfrak {g}(d)_0]) \\[5pt] &\quad\cong \textrm {Hom}_{Y(d)}(\pi _{Y\ast }j^r_{\ast }\mathcal {H}om_{\mathbb {T}}(\mathcal {E}_1, \mathcal {E}_2), \mathcal {O}_{Y(d)}[\dim \mathfrak {g}(d)_0]) \\[5pt] &\quad\cong \textrm {Hom}_{Y(d)}(\pi _{Y\ast }\eta _{\ast }\mathcal {Q}, \mathcal {O}_{Y(d)}[\dim \mathfrak {g}(d)_0]) \\[5pt] &\quad\cong \textrm {Hom}_{Y(d)}(\overline {\eta }_{\ast }\pi _{X\ast }\mathcal {Q}, \mathcal {O}_{Y(d)}[\dim \mathfrak {g}(d)_0]). \end{align*}

By Lemma 5.7(i), we have

\begin{align*} \omega _{X(d)}=\mathcal {O}_{X(d)}[\dim X(d)](-2\dim \mathfrak {g}(d)_0)= \mathcal {O}_{X(d)}[\dim Y(d)-\dim \mathfrak {g}(d)_0]. \end{align*}

Here, we denote by $(1)$ the grade shift with respect to the $\mathbb {C}^{\ast }$ -action on $X(d)$ , induced by the fiberwise weight-two $\mathbb {C}^{\ast }$ -action on $\mathscr {X}_0(d) \to \mathscr {Y}(d)$ , which is isomorphic to $[1]$ in $\mathrm {MF}^{\textrm {gr}}(X(d), 0)$ . As in the proof of Lemma 5.9, the complex $\pi _{X\ast }\mathcal {Q}$ has proper support over $Y(d)$ . Therefore, from $\eta ^{!}\omega _{Y(d)}=\omega _{X(d)}$ and (5.28), we have

\begin{align*} &\textrm {Hom}_{Y(d)}(\overline {\eta }_{\ast }\pi _{X\ast }\mathcal {Q}, \mathcal {O}_{Y(d)}[\dim \mathfrak {g}(d)_0]) \\[5pt] &\quad\cong \textrm {Hom}_{X(d)}(\pi _{X\ast }\mathcal {Q}, \overline {\eta }^{!}\mathcal {O}_{Y(d)}[\dim \mathfrak {g}(d)_0]) \\[5pt] &\quad\cong \textrm {Hom}_{X(d)}(\pi _{X\ast }\mathcal {Q}, \mathcal {O}_{X(d)}). \end{align*}

Note that $\overline {\eta }$ is not proper, but the first isomorphism above holds because $\pi _{X\ast }\mathcal {Q}$ has proper support over $Y(d)$ . In the above, $\textrm {Hom}_{X(d)}(-, -)$ denotes the space of homomorphisms in the category $\mathrm {MF}^{\textrm {gr}}(X(d), 0)$ ; note that $X(d)$ may not be smooth, but its definition and the definition of the functor $\overline {\eta }_*$ on the subcategory of matrix factorizations with proper support over $Y(d)$ are as in the smooth case [PV11, EP15].

By the definition of $\mathbb {S}^{\mathrm {gr}}(d)_v$ , the object $\mathcal {Q}$ is represented by

\begin{equation*}(\mathcal {Q}^0 \leftrightarrows \mathcal {Q}^1),\end{equation*}

where $\mathcal {Q}^0$ and $\mathcal {Q}^1$ are direct sums of vector bundles of the form $\Gamma _{G(d)}(\chi _1)^{\vee } \otimes \Gamma _{G(d)}(\chi _2) \otimes \mathcal {O}_{\mathscr {X}_0(d)}$ such that the weights $\chi _i$ satisfy, for $i\in \{1, 2\}$ ,

