2.1 Notations

All the spaces $\mathcal{X}$ considered are quasi-smooth (derived) stacks over $\mathbb{C}$ , see § 2.5. Its classical truncation is denoted by $t_0(\mathcal{X})$ . We denote by $\mathbb{L}_{\mathcal{X}}$ the cotangent complex of $\mathcal{X}$ . For a quasi-projective scheme $X$ over $\mathbb{C}$ , we denote by $\chi _c(X)$ the Euler characteristic of its compactly supported cohomology.

In this paper, any dg-category considered is a $\mathbb{C}$ -linear pre-triangulated dg-category, in particular its homotopy category is a triangulated category. For a dg-category $\mathscr{D}$ , we denote by $K(\mathscr{D})$ the Grothendieck group of the homotopy category of $\mathscr{D}$ . Let $\mathcal{X}$ be a stack with an action of an abelian group $T$ . We denote by $G_T({\mathcal{X}})$ the Grothendieck group of the derived category of bounded complexes of $T$ -equivariant coherent sheaves $D^b_T({\mathcal{X}})$ . We denote by $K_T({\mathcal{X}})$ the Grothendieck group of the category of perfect complexes ${\rm Perf}_T({\mathcal{X}})\subset D^b_T({\mathcal{X}})$ . If $T$ is trivial, we drop it from the notation. We let $\mathbb{K}:=K_0(BT)$ and $\mathbb{F}:={\rm Frac}\,\mathbb{K}$ . If $V$ is a $\mathbb{K}$ -module, we let $V_{\mathbb{F}}:=V\otimes _{\mathbb{K}}\mathbb{F}$ .

For $G$ a reductive group and $X$ a dg-scheme with an action of $G$ , denote by $X/G$ the corresponding quotient stack. When $X$ is affine, we denote by $X/\!\!/ G$ the quotient dg-scheme with dg-ring of regular functions $\mathcal{O}_X^G$ . For a morphism $f \colon X\to Y$ and for a closed point $y \in Y$ , we denote by $\widehat {X}_y$ the formal fiber of $f$ at $y$ , i.e.

\begin{align*} \widehat {X}_y := X \times _{Y} \textrm {Spec} \Big(\widehat {\mathcal{O}}_{Y, y} \Big). \end{align*}

When $X$ is a $G$ -representation, $f\colon X\to Y:=X/\!\!/ G$ and $y=0$ , we omit the subscript $y$ from the above notation.

Let $R$ be a set. Consider a set $O\subset R\times R$ such that for any $i, j\in R$ we have $(i,j)\in O$ , $(j,i)\in O$ , 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

\begin{align*}\mathbb{T}=\langle \mathbb{A}_i \mid i \in R \rangle \end{align*}

with summands 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 $\mathcal{A}_i\in \mathbb{A}_i$ , $\mathcal{A}_j\in \mathbb{A}_j$ , we have $\textrm {Hom}_{\mathbb{T}}(\mathcal{A}_i,\mathcal{A}_j)=0$ .

2.2 Matrix factorizations

Relevant references for this subsection are [Reference TodaTod24b, § 2.2], [Reference TodaTod23, § 2.2], [Reference Ballard, Favero and KatzarkovBFK19, § 2.3], [Reference Polishchuk and VaintrobPV11, § 1].

2.2.1 The definition of categories of matrix factorizations.

Let $G$ be a reductive group and let $Y$ be a smooth affine scheme with an action of $G$ and a trivial $\mathbb{Z}/2$ -action. Let $\mathcal{Y}=Y/G$ be the corresponding quotient stack and let $f$ be a regular function

\begin{align*}f \colon \mathcal{Y}\to \mathbb{C}.\end{align*}

We define the dg-category of matrix factorizations ${\rm MF}(\mathcal{Y}, f)$ . Its objects are $(\mathbb{Z}/2\mathbb{Z})\times G$ -equivariant factorizations $(P, d_P)$ , where $P$ is a $G$ -equivariant coherent sheaf on $Y$ , $\langle 1\rangle$ is the twist corresponding to a non-trivial $\mathbb{Z}/2\mathbb{Z}$ -character on $Y$ , and

\begin{align*}d_P \colon P\to P\langle 1\rangle \end{align*}

is a morphism satisfying $d_P\circ d_P=f$ . Alternatively, the objects of ${\rm MF}(\mathcal{Y}, f)$ are tuples

(2.1) \begin{align} \left (\alpha \colon E\rightleftarrows F \colon \beta \right ), \end{align}

where $E$ and $F$ are $ G$ -equivariant coherent sheaves on $Y$ , and $\alpha$ and $\beta$ are $G$ -equivariant morphisms such that $\alpha \circ \beta$ and $\beta \circ \alpha$ are multiplication by $f$ . See [Reference Ballard, Favero and KatzarkovBFK19, § 2.3], [Reference TodaTod23, § 2.2] and [Reference Polishchuk and VaintrobPV11, Definition 1.2] for the definition of the morphism spaces.

For a dg-subcategory $\mathscr{A}$ of $D^b(\mathcal{Y})$ , define ${\rm MF}(\mathscr{A}, f)$ as the full subcategory of ${\rm MF}(\mathcal{Y}, f)$ with objects pairs $(P, d_P)$ with $P$ in $\mathscr{A}$ .

2.2.2 Graded matrix factorizations.

Assume there exists an extra action of $\mathbb{C}^*$ on $Y$ which commutes with the action of $G$ on $Y$ , and trivial on $\mathbb{Z}/2 \subset \mathbb{C}^{\ast }$ . Assume that $f$ is weight 2 with respect to the above $\mathbb{C}^{\ast }$ -action. Denote by $(1)$ the twist by the character

\begin{align*}{\rm pr}_2 \colon G\times \mathbb{C}^*\to \mathbb{C}^*.\end{align*}

Consider the category of graded matrix factorizations ${\rm MF}^{{\rm gr}}(\mathcal{Y}, f)$ . Its objects are pairs $(P, d_P)$ with $P$ an equivariant $G\times \mathbb{C}^*$ -sheaf on $Y$ and $d_P \colon P\to P(1)$ a $G\times \mathbb{C}^*$ -equivariant morphism. Note that as the $\mathbb{C}^{\ast }$ -action is trivial on $\mathbb{Z}/2$ , we have the induced action of $\mathbb{C}^{\star }=\mathbb{C}^{\ast }/(\mathbb{Z}/2)$ on $Y$ and $f$ is weight 1 with respect to the above $\mathbb{C}^{\star }$ -action. The objects of ${\rm MF}^{{\rm gr}}(\mathcal{Y}, f)$ can be alternatively described as tuples

(2.2) \begin{align} (E, F, \alpha \colon E\to F(1)', \beta \colon F\to E), \end{align}

where $E$ and $F$ are $G\times \mathbb{C}^{\star }$ -equivariant coherent sheaves on $Y$ , $(1)'$ is the twist by the character $G \times \mathbb{C}^{\star } \to \mathbb{C}^{\star }$ , and $\alpha$ and $\beta$ are $\mathbb{C}^{\star }$ -equivariant morphisms such that $\alpha \circ \beta$ and $\beta \circ \alpha$ are multiplications by $f$ .

Functoriality of categories of (graded or ungraded) matrix factorizations for pullback and proper pushfoward is discussed in [Reference Polishchuk and VaintrobPV11].

2.3 The Koszul equivalence

Let $Y$ be a smooth affine scheme with an action of a reductive group $G$ , let $\mathcal{Y}=Y/G$ , and let $V$ be a $G$ -equivariant vector bundle on $Y$ . Let $\mathbb{C}^*$ act on the fibers of $V$ with weight $2$ and consider $s\in \Gamma (Y, V)$ a section of $V$ of $\mathbb{C}^*$ -weight $2$ . It induces a map $\partial \colon V^{\vee } \to \mathcal{O}_Y$ . Let $s^{-1}(0)$ be the derived zero locus of $s$ with ring of regular functions

(2.3) \begin{align} \mathcal{O}_{s^{-1}(0)}:=\mathcal{O}_Y\left [V^{\vee }[1];\partial \right ]\!. \end{align}

Consider the quotient

(2.4) \begin{align} \mathscr{P}:=s^{-1}(0)/G. \end{align}

We call $\mathscr{P}$ the Koszul stack associated with $(Y, V, s, G)$ . We denote by $D^b(\mathscr{P})$ the derived category of $G$ -equivariant dg-modules over $\mathcal{O}_{s^{-1}(0)}$ with bounded coherent cohomologies.

The section $s$ also induces the regular function

(2.5) \begin{align} f\colon \mathscr{V\;}^{\vee }:={\rm Tot}_Y\left (V^{\vee }\right )/G\to \mathbb{C} \end{align}

defined by $f(y,v)=\langle s(y), v \rangle$ for $y\in Y(\mathbb{C})$ and $v\in V^{\vee }|_y$ . Consider the category of graded matrix factorizations ${\rm MF}^{{\rm gr}}\left (\mathscr{V\;}^{\vee }, f\right )$ with respect to the group $\mathbb{C}^*$ mentioned above. The Koszul duality equivalence, also called dimensional reduction in the literature, says the following.

Theorem 2.1 [Reference IsikIsi13, Reference HiranoHir17, Reference TodaTod24a]. There is an equivalence

(2.6) \begin{align} \Phi \colon D^b(\mathscr{P}) \stackrel {\sim }{\to } {\rm MF}^{{\rm gr}}(\mathscr{V\;}^{\vee }, f). \end{align}

The equivalence $\Phi$ is given by $\Phi (-)=\mathcal{K}\otimes _{\mathcal{O}_{\mathscr{P}}}(-)$ for the Koszul factorization $\mathcal{K}$ , see [Reference TodaTod24a , Theorem 2.3.3].

2.4 Window categories

2.4.1 Attracting stacks.

Let $Y$ be an affine variety with an action of a reductive group $G$ . Let $\lambda$ be a cocharacter of $G$ . Let $G^\lambda$ and $G^{\lambda \geqslant 0}$ be the Levi and parabolic groups associated to $\lambda$ . Let $Y^\lambda \subset Y$ be the closed subvariety of $\lambda$ -fixed points. Consider the attracting variety

\begin{align*}Y^{\lambda \geqslant 0}:=\{y\in Y|\,\lim _{t\to 0}\lambda (t)\cdot y\in Y^\lambda \}\subset Y.\end{align*}

Consider the attracting and fixed stacks

(2.7) \begin{align} \mathscr{Z}:=Y^\lambda /G^\lambda \xleftarrow {q}\mathscr{S\;}:=Y^{\lambda \geqslant 0}/G^{\lambda \geqslant 0}\xrightarrow {p}\mathcal{Y}. \end{align}

The map $p$ is proper. Kempf–Ness strata are connected components of certain attracting stacks $\mathscr{S}$ , and the map $p$ restricted to a Kempf–Ness stratum is a closed immersion, see [Reference Halpern-LeistnerHL15, § 2.1]. The attracting stacks also appear in the definition of Hall algebras [Reference PădurariuPăd23a] (for $Y$ an affine space), where the Hall product is induced by the functor

(2.8) \begin{align} \ast :=p_*q^*\colon D^b(\mathscr{Z}\,)\to D^b(\mathcal{Y}). \end{align}

In this case, the map $p$ may not be a closed immersion.

Let $T \subset G$ be a maximal torus and $\lambda$ is a cocharacter $\lambda \colon \mathbb{C}^{\ast } \to T$ . For a $G$ -representation $Y$ , the attracting variety $Y^{\lambda \geqslant 0} \subset Y$ coincides with the sub $T$ -representation generated by weights which pair non-negatively with $\lambda$ . We denote by $\langle \lambda , Y^{\lambda \geqslant 0} \rangle := \langle \lambda , \det Y^{\lambda \geqslant 0} \rangle$ , where $\det Y^{\lambda \geqslant 0}$ is the sum of $T$ -weights of $Y^{\lambda \geqslant 0}$ .

2.4.2 The definition of window categories.

Let $Y$ be an affine variety with an action of a reductive group $G$ and a linearization $\mathscr{L}$ . Consider the stacks

\begin{align*}j\colon \mathcal{Y}^{\mathscr{L}{\rm -ss}}:=Y^{\mathscr{L}{\rm -ss}}/G\hookrightarrow \mathcal{Y}:=Y/G.\end{align*}

We review the construction of window categories of $D^b(\mathcal{Y})$ which are equivalent to $D^b(\mathcal{Y}^{\mathscr{L}{\rm -ss}})$ via the restriction map, due to Segal [Reference SegalSeg11], Halpern–Leistner [Reference Halpern-LeistnerHL15], and Ballard–Favero–Katzarkov [Reference Ballard, Favero and KatzarkovBFK19]. We follow the presentation from [Reference Halpern-LeistnerHL15].

By also fixing a Weyl-invariant norm on the cocharacter lattice, the unstable locus $\mathcal{Y}\setminus \mathcal{Y}^{\mathscr{L}{\rm -ss}}$ has a stratification in Kempf–Ness strata $\mathscr{S}_i$ for $i\in I$ a finite ordered set:

\begin{align*}\mathcal{Y}\setminus \mathcal{Y}^{\mathscr{L}{\rm -ss}}=\bigsqcup _{i\in I}\mathscr{S}_i.\end{align*}

A Kempf–Ness stratum $\mathscr{S}_i$ is the attracting stack in $\mathcal{Y} \setminus \sqcup _{j\lt i}\mathcal{S}_j$ for a cocharacter $\lambda _i$ , with the fixed stack $\mathscr{Z}_i:=\mathscr{S\;}_i^{\lambda _i}$ . Let $N_{\mathscr{S}_i/\mathcal{Y}}$ be the normal bundle of $\mathscr{S}_i$ in $\mathcal{Y}$ . Define the width of the window categories

\begin{align*}\eta _i:=\big \langle \lambda _i, N^{\vee }_{\mathscr{S}_i/\mathcal{Y}}|_{\mathscr{Z}_i} \big \rangle .\end{align*}

For a choice of real numbers $w=(w_i)_{i\in I}\in \mathbb{R}^I$ , define the category

(2.9) \begin{align} \mathbb{G}_w:=\{\mathcal{F}\in D^b(\mathcal{Y})\;\textrm{such that}\; {\rm wt}_{\lambda _i}(\mathcal{F}|_{\mathscr{Z}_i})\subset [ w_i, w_i+\eta _i) \;\textrm{for all}\; i\in I\}. \end{align}

In the above, ${\rm wt}_{\lambda _i}(\mathcal{F}|_{\mathscr{Z}_i})$ is the set of $\lambda _i$ -weights on $\mathcal{F}|_{\mathscr{Z}_i}$ . Then [Reference Halpern-LeistnerHL15, Theorem 2.10] says that the restriction functor $j^*$ induces an equivalence of categories:

(2.10) \begin{align} j^*\colon \mathbb{G}_w\xrightarrow {\sim } D^b\big (\mathcal{Y}^{\mathscr{L}{\rm -ss}}\big ) \end{align}

for any choice of real numbers $w=(w_i)_{i\in I}\in \mathbb{R}^I$ .

We discuss a slight extension of the above theorem for matrix factorizations. Let $f\colon \mathcal{Y}\to \mathbb{C}$ be a regular function and recall the definition of ${\rm MF}(\mathbb{G}_w, f)$ from § 2.2.1. Then there is an equivalence

(2.11) \begin{align} j^*\colon {\rm MF}(\mathbb{G}_w, f)\xrightarrow {\sim } {\rm MF}\big (\mathcal{Y}^{\mathscr{L}{\rm -ss}}, f\big ). \end{align}

The analogous statement holds for categories of graded matrix factorizations.

