# The categorical DT/PT correspondence and quasi-BPS categories for local
surfaces

Tudor Pădurariu and Yukinobu Toda

###### Abstract.

We construct semiorthogonal decompositions of Donaldson-Thomas (DT) categories
for reduced curve classes on local surfaces into products of quasi-BPS
categories and Pandharipande-Thomas (PT) categories, giving a categorical
analogue of the numerical DT/PT correspondence for Calabi-Yau 3-folds. The
main ingredient is a categorical wall-crossing formula for DT/PT quivers
(which appear as Ext-quivers in the DT/PT wall-crossing) proved in our
previous paper. We also study quasi-BPS categories of points on local surfaces
and propose conjectural computations of their K-theory analogous to formulas
already known for the three dimensional affine space.

##  1\. Introduction

###  1.1. The DT/PT correspondence for Calabi-Yau 3-folds

For a Calabi-Yau 3-fold $X$, the DT/PT correspondence is a formula which
relates rank one Donaldson-Thomas (DT) invariants counting one-dimensional
closed subschemes in $X$ with Pandharipande-Thomas (PT) invariants counting
stable pairs on $X$. The PT invariants provide a geometric description of the
contribution of sheaf counting invariants in the
Maulik–Nekrasov–Okounkov–Pandharipande conjecture [MNOP06]. More precisely,
for a curve class $\beta\in H_{2}(X,\mathbb{Z})$ and $n\in\mathbb{Z}$, we
denote by

(1.1) $\displaystyle I_{X}(\beta,n),\ (\mbox{resp.~{}}P_{X}(\beta,n))$

the moduli space of (compactly supported) closed subschemes $C\subset X$ with
$[C]=\beta$, $\chi(\mathcal{O}_{C})=n$, (resp. stable pairs $(F,s)$ with
$[F]=\beta$, $\chi(F)=n$). By taking the integration of the Behrend
constructible functions on the spaces (1.1), we obtain the DT invariant
$\mathrm{DT}_{\beta,n}\in\mathbb{Z}$ and the PT invariant
$\mathrm{PT}_{\beta,n}\in\mathbb{Z}$. The DT/PT correspondence is the formula

(1.2) $\text{DT}_{\beta,n}=\sum_{k\geqslant
0}\text{DT}_{0,k}\cdot\text{PT}_{\beta,n-k}.$

The above formula was conjectured in [PT09], its unweighted version was proved
in [Tod10, ST11] using Joyce’s motivic Hall algebra machinery [Joy06, Joy07a,
Joy07b, Joy08], and its version incorporating the Behrend function was proved
in [Bri11] (also see [Tod20a]) using the work of Joyce-Song [JS12] (see also
[KS]).

The purpose of this paper is to prove a categorical version of (1.2) for $X$ a
local surface, see Theorem 1.1. In [Toda], the second author defined dg-
categories categorifying DT and PT invariants for local surfaces, which are
total spaces of canonical line bundles on surfaces. For reduced curve classes,
we construct semiorthogonal decompositions of DT categories with summands
which are products of quasi-BPS categories on points on $X$ and PT categories.
By taking the Grothendieck groups of the categories appearing in these
semiorthogonal decompositions, we obtain a K-theoretic version of (1.2).

###  1.2. The categorical DT/PT correspondence

Let $S$ be a smooth projective surface. The local surface is the following
non-compact Calabi-Yau 3-fold

$\displaystyle X:=\mathrm{Tot}_{S}(K_{S}).$

Take $\beta\in H_{2}(S,\mathbb{Z})$ and $n\in\mathbb{Z}$. In [Toda, Definition
4.2.1], the second author defined dg-categories

(1.3) $\mathcal{DT}_{X}(\beta,n)\text{ and }\mathcal{PT}_{X}(\beta,n).$

These categories may be regarded as a gluing of categories of graded matrix
factorizations of super-potentials whose critical loci are locally isomorphic
to $I_{X}(\beta,n)$ and $P_{X}(\beta,n)$, respectively. The dg-categories
(1.3) should recover DT/PT invariants by taking their periodic cyclic
homologies, see [Efi18, Theorem 1.1], [Toda, Subsection 3.3], especially
[Toda, Conjecture 3.3.4]. The DT/PT categories (1.3) are constructed as
quotients of $D^{b}(\mathfrak{M}^{{\dagger}})$, where
$\mathfrak{M}^{{\dagger}}$ is a certain derived open substack of the moduli
stack of pairs $(F,s)$, where $F\in\operatorname{Coh}(S)$ is at most one-
dimensional and $s\colon\mathcal{O}_{S}\to F$ is a section, by the subcategory
of objects whose singular supports [AG15] are contained in unstable loci. We
will review their constructions in Subsections 2.5.4 and 5.1.

We say that $\beta$ is a reduced class if any effective divisor on $S$ with
class $\beta$ is a reduced divisor. In this case, the second author
constructed a fully-faithful functor from the PT category to the DT category,
see [Toda, Theorem 1.4.5]. The main theorem of this paper is the following,
which refines the above fully-faithful functor to a semiorthogonal
decomposition:

######  Theorem 1.1.

Suppose that $\beta$ is a reduced class. Then there is a semiorthogonal
decomposition

$\displaystyle\mathcal{DT}_{X}(\beta,n)=\left\langle\bigotimes_{i=1}^{k}\mathbb{T}_{X}(d_{i})_{v_{i}}\otimes\mathcal{PT}_{X}(\beta,n^{\prime})\,\Big{|}-1<\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}\leqslant
0\right\rangle,$

where the right hand side is after all partitions
$d_{1}+\cdots+d_{k}+n^{\prime}=n$.

There is also a version for arbitrary classes $\beta$ and for the DT/PT
categories of reduced supports over $S$, see Remark 5.10. In the above,
$\mathbb{T}_{X}(d)_{w}$ is the quasi-BPS category of $d$ points and weight $w$
of $X$, see Subsection 1.3 for more details.

The main ingredient in the proof of Theorem 1.1 is the local categorical DT/PT
correspondence for DT/PT quivers proved in [PTc, Theorem 1.1]. The wall
crossing between ideal sheaves and stable pairs (on an arbitrary Calabi-Yau
$3$-fold) can be described using Ext quivers with super-potential [Tod18], and
all these Ext quivers can be constructed from DT/PT quivers, see Subsection
3.1. We also make use of the Koszul equivalence [Isi13] between coherent
sheaves on a Koszul (quasi-smooth) stack and a category of matrix
factorizations, the window categories of Segal [Seg11], Halpern-Leistner
[HL15], and Ballard–Favero–Katzarkov [BFK19], the noncommutative resolutions
of Špenko–Van den Bergh [ŠVdB17], and categorical Hall algebras of surfaces
[PS] and of quivers with potential [Păda].

We expect that the semiorthogonal decomposition in Theorem 1.1 also holds for
an arbitrary class $\beta$. The reason we restrict to a reduced class is that,
in this case, one can take $\mathfrak{M}^{{\dagger}}$ to be the derived moduli
stack $\mathfrak{T}_{S}(\beta,n)$ of pairs $(F,s)$ such that $s$ has at most
zero-dimensional cokernel. Then its classical truncation
$\mathcal{T}_{S}(\beta,n)$ admits a good moduli space

$\mathcal{T}_{S}(\beta,n)\to T_{S}(\beta,n),$

and we can reduce the statement to be proved to the analogous formal local
statements on $T_{S}(\beta,n)$. For an arbitrary class $\beta$, we cannot find
such a $\mathfrak{M}^{{\dagger}}$ admitting a good moduli space in general.
Alternatively, we expect an argument reducing to a formal local statement on
$T_{X}(\beta,n)$, but the local structure of the categories (1.3) over
$T_{X}(\beta,n)$ is hard to investigate and we can prove Theorem 1.1 only for
reduced classes at this moment.

For $\beta=0$, we obtain the following semiorthogonal decomposition of the DT
category of points on $X$ in quasi-BPS categories:

######  Corollary 1.2.

Let $n\in\mathbb{N}$. There is a semiorthogonal decomposition

(1.4)
$\displaystyle\mathcal{DT}_{X}(0,n)=\left\langle\bigotimes_{i=1}^{k}\mathbb{T}_{X}(d_{i})_{v_{i}}\,\Big{|}-1<\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}\leqslant
0\right\rangle,$

where the right hand side contains all partitions $d_{1}+\cdots+d_{k}=n$.

We proved Corollary 1.2 for $\mathbb{C}^{3}$ in [PTa, Theorem 1.1]. We explain
in Subsection 1.3 that Corollary 1.2 is a categorical analogue of the
computation of DT invariants of points on a $3$-fold of Behrend–Fantechi
[BF08]. By taking the Grothendieck groups of the categories in Theorem 1.1 and
Corollary 1.2, we obtain an analogue of (1.2) in K-theory:

######  Corollary 1.3.

There is an isomorphism

(1.5) $K(\mathcal{DT}_{X}(\beta,n))\cong\bigoplus_{n^{\prime}\leqslant
n}K\left(\mathcal{DT}_{X}(0,n-n^{\prime})\otimes\mathcal{PT}_{X}(\beta,n^{\prime})\right).$

###  1.3. Quasi-BPS categories

Let $S$ be a smooth (not necessarily projective) surface. For the local
surface $X=\mathrm{Tot}_{S}(K_{S})$, the first author defined quasi-BPS
categories $\mathbb{T}_{X}(d)_{w}$ of $d$ points and weight $w$ in [Pădb]. We
briefly explain the relation of these categories to (numerical, cohomological)
BPS (also called Gopakumar-Vafa) invariants.

There are semiorthogonal decompositions of the Hall algebra of points on a
surface of Porta–Sala [PS] with summands which are products of quasi-BPS
categories [Pădb, Theorem 1.1]. This is a categorical analogue of the
Davison–Meinhardt PBW theorem for the cohomological Hall algebra (CoHA) of a
quiver with potential [DM20] which says that the CoHA is generated by the
(cohomological) BPS spaces of the quiver with potential.

The numerical BPS invariants are expected to be fundamental enumerative
invariants which determine DT/PT invariants and Gromow-Witten invariants
[PT14, Section 2 and a half]. The BPS invariants of $d$ points on $X$ are
given by

(1.6) $\Omega_{X,d}=-\chi_{c}(S),$

where $\chi_{c}(S)$ is the Euler characteristic of the compactly supported
cohomology of $S$. It is an interesting problem to compute the K-theory of
quasi-BPS categories. In [PTa], we discussed two K-theoretic computations for
quasi-BPS categories of $\mathbb{C}^{3}$: of localized equivariant K-theory
(with respect to the two dimensional Calabi-Yau torus of $\mathbb{C}^{3}$) and
of topological K-theory.

Assume $S$ is a toric surface. Let $T$ be the two dimensional torus acting on
$S$. Let $\mathbb{K}:=K_{0}(BT)$, $\mathbb{F}:=\mathrm{Frac}\,\mathbb{K}$, and
$K_{T}\left(\mathbb{T}_{X}(d)_{w}\right)_{\mathbb{F}}:=K_{T}\left(\mathbb{T}_{X}(d)_{w}\right)\otimes_{\mathbb{K}}\mathbb{F}$.
The following is the analogue of [PTa, Theorem 4.12]:

######  Theorem 1.4.

Let $S$ be a toric surface, let $X=\mathrm{Tot}_{S}(K_{S})$, let
$(d,w)\in\mathbb{N}\times\mathbb{Z}$, and let $n=\gcd(d,w)$. Then

$\dim_{\mathbb{F}}K_{T}\left(\mathbb{T}_{X}(d)_{w}\right)_{\mathbb{F}}=\chi_{c}(\mathrm{Hilb}(S,n)).$

We next discuss topological K-theoretic computations. For a dg-category
$\mathcal{D}$, consider its topological K-theory
$K_{i}^{\rm{top}}(\mathcal{D})$ defined by Blanc [Bla16], which is two-
periodic by its construction [Bla16, Definition 1.6]. Let

$\chi_{K}(\mathbb{T}_{X}(d)_{w}):=\dim_{\mathbb{Q}}K_{0}^{\mathrm{top}}(\mathbb{T}_{X}(d)_{w})_{\mathbb{Q}}-\dim_{\mathbb{Q}}K_{1}^{\mathrm{top}}(\mathbb{T}_{X}(d)_{w})_{\mathbb{Q}}.$

We propose the following K-theoretic computations, see [PTa, Corollary 5.13]
for the case of $\mathbb{C}^{3}$.

######  Conjecture 1.5.

Let $S$ be a smooth surface, let $X=\mathrm{Tot}_{S}(K_{S})$, let
$(d,w)\in\mathbb{N}\times\mathbb{Z}$, and let $n=\gcd(d,w)$. Then

$\chi_{K}(\mathbb{T}_{X}(d)_{w})=\chi_{c}(\mathrm{Hilb}(S,n)).$

When $\gcd(d,w)=1$, we regard $\mathbb{T}_{X}(d)_{w}$ as categorifications of
the BPS invariants $\Omega_{X,d}$, for example Conjecture 1.5 implies that
$\chi_{K}(\mathbb{T}_{X}(d)_{w})=|\Omega_{X,d}|$. Further, when $\gcd(d,w)=1$,
we show (using [PTb, Theorem 1.1]) that objects in $\mathbb{T}_{X}(d)_{w}$ are
supported over the small diagonal $X\hookrightarrow\mathrm{Sym}^{d}(X)$, which
is a categorical analogue of Davison’s support lemma [Dav, Lemma 4.2, Theorem
5.1].

For $\gcd(d,w)>1$, the Euler characteristic of the topological K-theory of
$\mathbb{T}_{X}(d)_{w}$ may be different from $|\Omega_{X,d}|$. In [PTb], we
constructed a coproduct on the K-theory of quasi-BPS categories of
$\mathbb{C}^{3}$ of fixed slope and we used it to define a K-theoretic version
of BPS invariants for all $(d,w)\in\mathbb{N}\times\mathbb{Z}$. We do not know
if this is possible for a general local surface $X$.