(5.29) \begin{align} \chi _i +\rho -v\tau _d \in \mathbf {W}:=\tfrac {1}{2}\mathrm {sum}[0, \beta ], \end{align}

where the above Minkowski sum is over all weights $\beta$ in $\overline {R}(d) \oplus \mathfrak {g}(d)_0$ . By Lemma 5.11 below, the weight $\chi _2-\chi _1$ lies in the interior of the polytope $-2\rho +2\mathbf {W}$ . Therefore, applying [ŠVdB17, Proposition 4.4.4] for the $PG(d):=G(d)/\mathbb {C}^{\ast }$ -action on $\overline {R}(d)\oplus \mathfrak {g}(d)_0$ , the sheaf $\pi _{X\ast }\mathcal {Q}^i$ is a maximal Cohen–Macaulay sheaf on $X(d)$ . It follows that

\begin{align*} \mathcal {H}om_{X(d)}(\pi _{X\ast }\mathcal {Q}^i, \mathcal {O}_{X(d)})=\pi _{X\ast }(\mathcal {Q}^i)^{\vee }. \end{align*}

The morphism $\pi _{X\ast }\mathcal {Q}^i \to \pi _{X\ast }\mathcal {Q}^{i+1}$ is uniquely determined by its restriction to the smooth locus of $X(d)$ , and so

\begin{align*} \mathcal {H}om_{X(d)}(\pi _{X\ast }\mathcal {Q}, \mathcal {O}_{X(d)}) \cong \pi _{X\ast }(\mathcal {Q}^{\vee }). \end{align*}

Therefore we have the isomorphisms

\begin{align*} \textrm {Hom}_{X(d)}(\pi _{X\ast }\mathcal {Q}, \mathcal {O}_{X(d)}) &\cong \textrm {Hom}_{X(d)}(\mathcal {O}_{X(d)}, \pi _{X\ast }(\mathcal {Q}^{\vee })) \\ &\cong \textrm {Hom}_{\mathrm {MF}(\mathscr {X}(d), \mathrm {Tr}\,W)}(\Theta _0(\mathcal {E}_2), \Theta _0(\mathcal {E}_1)) \\ &\cong \textrm {Hom}_{\mathscr {P}(d)}(\mathcal {E}_2, \mathcal {E}_1) \\ &\cong \textrm {Hom}_{\mathbb {T}}(\mathcal {E}_2, \mathcal {E}_1), \end{align*}

and the conclusion follows.

We are left with proving the following.

Lemma 5.11. In the setting of (5.29), let $\chi \in M(d)$ be a dominant weight such that $\chi +\rho -v\tau _d \in \mathbf {W}$ . If $\gcd (\underline {d}, v)=1$ , then $\chi +\rho -v\tau _d$ is not contained in the boundary of $\mathbf {W}$ .

Proof. Suppose that $\chi +\rho -v\tau _d$ is on the boundary of $\mathbf {W}$ . Use the decomposition (4.6) for $\delta _d=0$ to obtain the decomposition

(5.30) \begin{align} \chi +\rho =-\tfrac {1}{2}R(d)^{\lambda \gt 0}+\sum _{i=1}^k \psi _i +v\tau _d \end{align}

for some antidominant cocharacter $\lambda$ such that if $d=d_1+\cdots +d_k$ for $k\geqslant 2$ is the decomposition corresponding to $\lambda$ , then $\langle 1_{d_i}, \psi _i\rangle =0$ . We write

(5.31) \begin{align} v\tau _d=\sum _{i=1}^k v_i \tau _{d_i}, \quad \frac {v}{\underline {d}}=\frac {v_i}{\underline {d}_i} \end{align}

for $v_i \in \mathbb {Q}$ . Then the identity (5.30) is written as

(5.32) \begin{align} \chi =-\frac {1}{2}\overline {R}(d)^{\lambda \gt 0}+\sum _{i=1}^k v_i\tau _{d_i}+\sum _{i=1}^k (\psi _i-\rho _i). \end{align}

By Assumption 4.17, we have $\frac {1}{2}\overline {R}(d)^{\lambda \gt 0} \in M(d)^{W_d}$ . Therefore the identity (5.32) implies that

\begin{align*} v_i=\langle 1_{d_i}, \chi \rangle + \langle 1_{d_i}, \tfrac {1}{2}\overline {R}(d)^{\lambda \gt 0} \rangle \in \mathbb {Z} \end{align*}

for all $1\leqslant i\leqslant k$ . However, as $v/\underline {d}=v_i/\underline {d}_i$ , we obtain a contradiction with the assumption that $\gcd (\underline {d}, v)=1$ .