2.5 Quasi-smooth stacks and singular support

Relevant references for this section are [Reference TodaTod24a, § 3.1 and 3.2.1] and [Reference Arinkin and GaitsgoryAG15].

2.5.1 Quasi-smooth stacks.

Let $\mathfrak{M}$ be a derived stack over $\mathbb{C}$ and let $\mathcal{M}$ be its classical truncation. Let $\mathbb{L}_{\mathfrak{M}}$ be the cotangent complex of $\mathfrak{M}$ . The stack $\mathfrak{M}$ is called quasi-smooth if, for all closed points $x\to \mathcal{M}$ , the restriction $\mathbb{L}_{\mathfrak{M}}|_x$ has cohomological amplitude in $[-1, 1]$ . Examples of quasi-smooth stacks are the Koszul stacks $\mathscr{P}$ from (2.4) or the moduli stacks of Gieseker semistable compactly supported sheaves on a smooth surface $S$ .

By [Reference Ben-Bassat, Brav, Bussi and JoyceBBB⁺15, Theorem 2.8], a stack $\mathfrak{M}$ is quasi-smooth if and only if it is a $1$ -stack and any point of $\mathfrak{M}$ lies in the image of a $0$ -representable smooth morphism

(2.12) \begin{align} \alpha \colon \mathscr{U}\to \mathfrak{M} \end{align}

for a Koszul scheme $\mathscr{U}$ as in (2.3). The dg-category $D^b(\mathfrak{M})$ is defined to be the limit

\begin{align*} D^b(\mathfrak{M})=\lim _{\mathscr{U} \to \mathfrak{M}} D^b(\mathscr{U}). \end{align*}

Suppose that $\mathcal{M}$ admits a good moduli space $\mathcal{M} \to M$ , see [Reference AlperAlp13] for the notion of good moduli space. For each point in $M$ , there is an étale neighborhood $U \to M$ and Cartesian squares

(2.13)

where each vertical arrow is étale and $\mathfrak{M}_U$ is equivalent to a Koszul stack $\mathscr{P}=s^{-1}(0)/G$ as in (2.4), see [Reference TodaTod24a, § 3.1.4], [Reference Halpern-LeistnerHLa, Theorem 4.2.3] and [Reference Alper, Hall and DavidAHD20].

2.5.2 $(-1)$ -shifted cotangent stacks.

Let $\mathfrak{M}$ be a quasi-smooth stack. Let $\mathbb{T}_{\mathfrak{M}}$ be the tangent complex of $\mathfrak{M}$ , which is the dual complex to the cotangent complex $\mathbb{L}_{\mathfrak{M}}$ . We denote by $\Omega _{\mathfrak{M}}[-1]$ the $(-1)$ -shifted cotangent stack of $\mathfrak{M}$ :

\begin{align*} \Omega _{\mathfrak{M}}[-1]:={\rm Spec}_{\mathfrak{M}}\left ({\rm Sym}(\mathbb{T}_{\mathfrak{M}}[1])\right )\!. \end{align*}

Consider the projection map

(2.14) \begin{align} p_0\colon \mathcal{N}:=t_0\left (\Omega _{\mathfrak{M}}[-1]\right )\to \mathfrak{M}. \end{align}

For a Koszul stack $\mathscr{P}$ as in (2.4), recall the function $f$ from (2.5) and consider the critical locus ${\rm Crit}(f)\subset \mathscr{V\;}^{\vee }$ . In this case, the map $p_0$ is the natural projection

(2.15) \begin{align} p_0\colon {\rm Crit}(f)=t_0\left (\Omega _{\mathscr{P}}[-1]\right )\to \mathscr{P}. \end{align}

2.5.3 Singular support.

We continue with the notations from the previous subsection. Arinkin–Gaitsgory [Reference Arinkin and GaitsgoryAG15] defined the notion of singular support of an object $\mathcal{F}\in D^b(\mathfrak{M})$ , denoted by

\begin{align*} {\rm Supp}^{\textrm {sg}}(\mathcal{F}) \subset \mathcal{N}. \end{align*}

The definition is compatible with maps $\alpha$ as in (2.12), see [Reference TodaTod24a, Definition 3.2.1]. Consider the group $\mathbb{C}^*$ scaling the fibers of the map $p_0$ . A closed substack $\mathcal{Z}$ of $\mathcal{N}$ is called conical if it is closed under the action of $\mathbb{C}^*$ . The singular support ${\rm Supp}^{\textrm {sg}}(\mathcal{F})$ of $\mathcal{F}$ is a conical subset $\mathcal{Z}$ of $\mathcal{N}$ .

Consider a Koszul stack $\mathscr{P}$ as in (2.4) and recall the Koszul equivalence $\Phi$ from (2.6). Under $\Phi$ , the singular support of $\mathcal{F}$ corresponds to the support $\mathcal{Z}$ of the matrix factorization $\Phi (\mathcal{F})$ , namely the maximal closed substack $\mathcal{Z}\subset {\rm Crit}(w)$ such that $\mathcal{F}|_{\mathscr{V\;}^{\vee }\setminus \mathcal{Z}}=0$ in ${\rm MF}^{{\rm gr}}(\mathscr{V\;}^{\vee }\setminus \mathcal{Z}, w)$ , see [Reference TodaTod24a, § 2.3.9].

2.5.4 DT categories via singular support quotients.

We review the definition of DT categories for quasi-smooth stacks [Reference TodaTod24a, § 3.2]. Let $\mathfrak{M}$ be a quasi-smooth stack with classical truncation $\mathcal{M}$ and let $\mathcal{N}:=t_0(\Omega _{\mathfrak{M}}[-1])$ . Let $\mathscr{L}$ be a line bundle on $\mathcal{M}$ . We regard $\mathscr{L}$ as a line bundle on $\mathcal{N}$ by pulling it back via $p_0\colon \mathcal{N} \to \mathcal{M}$ . We let $\mathcal{N}^{\mathscr{L}{\rm -ss}} \subset \mathcal{N}$ be the open substack of $\mathscr{L}$ -semistable points, see [Reference Halpern-LeistnerHLb, § 2.1 and 2.2, Example 4.5.2] for the definition of semistable points with respect to maps from the $\Theta$ -stack $\Theta :=[\mathbb{A}^1/\mathbb{G}_m]$ . Its complement

\begin{align*}\mathcal{Z}:=\mathcal{N}\setminus \mathcal{N}^{\mathscr{L}{\rm -ss}}\subset \mathcal{N}\end{align*}

is a conical closed substack. Let $\mathcal{C}_{\mathcal{Z}} \subset D^b(\mathfrak{M})$ be the subcategory consisting of objects with singular supports contained in $\mathcal{Z}$ . The quotient category

(2.16) \begin{align} D^b(\mathfrak{M})/\mathcal{C}_{\mathcal{Z}} \end{align}

is a model of the DT category for the semistable locus on the $(-1)$ -shifted cotangent stack, see [Reference TodaTod24a, Definition 3.2.2].

Remark 2.2. If $\mathfrak{M}=\mathscr{P}$ for a Koszul stack $\mathscr{P}$ as in (2.4), then the quotient category (2.16) is equivalent to ${\rm MF}^{\textrm {gr}}((\mathscr{V\;}^{\vee })^{\mathscr{L}{\rm -ss}}, f)$ , see [Reference TodaTod24a, Proposition 2.3.9]. As a grading is involved on the matrix factorization side, the category (2.16) is called the $\mathbb{C}^{\ast }$ -equivariant DT category in [Reference TodaTod24a]. The (ungraded) $\mathbb{Z}/2$ -periodic version is introduced in [Reference TodaTod23] and it is given by the quotient

(2.17) \begin{align} D^b(\mathfrak{M}_{\varepsilon })/\mathcal{C}_{\mathcal{Z}_{\varepsilon }}. \end{align}

Here $\mathfrak{M}_{\varepsilon }=\mathfrak{M}\times \textrm {Spec} \mathbb{C}[\varepsilon ]$ with $\deg \varepsilon=-1$ , and

\begin{align*} \mathcal{Z}_{\varepsilon }=\mathbb{C}^{\ast }(\mathcal{Z} \times \{1\}) \sqcup (\mathcal{N} \times \{0\}) \subset \mathcal{N} \times \mathbb{C} =t_0(\Omega _{\mathfrak{M}_{\varepsilon }}[-1]). \end{align*}

In the case of a Koszul stack $\mathfrak{M}=\mathscr{P}$ , the category (2.17) is equivalent to the category of (ungraded) matrix factorizations ${\rm MF}((\mathscr{V\;}^{\vee })^{\mathscr{L}{\rm -ss}}, f)$ . The argument used to prove Theorem 1.1 can be also used to obtain its $\mathbb{Z}/2$ -periodic version.

2.6 The window theorem for DT categories

2.6.1 Window categories.

We review the theory of window categories for quasi-smooth stacks [Reference TodaTod24a, Chapter 5]. Let $\mathfrak{M}$ be quasi-smooth and assume throughout this subsection that $\mathcal{M}=t_0(\mathfrak{M})$ admits a good moduli space $\mathcal{M} \to M.$ Let $\mathscr{L}$ be a line bundle on $\mathcal{M}$ and take a positive definite class $b \in H^4(\mathcal{M}, \mathbb{Q})$ , see [Reference Halpern-LeistnerHLb, Definition 3.7.6]. We also use the same symbols $(\mathcal{L}, b)$ for $p_0^{\ast }\mathcal{L} \in {\rm Pic}(\mathcal{N})$ and $p_0^{\ast }b \in H^4(\mathcal{N}, \mathbb{Q})$ . Then there is a $\Theta$ -stratification with respect to $(\mathcal{L}, b)$

(2.18) \begin{align} \mathcal{N}=\mathscr{S}_1 \sqcup \cdots \sqcup \mathscr{S}_N\sqcup \mathcal{N}^{\mathscr{L}{\rm -ss}} \end{align}

with centers $\mathscr{Z}_i\subset \mathscr{S}_i$ , see [Reference Halpern-LeistnerHLb, Theorem 5.2.3, Proposition 5.3.3]. When $\mathcal{M}$ is a (global) quotient stack, $\Theta$ -stratifications are the same as Kempf–Ness stratifications [Reference Halpern-LeistnerHLb, Example 0.0.5]. The class $b$ is then constructed as the pullback of the class in $H^4(BG_y, \mathbb{Q})$ corresponding to chosen positive definite form [Reference Halpern-LeistnerHLb, Example 5.3.4]. In the above situation, an analogue of the window theorem is proved in [Reference TodaTod, Theorem 1.1] and [Reference TodaTod24a, Theorem 5.3.13].

Theorem 2.3 [Reference TodaTod24a, Reference TodaTod]. In addition to the above, suppose that $\mathcal{M} \to M$ satisfies the formal neighborhood theorem. Then for each $w=(w_i)_{i=1}^N\in \mathbb{R}^N$ , there is a subcategory $\mathbb{W}_w \subset D^b(\mathfrak{M})$ such that the composition

\begin{align*} \mathbb{W}_{w} \subset D^b(\mathfrak{M}) \twoheadrightarrow D^b(\mathfrak{M})/\mathcal{C}_{\mathcal{Z}} \end{align*}

is an equivalence.

We explain the meaning of ‘the formal neighborhood theorem’in the statement of the above theorem, see [Reference TodaTod24a, Definition 5.2.3]. For a closed point $y \in M$ , denote also by $y \in \mathcal{M}$ the unique closed point in the fiber of $\mathcal{M} \to M$ at $y$ . Set $G_y:={\rm Aut}(y)$ , which is a reductive algebraic group. Let $\widehat {\mathcal{M}}_y$ be the formal fiber along with $\mathcal{M} \to M$ at $y$ . Then the formal neighborhood theorem says that there is a $G_y$ -equivariant morphism

\begin{align*} \kappa _y \colon \widehat {\mathcal{H}}^0(\mathbb{T}_{\mathcal{M}}|_{y}) \to \mathcal{H}^1(\mathbb{T}_{\mathcal{M}}|_{y}) \end{align*}

such that, by setting $\mathcal{U}_y$ to be the classical zero locus of $\kappa _y$ , there is an isomorphism $\widehat {\mathcal{M}}_y \cong \mathcal{U}_y/G_y$ . In this case, there is a (unique up to equivalence) derived stack $\widehat {\mathfrak{M}}_y$ and Cartesian squares.

We call $\widehat {\mathfrak{M}}_y$ the formal fiber of $\mathfrak{M}$ at $y$ . Let $\mathfrak{U}_y$ be the derived zero locus of $\kappa _y$ . Then, by replacing $\kappa _y$ if necessary, $\widehat {\mathfrak{M}}_y$ is equivalent to $\mathfrak{U}_y/G$ , see [Reference TodaTod24a, Lemma 5.2.5].

2.6.2 Local description of window categories.

Below we give a formal local description of $\mathbb{W}_w$ . Consider the pair of a smooth stack and a regular function $(\mathcal{X}_y, f_y)$ :

\begin{align*} \mathcal{X}_y:=\left (\widehat {\mathcal{H}}^0(\mathbb{T}_{\mathfrak{M}}|_{y}) \oplus \mathcal{H}^1(\mathbb{T}_{\mathfrak{M}}|_{y})^{\vee }\right )/G_y \stackrel {f_y}{\to } \mathbb{C}, \end{align*}

where $f_y(u, v)=\langle \kappa _y(u), v \rangle$ . From (2.15), the critical locus of $f_y$ is isomorphic to the classical truncation of the $(-1)$ -shifted cotangent stack over $\widehat {\mathfrak{M}}_y$ , so it is isomorphic to the formal fiber $\widehat {\mathcal{N}}_y$ of $\mathcal{N} \to \mathcal{M} \to M$ at $y$ . The pullback of the $\Theta$ -stratification (2.18) to $\widehat {\mathcal{N}}_y$ gives a Kempf–Ness stratification

\begin{align*} \widehat {\mathcal{N}}_y=\;\;\widehat {\!\!\mathscr{S}}_{1, y} \sqcup \cdots \sqcup \;\;\widehat {\!\!\mathscr{S}}_{N, y} \sqcup \widehat {\mathcal{N}}_y^{\mathscr{L}{\rm -ss}} \end{align*}

with centers $\widehat {\mathscr{Z}}_{i, y}\subset \;\;\widehat {\!\!\mathscr{S}}_{i, y}$ and one parameter subgroups $\lambda _i \colon \mathbb{C}^{\ast } \to G_y$ . By Koszul duality, see Theorem2.1, there is an equivalence:

\begin{align*} \Phi _y \colon D^b(\widehat {\mathfrak{M}}_y) \stackrel {\sim }{\to } {\rm MF}^{{\rm gr}}(\mathcal{X}_y, f_y). \end{align*}

Then the subcategory $\mathbb{W}_{w}$ in Theorem2.3 is characterized as follows: an object $\mathcal{E} \in D^b(\mathfrak{M})$ is an object of $\mathbb{W}_w$ if and only if, for any closed point $y \in M$ , we have

(2.19) \begin{align} \Phi _{y}\big(\mathcal{E}|_{\widehat {\mathfrak{M}}_y}\big) \in {\rm MF}^{{\rm gr}}(\mathbb{G}_{w'}, f_y), \ w_i'=w_i-\langle \lambda _i, \mathcal{H}^1(\mathbb{T}_{\mathfrak{M}}|_{y})^{\lambda _i\gt 0} \rangle . \end{align}

The category $\mathbb{G}_w$ is the window category (2.9) for the weights $w'_i$ and the line bundle $\mathscr{L}$ . The difference of $w_i$ and $w_i'$ is due to the discrepancy of categorical Hall products on $\mathfrak{M}_y$ and $\mathcal{X}_y$ , see [Reference PădurariuPăd23b, Proposition 3.1].