Let $(d,w)\in\mathbb{N}\times\mathbb{Z}$ with $\gcd(d,w)=n$. Similarly to our
expectations for $S=\mathbb{C}^{2}$ from [PTa], [PTb], we believe the category
$\mathbb{T}_{X}(d)_{w}$ is related to the category
$\mathrm{MF}^{\mathrm{gr}}(X^{\times d}/\mathfrak{S}_{d},0)$ of zero matrix
factorizations on the orbifold $X^{\times d}/\mathfrak{S}_{d}$.

###  1.4. Computations of K-theoretic DT invariants via quasi-BPS categories

We revisit the computation of DT invariants from BPS invariants. By [BF08],
there is an equality

(1.7) $\displaystyle\sum_{d\geqslant 0}\mathrm{DT}_{X,d}\cdot
q^{d}=\prod_{d\geqslant 1}(1-(-q)^{d})^{d\Omega_{X,d}}=\prod_{d\geqslant
1}(1-(-q)^{d})^{-d\chi_{c}(S)}.$

In the spirit of [PTa], [PTb, Subsection 5.5], we regard Corollary 1.2 as a
(partial) categorical analogue of the above formula. To obtain a K-theoretic
version of (1.7) from Theorem 1.2, one needs a K-theoretic computation of
quasi-BPS categories as in Conjecture 1.5. More precisely, assuming Conjecture
1.5 and the following Künneth formula

(1.8)
$\chi_{K}\left(\bigotimes_{i=1}^{k}\mathbb{T}_{X}(d_{i})_{v_{i}}\right)=\prod_{i=1}^{k}\chi_{K}(\mathbb{T}_{X}(d_{i})_{v_{i}}),$

we obtain from Corollary 1.2 the following computation of the Euler
characteristic of the topological K-theory of DT categories of points on a
local surface:

(1.9) $\sum_{d\geqslant 0}\chi_{K}(\mathcal{DT}_{X}(0,d))\cdot
q^{d}=\prod_{d\geqslant 1}(1-q^{d})^{-d\chi_{c}(S)},$

see for example [PTa, Corollary 4.13], [PTb, Subsection 5.5]. We discuss an
analogous statement for the localized equivariant K-theory of toric surfaces
in Remark 5.12.

###  1.5. Acknowledgements

Y. T. is supported by World Premier International Research Center Initiative
(WPI initiative), MEXT, Japan, and Grant-in Aid for Scientific Research grant
(No. 19H01779) from MEXT, Japan.

##  2\. Preliminaries

###  2.1. Notations

All the spaces $\mathcal{X}$ considered are quasi-smooth (derived) stacks over
$\mathbb{C}$, see Subsection 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
$\mathrm{Perf}_{T}(\mathcal{X})\subset D^{b}_{T}(\mathcal{X})$. If $T$ is
trivial, we drop it from the notation. We let $\mathbb{K}:=K_{0}(BT)$ and
$\mathbb{F}:=\mathrm{Frac}\,\mathbb{K}$. 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.

$\displaystyle\widehat{X}_{y}:=X\times_{Y}\operatorname{Spec}\left(\widehat{\mathcal{O}}_{Y,y}\right).$

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 $(i,j)\in O$, 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

$\mathbb{T}=\langle\mathbb{A}_{i}\mid i\in R\rangle$

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 $\operatorname{Hom}_{\mathbb{T}}(\mathcal{A}_{i},\mathcal{A}_{j})=0$.

###  2.2. Matrix factorizations

Reference for this subsection are [Todc, Section 2.2], [Todb, Subsection 2.2],
[BFK19, Section 2.3], [PV11, Section 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

$f\colon\mathcal{Y}\to\mathbb{C}.$

We define the dg-category of matrix factorizations
$\mathrm{MF}(\mathcal{Y},f)$. 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

$d_{P}\colon P\to P\langle 1\rangle$

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

(2.1) $\displaystyle\left(\alpha\colon E\rightleftarrows F\colon\beta\right),$

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 [BFK19, Section 2.3], [Todb,
Subsection 2.2], [PV11, Definition 1.2] for the definition of the morphism
spaces.

For a dg-subcategory $\mathscr{A}$ of $D^{b}(\mathcal{Y})$, define
$\mathrm{MF}(\mathscr{A},f)$ as the full subcategory of
$\mathrm{MF}(\mathcal{Y},f)$ 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 two with
respect to the above $\mathbb{C}^{\ast}$-action. Denote by $(1)$ the twist by
the character

$\mathrm{pr}_{2}\colon G\times\mathbb{C}^{*}\to\mathbb{C}^{*}.$

Consider the category of graded matrix factorizations
$\mathrm{MF}^{\mathrm{gr}}(\mathcal{Y},f)$. 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 one with respect to the above $\mathbb{C}^{\star}$-action. The
objects of $\mathrm{MF}^{\mathrm{gr}}(\mathcal{Y},f)$ can be alternatively
described as tuples

(2.2) $\displaystyle(E,F,\alpha\colon E\to F(1)^{\prime},\beta\colon F\to E),$

where $E$ and $F$ are $G\times\mathbb{C}^{\star}$-equivariant coherent sheaves
on $Y$, $(1)^{\prime}$ 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 multiplication by $f$.

Functoriality of categories of (graded or ungraded) matrix factorizations for
pullback and proper pushfoward is discussed in [PV11].

###  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)
$\displaystyle\mathcal{O}_{s^{-1}(0)}:=\mathcal{O}_{Y}\left[V^{\vee}[1];\partial\right].$

Consider the quotient

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

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)
$f\colon\mathscr{V}^{\vee}:=\text{Tot}_{Y}\left(V^{\vee}\right)/G\to\mathbb{C}$

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
$\text{MF}^{\text{gr}}\left(\mathscr{V}^{\vee},f\right)$ 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.

_([Isi13, Hir17, Toda])_ There is an equivalence

(2.6) $\displaystyle\Phi\colon
D^{b}(\mathscr{P})\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{MF}^{\mathrm{gr}}(\mathscr{V}^{\vee},f).$

The equivalence $\Phi$ is given by
$\Phi(-)=\mathcal{K}\otimes_{\mathcal{O}_{\mathscr{P}}}(-)$ for the Koszul
factorization $\mathcal{K}$, see [Toda, 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

$Y^{\lambda\geqslant 0}:=\\{y\in Y|\,\lim_{t\to 0}\lambda(t)\cdot y\in
Y^{\lambda}\\}\subset Y.$

Consider the attracting and fixed stacks

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

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 [HL15, Section 2.1]. The attracting stacks
also appear in the definition of Hall algebras [Păda] (for $Y$ an affine
space), where the Hall product is induced by the functor

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

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

$j\colon\mathcal{Y}^{\mathscr{L}\text{-ss}}:=Y^{\mathscr{L}\text{-ss}}/G\hookrightarrow\mathcal{Y}:=Y/G.$

We review the construction of window categories of $D^{b}(\mathcal{Y})$ which
are equivalent to $D^{b}(\mathcal{Y}^{\mathscr{L}\text{-ss}})$ via the
restriction map, due to Segal [Seg11], Halpern-Leistner [HL15], and
Ballard–Favero–Katzarkov [BFK19]. We follow the presentation from [HL15].

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

$\mathcal{Y}\setminus\mathcal{Y}^{\mathscr{L}\text{-ss}}=\bigsqcup_{i\in
I}\mathscr{S}_{i}.$

A Kempf-Ness stratum $\mathscr{S}_{i}$ is the attracting stack in
$\mathcal{Y}\setminus\sqcup_{j<i}\mathcal{S}_{j}$ 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

$\eta_{i}:=\left\langle\lambda_{i},N^{\vee}_{\mathscr{S}_{i}/\mathcal{Y}}|_{\mathscr{Z}_{i}}\right\rangle.$

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

(2.9) $\mathbb{G}_{w}:=\\{\mathcal{F}\in D^{b}(\mathcal{Y})\text{ such that
}\mathrm{wt}_{\lambda_{i}}(\mathcal{F}|_{\mathscr{Z}_{i}})\subset[w_{i},w_{i}+\eta_{i})\text{
for all }i\in I\\}.$

In the above, $\mathrm{wt}_{\lambda_{i}}(\mathcal{F}|_{\mathscr{Z}_{i}})$ is
the set of $\lambda_{i}$-weights on $\mathcal{F}|_{\mathscr{Z}_{i}}$. Then
[HL15, Theorem 2.10] says that the restriction functor $j^{*}$ induces an
equivalence of categories:

(2.10)
$j^{*}\colon\mathbb{G}_{w}\xrightarrow{\sim}D^{b}\big{(}\mathcal{Y}^{\mathscr{L}\text{-ss}}\big{)}$

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 $\mathrm{MF}(\mathbb{G}_{w},f)$ from Subsection 2.2.1. Then
there is an equivalence

(2.11)
$j^{*}\colon\mathrm{MF}(\mathbb{G}_{w},f)\xrightarrow{\sim}\mathrm{MF}\big{(}\mathcal{Y}^{\mathscr{L}\text{-ss}},f\big{)}.$

The analogous statement holds for categories of graded matrix factorizations.

###  2.5. Quasi-smooth stacks and singular support

References for this section are [Toda, Subsections 3.1 and 3.2.1], [AG15].

####  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 [BBBBJ15, 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) $\alpha\colon\mathscr{U}\to\mathfrak{M}$

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

$\displaystyle
D^{b}(\mathfrak{M})=\lim_{\mathscr{U}\to\mathfrak{M}}D^{b}(\mathscr{U}).$

Suppose that $\mathcal{M}$ admits a good moduli space $\mathcal{M}\to M$, see
[Alp13] for the notion of good moduli space. For each point in $M$, there is
an étale neighborhood $U\to M$ and Cartesian squares

(2.17)

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 [Toda, Subsection
3.1.4], [HLa, Theorem 4.2.3], [AHD20].

####  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
$\mathfrak{M}$:

$\displaystyle\Omega_{\mathfrak{M}}[-1]:=\mathrm{Spec}_{\mathfrak{M}}\left(\mathrm{Sym}(\mathbb{T}_{\mathfrak{M}}[1])\right).$

Consider the projection map

(2.18)
$p_{0}\colon\mathcal{N}:=t_{0}\left(\Omega_{\mathfrak{M}}[-1]\right)\to\mathfrak{M}.$

For a Koszul stack $\mathscr{P}$ as in (2.4), recall the function $f$ from
(2.5) and consider the critical locus
$\mathrm{Crit}(f)\subset\mathscr{V}^{\vee}$. In this case, the map $p_{0}$ is
the natural projection

(2.19)
$p_{0}\colon\mathrm{Crit}(f)=t_{0}\left(\Omega_{\mathscr{P}}[-1]\right)\to\mathscr{P}.$

####  2.5.3. Singular support

We continue with the notations from the above subsection. Arinkin–Gaitsgory
[AG15] defined the notion of singular support of an object $\mathcal{F}\in
D^{b}(\mathfrak{M})$, denoted by

$\displaystyle\mathrm{Supp}^{\rm{sg}}(\mathcal{F})\subset\mathcal{N}.$

The definition is compatible with maps $\alpha$ as in (2.12), see [Toda,
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 $\mathrm{Supp}^{\rm{sg}}(\mathcal{F})$ 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
$\mathcal{Z}\subset\mathrm{Crit}(w)$ such that
$\mathcal{F}|_{\mathscr{V}^{\vee}\setminus\mathcal{Z}}=0$ in
$\mathrm{MF}^{\mathrm{gr}}(\mathscr{V}^{\vee}\setminus\mathcal{Z},w)$, see
[Toda, Subsection 2.3.9].

####  2.5.4. DT categories via singular support quotients

We review the definition of DT categories for quasi-smooth stacks [Toda,
Section 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 $\mathcal{N}^{\mathscr{L}\text{-ss}}\subset\mathcal{N}$ be the open
substack of $\mathscr{L}$-semistable points, see [HLb, Subsections 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

$\mathcal{Z}:=\mathcal{N}\setminus\mathcal{N}^{\mathscr{L}\text{-ss}}\subset\mathcal{N}$

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

(2.20) $\displaystyle D^{b}(\mathfrak{M})/\mathcal{C}_{\mathcal{Z}}$

is a model of the DT category for the semistable locus on the $(-1)$-shifted
cotangent stack, see [Toda, 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 (2.20) is equivalent to
$\mathrm{MF}^{\rm{gr}}((\mathscr{V}^{\vee})^{\mathscr{L}\text{-ss}},f)$, see
[Toda, Proposition 2.3.9]. As a grading is involved on the matrix
factorization side, the category (2.20) is called the
$\mathbb{C}^{\ast}$-equivariant DT category in [Toda]. The (ungraded)
$\mathbb{Z}/2$-periodic version is introduced in [Todb] and it is given by the
quotient

(2.21) $\displaystyle
D^{b}(\mathfrak{M}_{\varepsilon})/\mathcal{C}_{\mathcal{Z}_{\varepsilon}}.$

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

$\displaystyle\mathcal{Z}_{\varepsilon}=\mathbb{C}^{\ast}(\mathcal{Z}\times\\{1\\})\sqcup(\mathcal{N}\times\\{0\\})\subset\mathcal{N}\times\mathbb{C}=t_{0}(\Omega_{\mathfrak{M}_{\varepsilon}}[-1]).$

In the case of a Koszul stack $\mathfrak{M}=\mathscr{P}$, the category (2.21)
is equivalent to the category of (ungraded) matrix factorizations
$\mathrm{MF}((\mathscr{V}^{\vee})^{\mathscr{L}\text{-ss}},f)$. 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 [Toda,
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 [HLb,
Definition 3.7.6]. We also use the same symbols $(\mathcal{L},b)$ for
$p_{0}^{\ast}\mathcal{L}\in\mathrm{Pic}(\mathcal{N})$ and $p_{0}^{\ast}b\in
H^{4}(\mathcal{N},\mathbb{Q})$. Then there is a $\Theta$-stratification with
respect to $(\mathcal{L},b)$

(2.22)
$\displaystyle\mathcal{N}=\mathscr{S}_{1}\sqcup\cdots\sqcup\mathscr{S}_{N}\sqcup\mathcal{N}^{\mathscr{L}\text{-ss}}$

with centers $\mathscr{Z}_{i}\subset\mathscr{S}_{i}$, see [HLb, 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 [HLb,
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
[HLb, Example 5.3.4]. In the above situation, an analogue of the window
theorem is proved in [Todd, Theorem 1.1], [Toda, Theorem 5.3.13]:

######  Theorem 2.3.

_([Toda, Todd])_ 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

$\displaystyle\mathbb{W}_{w}\subset D^{b}(\mathfrak{M})\twoheadrightarrow
D^{b}(\mathfrak{M})/\mathcal{C}_{\mathcal{Z}}$

is an equivalence.

