# The local categorical DT/PT correspondence

Tudor Pădurariu and Yukinobu Toda

###### Abstract.

In this paper, we prove the categorical wall-crossing formula for certain
quivers containing the three loop quiver, which we call DT/PT quivers. These
quivers appear as Ext-quivers for the wall-crossing of DT/PT moduli spaces on
Calabi-Yau 3-folds. The resulting formula is a semiorthogonal decomposition
which involves quasi-BPS categories studied in our previous papers, and we
regard it as a categorical analogue of the numerical DT/PT correspondence. As
an application, we prove a categorical DT/PT correspondence for sheaves
supported on reduced plane curves in the affine three dimensional space.

##  1\. Introduction

The purpose of this paper is to give a categorical wall-crossing formula for
certain quivers, called DT/PT quivers. Our main result is motivated by our
pursuit of categorifying the DT/PT correspondence for curve counting theories
on Calabi-Yau 3-folds, see Subsection 1.2 for its review. As we will explain
in Subsection 1.3, the DT/PT quivers appear as Ext-quivers of wall-crossing of
DT/PT moduli spaces on Calabi-Yau 3-folds. We regard the resulting categorical
wall-crossing formula as a local categorical analogue of the DT/PT
correspondence. We apply it to obtain a categorical (and K-theoretic) DT/PT
correspondence in a geometric example, for sheaves supported on reduced plane
curves in the affine three dimensional space. The main result of this paper
will be applied in [PTc] to give a categorical DT/PT wall-crossing for reduced
curve classes on local surfaces.

###  1.1. The categorical wall-crossing formula for DT/PT quivers

For $a\in\mathbb{N}$, we define the DT/PT quiver $Q^{af}$ to be the quiver
with two vertices $0$ and $1$, three loops at $1$, $a+1$ edges from $0$ to
$1$, and $a$ edges from $1$ to $0$, see Figure 1 for the picture of $Q^{2f}$.

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quiver $Q^{af}$ for $a=2$

We denote by $R^{af}(1,d)$ the affine space of representations of $Q^{af}$ of
dimension $(1,d)$. Let $\chi_{0}\colon GL(V)\to\mathbb{C}^{\ast}$ be the
determinant character $g\mapsto\det g$. Consider the GIT semistable stacks,
which are smooth quasi-projective varieties:

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

As we explain in Subsection 1.3, these moduli spaces together with some super-
potentials give local models of the DT/PT moduli spaces on Calabi-Yau 3-folds.
The following is the main result of this paper:

######  Theorem 1.1.

Let $\mu\in\mathbb{R}$ such that $2\mu l\notin\mathbb{Z}$ for $1\leqslant
l\leqslant d$. There is a semiorthogonal decomposition

(1.1)
$D^{b}(I^{a}(d))=\Big{\langle}\left(\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\right)\boxtimes
D^{b}(P^{a}(d^{\prime}))\Big{\rangle}.$

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

(1.2)
$-1-\mu-\frac{a}{2}<\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}<-\mu-\frac{a}{2}.$

The order on the categories is discussed in Subsection 3.2.3, see also
Subsection 2.5.

In the above, the categories $\mathbb{M}(d)_{w}$ are certain (twisted)
noncommutative resolutions of singularities of $\mathfrak{gl}(d)^{\oplus
3}/\\!\\!/GL(d)$ introduced by Špenko–Van den Bergh [ŠVdB17].

We treated the case $a=0$ in [PTa, Theorem 1.1], when $P^{0}(d)$ is empty for
$d>0$ and is a point for $d=0$. We note that while the proof of [PTa, Theorem
1.1] is based on the constructions of [Pădc] (going back to [ŠVdB21]) and on
the use of decompositions of the weight space of
$T(d)\cong(\mathbb{C}^{*})^{d}$ in polytopes using the $(r,p)$-invariant of a
weight introduced by Špenko–Van den Bergh [ŠVdB17, ŠVdB21], a similar argument
does not produce the semiorthogonal decomposition (1.1). The proof of Theorem
1.1 is still based on decompositions of polytopes in the weight space of
$T(d)$, but these decompositions are different from the ones considered in
[ŠVdB21, Corollary 8.5], [Pădc, Theorem 1.1], [PTa, Theorem 1.1].

Using [PTa, Theorem 1.1], we also obtain the following decomposition in
K-theory, where $K$ denotes the Grothendieck group of a dg-category:

######  Corollary 1.2.

Let $d\in\mathbb{N}$. Then there is an isomorphism

$\displaystyle
K(I^{a}(d))\cong\bigoplus_{d^{\prime}=0}^{d}K(\mathrm{NHilb}(d-d^{\prime}))\otimes
K(P^{a}(d^{\prime})).$

Compare the above isomorphism with (1.3). We prove Theorem 1.1 and Corollary
1.2 in Section 3.

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

We now review the DT/PT correspondence for Calabi-Yau 3-folds. For a smooth
Calabi-Yau 3-fold $X$, let $\beta\in H_{2}(X,\mathbb{Z})$ and
$n\in\mathbb{Z}$. The moduli space which defines the (rank one) Donaldson-
Thomas (DT) invariant $I_{n,\beta}$ is the classical Hilbert scheme

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

which parametrizes closed subschemes $C\subset X$ with $\dim C\leqslant 1$ and
$([C],\chi(\mathcal{O}_{C}))=(\beta,n)\in
H_{2}(X,\mathbb{Z})\oplus\mathbb{Z}.$ The rank one DT invariant is defined by

$\displaystyle\mathrm{DT}_{\beta,n}=\int_{I_{X}(\beta,n)}\chi_{B}\ de,$

where $\chi_{B}$ is the Behrend constructible function [Beh09]. The DT
invariants are related to Gromov-Witten invariants by the Maulik-Nekrasov-
Okounkov-Pandharipande (MNOP) conjecture [MNOP06].

In [PT09], Pandharipande–Thomas defined virtual counts of stable pairs on a
smooth complex 3-fold, which we call PT invariants. In the Calabi-Yau case, PT
invariants provide a geometric construction of the contribution of DT
invariants in the statement of the MNOP conjecture. The moduli space of PT
stable pairs is denoted by

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

and it parametrizes pairs $(F,s)$, where $F$ is a pure one-dimensional
coherent sheaf on $X$ with $([F],\chi(F))=(\beta,n)\in
H_{2}(X,\mathbb{Z})\oplus\mathbb{Z}$ and $s\colon\mathcal{O}_{X}\to F$ is a
section whose cokernel is at most zero-dimensional. The PT invariant is
defined by

$\displaystyle\mathrm{PT}_{\beta,n}=\int_{P_{X}(\beta,n)}\chi_{B}\ de.$

The DT/PT correspondence is the formula

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

The above formula was conjectured in [PT09], its $\chi_{B}$-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] (see also [Tod20]) using the work of Joyce-Song
[JS12] (see also [KS]).

There are several refinements of DT invariants in the Calabi-Yau case:
motivic, cohomological, K-theoretic, or categorical, see [Sze16, Toda, PTa].
The motivic DT/PT correspondence was established in local cases by
Davison–Ricolfi [DR21]. It is an interesting problem to study the
cohomological DT/PT correspondence for quivers with potential (following
[DM20]), or for global geometries.

In [Toda, Subsection 1.3, Conjecture 1.3.2], the second author proposed the
existence of dg-categories $\mathcal{DT}_{X}(\beta,n)$ and
$\mathcal{PT}_{X}(\beta,n)$ which categorify DT and PT invariants, and the
existence of a fully-faithful functor

(1.4)
$\displaystyle\mathcal{PT}_{X}(\beta,n)\hookrightarrow\mathcal{DT}_{X}(\beta,n).$

We do not know how to construct such categories for a general Calabi-Yau
$3$-fold. In [Toda, Sections 3 and 4], the second author constructed such dg-
categories in the case of local surfaces $X=\text{Tot}_{S}(\omega_{S})$ (for
$S$ a smooth surface), and proved the existence of a fully-faithful functor
(1.4) when $\beta$ is a reduced curve class [Toda, Theorem 5.5.5]. It is
natural to try to obtain a finer statement than (1.4) such as a semiorthogonal
decomposition of $\mathcal{DT}_{X}(\beta,n)$ which categorifies the numerical
DT/PT correspondence (1.3).

As we explain in the next subsection, the motivation of the main result in
this paper is to establish such a categorical DT/PT wall-crossing for certain
local models of the (still to be defined) dg-categories in (1.4) via Ext-
quivers.

###  1.3. DT/PT wall-crossing and Ext-quivers

We now explain a construction of local models of the dg-categories (1.4) in
terms of Ext-quivers associated with a wall-crossing diagram. For a collection
of objects $\\{E_{1},\ldots,E_{k}\\}$, the associated Ext-quiver has set of
vertices $\\{1,\ldots,k\\}$ and the number of arrows from $i$ to $j$ is
$\dim\operatorname{Ext}^{1}(E_{i},E_{j})$.

The two moduli spaces $I_{X}(\beta,n)$ and $P_{X}(\beta,n)$ are related by
wall-crossing in the derived category of $X$ by regarding them as moduli
spaces of certain two-term complexes. We have open immersions

$\displaystyle I_{X}(\beta,n)\subset\mathcal{T}_{X}(\beta,n)\supset
P_{X}(\beta,n),$

where $\mathcal{T}_{X}(\beta,n)$ is the moduli stack of pairs $(F,s)$ such
that $F$ is a (not necessary pure) one-dimensional sheaf and
$s\colon\mathcal{O}_{X}\to F$ is a section with at most zero-dimensional
cokernel. The stack $\mathcal{T}_{X}(\beta,n)$ is the moduli stack of
semistable objects on the DT/PT wall. Consider its good moduli space

(1.5) $\displaystyle\pi\colon\mathcal{T}_{X}(\beta,n)\to T_{X}(\beta,n).$

We have the following wall-crossing diagram

which is an example of a d-critical flip as defined in [Tod22].

A closed point $p\in T_{X}(\beta,n)$ corresponds to a direct sum

$\displaystyle
I_{C}\oplus\bigoplus_{j=1}^{m}V^{(j)}\otimes\mathcal{O}_{x^{(j)}}[-1]$

where $C\subset X$ is a Cohen-Macaulay curve and $x^{(1)},\ldots,x^{(m)}$ are
distinct points in $X$. The formal fiber
$\widehat{\mathcal{T}}_{X}(\beta,n)_{p}$ of the morphism (1.5) at $p$ is
described in terms of the Ext-quiver $Q_{p}$ associated with the collection

$\displaystyle\\{I_{C},\mathcal{O}_{x^{(1)}}[-1],\ldots,\mathcal{O}_{x^{(m)}}[-1]\\}.$

The Ext-quiver $Q_{p}$ is given by gluing the quivers $Q^{a^{(j)}f}$ at the
vertex $0$, and adding $N$-loops at $0$, where
$a^{(j)}=\mathrm{ext}^{1}(I_{C},\mathcal{O}_{x^{(j)}})$ and
$N=\mathrm{ext}^{1}(I_{C},I_{C})$, see [Toda, Proposition 5.5.2]. The moduli
stack of the representations of the above Ext-quiver $Q_{p}$ with dimension
vector $(1,d^{(1)},\ldots,d^{(m)})$, where $d^{(j)}=\dim V^{(j)}$, is then
given by

$\displaystyle\mathcal{U}_{p}=\left(\mathbb{C}^{N}\times\prod_{j=1}^{m}R^{a^{(j)}f}(1,d^{(j)})\right)\Big{/}\prod_{j=1}^{m}GL(d^{(j)}).$

Then there is a super-potential $w_{p}$ on some formal completion
$\widehat{\mathcal{U}}_{p}$ of $\mathcal{U}_{p}$ such that
$\widehat{\mathcal{T}}_{X}(\beta,n)_{p}$ is isomorphic to
$\mathrm{Crit}(w_{p})$, see [Tod18], [Tod22, Proposition 9.11].

Let $\chi_{0}$ be the character of $\prod_{j=1}^{k}GL(d^{(j)})$ given by the
product of each determinant character of $GL(d^{(j)})$. Then the formal fibers
of $\pi^{\pm}$ are given by $\chi_{0}^{\pm}$-semistable loci on
$\mathrm{Crit}(w_{p})$. Therefore the categories

$\displaystyle\widehat{\mathcal{DT}}_{X}(\beta,n)_{p}:=\mathrm{MF}(\widehat{\mathcal{U}}_{p}^{+},w_{p}),\
\widehat{\mathcal{PT}}_{X}(\beta,n)_{p}:=\mathrm{MF}(\widehat{\mathcal{U}}_{p}^{-},w_{p})$

are natural candidates of formal local models of the (yet to be defined) DT/PT
categories in (1.4). Theorem 1.1 (more precisely, its version allowing loops
at $0$ and a super-potential, see Corollary 3.14) implies that there is a
semiorthogonal decomposition

(1.6)
$\displaystyle\widehat{\mathcal{DT}}_{X}(\beta,n)_{p}=\left\langle\boxtimes_{j=1}^{m}\boxtimes_{i=1}^{k^{(j)}}\widehat{\mathbb{S}}(d_{i}^{(j)})_{w_{i}^{(j)}}\boxtimes\widehat{\mathcal{PT}}_{X}(\beta,n^{\prime})_{p^{\prime}}\right\rangle,$

where $\sum d_{i}^{(j)}+n^{\prime}=n$, the weights $w_{i}^{(j)}$ satisfy a
condition analogous to (1.2), and the point $p^{\prime}\in
T_{n^{\prime}}(X,\beta)$ satisfies $\sum d_{i}^{(j)}x^{(j)}+p^{\prime}=p$. In
the above, $\widehat{\mathbb{S}}(d)_{w}$ is the formal analogue of the quasi-
BPS category $\mathbb{S}(d)_{w}$ of $\mathbb{C}^{3}$ considered in [PTa, PTb].

We expect that (1.6) can be globalized to give a semiorthogonal decomposition
of $\mathcal{DT}_{X}(\beta,n)$. An issue here is that (1.6) depends on local
data $a^{(j)}$ and a choice of $\mu$ in Theorem 1.1. In Theorem 1.1, if we
take $\mu=-a/2-\varepsilon$ for $0<\varepsilon\ll 1$, then these data cancel
and the condition (1.2) becomes

(1.7) $\displaystyle-1<\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}\leqslant
0.$

From the above consideration, if the DT/PT categories in (1.4) exist, we
expect a semiorthogonal decomposition of the form

(1.8)
$\displaystyle\mathcal{DT}_{X}(\beta,n)=\left\langle\boxtimes_{i=1}^{k}\mathbb{S}_{X}(d_{i})_{w_{i}}\boxtimes\mathcal{PT}_{X}(\beta,n^{\prime})\right\rangle$

where $d_{1}+\cdots+d_{k}+n^{\prime}=n$, the integers $w_{i}$ satisfy the
condition (1.7) under the transformation $w_{i}\mapsto v_{i}$ in Theorem 1.1,
and $\mathbb{S}_{X}(d)_{w}$ is an analogue of the quasi-BPS category
$\mathbb{S}(d)_{w}$ for $X$.

In the local surface case, the DT/PT categories have been defined in [Toda]
and the quasi-BPS categories in [Pădb]. In [PTc], we use the results of this
paper to construct the semiorthogonal decomposition (1.8) for reduced curve
classes on local surfaces.

###  1.4. The categorical DT/PT correspondence for $\mathbb{C}^{3}$

We will give an example of dg-categories of matrix factorizations with
explicit super-potentials which categorify DT/PT invariants on
$\mathbb{C}^{3}$ with reduced supports. Then Theorem 1.1 for $a=1$ implies the
categorical DT/PT correspondence in this case (not only formal locally as in
Subsection 1.2). We are able to define the categories $\mathcal{DT}$ and
$\mathcal{PT}$ in this case because of the global description as a critical
locus of the relevant moduli spaces, see Theorem 1.3 and the discussion in
Subsection 4.5.

Let $X=\mathrm{Tot}_{\mathbb{P}^{2}}(\omega_{\mathbb{P}^{2}})$, and let
$\mathcal{T}_{X}^{\rm{red}}(m,d)$ be the classical moduli stack of pairs
$(F,s)$, where $F$ is a one-dimensional sheaf on $X$ with support a reduced
plane curve of degree $m$ in $\mathbb{P}^{2}$, and $s\colon\mathcal{O}_{X}\to
F$ is a section with at most zero-dimensional cokernel. The open immersion
$\mathbb{C}^{2}\subset\mathbb{P}^{2}$ determines an open immersion
$\mathbb{C}^{3}\subset X$. We consider the open substack

$\displaystyle\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)\subset\mathcal{T}_{X}^{\rm{red}}(m,d)$

consisting of $(F,s)$ such that $\mathrm{Cok}(s)$ and the maximal zero-
dimensional subsheaf of $F$ are supported on $\mathbb{C}^{3}$, and the one-
dimensional support of $F$ is a plane curve from
$(\mathbb{C}^{I_{m}})^{\rm{red}}$, see (4.6) and (4.37).