2.6.3 Adjoints of window categories.

Let $\mathcal{N}^{\mathscr{L}{\rm -st}} \subset \mathcal{N}^{\mathscr{L}{\rm -ss}}$ be the substack of $\mathscr{L}$ -stable points. We will use the following lemma in the proof of Theorem5.9.

Lemma 2.4. Assume that $\mathcal{N}^{\mathscr{L}{\rm -st}} =\mathcal{N}^{\mathscr{L}{\rm -ss}}$ . Then the inclusion $\mathbb{W}_w \subset D^b(\mathfrak{M})$ has a right adjoint.

Proof. First assume that $\mathfrak{M}$ is a Koszul stack (2.4), so $\mathfrak{M}=s^{-1}(0)/G$ for a section $s$ of a vector bundle $V$ on a smooth affine scheme $Y$ and a reductive algebraic group $G$ . By Koszul duality and replacing $Y$ with ${\rm Tot}_Y(V^{\vee })$ , we can reduce the problem to the existence of a right adjoint of a window category $\mathbb{G}_w \subset D^b(\mathcal{Y})$ as in § 2.4, where $\mathcal{Y}=Y/G$ and $\mathcal{Y}^{\mathscr{L}{\rm -st}}=\mathcal{Y}^{\mathscr{L}{\rm -ss}}$ so that $\mathcal{Y}^{\mathscr{L}{\rm -ss}}$ is a projective scheme over $Y/\!\!/ G$ . The scheme $Y$ is smooth and affine, so the good moduli space morphism $\mathcal{Y} \to Y/\!\!/ G$ is cohomologically proper. The composition

(2.20) \begin{align} D^b\big(\mathcal{Y}^{\mathscr{L}{\rm -ss}}\big) \stackrel {(j^{\ast })^{-1}}{\longrightarrow } \mathbb{G}_w \subset D^b(\mathcal{Y}) \end{align}

is given by a Fourier–Mukai functor with kernel object $\mathcal{P} \in D^b(\mathcal{Y}^{\mathscr{L} {\rm -ss}} \times \mathcal{Y})$ supported on $\mathcal{Y}^{\mathscr{L}{\rm -ss}} \times _{Y/\!\!/ G}\mathcal{Y}$ , see [Reference TodaTod24b, Lemma 6.7]. The support of $\mathcal{P}$ is proper over $\mathcal{Y}$ and cohomologically proper over $\mathcal{Y}^{\mathscr{L}{\rm -ss}}$ . Thus the Fourier–Mukai functor $D^b(\mathcal{Y}) \to D^b(\mathcal{Y}^{\mathscr{L}{\rm -ss}})$ with kernel $\mathcal{P}^{\vee } \boxtimes \omega _{\mathcal{Y}^{\mathscr{L}{\rm -ss}}}[\dim \mathcal{Y}]$ is well-defined and gives a right adjoint of (2.20).

In general, for each point in $M$ , there is an étale neighborhood $U \to M$ and Cartesian squares (2.13) such that $\mathfrak{M}_U$ is of the form $s^{-1}(0)/G$ as above. Let $\mathbb{W}_{U, w} \subset D^b(\mathfrak{M}_U)$ be the window subcategory in Theorem2.3 for $\mathfrak{M}_U$ , which admits a right adjoint $R_U$ by the above argument. Since we have

\begin{align*} \mathbb{W}_{w}=\lim _{U \to M}\mathbb{W}_{U, w}, \ D^b(\mathfrak{M})=\lim _{U\to M}D^b(\mathfrak{M}_U), \end{align*}

and $R_U$ is compatible with base change, a right adjoint of $\mathbb{W}_w \subset D^b(\mathfrak{M})$ is obtained as $\lim _{U\to M}R_U$ . See Subsection [Reference TodaTod24a, § 3.1.4] for the definition of the category of smooth morphisms in which we are taking the above limits, and the proof of [Reference TodaTod24a, Theorem 5.3.13] for the above descriptions of $\mathbb{W}_w$ and $D^b(\mathfrak{M})$ as limits.

2.7 Intrinsic window subcategory

Let $\mathfrak{M}$ be a quasi-smooth derived stack such that $\mathcal{M}=t_0(\mathfrak{M})$ admits a good moduli space $\mathcal{M} \to M$ . We say that $\mathfrak{M}$ is symmetric if for any closed point $x \in \mathfrak{M}$ , the $G_x:={\rm Aut}(x)$ -representation

\begin{align*} \mathcal{H}^0(\mathbb{T}_{\mathfrak{M}}|_{x}) \oplus \mathcal{H}^1(\mathbb{T}_{\mathfrak{M}}|_{x})^{\vee } \end{align*}

is a self dual $G_x$ -representation. In this subsection, we assume that $\mathfrak{M}$ is symmetric. Let $\delta \in {\rm Pic}(\mathfrak{M})_{\mathbb{R}}$ . We define a different kind of window categories from the ones in § 2.6, called intrinsic window subcategories $\mathbb{W}_{\delta }^{\textrm {int}} \subset D^b(\mathfrak{M})$ , see [Reference TodaTod24a, Definition 5.2.12, 5.3.12]. These categories are the quasi-smooth version of ‘magic window categories’ from [Reference Špenko and Van den BerghŠVdB17] and [Reference Halpern-Leistner and SamHLS20].

First, assume that there is a smooth affine scheme $Y$ with an action of a reductive algebraic group $G$ , $V \to Y$ a $G$ -equivariant vector bundle, and $s$ is a $G$ -equivariant section of $V \to Y$ such that

(2.21) \begin{align} \mathfrak{M}=s^{-1}(0)/G. \end{align}

Let $\mathcal{Y}=Y/G$ , $i \colon \mathfrak{M} \hookrightarrow \mathcal{Y}$ the closed immersion, and $\mathscr{V\;} \to \mathcal{Y}$ the total space of $V/G \to Y/G$ . In this case, we define $\mathbb{W}_{\delta }^{\textrm {int}} \subset D^b(\mathfrak{M})$ to be consisting of $\mathcal{E} \in D^b(\mathfrak{M})$ such that for any map $\lambda \colon B\mathbb{C}^{\ast } \to \mathfrak{M}$ we have

\begin{align*} {\rm wt}(\lambda ^{\ast }i_{\ast }\mathcal{E}) \subset \left [\frac {1}{2}{\rm wt}\left (\det \lambda ^{\ast }(\mathbb{L}_{\mathscr{V\;}}|_{\mathcal{Y}})^{\lambda \lt 0} \right ), \frac {1}{2}{\rm wt}\left (\det \lambda ^{\ast }(\mathbb{L}_{\mathscr{V\;}}|_{\mathcal{Y}})^{\lambda \gt 0}\right ) \right ] +{\rm wt}(\lambda ^{\ast }\delta ). \end{align*}

Here, by abuse of notation, we have also denoted the composition $B\mathbb{C}^{\ast }\stackrel {\lambda }{\to } \mathfrak{M} \stackrel {i}{\hookrightarrow } \mathcal{Y}$ by $\lambda$ . The above subcategory $\mathbb{W}_{\delta }^{\textrm {int}}$ is intrinsic to $\mathfrak{M}$ , and independent of a choice of a presentation $\mathfrak{M}$ as (2.21) for $(Y, V, s, G)$ , see [Reference TodaTod24a, Lemma 5.3.14].

In general, the intrinsic window subcategory is defined as follows.

Definition 2.5 [Reference TodaTod24a, Definition 5.3.12]. We define the subcategory

\begin{align*} \mathbb{W}_{\delta }^{\textrm {int}} \subset D^b(\mathfrak{M}) \end{align*}

to be consisting of objects $\mathcal{E}$ satisfying the following: for any étale morphism $U \to M$ such that $\mathfrak{M}_U$ is of the form $s^{-1}(0)/G$ as above with étale morphism $\iota _U \colon \mathfrak{M}_U \to \mathfrak{M}$ , § 2.5.1 , we have $\iota _U^{\ast }\mathcal{E} \in \mathbb{W}_{\iota _U^{\ast }\delta }^{\textrm {int}} \subset D^b(\mathfrak{M}_U)$ .

3.1 DT/PT quivers

A quiver $Q=(I, E)$ is a direct graph with a set of vertices $I$ and a set of oriented edges $E$ . A super-potential is a linear combination of cycles in $Q$ .

We discuss some examples of quivers which model the DT/PT wall-crossing for any curve class on a Calabi–Yau $3$ -fold. These quivers have been studied in [Reference Pădurariu and TodaPT24b]. Let $Q=(I, E)$ be the triple loop quiver, that is, the quiver with vertex set $I=\{1\}$ and edge set $E=\{x, y, z\}$ .

For $a\in \mathbb{N}$ , let $Q^{af}=(J, E^{af})$ be the quiver with vertex set $J=\{0, 1\}$ , and edge set $E^{af}$ containing three loops $E=\{x, y, z\}$ at $1$ , $(a+1)$ -edges from $0$ to $1$ and $a$ -edges from $1$ to $0$ . A DT/PT quiver is a quiver of the form $Q^{af}$ for some $a\in \mathbb{N}$ , see the following picture for $a=2$ .

For $d\in \mathbb{N}$ , let $V$ be a $\mathbb{C}$ -vector space of dimension $d$ . We often write $GL(d)$ as $GL(V)$ . Its Lie algebra is denoted by $\mathfrak{gl}(d)=\mathfrak{gl}(V):={\rm End}(V)$ . When the dimension is clear from the context, we drop $d$ from its notation and write it as $\mathfrak{g}$ . Consider the $GL(V)\cong GL(d)$ representations:

\begin{align*} R(d) & :=\mathfrak{gl}(V)^{\oplus 3},\ R^{af}(1,d):=V^{\oplus \left (a+1\right )}\oplus \left (V^{\vee }\right )^{\oplus a}\oplus \mathfrak{gl}(V)^{\oplus 3}. \end{align*}

Define the following stacks:

\begin{align*} {\mathcal{X}}(d):=R(d)/GL(d),\ \mathcal{X}^{af}(1, d):=R^{af}(1, d)/GL(d). \end{align*}

3.2 Quasi-BPS categories for $\mathbb{C}^3$

3.2.1 Notations involving weights.

Recall the triple loop quiver $Q$ with moduli stack of representation

\begin{align*}{\mathcal{X}}(d)=R(d)/GL(d)=\mathfrak{gl}(d)^{\oplus 3}/GL(d)\end{align*}

of dimension $d$ . Let $T(d)$ be the maximal torus of $GL(d)$ , let $M(d)$ be its weight space, and let $M(d)_{\mathbb{R}}:=M(d)\otimes _{\mathbb{Z}}\mathbb{R}$ . Denote by $\beta _1,\ldots , \beta _d$ the simple roots of $T(d)$ . We choose the dominant chamber of $M(d)$ such that a weight $\chi =\sum _{i=1}^d c_i\beta _i$ is dominant if $c_1\leqslant \cdots \leqslant c_d.$ We also define the following elements in $M(d)_{\mathbb{R}}$ :

\begin{align*} \rho :=\frac {1}{2}\sum _{i\gt j}(\beta _i-\beta _j), \ \tau _d :=\frac {1}{d}\sum _{i=1}^d \beta _i. \end{align*}

Note that $\rho$ is half the sum of positive roots of $GL(d)$ .

For later use, we also generalize the above notation and construction. Let $d=d^{(1)}+\cdots +d^{(m)}$ and set $G=\prod _{j=1}^m GL(d^{(j)})$ . Its maximal torus is $T(d)=\prod _{j=1}^m T(d^{(j)})$ , and we denote the simple roots of $T(d^{(j)})$ by $\beta _1^{(j)}, \ldots , \beta _{d^{(j)}}^{(j)}$ . We write $\chi \in M(d)_{\mathbb{R}}$ as

(3.1) \begin{align} \chi =\sum _{j=1}^m \chi ^{(j)} \in \bigoplus _{j=1}^m M\!\left(d^{(j)}\right)_{\mathbb{R}}. \end{align}

We say that $\chi$ is $G$ -dominant if each $\chi ^{(j)}$ is dominant. Let $\rho ^{(j)}$ be half the sum of positive roots of $GL(d^{(j)})$ . We set $\rho _G=\sum _{j=1}^m \rho ^{(j)}$ , which is half the sum of positive roots of $G$ .

3.2.2 The Špenko–Van den Bergh construction.

We explain a construction of (twisted) noncommutative resolutions of singularities $\mathbb{M}(d)_w$ of the coarse quotient space $R(d)/\!\!/ GL(d)=\mathfrak{gl}(d)^{\oplus 3}/\!\!/ GL(d)$ due to Špenko and Van den Bergh [Reference Špenko and Van den BerghŠVdB17]. Define the polytope $\textbf {W}(d)$ as

(3.2) \begin{align} \textbf {W}(d):=\frac {3}{2}{\rm sum}[0, \beta _i-\beta _j]+\mathbb{R}\tau _d\subset M(d)_{\mathbb{R}}, \end{align}

where the Minkowski sum is after all $1\leqslant i, j\leqslant d$ . For $w\in \mathbb{Z}$ , consider the hyperplane:

(3.3) \begin{align} \textbf {W}(d)_w:=\frac {3}{2}{\rm sum}[0, \beta _i-\beta _j]+w\tau _d\subset \textbf {W}(d). \end{align}

For $w \in \mathbb{Z}$ , denote by $D^b(\mathcal{X}(d))_w$ the subcategory of $D^b(\mathcal{X}(d))$ consisting of objects of weight $w$ with respect to the diagonal cocharacter $1_d$ of $T(d)$ . There is a direct sum decomposition

\begin{align*} D^b(\mathcal{X}(d))=\bigoplus _{w\in \mathbb{Z}} D^b(\mathcal{X}(d))_w. \end{align*}

We define the dg-subcategories

\begin{align*} \mathbb{M}(d) \subset D^b({\mathcal{X}}(d)) \quad (\mbox{respectively } \mathbb{M}(d)_w \subset D^b({\mathcal{X}}(d))_w) \end{align*}

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

(3.4) \begin{align} \chi +\rho \in \textbf {W}(d) \quad (\textrm{respectively} \chi +\rho \in \textbf {W}(d)_w) \end{align}

and $\Gamma _{GL(d)}(\chi )$ is the irreducible $GL(d)$ -representation with highest weight $\chi$ . Note that $\mathbb{M}(d)$ decomposes into the direct sum of $\mathbb{M}(d)_w$ for $w \in \mathbb{Z}$ . Alternatively, the category $\mathbb{M}(d)_w$ is generated by the vector bundles ${\mathcal{O}}_{{\mathcal{X}}(d)} \otimes \Gamma$ for $\Gamma$ a $GL(d)$ -representation such that the $T(d)$ -weights of $\Gamma$ are contained in the set $\nabla _w$ defined by

(3.5) \begin{align} \nabla _{w} & := \left \{\chi \in M_{\mathbb{R}} \,\Big | -\frac {1}{2} \eta _{\lambda } \leqslant \langle \lambda , \chi \rangle \leqslant \frac {1}{2}\eta _{\lambda } \mbox{ for all } \lambda \colon \mathbb{C}^* \to T(d) \right \} +w\tau _d, \end{align}

where $\eta _{\lambda }:=\langle \lambda , \mathbb{L}_{\mathcal{X}(d)}^{\lambda \gt 0}\rangle=2\langle \lambda , \mathfrak{gl}(d)^{\lambda \gt 0}\rangle$ , see [Reference Halpern-Leistner and SamHLS20, Lemma 2.9].