We explain the meaning of “the formal neighborhood theorem” in the statement
of the above theorem, see [Toda, 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 $G_{y}:=\mathrm{Aut}(y)$, 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

$\displaystyle\kappa_{y}\colon\widehat{\mathcal{H}}^{0}(\mathbb{T}_{\mathcal{M}}|_{y})\to\mathcal{H}^{1}(\mathbb{T}_{\mathcal{M}}|_{y})$

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 [Toda, 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 $(\mathcal{X}_{y},f_{y})$:

$\displaystyle\mathcal{X}_{y}:=\left(\widehat{\mathcal{H}}^{0}(\mathbb{T}_{\mathfrak{M}}|_{y})\oplus\mathcal{H}^{1}(\mathbb{T}_{\mathfrak{M}}|_{y})^{\vee}\right)/G_{y}\stackrel{{\scriptstyle
f_{y}}}{{\to}}\mathbb{C},$

where $f_{y}(u,v)=\langle\kappa_{y}(u),v\rangle$. From (2.19), 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 pull-back of the
$\Theta$-stratification (2.22) to $\widehat{\mathcal{N}}_{y}$ gives a Kempf-
Ness stratification

$\displaystyle\widehat{\mathcal{N}}_{y}=\widehat{\mathscr{S}}_{1,y}\sqcup\cdots\sqcup\widehat{\mathscr{S}}_{N,y}\sqcup\widehat{\mathcal{N}}_{y}^{\mathscr{L}\text{-ss}}$

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 Theorem 2.1, there is an equivalence:

$\displaystyle\Phi_{y}\colon
D^{b}(\widehat{\mathfrak{M}}_{y})\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{MF}^{\text{gr}}(\mathcal{X}_{y},f_{y}).$

Then the subcategory $\mathbb{W}_{w}$ in Theorem 2.3 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

(2.23)
$\displaystyle\Phi_{y}(\mathcal{E}|_{\widehat{\mathfrak{M}}_{y}})\in\mathrm{MF}^{\text{gr}}(\mathbb{G}_{w^{\prime}},f_{y}),\
w_{i}^{\prime}=w_{i}-\langle\lambda_{i},\mathcal{H}^{1}(\mathbb{T}_{\mathfrak{M}}|_{y})^{\lambda_{i}>0}\rangle.$

The category $\mathbb{G}_{w}$ is the window category (2.9) for the weights
$w^{\prime}_{i}$ and the line bundle $\mathscr{L}$. The difference of $w_{i}$
and $w_{i}^{\prime}$ is due to the discrepancy of categorical Hall products on
$\mathfrak{M}_{y}$ and $\mathcal{X}_{y}$, see [Pădb, Proposition 3.1].

####  2.6.3. Adjoints of window categories

Let
$\mathcal{N}^{\mathscr{L}\text{-st}}\subset\mathcal{N}^{\mathscr{L}\text{-ss}}$
be the substack of $\mathscr{L}$-stable points. We will use the following
lemma in the proof of Theorem 5.9:

######  Lemma 2.4.

Assume that
$\mathcal{N}^{\mathscr{L}\text{-st}}=\mathcal{N}^{\mathscr{L}\text{-ss}}$.
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 $\mathrm{Tot}_{Y}(V^{\vee})$, 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 Subsection 2.4, where
$\mathcal{Y}=Y/G$ and
$\mathcal{Y}^{\mathscr{L}\text{-st}}=\mathcal{Y}^{\mathscr{L}\text{-ss}}$ so
that $\mathcal{Y}^{\mathscr{L}\text{-ss}}$ 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

(2.24) $\displaystyle
D^{b}(\mathcal{Y}^{\mathscr{L}\text{-ss}})\stackrel{{\scriptstyle(j^{\ast})^{-1}}}{{\longrightarrow}}\mathbb{G}_{w}\subset
D^{b}(\mathcal{Y})$

is given by a Fourier-Mukai functor with kernel object $\mathcal{P}\in
D^{b}(\mathcal{Y}^{\mathscr{L}\text{-ss}}\times\mathcal{Y})$ supported on
$\mathcal{Y}^{\mathscr{L}\text{-ss}}\times_{Y/\\!\\!/G}\mathcal{Y}$, see
[Todc, Lemma 6.7]. The support of $\mathcal{P}$ is proper over $\mathcal{Y}$
and cohomologically proper over $\mathcal{Y}^{\mathscr{L}\text{-ss}}$. Thus
the Fourier-Mukai functor $D^{b}(\mathcal{Y})\to
D^{b}(\mathcal{Y}^{\mathscr{L}\text{-ss}})$ with kernel
$\mathcal{P}^{\vee}\boxtimes\omega_{\mathcal{Y}^{\mathscr{L}\text{-ss}}}[\dim\mathcal{Y}]$
is well-defined and gives a right adjoint of (2.24).

In general, for each point in $M$, there is an étale neighborhood $U\to M$ and
Cartesian squares (2.17) 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 Theorem 2.3 for $\mathfrak{M}_{U}$, which admits
a right adjoint $R_{U}$ by the above argument. Since we have

$\displaystyle\mathbb{W}_{w}=\lim_{U\to M}\mathbb{W}_{U,w},\
D^{b}(\mathfrak{M})=\lim_{U\to M}D^{b}(\mathfrak{M}_{U}),$

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 [Toda, Subsection 3.1.4] for the definition of the
category of smooth morphisms in which we are taking the above limits, and the
proof of [Toda, 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 $G_{x}:=\mathrm{Aut}(x)$-representation

$\displaystyle\mathcal{H}^{0}(\mathbb{T}_{\mathfrak{M}}|_{x})\oplus\mathcal{H}^{1}(\mathbb{T}_{\mathfrak{M}}|_{x})^{\vee}$

is a self dual $G_{x}$-representation. In this subsection, we assume that
$\mathfrak{M}$ is symmetric. Let
$\delta\in\mathrm{Pic}(\mathfrak{M})_{\mathbb{R}}$. We define a different kind
of window categories from the ones in Subsection 2.6, called intrinsic window
subcategories $\mathbb{W}_{\delta}^{\rm{int}}\subset D^{b}(\mathfrak{M})$, see
[Toda, Definition 5.2.12, 5.3.12]. These categories are the quasi-smooth
version of “magic window categories” from [ŠVdB17, HLS20].

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

(2.25) $\displaystyle\mathfrak{M}=s^{-1}(0)/G.$

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 $\mathbb{W}_{\delta}^{\rm{int}}\subset
D^{b}(\mathfrak{M})$ 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

$\displaystyle\mathrm{wt}(\lambda^{\ast}i_{\ast}\mathcal{E})\subset\left[\frac{1}{2}\mathrm{wt}\left(\det\lambda^{\ast}(\mathbb{L}_{\mathscr{V}}|_{\mathcal{Y}})^{\lambda<0}\right),\frac{1}{2}\mathrm{wt}\left(\det\lambda^{\ast}(\mathbb{L}_{\mathscr{V}}|_{\mathcal{Y}})^{\lambda>0}\right)\right]+\mathrm{wt}(\lambda^{\ast}\delta).$

Here, by abuse of notation, we have also denoted the composition
$B\mathbb{C}^{\ast}\stackrel{{\scriptstyle\lambda}}{{\to}}\mathfrak{M}\stackrel{{\scriptstyle
i}}{{\hookrightarrow}}\mathcal{Y}$ by $\lambda$. The above subcategory
$\mathbb{W}_{\delta}^{\rm{int}}$ is intrinsic to $\mathfrak{M}$, and
independent of a choice of a presentation $\mathfrak{M}$ as (2.25) for
$(Y,V,s,G)$, see [Toda, Lemma 5.3.14].

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

######  Definition 2.5.

([Toda, Definition 5.3.12]) We define the subcategory

$\displaystyle\mathbb{W}_{\delta}^{\rm{int}}\subset D^{b}(\mathfrak{M})$

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
$\iota_{U}\colon\mathfrak{M}_{U}\to\mathfrak{M}$, see Subsection 2.5.1, we
have
$\iota_{U}^{\ast}\mathcal{E}\in\mathbb{W}_{\iota_{U}^{\ast}\delta}^{\rm{int}}\subset
D^{b}(\mathfrak{M}_{U})$.

##  3\. Categorical wall-crossing of DT/PT quivers: review

In this section, we review the categorical wall-crossing formula of DT/PT
quivers obtained in [PTc].

###  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
[PTc]. Let $Q=(I,E)$ be the triple loop quiver, that is, the quiver with
vertex set $I=\\{1\\}$ and edge set $E=\\{x,y,z\\}$:

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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 $a=2$:

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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
$\mathfrak{gl}(d)=\mathfrak{gl}(V):=\text{End}(V)$. 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:

$\displaystyle R(d)$ $\displaystyle:=\mathfrak{gl}(V)^{\oplus 3},\
R^{af}(1,d):=V^{\oplus\left(a+1\right)}\oplus\left(V^{\vee}\right)^{\oplus
a}\oplus\mathfrak{gl}(V)^{\oplus 3}.$

Define the following stacks:

$\displaystyle\mathcal{X}(d):=R(d)/GL(d),\
\mathcal{X}^{af}(1,d):=R^{af}(1,d)/GL(d).$

###  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

$\mathcal{X}(d)=R(d)/GL(d)=\mathfrak{gl}(d)^{\oplus 3}/GL(d)$

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 $M(d)_{\mathbb{R}}$:

$\displaystyle\rho:=\frac{1}{2}\sum_{i>j}(\beta_{i}-\beta_{j}),\
\tau_{d}:=\frac{1}{d}\sum_{i=1}^{d}\beta_{i}.$

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)
$\displaystyle\chi=\sum_{j=1}^{m}\chi^{(j)}\in\bigoplus_{j=1}^{m}M(d^{(j)})_{\mathbb{R}}.$

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–Van
den Bergh [ŠVdB17]. Define the polytope $\textbf{W}(d)$ as

(3.2)
$\textbf{W}(d):=\frac{3}{2}\text{sum}[0,\beta_{i}-\beta_{j}]+\mathbb{R}\tau_{d}\subset
M(d)_{\mathbb{R}},$

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

(3.3)
$\textbf{W}(d)_{w}:=\frac{3}{2}\text{sum}[0,\beta_{i}-\beta_{j}]+w\tau_{d}\subset\textbf{W}(d).$

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

$\displaystyle
D^{b}(\mathcal{X}(d))=\bigoplus_{w\in\mathbb{Z}}D^{b}(\mathcal{X}(d))_{w}.$

We define the dg-subcategories

$\displaystyle\mathbb{M}(d)\subset D^{b}(\mathcal{X}(d)),\ (\mbox{resp.
}\mathbb{M}(d)_{w}\subset D^{b}(\mathcal{X}(d))_{w})$

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) $\chi+\rho\in\textbf{W}(d),\ (\mbox{resp.
}\chi+\rho\in\textbf{W}(d)_{w}).$

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) $\displaystyle\nabla_{w}$ $\displaystyle:=\left\\{\chi\in
M_{\mathbb{R}}\,\Big{|}-\frac{1}{2}\eta_{\lambda}\leqslant\langle\lambda,\chi\rangle\leqslant\frac{1}{2}\eta_{\lambda}\mbox{
for all }\lambda\colon\mathbb{C}^{*}\to T(d)\right\\}+w\tau_{d},$

where
$\eta_{\lambda}:=\langle\lambda,\mathbb{L}_{\mathcal{X}(d)}^{\lambda>0}\rangle=2\langle\lambda,\mathfrak{gl}(d)^{\lambda>0}\rangle$,
see [HLS20, Lemma 2.9].

####  3.2.3. Quasi-BPS categories via matrix factorizations

Consider the regular function

(3.6)
$\displaystyle\mathrm{Tr}\,W_{d}\colon\mathcal{X}(d)\to\mathbb{C},\,(X,Y,Z)\mapsto\mathrm{Tr}\left(X[Y,Z]\right)$

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

$\mathbb{S}(d):=\text{MF}(\mathbb{M}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d})\subset\mathrm{MF}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits W_{d})$

be the subcategory of matrix factorizations $\left(\alpha\colon
F\rightleftarrows G\colon\beta\right)$ 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

$\mathbb{S}^{\bullet}_{\ast}(d):=\text{MF}^{\bullet}_{\ast}(\mathbb{M}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d})\subset\text{MF}^{\bullet}_{\ast}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d})$

for $\ast\in\\{\emptyset,T\\}$, $\bullet\in\\{\emptyset,\text{gr}\\}$, 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 two $\mathbb{C}^{\ast}$-action on $Z$.