5.6 Indecomposability of quasi-BPS categories

Let $Q^\circ =(I, E^\circ )$ be a quiver and let $d\in \mathbb {N}^I$ . Recall the good moduli space map

\begin{equation*}\pi _P\colon \mathscr {P}(d)^{\mathrm {red}}=\mu _0^{-1}(0)/G(d)\to P(d)\end{equation*}

from the reduced stack of dimension- $d$ representations of the preprojective algebra of $Q^\circ$ .

Proposition 5.12. Let $Q^\circ$ be a quiver and let $d\in \mathbb {N}^I$ be such that $\mathscr {P}(d)^{\mathrm {red}}$ is a classical stack, e.g. $\alpha _{Q^{\circ }} \geqslant 2$ . Let $v\in \mathbb {Z}$ . Then $\mathbb {T}(d)_v^{\mathrm {red}}$ does not have a non-trivial orthogonal decomposition.

Recall the following standard lemma.

Lemma 5.13. Let $\mathcal {D}$ be a triangulated category with Serre functor equal to a shift. Then a semiorthogonal decomposition $\mathcal {D}=\langle \mathbb {A}, \mathbb {B}\rangle$ is an orthogonal decomposition $\mathcal {D}=\mathbb {A}\oplus \mathbb {B}$ .

We note the following corollary of Proposition 5.12 and Lemma 5.13.

Corollary 5.14. Let $Q^\circ$ be a quiver satisfying Assumption 4.17 and let $d\in \mathbb {N}^I$ be such that $\mathscr {P}(d)^{\mathrm {red}}$ is a classical stack. Let $v\in \mathbb {Z}$ be such that $\gcd (v, \underline {d})=1$ . Then the category $\mathbb {T}(d)_v^{\mathrm {red}}$ does not have any non-trivial semiorthogonal decompositions.

Proof. Assume that there is a semiorthogonal decomposition $\mathbb {T}(d)_v^{\mathrm {red}}=\langle \mathbb {A}, \mathbb {B}\rangle$ . By Theorem 5.10 and Lemma 5.13, there is an orthogonal decomposition $\mathbb {T}(d)_v^{\mathrm {red}}=\mathbb {A}\oplus \mathbb {B}$ . By Proposition 5.12, one of the categories $\mathbb {A}$ and $\mathbb {B}$ is zero.

Before we begin the proof of Proposition 5.12, we note a few preliminary results. We assume in the remaining of the subsection that $\mathscr {P}(d)^{\mathrm {red}}$ is a classical stack. We say that a representation $V$ of $G(d)$ has weight $v\in \mathbb {Z}$ if $1_d$ acts on $V$ with weight $v$ .

Proposition 5.15. Let $V$ be a representation of $G(d)$ of weight zero. Then the sheaf $\pi _*(\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V)$ is non-zero and torsion-free.

Proof. The module $\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V$ is a non-zero torsion-free $\mathcal {O}_{\mu _0^{-1}(0)}$ -module of weight zero, thus it is also a torsion-free $\mathcal {O}_{P(d)}=\mathcal {O}^{G(d)}_{\mu _0^{-1}(0)}\subset \mathcal {O}_{\mu _0^{-1}(0)}$ -module.