3.2.3 Quasi-BPS categories via matrix factorizations.

Consider the regular function

(3.6) \begin{align} {\rm Tr}\,W_d\colon {\mathcal{X}}(d)\to \mathbb{C},\, (X, Y, Z)\mapsto {\rm Tr}\left (X[Y, Z]\right ) \end{align}

induced by the potential $W=x[y, z]=xyz-xzy$ of the triple loop quiver $Q$ . Let

\begin{align*}\mathbb{S}(d):={\rm MF}(\mathbb{M}(d), \mathop {{\rm Tr}}\nolimits W_d) \subset {\rm MF}(\mathcal{X}(d), \mathop {{\rm Tr}}\nolimits W_d) \end{align*}

be the subcategory of matrix factorizations $ (\alpha \colon Fleftarrows G\colon \beta )$ with $F$ and $G$ in $\mathbb{M}(d)$ . It decomposes into the direct sum of $\mathbb{S}(d)_w$ for $w \in \mathbb{Z}$ , where $\mathbb{S}(d)_w$ is defined similarly to $\mathbb{S}(d)$ using $\mathbb{M}(d)_w$ . We also consider analogous categories of graded and/or equivariant matrix factorizations

\begin{align*}\mathbb{S}^{\bullet }_{\ast }(d):= {\rm MF}^{\bullet }_{\ast }(\mathbb{M}(d), \mathop {{\rm Tr}}\nolimits W_d) \subset {\rm MF}^{\bullet }_{\ast }({\mathcal{X}}(d), \mathop {{\rm Tr}}\nolimits W_d) \end{align*}

for $\ast \in \{\emptyset , T\}$ , $\bullet \in \{\emptyset , {\rm gr}\}$ , where $T=(\mathbb{C}^{\ast })^2$ acts on $\mathbb{C}^3$ by $(t_1, t_2)(x, y, z)=(t_1 x, t_2 y, t_1^{-1}t_2^{-1}z)$ and the grading is given by the weight 2 $\mathbb{C}^{\ast }$ -action on $Z$ , where $Z\in \mathfrak{gl}(d)$ is the last factor of ${\mathcal{X}}(d)=\mathfrak{gl}(d)^{\oplus 3}/GL(d)$ .

3.2.4 Quasi-BPS categories via the Koszul equivalence.

Let $V$ be a $d$ -dimensional complex vector space and denote by $\mathfrak{g}=\textrm {Hom}(V, V)$ the Lie algebra of $GL(V)$ . We set

\begin{align*} \mathcal{Y}(d):=\mathfrak{g}^{\oplus 2}/GL(V), \end{align*}

where $GL(V)$ acts on $\mathfrak{g}$ by conjugation. The stack $\mathcal{Y}(d)$ is the moduli stack of representations of dimension $d$ of the quiver with one vertex and two loops. Let $\mu$ be the morphism

(3.7) \begin{align} \mu \colon \mathcal{Y}(d) \to \mathfrak{g}, \ (X, Y) \mapsto [X, Y]. \end{align}

The morphism $\mu$ induces a morphism of vector bundles $\partial : \mathfrak{g}^{\vee }\otimes \mathcal{O}_{\mathfrak{g}^{\oplus 2}}\to \mathcal{O}_{\mathfrak{g}^{\oplus 2}}$ . Let $\mu ^{-1}(0)$ be the derived scheme with the dg-ring of regular functions

(3.8) \begin{align} \mathcal{O}_{\mu ^{-1}(0)}:=\mathcal{O}_{\mathfrak{g}^{\oplus 2}}\left [\mathfrak{g}^{\vee }\otimes \mathcal{O}_{\mathfrak{g}^{\oplus 2}}[1]; d_{\mu }\right ]\!, \end{align}

where the differential $d_\mu$ is induced by the morphism $\partial$ . Consider the (derived) stack

(3.9) \begin{align} i \colon \mathscr{C}(d):=\mu ^{-1}(0)/GL(V) \hookrightarrow \mathcal{Y}(d). \end{align}

For $v\in \mathbb{Z}$ , define the full dg-subcategory $\widetilde {\mathbb{T}}(d)_v\subset D^b(\mathcal{Y}(d))$ generated by the vector bundles $\mathcal{O}_{\mathcal{Y}(d)}\otimes \Gamma _{GL(d)}(\chi )$ for a dominant weight $\chi$ satisfying $\chi +\rho \in \textbf {W}(d)_{v}$ . Define the full dg-subcategory

(3.10) \begin{align} \mathbb{T}(d)_v \subset D^b(\mathscr{C}(d)) \end{align}

with objects $\mathcal{E}$ such that $i_{\ast }\mathcal{E}$ is in $\widetilde {\mathbb{T}}(d)_v$ .

Consider the grading induced by the action of $\mathbb{C}^*$ on ${\mathcal{X}}(d)$ scaling the linear map corresponding to $Z$ with weight $2$ . By Koszul duality in Theorem2.1, there is an equivalence

(3.11) \begin{align} \Phi \colon D^b(\mathscr{C}(d)) \stackrel {\sim }{\to } {\rm MF}^{{\rm gr}}(\mathcal{X}(d), \mathop {{\rm Tr}}\nolimits W_d). \end{align}

The above equivalence restricts to the equivalence, see [Reference Pădurariu and TodaPT24a, Equation (4.22)]:

(3.12) \begin{align} \Phi \colon \mathbb{T}(d)_v\stackrel {\sim }{\to } \mathbb{S}^{{\rm gr}}(d)_v. \end{align}

3.3 The local categorical DT/PT correspondence

Let $Q^{af, N}$ be the quiver obtained by adding $N$ -loops at the vertex $0$ of the DT/PT quiver $Q^{af}$ . Then the moduli stack of representations of $Q^{af, N}$ of dimension $(1, d)$ is

\begin{align*} \mathcal{X}^{af, N}(1, d)=\mathbb{C}^N \times \mathcal{X}^{af}(1, d) =(\mathbb{C}^N \times R^{af}(1, d))/GL(d) \end{align*}

where $GL(d)$ acts on $\mathbb{C}^N$ trivially. Let $\widetilde {W}$ be a super-potential on $Q^{af, N}$ satisfying

\begin{align*}\widetilde {W}|_{Q}=X[Y, Z],\end{align*}

where $Q \subset Q^{af, N}$ is the full subquiver consisting of the vertex $\{1\}$ , i.e. it is the triple loop quiver. Then we have the function

\begin{align*} \mathop {{\rm Tr}}\nolimits \widetilde {W}_d \colon \mathcal{X}^{af, N}(1, d) \to \mathbb{C}. \end{align*}

Let $\lambda \colon \mathbb{C}^{\ast } \to T(d)$ be the cocharacter given by

\begin{align*} \lambda (t)=(\overbrace {t^k, \ldots , t^k}^{d_1}, \ldots , \overbrace {t, \ldots , t}^{d_k}, \overbrace {1, \ldots , 1}^{d'}). \end{align*}

We have the diagram of attracting and fixed stacks

(3.13) \begin{align} \mathcal{X}^{af, N}(1, d)^{\lambda } \stackrel {q_{\lambda }}{\leftarrow } \mathcal{X}^{af, N}(1, d)^{\lambda \geqslant 0} \stackrel {p_{\lambda }}{\to } \mathcal{X}^{af, N}(1, d) \end{align}

where the fixed stack is

\begin{align*} \mathcal{X}^{af, N}(1, d)^{\lambda }=\prod _{i=1}^k \mathcal{X}(d_i) \times \mathcal{X}^{af, N}(1, d'). \end{align*}

The pullback/pushforward with respect to the diagram (3.13) gives the categorical Hall product for DT/PT quivers with super-potentials, see [Reference PădurariuPăd23a, § 3]:

(3.14) \begin{align} \ast =q_{\lambda \ast }p_{\lambda }^{\ast } \colon \boxtimes _{i=1}^k {\rm MF}(\mathcal{X}(d_i), \mathop {{\rm Tr}}\nolimits W_{d_i}) \boxtimes & \,{\rm MF}(\mathcal{X}^{af, N}(1, d'), \mathop {{\rm Tr}}\nolimits \widetilde {W}_{d'}) \\ & \notag \to {\rm MF}(\mathcal{X}^{af, N}(1, d), \mathop {{\rm Tr}}\nolimits \widetilde {W}_d). \end{align}

Let $\delta =\mu \chi _0=\mu \sum _{i=1}^d \beta _i$ for $\mu \in \mathbb{R}$ . We define the subcategories

\begin{align*} & \mathbb{E}^a(1, d; \delta ) \subset {\rm MF}(\mathcal{X}^{af, N}(1, d), \mathop {{\rm Tr}}\nolimits \widetilde {W}_d), \\ & \mathbb{F}^a(1, d; \delta ) \subset {\rm MF}(\mathcal{X}^{af, N}(1, d), \mathop {{\rm Tr}}\nolimits \widetilde {W}_d) \end{align*}

of matrix factorizations whose factors are generated by $\Gamma _{GL(d)}(\chi ) \otimes \mathcal{O}_{\mathcal{X}^{af, N}(1, d)}$ for $\chi$ a dominant weight of $T(d)$ satisfying

\begin{align*} & \chi +\rho +\delta \in \frac {3}{2}{\rm sum}[0, \beta _i-\beta _j]+ \frac {a}{2}{\rm sum}[-\beta _k, \beta _k]+{\rm sum}[-\beta _k, 0], \\ & \chi +\rho +\delta \in \frac {3}{2}{\rm sum}[0, \beta _i-\beta _j]+ \frac {a}{2}{\rm sum}(-\beta _k, \beta _k] \end{align*}

respectively, where $1\leqslant i,j,k\leqslant d$ .

Let $R$ be the following set

\begin{align*}R=\{(d_i, v_i)_{i=1}^k\mid d_i\in \mathbb{N}, v_i \in \mathbb{Z}\}.\end{align*}

We define a subset $O \subset R \times R$ which will be used to compare summands in the semiorthogonal decomposition of Theorems3.2 and 1.1, see § 2.1. This order is a particular example of an order from [Reference Pădurariu and TodaPT24b, § 3.2.3], which depends on a choice of $\mu \in \mathbb{R}$ .

The following is the local categorical DT/PT correspondence in terms of the above subcategories, see [Reference Pădurariu and TodaPT24b, Proposition 3.12, Corollary 3.14].

Theorem 3.2. Suppose that $2 \mu l \notin \mathbb{Z}$ for $1\leqslant l \leqslant d$ . There is a semiorthogonal decomposition

\begin{align*} \mathbb{E}^a(1, d; \delta )=\left \langle \boxtimes _{i=1}^k \mathbb{S}(d_i)_{w_i} \boxtimes \mathbb{F}(1, d'; \delta ') \right \rangle\!. \end{align*}

The right-hand side is after all $d'\leqslant d$ , partitions $(d_i)_{i=1}^k$ of $d-d'$ and integers $(w_i)_{i=1}^k$ such that for

(3.15) \begin{align} v_i:=w_i+d_i\bigg (d'+\sum _{j\gt i} d_j-\sum _{j\lt i}d_j\bigg ), \end{align}

we have

(3.16) \begin{align} -1-\mu -\frac {a}{2}\lt \frac {v_1}{d_1}\lt \cdots \lt \frac {v_k}{d_k}\lt -\mu -\frac {a}{2}. \end{align}

In the above, we let $\delta ':=\left (\mu -d+d'\right )\sum _{i\gt d-d'}^d\beta _i$ . Moreover, each fully-faithful functor

\begin{align*}\boxtimes _{i=1}^k \mathbb{S}(d_i)_{w_i} \boxtimes \mathbb{F}(1, d'; \delta ') \to \mathbb{E}^a(1, d; \delta )\end{align*}

is given by the restriction of the categorical Hall product ( 3.14 ).

The order of the semiorthogonal summands is discussed in [Reference Pădurariu and TodaPT24b , § 3.2.3] and depends on $\mu \in \mathbb{R}$ . For $0\lt \varepsilon \ll 1$ , let $\mu =-a/2-\varepsilon$ . Then the order is that of $(d_i, v_i)_{i=1}^k \in R$ in Definition 3.1 , see also § 2.1 .

The analogous conclusion holds if we replace $\mathbb{C}^N$ by an open subset in all the constructions and statements above, and also for graded categories of matrix factorizations.

Remark 3.3. One also obtains similar semiorthogonal decompositions for different super-potentials $\widetilde {W}$ . Assume $N=0$ and $\widetilde {W}=0$ . Let $\chi _0 \colon GL(V) \to {\mathbb{C}}^{\ast }$ be the determinant character $g \mapsto \det g$ , and define the following DT and PT spaces for the quivers $Q^{af}$ defined by the GIT quotient stacks (which are smooth quasi-projective varieties):

\begin{align*} I^a(d):=\mathcal{X}^{af}(1, d)^{\chi _0{\rm -ss}},\ P^a(d):=\mathcal{X}^{af}(1, d)^{\chi ^{-1}_0{\rm -ss}}. \end{align*}

Using the window theorem, we obtain the local categorical DT/PT correspondence [Reference Pădurariu and TodaPT24b, Theorem 1.1], which says that there is a semiorthogonal decomposition

\begin{align*}D^b(I^a(d))=\Big \langle \boxtimes _{i=1}^k \mathbb{M}(d_i)_{w_i}\boxtimes D^b(P^a(d'))\Big \rangle ,\end{align*}

where the right-hand side is indexed as in Theorem 3.2. The above semiorthogonal decomposition is regarded as a categorical wall-crossing formula for the one parameter family of stability conditions $\chi _t=\chi _0^t$ for $t\in \mathbb{R}$ (which is a $\mathbb{R}$ -character), where the wall is at $t=0$ , the DT-chamber is $t\gt 0$ and the PT-chamber is $t\lt 0$ .

4.1 The definition of quasi-BPS category

Let $S$ be a smooth surface. We denote by $\mathfrak{M}_S(d)$ the derived moduli stack of zero-dimensional sheaves on $S$ of length $d$ . The derived stack $\mathfrak{M}_S(d)$ is quasi-smooth, and its classical truncation $\mathcal{M}_S(d)=t_0(\mathfrak{M}_S(d))$ admits a good moduli space

\begin{align*} \mathcal{M}_S(d) \to {\rm Sym}^d(S) \end{align*}

by sending a zero-dimensional sheaf to its support. Moreover, the stack $\mathfrak{M}_S(d)$ is symmetric, see [Reference TodaTod24a, Lemma 5.4.1].

Let $\mathcal{Q}\in D^b(S \times \mathfrak{M}_S(d))$ be the universal zero-dimensional sheaf. We define the following line bundle on $\mathfrak{M}_S(d)$ :

\begin{align*} \mathscr{L}\;:= {\rm det}(p_{\mathfrak{M}\ast }\mathcal{Q}) \in {\rm Pic}(\mathfrak{M}_S(d)), \end{align*}

where $p_{\mathfrak{M}} \colon S \times \mathfrak{M}_S(d) \to \mathfrak{M}_S(d)$ is the natural projection. Recall the definition of an intrinsic window subcategory for a quasi-smooth and symmetric derived stack, see Definition 2.5. We define the quasi-BPS category as follows.