####  3.2.4. Quasi-BPS categories via the Koszul equivalence

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

$\displaystyle\mathcal{Y}(d):=\mathfrak{g}^{\oplus 2}/GL(V),$

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) $\displaystyle\mu\colon\mathcal{Y}(d)\to\mathfrak{g},\
(X,Y)\mapsto[X,Y].$

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

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

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

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) $\displaystyle\mathbb{T}(d)_{v}\subset D^{b}(\mathscr{C}(d))$

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 Theorem 2.1, there is an equivalence

(3.11) $\displaystyle\Phi\colon
D^{b}(\mathscr{C}(d))\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{MF}^{\mathrm{gr}}(\mathcal{X}(d),\mathop{\mathrm{Tr}}\nolimits
W_{d}).$

The above equivalence restricts to the equivalence, see [PTa, Equation
(4.22)]:

(3.12)
$\displaystyle\Phi\colon\mathbb{T}(d)_{v}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{S}^{\mathrm{gr}}(d)_{v}.$

###  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

$\displaystyle\mathcal{X}^{af,N}(1,d)=\mathbb{C}^{N}\times\mathcal{X}^{af}(1,d)=(\mathbb{C}^{N}\times
R^{af}(1,d))/GL(d)$

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

$\widetilde{W}|_{Q}=X[Y,Z],$

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

$\displaystyle\mathop{\mathrm{Tr}}\nolimits\widetilde{W}_{d}\colon\mathcal{X}^{af,N}(1,d)\to\mathbb{C}.$

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

$\displaystyle\lambda(t)=(\overbrace{t^{k},\ldots,t^{k}}^{d_{1}},\ldots,\overbrace{t,\ldots,t}^{d_{k}},\overbrace{1,\ldots,1}^{d^{\prime}}).$

We have the diagram of attracting and fixed stacks

(3.13) $\displaystyle\mathcal{X}^{af,N}(1,d)^{\lambda}\stackrel{{\scriptstyle
q_{\lambda}}}{{\leftarrow}}\mathcal{X}^{af,N}(1,d)^{\lambda\geqslant
0}\stackrel{{\scriptstyle p_{\lambda}}}{{\to}}\mathcal{X}^{af,N}(1,d)$

where the fixed stack is

$\displaystyle\mathcal{X}^{af,N}(1,d)^{\lambda}=\prod_{i=1}^{k}\mathcal{X}(d_{i})\times\mathcal{X}^{af,N}(1,d^{\prime}).$

The pull-back/push-forward with respect to the diagram (3.13) gives the
categorical Hall product for DT/PT quivers with super-potentials, see [Păda,
Section 3]:

(3.14)
$\displaystyle\ast=q_{\lambda\ast}p_{\lambda}^{\ast}\colon\boxtimes_{i=1}^{k}\mathrm{MF}(\mathcal{X}(d_{i}),\mathop{\mathrm{Tr}}\nolimits
W_{d_{i}})\boxtimes$
$\displaystyle\,\mathrm{MF}(\mathcal{X}^{af,N}(1,d^{\prime}),\mathop{\mathrm{Tr}}\nolimits\widetilde{W}_{d^{\prime}})$
$\displaystyle\to\mathrm{MF}(\mathcal{X}^{af,N}(1,d),\mathop{\mathrm{Tr}}\nolimits\widetilde{W}_{d}).$

Let $\delta=\mu\chi_{0}=\mu\sum_{i=1}^{d}\beta_{i}$ for $\mu\in\mathbb{R}$. We
define the subcategories

$\displaystyle\mathbb{E}^{a}(1,d;\delta)\subset\mathrm{MF}(\mathcal{X}^{af,N}(1,d),\mathop{\mathrm{Tr}}\nolimits\widetilde{W}_{d}),$
$\displaystyle\mathbb{F}^{a}(1,d;\delta)\subset\mathrm{MF}(\mathcal{X}^{af,N}(1,d),\mathop{\mathrm{Tr}}\nolimits\widetilde{W}_{d})$

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

$\displaystyle\chi+\rho+\delta\in\frac{3}{2}\mathrm{sum}[0,\beta_{i}-\beta_{j}]+\frac{a}{2}\mathrm{sum}[-\beta_{k},\beta_{k}]+\mathrm{sum}[-\beta_{k},0],$
$\displaystyle\chi+\rho+\delta\in\frac{3}{2}\mathrm{sum}[0,\beta_{i}-\beta_{j}]+\frac{a}{2}\mathrm{sum}(-\beta_{k},\beta_{k}]$

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

Let $R$ be the following set

$R=\\{(d_{i},v_{i})_{i=1}^{k}\mid d_{i}\in\mathbb{N},v_{i}\in\mathbb{Z}\\}.$

We define a subset $O\subset R\times R$ which will be used to compare summands
in the semiorthogonal decomposition of Theorem 3.2 and Theorem 1.1, see
Subsection 2.1. This order is a particular example of an order from [PTc,
Subsection 3.2.3], which depends on a choice of $\mu\in\mathbb{R}$.

######  Definition 3.1.

A pair of elements $S=(d_{i},v_{i})_{i=1}^{k}$ and
$S^{\prime}=(d_{i}^{\prime},v_{i}^{\prime})_{i=1}^{k^{\prime}}$ in $R$ is an
element of $O\subset R\times R$ if either

  * •

$\sum_{i=1}^{k}v_{i}>\sum_{i=1}^{k^{\prime}}v_{i}^{\prime}$, or

  * •

$\sum_{i=1}^{k}v_{i}=\sum_{i=1}^{k^{\prime}}v_{i}^{\prime}$ and
$\sum_{i=1}^{k}d_{i}<\sum_{i=1}^{k^{\prime}}d_{i}^{\prime}$, or

  * •

$\sum_{i=1}^{k}v_{i}=\sum_{i=1}^{k^{\prime}}v_{i}^{\prime}$, and
$\sum_{i=1}^{k}d_{i}=\sum_{i=1}^{k^{\prime}}d_{i}^{\prime}$, and
$(S,S^{\prime})$ is in the set $O$ from [PTa, Subsection 3.4].

The following is the local categorical DT/PT correspondence in terms of the
above subcategories, see [PTc, 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

$\displaystyle\mathbb{E}^{a}(1,d;\delta)=\left\langle\boxtimes_{i=1}^{k}\mathbb{S}(d_{i})_{w_{i}}\boxtimes\mathbb{F}(1,d^{\prime};\delta^{\prime})\right\rangle.$

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

(3.15)
$v_{i}:=w_{i}+d_{i}\left(d^{\prime}+\sum_{j>i}d_{j}-\sum_{j<i}d_{j}\right),$

we have

(3.16)
$-1-\mu-\frac{a}{2}<\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}<-\mu-\frac{a}{2}.$

In the above, we let
$\delta^{\prime}:=\left(\mu-d+d^{\prime}\right)\sum_{i>d-d^{\prime}}^{d}\beta_{i}$.
Moreover, each fully-faithful functor

$\boxtimes_{i=1}^{k}\mathbb{S}(d_{i})_{w_{i}}\boxtimes\mathbb{F}(1,d^{\prime};\delta^{\prime})\to\mathbb{E}^{a}(1,d;\delta)$

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

The order of the semiorthogonal summands is discussed in [PTc, Subsection
3.2.3] and depends on $\mu\in\mathbb{R}$. For $0<\varepsilon\ll 1$, 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 Subsection 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):

$\displaystyle I^{a}(d):=\mathcal{X}^{af}(1,d)^{\chi_{0}\text{-ss}},\
P^{a}(d):=\mathcal{X}^{af}(1,d)^{\chi^{-1}_{0}\text{-ss}}.$

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

$D^{b}(I^{a}(d))=\Big{\langle}\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\boxtimes
D^{b}(P^{a}(d^{\prime}))\Big{\rangle},$

where the right hand side is indexed as in Theorem 3.2.

##  4\. Quasi-BPS categories for points on a surface

In this section, we introduce quasi-BPS categories for points on surfaces and
study their properties.

###  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

$\displaystyle\mathcal{M}_{S}(d)\to\mathrm{Sym}^{d}(S)$

by sending a zero-dimensional sheaf to its support. Moreover, the stack
$\mathfrak{M}_{S}(d)$ is symmetric, see [Toda, 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
$\mathfrak{M}_{S}(d)$:

$\displaystyle\mathscr{L}:=\mathrm{det}(p_{\mathfrak{M}\ast}\mathcal{Q})\in\mathrm{Pic}(\mathfrak{M}_{S}(d)),$

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

$\displaystyle\mathbb{T}_{X}(d)_{w}:=\mathbb{W}_{\delta=w\mathscr{L}/d}^{\rm{int}}\subset
D^{b}(\mathfrak{M}_{S}(d)).$

We see that the above definition coincides with the one introduced in [Pădb].
Let us take a point $p=\sum_{j=1}^{m}d^{(j)}x^{(j)}\in\mathrm{Sym}^{d}(S)$ 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 $q=\sum_{j=1}^{m}d^{(j)}y^{(j)}\in\mathrm{Sym}^{d}(\mathbb{C}^{2})$. Then
there is an equivalence of derived stacks

$\displaystyle\iota_{p}\colon\widehat{\mathscr{C}}(d)_{q}\stackrel{{\scriptstyle\sim}}{{\to}}\widehat{\mathfrak{M}}_{S}(d)_{p}.$

Let $\mathbb{T}(d)_{w}\subset D^{b}(\mathscr{C}(d))$ be the quasi-BPS category
from Subsection 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 [Pădb, Section
4.1.6]:

######  Lemma 4.2.

An object $\mathcal{E}\in D^{b}(\mathfrak{M}_{S}(d))$ is an object in
$\mathbb{T}_{X}(d)_{w}$ if and only if for any $p\in\mathrm{Sym}^{d}(S)$, we
have
$\iota_{p}^{\ast}\mathcal{E}|_{\widehat{\mathfrak{M}}_{S}(d)_{p}}\in\widehat{\mathbb{T}}(d)_{w,q}$.

###### 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
$\mathbb{W}_{\delta}^{\rm{int}}$ if and only if, for any
$p\in\mathrm{Sym}^{d}(S)$, we have
$\mathcal{E}|_{\widehat{\mathfrak{M}}_{S}(d)_{p}}\in\mathbb{W}_{\delta_{p}}^{\rm{int}}$
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 [Toda, Lemma 5.3.14], this
is also equivalent to
$\mathcal{F}_{q}:=\iota_{p}^{\ast}\left(\mathcal{E}\big{|}_{\widehat{\mathfrak{M}}_{S}(d)_{p}}\right)$
being an object of $\mathbb{W}_{w\chi_{0}/d}^{\rm{int}}$ 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

$\displaystyle
j_{q}\colon\widehat{\mathscr{C}}(d)_{q}=\mu_{q}^{-1}(0)/GL(d)\hookrightarrow\widehat{\mathcal{Y}}(d)_{q},$

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

$\displaystyle\mathbb{L}_{\mathscr{V}}|_{\widehat{\mathcal{Y}}(d)_{q}}=\left(\mathfrak{gl}(d)\otimes\mathcal{O}_{\widehat{\mathcal{Y}}(d)_{q}}\to\mathfrak{gl}(d)^{\oplus
3}\otimes\mathcal{O}_{\widehat{\mathcal{Y}}(d)_{q}}\right).$

Therefore, the condition that
$\mathcal{F}_{q}\in\mathbb{W}_{w\chi_{0}/d}^{\rm{int}}$ means that, for any
$\lambda\colon B\mathbb{C}^{\ast}\to\widehat{\mathscr{C}}(d)_{q}$, we have

(4.1)
$\displaystyle\mathrm{wt}(\lambda^{\ast}j_{q\ast}\mathcal{F}_{q})\subset\left[-\langle\lambda,\mathfrak{gl}(d)^{\lambda>0}\rangle,\langle\lambda,\mathfrak{gl}(d)^{\lambda>0}\rangle\right]+w\tau_{d},$

where $\mathrm{wt}(\lambda^{\ast}j_{q\ast}\mathcal{F}_{q})$ 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 Subsection 3.2.1. ∎

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

$\displaystyle F=\bigoplus_{j=1}^{m}V^{(j)}\otimes\mathcal{O}_{x^{(j)}}.$

We set

$\displaystyle
G_{p}:=\mathrm{Aut}(F)=\prod_{j=1}^{m}\operatorname{GL}(V^{(j)}),\
\mathcal{Y}^{\prime}(d)_{p}$
$\displaystyle:=\left[\widehat{\operatorname{Ext}}_{S}^{1}(F,F)/G_{p}\right].$

Let $\kappa_{p}$ be the following $G_{p}$-equivariant morphism

$\displaystyle\kappa_{p}\colon\widehat{\operatorname{Ext}}_{S}^{1}(F,F)\to\operatorname{Ext}_{S}^{2}(F,F),\
u\mapsto[u,u].$

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)
$\displaystyle\kappa_{p}^{-1}(0)/G_{p}\stackrel{{\scriptstyle\sim}}{{\to}}\widehat{\mathfrak{M}}_{S}(d)_{p}.$

We define $\widetilde{\mathbb{T}}_{X}(d)_{w,p}\subset
D^{b}(\mathcal{Y}^{\prime}(d)_{p})$ to be the subcategory generated by
$\Gamma_{G_{p}}(\chi)\otimes\mathcal{O}_{\mathcal{Y}^{\prime}(d)_{p}}$, where
$\chi$ is a $G_{p}$-dominant weight of $T(d)$ satisfying

(4.3)
$\displaystyle\chi+\rho_{G_{p}}\in\frac{3}{2}\mathrm{sum}[0,\beta_{i}^{(j)}-\beta_{i^{\prime}}^{(j)}]+w\tau_{d},$

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

$\displaystyle\mathbb{T}_{X}(d)_{w,p}\subset
D^{b}(\widehat{\mathfrak{M}}_{S}(d)_{p})$

consisting of objects $\mathcal{E}$ such that
$i_{p\ast}\mathcal{E}\in\widetilde{\mathbb{T}}_{X}(d)_{w,p}$, where
$i_{p}\colon\widehat{\mathfrak{M}}_{S}(d)_{p}\hookrightarrow\mathcal{Y}^{\prime}(d)_{p}$
is the closed immersion through the equivalence (4.2).

######  Lemma 4.3.

An object $\mathcal{E}\in D^{b}(\mathfrak{M}_{S}(d))_{w}$ lies in
$\mathbb{T}_{X}(d)_{w}$ if and only if, for any closed point
$p\in\mathrm{Sym}^{d}(S)$, we have
$\mathcal{E}|_{\widehat{\mathfrak{M}}_{S}(d)_{p}}\in\mathbb{T}_{X}(d)_{w,p}$.