######  Theorem 1.3.

_(Theorem 4.5)_ Let $V$ be a vector space of dimension $d$. Then
$\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)$ is the global critical locus of
a regular function

$\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)=\mathrm{Crit}\left(\mathop{\mathrm{Tr}}\nolimits
W_{m,d}\colon\left(V^{\oplus 2}\oplus V^{\vee}\oplus\mathfrak{g}^{\oplus
3}\times(\mathbb{C}^{I_{m}})^{\rm{red}}\right)\big{/}GL(V)\to\mathbb{C}\right),$

where $\mathop{\mathrm{Tr}}\nolimits W_{m,d}$ is explicitly given by, see
(4.5) and (4.38):

$\displaystyle\mathop{\mathrm{Tr}}\nolimits
W_{m,d}(u_{1},u_{2},v,A,B,C,(\alpha_{ij}))=\sum_{1\leqslant i+j\leqslant
m}\alpha_{ij}vA^{i}B^{j}(u_{2})+\mathop{\mathrm{Tr}}\nolimits C(u_{1}\circ
v+[A,B]).$

Using the above presentation of $\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)$
as a global critical locus, we define dg-categories

$\mathcal{DT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d),\
(\text{resp.}~{}\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d))$

categorifying DT (resp. PT) invariants for the Hilbert schemes of
1-dimensional subschemes (resp. PT stable pair moduli spaces) with reduced
supports. Using Theorem 1.1 for $a=1$ and $\mu=-1/2-\varepsilon$ for
$0<\varepsilon\ll 1$ and applying matrix factorizations for
$\mathop{\mathrm{Tr}}\nolimits W_{d,m}$, see Corollary 3.14, we obtain the
following:

######  Theorem 1.4.

There is a semiorthogonal decomposition

(1.9)
$\displaystyle\mathcal{DT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)=\left\langle\left(\boxtimes_{i=1}^{k}\mathbb{S}(d_{i})_{w_{i}}\right)\boxtimes\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d^{\prime})\right\rangle.$

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

$-1<\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}\leqslant 0.$

The quasi-BPS categories $\mathbb{S}(d)_{w}$ are defined as categories of
matrix factorizations for the potential $\mathrm{Tr}\,X[Y,Z]$ with factors in
$\mathbb{M}(d)_{w}$, see [PTb] for results on these categories. Fix
$0\leqslant d^{\prime}\leqslant d$. By [PTa, Theorem 1.1], the categories with
which we tensor $\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d^{\prime})$ in
(1.9) form a semiorthogonal decomposition of $\mathcal{DT}(d-d^{\prime})$, the
DT category of $d-d^{\prime}$ points in $\mathbb{C}^{3}$. We thus obtain the
following decomposition in K-theory, compare with (1.3):

######  Corollary 1.5.

Let $d\in\mathbb{N}$. Then there is an isomorphism

$\displaystyle
K\left(\mathcal{DT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)\right)\cong\bigoplus_{d^{\prime}=0}^{d}K\left(\mathcal{DT}(d-d^{\prime})\otimes\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d^{\prime})\right).$

We do not know whether the Künneth isomorphism holds above. In Corollary 4.8,
we discuss the analogous statement in the localized equivariant situation
(with respect to the two dimensional Calabi-Yau torus of $\mathbb{C}^{3}$),
when the Künneth isomorphism holds.

###  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

The spaces considered in this paper are defined over the complex field
$\mathbb{C}$ and are quotient stacks $\mathcal{X}=A/G$, where $A$ is a dg-
scheme (or also called derived scheme), the derived zero locus of a section
$s$ of a finite rank vector bundle $\mathcal{E}$ on a finite type separated
scheme $X$ over $\mathbb{C}$, and $G$ is a reductive group. For such a dg-
scheme $A$, let $\dim A:=\dim X-\text{rank}(\mathcal{E})$, let
$A^{\mathrm{cl}}:=Z(s)\subset X$ be the (classical) zero locus, and let
$\mathcal{X}^{\mathrm{cl}}:=A^{\mathrm{cl}}/G$. We denote by
$\mathcal{O}_{\mathcal{X}}$ or $\mathcal{O}_{A}$ the structure sheaf of
$\mathcal{X}$. We denote by $D^{b}(\mathcal{X})$ the bounded derived category
of coherent sheaves on $\mathcal{X}$.

For $G$ a reductive group and $A$ a dg-scheme as above, denote by $A/G$ the
corresponding quotient stack and by $A/\\!\\!/G$ the quotient dg-scheme with
dg-ring of regular functions $\mathcal{O}_{A}^{G}$.

In this paper, all the dg-categories are $\mathbb{C}$-linear pre-triangulated
dg-categories, in particular their homotopy categories are triangulated
categories. For $\mathcal{D}$ a dg-category, we denote by $K(\mathcal{D})$ the
Grothendieck group of the homotopy category of $\mathcal{D}$. For
$\mathcal{X}$ a stack as above, we denote by
$G(\mathcal{X})=G(\mathcal{X}^{\text{cl}})$ the Grothendieck group of
$D^{b}\text{Coh}(\mathcal{X})$ and by $K(\mathcal{X})$ the Grothendieck group
of the category of perfect complexes $\text{Perf}(\mathcal{X})\subset
D^{b}(\mathcal{X})$.

###  2.2. Weights and partitions

#### 2.2.1.

Let $Q=(I,E)$ be the quiver with vertex set $I=\\{1\\}$ and edge set
$E=\\{x,y,z\\}$. For $a\in\mathbb{N}$, let $Q^{a}=(J,E^{a})$ be the quiver
with vertex set $J=\\{0,1\\}$ and edge set $E^{a}$ containing three loops
$E=\\{x,y,z\\}$ at $1$, $a$ edges from $0$ to $1$, and $a$ edges from $1$ to
$0$. Then $Q^{0}$ is the disjoint union of $Q$ and the quiver with one vertex
$0$. Let $Q^{af}=(J,E^{af})$ be the quiver with edges $E^{af}=e\sqcup E^{a}$,
where $e$ is an edge from $0$ to $1$, see Figure 1 for a picture of $Q^{2f}$.

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:

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

Define the following stacks:

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

#### 2.2.2.

Fix $T(d)\subset GL(d)$ the maximal torus consisting of diagonal matrices.
Denote by $M(d)$ the weight space of $T(d)$ and let
$M(d)_{\mathbb{R}}:=M(d)\otimes_{\mathbb{Z}}\mathbb{R}$. Let
$\beta_{1},\ldots,\beta_{d}$ be the simple roots of $GL(d)$. A weight
$\chi=\sum_{i=1}^{d}c_{i}\beta_{i}$ is dominant (resp. strictly dominant) if

$\displaystyle c_{1}\leqslant\cdots\leqslant c_{d},\
(\mbox{resp.~{}}c_{1}<\cdots<c_{d}).$

We denote by $M^{+}\subset M$ and $M^{+}_{\mathbb{R}}\subset M_{\mathbb{R}}$
the dominant chambers. When we want to emphasize the dimension vector, we
write $M(d)$ etc. Denote by $N$ the coweight lattice of $T(d)$ and by
$N_{\mathbb{R}}:=N\otimes_{\mathbb{Z}}\mathbb{R}$. Let $\langle\,,\,\rangle$
be the natural pairing between $N_{\mathbb{R}}$ and $M_{\mathbb{R}}$.

Let $W=\mathfrak{S}_{d}$ be the Weyl group of $GL(d)$. For $\chi\in M(d)^{+}$,
let $\Gamma_{GL(d)}(\chi)$ be the irreducible representation of $GL(d)$ of
highest weight $\chi$. We drop $GL(d)$ from the notation if the dimension
vector $d$ is clear from the context. Let $w*\chi:=w(\chi+\rho)-\rho$ be the
Weyl-shifted action of $w\in W$ on $\chi\in M(d)_{\mathbb{R}}$. We denote by
$\ell(w)$ the length of $w\in W$.

#### 2.2.3.

Denote by $\mathcal{W}^{a}$ the multiset of $T(d)$-weights of $R^{a}(1,d)$ and
by $\mathcal{W}^{af}$ the multiset of $T(d)$-weights of $R^{af}(1,d)$. Namely,
we have

$\displaystyle\mathcal{W}^{a}=\\{(\beta_{i}-\beta_{j})^{\times
3},(\pm\beta_{i})^{\times a}\mid 1\leqslant i,j\leqslant d\\},\
\mathcal{W}^{af}=\mathcal{W}^{a}\cup\\{\beta_{i}\mid 1\leqslant i\leqslant
d\\}.$

We denote by $\rho$ half the sum of positive roots of $GL(d)$. In our
convention of the dominant chamber, it is given by

$\displaystyle\rho=\frac{1}{2}\mathfrak{g}^{\lambda<0}=\frac{1}{2}\sum_{j<i}(\beta_{i}-\beta_{j}),$

where $\lambda$ is the antidominant cocharacter
$\lambda(t)=(t^{d},t^{d-1},\ldots,t)$. We denote by $1_{d}:=z\cdot\text{Id}$
the diagonal cocharacter of $T(d)$. We define the weights in $M_{\mathbb{R}}$:

(2.1) $\displaystyle\sigma_{d}:=\sum_{j=1}^{d}\beta_{j},\
\tau_{d}:=\frac{1}{d}\sigma_{d}.$

#### 2.2.4.

Let $G$ be a reductive group (in this paper, one may assume $G=GL(d)$), let
$X$ be a $G$-representation, and let $\mathcal{X}=X/G$ be the corresponding
quotient stack. Let $\mathcal{V}$ be the multiset of $T(d)$-weights of $X$.
For a cocharacter $\lambda$ of $T(d)$, let $X^{\lambda}\subset X$ be the
subspace generated by weights $\beta\in\mathcal{V}$ such that
$\langle\lambda,\beta\rangle=0$, let $X^{\lambda\geqslant 0}\subset X$ be the
subspace generated by weights $\beta\in\mathcal{V}$ such that
$\langle\lambda,\beta\rangle\geqslant 0$, and let $G^{\lambda}$ and
$G^{\lambda\geqslant 0}$ be the Levi and parabolic groups associated to
$\lambda$. Consider the fixed and attracting stacks

$\displaystyle\mathcal{X}^{\lambda}:=X^{\lambda}/G^{\lambda},\
\mathcal{X}^{\lambda\geqslant 0}:=X^{\lambda\geqslant 0}/G^{\lambda\geqslant
0}$

with maps

(2.2)
$\displaystyle\mathcal{X}^{\lambda}\xleftarrow{q_{\lambda}}\mathcal{X}^{\lambda\geqslant
0}\xrightarrow{p_{\lambda}}\mathcal{X}.$

We abuse notation and denote by $\mathcal{X}^{\lambda\geqslant 0}$ the class

$\displaystyle\left[X^{\lambda\geqslant
0}\right]-\left[\mathfrak{g}^{\lambda\geqslant 0}\right]\in K_{0}(T(d))=M$

and by $\langle\lambda,\mathcal{X}^{\lambda\geqslant 0}\rangle$ the
corresponding integer. We use the similar notation for
$\mathcal{X}^{\lambda>0}$, $\mathcal{X}^{\lambda\leqslant 0}$ etc.

#### 2.2.5.

Let $d\in\mathbb{N}$. We call $\underline{d}:=(d_{i})_{i=1}^{k}$ a partition
of $d$ if $d_{i}\in\mathbb{N}$ are all non-zero and $\sum_{i=1}^{k}d_{i}=d$.
We similarly define partitions of $(d,w)\in\mathbb{N}\times\mathbb{Z}$. For a
partition $(d_{i})_{i=1}^{k}$ of $d$, there is an antidominant cocharacter
$\lambda$ of $T(d)$ such that
$\mathcal{X}(d)^{\lambda}\cong\times_{i=1}^{k}\mathcal{X}(d_{i})$. For example
we can take

(2.3)
$\displaystyle\lambda=(\overbrace{t^{k},\ldots,t^{k}}^{d_{1}},\overbrace{t^{k-1},\ldots,t^{k-1}}^{d_{2}},\ldots,\overbrace{t,\ldots,t}^{d_{k}}).$

We have the maps from (2.2)

$\times_{i=1}^{k}\mathcal{X}(d_{i})\xleftarrow{q_{\lambda}}\mathcal{X}(d)^{\lambda\geqslant
0}\xrightarrow{p_{\lambda}}\mathcal{X}(d).$

We also use the notations $p_{\lambda}=p_{\underline{d}}$,
$q_{\lambda}=q_{\underline{d}}$. Conversely, given an antidominant cocharacter
$\lambda$, there is an associated partition $(d_{i})_{i=1}^{k}$ inducing the
diagram above. Define the length $\ell(\lambda):=k$.

The stack $\mathcal{X}(d)^{\lambda\geqslant 0}$ is isomorphic to the moduli
stack of filtrations of $Q$-representations

$\displaystyle 0=R_{0}\subset R_{1}\subset\cdots\subset R_{k}$

such that $R_{i}/R_{i-1}$ has dimension $d_{i}$. The morphism $q_{\lambda}$
sends the above filtration to its associated graded, and $p_{\lambda}$ sends
it to $R_{k}$. The categorical Hall product for $Q$ is given by the functor
$p_{\lambda*}q_{\lambda}^{*}=p_{\underline{d}*}q_{\underline{d}}^{*}$ and
denoted by

(2.4) $\displaystyle\ast\colon
D^{b}(\mathcal{X}(d_{1}))\boxtimes\cdots\boxtimes D^{b}(\mathcal{X}(d_{k}))\to
D^{b}(\mathcal{X}(d)).$

We may drop the subscript $\lambda$ or $\underline{d}$ in the functors $p_{*}$
and $q^{*}$ when the cocharacter $\lambda$ or the partition $\underline{d}$ is
clear.

We will also use the Hall products for the quivers $Q^{a}$ and $Q^{af}$. For
$e\leqslant d$, let $(d_{i})_{i=1}^{k}$ be a partition of $e$ and set
$d^{\prime}=d-e$. Let $\lambda$ be the antidominant cocharacter of $T(e)$
given by (2.3), and set $\lambda^{\prime}=(\lambda,1_{d^{\prime}})$ which is
an antidominant cocharacter of $T(d)$. Then the diagrams of (2.2) for
$\mathcal{X}^{a}(1,d)$, $\mathcal{X}^{af}(1,d)$ are given by

(2.5)
$\displaystyle\times_{i=1}^{k}\mathcal{X}(d_{i})\times\mathcal{X}^{a}(1,d^{\prime})\stackrel{{\scriptstyle
q_{\lambda}}}{{\leftarrow}}\mathcal{X}^{a}(1,d)^{\lambda\geqslant
0}\stackrel{{\scriptstyle p_{\lambda}}}{{\to}}\mathcal{X}^{a}(1,d),$
$\displaystyle\times_{i=1}^{k}\mathcal{X}(d_{i})\times\mathcal{X}^{af}(1,d^{\prime})\stackrel{{\scriptstyle
q_{\lambda}}}{{\leftarrow}}\mathcal{X}^{af}(1,d)^{\lambda\geqslant
0}\stackrel{{\scriptstyle p_{\lambda}}}{{\to}}\mathcal{X}^{af}(1,d)$

respectively. The functors $p_{\lambda\ast}q_{\lambda}^{\ast}$ give
categorical Hall products

(2.6) $\displaystyle\ast\colon
D^{b}(\mathcal{X}(d_{1}))\boxtimes\cdots\boxtimes
D^{b}(\mathcal{X}(d_{k}))\boxtimes D^{b}(\mathcal{X}^{a}(1,d^{\prime}))\to
D^{b}(\mathcal{X}^{a}(1,d)),$ $\displaystyle\ast\colon
D^{b}(\mathcal{X}(d_{1}))\boxtimes\cdots\boxtimes
D^{b}(\mathcal{X}(d_{k}))\boxtimes D^{b}(\mathcal{X}^{af}(1,d^{\prime}))\to
D^{b}(\mathcal{X}^{af}(1,d)).$

#### 2.2.6.

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

$\bigoplus_{i=1}^{k}M(d_{i})\cong M(d),$

where the simple roots $\beta_{j}$ in $M(d_{1})$ correspond to the first
$d_{1}$ simple roots $\beta_{j}$ of $M(d)$ etc.

###  2.3. Polytopes

We construct several polytopes in $M(d)_{\mathbb{R}}$ which will be used to
define categories in Subsection 2.7. The polytope $\mathbf{W}(d)$ is defined
as

(2.7)
$\mathbf{W}(d):=\frac{3}{2}\text{sum}[0,\beta_{i}-\beta_{j}]+\mathbb{R}\tau_{d}\subset
M(d)_{\mathbb{R}},$

where the Minkowski sum is after all $1\leqslant i,j\leqslant d$ and where
$\tau_{d}$ is given by (2.1). Consider the hyperplane

(2.8)
$\mathbf{W}(d)_{w}:=\frac{3}{2}\text{sum}[0,\beta_{i}-\beta_{j}]+w\tau_{d}\subset\mathbf{W}(d).$

The polytope $\mathbf{V}(d)$ is defined as

(2.9)
$\mathbf{V}(d):=\frac{3}{2}\text{sum}[0,\beta_{i}-\beta_{j}]+\text{sum}[-\beta_{k},0]\subset
M(d)_{\mathbb{R}},$

where the Minkowski sum is after all $1\leqslant i,j,k\leqslant d$. Note that
the definition of the polytope $\mathbf{V}(d)$ differs by the one used in
[PTa] by a translation by $\sigma_{d}$. We let
$\mathbf{V}(d)_{w}\subset\mathbf{V}(d)$ be the subspace of weights $\chi$ such
that $\langle 1_{d},\chi\rangle=w$.

Finally, define the (not necessarily closed) polytopes in $M(d)_{\mathbb{R}}$:

$\displaystyle\mathbf{W}^{a}(1,d)$
$\displaystyle:=\frac{3}{2}\text{sum}[0,\beta_{i}-\beta_{j}]+\frac{a}{2}\text{sum}(-\beta_{k},\beta_{k}],$
$\displaystyle\mathbf{V}^{a}(1,d)$
$\displaystyle:=\frac{3}{2}\text{sum}[0,\beta_{i}-\beta_{j}]+\frac{a}{2}\text{sum}[-\beta_{k},\beta_{k}]+\text{sum}[-\beta_{k},0],$

where the Minkowski sums are after all $1\leqslant i,j,k\leqslant d$.

###  2.4. A corollary of the Borel-Weyl-Bott theorem

For future reference, we state a result from [HLS20, Section 3.2]. We continue
with the notations from Subsection 2.2.4. For a weight $\chi\in M$, let
$\chi^{+}$ be the dominant Weyl-shifted conjugate of $\chi$ if it exists, and
let $\chi^{+}=0$ otherwise. Let $\mathcal{V}$ be the multiset of weights in
$X$. For a multiset $J\subset\mathcal{V}$, let

$\sigma_{J}:=\sum_{\beta\in J}\beta.$

For a weight $\chi\in M$, let $w$ be the element of the Weyl group such that
$w*(\chi-\sigma_{J})$ is dominant or zero. It has length $\ell(w)=:\ell(J)$.