Definition 4.1. For $(d, w) \in \mathbb{N} \times \mathbb{Z}$ , we define the quasi-BPS category $\mathbb{T}_X(d)_w \subset D^b$ $(\mathfrak{M}_S(d))$ by

\begin{align*} \mathbb{T}_X(d)_w:=\mathbb{W}_{\delta =w \mathscr{L}/d}^{\textrm {int}} \subset D^b(\mathfrak{M}_S(d)). \end{align*}

Here $w \mathscr{L}/d:=w/d \cdot \mathscr{L} \in {\rm Pic}(\mathfrak{M}_S(d))_{\mathbb{Q}}.$

We see that the above definition coincides with the one introduced in [Reference PădurariuPăd23b]. Let us take a point $p=\sum _{j=1}^m d^{(j)}x^{(j)} \in {\rm Sym}^d(S)$ for distinct points $x^{(1)}, \ldots , x^{(m)} \in S$ , and denote by $\widehat {\mathfrak{M}}_S(d)_p$ the formal fiber of $\mathfrak{M}_S(d)$ at $p$ . Let $y^{(1)}, \ldots , y^{(m)} \in \mathbb{C}^2$ be distinct points and set $q=\sum _{j=1}^m d^{(j)}y^{(j)} \in {\rm Sym}^d(\mathbb{C}^2)$ . Then there is an equivalence of derived stacks

\begin{align*} \iota _p \colon \widehat {\mathscr{C}}(d)_q \stackrel {\sim }{\to } \widehat {\mathfrak{M}}_S(d)_p. \end{align*}

Let $\mathbb{T}(d)_w \subset D^b(\mathscr{C}(d))$ be the quasi-BPS category from § 3.2.4 and define $\widehat {\mathbb{T}}(d)_{w, q} \subset D^b(\widehat {\mathscr{C}}(d)_q)$ to be the subcategory split generated by $\mathbb{T}(d)_{w}|_{\widehat {\mathscr{C}}(d)_q}$ . The following lemma shows that $\mathbb{T}_X(d)_w$ coincides with the one defined in [Reference PădurariuPăd23b, § 4.1.6].

Proof. The defining property of the intrinsic window subcategory is local on the good moduli space, so $\mathcal{E} \in D^b(\mathfrak{M}_S(d))$ is an object in $\mathbb{W}_{\delta }^{\textrm {int}}$ if and only if, for any $p\in {\rm Sym}^d(S)$ , we have $\mathcal{E}|_{\widehat {\mathfrak{M}}_S(d)_p} \in \mathbb{W}_{\delta _p}^{\textrm {int}}$ in $D^b(\widehat {\mathfrak{M}}_S(d)_p)$ , where $\delta _p=\delta |_{\widehat {\mathfrak{M}}_S(d)_p}$ . By the presentation invariance of the intrinsic window subcategory, see [Reference TodaTod24a, Lemma 5.3.14], this is also equivalent to $\mathcal{F}_q:=\iota _p^{\ast } (\mathcal{E}\big |_{\widehat {\mathfrak{M}}_S(d)_p})$ being an object of $\mathbb{W}_{w \chi _0/d}^{\textrm {int}}$ in $D^b(\widehat {\mathscr{C}}(d)_q)$ . Here, note that the line bundle $\mathscr{L}$ restricted to $\widehat {\mathfrak{M}}_S(d)_p$ corresponds to the determinant character $\chi _0=\det \colon GL(d) \to \mathbb{C}^{\ast }$ on $\widehat {\mathscr{C}}(d)_q$ .

We have the closed immersion

\begin{align*} j_q \colon \widehat {\mathscr{C}}(d)_q= \mu _q^{-1}(0)/GL(d) \hookrightarrow \widehat {\mathcal{Y}}(d)_q, \end{align*}

where $\mu _q$ is the commutator map $\widehat {\mathcal{Y}}(d)_q \to \mathfrak{gl}(d)/GL(d)$ . Let $\mathscr{V\;} \to \widehat {\mathcal{Y}}(d)_q$ be the total space of the vector bundle induced by the $GL(d)$ -representation $\mathfrak{gl}(d)$ . Then we have

\begin{align*} \mathbb{L}_{\mathscr{V\;}}|_{\widehat {\mathcal{Y}}(d)_q} =\left (\mathfrak{gl}(d) \otimes \mathcal{O}_{\widehat {\mathcal{Y}}(d)_q} \to \mathfrak{gl}(d)^{\oplus 3} \otimes \mathcal{O}_{\widehat {\mathcal{Y}}(d)_q} \right )\!. \end{align*}

Therefore, the condition that $\mathcal{F}_q \in \mathbb{W}_{w\chi _0/d}^{\textrm {int}}$ means that, for any $\lambda \colon B\mathbb{C}^{\ast } \to \widehat {\mathscr{C}}(d)_q$ , we have

(4.1) \begin{align} {\rm wt}(\lambda ^{\ast }j_{q\ast }\mathcal{F}_q) \subset \left [-\langle \lambda , \mathfrak{gl}(d)^{\lambda \gt 0} \rangle , \langle \lambda , \mathfrak{gl}(d)^{\lambda \gt 0}\rangle \right ]+w\tau _d, \end{align}

where ${\rm wt}(\lambda ^{\ast }j_{q\ast }\mathcal{F}_q)$ denotes the set of $\mathbb{C}^*$ -weights of $\lambda ^{\ast }j_{q\ast }\mathcal{F}_q$ . Here, in the right-hand side, we have denoted the cocharacter $\mathbb{C}^{\ast }\to GL(d)$ associated with $\lambda \colon B\mathbb{C}^{\ast } \to \widehat {\mathscr{C}}(d)_q$ by the same symbol $\lambda$ . The condition (4.1) for all $\lambda$ is equivalent to $j_{q\ast }\mathcal{F}_q$ being generated by the vector bundles $\mathcal{O}_{\widehat {\mathcal{Y}}(d)_q}\otimes \Gamma$ for $GL(d)$ -representations $\Gamma$ whose $T(d)$ -weights are contained in $\nabla _w$ , see (3.5). The last condition is also equivalent to $\mathcal{F}_q \in \widehat {\mathbb{T}}(d)_{w, q}$ , see § 3.2.1.

We also give another characterization of quasi-BPS categories in terms of Ext quivers. Let $p \in {\rm Sym}^d(S)$ be as above. The unique closed point in the fiber of $\mathcal{M}_S(d) \to {\rm Sym}^d(S)$ at $p$ corresponds to the semisimple sheaf

\begin{align*} F=\bigoplus _{j=1}^m V^{(j)} \otimes \mathcal{O}_{x^{(j)}}. \end{align*}

We set

\begin{align*} G_p :={\rm Aut}(F)=\prod _{j=1}^m \textrm {GL}(V^{(j)}), \ \mathcal{Y}'(d)_p & := \left [\widehat {\textrm {Ext}}_S^1(F, F)/G_p\right ]\!. \end{align*}

Let $\kappa _p$ be the following $G_p$ -equivariant morphism

\begin{align*} \kappa _p \colon \widehat {\textrm {Ext}}_S^1(F, F) \to \textrm {Ext}_S^2(F, F), \ u \mapsto [u, u]. \end{align*}

Then the formal fiber $\widehat {\mathfrak{M}}_{S}(d)_p$ of $\mathfrak{M}_S(d)$ at $p$ is also equivalent to the derived zero locus of $\kappa _p$ :

(4.2) \begin{align} \kappa _p^{-1}(0)/G_p \stackrel {\sim }{\to } \widehat {\mathfrak{M}}_S(d)_p. \end{align}

We define $\widetilde {\mathbb{T}}_{X}(d)_{w, p} \subset D^b(\mathcal{Y}'(d)_p)$ as the subcategory generated by $\Gamma _{G_p}(\chi ) \otimes \mathcal{O}_{\mathcal{Y}'(d)_p}$ , where $\chi$ is a $G_p$ -dominant weight of $T(d)$ satisfying

(4.3) \begin{align} \chi +\rho _{G_p} \in \frac {3}{2}{\rm sum}\Big[0, \beta _{i}^{(j)}-\beta _{i'}^{(j)}\Big] +w\tau _d, \end{align}

where the sum is after all $1\leqslant j\leqslant m$ and $1\leqslant i, i'\leqslant d^{(j)}$ , see also § 3.2.1 for the definition of a $G_p$ -dominant weight. Define the subcategory

\begin{align*} \mathbb{T}_{X}(d)_{w, p} \subset D^b(\widehat {\mathfrak{M}}_S(d)_p) \end{align*}

consisting of objects $\mathcal{E}$ such that $i_{p\ast }\mathcal{E} \in \widetilde {\mathbb{T}}_{X}(d)_{w, p}$ , where $i_p \colon \widehat {\mathfrak{M}}_S(d)_p \hookrightarrow \mathcal{Y}'(d)_p$ is the closed immersion through the equivalence (4.2).

As before, we can also characterize $\mathbb{T}_X(d)_{w, p}$ via Koszul duality. Let $(\mathcal{X}(d)_p, f_p)$ be defined by

\begin{align*} \mathcal{X}(d)_p=\left (\widehat {\textrm {Ext}}_S^1(F, F) \oplus \textrm {Ext}_S^2(F, F)^{\vee } \right )/G_p \stackrel {f_p}{\to } \mathbb{C} \end{align*}

where $f_p(u, v)=\langle \kappa _p(u), v \rangle$ . Note that we have

(4.4) \begin{align} (\textrm {Ext}_S^1(F, F) \oplus \textrm {Ext}_S^2(F, F)^{\vee })/G_p =\prod _{j=1}^m {\rm End}(V^{(j)})^{\oplus 3}/\textrm {GL}(V^{(j)}), \end{align}

such that $f_p=\sum _{j=1}^m \mathop {{\rm Tr}}\nolimits W_{d^{(j)}}$ , where $\mathop {{\rm Tr}}\nolimits W_d$ is given in (3.6). We denote by

\begin{align*} \mathbb{S}^{\textrm {gr}}(d)_{w, p} \subset {\rm MF}^{{\rm gr}}(\mathcal{X}(d)_p, f_p) \end{align*}

the dg-subcategory generated by matrix factorizations whose factors are generated by $\Gamma _{G_p}(\chi ) \otimes \mathcal{O}_{\mathcal{X}(d)_p}$ , where $\chi$ is a $G_p$ -dominant weight satisfying (4.3). The Koszul duality gives an equivalence (see Theorem2.1):

(4.5) \begin{align} \Phi _p \colon D^b(\widehat {\mathfrak{M}}_{S}(d)_p) \stackrel {\sim }{\to } {\rm MF}^{{\rm gr}}(\mathcal{X}(d)_p, f_p). \end{align}

Then similarly to (3.12), it restricts to the equivalence

\begin{align*} \Phi _p \colon \mathbb{T}_X(d)_{w, p} \stackrel {\sim }{\to }\mathbb{S}^{\textrm {gr}}(d)_{w, p}. \end{align*}

Remark 4.4. We set $p^{(j)}=d^{(j)}x^{(j)} \in {\rm Sym}^{d^{(j)}}(S)$ for $1\leqslant j\leqslant m$ . We then have

(4.6) \begin{align} \widehat {\mathfrak{M}}_{S}(d)_p=\prod _{j=1}^m \widehat {\mathfrak{M}}_{S}(d^{(j)})_{p^{(j)}}. \end{align}

Note that under the product (4.6), we have

\begin{align*} \mathbb{T}_{X}(d)_{w, p}=\boxtimes _{j=1}^m \mathbb{T}_{X}(d^{(j)})_{w^{(j)}, p^{(j)}} \end{align*}

where $w^{(j)}/d^{(j)}=w/d$ . Similarly under the product (4.4), we have

(4.7) \begin{align} \mathbb{S}^{\textrm {gr}}(d)_{w, p}= \boxtimes _{j=1}^m \mathbb{S}^{\textrm {gr}}(d^{(j)})_{w^{(j)}, p^{(j)}}. \end{align}

4.2 The semiorthogonal decomposition of $D^b(\mathfrak{M}_S(d))$ into quasi-BPS categories

The main result of [Reference PădurariuPăd23b] is that quasi-BPS categories provide a semiorthogonal decomposition of $D^b(\mathfrak{M}_{S}(d))$ , alternatively of the Hall algebra of points on a surface. Let

(4.8) \begin{align} \mathfrak{M}_S(d_1, \ldots , d_k) \end{align}

be the derived moduli stack of filtrations of coherent sheaves on $S$ :

(4.9) \begin{align} 0=Q_0 \subset Q_1 \subset Q_2 \subset \cdots \subset Q_k \end{align}

such that each subquotient $Q_i/Q_{i-1}$ is a zero-dimensional sheaf on $S$ with length $d_i$ . There exist evaluation morphisms

\begin{align*} \mathfrak{M}_S(d_1) \times \cdots \times \mathfrak{M}_S(d_k) \stackrel {q}{\leftarrow } \mathfrak{M}_S(d_1, \ldots , d_k) \stackrel {p}{\to } \mathfrak{M}_S(d), \end{align*}

where $p$ is proper, $q$ is quasi-smooth and $d=d_1+\cdots +d_k$ . The above diagram for $k=2$ defines the categorical Hall product due to Porta–Sala [Reference Porta and SalaPS23]:

(4.10) \begin{align} \ast =p_{\ast }q^{\ast } \colon D^b(\mathfrak{M}_S(d_1)) \boxtimes D^b(\mathfrak{M}_S(d_2)) \to D^b(\mathfrak{M}_S(d)). \end{align}

Theorem 4.5 [Reference PădurariuPăd23b, Theorem 1.1]. There is a semiorthogonal decomposition

(4.11) \begin{align} D^b(\mathfrak{M}_{S}(d))=\left \langle \bigotimes _{i=1}^k \mathbb{T}_X(d_i)_{v_i} \,\Big |\, \frac {v_1}{d_1}\lt \cdots \lt \frac {v_k}{d_k}, d_1+\cdots +d_k=d\right \rangle \!. \end{align}

Each fully-faithful functor $\bigotimes _{i=1}^k \mathbb{T}_X(d_i)_{v_i} \hookrightarrow D^b(\mathfrak{M}_{S}(d))$ is given by the categorical Hall product. The order is given by the subset $O \subset R \times R$ from [Reference Pădurariu and TodaPT24a , § 3.4].

4.3 Objects in quasi-BPS categories for $\mathbb{C}^3$

If $S=\mathbb{C}^2$ , the stack $\mathfrak{M}_S(d)$ is identified with the derived commuting stack defined in § 3.2.4, i.e.

\begin{align*} \mathfrak{M}_{\mathbb{C}^2}(d)=\mathscr{C}(d). \end{align*}

Let $(d, v)\in \mathbb{N}\times \mathbb{Z}$ . We revisit the construction of the objects $\mathcal{E}_{d,v}\in D^b(\mathscr{C}(d))$ from [Reference Pădurariu and TodaPT24a, Definition 4.2], which are objects of quasi-BPS categories. Let $\mathfrak{b}$ be the Lie algebra of the Borel subgroup $B(d) \subset GL(d)$ consisting of upper triangular matrices, let $\mathfrak{n}$ be the nilpotent subalgebra of $\mathfrak{b}$ and let $\overline {\mathfrak{n}}:=[\mathfrak{n}, \mathfrak{n}]$ . Consider the commutator map

\begin{align*}\nu \colon \mathfrak{n}^{\oplus 2}\to \overline {\mathfrak{n}}.\end{align*}

Let $B(d)$ act naturally on the derived zero locus $\nu ^{-1}(0)$ and let $T(d)\subset B(d)$ act trivially on $\mathbb{C}^2$ . Consider the stack $\mathcal{Z}$ and the maps

\begin{align*}\mathbb{C}^2/T(d)\xleftarrow {q} \mathcal{Z}:=\left (\nu ^{-1}(0)\times \mathbb{C}^2\right )/B(d)\xrightarrow {p}\mathscr{C}(d),\end{align*}

where $p$ is a proper map induced by the inclusion $\mathfrak{n}^{\oplus 2} \subset \mathfrak{g}^{\oplus 2}$ and the diagonal matrices $\mathbb{C} \subset \mathfrak{g}$ , and $q$ is induced by the natural projections $\nu ^{-1}(0) \times \mathbb{C}^2 \to \mathbb{C}^2$ , $B(d) \to T(d)$ , see [Reference Pădurariu and TodaPT24a, § 4.3]. The following lemma is implicitly proved in [Reference Pădurariu and TodaPT24a, § 5].