###### Proof.

The proof is the same as the one of Lemma 4.2, using the presentation
independence of the intrinsic window subcategory and the presentation
$\widehat{\mathfrak{M}}_{S}(d)_{p}=\kappa_{p}^{-1}(0)/G_{p}$ instead of
$\mu_{q}^{-1}(0)/GL(d)$. ∎

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

$\displaystyle\mathcal{X}(d)_{p}=\left(\widehat{\operatorname{Ext}}_{S}^{1}(F,F)\oplus\operatorname{Ext}_{S}^{2}(F,F)^{\vee}\right)/G_{p}\stackrel{{\scriptstyle
f_{p}}}{{\to}}\mathbb{C}$

where $f_{p}(u,v)=\langle\kappa_{p}(u),v\rangle$. Note that we have

(4.4)
$\displaystyle(\operatorname{Ext}_{S}^{1}(F,F)\oplus\operatorname{Ext}_{S}^{2}(F,F)^{\vee})/G_{p}=\prod_{j=1}^{m}\mathrm{End}(V^{(j)})^{\oplus
3}/\operatorname{GL}(V^{(j)}),$

such that $f_{p}=\sum_{j=1}^{m}\mathop{\mathrm{Tr}}\nolimits W_{d^{(j)}}$,
where $\mathop{\mathrm{Tr}}\nolimits W_{d}$ is given in (3.6). We denote by

$\displaystyle\mathbb{S}^{\rm{gr}}(d)_{w,p}\subset\text{MF}^{\text{gr}}(\mathcal{X}(d)_{p},f_{p})$

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 Theorem 2.1):

(4.5) $\displaystyle\Phi_{p}\colon
D^{b}(\widehat{\mathfrak{M}}_{S}(d)_{p})\stackrel{{\scriptstyle\sim}}{{\to}}\text{MF}^{\text{gr}}(\mathcal{X}(d)_{p},f_{p}).$

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

$\displaystyle\Phi_{p}\colon\mathbb{T}_{X}(d)_{w,p}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{S}^{\rm{gr}}(d)_{w,p}.$

######  Remark 4.4.

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

(4.6)
$\displaystyle\widehat{\mathfrak{M}}_{S}(d)_{p}=\prod_{j=1}^{m}\widehat{\mathfrak{M}}_{S}(d^{(j)})_{p^{(j)}}.$

Note that under the product (4.6), we have

$\displaystyle\mathbb{T}_{X}(d)_{w,p}=\boxtimes_{j=1}^{m}\mathbb{T}_{X}(d^{(j)})_{w^{(j)},p^{(j)}}$

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

(4.7)
$\displaystyle\mathbb{S}^{\rm{gr}}(d)_{w,p}=\boxtimes_{j=1}^{m}\mathbb{S}^{\rm{gr}}(d^{(j)})_{w^{(j)},p^{(j)}}.$

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

The main result of [Pădb] 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) $\displaystyle\mathfrak{M}_{S}(d_{1},\ldots,d_{k})$

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

(4.9) $\displaystyle 0=Q_{0}\subset Q_{1}\subset Q_{2}\subset\cdots\subset
Q_{k}$

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

$\displaystyle\mathfrak{M}_{S}(d_{1})\times\cdots\times\mathfrak{M}_{S}(d_{k})\stackrel{{\scriptstyle
q}}{{\leftarrow}}\mathfrak{M}_{S}(d_{1},\ldots,d_{k})\stackrel{{\scriptstyle
p}}{{\to}}\mathfrak{M}_{S}(d),$

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
[PS]:

(4.10) $\displaystyle\ast=p_{\ast}q^{\ast}\colon
D^{b}(\mathfrak{M}_{S}(d_{1}))\boxtimes D^{b}(\mathfrak{M}_{S}(d_{2}))\to
D^{b}(\mathfrak{M}_{S}(d)).$

######  Theorem 4.5.

_([Pădb, Theorem 1.1])_ There is a semiorthogonal decomposition

(4.11) $\displaystyle
D^{b}(\mathfrak{M}_{S}(d))=\left\langle\bigotimes_{i=1}^{k}\mathbb{T}_{X}(d_{i})_{v_{i}}\,\Big{|}\,\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}},d_{1}+\cdots+d_{k}=d\right\rangle.$

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 [PTa, Subsection 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 Subsection 3.2.4, i.e.

$\displaystyle\mathfrak{M}_{\mathbb{C}^{2}}(d)=\mathscr{C}(d).$

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 [PTa, 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

$\nu\colon\mathfrak{n}^{\oplus 2}\to\overline{\mathfrak{n}}.$

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

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

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 [PTa,
Subsection 4.3]. The following lemma is implicitly proved in [PTa, Section 5].

######  Lemma 4.6.

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

(4.12)
$\mathcal{Z}^{\mathrm{cl}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{Z}.$

###### 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) $\dim\nu^{-1}(0)=\dim\nu^{-1}(0)^{\mathrm{cl}},$

see for example [AN10, Subsection 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 [PTa, Proposition 5.17]. ∎

We set

(4.14) $\displaystyle
m_{i}:=\left\lceil\frac{vi}{d}\right\rceil-\left\lceil\frac{v(i-1)}{d}\right\rceil+\delta_{i}^{d}-\delta_{i}^{1}\in\mathbb{Z},$

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) $\displaystyle\lambda\colon\mathbb{C}^{\ast}\to GL(d),\
t\mapsto(t^{d},t^{d-1},\ldots,t).$

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

$\displaystyle GL(d)^{\lambda\geqslant 0}=B(d)\to
GL(d)^{\lambda}=T(d)\stackrel{{\scriptstyle\chi}}{{\to}}\mathbb{C}^{\ast},$

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

(4.16)
$\displaystyle\mathcal{E}_{d,v}:=p_{\ast}\left(\mathcal{O}_{\mathcal{Z}}\otimes\mathbb{C}(m_{1},\ldots,m_{d})\right)\in
D^{b}(\mathscr{C}(d)).$

By [PTa, Lemma 4.3], we have

(4.17) $\mathcal{E}_{d,v}\in\mathbb{T}_{\mathbb{C}^{3}}(d)_{v}.$

###  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
$d_{i}=1$:

$\mathcal{M}_{S}(1,\ldots,1)=t_{0}(\mathfrak{M}_{S}(1,\ldots,1))$

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

$\mathcal{Z}_{S}:=\mathcal{M}_{S}(1,\ldots,1)\times_{S^{\times d}}\Delta,$

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

$\mathcal{Z}_{\mathbb{C}^{2}}\cong\mathcal{Z}.$

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

(4.22)

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

$\displaystyle\Phi_{d,v}\colon D^{b}(S)$ $\displaystyle\to
D^{b}(\mathfrak{M}_{S}(d))_{v},$ $\displaystyle\mathcal{F}$
$\displaystyle\mapsto
p_{S*}\left(q_{S}^{*}(\mathcal{F}\otimes\mathbb{C}(m_{1},\ldots,m_{d}))\right).$

######  Proposition 4.7.

The image of the functor $\Phi_{d,v}$ is in the category
$\mathbb{T}_{X}(d)_{v}$.

###### Proof.

It suffices to check the claim formally locally on $\mathrm{Sym}^{d}(S)$. The
objects in the image of $\Phi_{d,v}$ are supported over the small diagonal
$\Delta\cong S\hookrightarrow\mathrm{Sym}^{d}(S)$ by construction. Let $x\in
S$ and consider the formal completions of spaces along the fiber over
$d[x]\in\mathrm{Sym}^{d}(S)$ or $d[0]\in\mathrm{Sym}^{d}(\mathbb{C}^{2})$. The
conclusion then follows from the diagrams (4.22) and (4.17). ∎

###  4.5. Dimensions of quasi-BPS categories

In this section, we prove Theorem 1.4. 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
$\mathbb{K}:=K_{0}(BT)$, $\mathbb{F}:=\mathrm{Frac}\,\mathbb{K}$, 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

$\displaystyle\mathrm{KHA}_{S,T}:=\bigoplus_{d\geqslant
0}G_{T}(\mathfrak{M}_{S}(d))_{\mathbb{F}}\stackrel{{\scriptstyle\cong}}{{\leftarrow}}\bigoplus_{d\geqslant
0}G_{T}(\mathcal{M}_{S}(d))_{\mathbb{F}}.$

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

######  Lemma 4.8.

There is an isomorphism

(4.23) $\displaystyle\mathrm{KHA}_{S,T}\cong\bigotimes_{x\in
S^{T}}\left(\bigoplus_{d\geqslant
0}G_{T}\big{(}\mathcal{M}_{S}(d)_{d[x]}\big{)}_{\mathbb{F}}\right).$

###### Proof.

The $T$-fixed loci of $\mathcal{M}_{S}(d)$ are included in fibers over
$T$-fixed loci of $\mathrm{Sym}^{d}(S)$. The $T$-fixed locus
$\mathrm{Sym}^{d}(S)^{T}$ 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 [Tak94],
there is an isomorphism

$\displaystyle G_{T}\left(\mathcal{M}_{S}(d)\right)_{\mathbb{F}}$
$\displaystyle\cong
G_{T}\left(\bigsqcup_{p\in\mathrm{Sym}^{d}(S)^{T}}\mathcal{M}_{S}(d)_{p}\right)_{\mathbb{F}}.$

Moreover, we have

$\displaystyle\mathcal{M}_{S}(d)_{p}=\prod_{j=1}^{m}\mathcal{M}_{S}(d^{(j)})_{d^{(j)}[x^{(j)}]}.$

We thus obtain the desired isomorphism. ∎

For $x\in S^{T}$, define

$\Phi(d)_{v,x}:=\Phi_{d,v}(\mathcal{O}_{x})\in\mathbb{T}_{X}(d)_{v}.$

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

######  Proposition 4.9.

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

(4.24)
$\Phi(d)_{v,x}\Phi(d^{\prime})_{v^{\prime},x^{\prime}}=\Phi(d^{\prime})_{v^{\prime},x^{\prime}}\Phi(d)_{v,x}.$

(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

(4.25) $\Phi(d_{1})_{v_{1},x_{1}}\cdots\Phi(d_{k})_{v_{k},x_{k}},$

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 (4.25) are equivalent if they
differ by relations (4.24).

###### Proof.

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

(4.26) $\mathrm{KHA}_{S,T,x}:=\bigoplus_{d\geqslant
0}G_{T}(\mathcal{M}_{S}(d)_{d[x]})_{\mathbb{F}}.$

First, by the localization theorem in K-theory [Tak94], 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

$G_{T}(\mathcal{M}_{S}(d)_{d[x]})_{\mathbb{F}}=\mathrm{Ker}\left(G_{T}(\mathcal{M}_{S}(d))_{\mathbb{F}}\xrightarrow{\mathrm{res}}G_{T}(\mathcal{M}_{S}(d)\setminus\mathcal{M}_{S}(d)_{d[x]})_{\mathbb{F}}\right).$

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

$\displaystyle G_{T}(\mathcal{M}_{S}(d)_{d[x]})_{\mathbb{F}}\otimes
G_{T}(\mathcal{M}_{S}(e)_{e[x]})_{\mathbb{F}}$ $\displaystyle\to
G_{T}(\mathcal{M}_{S}(d))_{\mathbb{F}}\otimes
G_{T}(\mathcal{M}_{S}(e))_{\mathbb{F}}$
$\displaystyle\xrightarrow{\ast}G_{T}(\mathcal{M}_{S}(d+e))_{\mathbb{F}}$
$\displaystyle\to
G_{T}(\mathcal{M}_{S}(d+e)\setminus\mathcal{M}_{S}(d+e)_{(d+e)[x]})_{\mathbb{F}}.$

There is thus a Hall product

$\ast\colon G_{T}(\mathcal{M}_{S}(d)_{d[x]})_{\mathbb{F}}\otimes
G_{T}(\mathcal{M}_{S}(e)_{e[x]})_{\mathbb{F}}\to
G_{T}(\mathcal{M}_{S}(d+e)_{(d+e)[x]})_{\mathbb{F}}$

and (4.26) is indeed an algebra. The algebras $\mathrm{KHA}_{S,T,x}$ and
$\mathrm{KHA}_{S,T,x^{\prime}}$ commute for $x\neq x^{\prime}\in S^{T}$ inside
$\mathrm{KHA}_{S,T}$. Further, by the isomorphism (4.22) and the localization
theorem in K-theory [Tak94], we have that

$\mathrm{KHA}_{S,T,x}\cong\mathrm{KHA}_{\mathbb{C}^{2},T,0}\cong\mathrm{KHA}_{\mathbb{C}^{2},T}.$

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

$\Phi(d_{1})_{v_{1},0}\ldots\Phi(d_{k})_{v_{k},0}$

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. ∎

######  Corollary 4.10.

The $\mathbb{F}$-vector space
$K_{T}\big{(}\mathbb{T}_{X}(d)_{v}\big{)}_{\mathbb{F}}$ has an
$\mathbb{F}$-basis given by equivalence classes of monomials
$\Phi(d_{1})_{v_{1},x_{1}}\cdots\Phi(d_{k})_{v_{k},x_{k}}$ for partition
$\sum_{i=1}^{k}(d_{i},v_{i})=(d,v)$, points $x_{i}\in S^{T}$ for $1\leqslant
i\leqslant k$, and $v_{1}/d_{1}=\cdots=v_{k}/d_{k}$, where two monomials are
equivalent if they differ by relations (4.24).