######  Proposition 2.1.

Let $G$ be a reductive group, let $X$ be a $G$-representation, and let
$\lambda$ be a cocharacter of the maximal torus $T\subset G$. Recall the fixed
and attracting stacks and the corresponding maps

$X^{\lambda}/G^{\lambda}\xleftarrow{q_{\lambda}}X^{\lambda\geqslant
0}/G^{\lambda\geqslant 0}\xrightarrow{p_{\lambda}}X/G.$

Let $\chi$ be a weight of $T$. Then there is a quasi-isomorphism

$\left(\bigoplus_{J}\mathcal{O}_{X}\otimes\Gamma_{G}\left((\chi-\sigma_{J})^{+}\right)\left[|J|-\ell(J)\right],d\right)\xrightarrow{\sim}p_{\lambda*}q_{\lambda}^{*}\left(\mathcal{O}_{X^{\lambda}}\otimes\Gamma_{G^{\lambda}}(\chi)\right),$

where the complex on the left hand side has terms (shifted) vector bundles for
all multisets
$J\subset\\{\beta\in\mathcal{V}\mid\langle\lambda,\beta\rangle<0\\}$.

###  2.5. Semiorthogonal decompositions

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 dg-category. We will construct semiorthogonal
decompositions

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

with summands dg-subcategories $\mathbb{A}_{i}$ indexed by $i\in R$ such that
for any $i,j\in R$ with $(i,j)\in O$ and objects
$\mathcal{A}_{i}\in\mathbb{A}_{i}$, $\mathcal{A}_{j}\in\mathbb{A}_{j}$, we
have

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

###  2.6. Matrix factorizations

Let $\mathcal{X}$ be a smooth and let $w\colon\mathcal{X}\to\mathbb{C}$ be a
regular function. We denote by

$\displaystyle\mathrm{MF}(\mathcal{X},w)$

the category of matrix factorizations of $w$. It consists of objects
$\left(\alpha\colon F\rightleftarrows G\colon\beta\right)$ with
$F,G\in\mathrm{Coh}(\mathcal{X})$ such that both of $\alpha\circ\beta$ and
$\beta\circ\alpha$ are multiplications by $w$. We refer to [PTa, Subsection
2.6] for details about categories of matrix factorizations.

###  2.7. Categories of generators

#### 2.7.1.

For $w\in\mathbb{Z}$, we 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 $T(d)$. We have the 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

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

Note that $\mathbb{M}(d)$ decomposes into the direct sum of
$\mathbb{M}(d)_{w}$ for $w\in\mathbb{Z}$. Moreover, taking the tensor product
with the determinant character $\det\colon GL(d)\to\mathbb{C}^{\ast}$ gives an
equivalence

(2.11)
$\displaystyle\otimes\det\colon\mathbb{M}(d)_{w}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{M}(d)_{d+w}.$

Let $\mu\in\mathbb{R}$ and let $\delta:=\mu\sigma_{d}\in M(d)_{\mathbb{R}}$.
We define the dg-subcategories

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

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

(2.12) $\chi+\rho+\delta\in\mathbf{V}(d),\ (\mbox{resp.
}\chi+\rho+\delta\in\mathbf{V}(d)_{w}).$

Note that $\mathbb{D}(d;\delta)$ decomposes into the direct sum of
$\mathbb{D}(d;\delta)_{w}$ for $w\in\mathbb{Z}$ defined as above. We will use
the following proposition proved in [PTa]:

######  Proposition 2.2.

_([PTa, Proposition 3.9])_ There is a semiorthogonal decomposition

$\displaystyle\mathbb{D}(d;\delta)=\left\langle\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\right\rangle.$

Here, the right hand side is after all partitions $d_{1}+\cdots+d_{k}=d$ such
that, by setting $v_{i}:=w_{i}+d_{i}(\sum_{j>i}d_{j}-\sum_{j<i}d_{j})$, we
have

$\displaystyle-1-\mu\leqslant\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}\leqslant-\mu.$

Moreover, each fully-faithful functor
$\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\to\mathbb{D}(d;\delta)$ is given
by the restriction of the categorical Hall product (2.4).

#### 2.7.2.

We fix $\mu\in\mathbb{R}$ and set $\delta:=\mu\sigma_{d}$. We define the dg-
subcategories

$\displaystyle\mathbb{M}^{a}(1,d;\delta)\subset D^{b}(\mathcal{X}^{a}(1,d)),\
\mbox{(resp. }\mathbb{F}^{a}(1,d;\delta)\subset D^{b}(\mathcal{X}^{af}(1,d)))$

to be generated by the vector bundles
$\mathcal{O}_{\mathcal{X}^{a}(1,d)}\otimes\Gamma_{GL(d)}(\chi)$ (resp.
$\mathcal{O}_{\mathcal{X}^{af}(1,d)}\otimes\Gamma_{GL(d)}(\chi)$), where
$\chi$ is a dominant weight of $T(d)$ such that

(2.13) $\chi+\rho+\delta\in\mathbf{W}^{a}(1,d).$

We define the dg-subcategories

(2.14) $\displaystyle\mathbb{D}^{a}(1,d;\delta)\subset
D^{b}(\mathcal{X}^{a}(1,d)),\ \mbox{(resp. }\mathbb{E}^{a}(1,d;\delta)\subset
D^{b}(\mathcal{X}^{af}(1,d)))$

to be generated by the vector bundles
$\mathcal{O}_{\mathcal{X}^{a}(1,d)}\otimes\Gamma_{GL(d)}(\chi)$ (resp.
$\mathcal{O}_{\mathcal{X}^{af}(1,d)}\otimes\Gamma_{GL(d)}(\chi)$), where
$\chi$ is a dominant weight of $T(d)$ such that

(2.15) $\chi+\rho+\delta\in\mathbf{V}^{a}(1,d).$

Note that for the projection
$b\colon\mathcal{X}^{af}(1,d)\to\mathcal{X}^{a}(1,d)$, the pull-back
$b^{\ast}$ restricts to the functors

$\displaystyle
b^{\ast}\colon\mathbb{M}^{a}(1,d;\delta)\to\mathbb{F}^{a}(1,d;\delta),\,b^{\ast}\colon\mathbb{D}^{a}(1,d;\delta)\to\mathbb{E}^{a}(1,d;\delta)$

whose essential images generate $\mathbb{F}^{a}(1,d;\delta)$ and
$\mathbb{E}^{a}(1,d;\delta)$, respectively.

#### 2.7.3.

Let $\mathop{\mathrm{Tr}}\nolimits W$ be the regular function

$\mathop{\mathrm{Tr}}\nolimits W:=\mathop{\mathrm{Tr}}\nolimits
X[Y,Z]\colon\mathcal{X}(d)=\mathfrak{gl}(d)^{\oplus 3}/GL(d)\to\mathbb{C}.$

We define the subcategory

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

to 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
$\mathbb{M}(d)_{w}$. There is an equivalence:

(2.16)
$\displaystyle\otimes\det\colon\mathbb{S}(d)_{w}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{S}(d)_{d+w}.$

The category $\mathbb{S}(d)_{w}$ is called the quasi-BPS category of
$\mathbb{C}^{3}$ for $(d,w)\in\mathbb{N}\times\mathbb{Z}$ in [PTa], see [PTb]
for its properties and computations of its K-theory.

###  2.8. More on weights

#### 2.8.1.

Let $A$ be a partition $(d_{i},w_{i})_{i=1}^{k}$ of $(d,w)$ and consider its
corresponding antidominant cocharacter $\lambda$. Define the weights

$\displaystyle\chi_{A}:=\sum_{i=1}^{k}w_{i}\tau_{d_{i}},\
\chi^{\prime}_{A}:=\chi_{A}+\mathfrak{g}^{\lambda>0}.$

Consider weights $\chi^{\prime}_{i}\in M(d_{i})_{\mathbb{R}}$ such that

$\chi^{\prime}_{A}=\sum_{i=1}^{k}\chi^{\prime}_{i}.$

Let $v_{i}$ be the sum of coefficients of $\chi^{\prime}_{i}$ for $1\leqslant
i\leqslant k$; alternatively, $v_{i}:=\langle
1_{d_{i}},\chi^{\prime}_{i}\rangle$. We denote the above transformation by

(2.17) $\displaystyle A\mapsto A^{\prime},\
(d_{i},w_{i})_{i=1}^{k}\mapsto(d_{i},v_{i})_{i=1}^{k}.$

Explicitly, the weights $v_{i}$ for $1\leqslant i\leqslant k$ are given by

(2.18) $\displaystyle
v_{i}=w_{i}+d_{i}\left(\sum_{j>i}d_{j}-\sum_{j<i}d_{j}\right).$

This transformation explains how to change weights under the Koszul
equivalence [Pădb, Proposition 3.1].

#### 2.8.2.

We recall a construction from [PTa], see [PTa, Subsections 3.3.2 and 3.3.3 and
Propositions 3.4 and 3.7] which is also used in the proof of Proposition 2.2.
Let $\mu\in\mathbb{R}$, and let $\chi$ be a dominant weight such that
$\chi+\rho+\mu\sigma_{d}\in\mathbf{V}(d)$. Then there exists a unique
partition $(d_{i})_{i=1}^{k}$ of $d$, integers $(w_{i})_{i=1}^{k}$ such that
for $v_{i}$ as in (2.18), we have

(2.19)
$-1-\mu\leqslant\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}\leqslant-\mu,$

and weights $\chi_{i}\in M(d_{i})$ such that

$\chi=\sum_{i=1}^{k}\chi_{i}\text{ with
}\chi_{i}+\rho_{i}+\mu\sigma_{d_{i}}\in\mathbf{W}(d_{i})_{w_{i}}\text{ for
}1\leqslant i\leqslant k.$

Here, we denote by $\rho_{i}$ half the sum of positive roots of $GL(d_{i})$
for $1\leqslant i\leqslant k$.

The converse is also true, and it follows from [PTa, Proposition 3.8].
Consider a partition $(d_{i})_{i=1}^{k}$ of $d$, integers $(w_{i})_{i=1}^{k}$
such that for $v_{i}$ as in (2.18), the inequality (2.19) holds, and weights
$\chi_{i}$ with
$\chi_{i}+\rho_{i}+\mu\sigma_{d_{i}}\in\mathbf{W}(d_{i})_{w_{i}}$ for
$1\leqslant i\leqslant k$. Let $\chi:=\sum_{i=1}^{k}\chi_{i}$. Then we have

$\chi+\rho+\mu\sigma_{d}\in\mathbf{V}(d).$

##  3\. The categorical wall-crossing formula for DT/PT quivers

Fix $a\geqslant 1$. Let $\chi_{0}\colon GL(V)\to\mathbb{C}^{\ast}$ be the
determinant character $g\mapsto\det g$. Recall the varieties:

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

######  Remark 3.1.

It is easy to describe the $\chi_{0}^{\pm}$-semistable loci on $R^{af}(1,d)$:
a $\chi_{0}$-semistable representation (resp. $\chi_{0}^{-1}$-semistable
representation) consists of tuples

$\displaystyle(u_{1},\ldots,u_{a+1},v_{1},\ldots,v_{a},A,B,C)\in
V^{\oplus(a+1)}\oplus(V^{\vee})^{\oplus a}\oplus\mathfrak{gl}(V)^{\oplus 3}$

such that (see [Tod22, Lemma 7.10])

$\displaystyle\mathbb{C}\langle A,B,C\rangle(u_{1},\ldots,u_{a+1})=V,\
(\mathrm{resp}.~{}\mathbb{C}\langle
A,B,C\rangle(v_{1},\ldots,v_{a})=V^{\vee}).$

The main result of this section is Theorem 1.1. The most important ingredient
in its proof is the following semiorthogonal decomposition for subcategories
of $D^{b}(\mathcal{X}^{a}(1,d))$:

######  Theorem 3.2.

Let $\mu\in\mathbb{R}$ and consider the weight $\delta:=\mu\sigma_{d}\in
M(d)_{\mathbb{R}}$. There is a semiorthogonal decomposition

(3.1)
$\mathbb{D}^{a}(1,d;\delta)=\Big{\langle}\left(\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\right)\boxtimes\mathbb{M}^{a}\left(1,d^{\prime};\delta^{\prime}\right)\Big{\rangle}.$

The right hand side is after all $d^{\prime}\leqslant d$, all decompositions
$\sum_{i=1}^{k}d_{i}=d-d^{\prime}$, and all integers $w_{i}\in\mathbb{Z}$ for
$1\leqslant i\leqslant k$ such that for

(3.2)
$v_{i}:=w_{i}+d_{i}\left(d^{\prime}+\sum_{j>i}d_{j}-\sum_{j<i}d_{j}\right),$

we have

(3.3)
$-1-\mu-\frac{a}{2}\leqslant\frac{v_{1}}{d_{1}}<\cdots<\frac{v_{k}}{d_{k}}\leqslant-\mu-\frac{a}{2},$

and where $\delta^{\prime}:=\left(\mu-d+d^{\prime}\right)\sigma_{d^{\prime}}$.
The orthogonality of the categories is as in Subsections 3.2.3. The functors

$\left(\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\right)\boxtimes\mathbb{M}^{a}\left(1,d^{\prime};\delta^{\prime}\right)\to\mathbb{D}^{a}(1,d;\delta)$

are given by the Hall product (2.6).

Given $\mu\in\mathbb{R}$ and a partition $(d_{i})_{i=1}^{k}$ of
$d^{\prime}\leqslant d$, we refer to (3.3) as a condition on the tuple of
integers $(w_{i})_{i=1}^{k}$.

###  3.1. Weight decompositions

We discuss several preliminary results about decompositions of weights, or,
alternatively, about decompositions of polytopes in $M(d)_{\mathbb{R}}$.

######  Proposition 3.3.

Let $\chi$ be a strictly dominant weight such that
$\chi\in\mathbf{V}^{a}(1,d)$. There exists a decomposition

(3.4) $D:\chi=\sum_{1\leqslant j<i\leqslant
d}c_{ij}(\beta_{i}-\beta_{j})+\sum_{i=1}^{d}c_{i}\beta_{i}$

with coefficients

(3.5) $0\leqslant c_{ij}\leqslant\frac{3}{2}\text{ and
}-\frac{a+2}{2}\leqslant c_{i}\leqslant\frac{a}{2}\text{ for }1\leqslant
j<i\leqslant d.$

###### Proof.

The argument used in [PTa, Proposition 3.5] also applies here. ∎

######  Proposition 3.4.

Let $\chi$ be a strictly dominant weight such that
$\chi\in\mathbf{V}^{a}(1,d)$. Then there exists a unique $e\leqslant d$ such
that

(3.6) $\chi=\sum_{j<i\leqslant
e}c_{ij}(\beta_{i}-\beta_{j})+\sum_{e<j<i}c_{ij}(\beta_{i}-\beta_{j})+\sum_{j\leqslant
e<i}\frac{3}{2}(\beta_{i}-\beta_{j})+\sum_{1\leqslant i\leqslant
d}c_{i}\beta_{i},$

where $c_{ij}$ and $c_{i}$ satisfy

$\displaystyle 0\leqslant c_{ij}\leqslant\frac{3}{2}\mbox{ for }j<i,\
-\frac{a+2}{2}\leqslant c_{i}\leqslant-\frac{a}{2}\mbox{ if }i\leqslant e,\
-\frac{a}{2}<c_{i}\leqslant\frac{a}{2}\mbox{ if }i>e.$

Let $f:=d-e$. Alternatively, we have that

(3.7)
$\chi\in\left(-\frac{a+3f}{2}\sigma_{e}+\mathbf{V}(e)\right)+\left(\frac{3e}{2}\sigma_{f}+\mathbf{W}^{a}(1,f)\right).$

######  Step 1.

There exists such $e\leqslant d$.

###### Proof.

Consider a decomposition

(3.8) $\displaystyle D\colon\chi=\sum_{1\leqslant j<i\leqslant
d}c_{ij}(\beta_{i}-\beta_{j})+\sum_{1\leqslant i\leqslant d}c_{i}\beta_{i}$

as in Proposition 3.3. A choice of the decomposition $D$ corresponds to a
point in the fiber of the following continuous map at $\chi\in M(d)$:

$\displaystyle\gamma\colon\left[0,\frac{3}{2}\right]^{\times
d(d-1)/2}\times\left[-\frac{a+2}{2},\frac{a}{2}\right]^{\times d}\to
M(d)_{\mathbb{R}}$

defined by sending $(c_{ij},c_{i})$ to the right hand side in (3.8). For each
decomposition $D$, we set

(3.9) $\displaystyle
I:=\left\\{i\mathrel{}\middle|\mathrel{}c_{i}>-\frac{a}{2}\right\\},\
J:=\left\\{i\mathrel{}\middle|\mathrel{}c_{i}\leqslant-\frac{a}{2}\right\\}.$

Define the function $\sigma\colon\gamma^{-1}(\chi)\to\mathbb{R}$:

$\sigma(D):=\sum_{i\in I}\left(c_{i}+\frac{a}{2}\right).$

The function $\sigma$ is continuous and the set $\gamma^{-1}(\chi)$ is
compact. Among all decompositions $D$ for which $\sigma$ is minimal, choose
one for which $|I|$ is as large as possible. Let $m:=\sigma(D)$.

We first show that

(3.10) $\displaystyle c_{ij}=\frac{3}{2}\mbox{ if }j<i,(i,j)\in I\times J.$

Assume there exists $j<i$ such that $(i,j)\in I\times J$ and $c_{ij}<3/2$. We
take $0<\varepsilon\ll 1$ and set $c_{i}^{\prime}=c_{i}-\varepsilon$,
$c_{j}^{\prime}=c_{j}+\varepsilon$, $c_{ij}^{\prime}=c_{ij}+\varepsilon$.
Consider the decomposition $D^{\prime}$ with the same coefficients as $D$
except for $c^{\prime}_{ij}$, $c^{\prime}_{i}$, and $c^{\prime}_{j}$. Then
$D^{\prime}$ is a decomposition of $\chi$ because

$c_{ij}(\beta_{i}-\beta_{j})+c_{i}\beta_{i}+c_{j}\beta_{j}=(c_{ij}+\varepsilon)(\beta_{i}-\beta_{j})+(c_{i}-\varepsilon)\beta_{i}+(c_{j}+\varepsilon)\beta_{j}.$

The decomposition $D^{\prime}$ satisfies (3.5). If $c^{\prime}_{j}<-a/2$, then
we may assume that $I^{\prime}=I$ and then $\sigma(D^{\prime})<\sigma(D)$. If
$c^{\prime}_{j}=-a/2$, then $I^{\prime}=I\cup\\{j\\}$ and
$\sigma(D^{\prime})=\sigma(D)$, which contradicts the maximality of $|I|$
among all decompositions with $\sigma(D)=m$. Thus (3.10) holds.