Lemma 4.6. The derived stack $\mathcal{Z}$ is classical, i.e. we have an equivalence

(4.12) \begin{align} \mathcal{Z}^{{\rm cl}}\stackrel {\sim }{\to }\mathcal{Z}. \end{align}

Proof. It suffices to show that $\nu ^{-1}(0)$ is a classical scheme. The structure complex $\mathcal{O}_{\nu ^{-1}(0)}$ is the Koszul complex of a section in $\Gamma (\mathfrak{n}^{\oplus 2}, \mathcal{O}_{\mathfrak{n}^{\oplus 2}}\otimes \overline {\mathfrak{n}})$ , so it suffices to check that

(4.13) \begin{align} \dim \nu ^{-1}(0)=\dim \nu ^{-1}(0)^{{\rm cl}}, \end{align}

see for example [Reference Aprodu and NagelAN10, § 1.4], where $\dim \nu ^{-1}(0)=2\dim _{\mathbb{C}}\mathfrak{n}-\dim _{\mathbb{C}}\overline {\mathfrak{n}}$ is the dimension of $\nu ^{-1}(0)$ as a quasi-smooth scheme. The equality (4.13) follows from [Reference Pădurariu and TodaPT24a, Proposition 5.17].

We set

(4.14) \begin{align} m_i :=\left \lceil \frac {vi}{d} \right \rceil -\left \lceil \frac {v(i-1)}{d} \right \rceil +\delta _i^d -\delta _i^1 \in \mathbb{Z}, \end{align}

where $\delta ^j_i$ is the Kronecker delta function: $\delta ^j_i=1$ if $i=j$ and $\delta ^j_i=0$ otherwise. Let $\lambda$ be the cocharacter

(4.15) \begin{align} \lambda \colon \mathbb{C}^{\ast } \to GL(d), \ t \mapsto \big(t^d, t^{d-1}, \ldots , t\big). \end{align}

For a weight $\chi =\sum _{i=1}^d n_i \beta _i$ with $n_i \in \mathbb{Z}$ , we denote by $\mathbb{C}(\chi )$ the one-dimensional $GL(d)^{\lambda \geqslant 0}$ -representation given by

\begin{align*} GL(d)^{\lambda \geqslant 0}= B(d) \to GL(d)^{\lambda }=T(d) \stackrel {\chi }{\to } \mathbb{C}^{\ast }, \end{align*}

where the first morphism is the natural projection. We define the object $\mathcal{E}_{d, v}$ by

(4.16) \begin{align} \mathcal{E}_{d, v}:=p_{\ast }\left (\mathcal{O}_{\mathcal{Z}} \otimes \mathbb{C}(m_1, \ldots , m_d)\right ) \in D^b(\mathscr{C}(d)). \end{align}

By [Reference Pădurariu and TodaPT24a, Lemma 4.3], we have

(4.17) \begin{align} \mathcal{E}_{d,v}\in \mathbb{T}_{\mathbb{C}^3}(d)_v. \end{align}

4.4 Objects in quasi-BPS categories for a local surface

Let $S$ be a smooth surface. The classical truncation of (4.8) for all $d_i=1$ :

\begin{align*}\mathcal{M}_S(1, \ldots , 1)=t_0(\mathfrak{M}_S(1, \ldots , 1))\end{align*}

is the classical moduli stack of filtrations (4.9) such that $Q_i/Q_{i-1}=\mathcal{O}_{x_i}$ for some $x_i \in S$ . It admits a natural morphism to $S^{\times d}$ by sending the above filtration to $(x_1, \ldots , x_d)$ . Define

\begin{align*}\mathcal{Z}_S:=\mathcal{M}_S(1, \ldots , 1) \times _{S^{\times d}} \Delta ,\end{align*}

where $\Delta \cong S \subset S^{\times d}$ is the small diagonal and the fiber product is in the classical sense. Observe that (4.12) implies that

\begin{align*}\mathcal{Z}_{\mathbb{C}^2}\cong \mathcal{Z}.\end{align*}

For $d\in \mathbb{N}$ , let $T(d)$ act trivially on $S$ . We have the following commutative diagram.

Let $x\in S(\mathbb{C})$ . Let $\widehat {\mathbb{C}}^2$ be the formal completion of $0$ in $\mathbb{C}^2$ . For a choice of an isomorphism $\widehat {\mathcal{O}}_{S, x} \cong \widehat {\mathcal{O}}_{\mathbb{C}^2, 0}$ , there is the following commutative diagram.

(4.18)

Let $(d, v)\in \mathbb{N}\times \mathbb{Z}$ . Recall the definition of the integers $m_i$ from (4.14). Define the functor

\begin{align*} \Phi _{d,v}\colon D^b(S) & \to D^b(\mathfrak{M}_{S}(d))_v,\\ \mathcal{F} & \mapsto p_{S*}\left (q_S^*(\mathcal{F}\otimes \mathbb{C}(m_1, \ldots , m_d))\right )\!. \end{align*}

4.5 Dimensions of quasi-BPS categories

In this section, we prove Theorem1.4. Let $S$ be a smooth toric surface. Then the two-dimensional torus $T=(\mathbb{C}^{\ast })^2$ acts on $S$ , and the categorical Hall product (4.10) can be defined $T$ -equivariantly. Recall that $\mathbb{K}:=K_0(BT)$ , $\mathbb{F}:={\rm Frac}\,\mathbb{K}$ and that for $V$ a $\mathbb{K}$ -module we let $V_{\mathbb{F}}:=V\otimes _{\mathbb{K}}\mathbb{F}$ . There is an associative algebra structure on

\begin{align*} {\rm KHA}_{S, T}:=\bigoplus _{d \geqslant 0} G_T(\mathfrak{M}_S(d))_{\mathbb{F}} \stackrel {\cong }{\leftarrow }\bigoplus _{d \geqslant 0} G_T(\mathcal{M}_S(d))_{\mathbb{F}} . \end{align*}

We begin with a few preliminaries. Let $S^T$ be the (finite) set of $T$ -fixed points on $S$ . For $p \in {\rm Sym}^d(S)$ , denote by $\mathcal{M}_S(d)_p$ the fiber over $p$ along the map $\pi _d\colon \mathcal{M}_{S}(d)\to {\rm Sym}^d(S)$ . We have the following lemma.

Lemma 4.8. There is an isomorphism

(4.19) \begin{align} {\rm KHA}_{S, T} \cong \bigotimes _{x\in S^T} \left (\bigoplus _{d\geqslant 0}G_T\big (\mathcal{M}_{S}(d)_{d[x]}\big )_{\mathbb{F}}\right )\!. \end{align}

Proof. The $T$ -fixed loci of $\mathcal{M}_{S}(d)$ are included in fibers over $T$ -fixed loci of ${\rm Sym}^d(S)$ . The $T$ -fixed locus ${\rm Sym}^d(S)^T$ consists of points $p=\sum _{j=1}^m d^{(j)} x^{(j)}$ such that $x^{(j)} \in S^T$ , $x^{(1)}, \ldots , x^{(m)}$ are distinct, and $d^{(1)}+\cdots +d^{(m)}=d$ . By the localization theorem in K-theory [Reference TakedaTak94], there is an isomorphism

\begin{align*} G_T\left (\mathcal{M}_{S}(d)\right )_{\mathbb{F}} & \cong G_T\left (\bigsqcup _{p\in {\rm Sym}^d(S)^T}\mathcal{M}_{S}(d)_p\right )_{\mathbb{F}}. \end{align*}

Moreover, we have

\begin{align*} \mathcal{M}_S(d)_p=\prod _{j=1}^m \mathcal{M}_S(d^{(j)})_{d^{(j)}[x^{(j)}]}. \end{align*}

We thus obtain the desired isomorphism.

For $x\in S^T$ , define

\begin{align*}\Phi (d)_{v,x}:=\Phi _{d,v}(\mathcal{O}_x)\in \mathbb{T}_X(d)_v.\end{align*}

We abuse notation and also denote by $\Phi (d)_{v,x}$ its class in K-theory.

Proposition 4.9. (a) Let $x$ and $x'$ be points in $S^T$ , not necessarily distinct, and consider two pairs $(d, v)$ and $(d', v')$ such that $v/d=v'/d'$ . Then

(4.20) \begin{align} \Phi (d)_{v,x}\Phi (d')_{v',x'}=\Phi (d')_{v',x'}\Phi (d)_{v,x}. \end{align}

(b) The $\mathbb{F}$ -vector space $G_T\big (\mathfrak{M}_{S}(d)\big )_{\mathbb{F}}$ has an $\mathbb{F}$ -basis given by equivalences classes of monomials

(4.21) \begin{align} \Phi (d_1)_{v_1,x_1}\cdots \Phi (d_k)_{v_k,x_k}, \end{align}

where $x_i\in S^T$ for $1\leqslant i\leqslant k$ , the inequality $v_1/d_1 \leqslant \cdots \leqslant v_k/d_k$ holds and $\sum _{i=1}^k d_i=d$ , where two monomials ( 4.21 ) are equivalent if they differ by relations ( 4.20 ).

Proof. Let $x\in S^T$ . We argue that the Hall product (4.10) induces an algebra structure on

(4.22) \begin{align} {\rm KHA}_{S, T, x}:=\bigoplus _{d\geqslant 0}G_T(\mathcal{M}_S(d)_{d[x]})_{\mathbb{F}}. \end{align}

First, by the localization theorem in K-theory [Reference TakedaTak94], there are injective maps $G_T(\mathcal{M}_S(d)_{d[x]})_{\mathbb{F}}\hookrightarrow G_T(\mathcal{M}_S(d))_{\mathbb{F}}$ , and thus

\begin{align*}G_T(\mathcal{M}_S(d)_{d[x]})_{\mathbb{F}}={\rm Ker}\left (G_T(\mathcal{M}_S(d))_{\mathbb{F}}\xrightarrow {{\rm res}} G_T(\mathcal{M}_S(d)\setminus \mathcal{M}_S(d)_{d[x]})_{\mathbb{F}} \right )\!.\end{align*}

The following composition of the natural pushforward, of the Hall product and of the natural restriction is a zero map:

\begin{align*} G_T(\mathcal{M}_S(d)_{d[x]})_{\mathbb{F}}\otimes G_T(\mathcal{M}_S(e)_{e[x]})_{\mathbb{F}} & \to G_T(\mathcal{M}_S(d))_{\mathbb{F}}\otimes G_T(\mathcal{M}_S(e))_{\mathbb{F}}\\ & \xrightarrow {\ast }G_T(\mathcal{M}_S(d+e))_{\mathbb{F}} \\ & \to G_T(\mathcal{M}_S(d+e)\setminus \mathcal{M}_S(d+e)_{(d+e)[x]})_{\mathbb{F}}. \end{align*}

There is thus a Hall product

\begin{align*}\ast \colon G_T(\mathcal{M}_S(d)_{d[x]})_{\mathbb{F}}\otimes G_T(\mathcal{M}_S(e)_{e[x]})_{\mathbb{F}}\to G_T(\mathcal{M}_S(d+e)_{(d+e)[x]})_{\mathbb{F}}\end{align*}

and (4.22) is indeed an algebra. The algebras ${\rm KHA}_{S, T, x}$ and ${\rm KHA}_{S, T, x'}$ commute for $x\neq x'\in S^T$ inside ${\rm KHA}_{S, T}$ . Furthermore, by the isomorphism (4.18) and the localization theorem in K-theory [Reference TakedaTak94], we have that

\begin{align*}{\rm KHA}_{S, T, x}\cong {\rm KHA}_{\mathbb{C}^2, T, 0}\cong {\rm KHA}_{\mathbb{C}^2, T}.\end{align*}

Part (a) then follows by [Reference Pădurariu and TodaPT23, Proposition 5.7]. By [Reference Pădurariu and TodaPT24a, (4.31) and (4.36)], the $\mathbb{F}$ -algebra ${\rm KHA}_{\mathbb{C}^2, T, 0}\cong {\rm KHA}_{\mathbb{C}^2, T}$ has an $\mathbb{F}$ -basis with unordered monomials

\begin{align*}\Phi (d_1)_{v_1,0}\ldots \Phi (d_k)_{v_k,0}\end{align*}

for decompositions $\sum _{i=1}^k d_i=d$ and integers $v_i$ for $1\leqslant i\leqslant k$ satisfying the inequality $v_1/d_1 \leqslant \cdots \leqslant v_k/d_k$ . By Lemma 4.8, the conclusion of (b) follows.

Proof of Theorem 1.4. Let $(d', v')\in \mathbb{N}\times \mathbb{Z}$ be coprime integers such that $n(d', v')=(d, v)$ . Using [Reference Pădurariu and TodaPT24a, Lemma 4.8], the Hall product induces an algebra structure on

(4.23) \begin{align} \bigoplus _{k\geqslant 0} K_T(\mathbb{T}_X(kd')_{kv'})_{\mathbb{F}}. \end{align}

By Corollary 4.10, the above algebra has an $\mathbb{F}$ -basis with equivalences classes of monomials $\Phi (k_1d')_{k_1v', x_1}\ldots \Phi (k_rd')_{k_rv', x_r}$ for all partitions $k_1+\cdots +k_r=k$ and choice of points $x_i\in S^T$ for $1\leqslant i\leqslant r$ . Let $a_k:=\dim _{\mathbb{F}} K_T(\mathbb{T}_X(kd')_{kv'})_{\mathbb{F}}$ and let $b_k$ be the number of partitions of $k$ . Then

\begin{align*} \sum _{k\geqslant 0}a_kq^k=\left (\sum _{k\geqslant 0}b_kq^k\right )^{|S^T|} =\left (\prod _{b\geqslant 1}\frac {1}{1-q^b}\right )^{\chi _c(S)} =\sum _{k\geqslant 0}\chi _c({\rm Hilb}(S, k))q^k. \end{align*}

The first equality holds from the description of (4.23), the second equality holds because $|S^T|=\chi _c(S)$ and the known product formula for the number of partitions, and the last equality follows from [Reference GöttscheGö90].