###### Proof.

The same argument as the one in [PTa, Proof of Theorem 4.12] (which treats the
case $S=\mathbb{C}^{2}$) applies here and is based on the isomorphism (4.23),
Proposition 4.9, and on the $T$-equivariant version of Theorem 4.5. ∎

###### Proof of Theorem 1.4.

Let $(d^{\prime},v^{\prime})\in\mathbb{N}\times\mathbb{Z}$ be coprime integers
such that $n(d^{\prime},v^{\prime})=(d,v)$. Using [PTa, Lemma 4.8], the Hall
product induces an algebra structure on

(4.27) $\displaystyle\bigoplus_{k\geqslant
0}K_{T}(\mathbb{T}_{X}(kd^{\prime})_{kv^{\prime}})_{\mathbb{F}}.$

By Corollary 4.10, the above algebra has an $\mathbb{F}$-basis with
equivalences classes of monomials
$\Phi(k_{1}d^{\prime})_{k_{1}v^{\prime},x_{1}}\ldots\Phi(k_{r}d^{\prime})_{k_{r}v^{\prime},x_{r}}$
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
$a_{k}:=\dim_{\mathbb{F}}K_{T}(\mathbb{T}_{X}(kd^{\prime})_{kv^{\prime}})_{\mathbb{F}}$
and let $b_{k}$ be the number of partitions of $k$. Then

$\displaystyle\sum_{k\geqslant 0}a_{k}q^{k}=\left(\sum_{k\geqslant
0}b_{k}q^{k}\right)^{|S^{T}|}=\left(\prod_{b\geqslant
1}\frac{1}{1-q^{b}}\right)^{\chi_{c}(S)}=\sum_{k\geqslant
0}\chi_{c}(\mathrm{Hilb}(S,k))q^{k}.$

The first equality holds from the description of (4.27), 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 [Gö90]. ∎

######  Remark 4.11.

Recall Conjecture 1.5. We may also conjecture the analogous statement

$\chi_{\mathrm{HP}}(\mathbb{T}_{X}(d)_{w})=\chi_{c}(\mathrm{Hilb}(S,n)),$

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

$\chi_{\mathrm{HP}}(\mathbb{T}_{X}(d)_{w}):=\dim_{\mathbb{C}(\\!(u)\\!)}\mathrm{HP}_{0}(\mathbb{T}_{X}(d)_{w})-\dim_{\mathbb{C}(\\!(u)\\!)}\mathrm{HP}_{1}(\mathbb{T}_{X}(d)_{w}).$

###  4.6. Singular support of objects in quasi-BPS categories

Let $S$ be a smooth surface, let $X=\mathrm{Tot}_{S}(K_{S})$, 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 [Toda, Lemma 3.4.1], we
have

$\displaystyle\mathcal{M}_{X}(d)=t_{0}(\Omega_{\mathfrak{M}_{S}(d)}[-1]).$

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

${\pi_{d}^{-1}(X)}$${\mathcal{M}_{X}(d)}$${X}$${\mathrm{Sym}^{d}(X),}$$\scriptstyle{\pi_{d}}$$\scriptstyle{\pi_{d}}$$\scriptstyle{\Delta}$

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

$\mathcal{B}PS(d)=\Delta_{*}\mathrm{IC}_{\mathbb{C}^{3}}.$

In [PTa, 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 [PTa, Theorem 3.1], we obtain the analogous result
for $X$:

######  Proposition 4.12.

Let $w\in\mathbb{Z}$ be such that $\gcd(d,w)=1$ and let
$F\in\mathbb{T}_{X}(d)_{w}$. Then
$\mathrm{Supp}^{\rm{sg}}(F)\subset\pi_{d}^{-1}(X)$.

###### Proof.

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

(4.28)
${\mathcal{M}_{X}(d)}$${\mathrm{Sym}^{d}(X)}$${\mathcal{M}_{S}(d)}$${\mathrm{Sym}^{d}(S).}$$\scriptstyle{\tau}$$\scriptstyle{\pi^{3}_{d}}$$\scriptstyle{\tau}$$\scriptstyle{\pi_{d}^{2}}$

We also use the notations $\pi^{2}_{d}$ and $\pi^{3}_{d}$ for maps of moduli
spaces constructed from $\mathbb{C}^{3}$. Let $p\in\mathrm{Sym}^{d}(S)$ be
such that
$\mathrm{Supp}^{\rm{sg}}(F)\cap(\pi^{3}_{d}\tau)^{-1}(p)\neq\emptyset$. 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 $q=\sum_{i=1}^{m}d^{(i)}y^{(i)}\in\mathrm{Sym}^{d}(\mathbb{C}^{2})$
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

(4.35)

In the above, $\widehat{\mathrm{Sym}}^{d}(S)_{p}$ is the completion of
$\mathrm{Sym}^{d}(S)$ at $p$ and all the spaces in the front square is a base-
change of the diagram (4.28) by
$\widehat{\mathrm{Sym}}^{d}(S)_{p}\to\mathrm{Sym}^{d}(S)$, and similarly for
the back square. The commutativity of the bottom square is classical (see the
constructions in Subsection 4.1). The left square is commutative since the
isomorphism
$\widehat{\mathcal{M}}_{S}(d)_{p}\stackrel{{\scriptstyle\cong}}{{\to}}\widehat{\mathcal{M}}_{\mathbb{C}^{2}}(d)_{q}$
extends to an equivalence

(4.36)
$\displaystyle\widehat{\mathfrak{M}}_{S}(d)_{p}\stackrel{{\scriptstyle\sim}}{{\to}}\widehat{\mathfrak{M}}_{\mathbb{C}^{2}}(d)_{q}$

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

The support of sheaves is respected under the isomorphisms in (4.35). Further,
(4.36) induces an equivalence

$\mathbb{T}_{X}(d)_{w,p}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{T}_{\mathbb{C}^{3}}(d)_{w,q}.$

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

$\alpha\left(\mathrm{Supp}^{\rm{sg}}(F|_{\widehat{\mathfrak{M}}_{S}(d)_{p}})\right)=\alpha\left(\mathrm{Supp}^{\rm{sg}}(F)\times_{\mathcal{M}_{S}(d)}\widehat{\mathcal{M}}_{S}(d)_{p}\right)=\mathrm{Supp}^{\rm{sg}}(E).$

In the proof of [PTb, Theorem 3.1], we showed that
$\mathrm{Supp}^{\rm{sg}}(E)\subset(\pi^{3}_{d})^{-1}(\Delta)$. The equivalence
(4.36) respect the singular support, and we thus obtain the desired
conclusion. ∎

##  5\. The categorical DT/PT correspondence for local surfaces

In this section, we prove Theorem 1.1.

###  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

$\displaystyle
X:=\mathrm{Tot}_{S}(K_{S})\stackrel{{\scriptstyle\tau}}{{\to}}S.$

Here, $\tau$ is the natural projection. Let $\beta\in H_{2}(S,\mathbb{Z})$ and
$n\in\mathbb{Z}$. Recall the DT moduli space

$\displaystyle I_{X}(\beta,n)$

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

$\displaystyle P_{X}(\beta,n)$

of PT stable pairs $(\mathcal{O}_{X}\stackrel{{\scriptstyle s}}{{\to}}F)$,
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 $F\in\operatorname{Coh}_{\leqslant 1}(S)$ has one-
dimensional support $\beta$, $\chi(F)=n$, and $s\colon\mathcal{O}_{S}\to F$ is
a morphism. We refer to [Toda, Section 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

$\displaystyle\mathcal{M}_{X}^{{\dagger}}(\beta,n):=t_{0}(\Omega_{\mathfrak{M}_{S}^{{\dagger}}(\beta,n)}[-1])\stackrel{{\scriptstyle\tau_{\ast}}}{{\to}}\mathfrak{M}_{S}^{{\dagger}}(\beta,n)$

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

$\displaystyle\mathcal{A}_{X}=\langle\mathcal{O}_{X},\operatorname{Coh}_{\leqslant
1}(X)[-1]\rangle_{\rm{ex}},$

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

$\displaystyle
I_{X}(\beta,n)\subset\mathcal{M}_{X}^{{\dagger}}(\beta,n)\supset
P_{X}(\beta,n).$

In general, the stack $\mathfrak{M}_{S}^{{\dagger}}(\beta,n)$ is too big, for
example it is not of finite type. Let
$\mathfrak{M}_{S}^{{\dagger}}(\beta,n)_{\rm{fin}}\subset\mathfrak{M}_{S}^{{\dagger}}(\beta,n)$
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

$\displaystyle I_{X}(\beta,n)\subset
t_{0}(\Omega_{\mathfrak{M}_{S}^{{\dagger}}(\beta,n)_{\rm{fin}}}[-1])\supset
P_{X}(\beta,n).$

Let $\mathcal{Z}_{I},\mathcal{Z}_{P}$ be the complements of the above open
immersions. Following the construction in Subsection 2.5.4, the DT/PT
categories are defined as follows:

######  Definition 5.1.

([Toda, Definition 4.2.1]) The DT/PT categories are defined to be the
following singular support quotients:

$\displaystyle\mathcal{DT}_{X}(\beta,n):=D^{b}(\mathfrak{M}_{S}^{{\dagger}}(\beta,n)_{\rm{fin}})/\mathcal{C}_{\mathcal{Z}_{I}},$
$\displaystyle\mathcal{PT}_{X}(\beta,n):=D^{b}(\mathfrak{M}_{S}^{{\dagger}}(\beta,n)_{\rm{fin}})/\mathcal{C}_{\mathcal{Z}_{P}}.$

######  Remark 5.2.

The DT/PT categories in Definition 5.1 are independent of a choice of
$\mathfrak{M}_{S}^{{\dagger}}(\beta,n)_{\rm{fin}}$ up to equivalence, see
[Toda, Lemma 3.2.9].

###  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
$\mathfrak{M}_{S}^{{\dagger}}(\beta,n)_{\rm{fin}}$ which admits a good moduli
space as we explain below. Let

$\displaystyle\mathfrak{T}_{S}(\beta,n)\subset\mathfrak{M}_{S}^{{\dagger}}(\beta,n)$

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 [Toda, Lemma 5.5.4] we have that

$\displaystyle\mathcal{T}_{X}(\beta,n):=t_{0}(\Omega_{\mathfrak{T}_{S}(\beta,n)}[-1])$

is isomorphic to the moduli stack of pairs
$(\mathcal{O}_{X}\stackrel{{\scriptstyle s}}{{\to}}E)$, where
$E\in\operatorname{Coh}_{\leqslant 1}(X)$ 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)
$\displaystyle\mathfrak{M}_{S}^{{\dagger}}(\beta,n)_{\rm{fin}}=\mathfrak{T}_{S}(\beta,n).$

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

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

(5.2) $\displaystyle\pi_{S}\colon\mathcal{T}_{S}(\beta,n)\to
T_{S}(\beta,n)=\mathrm{Chow}^{\beta}(S)\times\mathrm{Sym}^{d}(S).$

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

(5.3) $\displaystyle-\frac{1}{2}\beta^{2}-\frac{1}{2}K_{S}\beta+d=n.$

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

$\displaystyle(\mathcal{O}_{S}\stackrel{{\scriptstyle
s}}{{\to}}F)\mapsto(\mathrm{Supp}(F),\mathrm{Supp}(F_{\rm{tor}})+\mathrm{Supp}(\mathrm{Cok}(s))),$

where $F_{\rm{tor}}\subset F$ 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) $\displaystyle I=(\mathcal{O}_{S}\to
F)=(\mathcal{O}_{S}\twoheadrightarrow\mathcal{O}_{C})\oplus\bigoplus_{j=1}^{m}V^{(j)}\otimes(0\to\mathcal{O}_{x^{(j)}}),$

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

$\displaystyle\mathcal{I}=(\mathcal{O}_{S\times\mathcal{T}_{S}(\beta,n)}\to\mathcal{F})\in
D^{b}(S\times\mathcal{T}_{S}(\beta,n)).$

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

$\mathscr{L}:=\det\left(Rp_{\mathcal{T}*}\mathcal{F}\right)\in\mathrm{Pic}(\mathcal{T}_{S}(\beta,n)).$

By [Toda, Proposition 5.5.2], we have that

$I_{X}(\beta,n)=\mathcal{T}_{X}(\beta,n)^{\mathscr{L}\text{-ss}},\,P_{X}(\beta,n)=\mathcal{T}_{X}(\beta,n)^{\mathscr{L}^{-1}\text{-ss}}.$

We further take

$\displaystyle b:=2\operatorname{ch}_{2}(Rp_{\mathcal{T}*}\mathcal{F})\in
H^{4}(\mathcal{T}_{S}(\beta,n),\mathbb{Q}).$

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

(5.5) $\displaystyle\mathcal{T}_{X}(\beta,n)$
$\displaystyle=\mathcal{S}_{0}^{I}\sqcup\cdots\sqcup\mathcal{S}_{d}^{I}\sqcup
I_{X}(\beta,n)$
$\displaystyle=\mathcal{S}_{0}^{P}\sqcup\cdots\sqcup\mathcal{S}_{d}^{P}\sqcup
P_{X}(\beta,n),$

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

$\displaystyle\mathcal{S}_{i}^{I}=\\{(\mathcal{O}_{X}\stackrel{{\scriptstyle
s}}{{\to}}F):\chi(\mathrm{Cok}(s))=d-i\\},$
$\displaystyle\mathcal{S}_{i}^{P}=\\{(\mathcal{O}_{X}\stackrel{{\scriptstyle
s}}{{\to}}F):\chi(F_{\rm{tor}})=d-i\\},$

where $d$ is given by (5.3). The same stratifications are constructed in
[Toda, Subsection 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
Theorem 2.3 to obtain the following window theorem for DT/PT categories (which
is also obtained in [Toda, Theorem 1.4.6]):

######  Theorem 5.3.

_([Toda, 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) $\displaystyle\mathbb{W}_{I}\subset
D^{b}(\mathfrak{T}_{S}(\beta,n)),\mathbb{W}_{P}\subset
D^{b}(\mathfrak{T}_{S}(\beta,n))$

such that the following composition functors are equivalences

$\displaystyle\mathbb{W}_{I}\hookrightarrow
D^{b}(\mathfrak{T}_{S}(\beta,n))\twoheadrightarrow\mathcal{DT}_{X}(\beta,n)$
$\displaystyle\mathbb{W}_{P}\hookrightarrow
D^{b}(\mathfrak{T}_{S}(\beta,n))\twoheadrightarrow\mathcal{PT}_{X}(\beta,n).$

######  Remark 5.4.

In [Toda, Section 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

$\displaystyle
G_{y}:=\mathrm{Aut}(y)=\prod_{j=1}^{m}\operatorname{GL}(V^{(j)}),\
T_{y}=\prod_{j=1}^{m}T(d^{(j)})\subset G_{y}$

where $T_{y}$ is the maximal torus of $G_{y}$. The derived stack
$\mathfrak{T}_{S}(\beta,n)$ satisfies the formal neighborhood theorem, see
[Toda, Lemma 7.4.3]. Moreover, we have

(5.7)
$\displaystyle\mathcal{H}^{i}(\mathbb{T}_{\mathfrak{T}_{S}(\beta,n)}|_{y})=\operatorname{Hom}^{i}_{S}(I,F)$

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
$\operatorname{Hom}$ 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

$\displaystyle\kappa_{y}\colon\widehat{\operatorname{Hom}}_{S}(I,F)\to\operatorname{Hom}^{1}_{S}(I,F).$

We set

$\displaystyle\mathcal{X}_{y}:=\left(\widehat{\operatorname{Hom}}_{S}(I,F)\oplus\operatorname{Hom}^{1}_{S}(I,F)^{\vee}\right)/G_{y}.$

Consider the following regular function

$\displaystyle f_{y}\colon\mathcal{X}_{y}\to\mathbb{C},\
(x,u)\mapsto\langle\kappa_{y}(x),u\rangle.$

The Koszul duality in Theorem 2.1 gives the equivalence

(5.8) $\displaystyle\Phi_{y}\colon
D^{b}(\widehat{\mathfrak{T}}_{S}(\beta,n)_{y})\stackrel{{\scriptstyle\sim}}{{\to}}\text{MF}^{\text{gr}}(\mathcal{X}_{y},f_{y}).$

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
[PTc].