The category $\mathbb {T}(d)_v^{\mathrm {red}}$ is admissible in $D^b(\mathscr {P}(d)^{\mathrm {red}})_v$ by an immediate modification of the argument for the admissibility of $\mathbb {T}(d)_v$ in $D^b(\mathscr {P}(d))_v$ , which follows from the Koszul equivalence (2.15) and [Păd21, Theorem 1.1]. Then there exists a left adjoint of the inclusion $\mathbb {T}(d)_v^{\mathrm {red}} \hookrightarrow D^b(\mathscr {P}(d)^{\mathrm {red}})_v$ , which we denote by

\begin{equation*}\Phi \colon D^b(\mathscr {P}(d)^{\mathrm {red}} )_v\to \mathbb {T}(d)_v^{\mathrm {red}}.\end{equation*}

Proposition 5.16. Let $V$ be a representation of $G(d)$ of weight $v$ and let $\mathscr {A}$ be a direct summand of $\Phi (\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V)$ . Then $\pi _*(\mathscr {A})$ has support $P(d)$ .

Proof. Write $\Phi (\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V)=\mathscr {A}\oplus \mathscr {B}$ for $\mathscr {A}, \mathscr {B}\in D^b(\mathscr {P}(d)^{\mathrm {red}})_v$ . There exists a representation $V'$ of $G(d)$ of weight $v$ such that $\mathrm {Hom}(\mathscr {A}, \mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V' )\neq 0$ . Then $\mathrm {Hom}(\mathscr {A}, \mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V')$ is a direct summand of

\begin{align*} \mathrm {Hom}(\Phi (\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V), \mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V' )&=\mathrm {Hom}(\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V, \mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V')\\&=\pi _*(\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V'\otimes V^\vee ), \end{align*}

which is non-zero and torsion-free over $P(d)$ by Proposition 5.15, and thus the $P(d)$ -sheaf $\mathrm {Hom}(\mathscr {A}, \mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V')$ has support $P(d)$ . Then also $\pi _*(\mathscr {A})$ has support $P(d)$ .

Proposition 5.17. For $\mathscr {A}, \mathscr {A}' \in D^b(\mathscr {P}(d)^{\mathrm {red}} )_v$ , suppose that $\pi _{\ast }\mathscr {A}$ and $\pi _{\ast }\mathscr {A}'$ have support $P(d)$ . Then $\mathrm {Hom}^{i}(\mathscr {A}, \mathscr {A}')\neq 0$ for some $i\in \mathbb {Z}$ .

Proof. The object $\pi _{P\ast }\mathcal {H}om(\mathscr {A}, \mathscr {A}')\in D_{\textrm {qc}}(P(d))$ is non-zero because it is non-zero over a generic point. Then $R\textrm {Hom}(\mathscr {A}, \mathscr {A}')=R\Gamma (\pi _{P\ast }\mathcal {H}om(\mathscr {A}, \mathscr {A}'))$ is also non-zero as $P(d)$ is affine, and the conclusion follows.

Proof of Proposition 5.12. Assume that $\mathbb {T}(d)_v^{\mathrm {red}}$ has an orthogonal decomposition in categories $\mathbb {A}$ and $\mathbb {B}$ . By Propositions 5.16 and 5.17, all summands of $\Phi (\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V)$ are in the same category, say $\mathbb {A}$ , for all representations $V$ of $G(d)$ of weight $v$ . Then the complexes $\Phi (\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V)$ are in $\mathbb {A}$ for all representations $V$ of $G(d)$ of weight $v$ .

Let $\mathscr {A}\in \mathbb {T}(d)_v^{\mathrm {red}}$ be non-zero and indecomposable. Then there exists $V$ such that $\mathrm {Hom}(\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V, \mathscr {A})\neq 0$ , and so $\mathrm {Hom}(\Phi (\mathcal {O}_{\mathscr {P}(d)^{\mathrm {red}}}\otimes V), \mathscr {A} )\neq 0$ . The complex $\mathscr {A}$ is indecomposable, so $\mathscr {A}\in \mathbb {A}$ , and thus $\mathbb {B}=0$ .