Remark 4.11. Recall Conjecture 1.5. We may also conjecture the analogous statement

\begin{align*}\chi _{{\rm HP}}(\mathbb{T}_X(d)_w)=\chi _c({\rm Hilb}(S, n)),\end{align*}

where the left-hand side is the Euler characteristic of the periodic cyclic homology of $\mathbb{T}_X(d)_w$ , which is two-periodic:

\begin{align*}\chi _{{\rm HP}}(\mathbb{T}_X(d)_w):=\dim _{\mathbb{C}(\!(u)\!)} {\rm HP}_0(\mathbb{T}_X(d)_w)-\dim _{\mathbb{C}(\!(u)\!)} {\rm HP}_1(\mathbb{T}_X(d)_w).\end{align*}

4.6 Singular support of objects in quasi-BPS categories

Let $S$ be a smooth surface, let $X={\rm Tot}_S(K_S)$ and let $d\in \mathbb{N}$ . Let $\mathcal{M}_X(d)$ be the classical moduli stack of zero-dimensional sheaves on $X$ with length $d$ . By [Reference TodaTod24a, Lemma 3.4.1], we have

\begin{align*} \mathcal{M}_X(d)= t_0(\Omega _{\mathfrak{M}_S(d)}[-1]). \end{align*}

In particular, an object $F\in D^b(\mathfrak{M}_{S}(d))$ has a singular support ${\rm Supp}^{\textrm {sg}}(F)\subset \mathcal{M}_{X}(d)$ , see § 2.5.3. Consider the diagram

where $\Delta \colon X\hookrightarrow {\rm Sym}^d(X)$ is the small diagonal map $x\mapsto (x,\ldots , x)$ . Davison [Reference DavisonDav, Theorem 5.1] showed that the BPS sheaf for the moduli stack of degree $d$ sheaves on $\mathbb{C}^3$ is

\begin{align*}\mathcal{B}PS(d)=\Delta _*{\rm IC}_{\mathbb{C}^3}.\end{align*}

In [Reference Pădurariu and TodaPT24a, Theorem 1.1], we proved a categorical version of the above result, namely that the singular support of an object in $\mathbb{T}_{\mathbb{C}^3}(d)_w$ is contained in $\pi _d^{-1}(\mathbb{C}^3)$ . Using the formal local description of $\mathbb{T}_X(d)_w$ and [Reference Pădurariu and TodaPT24a, Theorem 3.1], we obtain the analogous result for $X$ .

Proof. Let $\tau \colon X\to S$ , $\tau \colon \mathbb{C}^3={\rm Tot}_{\mathbb{C}^2}(K_{\mathbb{C}^2})\to \mathbb{C}^2$ be the natural projection maps. Consider the following maps.

(4.24)

We also use the notations $\pi^2_d$ and $\pi^3_d$ for maps of moduli spaces constructed from $\mathbb{C}^3$ . Let $p\in {\rm Sym}^d(S)$ be such that ${\rm Supp}^{\textrm {sg}}(F)\cap (\pi^3_d\tau )^{-1}(p)\neq \emptyset$ . Write $p=\sum _{i=1}^m d^{(i)}x^{(i)}$ for $m\geqslant 1$ , $x^{(i)}\neq x^{(j)}$ points in $S$ for $1\leqslant i, j\leqslant m$ and $d^{(i)}\geqslant 1$ . There is a point $q=\sum _{i=1}^m d^{(i)}y^{(i)}\in {\rm Sym}^d(\mathbb{C}^2)$ for points $y^{(i)}\neq y^{(j)}$ in $\mathbb{C}^2$ for $1\leqslant i\neq j\leqslant m$ , together with the following commutative diagram.

(4.25)

In the above, $\widehat {{\rm Sym}}^d(S)_p$ is the completion of ${\rm Sym}^d(S)$ at $p$ and all the spaces in the front square is a base change of the diagram (4.24) by $\widehat {{\rm Sym}}^d(S)_p \to {\rm Sym}^d(S)$ , and similarly for the back square. The commutativity of the bottom square is classical (see the constructions in § 4.1). The left square is commutative since the isomorphism $\widehat {\mathcal{M}}_S(d)_p \stackrel {\cong }{\to } \widehat {\mathcal{M}}_{\mathbb{C}^2}(d)_q$ extends to an equivalence

(4.26) \begin{align} \widehat {\mathfrak{M}}_S(d)_p \stackrel {\sim }{\to }\widehat {\mathfrak{M}}_{\mathbb{C}^2}(d)_q \end{align}

and we have the induced isomorphisms $\alpha$ on $(-1)$ -shifted cotangent stacks.

The support of sheaves is respected under the isomorphisms in (4.25). Furthermore, (4.26) induces an equivalence

\begin{align*}\mathbb{T}_X(d)_{w, p}\stackrel {\sim }{\to } \mathbb{T}_{\mathbb{C}^3}(d)_{w, q}.\end{align*}

Let $E\in \mathbb{T}_{\mathbb{C}^3}(d)_{w, q}$ be the image of $F|_{\widehat {\mathfrak{M}}_S(d)_p}$ under the above equivalence. Then

\begin{align*}\alpha \left ({\rm Supp}^{\textrm {sg}}(F|_{\widehat {\mathfrak{M}}_S(d)_p})\right )= \alpha \left ({\rm Supp}^{\textrm {sg}}(F) \times _{\mathcal{M}_S(d)} \widehat {\mathcal{M}}_S(d)_p \right )= {\rm Supp}^{\textrm {sg}}(E).\end{align*}

In the proof of [Reference Pădurariu and TodaPT23, Theorem 3.1], we showed that ${\rm Supp}^{\textrm {sg}}(E)\subset (\pi^3_d)^{-1}(\Delta )$ . The equivalence (4.26) respect the singular support, and we thus obtain the desired conclusion.

5.1 DT/PT categories for local surfaces

Let $S$ be a smooth projective surface. Its corresponding local surface is the non-compact Calabi–Yau 3-fold

\begin{align*} X:= {\rm Tot}_S(K_S) \stackrel {\tau }{\to } S. \end{align*}

Here, $\tau$ is the natural projection. Let $\beta \in H_2(S, \mathbb{Z})$ and $n \in \mathbb{Z}$ . Recall the DT moduli space

\begin{align*} I_X(\beta , n) \end{align*}

which parametrizes ideal sheaves $I_C$ for $C \subset X$ compactly supported and with $\dim C \leqslant 1$ , $\tau _{\ast }[C]=\beta$ and $\chi (\mathcal{O}_C)=n$ . Recall also the PT moduli space

\begin{align*} P_X(\beta , n) \end{align*}

of PT stable pairs $(\mathcal{O}_X \stackrel {s}{\to } F)$ , where $F$ is a compactly supported pure one-dimensional sheaf on $X$ and $s$ has at most zero-dimensional cokernel satisfying $\tau _{\ast }[F]=\beta$ and $\chi (F)=n$ , where $[F]$ is the fundamental one-cycle associated with $F$ .

We denote by $\mathfrak{M}_S^{\dagger }(\beta , n)$ the derived moduli stack of pairs $(F, s)$ , where $F \in \textrm {Coh}_{\leqslant 1}(S)$ has one-dimensional support $\beta$ , $\chi (F)=n$ and $s \colon \mathcal{O}_S \to F$ is a morphism. We refer to [Reference TodaTod24a, § 4.1.1] for the derived structure on $\mathfrak{M}_S^{\dagger }(\beta , n)$ . The derived stack $\mathfrak{M}_S^{\dagger }(\beta , n)$ is quasi-smooth, and its $(-1)$ -shifted cotangent stack

\begin{align*} \mathcal{M}_X^{\dagger }(\beta , n) :=t_0(\Omega _{\mathfrak{M}_S^{\dagger }(\beta , n)}[-1]) \stackrel {\tau _{\ast }}{\to } \mathfrak{M}_S^{\dagger }(\beta , n) \end{align*}

is isomorphic to the moduli stack of objects in the extension closure

\begin{align*} \mathcal{A}_X=\langle \mathcal{O}_X, \textrm {Coh}_{\leqslant 1}(X)[-1] \rangle _{\textrm {ex}}, \end{align*}

see [Reference TodaTod24a, Theorem 4.1.3.(ii)]. The category $\mathcal{A}_X$ is called the category of D0-D2-D6 bound states in [Reference TodaTod24a, § 4.1.2]. In particular, there are open immersions

\begin{align*} I_X(\beta , n) \subset \mathcal{M}_X^{\dagger }(\beta , n) \supset P_X(\beta , n). \end{align*}

In general, the stack $\mathfrak{M}_S^{\dagger }(\beta , n)$ is too big, for example it is not of finite type. Let $\mathfrak{M}_S^{\dagger }(\beta , n)_{\textrm {fin}} \subset \mathfrak{M}_S^{\dagger }(\beta , n)$ be an open substack of finite type which contains both of $\tau _{\ast }(I_X(\beta , n))$ and $\tau _{\ast }(P_X(\beta , n))$ . Then there are open immersions

\begin{align*} I_X(\beta , n) \subset t_0(\Omega _{\mathfrak{M}_S^{\dagger }(\beta , n)_{\textrm {fin}}}[-1]) \supset P_X(\beta , n). \end{align*}

Let $\mathcal{Z}_I, \mathcal{Z}_P$ be the complements of the above open immersions. Following the construction in § 2.5.4, the DT/PT categories are defined as follows.

Definition 5.1 [Reference TodaTod24a, Definition 4.2.1]. The DT/PT categories are defined to be the following singular support quotients:

\begin{align*} & \mathcal{DT}_X(\beta , n):= D^b(\mathfrak{M}_S^{\dagger }(\beta , n)_{\textrm {fin}})/\mathcal{C}_{\mathcal{Z}_I}, \\ & \mathcal{PT}_X(\beta , n):= D^b(\mathfrak{M}_S^{\dagger }(\beta , n)_{\textrm {fin}})/\mathcal{C}_{\mathcal{Z}_P}. \end{align*}

5.2 The moduli of pairs for reduced curve classes

Suppose that $\beta$ is a reduced class, i.e. any effective divisor on $S$ with class $\beta$ is a reduced divisor. In this case, we can take $\mathfrak{M}_S^{\dagger }(\beta , n)_{\textrm {fin}}$ which admits a good moduli space as we explain below. Let

\begin{align*} \mathfrak{T}_S(\beta , n) \subset \mathfrak{M}_S^{\dagger }(\beta , n) \end{align*}

be the open substack consisting of $(F, s)$ such that $s$ has at most zero-dimensional cokernel. If $\beta$ is reduced, then both of $\tau _{\ast }(I_n(X, \beta ))$ and $\tau _{\ast }(P_n(X, \beta ))$ are contained in $\mathfrak{T}_S(\beta , n)$ . Indeed, when $\beta$ is a reduced curve class, by [Reference TodaTod24a, Lemma 5.5.4] we have that

\begin{align*} \mathcal{T}_X(\beta , n):=t_0(\Omega _{\mathfrak{T}_S(\beta , n)}[-1]) \end{align*}

is isomorphic to the moduli stack of pairs $(\mathcal{O}_X \stackrel {s}{\to } E)$ , where $E \in \textrm {Coh}_{\leqslant 1}(X)$ is compactly supported and $s$ has at most zero-dimensional cokernel. Moreover, the stack $\mathfrak{T}_S(\beta , n)$ is of finite type, as the pairs $(F, s)$ with zero-dimensional cokernel are bounded. Therefore, if $\beta$ is reduced, we can take

(5.1) \begin{align} \mathfrak{M}_S^{\dagger }(\beta , n)_{\textrm {fin}}=\mathfrak{T}_S(\beta , n). \end{align}

In what follows, we take $\beta$ to be reduced and $\mathfrak{M}_S^{\dagger }(\beta , n)_{\textrm {fin}}$ to be (5.1).

Let $\mathcal{T}_S(\beta , n)$ be the classical truncation of $\mathfrak{T}_S(\beta , n)$ . As discussed in [Reference TodaTod24a, § 4.2.1, 6.3.2], the stack $\mathcal{T}_S(\beta , n)$ admits a good moduli space

(5.2) \begin{align} \pi _S\colon \mathcal{T}_S(\beta , n) \to T_S(\beta , n) ={\rm Chow}^{\beta }(S) \times {\rm Sym}^d(S). \end{align}

In the above, ${\rm Chow}^{\beta }(S)$ is the moduli space of effective divisors in $S$ with class $\beta$ (which is a stratified projective bundle over ${\rm Pic}^{\beta }(S)$ ), and $d$ is determined by

(5.3) \begin{align} -\frac {1}{2}\beta^2-\frac {1}{2}K_S \beta +d=n. \end{align}

The first two terms correspond to $\chi (\mathcal{O}_C)$ for $[C] \in {\rm Chow}^{\beta }(S)$ . The morphism (5.2) is given by

\begin{align*} (\mathcal{O}_S \stackrel {s}{\to } F) \mapsto ({\rm Supp}(F), {\rm Supp}(F_{\textrm {tor}}) +{\rm Supp}({\rm Cok}(s))), \end{align*}

where $F_{\textrm {tor}} \subset F$ is the maximal zero-dimensional subsheaf.

For $y=(C, p) \in T_S(\beta , n)$ with $p=\sum _{j=1}^m d^{(j)}x^{(j)}$ such that $x^{(1)}, \ldots , x^{(m)} \in S$ are mutually distinct, the unique closed point in the fiber of (5.2) at $y$ corresponds to a direct sum

(5.4) \begin{align} I=(\mathcal{O}_S \to F)=(\mathcal{O}_S \twoheadrightarrow \mathcal{O}_C) \oplus \bigoplus _{j=1}^m V^{(j)} \otimes (0 \to \mathcal{O}_{x^{(j)}}), \end{align}

where $V^{(j)}$ is a finite dimensional vector space with $\dim V^{(j)}=d^{(j)}$ .

5.3 Window categories for DT/PT categories

5.3.1 Construction of window categories.

Let $\beta$ be a reduced class. Consider the universal pair

\begin{align*} \mathcal{I}=(\mathcal{O}_{S\times \mathcal{T}_S(\beta ,n)}\to \mathcal{F}) \in D^b(S \times \mathcal{T}_S(\beta , n)). \end{align*}

Denote by $\pi _{\mathcal{T}}\colon S\times \mathcal{T}_S(\beta ,n)\to \mathcal{T}_S(\beta ,n)$ the projection onto the second factor. Let

\begin{align*}\mathscr{L}:=\det \left (Rp_{\mathcal{T}*}\mathcal{F}\right )\in {\rm Pic}(\mathcal{T}_S(\beta ,n)).\end{align*}

By [Reference TodaTod24a, Proposition 5.5.2], we have that

\begin{align*}I_X(\beta , n)=\mathcal{T}_X(\beta , n)^{\mathscr{L}{\rm -ss}},\, P_X(\beta , n)=\mathcal{T}_X(\beta , n)^{\mathscr{L}^{-1}{\rm -ss}}.\end{align*}

We further take

\begin{align*} b:=2\textrm {ch}_2(Rp_{\mathcal{T}*}\mathcal{F}) \in H^4(\mathcal{T}_S(\beta , n), \mathbb{Q}). \end{align*}

Its pullback to $H^4(\mathcal{T}_X(\beta , n), \mathbb{Q})$ is also denoted by $b$ and it is positive definite, see the argument of [Reference TodaTod, Lemma 7.25]. Then $(\mathcal{L}, b)$ determine $\Theta$ -stratifications

(5.5) \begin{align} \mathcal{T}_X(\beta , n) & =\mathcal{S}_0^I \sqcup \cdots \sqcup \mathcal{S}_d^I \sqcup I_X(\beta , n) \\ \notag & =\mathcal{S}_0^P \sqcup \cdots \sqcup \mathcal{S}_d^P \sqcup P_X(\beta , n), \end{align}

where $\mathcal{S}_i^I$ and $\mathcal{S}_i^P$ are given by

\begin{align*} & \mathcal{S}_i^I=\{(\mathcal{O}_X \stackrel {s}{\to } F) : \chi ({\rm Cok}(s))=d-i\}, \\ & \mathcal{S}_i^P=\{(\mathcal{O}_X \stackrel {s}{\to } F) : \chi (F_{\textrm {tor}})=d-i\}, \end{align*}

where $d$ is given by (5.3). The same stratifications are constructed in [Reference TodaTod24a, § 6.2.1] without using $b$ , and it is straightforward to show that they are $\Theta$ -stratifications with respect to $(\mathcal{L}, b)$ .