Similarly, we show that

(3.11) $\displaystyle c_{ij}=0\mbox{ if }j<i,(i,j)\in J\times I.$

Assume there exists $i>j$ such that $(i,j)\in J\times I$ and $c_{ij}>0$. We
take $0<\varepsilon\ll 1$ and set $c_{i}^{\prime}=c_{i}+\varepsilon$,
$c_{j}^{\prime}=c_{j}-\varepsilon$, $c^{\prime}_{ij}=c_{ij}-\varepsilon$.
Consider the decomposition $D^{\prime}$ with the same coefficients as $D$
except for $c^{\prime}_{ij}$, $c^{\prime}_{i}$, and $c^{\prime}_{j}$. Then
$D^{\prime}$ is a decomposition of $\chi$ because

$c_{ij}(\beta_{i}-\beta_{j})+c_{i}\beta_{i}+c_{j}\beta_{j}=(c_{ij}-\varepsilon)(\beta_{i}-\beta_{j})+(c_{i}+\varepsilon)\beta_{i}+(c_{j}-\varepsilon)\beta_{j}$

and $D^{\prime}$ satisfies (3.5). If $c_{i}<-a/2$, then we can choose
$\varepsilon$ such that $I^{\prime}=I$ and then
$\sigma(D^{\prime})<\sigma(D)$. If $c_{i}=-a/2$, then
$I^{\prime}=I\cup\\{i\\}$ and $\sigma(D^{\prime})=\sigma(D)$, which
contradicts the maximality of $|I|$ among decompositions $D$ with
$\sigma(D)=m$. Thus (3.11) holds.

We denote $f=|I|$, $e=|J|=d-f$, and write

$\chi=\sum_{i=1}^{d}b_{i}\beta_{i},\
b_{i}=\sum_{j<i}c_{ij}-\sum_{i<j}c_{ji}+c_{i}.$

Let $i^{\prime}\in I$ and $j^{\prime}\in J$. Then from (3.10) and (3.11) and
noting that $c_{ij}\in[0,3/2]$ for all $j<i$, we have

(3.12) $\displaystyle b_{i^{\prime}}$
$\displaystyle>-\frac{a}{2}+\frac{3}{2}|J\cap\\{1,\ldots,i^{\prime}-1\\}|-\frac{3}{2}|I\cap\\{i^{\prime}+1,\ldots,d\\}|,$
$\displaystyle b_{j^{\prime}}$
$\displaystyle\leqslant-\frac{a}{2}+\frac{3}{2}|J\cap\\{1,\ldots,j^{\prime}-1\\}|-\frac{3}{2}|I\cap\\{j^{\prime}+1,\ldots,d\\}|.$

We claim that

(3.13) $J=\\{1,\ldots,e\\}\text{ and }I=\\{e+1,\ldots,d\\}.$

Assume the claim in (3.13) is false. Thus there exist $i^{\prime}\in I$,
$j^{\prime}\in J$ with $i^{\prime}+1=j^{\prime}$. Then
$b_{i^{\prime}}<b_{j^{\prime}}$ because $\chi$ is strictly dominant. Combining
inequalities (3.12), we obtain:

$-\frac{a}{2}+\frac{3}{2}|J\cap\\{1,\ldots,j^{\prime}-1\\}|-\frac{3}{2}|I\cap\\{j^{\prime}+1,\ldots,d\\}|\\\
>-\frac{a}{2}+\frac{3}{2}|J\cap\\{1,\ldots,i^{\prime}-1\\}|-\frac{3}{2}|I\cap\\{i^{\prime}+1,\ldots,d\\}|,$

and so

(3.14)
$|J\cap\\{i^{\prime},\ldots,j^{\prime}-1\\}|+|I\cap\\{i^{\prime}+1,\ldots,j^{\prime}\\}|>0.$

However,
$I\cap\\{i^{\prime}+1,\ldots,j^{\prime}\\}=I\cap\\{j^{\prime}\\}=\emptyset$
and
$J\cap\\{i^{\prime},\ldots,j^{\prime}-1\\}=J\cap\\{i^{\prime}\\}=\emptyset$.
Thus (3.14) is false and so the claim in (3.13) is true. Consider the weights
in $\mathbf{V}(e)$ and $\mathbf{W}^{a}(1,f)$, respectively:

$\displaystyle\chi_{1}$ $\displaystyle=\sum_{j<i\leqslant
e}c_{ij}(\beta_{i}-\beta_{j})+\sum_{i\leqslant
e}\left(c_{i}+\frac{a}{2}\right)\beta_{i},$ $\displaystyle\chi_{2}$
$\displaystyle=\sum_{e<j<i}c_{ij}(\beta_{i}-\beta_{j})+\sum_{i\geqslant
e+1}c_{i}\beta_{i}.$

Then

$\chi=\left(-\frac{a+3f}{2}\sigma_{e}+\chi_{1}\right)+\left(\frac{3e}{2}\sigma_{f}+\chi_{2}\right).$

The conclusion thus follows. ∎

######  Step 2.

The $e\leqslant d$ with the desired property is unique.

###### Proof.

Assume there exist $e^{\prime}<e$ such that (3.7) is satisfied for $e$ and
$e^{\prime}$. Let $f=d-e$. Recall that $\chi=\sum_{i=1}^{d}b_{i}\beta_{i}$.
The weight $\chi$ has a description (3.6) for $e^{\prime}$, so

(3.15) $\displaystyle\sum_{e^{\prime}<i\leqslant e}b_{i}$
$\displaystyle=\sum_{e^{\prime}<i\leqslant
e}\left(\sum_{e^{\prime}<j<i}c_{ij}-\sum_{i<j}c_{ji}+\frac{3}{2}e^{\prime}+c_{i}\right)$
$\displaystyle=-\sum_{e^{\prime}<i\leqslant
e<j}c_{ji}+\sum_{e^{\prime}<i\leqslant
e}\left(\frac{3}{2}e^{\prime}+c_{i}\right)$
$\displaystyle>\left(-\frac{3}{2}f+\frac{3}{2}e^{\prime}-\frac{1}{2}a\right)(e-e^{\prime}).$

Next, $\chi$ has a description (3.6) for $e$. We abuse notation and denote the
coefficients of this description also by $c_{ij}$ and $c_{i}$. Similarly to
(3.15), we have

(3.16) $\displaystyle\sum_{e^{\prime}<i\leqslant e}b_{i}$
$\displaystyle=\sum_{e^{\prime}<i\leqslant e}\left(-\sum_{i<j\leqslant
e}c_{ji}+\sum_{j<i}c_{ij}-\frac{3}{2}f+c_{i}\right)$
$\displaystyle=-\sum_{j\leqslant e^{\prime}<i\leqslant
e}c_{ij}+\sum_{e^{\prime}<i\leqslant e}\left(-\frac{3}{2}f+c_{i}\right)$
$\displaystyle\leqslant\left(\frac{3}{2}e^{\prime}-\frac{3}{2}f-\frac{1}{2}a\right)(e-e^{\prime}).$

We obtain a contradiction by comparing (3.15) and (3.16). ∎

The decomposition of $\mathbf{V}^{a}(1,2)$ into chambers constructed in
Proposition 3.4 is depicted in Figure 2.

Figure 2. Decomposition of $\mathbf{V}^{a}(1,2)$

###  3.2. Invariants of weights

#### 3.2.1.

Let $e\leqslant d$ be two natural numbers and let $f=d-e$. Consider the
antidominant cocharacter

$\tau_{e}:=(\overbrace{t,\ldots,t}^{e},\overbrace{1,\ldots,1}^{f}):\mathbb{C}^{*}\to
T(d).$

Fix $\mu\in\mathbb{R}$ and let $\delta:=\mu\sigma_{d}$. Let $\chi$ be a
dominant weight such that $\chi+\rho+\delta\in\mathbf{V}^{a}(1,d)$. Let
$e=e(\chi)$ be the integer $0\leqslant e\leqslant d$ such that

(3.17) $\displaystyle\chi+\rho+\delta=\sum_{j<i\leqslant
e}c_{ij}(\beta_{i}-\beta_{j})+\sum_{e<j<i}c_{ij}(\beta_{i}-\beta_{j})+\sum_{j\leqslant
e<i}\frac{3}{2}(\beta_{i}-\beta_{j})+\sum_{1\leqslant i\leqslant
d}c_{i}\beta_{i},$

where $c_{ij}$ and $c_{i}$ are as in Proposition 3.4. We define $p(\chi)$ to
be

$\displaystyle p(\chi)$
$\displaystyle:=\langle\tau_{e},\chi+\rho+\delta\rangle+\frac{3}{2}ef+\frac{a}{2}e=\sum_{i=1}^{e}\left(c_{i}+\frac{a}{2}\right)\leqslant
0.$

The functions $e(\chi),p(\chi)$ depend on $\mu$.

#### 3.2.2.

The following is a consequence of Proposition 3.4 and the discussion in
Subsection 2.8.2.

######  Proposition 3.5.

Let $\mu\in\mathbb{R}$, let $\delta=\mu\sigma_{d}$, and let $\chi$ be a
dominant (integral) weight such that $\chi+\rho+\delta\in\mathbf{V}^{a}(1,d)$.
Then there exists a unique $d^{\prime}\leqslant d$, partition
$(d_{i})_{i=1}^{k}$ of $e:=d-d^{\prime}$, integers $(w_{i})_{i=1}^{k}$
satisfying (3.3), and weights $\chi_{i}\in M(d_{i})$, $\chi^{\prime}\in
M(d^{\prime})$ such that $\chi=\chi_{1}+\cdots+\chi_{k}+\chi^{\prime}$ and

$\displaystyle\chi_{i}+\rho_{i}$ $\displaystyle\in\mathbf{W}(d_{i})_{w_{i}},\
\chi^{\prime}+\rho^{\prime}+(\mu-e)\sigma_{d^{\prime}}\in\mathbf{W}^{a}(1,d^{\prime}).$

In the above, $\rho_{i}$ and $\rho^{\prime}$ are half the sums of positive
roots of $GL(d_{i})$ and $GL(d^{\prime})$, respectively, where $1\leqslant
i\leqslant k$.

Let $\mu\in\mathbb{R}$, let $\delta=\mu\sigma_{d}$, and let $\chi$ be a
dominant weight such that $\chi+\rho+\delta\in\mathbf{V}^{a}(1,d)$. Let
$S(\chi)=S$ be the ordered collection of pairs $S=(d_{i},w_{i})_{i=1}^{k}$
satisfying the condition in Proposition 3.5. We say $S$ is the type of $\chi$.
Then $p(\chi)$ is alternatively written as

(3.18) $\displaystyle p(\chi)$ $\displaystyle=\sum_{1\leqslant i\leqslant
k}w_{i}-(d-d^{\prime})\left(-\mu-d^{\prime}-\frac{a}{2}\right)$
$\displaystyle=\sum_{1\leqslant i\leqslant
k}v_{i}-(d-d^{\prime})\left(-\mu-\frac{a}{2}\right)\leqslant 0.$

We also denote the above number by $p(S)$.

####  3.2.3. Comparison of ordered pairs

Fix $d\in\mathbb{N}$ and $\mu\in\mathbb{R}$. Let $R$ be the set

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

We define a subset $O\subset R\times R$ depending on $\mu$. A pair of types
$(S^{\prime},S)$ is in $O$ if either

  * •

$p(S^{\prime})>p(S)$, or

  * •

$p(S^{\prime})=p(S)$ and
$\sum_{i=1}^{k^{\prime}}d^{\prime}_{i}<\sum_{i=1}^{k}d_{i}$, or

  * •

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

If $(S^{\prime},S)$ is in $O$, we also write $S>S^{\prime}$.

For $0<\varepsilon\ll 1$, let $\mu=-a/2-\varepsilon$. Recall the definition
(3.2) of the integers $v_{i}$ from $w_{i}$. Then

$p(S)=\sum_{i=1}^{k}v_{i}-\varepsilon\sum_{i=1}^{k}d_{i}.$

In this case, the set $O$ contains pairs of types $(S^{\prime},S)$ such that

  * •

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

  * •

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

  * •

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

###  3.3. Weight comparison

We discuss some preliminary results on weights. We fix $\mu\in\mathbb{R}$ and
let $\delta:=\mu\sigma_{d}\in M(d)_{\mathbb{R}}$. For $\chi$ a weight and for
$1\leqslant l\leqslant d$, define

$\displaystyle
p_{l}(\chi):=\langle\tau_{l},\chi+\rho+\delta\rangle+\frac{3}{2}l(d-l)+\frac{a}{2}l.$

######  Lemma 3.6.

Let $\chi$ be a dominant weight such that
$\chi+\rho+\delta\in\mathbf{V}^{a}(1,d)$ and let $e=e(\chi)$. Then
$p_{l}(\chi)\geqslant p(\chi)$ and the inequality is strict if $l>e$.

###### Proof.

Note that $p_{e}(\chi)=p(\chi)$. We write $\chi+\rho+\delta$ as (3.17), where
$c_{ij}$ and $c_{i}$ are as in Proposition 3.4. We have

$\displaystyle p_{l}(\chi)$ $\displaystyle=-\sum_{i\leqslant
l<j}c_{ji}+\sum_{i\leqslant l}c_{i}+\frac{3}{2}l(d-l)+\frac{a}{2}l$
$\displaystyle\geqslant\sum_{i\leqslant
l}\left(c_{i}+\frac{a}{2}\right)\stackrel{{\scriptstyle(\star)}}{{\geqslant}}\sum_{i\leqslant
e}\left(c_{i}+\frac{a}{2}\right)=p(\chi).$

The inequality $(\star)$ holds from Lemma 3.7, and is strict if $l>e$. ∎

We have used the following lemma, whose proof is obvious:

######  Lemma 3.7.

Let $x_{1},\ldots,x_{d}\in\mathbb{R}$ and let $S:=\\{i\mid x_{i}\leqslant
0\\}\subset\\{1,\ldots,d\\}$. Then for any subset $T\subset\\{1,\ldots,d\\}$,
we have

$\displaystyle\sum_{i\in T}x_{i}\geqslant\sum_{i\in S}x_{i}.$

If equality holds, then $T\subset S$. If equality holds and $\lvert
T\rvert=\lvert S\rvert$, then $T=S$.

For $e\leqslant d$, consider the multiset of weights
$\mathcal{W}^{a}_{e}:=R^{a}(1,d)^{\tau_{e}<0}$, or explicitly

$\displaystyle\mathcal{W}^{a}_{e}=\\{(\beta_{i}-\beta_{j})^{\times
3},(-\beta_{j})^{\times a}\mid j\leqslant e<i\\}.$

######  Proposition 3.8.

Let $\chi$, $\chi^{\prime}$ be dominant weights such that
$\chi+\rho+\delta\in\mathbf{V}^{a}(1,d)$ and
$\chi^{\prime}+\rho+\delta\in\mathbf{V}^{a}(1,d)$. Let $e=e(\chi)$,
$e^{\prime}=e(\chi^{\prime})$, and let $I$ be a subset
$\mathcal{W}_{e^{\prime}}^{a}$. Let
$e^{\prime\prime}=e((\chi^{\prime}-\sigma_{I})^{+})$. Suppose that either

  1. (i)

$p(\chi^{\prime})>p(\chi)$ or

  2. (ii)

$p(\chi^{\prime})=p(\chi),e^{\prime}<e$ or

  3. (iii)

$p(\chi^{\prime})=p(\chi)$, $e^{\prime}=e$, and $I\neq\emptyset$.

Then we have

(3.19) $\displaystyle p_{e}((\chi^{\prime}-\sigma_{I})^{+})\geqslant
p((\chi^{\prime}-\sigma_{I})^{+})\geqslant p(\chi).$

Moreover, if the second inequality is an equality, then
$p(\chi^{\prime})=p(\chi)$ and $e^{\prime\prime}\leqslant e^{\prime}$, and
$e^{\prime\prime}<e^{\prime}$ if $I\neq\emptyset$. Hence $e^{\prime\prime}<e$,
and thus the first inequality is strict by Lemma 3.6. In particular, in each
of the above three cases, we have

$\displaystyle\langle\tau_{e},(\chi^{\prime}-\sigma_{I})^{+}\rangle>\langle\tau_{e},\chi\rangle.$

###### Proof.

As the first inequality in (3.19) follows from Lemma 3.6, it is enough to
prove the second inequality. Write

$\displaystyle\chi^{\prime}+\rho+\delta$ $\displaystyle=\sum_{1\leqslant
i,j\leqslant d}c^{\prime}_{ij}(\beta_{i}-\beta_{j})+\sum_{1\leqslant
i\leqslant d}c^{\prime}_{i}\beta_{i},$
$\displaystyle\chi^{\prime}-\sigma_{I}+\rho+\delta$
$\displaystyle=\sum_{1\leqslant i,j\leqslant
d}\widetilde{c}_{ij}(\beta_{i}-\beta_{j})+\sum_{1\leqslant i\leqslant
d}\widetilde{c}_{i}\beta_{i},$

where $c^{\prime}_{ij},c^{\prime}_{i}$ are as in Proposition 3.4. Then
$|\widetilde{c}_{ij}|\leqslant 3/2$ for all $1\leqslant i,j\leqslant d$, and
$\widetilde{c}_{i}\geqslant c^{\prime}_{i}$ for $1\leqslant i\leqslant d$. Let
$w\in\mathfrak{S}_{d}$ be such that
$(\chi^{\prime}-\sigma_{I})^{+}+\rho=w(\chi^{\prime}-\sigma_{I}+\rho)$. Then
using Lemma 3.7 and the assumption (i) or (ii) or (iii), we have

(3.20) $\displaystyle p((\chi^{\prime}-\sigma_{I})^{+})$
$\displaystyle=\sum_{w(i)\leqslant
e^{\prime\prime}}\left(\widetilde{c}_{i}+\frac{a}{2}\right)\stackrel{{\scriptstyle(\star)}}{{\geqslant}}\sum_{i=1}^{e^{\prime\prime}}\left(\widetilde{c}_{i}+\frac{a}{2}\right)\geqslant\sum_{i=1}^{e^{\prime\prime}}\left(c_{i}^{\prime}+\frac{a}{2}\right)$
$\displaystyle\stackrel{{\scriptstyle(\diamondsuit)}}{{\geqslant}}\sum_{i=1}^{e^{\prime}}\left(c_{i}^{\prime}+\frac{a}{2}\right)=p(\chi^{\prime})\geqslant
p(\chi).$

Therefore the second inequality in (3.19) also holds. Suppose that every
inequality in (3.20) is an equality. Then $p(\chi)=p(\chi^{\prime})$ and
$e^{\prime\prime}\leqslant e^{\prime}$ from the equality of $(\diamondsuit)$
and Lemma 3.7. Assume that $e^{\prime\prime}=e^{\prime}$ and $I\neq\emptyset$.
From the equality of $(\star)$ and $e^{\prime\prime}=e^{\prime}$, by Lemma 3.7
we have $\\{w(1),\ldots,w(e^{\prime})\\}=\\{1,\ldots,e^{\prime}\\}$. Then we
have

$\displaystyle\langle\tau_{e^{\prime}},(\chi^{\prime}-\sigma_{I})^{+}\rangle=\langle\tau_{e^{\prime}},\chi^{\prime}-\sigma_{I}\rangle=\langle\tau_{e^{\prime}},\chi^{\prime}\rangle-\langle\tau_{e^{\prime}},\sigma_{I}\rangle.$

As $I\subset\mathcal{W}_{e^{\prime}}^{a}$ is non-empty, we have
$\langle\tau_{e^{\prime}},\sigma_{I}\rangle<0$. Then
$p((\chi^{\prime}-\sigma_{I})^{+})>p(\chi^{\prime})$, which contradicts that
(3.20) are equalities. ∎

###  3.4. Generation

Fix $\mu\in\mathbb{R}$ and let $\delta:=\mu\sigma_{d}\in M(d)_{\mathbb{R}}$.