######  Lemma 5.5.

As a $G_{y}$-representation, we have

$\displaystyle\operatorname{Hom}_{S}(I,F)\oplus\operatorname{Hom}^{1}_{S}(I,F)^{\vee}$
$\displaystyle=H^{0}(\mathcal{O}_{C}(C))\oplus
H^{1}(\mathcal{O}_{C}(C))^{\vee}\oplus\bigoplus_{j=1}^{m}\left((V^{(j)})^{\oplus
a^{(j)}+1}\oplus(V^{(j)\vee})^{\oplus
a^{(j)}}\oplus\mathrm{End}(V^{(j)})^{\oplus 3}\right)$

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

$\displaystyle
F=\mathcal{O}_{C}\oplus\bigoplus_{j=1}^{m}V^{(j)}\otimes\mathcal{O}_{x^{(j)}}.$

By direct calculation, we have

(5.9) $\displaystyle\operatorname{Hom}_{S}(I,F)=$ $\displaystyle
H^{0}(\mathcal{O}_{C}(C))\oplus$
$\displaystyle\bigoplus_{j=1}^{m}\left(V^{(j)}\oplus(V^{(j)\vee}\otimes\operatorname{Hom}_{S}^{1}(\mathcal{O}_{x^{(j)}},\mathcal{O}_{C}))\oplus\mathrm{End}(V^{(j)})^{\oplus
2}\right),$ $\displaystyle\operatorname{Hom}^{1}_{S}(I,F)=$ $\displaystyle
H^{1}(\mathcal{O}_{C}(C))\oplus$
$\displaystyle\bigoplus_{j=1}^{m}\left((V^{(j)\vee}\otimes\operatorname{Hom}_{S}^{2}(\mathcal{O}_{x^{(j)}},\mathcal{O}_{C}))\oplus\mathrm{End}(V^{(j)})\right).$

Since $C$ is a reduced divisor, we have

$\displaystyle\operatorname{Hom}_{S}^{1}(\mathcal{O}_{x^{(j)}},\mathcal{O}_{C})=\operatorname{Hom}_{S}^{2}(\mathcal{O}_{x^{(j)}},\mathcal{O}_{C})=\begin{cases}\mathbb{C},&x^{(j)}\in
C,\\\ 0,&x^{(j)}\notin C.\end{cases}$

∎

We denote by $\chi_{0}^{(j)}$ and $\chi_{0}$ the determinant characters

(5.10)
$\displaystyle\chi_{0}^{(j)}\colon\operatorname{GL}(V^{(j)})\to\mathbb{C}^{\ast},\
g^{(j)}\mapsto\det(g^{(j)}),\ \chi_{0}=\prod_{j=1}^{m}\chi_{0}^{(j)}\colon
G_{y}\to\mathbb{C}^{\ast}.$

The data $(\mathcal{L},b)$ pulled back to $\mathrm{Crit}(f_{y})$ 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)
$\displaystyle\lvert\lambda\rvert^{2}=\sum_{j=1}^{m}\sum_{i=1}^{d^{(j)}}(\lambda_{i}^{(j)})^{2}.$

We have the Kempf-Ness stratifications

(5.12)
$\displaystyle\mathcal{X}_{y}=\mathscr{S}_{1}^{\pm}\sqcup\cdots\sqcup\mathscr{S}_{N^{\pm}}^{\pm}\sqcup\mathcal{X}_{y}^{\chi_{0}^{\pm}\mathchar
45\relax\rm{ss}}$

of $\mathcal{X}_{y}$ with respect to $\chi_{0}^{\pm 1}$ and a Weyl-invariant
norm (5.11), see [Toda, Section 5.1.5]. The restrctions of the above
stratifications to $\mathrm{Crit}(f_{y})$ coincides with the restrictions of
(5.5). We note that the restrictions of (5.5) to $\mathrm{Crit}(f_{y})$ 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 [Toda, Lemma 5.1.9], for each $1\leqslant
i\leqslant N^{\pm}$ and $1\leqslant j\leqslant m$, there is a decomposition

(5.13) $\displaystyle V^{(j)}=V_{1}^{(j)}\oplus V_{2}^{(j)},\ 1\leqslant
j\leqslant m,$

such that $\lambda_{i}^{\pm}$ is the cocharacter

$\displaystyle\lambda_{i}^{\pm}(t)=\\{(\operatorname{id},t^{\mp
1}\operatorname{id})\\}_{1\leqslant j\leqslant
m}\subset\prod_{j=1}^{m}\operatorname{GL}(V_{1}^{(j)})\times\operatorname{GL}(V_{2}^{(j)})\subset
G_{y}.$

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

(5.14) $\displaystyle I_{1}=(\mathcal{O}_{S}\to F_{1})$
$\displaystyle:=(\mathcal{O}_{S}\twoheadrightarrow\mathcal{O}_{C})\oplus\bigoplus_{j=1}^{m}V_{1}^{(j)}\otimes(0\to\mathcal{O}_{x^{(j)}}),$
$\displaystyle F_{2}$
$\displaystyle:=\bigoplus_{j=1}^{m}V_{2}^{(j)}\otimes\mathcal{O}_{x^{(j)}}.$

Then we have the decompositions

(5.15) $\displaystyle I=I_{1}\oplus(0\to F_{2}),\ F=F_{1}\oplus F_{2}.$

We also set $n_{2}:=\chi(F_{2})=\sum_{j=1}^{m}\dim V_{2}^{(j)}$ and let

$\displaystyle O_{12}:=\operatorname{Hom}^{1}_{S}(I_{1},F_{2}),\
O_{21}:=\operatorname{Hom}^{2}_{S}(F_{2},F_{1}).$

Define the widths $\eta_{i}^{\pm}$ of window categories by

$\displaystyle\eta_{i}^{\pm}:=\langle\lambda_{i}^{\pm},N^{\vee}_{\mathscr{S}_{i}^{\pm}/\mathcal{X}_{y}}|_{\mathscr{Z}_{i}^{\pm}}\rangle\in\mathbb{Z}.$

We define the subcategories

$\displaystyle\mathbb{V}_{I,y}\subset\text{MF}^{\text{gr}}(\mathcal{X}_{y},f_{y}),\
\mathbb{V}_{P,y}\subset\text{MF}^{\text{gr}}(\mathcal{X}_{y},f_{y})$

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

(5.16)
$\displaystyle\mathrm{wt}_{\lambda_{i}^{+}}(\mathcal{F}|_{\mathscr{Z}_{i}^{+}})\subset[-k_{I}(n_{2})-\dim
O_{21},-k_{I}(n_{2})-\dim O_{21}+\eta_{i}^{+}),$
$\displaystyle\mathrm{wt}_{\lambda_{i}^{-}}(\mathcal{F}|_{\mathscr{Z}_{i}^{-}})\subset[k_{P}(n_{2})-\dim
O_{12},k_{P}(n_{2})-\dim O_{12}+\eta_{i}^{-}).$

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) $\displaystyle\eta_{i}^{+}$
$\displaystyle=\hom(I_{1},F_{2})+\hom^{2}(F_{2},F_{1})-\hom^{-1}(I_{1},F_{2})$
$\displaystyle=\sum_{j=1}^{m}n_{2}^{(j)}(a^{(j)}+1+2n_{1}^{(j)})$
$\displaystyle\eta_{i}^{-}$
$\displaystyle=\hom^{1}(F_{2},F_{1})+\hom^{1}(I_{1},F_{2})-\hom(F_{2},F_{1})$
$\displaystyle=\sum_{j=1}^{m}n_{2}^{(j)}(a^{(j)}+2n_{1}^{(j)}).$

###### Proof.

By the definition of Kempf-Ness stratifications, we have

$\displaystyle\eta_{i}^{\pm}$
$\displaystyle=\langle\lambda_{i}^{\pm},N^{\vee}_{\mathscr{S}_{i}^{\pm}/\mathcal{X}_{y}}|_{\mathscr{Z}_{i}^{\pm}}\rangle=\langle\lambda_{i}^{\pm},\mathbb{L}_{\mathcal{X}_{y}}^{\lambda_{i}^{\pm}>0}\rangle$
$\displaystyle=\langle\lambda_{i}^{\pm},(\operatorname{Hom}_{S}(I,F)^{\vee}+\operatorname{Hom}^{1}_{S}(I,F)-\operatorname{Hom}^{-1}_{S}(I,F)^{\vee})^{\lambda_{i}^{\pm}>0}\rangle,$

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)
$\displaystyle\mathbb{W}_{\ast,y}:=\Phi_{y}^{-1}(\mathbb{V}_{\ast,y})\subset
D^{b}(\widehat{\mathfrak{T}}_{S}(\beta,n)_{y}).$

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
$\mathbb{W}_{I}$ (resp. $\mathbb{W}_{P})$ if and only if for any closed point
$y\in T_{S}(\beta,n)$, we have

$\displaystyle\mathcal{E}|_{\widehat{\mathfrak{T}}_{S}(\beta,n)_{y}}\in\mathbb{W}_{I,y},\quad(\mbox{resp.~{}}\mathcal{E}|_{\widehat{\mathfrak{T}}_{S}(\beta,n)_{y}}\in\mathbb{W}_{P,y}).$

###### Proof.

The proposition is proved in [Toda, Theorem 6.3.13]. It is also a direct
application of (2.23). From (5.7) and (5.15), we have

$\displaystyle\mathcal{H}^{1}(\mathbb{T}_{\mathfrak{T}_{S}(\beta,n)}|_{y})=\operatorname{Hom}^{1}_{S}(I_{1},F_{1})\oplus\operatorname{Hom}^{1}_{S}(I_{1},F_{2})\oplus\operatorname{Hom}^{2}_{S}(F_{2},F_{1})\oplus\operatorname{Hom}^{2}_{S}(F_{2},F_{2}).$

Therefore we have

$\displaystyle\mathcal{H}^{1}(\mathbb{T}_{\mathfrak{T}_{S}(\beta,n)}|_{y})^{\lambda_{i}^{+}>0}=O_{21},\
\mathcal{H}^{1}(\mathbb{T}_{\mathfrak{T}_{S}(\beta,n)}|_{y})^{\lambda_{i}^{-}>0}=O_{12}.$

Then the proposition follows from (2.23) and (5.16). ∎

####  5.3.3. Generators of window categories

Below, let $0<\varepsilon\ll 1$ and let $k_{I}$ and $k_{P}$ be the constant
functions

$k_{I}(-)=0,\,k_{P}(-)=\varepsilon.$

Recall that $\chi_{0}$ is the determinant character (5.10), so
$\chi_{0}=\sum_{j=1}^{m}\sum_{i=1}^{d^{(j)}}\beta_{i}^{(j)}$ in the notation
of Subsection 3.2.1. In this case, we have another description of
$\mathbb{V}_{I,y}$, $\mathbb{V}_{P,y}$ as follows (see [KT21, Proposition
2.6], [PTa, Proposition 3.13], and especially the “magic window theorem” in
[HLS20] for similar statements where window categories are described by
generating vector bundles):

######  Lemma 5.8.

The subcategories $\mathbb{V}_{I,y}$, $\mathbb{V}_{P,y}$ are generated by
matrix factorizations whose factors are generated by
$\Gamma_{G_{y}}(\chi)\otimes\mathcal{O}_{\mathcal{X}_{y}}$ for a
$G_{y}$-dominant weight $\chi$ of the maximal torus $T_{y}\subset G_{y}$
satisfying

(5.19)
$\displaystyle\chi+\rho_{G_{y}}+\delta\in\frac{3}{2}\mathrm{sum}\left[0,\beta_{i}^{(j)}-\beta_{i^{\prime}}^{(j)}\right]+\mathrm{sum}\left[-\frac{a^{(j)}}{2}\beta_{i}^{(j)},\frac{a^{(j)}}{2}\beta_{i}^{(j)}\right]+\mathrm{sum}\left[-\beta_{i}^{(j)},0\right],$
$\displaystyle\chi+\rho_{G_{y}}+\delta\in\frac{3}{2}\mathrm{sum}\left[0,\beta_{i}^{(j)}-\beta_{i^{\prime}}^{(j)}\right]+\mathrm{sum}\left[-\frac{a^{(j)}}{2}\beta_{i}^{(j)},\frac{a^{(j)}}{2}\beta_{i}^{(j)}\right]$

respectively, where the Minkowski sums above are after $1\leqslant j\leqslant
m$, $1\leqslant i,i^{\prime}\leqslant d^{(j)}$, and the weight $\delta$ is
defined by

$\displaystyle\delta=-\sum_{j=1}^{m}\left(\frac{a^{(j)}}{2}+\varepsilon\right)\chi_{0}^{(j)},\
0<\varepsilon\ll 1.$

###### Proof.

We only prove the case of $\mathbb{V}_{I,y}$. Let $\delta_{0}$ be the
following $G_{y}$-character

$\displaystyle\delta_{0}:=\det\operatorname{Hom}^{1}_{S}(I,F)=-\sum_{j=1}^{m}a^{(j)}\chi_{0}^{(j)},$

where the second equality follows from (5.9). In particular, we have
$\delta=\delta_{0}/2-\varepsilon\chi_{0}$. From the computation in [Toda,
Theorem 6.3.18, (6.3.37)], the condition (5.16) is equivalent to

(5.20)
$\displaystyle\mathrm{wt}_{\lambda_{i}^{+}}(\mathcal{F}|_{\mathscr{Z}_{i}^{+}})\subset\left[-\frac{1}{2}\eta_{i}^{+}-\frac{1}{2}\langle\lambda_{i}^{+},\delta_{0}\rangle+\frac{1}{2}n_{2},\frac{1}{2}\eta_{i}^{+}-\frac{1}{2}\langle\lambda_{i}^{+},\delta_{0}\rangle+\frac{1}{2}n_{2}\right).$