Since $\mathcal{T}_S(\beta , n)$ admits a good moduli space, we can apply Theorem2.3 to obtain the following window theorem for DT/PT categories (which is also obtained in [Reference TodaTod24a, Theorem 1.4.6]).

Theorem 5.3 [Reference TodaTod24a, Theorem 1.4.6]. For each maps $k_I, k_P \colon \mathbb{Z}_{\geqslant 1} \to \mathbb{R}$ , there exist dg-subcategories

(5.6) \begin{align} \mathbb{W}_I \subset D^b(\mathfrak{T}_S(\beta , n)), \mathbb{W}_P \subset D^b(\mathfrak{T}_S(\beta , n)) \end{align}

such that the following composition functors are equivalences

\begin{align*} & \mathbb{W}_I \hookrightarrow D^b(\mathfrak{T}_S(\beta , n)) \twoheadrightarrow \mathcal{DT}_X(\beta , n) \\ & \mathbb{W}_P \hookrightarrow D^b(\mathfrak{T}_S(\beta , n)) \twoheadrightarrow \mathcal{PT}_X(\beta , n). \end{align*}

Remark 5.4. In [Reference TodaTod24a, § 6], the existence of stratifications (5.5) and Theorem 5.3 are proved without using $b$ , so in the arguments below the use of $b$ is not relevant.

5.3.2 The formal local descriptions of window categories.

The subcategories (5.6) are characterized in terms of Koszul duality equivalences on each formal fiber along with the good moduli space $\mathcal{T}_S(\beta , n)\to T_S(\beta , n)$ , see (5.16), (5.18) and Proposition 5.7. Below we make it more explicit. Let $y \in T_S(\beta , n)$ be a closed point represented by a direct sum (5.4). We also denote by $y \in \mathcal{T}_S(\beta , n)$ the corresponding closed point. Then

\begin{align*} G_y := {\rm Aut}(y)=\prod _{j=1}^m \textrm {GL}(V^{(j)}), \ T_y=\prod _{j=1}^m T(d^{(j)}) \subset G_y \end{align*}

where $T_y$ is the maximal torus of $G_y$ . The derived stack $\mathfrak{T}_S(\beta , n)$ satisfies the formal neighborhood theorem, see [Reference TodaTod24a, Lemma 7.4.3]. Moreover, we have

(5.7) \begin{align} \mathcal{H}^i(\mathbb{T}_{\mathfrak{T}_S(\beta , n)}|_{y})=\textrm {Hom}^i_S(I, F) \end{align}

for every $i\in \mathbb{Z}$ . Here, we regard $I$ as a two term complex $(\mathcal{O}_S \to F)$ with $\mathcal{O}_S$ located in degree zero, and $\textrm {Hom}$ denotes morphisms in the derived category $D^b(S)$ . Then the formal fiber $\widehat {\mathfrak{T}}_S(\beta , n)_y$ of $\mathfrak{T}_S(\beta , n)$ at $y \in T_S(\beta , n)$ is a quotient stack by the $G_y$ -action of the derived zero locus of a $G_y$ -equivariant morphism

\begin{align*} \kappa _y \colon \widehat {\textrm {Hom}}_S(I, F) \to \textrm {Hom}^1_S(I, F). \end{align*}

We set

\begin{align*} \mathcal{X}_y:=\left (\widehat {\textrm {Hom}}_S(I, F) \oplus \textrm {Hom}^1_S(I, F)^{\vee }\right )/G_y. \end{align*}

Consider the following regular function

\begin{align*} f_y \colon \mathcal{X}_y \to \mathbb{C}, \ (x, u) \mapsto \langle \kappa _y(x), u \rangle . \end{align*}

The Koszul duality in Theorem2.1 gives the equivalence

(5.8) \begin{align} \Phi _y \colon D^b(\widehat {\mathfrak{T}}_S(\beta , n)_y) \stackrel {\sim }{\to } {\rm MF}^{{\rm gr}}(\mathcal{X}_y, f_y). \end{align}

The following lemma shows that the stack $\mathcal{X}_y$ is a completion of the product of the stack of representations of DT/PT quivers considered in [Reference Pădurariu and TodaPT24b].

Lemma 5.5. As a $G_y$ -representation, we have

\begin{align*} & \textrm {Hom}_S(I, F) \oplus \textrm {Hom}^1_S(I, F)^{\vee } \\[5pt] & =H^0(\mathcal{O}_C(C)) \oplus H^1(\mathcal{O}_C(C))^{\vee } \oplus \bigoplus _{j=1}^m\left ( (V^{(j)})^{\oplus a^{(j)}+1} \oplus (V^{(j)\vee })^{\oplus a^{(j)}} \oplus {\rm End}(V^{(j)})^{\oplus 3}\right ) \end{align*}

where $a^{(j)}=1$ if $x^{(j)} \in C$ and $a^{(j)}=0$ if $x^{(j)} \notin C$ and where $G_y$ acts trivially on the first two summands and acts naturally on the other summands.

Proof. Note that from (5.4) we have

\begin{align*} F=\mathcal{O}_C \oplus \bigoplus _{j=1}^m V^{(j)} \otimes \mathcal{O}_{x^{(j)}}. \end{align*}

By direct calculation, we have

(5.9) \begin{align} \textrm {Hom}_S(I, F)= & H^0(\mathcal{O}_C(C))\oplus \\ \notag & \bigoplus _{j=1}^m\big( V^{(j)} \oplus (V^{(j)\vee } \otimes \textrm {Hom}_S^1(\mathcal{O}_{x^{(j)}}, \mathcal{O}_C)) \oplus {\rm End}(V^{(j)})^{\oplus 2}\big), \\ \notag \textrm {Hom}^1_S(I, F)= & H^1(\mathcal{O}_C(C))\oplus \\ \notag & \bigoplus _{j=1}^m\big((V^{(j)\vee } \otimes \textrm {Hom}_S^2(\mathcal{O}_{x^{(j)}}, \mathcal{O}_C)) \oplus {\rm End}(V^{(j)})\big). \end{align}

Since $C$ is a reduced divisor, we have

\begin{align*} \textrm {Hom}_S^1(\mathcal{O}_{x^{(j)}}, \mathcal{O}_C)= \textrm {Hom}_S^2(\mathcal{O}_{x^{(j)}}, \mathcal{O}_C)=\begin{cases} \mathbb{C}, & x^{(j)} \in C, \\ 0, & x^{(j)} \notin C. \end{cases} \end{align*}

We denote by $\chi _0^{(j)}$ and $\chi _0$ the determinant characters

(5.10) \begin{align} \chi _0^{(j)} \colon \textrm {GL}(V^{(j)}) \to \mathbb{C}^{\ast }, \ g^{(j)} \mapsto \det (g^{(j)}), \ \chi _0=\prod _{j=1}^m \chi _0^{(j)} \colon G_y \to \mathbb{C}^{\ast }. \end{align}

The data $(\mathcal{L}, b)$ pulled back to ${\rm Crit}(f_y)$ is induced from $BG_y$ . By writing it as $(\mathcal{L}_y, b_y)$ , we have $\mathcal{L}_y=\chi _0$ and $b_y \in H^4(BG_y, \mathbb{Q})$ corresponds to the Weyl-invariant norm on the set of cocharacters $\lambda =\prod _{j=1}^m (\lambda _i^{(j)})_{i=1}^{d^{(j)}}$ of $\prod _{j=1}^m T(d^{(j)})$ given by

(5.11) \begin{align} \lvert \lambda \rvert^2=\sum _{j=1}^m \sum _{i=1}^{d^{(j)}} \left(\lambda _i^{(j)}\right)^2. \end{align}

We have the Kempf–Ness stratifications

(5.12) \begin{align} \mathcal{X}_y=\mathscr{S\;}_1^{\pm } \sqcup \cdots \sqcup \mathscr{S\;}_{N^{\pm }}^{\pm } \sqcup \mathcal{X}_y^{\chi _0^{\pm }-\textrm {ss}} \end{align}

of $\mathcal{X}_y$ with respect to $\chi _0^{\pm 1}$ and a Weyl-invariant norm (5.11), see [Reference TodaTod24a, § 5.1.5]. The restrctions of the above stratifications to ${\rm Crit}(f_y)$ coincides with the restrictions of (5.5). We note that the restrictions of (5.5) to ${\rm Crit}(f_y)$ may be decomposed into several connected components, and these components are distinguished in (5.12), so $N^{\pm }$ may be larger than $d$ . Let $\lambda _i^{\pm } \colon \mathbb{C}^{\ast } \to G_y$ be the associated cocharacter and $\mathscr{Z\;}_i^{\pm } \subset \mathscr{S\;}_i^{\pm }$ the center of $\mathscr{S\;}_i^{\pm }$ . By [Reference TodaTod24a, Lemma 5.1.9], for each $1\leqslant i\leqslant N^{\pm }$ and $1\leqslant j\leqslant m$ , there is a decomposition

(5.13) \begin{align} V^{(j)}=V_1^{(j)} \oplus V_2^{(j)}, \ 1\leqslant j\leqslant m, \end{align}

such that $\lambda _i^{\pm }$ is the cocharacter

\begin{align*} \lambda _i^{\pm }(t)=\{(\textrm {id}, t^{\mp 1} \textrm {id})\}_{1\leqslant j\leqslant m} \subset \prod _{j=1}^m \textrm {GL}\left(V_1^{(j)}\right) \times \textrm {GL}\left(V_2^{(j)}\right) \subset G_y. \end{align*}

Note that the decompositions (5.13) depend on $i$ . We set

(5.14) \begin{align} I_1=(\mathcal{O}_S \to F_1) & := (\mathcal{O}_S \twoheadrightarrow \mathcal{O}_C) \oplus \bigoplus _{j=1}^{m}V_1^{(j)} \otimes (0 \to \mathcal{O}_{x^{(j)}}), \\ \notag F_2 & :=\bigoplus _{j=1}^m V_2^{(j)} \otimes \mathcal{O}_{x^{(j)}}. \end{align}

Then we have the decompositions

(5.15) \begin{align} I=I_1 \oplus (0 \to F_2), \ F=F_1 \oplus F_2. \end{align}

We also set $n_2 :=\chi (F_2)=\sum _{j=1}^m \dim V_2^{(j)}$ and let

\begin{align*} O_{12}:= \textrm {Hom}^1_S(I_1, F_2), \ O_{21} := \textrm {Hom}^2_S(F_2, F_1). \end{align*}

Define the widths $\eta _i^{\pm }$ of window categories by

\begin{align*} \eta _i^{\pm }:=\langle \lambda _i^{\pm }, N^{\vee }_{\mathscr{S\;}_i^{\pm }/\mathcal{X}_y}|_{\mathscr{Z\;}_i^{\pm }}\rangle \in \mathbb{Z}. \end{align*}

We define the subcategories

\begin{align*} \mathbb{V}_{I, y} \subset {\rm MF}^{{\rm gr}}(\mathcal{X}_y, f_y), \ \mathbb{V}_{P, y} \subset {\rm MF}^{{\rm gr}}(\mathcal{X}_y, f_y) \end{align*}

to be consisting of objects $\mathcal{F}$ such that for each center $\mathscr{Z\;}_i^{\pm }$ we have

(5.16) \begin{align} & {\rm wt}_{\lambda _i^+}(\mathcal{F}|_{\mathscr{Z\;}_i^+}) \subset [ -k_I(n_2)-\dim O_{21}, -k_I(n_2)-\dim O_{21}+\eta _i^+), \\ \notag & {\rm wt}_{\lambda _i^-}(\mathcal{F}|_{\mathscr{Z\;}_i^-}) \subset [ k_P(n_2)-\dim O_{12}, k_P(n_2)-\dim O_{12}+\eta _i^-). \end{align}

The widths $\eta ^{\pm }_i$ are computed as follows.

Lemma 5.6. We set $n_1^{(j)}:= \dim V_1^{(j)}$ and $n_2^{(j)}:= \dim V_2^{(j)}$ . We have the equalities

(5.17) \begin{align} \eta _i^+ & = \hom (I_1, F_2)+\hom^2(F_2, F_1)-\hom ^{-1}(I_1, F_2) \\ \notag & = \sum _{j=1}^m n_2^{(j)}(a^{(j)}+1+2n_1^{(j)}) \\ \notag \eta _i^- & = \hom^1(F_2, F_1)+\hom^1(I_1, F_2)-\hom (F_2, F_1) \\ \notag & = \sum _{j=1}^m n_2^{(j)}(a^{(j)}+2n_1^{(j)}). \end{align}

Proof. By the definition of Kempf–Ness stratifications, we have

\begin{align*} \eta _i^{\pm } & =\Big\langle \lambda _i^{\pm }, N^{\vee }_{\mathscr{S\;}_i^{\pm }/\mathcal{X}_y}|_{\mathscr{Z\;}_i^{\pm }}\rangle =\langle \lambda _i^{\pm }, \mathbb{L}_{\mathcal{X}_y}^{\lambda _i^{\pm }\gt 0}\Big\rangle \\ & =\Big\langle \lambda _i^{\pm }, (\textrm {Hom}_S(I, F)^{\vee }+\textrm {Hom}^1_S(I, F)-\textrm {Hom}^{-1}_S(I, F)^{\vee })^{\lambda _i^{\pm }\gt 0} \Big\rangle , \end{align*}

where the third identity holds from the definition of $\mathcal{X}_y$ as a quotient stack. By substituting the decompositions (5.15) and taking account of the $\lambda _i^{\pm }$ -weights, we obtain the first identities for $\eta ^{\pm }_i$ . The second identities follow by substituting the decompositions (5.14) and straightforward calculations.

Recall the Koszul duality equivalence (5.8). For $\ast \in \{I, P\}$ , we define

(5.18) \begin{align} \mathbb{W}_{\ast , y} :=\Phi _y^{-1}(\mathbb{V}_{\ast , y}) \subset D^b(\widehat {\mathfrak{T}}_S(\beta , n)_y). \end{align}

We have the following characterizations of the subcategories (5.6) in terms of formal fibers.

Proposition 5.7. An object $\mathcal{E} \in D^b(\mathfrak{T}_S(\beta , n))$ lies in $\mathbb{W}_I$ (respectively $\mathbb{W}_P)$ if and only if for any closed point $y \in T_S(\beta , n)$ , we have

\begin{align*} \mathcal{E}|_{\widehat {\mathfrak{T}}_S(\beta , n)_y} \in \mathbb{W}_{I, y}, \quad (\mbox{respectively } \mathcal{E}|_{\widehat {\mathfrak{T}}_S(\beta , n)_y} \in \mathbb{W}_{P, y}). \end{align*}

Proof. The proposition is proved in [Reference TodaTod24a, Theorem 6.3.13]. It is also a direct application of (2.19). From (5.7) and (5.15), we have

\begin{align*} \mathcal{H}^1(\mathbb{T}_{\mathfrak{T}_S(\beta , n)}|_{y})= \textrm {Hom}^1_S(I_1, F_1) \oplus \textrm {Hom}^1_S(I_1, F_2) \oplus \textrm {Hom}^2_S(F_2, F_1) \oplus \textrm {Hom}^2_S(F_2, F_2). \end{align*}

Therefore we have

\begin{align*} \mathcal{H}^1(\mathbb{T}_{\mathfrak{T}_S(\beta , n)}|_{y})^{\lambda _i^+\gt 0}=O_{21}, \ \mathcal{H}^1(\mathbb{T}_{\mathfrak{T}_S(\beta , n)}|_{y})^{\lambda _i^-\gt 0}=O_{12}. \end{align*}

Then the proposition follows from (2.19) and (5.16).