######  Proposition 3.9.

The category $\mathbb{D}^{a}(1,d;\delta)$ is generated by the images of the
Hall products

(3.21)
$\left(\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\right)\boxtimes\mathbb{M}^{a}\left(1,d^{\prime};\delta^{\prime}\right)\to\mathbb{D}^{a}(1,d;\delta)$

for all $d^{\prime}\leqslant d$, all decompositions $(d_{i})_{i=1}^{k}$ of
$d-d^{\prime}$, and tuples of integers $(w_{i})_{i=1}^{k}$ satisfying (3.3),
and where $\delta^{\prime}:=\left(\mu-d+d^{\prime}\right)\sigma_{d^{\prime}}$.

We first explain that the images of the functors (3.21) are indeed in the
category $\mathbb{D}^{a}(1,d;\delta)$.

######  Proposition 3.10.

Let $(d_{i},w_{i})_{i=1}^{k}$ and $d^{\prime}$ be as above. Then the image of
the Hall product

$\left(\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\right)\boxtimes\mathbb{M}^{a}\left(1,d^{\prime};\delta^{\prime}\right)\to
D^{b}(\mathcal{X}^{a}(1,d))$

is inside $\mathbb{D}^{a}(1,d;\delta)$.

###### Proof.

By Proposition 2.2, it suffices to show that the image of the functor

$p_{e,d^{\prime}*}q_{e,d^{\prime}}^{*}\colon\mathbb{D}(e;\delta_{e})\boxtimes\mathbb{M}^{a}(1,d^{\prime};\delta^{\prime})\to
D^{b}(\mathcal{X}^{a}(1,d))$

lies in $\mathbb{D}^{a}(1,d;\delta)$, where
$\delta_{e}:=\left(\mu+d^{\prime}+a/2\right)\sigma_{e}$. Consider weights
$\chi_{e}\in M(e)$ and $\chi_{d^{\prime}}\in M(d^{\prime})$ such that
$\chi_{e}+\rho_{e}+\delta_{e}\in\mathbf{V}(e)$ and
$\chi_{d^{\prime}}+\rho_{d^{\prime}}+\delta^{\prime}\in\mathbf{W}^{a}(1,d^{\prime})$.
It suffices to check that

(3.22) $\displaystyle
p_{e,d^{\prime}*}q_{e,d^{\prime}}^{*}\left(\mathcal{O}_{\mathcal{Y}}\otimes\Gamma_{L}(\chi_{e}+\chi_{d^{\prime}})\right)\in\mathbb{D}^{a}(1,d;\delta),$

where $L:=GL(e)\times GL(d^{\prime})$ and
$\mathcal{Y}:=\mathcal{X}(e)\times\mathcal{X}(d^{\prime})$. Let
$\chi:=\chi_{e}+\chi_{d^{\prime}}$. We have

$\displaystyle\chi+\rho+\delta$
$\displaystyle=(\chi_{e}+\rho_{e}+\delta_{e})+(\chi_{d^{\prime}}+\rho_{d^{\prime}}+\delta^{\prime})+(\rho-\rho_{e}-\rho_{d^{\prime}})+(\delta-\delta_{e}-\delta^{\prime})$
$\displaystyle\in\mathbf{V}(e)+\mathbf{W}^{a}(1,d^{\prime})-\left(\frac{3}{2}d^{\prime}+\frac{a}{2}\right)\sigma_{e}+\frac{3}{2}e\sigma_{d^{\prime}}.$

Then we can write

$\chi+\rho+\delta=\sum_{i,j\leqslant
d}c_{ij}(\beta_{i}-\beta_{j})+\sum_{i\leqslant d}c_{i}\beta_{i}$

where $c_{ij}$ and $c_{i}$ satisfy $|c_{ij}|\leqslant 3/2$ with $c_{ij}=3/2$
and $c_{ji}=0$ for $j\leqslant e<i$, $-1\leqslant c_{i}+a/2\leqslant 0$ for
$1\leqslant i\leqslant e$, $|c_{i}|\leqslant a/2$ for $e<i\leqslant d$. We
take $I\subset\mathcal{W}^{a}_{e}$ and write $\sigma_{I}$ as

$\displaystyle\sigma_{I}=\sum_{j\leqslant
e<i}f_{ij}(\beta_{i}-\beta_{j})-\sum_{i\leqslant e}f_{i}\beta_{i}$

for $0\leqslant f_{ij}\leqslant 3$ and $0\leqslant f_{i}\leqslant a$. Then
$\chi-\sigma_{I}+\rho+\delta$ is in $\mathbf{V}^{a}(1,d)$. The polytope
$\mathbf{V}^{a}(1,d)$ is Weyl invariant, so
$(\chi-\sigma_{I})^{+}+\rho+\delta$ is in $\mathbf{V}^{a}(1,d)$. Thus we have
(3.22) by Proposition 2.1. ∎

###### Proof of Proposition 3.9.

By Proposition 2.2, it suffices to check that the category
$\mathbb{D}^{a}(1,d;\delta)$ is generated by the images of the Hall products

$p_{e,d^{\prime}*}q_{e,d^{\prime}}^{*}\colon\mathbb{D}(e;\delta_{e})\boxtimes\mathbb{M}^{a}(1,d^{\prime};\delta^{\prime})\to\mathbb{D}^{a}(1,d;\delta)$

for all $e\leqslant d$, $d^{\prime}=d-e$, and
$\delta_{e}:=\left(\mu+d^{\prime}+a/2\right)\sigma_{e}$. Let $\chi$ be a
dominant weight such that $\chi+\rho+\delta\in\mathbf{V}^{a}(1,d;\delta)$. We
need to show that $\mathcal{O}_{\mathcal{X}(d)}\otimes\Gamma_{GL(d)}(\chi)$ is
generated by the images of these functors. The function $p(\chi)$ takes
finitely many values. We use induction on the pair $(-p(\chi),e(\chi))$
ordered lexicographically. Use Proposition 3.4 for $\chi+\rho+\delta$ and
write $\chi=\chi_{e}+\chi_{d^{\prime}}$ such that

$\chi_{e}+\rho_{e}+\delta_{e}\in\mathbf{V}(e),\
\chi_{d^{\prime}}+\rho_{d^{\prime}}+\delta^{\prime}\in\mathbf{W}^{a}(1,d^{\prime}).$

By Proposition 2.1, the cone of the map

$\mathcal{O}_{\mathcal{X}(d)}\otimes\Gamma_{GL(d)}(\chi)\to
p_{e,d^{\prime}*}q_{e,d^{\prime}}^{*}\left(\mathcal{O}_{\mathcal{Y}}\otimes\Gamma_{L}(\chi_{e}+\chi_{d^{\prime}})\right)$

is generated by the vector bundles
$\mathcal{O}_{\mathcal{X}(d)}\otimes\Gamma_{GL(d)}((\chi-\sigma_{I})^{+})$ for
non-empty multisets $I\subset\mathcal{W}^{a}_{e}$.

If $(-p(\chi),e(\chi))=(0,0)$, then
$\chi+\rho+\delta\in\mathbf{W}^{a}(1,d;\delta)$. Assume $(-p(\chi),e(\chi))$
is different from $(0,0)$ and let $I$ be a non-empty subset of
$\mathcal{W}^{a}_{e}$. Then as in the proof of Proposition 3.10, we have
$(\chi-\sigma_{I})^{+}+\rho+\delta\in\mathbf{V}^{a}(1,d)$. By Proposition 3.8,
either $-p((\chi-\sigma_{I})^{+})<-p(\chi)$ or
$p((\chi-\sigma_{I})^{+})=p(\chi)$ and $e((\chi-\sigma_{I})^{+})<e(\chi)$. The
claim thus follows by the induction of $(-p(\chi),e(\chi))$. ∎

###  3.5. Orthogonality

In this subsection, we discuss orthogonality of the categories appearing on
the right hand side of (3.1). For $S=(d_{i},w_{i})_{i=1}^{k}$, let
$e=\sum_{i=1}^{k}d_{i}$ and $f:=d-e$. We denote by $\lambda_{S}$ the
antidominant cocharacter

(3.23)
$\lambda_{S}:=(\overbrace{t^{k},\ldots,t^{k}}^{d_{1}},\ldots,\overbrace{t,\ldots,t}^{d_{k}},\overbrace{1,\ldots,1}^{f}):\mathbb{C}^{*}\to
T(d).$

In order to simplify the notation, we set $\mathcal{X}:=\mathcal{X}^{a}(1,d)$,
$\mathcal{Y}:=\mathcal{X}^{a}(1,d)^{\lambda_{S}\geqslant 0}$ and
$\mathcal{Z}:=\mathcal{X}^{a}(1,d)^{\lambda_{S}}$. We have the maps from the
attracting stacks for the antidominant cocharacter $\lambda_{S}$, see (2.5)

$\displaystyle\mathcal{Z}\stackrel{{\scriptstyle
q}}{{\leftarrow}}\mathcal{Y}\stackrel{{\scriptstyle p}}{{\to}}\mathcal{X}.$

For another $S^{\prime}=(d_{i}^{\prime},w_{i}^{\prime})$, we define
$(e^{\prime},f^{\prime},\lambda_{S^{\prime}},\mathcal{X}^{\prime},\mathcal{Y}^{\prime},\mathcal{Z}^{\prime},p^{\prime},q^{\prime})$
in a similar way.

######  Proposition 3.11.

Suppose that $(S^{\prime},S)\in O$, see Subsection 3.2.3. Let
$\mu\in\mathbb{R}$, and set $\delta_{f}=(\mu-e)\sigma_{f}\in
M(f)_{\mathbb{R}}$ and
$\delta_{f^{\prime}}=(\mu-e^{\prime})\sigma_{f^{\prime}}\in
M(f^{\prime})_{\mathbb{R}}$. Let $A,\widetilde{A}$ and $B$ be objects

$\displaystyle
A,\widetilde{A}\in\left(\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\right)\boxtimes\mathbb{M}^{a}\left(1,f;\delta_{f}\right),\
B\in\left(\boxtimes_{i=1}^{k^{\prime}}\mathbb{M}(d^{\prime}_{i})_{w^{\prime}_{i}}\right)\boxtimes\mathbb{M}^{a}\left(1,f^{\prime};\delta_{f^{\prime}}\right).$

We have

$\displaystyle\mathrm{Hom}_{\mathcal{X}}(p^{\prime}_{*}q^{\prime*}B,p_{*}q^{*}A)=0,\
\mathrm{Hom}_{\mathcal{Z}}(\widetilde{A},A)\stackrel{{\scriptstyle\cong}}{{\to}}\mathrm{Hom}_{\mathcal{X}}(p_{*}q^{*}\widetilde{A},p_{*}q^{*}A).$

###### Proof.

We discuss the first equality. Let $\chi$ and $\chi^{\prime}$ be dominant
weights such that

$\displaystyle\chi+\rho+\delta\in\mathbf{V}^{a}(1,d),\
\chi^{\prime}+\rho+\delta\in\mathbf{V}^{a}(1,d),$

which are of types $S$, $S^{\prime}$ respectively. Consider the Levi group
$L:=GL(d)^{\lambda_{S}}$, and define $L^{\prime}$ analogously for
$S^{\prime}$. We may assume that
$A=\mathcal{O}_{\mathcal{Z}}\otimes\Gamma_{L}(\chi)$ and
$B=\mathcal{O}_{\mathcal{Z}^{\prime}}\otimes\Gamma_{L^{\prime}}(\chi^{\prime})$.
By adjunction, it suffices to check that

$\text{Hom}_{\mathcal{Y}}(p^{*}p^{\prime}_{*}q^{\prime*}B,q^{*}A)=0.$

Assume first that $e=e^{\prime}$ and
$\sum_{i=1}^{k}w_{i}=\sum_{i=1}^{k^{\prime}}w^{\prime}_{i}$. Then the
conclusion follows as in [PTa, Proposition 3.9]. More explicitly, by [Pădc,
Proposition 4.2] and Proposition 2.1, it suffices to check that the
$\lambda_{S}$-weights of the vector bundles in the resolution of
$p^{*}p^{\prime}_{*}q^{\prime*}B$ are strictly greater than the
$\lambda_{S}$-weight of $\chi$, and this is verified in [Pădc, Proposition
4.3].

Assume next that either $p(S^{\prime})>p(S)$ or $p(S)=p(S^{\prime})$ and
$e>e^{\prime}$. By [Pădc, Proposition 4.2] and Proposition 2.1, it suffices to
check that

(3.24)
$\displaystyle\langle\tau_{e},(\chi^{\prime}-\sigma_{I})^{+}\rangle>\langle\tau_{e},\chi\rangle$

for all $I\subset\mathcal{W}^{a}_{S^{\prime}}$, where
$\mathcal{W}^{a}_{S^{\prime}}$ be the multiset of weights of
$R^{a}(1,d)^{\lambda_{S^{\prime}}<0}$. Since
$\mathcal{W}_{S^{\prime}}^{a}\subset\mathcal{W}_{e}^{a}$, the inequality
(3.24) follows from Proposition 3.8. Note that [Pădc, Proposition 4.2] can be
applied ad litteram when comparing $\lambda_{S}$-weights, however the proof in
loc. cit. applies to $\tau_{e}$-weights as well because $\tau_{e}$ fixes
$R^{a}(d)^{\lambda_{S}}$ and acts with non-negative weights on
$R^{a}(d)^{\lambda_{S}<0}$.

We next discuss the second isomorphism. By [PTa, Proposition 3.9], it suffices
to show that the functor

$p_{*}q^{*}\colon\mathbb{D}(e;\delta_{e})\boxtimes\mathbb{M}^{a}(1,f;\delta_{f})\to\mathbb{D}^{a}(1,d;\delta)$

is fully faithful, where $\delta_{e}:=\left(\mu+f+a/2\right)\sigma_{e}$. We
may assume that
$\widetilde{A}=\mathcal{O}_{\mathcal{Z}}\otimes\Gamma_{L}(\widetilde{\chi})$
for $\widetilde{\chi}$ a dominant weight of type $S$. By adjunction, it
suffices to show that

$\operatorname{Hom}_{\mathcal{Y}}(p^{*}p_{*}q^{*}\widetilde{A},q^{*}A)\cong\operatorname{Hom}_{\mathcal{Y}}(q^{*}\widetilde{A},q^{*}A)\cong\operatorname{Hom}_{\mathcal{Z}}(\widetilde{A},A).$

Let $C$ be the cone of the morphism $p^{*}p_{*}q^{*}\widetilde{A}\to
q^{*}\widetilde{A}$. It suffices to check that
$\operatorname{Hom}_{\mathcal{Y}}(C,q^{*}A)=0$. By [Pădc, Proposition 4.2] and
Proposition 2.1, it suffices to check that

$\langle\tau_{e},(\widetilde{\chi}-\sigma_{I})^{+}\rangle>\langle\tau_{e},\chi\rangle$

for all non-empty $I\subset\mathcal{W}^{a}_{S^{\prime}}$. This follows from
Proposition 3.8. ∎

###  3.6. Proofs of the theorems

In this subsection, we prove Theorems 3.2 and 1.1 and Corollary 1.2.

###### Proof of Theorem 3.2.

The statement follows from Propositions 3.9 and 3.11. ∎

Recall the subcategory $\mathbb{E}^{a}(1,d;\delta)$ of
$D^{b}(\mathcal{X}^{af}(1,d))$ from Subsection 2.7.2. Consider the projection
map $b\colon\mathcal{X}^{af}(1,d)\to\mathcal{X}^{a}(1,d)$.

######  Proposition 3.12.

There is a semiorthogonal decomposition

(3.25)
$\mathbb{E}^{a}(1,d;\delta)=\Big{\langle}\left(\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\right)\boxtimes\mathbb{F}^{a}(1,d^{\prime};\delta^{\prime})\Big{\rangle},$

where the right hand side is as in Theorem 3.2.

###### Proof.

By Theorem 3.2, the result follows as [PTa, Corollary 3.11] using [PTa,
Proposition 3.10]. ∎

######  Proposition 3.13.

Let $\mu\in\mathbb{R}$ with $2\mu l\notin\mathbb{Z}$ for $1\leqslant
l\leqslant d$ and let $\delta:=\mu\sigma_{d}$. Consider the inclusions

$\iota\colon
I^{a}(d)\hookrightarrow\mathcal{X}^{af}(1,d),\,\iota^{\prime}\colon
P^{a}(d)\hookrightarrow\mathcal{X}^{af}(1,d).$

Then the following functors are equivalences

$\displaystyle\mathbb{E}^{a}(1,d;\delta)$ $\displaystyle\hookrightarrow
D^{b}(\mathcal{X}^{af}(1,d))\xrightarrow{\iota^{*}}D^{b}(I^{a}(d)),$
$\displaystyle\mathbb{F}^{a}(1,d;\delta)$ $\displaystyle\hookrightarrow
D^{b}(\mathcal{X}^{af}(1,d))\xrightarrow{\iota^{\prime*}}D^{b}(P^{a}(d)).$

###### Proof.

The same proof as in [PTa, Proposition 3.13] applies here. We refer to loc.
cit. for full details. The main tools used in the proof are window categories
[HL15] and the magic window theorem for symmetric representations [HLS20].