In fact, by Lemma 5.6 and the decompositions (5.15) at each
$\mathscr{Z}_{i}^{+}$, we have

$\displaystyle\frac{1}{2}\eta_{i}^{+}+\frac{1}{2}\langle\delta_{0},\lambda_{i}^{+}\rangle-\frac{1}{2}n_{2}$
$\displaystyle=\frac{1}{2}(\hom(I_{1},F_{2})+\hom^{2}(F_{2},F_{1})-\hom^{-1}(I_{1},F_{2}))$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{1}{2}(\hom^{2}(F_{2},F_{1})-\hom^{1}(I_{1},F_{2}))-\frac{1}{2}n_{2}$
$\displaystyle=\hom^{2}(F_{2},F_{1})+\frac{1}{2}\chi(I_{1},F_{2})-\frac{1}{2}n_{2}=\dim
O_{21}.$

Therefore (5.16) is equivalent to (5.20). Since
$n_{2}=-\langle\lambda_{i}^{+},\chi_{0}\rangle$, the condition (5.20) is also
equivalent to

(5.21)
$\displaystyle\mathrm{wt}_{\lambda_{i}^{+}}(\mathcal{F}|_{\mathscr{Z}_{i}^{+}})\subset\left[-\frac{1}{2}\eta_{i}^{+}-\big{\langle}\lambda_{i}^{+},\frac{1}{2}\delta_{0}+\frac{1}{2}\chi_{0}-\varepsilon\chi_{0}\big{\rangle},\frac{1}{2}\eta_{i}^{+}-\big{\langle}\lambda_{i}^{+},\frac{1}{2}\delta_{0}+\frac{1}{2}\chi_{0}-\varepsilon\chi_{0}\big{\rangle}\right].$

Let $U_{y}$ be the following symmetric $G_{y}$-representation

$\displaystyle U_{y}=H^{0}(\mathcal{O}_{C}(C))\oplus
H^{1}(\mathcal{O}_{C}(C))^{\vee}\oplus\bigoplus_{j=1}^{m}(V^{(j)})^{\oplus
a^{(j)}+1}\oplus(V^{(j)\vee})^{\oplus
a^{(j)}+1}\oplus\mathrm{End}(V^{(j)})^{\oplus 3},$

where $G_{y}$ acts trivially on the first two summands and acts naturally on
the other summands. Denote by
$\mathfrak{g}_{y}=\bigoplus_{j=1}^{m}\mathrm{End}(V^{(j)})$ the Lie algebra of
$G_{y}$. For each cocharacter $\lambda\colon\mathbb{C}^{\ast}\to T_{y}$, we
set

$\displaystyle\eta_{\lambda}:=\langle\lambda,U_{y}^{\lambda>0}-\mathfrak{g}_{y}^{\lambda>0}\rangle.$

A straightforward computation shows that

(5.22)
$\displaystyle\eta_{\lambda_{i}^{+}}=\sum_{j=1}^{m}n_{2}^{(j)}(a^{(j)}+1+2n_{1}^{(j)})=\eta_{i}^{+}.$

Let $\mathbb{V}_{I,y}^{\prime}$ be the triangulated subcategory of
$\text{MF}^{\text{gr}}(\mathcal{X}_{y},f_{y})$ of matrix factorizations whose
factors are generated by
$\Gamma_{G_{y}}(\chi)\otimes\mathcal{O}_{\mathcal{X}_{y}}$ satisfying the
condition (5.19), or equivalently

(5.23)
$\displaystyle\chi+\rho_{G_{y}}+\frac{1}{2}\delta_{0}+\frac{1}{2}\chi_{0}-\varepsilon\chi_{0}$
$\displaystyle\in\frac{3}{2}\mathrm{sum}\left[0,\beta_{i}^{(j)}-\beta_{i^{\prime}}^{(j)}\right]+\mathrm{sum}\left[-\frac{a^{(j)}+1}{2}\beta_{i}^{(j)},\frac{a^{(j)}+1}{2}\beta_{i}^{(j)}\right].$

Note that the right hand side is half of the convex hull of weights in
$\wedge^{\bullet}(U_{y})$. By [HLS20, Lemma 2.9], for a weight $\chi$
satisfying the above condition (5.23), any weight $\chi^{\prime}$ in
$\Gamma_{G_{y}}(\chi)$ satisfies

$\displaystyle\langle\lambda,\chi^{\prime}\rangle\in\left[-\frac{1}{2}\eta_{\lambda}-\big{\langle}\lambda,\frac{1}{2}\delta_{0}+\frac{1}{2}\chi_{0}-\varepsilon\chi_{0}\big{\rangle},\frac{1}{2}\eta_{\lambda}-\big{\langle}\lambda,\frac{1}{2}\delta_{0}+\frac{1}{2}\chi_{0}-\varepsilon\chi_{0}\big{\rangle}\right]$

for any cocharacter $\lambda\colon\mathbb{C}^{\ast}\to T_{y}$. Therefore, by
(5.22), the vector bundle
$\Gamma_{G_{y}}(\chi)\otimes\mathcal{O}_{\mathcal{X}_{y}}$ for a weight $\chi$
satisfying (5.19) satisfies the condition (5.21), i.e.
$\mathbb{V}_{I,y}^{\prime}\subset\mathbb{V}_{I,y}$ holds.

It remains to show that $\mathbb{V}_{I,y}^{\prime}\subset\mathbb{V}_{I,y}$ is
essentially surjective. From the construction of $\mathbb{V}_{I,y}$, the
“window theorem” [HL15, Theorem 2.10] (see Subsection 2.4) shows that the
composition

$\displaystyle\mathbb{V}_{I,y}\hookrightarrow\text{MF}^{\text{gr}}(\mathcal{X}_{y},f_{y})\twoheadrightarrow\text{MF}^{\text{gr}}(\mathcal{X}_{y}^{\chi_{0}\mathchar
45\relax\rm{ss}},f_{y})$

is an equivalence. On the other hand, by the “magic window theorem” [HLS20],
the category $\text{MF}^{\text{gr}}\left(U^{\chi_{0}\mathchar
45\relax\rm{ss}}_{y}/G_{y},f_{y}\right)$ is generated by matrix factorizations
whose factors are generated by
$\Gamma_{G_{y}}(\chi)\otimes\mathcal{O}_{\mathcal{X}_{y}}$ for a weight $\chi$
satisfying (5.23). Consider an embedding
$\iota\colon\mathcal{X}_{y}\hookrightarrow U_{y}/G_{y}$ given by linear
inclusions $(V^{(j)\vee})^{\oplus a^{(j)}}\hookrightarrow(V^{(j)\vee})^{\oplus
a^{(j)}+1}$ into the first $a^{(j)}$-th factors for each $j$. Then
$\mathcal{X}_{y}^{\chi_{0}\mathchar 45\relax\rm{ss}}$ is the pull-back of
$U^{\chi_{0}\mathchar 45\relax\rm{ss}}_{y}/G_{y}\subset U_{y}/G_{y}$ under
$\iota$, see [PTc, Remark 3.1] or [Toda, Lemma 7.10]. In particular,
$\mathrm{MF}^{\rm{gr}}(\mathcal{X}_{y}^{\chi_{0}\text{-ss}},f_{y})$ is
generated by restrictions from
$\mathrm{MF}^{\rm{gr}}(U_{y}^{\chi_{0}\text{-ss}}/G_{y},f_{y})$. So the
composition

$\displaystyle\mathbb{V}_{I,y}^{\prime}\hookrightarrow\text{MF}^{\text{gr}}(\mathcal{X}_{y},f_{y})\twoheadrightarrow\text{MF}^{\text{gr}}(\mathcal{X}_{y}^{\chi_{0}\mathchar
45\relax\rm{ss}},f_{y})$

is essentially surjective, hence
$\mathbb{V}_{I,y}^{\prime}\subset\mathbb{V}_{I,y}$ is essentially surjective.
∎

If we write $\chi=\sum_{j=1}^{m}\chi^{(j)}$ as in (3.1), the conditions (5.19)
are equivalent to

$\displaystyle\chi^{(j)}+\rho^{(j)}+\delta^{(j)}\in\frac{3}{2}\mathrm{sum}\left[0,\beta_{i}^{(j)}-\beta_{i^{\prime}}^{(j)}\right]+\mathrm{sum}\left[-\frac{a^{(j)}}{2}\beta_{i}^{(j)},\frac{a^{(j)}}{2}\beta_{i}^{(j)}\right]+\mathrm{sum}\left[-\beta_{i}^{(j)},0\right],$
$\displaystyle\chi^{(j)}+\rho^{(j)}+\delta^{(j)}\in\frac{3}{2}\mathrm{sum}\left[0,\beta_{i}^{(j)}-\beta_{i^{\prime}}^{(j)}\right]+\mathrm{sum}\left[-\frac{a^{(j)}}{2}\beta_{i}^{(j)},\frac{a^{(j)}}{2}\beta_{i}^{(j)}\right]$

for each $1\leqslant j\leqslant m$ respectively. Let $y^{(j)}\in
T_{S}(\beta,\chi(\mathcal{O}_{C})+d^{(j)})$ be the point corresponding to

(5.24)
$\displaystyle(\mathcal{O}_{S}\twoheadrightarrow\mathcal{O}_{C})\oplus(V^{(j)}\otimes(0\to\mathcal{O}_{x^{(j)}})).$

Then $\mathcal{X}_{y}$ is written as

$\displaystyle\mathcal{X}_{y}=\prod_{j=1}^{m}\mathcal{X}_{y^{(j)}}$

where the product is the fiber product over $H^{0}(\mathcal{O}_{C}(C))\oplus
H^{1}(\mathcal{O}_{C}(C))^{\vee}$. Under the above product, by Lemma 5.8 we
have decompositions of window categories

(5.25)
$\displaystyle\mathbb{V}_{I,y}=\boxtimes_{j=1}^{m}\mathbb{V}_{I,y^{(j)}},\
\mathbb{V}_{P,y}=\boxtimes_{j=1}^{m}\mathbb{V}_{P,y^{(j)}}.$

###  5.4. Semiorthogonal decomposition of DT categories

Recall the categorical Hall product (4.10) on $D^{b}(\mathfrak{M}_{S}(d))$.
There is a left action of the categorical Hall algebra on
$D^{b}(\mathfrak{T}_{S}(\beta,n))$ via the stack of short exact sequences of
framed sheaves, see [Tod20b, Section 7.2]. For $\widetilde{d}+n^{\prime}=n$,
let $\mathfrak{T}_{S}(\widetilde{d},(\beta,n^{\prime}))$ be the derived stack
of objects $(\mathcal{O}_{S}\to F)$ in $\mathfrak{T}_{S}(\beta,n)$ together
with $Q\subset F$ such that $Q$ is zero-dimensional with length
$\overline{d}$. We have evaluation morphisms

$\displaystyle\mathfrak{M}_{S}(\widetilde{d})\times\mathfrak{T}_{S}(\beta,n^{\prime})\stackrel{{\scriptstyle
q}}{{\leftarrow}}\mathfrak{T}_{S}(\widetilde{d},(\beta,n^{\prime}))\stackrel{{\scriptstyle
p}}{{\to}}\mathfrak{T}_{S}(\beta,n),$

where $q$ sends $(\mathcal{O}_{S}\to F)$ together with $Q\subset F$ to
$(Q,(\mathcal{O}_{S}\to F/Q))$. The morphism $q$ is quasi-smooth and $p$ is
proper, see the argument of [Tod20b, Section 7.1]. The above mentioned left
action is given by the functor

(5.26) $\displaystyle\ast=p_{\ast}q^{\ast}\colon
D^{b}(\mathfrak{M}_{S}(\widetilde{d}))\otimes
D^{b}(\mathfrak{T}_{S}(\beta,n^{\prime}))\to
D^{b}(\mathfrak{T}_{S}(\beta,n)).$

Its composition gives a functor

(5.27) $\displaystyle\ast\colon
D^{b}(\mathfrak{M}_{S}(d_{1}))\otimes\cdots\otimes
D^{b}(\mathfrak{M}_{S}(d_{k}))\otimes
D^{b}(\mathfrak{T}_{S}(\beta,n^{\prime}))\to D^{b}(\mathfrak{T}_{S}(\beta,n))$

where $d_{1}+\cdots+d_{k}+n^{\prime}=n$. The main result of this paper,
Theorem 1.1, follows from the following result via the window theorem:

######  Theorem 5.9.

The categorical Hall product (5.27) induces a fully-faithful functor

(5.28)
$\displaystyle\ast\colon\mathbb{T}_{X}(d_{1})_{v_{1}}\otimes\cdots\otimes\mathbb{T}_{X}(d_{k})_{v_{k}}\otimes\mathbb{W}_{P}\to\mathbb{W}_{I}$

for $-1<v_{1}/d_{1}<\cdots<v_{k}/d_{k}\leqslant 0$ such that there is a
semiorthogonal decomposition

$\displaystyle\mathbb{W}_{I}=\left\langle\mathbb{T}_{X}(d_{1})_{v_{1}}\otimes\cdots\otimes\mathbb{T}_{X}(d_{k})_{v_{k}}\otimes\mathbb{W}_{P}\,\Big{|}-1<\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}\leqslant
0\right\rangle.$

The order of the semiorthogonal summands is that of
$(d_{i},v_{i})_{i=1}^{k}\in R$ given in Definition 3.1, see Subsection 2.1.

######  Step 1.

It suffices to prove the statement formally locally on $T_{S}(\beta,n)$.

###### Proof.

Reducing the proof of global semiorthogonal decompositions to the proof of
formal local semiorthogonal decompositions is standard (in the presence of
right adjoints), see for example [Toda, Lemma 6.3.12], [Tod21, Theorem 4.5],
[Todc, Theorem 5.16].

We first note that the functor (5.27) is given by the correspondence

$\displaystyle\prod_{i=1}^{k}\mathfrak{M}_{S}(d_{i})\times\mathfrak{T}_{S}(\beta,n^{\prime})\leftarrow\mathfrak{T}_{S}(d_{\bullet},(\beta,n^{\prime}))\to\mathfrak{T}_{S}(\beta,n)$

where the middle term is the derived stack of objects $(\mathcal{O}_{S}\to F)$