The Kempf-Ness loci for $I^{a}(d)$ and $P^{a}(d)$ are attracting loci for the
cocharacters $\tau_{e}^{-1}$ and $\tau_{e}$, respectively, see [PTa, Lemma
3.14]. We first explain fully faithfulness of the first functor. Let $I$ be
the set of Kempf-Ness strata and consider the stratification

$\mathcal{X}^{af}(1,d)=I^{a}(d)\sqcup\bigsqcup_{i\in I}\mathcal{S}_{i}.$

Let $\lambda_{i}=\tau_{e_{i}}^{-1}$ be the cocharacter corresponding to
$\mathcal{S}_{i}$, let $\mathcal{Z}_{i}:=\mathcal{S}_{i}^{\lambda_{i}}$, and
let

$\displaystyle\eta_{i}:=\langle\lambda_{i},(\mathcal{X}^{af}(1,d)^{\vee})^{\lambda_{i}>0}\rangle=-\langle\lambda_{i},\mathcal{X}^{af}(1,d)^{\lambda_{i}<0}\rangle=(a+1)e_{i}+2e_{i}(d-e_{i}).$

Let $\mathbb{G}_{\eta}\subset D^{b}\left(\mathcal{X}^{af}(1,d)\right)$ be the
subcategory of complexes $\mathcal{F}$ such that

(3.26)
$\displaystyle\mathrm{wt}_{\lambda_{i}}(\mathcal{F}|_{\mathcal{Z}_{i}})+\left\langle\lambda_{i},\delta+\frac{1}{2}d\tau_{d}\right\rangle\subset\left[-\frac{1}{2}\eta_{i},\frac{1}{2}\eta_{i}\right)$

for all $i\in I$. By [HL15, Theorem 2.10], the restriction $\iota^{*}$ induces
an equivalence

(3.27) $\iota^{*}\colon\mathbb{G}_{\eta}\xrightarrow{\sim}D^{b}(I^{a}(d)).$

On the other hand, as $\lambda_{i}=\tau_{e_{i}}^{-1}$, we have that
$\mathcal{X}^{af}(1,d)^{\lambda_{i}<0}=\mathcal{X}^{a+1}(1,d)^{\lambda_{i}<0}$,
and thus

(3.28)
$\displaystyle\eta_{i}=\langle\lambda_{i},\mathcal{X}^{a+1}(1,d)^{\lambda_{i}>0}\rangle.$

By the assumption $2\mu l\notin\mathbb{Z}$ for $1\leqslant l\leqslant d$ and
$\langle\lambda_{i},\delta\rangle=-\mu e$,
$\langle\lambda_{i},d\tau_{d}\rangle=-e$, the condition (3.26) is equivalent
to

$\displaystyle\mathrm{wt}_{\lambda_{i}}(\mathcal{F}|_{\mathcal{Z}_{i}})+\left\langle\lambda_{i},\delta+\frac{1}{2}d\tau_{d}\right\rangle\subset\left[-\frac{1}{2}\eta_{i},\frac{1}{2}\eta_{i}\right].$

Thus $\mathbb{E}(d;\delta)\subset\mathbb{G}_{\eta}$ by (3.28) and [HLS20,
Lemma 2.9], noting that $R^{a+1}(1,d)$ is a symmetric $GL(d)$-representation.
Fully faithfulness of the second functor follows similarly.

We next discuss essential surjectivity. We have the projection maps

$\displaystyle\mathcal{X}^{a+1}(1,d)\stackrel{{\scriptstyle
c}}{{\to}}\mathcal{X}^{af}(1,d)\stackrel{{\scriptstyle
b}}{{\to}}\mathcal{X}^{a}(1,d)$

where $c$ (resp. $b$) forgets one arrow from $1$ to $0$, (resp. $0$ to $1$).
The maps $c$, $b$ are affine bundles of relative dimension $d$. By Remark 3.1,
the $\chi_{0}$-stability (resp. $\chi_{0}^{-1}$-stability) on $R^{af}(1,d)$
does not impose constraint on maps corresponding to edges from $1$ to $0$
(resp. $0$ to $1$). We have similar descriptions of $\chi_{0}$-semistable loci
for $R^{a+1}(1,d)$ (resp. $\chi^{-1}_{0}$-semistable loci for $R^{a}(1,d)$),
see [Tod22, Lemma 7.10]. Therefore the projection maps $c$, $b$ restrict to
affine bundles of relative dimension $d$

$\displaystyle c\colon\mathcal{X}^{a+1}(1,d)^{\chi_{0}\text{-ss}}\to
I^{a}(d),\ b\colon P^{a}(d)\to\mathcal{X}^{a}(1,d)^{\chi_{0}^{-1}\text{-ss}}.$

Then the essential surjectivity follows from the magic window theorem [HLS20,
Theorem 3.2] for the symmetric stacks $\mathcal{X}^{a+1}(a,d)$,
$\mathcal{X}^{a}(1,d)$ as in [PTa, Proof of Proposition 3.13]. ∎

###### Proof of Theorem 1.1.

The claim follows from Propositions 3.12 and 3.13, noting that the most left
and right inequalities in (3.3) are never equalities by the condition $2\mu
l\notin\mathbb{Z}$ for $1\leqslant l\leqslant d$. ∎

###### Proof of Corollary 1.2.

Taking the Grothendieck group of the categories in (1.1), we obtain the
isomorphism

(3.29) $K(I^{a}(d))\cong\bigoplus
K\left((\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}})\boxtimes
P^{a}(d^{\prime})\right),$

where the sum of the right hand side is as in Theorem 1.1. Choose
$d^{\prime}\leqslant d$. By [PTa, Theorem 1.1], there is a semiorthogonal
decomposition

(3.30)
$D^{b}(\mathrm{NHilb}(d-d^{\prime}))=\Big{\langle}\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}}\Big{\rangle}$

where the right hand side is after all partitions $(d_{i})_{i=1}^{k}$ of
$d-d^{\prime}$ and integers $(w_{i})_{i=1}^{k}$ such that for
$v_{i}:=w_{i}+d_{i}\left(d^{\prime}+\sum_{j>i}d_{j}-\sum_{j<i}d_{j}\right)$,
the inequality (1.2) is satisfied. There is thus a semiorthogonal
decomposition

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

and thus a decomposition in K-theory:

$K\left(\mathrm{NHilb}(d-d^{\prime})\times
P^{a}(d^{\prime})\right)\cong\bigoplus
K\left((\boxtimes_{i=1}^{k}\mathbb{M}(d_{i})_{w_{i}})\boxtimes
P^{a}(d^{\prime})\right),$

where the right hand side is as for (3.30). By (3.29), we obtain

(3.31)
$K(I^{a}(d))\cong\bigoplus_{d^{\prime}=0}^{d}K\left(\mathrm{NHilb}(d-d^{\prime})\times
P^{a}(d^{\prime})\right).$

We finally show that there is a Künneth isomorphism

(3.32) $K\left(\mathrm{NHilb}(d-d^{\prime})\times
P^{a}(d^{\prime})\right)\cong K(\mathrm{NHilb}(d-d^{\prime}))\otimes
K(P^{a}(d^{\prime})),$

which implies the conclusion by (3.31). We use [HL15, Theorem 2.10] to choose
window subcategories $\mathbb{G}\subset
D^{b}\left(\mathcal{X}^{f}(1,d-d^{\prime})\right)$ and $\mathbb{H}\subset
D^{b}\left(\mathcal{X}^{af}(1,d^{\prime})\right)$ such that the restriction
maps to the stable locus give equivalences

$\mathbb{G}\xrightarrow{\sim}D^{b}(\mathrm{NHilb}(d-d^{\prime})),\,\mathbb{H}\xrightarrow{\sim}D^{b}(P^{a}(d^{\prime})).$

Then, by loc. cit., the category $\mathbb{G}\boxtimes\,\mathbb{H}$ is part of
a semiorthogonal decomposition of
$D^{b}\left(\mathcal{X}^{f}(1,d-d^{\prime})\times\mathcal{X}^{af}(1,d^{\prime})\right)$
such that the restriction map gives an equivalence

$\mathbb{G}\boxtimes\mathbb{H}\xrightarrow{\sim}D^{b}\left(\mathrm{NHilb}(d-d^{\prime})\times
P^{a}(d^{\prime})\right).$

The following diagram commutes and the spaces on the left column are summands
of the spaces of the right column:

${K(\mathbb{G})\otimes
K(\mathbb{H})}$${K(\mathcal{X}^{f}(d-d^{\prime}))\otimes
K(\mathcal{X}^{af}(d^{\prime}))}$${K(\mathbb{G}\boxtimes\,\mathbb{H})}$${K(\mathcal{X}^{f}(d-d^{\prime})\times\mathcal{X}^{af}(d^{\prime})).}$$\scriptstyle{a}$$\scriptstyle{b}$

To show that $a$ is an isomorphism, it suffices to show that $b$ is an
isomorphism, which is clear because $K(\mathcal{X}^{af}(1,d^{\prime}))\cong
K_{GL(d^{\prime})}(\text{pt})$, $K(\mathcal{X}^{f}(1,d-d^{\prime}))\cong
K_{GL(d-d^{\prime})}(\text{pt})$, and
$K(\mathcal{X}^{f}(1,d-d^{\prime})\times\mathcal{X}^{af}(d^{\prime}))\cong
K_{GL(d-d^{\prime})\times GL(d^{\prime})}(\text{pt})$. ∎

###  3.7. A variant of Theorem 1.1

We discuss a slight extension of Theorem 1.1 for quivers obtained from
$Q^{af}$ by adding loops at the vertex $0$ and imposing a super-potential. We
will use this extension in the proof of Theorem 1.4.

Let $Q^{af,N}$ be the quiver adding $N$-loops at the vertex $0$. Then the
affine space of representations of $Q^{af,N}$ of dimension $(1,d)$ is

$\displaystyle R^{af,N}(1,d)=\mathbb{C}^{N}\times R^{af}(1,d).$

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. We define

$\displaystyle\mathcal{DT}^{a,N}_{\widetilde{W}}(d)$
$\displaystyle:=\mathrm{MF}\left(R^{af,N}(1,d)^{\chi_{0}\text{-ss}}/GL(d),\mathop{\mathrm{Tr}}\nolimits\widetilde{W}\right),$
$\displaystyle\mathcal{PT}^{a,N}_{\widetilde{W}}(d)$
$\displaystyle:=\mathrm{MF}\left(R^{af,N}(1,d)^{\chi_{0}^{-1}\text{-ss}}/GL(d),\mathop{\mathrm{Tr}}\nolimits\widetilde{W}\right).$

By applying $D^{b}(\mathbb{C}^{N})\boxtimes$ to the semiorthogonal
decomposition in Theorem 1.1, and the super-potential
$\mathop{\mathrm{Tr}}\nolimits\widetilde{W}$, see [Păda, Proposition 2.1], we
have the following corollary:

######  Corollary 3.14.

Let $\mu\in\mathbb{R}$ such that $2\mu l\notin\mathbb{Z}$ for $1\leqslant
l\leqslant d$. There is a semiorthogonal decomposition

$\mathcal{DT}^{a,N}_{\widetilde{W}}(d)=\Big{\langle}\left(\boxtimes_{i=1}^{k}\mathbb{S}(d_{i})_{w_{i}}\right)\boxtimes\mathcal{PT}^{a,N}_{\widetilde{W}}(d^{\prime})\Big{\rangle}.$

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

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

The order of the semiorthogonal summands is the same as in Theorem 1.1.

The analogous conclusion holds if we replace $\mathbb{C}^{N}$ by an open
subset in all the constructions and statements above.

##  4\. The categorical DT/PT correspondence for $\mathbb{C}^{3}$

In this section, we prove Theorems 1.3 and 1.4 and Corollary 1.5. We give
explicit descriptions of DT/PT moduli spaces on $\mathbb{C}^{3}$ with reduced
supports via extended ADHM quivers. The ADHM quiver is a quiver with a
relation (depicted in Figure 3) which was used to construct framed moduli
spaces of instantons on $\mathbb{P}^{2}$, see [Nak99].

$\displaystyle[A,B]+u\circ v=0.$ Figure 3. ADHM quiver

We use an interpretation of representations of the ADHM quiver in terms of
perverse coherent sheaves, and then relate them to moduli spaces of pairs of
one-dimensional sheaves with sections.

###  4.1. Perverse coherent sheaves via the ADHM construction

Below we regard $\mathbb{C}^{2}$ as an open subset of $\mathbb{P}^{2}$
consisting of $[X\colon Y\colon Z]$ such that $Z\neq 0$, and set
$l_{\infty}=(Z=0)\subset\mathbb{P}^{2}$.

Let $\mathcal{T}\subset\operatorname{Coh}(\mathbb{P}^{2})$ be the subcategory
of zero-dimensional sheaves and let
$\mathcal{F}\subset\operatorname{Coh}(\mathbb{P}^{2})$ be the subcategory of
sheaves $F$ such that $\operatorname{Hom}(\mathcal{T},F)=0$. The pair
$(\mathcal{T},\mathcal{F})$ is a torsion pair on
$\operatorname{Coh}(\mathbb{P}^{2})$, and we define $\mathcal{A}$ to be the
associated tilting abelian category

$\displaystyle\mathcal{A}=\langle\mathcal{F},\mathcal{T}[-1]\rangle\subset
D^{b}(\mathbb{P}^{2}).$

An object in $\mathcal{A}$ is an example of a perverse coherent sheaf on
$\mathbb{P}^{2}$ introduced in [Kas04, AB10]. We denote by
$\mathfrak{U}_{\mathbb{C}^{2}}(d)$ the derived moduli stack of objects
$I\in\mathcal{A}$ with an isomorphism
$I|_{l_{\infty}}\cong\mathcal{O}_{l_{\infty}}$ such that
$\operatorname{ch}_{2}(I)=-d$. Note that $I$ fits into an exact sequence in
$\mathcal{A}$:

(4.1) $\displaystyle 0\to I_{Z}\to I\to Q[-1]\to 0,$

where $I_{Z}=\mathcal{H}^{0}(I)$ is an ideal sheaf of a zero-dimensional
subscheme $Z\subset\mathbb{C}^{2}$ and $Q=\mathcal{H}^{1}(I)$ is zero-
dimensional on $\mathbb{C}^{2}$ such that $\lvert Q\rvert+\lvert Z\rvert=d$.
The stack $\mathfrak{U}_{\mathbb{C}^{2}}(d)$ is an Artin stack of finite type
whose classical truncation $\mathfrak{U}^{\mathrm{cl}}_{\mathbb{C}^{2}}(d)$
admits a good moduli space

$\displaystyle\mathfrak{U}^{\mathrm{cl}}_{\mathbb{C}^{2}}(d)\to\mathrm{Sym}^{d}(\mathbb{C}^{2}),\,I\mapsto\mathrm{Supp}(Q)+\mathrm{Supp}(Z).$

There is an explicit description of
$\mathfrak{U}^{\mathrm{cl}}_{\mathbb{C}^{2}}(d)$ via the ADHM quiver. Let $V$
be a $d$-dimensional vector space and let
$\mathfrak{g}=\operatorname{Hom}(V,V)$. We consider the following
$GL(V)$-equivariant morphism:

$\displaystyle\nu\colon V\oplus V^{\vee}\oplus\mathfrak{g}^{\oplus 2}$
$\displaystyle\to\mathfrak{g},$ $\displaystyle(u,v,A,B)$
$\displaystyle\mapsto[A,B]+u\circ v.$

Let $\nu^{-1}(0)$ be the derived zero locus of $\nu$. The derived moduli stack
of the ADHM quiver with relation is given by

$\displaystyle\mathfrak{N}_{\mathbb{C}^{2}}(d):=\nu^{-1}(0)/GL(V).$

Then there is an equivalence of derived stacks, see [BFG06, Theorem 5.7]

(4.2)
$\displaystyle\Upsilon\colon\mathfrak{N}_{\mathbb{C}^{2}}(d)\stackrel{{\scriptstyle\sim}}{{\to}}\mathfrak{U}_{\mathbb{C}^{2}}(d).$

For a point $(u,v,A,B)\in\nu^{-1}(0)$, the corresponding object in
$\mathfrak{U}_{\mathbb{C}^{2}}(d)$ is given by the complex on
$\mathbb{P}^{2}$:

(4.3) $\displaystyle 0\to
V\otimes\mathcal{O}_{\mathbb{P}^{2}}(-1)\stackrel{{\scriptstyle\phi}}{{\to}}(V^{\oplus
2}\oplus\mathbb{C})\otimes\mathcal{O}_{\mathbb{P}^{2}}\stackrel{{\scriptstyle\psi}}{{\to}}V\otimes\mathcal{O}_{\mathbb{P}^{2}}(1)\to
0.$

Here, the maps $\phi$ and $\psi$ are given by

(4.4) $\displaystyle\phi=\begin{pmatrix}ZA-X\operatorname{id}\\\
ZB-Y\operatorname{id}\\\ Zv\end{pmatrix},\
\psi=\begin{pmatrix}-ZB+Y\operatorname{id},ZA-X\operatorname{id},Zu\end{pmatrix}.$

It is proved in [BFG06, Theorem 5.7] that the above correspondence gives an
isomorphism (4.2) on classical truncations. The above construction and its
inverse in [BFG06, Theorem 5.7] can be generalized to families over dg-rings,
so giving the claimed equivalence (4.2).

###  4.2. The extended ADHM stack

For each $m\in\mathbb{N}$, let $I_{m}$ be the set

$\displaystyle I_{m}:=\\{(i,j)\in\mathbb{Z}_{\geqslant 0}^{\oplus
2}:1\leqslant i+j\leqslant m\\}.$

Note that we have $\lvert
I_{m}\rvert=m^{2}/2+3m/2=\dim\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert$. For
each $\alpha=(\alpha_{ij})\in\mathbb{C}^{I_{m}}$, consider the polynomial

(4.5) $\displaystyle f_{\alpha}=1+\sum_{(i,j)\in
I_{m}}\alpha_{ij}x^{i}y^{j}\in\mathbb{C}[x,y]$

and let $C_{\alpha}\subset\mathbb{C}^{2}$ be the plane curve defined by
$f_{\alpha}=0$. There is an open immersion

(4.6)
$\displaystyle\mathbb{C}^{I_{m}}\hookrightarrow\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert,\
\alpha\mapsto\overline{f}_{\alpha}:=Z^{m}f_{\alpha}(X/Z,Y/Z).$

whose image consists of polynomials $\sum c_{ijk}X^{i}Y^{j}Z^{k}$ such that
$c_{00m}\neq 0$.

Let $V$ be a $d$-dimensional vector space and
$\mathfrak{g}=\operatorname{Hom}(V,V)$. We consider the map

$\displaystyle\mu\colon V\oplus V^{\vee}\oplus\mathfrak{g}^{\oplus
2}\oplus\mathbb{C}^{I_{m}}$ $\displaystyle\to\mathfrak{g}\oplus V^{\vee}$
$\displaystyle(u,v,A,B,\alpha)$ $\displaystyle\mapsto([A,B]+u\circ v,v\circ
f_{\alpha}(A,B)).$

The above map $\mu$ is $GL(V)$-equivariant, where $GL(V)$ acts on
$\mathfrak{g}$ by conjugation and on $\mathbb{C}^{I_{m}}$ trivially. Let
$\mu^{-1}(0)$ be the derived zero locus of $\mu$. We define the derived stack

$\displaystyle\mathfrak{M}_{\mathbb{C}^{2}}(m,d):=\mu^{-1}(0)/GL(V).$

By forgetting $\alpha$, there is a morphism of derived stacks

(4.7) $\displaystyle
g\colon\mathfrak{M}_{\mathbb{C}^{2}}(m,d)\to\mathfrak{N}_{\mathbb{C}^{2}}(d).$

In what follows, we explain that $\mathfrak{M}_{\mathbb{C}^{2}}(m,d)$ can be
interpreted as a derived moduli stack of pairs of a one-dimensional sheaf on
$\mathbb{C}^{2}$ with a section.

###  4.3. The derived moduli stack of pairs on $\mathbb{C}^{2}$

We consider pairs

(4.8) $\displaystyle(F,s),\
F\in\operatorname{Coh}(\mathbb{P}^{2}),s\colon\mathcal{O}_{\mathbb{P}^{2}}\to
F,$

where $F$ is one-dimensional and the cokernel of $s$ is at most zero-
dimensional. Let $l\subset\mathbb{P}^{2}$ be a line. We denote by

$\displaystyle\mathfrak{T}_{\mathbb{P}^{2}}(m,d)\hookleftarrow\mathfrak{T}^{\mathrm{cl}}_{\mathbb{P}^{2}}(m,d)$

the derived moduli stack of pairs (4.8) and its classical truncation
satisfying

$\displaystyle[F]=m[l],\chi(F)=3m/2-m^{2}/2+d.$

The stack $\mathfrak{T}^{\mathrm{cl}}_{d}(\mathbb{P}^{2},m)$ is an Artin stack
of finite type which admits a good moduli space, see [Toda, Subsection 4.2.4].

Note that a pair $(\mathcal{O}_{\mathbb{P}^{2}}\to F)$ in
$\mathfrak{T}_{\mathbb{P}^{2}}(m,d)$ admits a filtration

$\displaystyle(0\to Q)\subset(\mathcal{O}_{\mathbb{P}^{2}}\twoheadrightarrow
F^{\prime})\subset(\mathcal{O}_{\mathbb{P}^{2}}\to F)$

such that $Q$ and $Q^{\prime}:=F/F^{\prime}$ are zero-dimensional sheaves with
$\lvert Q\rvert+\lvert Q^{\prime}\rvert=d$ and $F^{\prime}/Q=\mathcal{O}_{C}$
for $C\in\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert$. The good moduli space
morphism is

$\displaystyle\mathfrak{T}^{\mathrm{cl}}_{\mathbb{P}^{2}}(m,d)$
$\displaystyle\to
T_{\mathbb{P}^{2}}(m,d)=\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert\times\mathrm{Sym}^{d}(\mathbb{P}^{2}),$
$\displaystyle(F,s)$
$\displaystyle\mapsto(C,\mathrm{Supp}(Q)+\mathrm{Supp}(Q^{\prime})).$

We define the derived open substack
$\mathfrak{T}_{\mathbb{C}^{2}}(m,d)\subset\mathfrak{T}_{\mathbb{P}^{2}}(m,d)$
whose classical truncation fits into the Cartesian square

######  Lemma 4.1.

There is a natural morphism

(4.9) $\displaystyle
h\colon\mathfrak{T}_{\mathbb{C}^{2}}(m,d)\to\mathfrak{U}_{\mathbb{C}^{2}}(d)$

sending a pair $(F,s)$ to a two term complex
$(\mathcal{O}_{\mathbb{P}^{2}}(m)\stackrel{{\scriptstyle s}}{{\to}}F(m))$.

###### Proof.

By the definition of $\mathfrak{T}_{\mathbb{C}^{2}}(m,d)$, for a point $(F,s)$
in $\mathfrak{T}_{\mathbb{C}^{2}}(m,d)$, the two term complex

$I=(\mathcal{O}_{\mathbb{P}^{2}}(m)\stackrel{{\scriptstyle s}}{{\to}}F(m))$

is an object in $\mathcal{A}$ such that $\mathcal{H}^{0}(I)=I_{Z}$ for a zero-
dimensional subscheme $Z\subset\mathbb{C}^{2}$ and $\mathcal{H}^{1}(I)$ is
zero-dimensional with support contained in $\mathbb{C}^{2}$. In particular,
there is an open neighborhood $l_{\infty}\subset U\subset\mathbb{P}^{2}$ such
that $I|_{U}=(\mathcal{O}_{U}(m)\stackrel{{\scriptstyle
s|_{U}}}{{\to}}\mathcal{O}_{C}(m)|_{U})$ for
$C\in\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert$. Then $I|_{U}$ is naturally
isomorphic to $\mathcal{O}_{U}$, giving a trivialization
$I|_{l_{\infty}}\cong\mathcal{O}_{l_{\infty}}$. ∎

######  Lemma 4.2.

For an object $I$ in $\mathfrak{U}_{\mathbb{C}^{2}}(d)$, we have

$\displaystyle\mathrm{Hom}_{\mathbb{P}^{2}}(I,\mathcal{O}_{\mathbb{P}^{2}}(m))=\operatorname{Ker}\left(H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(m))\stackrel{{\scriptstyle\eta}}{{\to}}\mathrm{Ext}^{2}_{\mathbb{P}^{2}}(\mathcal{H}^{1}(I),\mathcal{O}_{\mathbb{P}^{2}}(m))\right).$

In the above, $\eta$ sends $a\in H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(m))$ to
$\mathcal{H}^{1}(I)\to I_{Z}[2]\to\mathcal{O}_{\mathbb{P}^{2}}(m)[2]$ where
the first arrow is the extension class of (4.1) and the second arrow is given
by multiplication with $a$.

###### Proof.

The lemma is straightforward by applying
$\mathrm{RHom}_{\mathbb{P}^{2}}(-,\mathcal{O}_{\mathbb{P}^{2}}(m))$ to (4.1)
and observing that
$\mathrm{Hom}(I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(m))=H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(m))$.
∎

By Lemma 4.2, we have an injection

(4.10)
$\displaystyle\operatorname{Hom}(I,\mathcal{O}_{\mathbb{P}^{2}}(m))\hookrightarrow
H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(m))$

which sends $t\colon I\to\mathcal{O}_{\mathbb{P}^{2}}(m)$ to
$\det(t)\colon\det
I=\mathcal{O}_{\mathbb{P}^{2}}\to\mathcal{O}_{\mathbb{P}^{2}}(m)$. We set

$\displaystyle\mathbb{P}(\operatorname{Hom}(I,\mathcal{O}_{\mathbb{P}^{2}}(m)))^{\circ}:=\mathbb{P}(\operatorname{Hom}(I,\mathcal{O}_{\mathbb{P}^{2}}(m)))\cap\mathbb{C}^{I_{m}}.$

######  Lemma 4.3.

For an object $I$ in $\mathfrak{U}_{\mathbb{C}^{2}}(d)$, we have
$h^{-1}(I)^{\mathrm{cl}}\cong\mathbb{P}(\operatorname{Hom}(I,\mathcal{O}_{\mathbb{P}^{2}}(m)))^{\circ}$.

###### Proof.

Given $I$, let us take a non-zero morphism $t\colon
I\to\mathcal{O}_{\mathbb{P}^{2}}(m)$ corresponding to a point in
$\mathbb{P}(\operatorname{Hom}(I,\mathcal{O}_{\mathbb{P}^{2}}(m)))^{\circ}$.
Let $C(t)$ be the cone of $t$ and consider the distinguished triangle

$\displaystyle I\to\mathcal{O}_{\mathbb{P}^{2}}(m)\to C(t)\to I[1].$

By taking the associated long exact sequence of cohomologies, we see that
$C(t)\in\mathrm{Coh}(\mathbb{P}^{2})$ and it fits into the exact sequence

$\displaystyle 0\to\mathcal{O}_{\mathbb{P}^{2}}(m)/\mathcal{H}^{0}(I)\to
C(t)\to\mathcal{H}^{1}(I)\to 0.$

It follows that $C(t)$ is one-dimensional with support a curve represented by
an element in $\mathbb{C}^{I_{m}}$. Moreover, $I$ is isomorphic to the two-
term complex

$I\cong(\mathcal{O}_{\mathbb{P}^{2}}(m)\xrightarrow{s}C(t))$

such that $\mathrm{Cok}(s)=\mathcal{H}^{1}(I)$ is zero-dimensional. Suppose
that for another morphism $t^{\prime}:I\to\mathcal{O}_{\mathbb{P}^{2}}(m)$,
there is a commutative diagram

Then there is an isomorphism $\phi\colon
I\stackrel{{\scriptstyle\cong}}{{\to}}I$ which makes the above diagram
commutative. Then $\det t\det\phi=\det t^{\prime}$. As $0\neq\det\phi\in
H^{0}(\mathcal{O}_{\mathbb{P}^{2}})=\mathbb{C}$, and (4.10) is injective, we
see that $t$ and $t^{\prime}$ differ by a non-zero scalar multiplication. It
follows that

(4.11)
$\displaystyle\mathbb{P}(\operatorname{Hom}(I,\mathcal{O}_{\mathbb{P}^{2}}(m)))^{\circ}\subset
h^{-1}(I)^{\mathrm{cl}}.$

Conversely, let $(\mathcal{O}_{\mathbb{P}^{2}}\xrightarrow{s}F)$ be a point in
$\mathfrak{T}_{\mathbb{C}^{2}}(m,d)$ such that the two-term complex
$(\mathcal{O}_{\mathbb{P}^{2}}(m)\stackrel{{\scriptstyle s}}{{\to}}F(m))$ is
isomorphic to $I$ in $D^{b}(\mathbb{P}^{2})$. Then there is a non-zero
morphism $I\to\mathcal{O}_{\mathbb{P}^{2}}(m)$ corresponding to a point in
$\mathbb{P}(\operatorname{Hom}(I,\mathcal{O}_{\mathbb{P}^{2}}(m)))^{\circ}$.
Therefore (4.11) is an isomorphism. ∎

###  4.4. Explicit description of the moduli stack of pairs on
$\mathbb{C}^{2}$

For $(i,j)\in I_{m}$ and $(A,B)\in\mathfrak{g}^{\oplus 2}$, we have

$\displaystyle A^{i}B^{j}-x^{i}y^{j}\operatorname{id}=$ $\displaystyle
A^{i}(B^{j-1}+yB^{j-2}+\cdots+y^{j-1}\operatorname{id})(B-y\operatorname{id})$
$\displaystyle+y^{j}(A^{i-1}+xA^{i-2}+\cdots+x^{i-1}\operatorname{id})(A-x\operatorname{id}).$

Let $\alpha\in\mathbb{C}^{I_{m}}$. By the above equality, we can write

$\displaystyle
f_{\alpha}(A,B)-f_{\alpha}\operatorname{id}=g_{\alpha}(A-x\operatorname{id})+h_{\alpha}(B-y\operatorname{id})$

for $g_{\alpha},h_{\alpha}\in\mathbb{C}[x,y]\otimes\mathfrak{g}$ with degree
less than $m$. Then for each element

$\displaystyle(u,v,A,B,\alpha)\in V\oplus V^{\vee}\oplus\mathfrak{g}^{\oplus
2}\oplus\mathbb{C}^{I_{m}}$

such that $[A,B]+u\circ v=0$ and $v\circ f_{\alpha}(A,B)=0$, we have the
commutative diagram

(4.16)

In the above diagram, $\phi,\psi$ are given as (4.4), and $\gamma$ is given by

$\displaystyle\gamma=(v\circ\overline{g}_{\alpha},v\circ\overline{h}_{\alpha},\overline{f}_{\alpha}).$

Here
$\overline{g}_{\alpha}=Z^{m}g_{\alpha}(X/Z,Y/Z)\in\mathbb{C}[X,Y,Z]\otimes\mathfrak{g}$,
etc. We note that the equation $v\circ f_{\alpha}(A,B)=0$ is used for the
commutativity of the left square in (4.16).

The diagram (4.16) determines a morphism
$I\to\mathcal{O}_{\mathbb{P}^{2}}(m)$, where $I$ is a perverse coherent sheaf
represented by (4.3). From the proof of Lemma 4.3, the totalization of the
diagram (4.16) is quasi-isomorphic to a one-dimensional sheaf $F$ on
$\mathbb{P}^{2}$ with curve class $(\overline{f}_{\alpha})=0$ and with a
morphism $\mathcal{O}_{\mathbb{P}^{2}}(m)\to F$. Therefore, by applying
$\otimes\mathcal{O}_{\mathbb{P}^{2}}(-m)$, we obtain the morphism

$\displaystyle\Theta\colon\mathfrak{M}_{\mathbb{C}^{2}}(m,d)\to\mathfrak{T}_{\mathbb{C}^{2}}(m,d)$

such that the following diagram commutes:

(4.21)

######  Proposition 4.4.

The morphism $\Theta$ is an equivalence of derived stacks.

###### Proof.

We first prove that $\Theta$ is an isomorphism on classical truncations. Let
$(u,v,A,B)$ be a point in $\mathfrak{U}_{\mathbb{C}^{2}}(d)$ and $I$ be the
associated perverse coherent sheaf via the construction (4.3). From the
commutative diagram (4.21), it is enough to show that $\Theta$ induces the
isomorphism

(4.22) $\displaystyle\Theta^{\rm{cl}}\colon
g^{-1}(u,v,A,B)^{\rm{cl}}\stackrel{{\scriptstyle\cong}}{{\to}}h^{-1}(I)^{\rm{cl}}.$

Here, the right hand side is given by
$\mathbb{P}(\operatorname{Hom}(I,\mathcal{O}_{\mathbb{P}^{2}}(m)))^{\circ}$ by
Lemma 4.3 and the left hand side is given by the zero-locus of the map

(4.23) $\displaystyle\mathbb{C}^{I_{m}}\to V^{\vee},\ \alpha\mapsto v\circ
f_{\alpha}(A,B).$

Let $\Sigma\subset V$ be the intersection of all subspaces $S\subset V$
satisfying $A(S)\subset S$, $B(S)\subset S$ and $u\in S$. We set $N=V/\Sigma$.
Then we have the commutative diagram

(4.30)

Recall that the middle horizontal complex is isomorphic to $I$. By [JM11,
Theorem 4.10], the top and bottom horizontal complexes are isomorphic to
$I_{Z}$ and $Q[-1]$, respectively, and the vertical arrows between these
complexes are identified with the exact sequence (4.1) in $\mathcal{A}$.
Moreover, we have $v|_{\Sigma}=0$ by [Nak99, Proposition 2.8]. Take a non-zero
element

$\displaystyle c=\sum_{i+j+k=m}c_{ijk}X^{i}Y^{j}Z^{k}\in
H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(m)).$

Under the identification
$\operatorname{Hom}(I_{Z},\mathcal{O}_{\mathbb{P}^{2}}(m))=H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(m))$,
the morphism $I_{Z}\to\mathcal{O}_{\mathbb{P}^{2}}(m)$ corresponding to $c$ is
represented by the commutative diagram

(4.35)

We compute $\eta(c)$ in Lemma 4.2 using the diagram (4.30). Since
$\mathcal{H}^{1}(I)$ is supported on $\mathbb{C}^{2}$, via the isomorphism
$Z^{m}\colon\mathcal{O}_{\mathbb{C}^{2}}\stackrel{{\scriptstyle\cong}}{{\to}}\mathcal{O}_{\mathbb{P}^{2}}(m)|_{\mathbb{C}^{2}}$
we have

(4.36)
$\displaystyle\operatorname{Ext}^{2}_{\mathbb{P}^{2}}(\mathcal{H}^{1}(I),\mathcal{O}_{\mathbb{P}^{2}}(m))\cong
H^{0}(\mathcal{E}xt^{2}_{\mathbb{C}^{2}}(\mathcal{H}^{1}(I),\mathcal{O}_{\mathbb{C}^{2}}))\cong
N^{\vee}\subset V^{\vee}.$

In order to compute $\eta(c)$ as an element of $V^{\vee}$, we extend the
middle vertical arrow in (4.35) to $\varepsilon$ as follows

We then compose it with the morphism $\phi$ in (4.30) to obtain the morphism

$\displaystyle
V\otimes\mathcal{O}_{\mathbb{P}^{2}}(-1)\stackrel{{\scriptstyle\phi}}{{\to}}(V^{\oplus
2}\oplus\mathbb{C})\otimes\mathcal{O}_{\mathbb{P}^{2}}\stackrel{{\scriptstyle\varepsilon}}{{\to}}\mathcal{O}_{\mathbb{P}^{2}}(m).$

The morphism above factors through $N\otimes\mathcal{O}_{\mathbb{P}^{2}}(-1)$,
giving a morphism

$\displaystyle v\circ Zc\colon
N\otimes\mathcal{O}_{\mathbb{P}^{2}}(-1)\to\mathcal{O}_{\mathbb{P}^{2}}(m).$

The above morphism represents $\eta(c)$. Under the isomorphism (4.36), we have

$\displaystyle\eta(c)=v\circ\sum_{i+j+k=m}c_{ijk}A^{i}B^{j}.$

Note that we have $\alpha_{ij}=c_{ijk}/c_{00m}$ under the embedding (4.6),
where $k=m-i-j$. Therefore comparing with (4.23), we see the isomorphism
(4.22).

Finally, we show that $\Theta$ induces an equivalence of derived stacks. As
$\Theta$ is an isomorphism on classical truncations and the bottom arrow in
(4.21) is an equivalence, it is enough to show that $\Theta$ induces a quasi-
isomorphism on relative tangent complexes
$\mathbb{T}_{g}\stackrel{{\scriptstyle\sim}}{{\to}}\Theta^{\ast}\mathbb{T}_{h}$.
Let us take a point $x=(u,v,A,B,\alpha)$ in
$\mathfrak{M}_{\mathbb{C}^{2}}(m,d)$ such that $[A,B]+u\circ v=0$ and $v\circ
f_{\alpha}(A,B)=0$. Then we have

$\displaystyle\mathbb{T}_{g}|_{x}=(\mathbb{C}^{I_{m}}\stackrel{{\scriptstyle\iota}}{{\to}}V^{\vee}),\
\iota\colon(e_{ij})\mapsto\sum_{(i,j)\in I_{m}}e_{ij}v\circ A^{i}B^{j}.$

Let $y=(\mathcal{O}_{\mathbb{P}^{2}}\to F)$ be the point in
$\mathfrak{T}_{\mathbb{C}^{2}}(m,d)$ corresponding to $\Theta(x)$. The tangent
complex of $\mathfrak{T}_{\mathbb{C}^{2}}(m,d)$ at $y$ is

$\displaystyle\mathrm{RHom}(\mathcal{O}_{\mathbb{P}^{2}}\to
F,F)=\mathrm{RHom}(I,F(m)),$

where $I=(\mathcal{O}_{\mathbb{P}^{2}}(m)\to F(m))$. The tangent complex
$\mathbb{T}_{h}|_{y}$ is

$\displaystyle\mathrm{Cone}\left(\mathrm{RHom}(I,F(m))\to\mathrm{RHom}(I,I)_{0}[1]\right)\cong\mathrm{Cone}\left(\mathbb{C}\to\mathrm{RHom}(I,\mathcal{O}_{\mathbb{P}^{2}}(m))\right).$

Here, $\mathrm{RHom}(I,I)_{0}$ is the cone of the natural morphism
$\mathbb{C}\to\mathrm{RHom}(I,I)$, and the map
$\mathbb{C}\to\mathrm{RHom}(I,\mathcal{O}_{\mathbb{P}^{2}}(m))$ is the
composition of the above natural morphism with
$I\to\mathcal{O}_{\mathbb{P}^{2}}(m)$. From the calculation of $\eta$, we have

$\displaystyle\mathbb{T}_{h}|_{y}=\left(H^{0}(\mathcal{O}_{\mathbb{P}^{2}}(m))/\mathbb{C}\overline{f}_{\alpha}\stackrel{{\scriptstyle\iota^{\prime}}}{{\to}}V^{\vee}\right),\
\iota^{\prime}\colon(c_{ijk})\mapsto v\circ\sum_{i+j+k=m}c_{ijk}A^{i}B^{j}.$

The required quasi-isomorphism is

Here the left vertical arrow sends $(e_{ij})$ to $(c_{ijk})$ for
$c_{ijk}=e_{ij}$ when $i+j>0$ and $c_{00m}=0$. ∎

###  4.5. The categorical DT/PT correspondence for $\mathbb{C}^{3}$

We define Zariski open subsets

(4.37)
$\displaystyle\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert^{\rm{red}}\subset\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert,\
(\mathbb{C}^{I_{m}})^{\rm{red}}:=\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert^{\rm{red}}\cap\mathbb{C}^{I_{m}},$

where $\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert^{\rm{red}}$ is the locus of
reduced plane curves in $\mathbb{P}^{2}$.

Let $X=\mathrm{Tot}_{\mathbb{P}^{2}}(\omega_{\mathbb{P}^{2}})$. Let

$\displaystyle\mathcal{T}_{X}^{\rm{red}}(m,d)$

be the classical moduli stack of pairs $(F,s)$, where $F$ is a one-dimensional
sheaf on $X$ with support a reduced plane curve of degree $m$ in
$\mathbb{P}^{2}$, and $s\colon\mathcal{O}_{X}\to F$ is surjective in dimension
one. The open immersion $\mathbb{C}^{2}\subset\mathbb{P}^{2}$ determines the
open immersion $\mathbb{C}^{3}\subset X$. We have the open substack

$\displaystyle\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)\subset\mathcal{T}_{X}^{\rm{red}}(m,d)$

to be consisting of $(F,s)$ such that $\mathrm{Cok}(s)$ and $F_{\rm{tor}}$ are
supported on $\mathbb{C}^{3}$, and the one-dimensional support of $F$ is an
element from $(\mathbb{C}^{I_{m}})^{\rm{red}}$. Here $F_{\rm{tor}}\subset F$
is the maximal zero-dimensional subsheaf.

Let $\mathop{\mathrm{Tr}}\nolimits W_{m,d}$ be the function

$\displaystyle\mathop{\mathrm{Tr}}\nolimits W_{m,d}\colon\left(V^{\oplus
2}\oplus V^{\vee}\oplus\mathfrak{g}^{\oplus
3}\times(\mathbb{C}^{I_{m}})^{\rm{red}}\right)\big{/}GL(V)\to\mathbb{C}$

defined by the formula

(4.38) $\displaystyle(u_{1},u_{2},v,A,B,C,\alpha)\mapsto v\circ
f_{\alpha}(A,B)(u_{2})+\mathop{\mathrm{Tr}}\nolimits C(u_{1}\circ v+[A,B]).$

The next theorem describes $\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)$ as a
critical locus. Here it is essential to restrict to reduced curves, see Remark
4.6.

######  Theorem 4.5.

We have

(4.39)
$\displaystyle\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)\cong\mathrm{Crit}(\mathop{\mathrm{Tr}}\nolimits
W_{m,d}).$

###### Proof.

We set

$\displaystyle\mathfrak{T}_{\mathbb{P}^{2}}^{\rm{red}}(m,d)$
$\displaystyle:=\mathfrak{T}_{\mathbb{P}^{2}}(m,d)\times_{\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert}\lvert\mathcal{O}_{\mathbb{P}^{2}}(m)\rvert^{\rm{red}},$
$\displaystyle\mathfrak{T}_{\mathbb{C}^{2}}^{\rm{red}}(m,d)$
$\displaystyle:=\mathfrak{T}_{\mathbb{C}^{2}}(m,d)\times_{\mathbb{C}^{I_{m}}}(\mathbb{C}^{I_{m}})^{\rm{red}}.$

By the proof of [Toda, Lemma 5.5.4], we have an isomorphism

(4.40)
$\displaystyle\mathcal{T}_{X}^{\rm{red}}(m,d)\cong\Omega_{\mathfrak{T}_{\mathbb{P}^{2}}^{\rm{red}}(m,d)}[-1]^{\rm{cl}}.$

The above isomorphism restricts to the isomorphism of open substacks

$\displaystyle\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)\cong\Omega_{\mathfrak{T}_{\mathbb{C}^{2}}^{\rm{red}}(m,d)}[-1]^{\rm{cl}}.$

The lemma follows from the explicit description of the $(-1)$-shifted
cotangent stack, see [Toda, Section 2.1.1]. ∎

######  Remark 4.6.

We need to restrict to reduced curves in the above theorem as we use the
isomorphism (4.40) from [Toda, Lemma 5.5.4]. Otherwise, a pair $(F,s)$ from
$X$ with zero-dimensional cokernel may push-forward to a pair on
$\mathbb{P}^{2}$ with one-dimensional cokernel, so the isomorphism (4.40) may
not hold.

We have the open substacks, which are quasi-projective schemes:

$\displaystyle
I_{\mathbb{C}^{3}}^{\rm{red}}(m,d)\subset\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)\supset
P_{d}^{\rm{red}}(\mathbb{C}^{3},m).$

Here $I_{d}^{\rm{red}}(\mathbb{C}^{3},m)$ is the DT moduli space consisting of
$(F,s)$ such that $s$ is surjective, and $P_{d}^{\rm{red}}(\mathbb{C}^{3},m)$
is the PT moduli space consisting of $(F,s)$ such that $F$ is pure. Let
$\chi_{0}\colon GL(V)\to\mathbb{C}^{\ast}$ be the determinant character
$g\mapsto\det g$. Under the isomorphism (4.39), the above open substacks
correspond to GIT-semistable loci, see [Toda, Proposition 5.5.2]:

$\displaystyle
I_{\mathbb{C}^{3}}^{\rm{red}}(m,d)=\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)^{\chi_{0}\text{-ss}},\
P_{\mathbb{C}^{3}}^{\rm{red}}(m,d)=\mathcal{T}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)^{\chi_{0}^{-1}\text{-ss}}.$

We define

(4.41) $\displaystyle\mathop{\mathrm{Tr}}\nolimits
W_{m,d}^{\pm}\colon\left((V^{\oplus 2}\oplus
V^{\vee}\oplus\mathfrak{g}^{\oplus
3})^{\chi_{0}^{\pm}\text{-ss}}\times(\mathbb{C}^{I_{m}})^{\rm{red}}\right)\big{/}GL(V)\to\mathbb{C}$

to be the restriction of $\mathop{\mathrm{Tr}}\nolimits W_{m,d}$ to the GIT
semistable loci. Then we have

$\displaystyle
I_{\mathbb{C}^{3}}^{\rm{red}}(m,d)=\mathrm{Crit}(\mathop{\mathrm{Tr}}\nolimits
W_{m,d}^{+}),\
P_{\mathbb{C}^{3}}^{\rm{red}}(m,d)=\mathrm{Crit}(\mathop{\mathrm{Tr}}\nolimits
W_{m,d}^{-}).$

We define the following dg-categories

$\displaystyle\mathcal{DT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)$
$\displaystyle:=\mathrm{MF}\left((V^{\oplus 2}\oplus
V^{\vee}\oplus\mathfrak{g}^{\oplus
3})^{\chi_{0}\text{-ss}}\times(\mathbb{C}^{I_{m}})^{\rm{red}}\big{/}GL(V),\mathop{\mathrm{Tr}}\nolimits
W_{m,d}^{+}\right),$
$\displaystyle\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)$
$\displaystyle:=\mathrm{MF}\left((V^{\oplus 2}\oplus
V^{\vee}\oplus\mathfrak{g}^{\oplus
3})^{\chi_{0}^{-1}\text{-ss}}\times(\mathbb{C}^{I_{m}})^{\rm{red}}\big{/}GL(V),\mathop{\mathrm{Tr}}\nolimits
W_{m,d}^{-}\right).$

The above dg-categories categorify DT invariants (resp. PT invariants) for
Hilbert schemes of 1-dimensional subschemes (resp. PT stable pair moduli
spaces) with reduced supports, see [Efi18, Theorem 1.1], [Toda, Subsection
3.3].

######  Remark 4.7.

Following the definition of DT and PT categories in [Toda], one can also
consider graded categories of matrix factorizations, with grading given by the
weight two $\mathbb{C}^{\ast}$-action on $v$ and $C$. The graded version also
recovers DT and PT invariants and they have the same Grothendieck group as the
ungraded one, see [Toda, Proposition 3.3.6], [Todb, Corollary 3.13].

###### Proof of Theorem 1.4.

The statement follows from Corollary 3.14 for $a=1$ and $\mu=-1/2-\varepsilon$
for $0<\varepsilon\ll 1$. ∎

###### Proof of Corollary 1.5.

The claim follows as the isomorphism (3.31) from the proof of Corollary 1.2. ∎

In the context of Corollary 1.5, we do not know whether the following is an
isomorphism

$K\left(\mathcal{DT}(d-d^{\prime})\otimes\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d^{\prime})\right)\cong
K\left(\mathcal{DT}(d-d^{\prime})\right)\otimes
K\left(\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d^{\prime})\right).$

To obtain a result analogous to the one of Corollary 1.2, we use equivariant
K-theory, see [PTa, Section 4] for similar constructions and computations. Let
$T\cong(\mathbb{C}^{*})^{2}$ be the natural two dimensional torus acting on
$\mathbb{C}^{2}$. We let $T$ act on $\mathbb{C}^{3}$ as follows:

$(t_{1},t_{2})\cdot(x,y,z)=(t_{1}x,t_{2}y,t_{1}^{-1}t_{2}^{-1}z).$

It induces the action on $V^{\oplus 2}\oplus
V^{\vee}\oplus\mathfrak{g}^{\oplus 3}\otimes\mathbb{C}^{I_{m}}$ by

$\displaystyle(t_{1},t_{2})\cdot(u_{1},u_{2},v,A,B,C,\alpha_{ij})=(u_{1},u_{2},v,t_{1}A,t_{2}B,t_{1}^{-1}t_{2}^{-1}C,t_{1}^{-i}t_{2}^{-j}\alpha_{ij}).$

The function (4.38) is invariant under the above $T$-action. We consider the
equivariant categories $\mathcal{DT}^{\rm{red}}_{\mathbb{C}^{3},T}(m,d)$ and
$\mathcal{PT}^{\rm{red}}_{\mathbb{C}^{3},T}(m,d)$ and we denote their
Grothendieck groups by

$K_{T}(\mathcal{DT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d))\text{ and
}K_{T}(\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)).$

Let $\mathbb{K}:=K_{T}(\text{pt})$ and let
$\mathbb{F}:=\text{Frac}\,\mathbb{K}$. For $V$ a $\mathbb{K}$-module, let
$V_{\mathbb{F}}:=V\otimes_{\mathbb{K}}\mathbb{F}$.

######  Corollary 4.8.

Let $d\in\mathbb{N}$. Then there is an isomorphism of $\mathbb{F}$-vector
spaces:

$\displaystyle
K_{T}\left(\mathcal{DT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)\right)_{\mathbb{F}}\cong\bigoplus_{d^{\prime}=0}^{d}K_{T}\left(\mathcal{DT}(d-d^{\prime})\right)_{\mathbb{F}}\otimes_{\mathbb{F}}K_{T}\left(\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d^{\prime})\right)_{\mathbb{F}}.$

###### Proof.

As in the proof of Corollary 1.5, there is an isomorphism of
$\mathbb{F}$-vector spaces:

$K_{T}\left(\mathcal{DT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d)\right)_{\mathbb{F}}\cong\bigoplus_{d^{\prime}=0}^{d}K_{T}\left(\mathcal{DT}(d-d^{\prime})\otimes\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d^{\prime})\right)_{\mathbb{F}}.$

It suffices to show that there is a Künneth isomorphism:

(4.42) $\displaystyle
K_{T}\left(\mathcal{DT}(d-d^{\prime})\otimes\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d^{\prime})\right)_{\mathbb{F}}\cong$
$\displaystyle
K_{T}\left(\mathcal{DT}(d-d^{\prime})\right)_{\mathbb{F}}\otimes_{\mathbb{F}}K_{T}\left(\mathcal{PT}_{\mathbb{C}^{3}}^{\rm{red}}(m,d^{\prime})\right)_{\mathbb{F}}.$

The category $\mathcal{DT}_{T}(d)$ has a semiorthogonal decomposition with
summands Hall products of $\mathbb{S}_{T}(e)_{v}$ for various
$(e,v)\in\mathbb{N}\times\mathbb{Z}$ by [PTa, Theorem 1.1]. Let
$\mathbb{S}^{\text{gr}}_{T}(d)_{w}$ be the graded category of matrix
factorizations, defined similarly to $\mathbb{S}_{T}(d)_{w}$, where the linear
map corresponding to $X$ is scaled by $2$. Then
$\mathbb{S}^{\text{gr}}_{T}(d)_{w}$ and $\mathbb{S}_{T}(d)_{w}$ have the same
Grothendieck group [Todb, Corollary 3.13]. Further,
$\mathbb{S}^{\text{gr}}_{T}(d)_{w}$ appears as summands in
$D^{b}_{T}(\mathscr{C}(d))_{w}$ [Pădb, Corollary 3.3], where $\mathscr{C}(d)$
is the (derived) stack of pairs of commuting matrices of size $d$. Consider
the analogous graded category
$\mathcal{PT}^{\rm{red},\rm{gr}}_{\mathbb{C}^{3},T}(m,d)$ with the same
Grothendieck group as $\mathcal{PT}^{\rm{red}}_{\mathbb{C}^{3},T}(m,d)$. By
[HL15, Theorem 2.10] and the Koszul equivalence, see for example [Toda,
Theorem 2.3.3], the category
$\mathcal{PT}^{\rm{red},\rm{gr}}_{\mathbb{C}^{3},T}(m,d)$ is admissible in
$D^{b}_{T}(\mathfrak{T}_{\mathbb{C}^{2}}^{\rm{red}}(m,d))$. By the argument in
the proof of Corollary 1.2, it suffices to check the Künneth isomorphism

(4.43)
$G_{T}\left(\mathscr{D}\times\mathfrak{T}_{\mathbb{C}^{2}}^{\rm{red}}(m,d^{\prime})\right)_{\mathbb{F}}\cong
G_{T}\left(\mathscr{D}\right)_{\mathbb{F}}\otimes_{\mathbb{F}}G_{T}\left(\mathfrak{T}_{\mathbb{C}^{2}}^{\rm{red}}(m,d^{\prime})\right)_{\mathbb{F}},$

where $\mathscr{D}$ is the product of finitely many commuting stacks
$\mathscr{C}(e)$ for various $e$. The statement follows from the localization
theorem in K-theory (see [Tak94]) with respect to $T$ because
$\mathscr{D}^{T}=\text{pt}/L$, where $L$ is a product of finitely many groups
$GL(e)$. ∎

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Tudor Pădurariu: Columbia University, Mathematics Hall, 2990 Broadway, New
York, NY 10027, USA.
E-mail address<EMAIL_ADDRESS>

Yukinobu Toda: Kavli Institute for the Physics and Mathematics of the Universe
(WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan.
E-mail address<EMAIL_ADDRESS>