$\mathrm{MF}(p,0)\simeq D^{b}(r)/\mathrm{Perf}(r)$), the equivalence (2.14)
induces an equivalence:

$\kappa^{\prime}\colon
D^{b}(\mathscr{K})\otimes_{D^{b}(p)}\mathrm{MF}(p,0)\xrightarrow{\sim}\mathrm{MF}(\mathscr{E}^{\vee},f).$

Let $\mathscr{K}_{0}:=\mathscr{K}\times r$ and let
$\pi\colon\mathscr{K}_{0}\to\mathscr{K}$ and $t\colon r\to p$ be the natural
projections. We have that $\mathrm{MF}(p,0)\cong D^{b}(r)/t^{*}(D^{b}(p))$.
Then

$D^{b}(\mathscr{K})\otimes_{D^{b}(p)}\mathrm{MF}(p,0)\cong
D^{b}(\mathscr{K}_{0})/\pi^{*}(D^{b}(\mathscr{K})).$

It suffices to show that the map

$\pi^{*}\colon G^{\mathrm{top}}_{i}(\mathscr{K})\to
G^{\mathrm{top}}_{i}(\mathscr{K}_{0})\cong G^{\mathrm{top}}_{i}(\mathscr{K})$

is zero, which follows as in the proof of Proposition 4.1. ∎

##  6\. Topological K-theory of quasi-BPS categories for quivers with
potential

In this section, we compute the topological K-theory of quasi-BPS categories
for symmetric quivers satisfying Assumption 2.1 with a quasi-homogeneous
potential in terms of BPS cohomology, see Theorem 6.2. The main step in the
proof of Theorem 6.2 is the construction of the cycle map from topological
K-theory of quasi-BPS categories to BPS cohomology, see Theorem 6.3 (which
holds for all symmetric quivers). The conclusion then follows by comparing the
decomposition of DT invariants in BPS invariants of Meinhardt–Reineke (which
also holds for all symmetric quivers) and and the semiorthogonal decomposition
of the variety of framed representations from Theorem 2.8. We note that there
is a version of Theorem 2.8 for all symmetric quivers, see [PTe]. However,
under Assumption 2.1, all quasi-BPS categories appearing in the semiorthogonal
decomposition are of the form $\mathbb{S}(d)_{v}$, which is used crucially in
the computation in Subsection 6.3.

The construction of the cycle map from Theorem 6.3 holds for all quasi-BPS
categories $\mathbb{S}(d;\delta)$ for $\delta\in M(d)^{W_{d}}_{\mathbb{R}}$.
Theorem 6.3 is proved in Subsections 6.6 and is based on the fact that the
weight conditions for complexes in $\mathbb{S}(d;\delta)$ restrict the
possible perverse degree of their image under the cycle map, see Proposition
6.15 and Corollaries 6.18 and 6.19.

In view of the assumptions in Section 4, we assume throughout this section
that the potential $W$ of $Q=(I,E)$ is quasi-homogeneous, that is, there
exists a weight function $w\colon E\to\mathbb{Z}_{\geqslant 0}$ such that $W$
is homogeneous of weight $d>0$ with respect to the function $w$.

###  6.1. Statement of the main theorem

Before we state Theorem 6.2, we introduce notation related to quasi-BPS
categories and BPS sheaves.

####  6.1.1. Quasi-BPS categories

Let $\delta\in M(d)_{\mathbb{R}}^{W_{d}}$. Consider the category
$\mathbb{M}(d;\delta)$ defined in (2.19) and recall the definition of quasi-
BPS categories $\mathbb{S}(d;\delta)$ from (2.21). For $\lambda$ a cocharacter
of $T(d)$, recall the definition of $n_{\lambda}$ from (2.20). For $\lambda$ a
dominant cocharacter of $T(d)$, define

(6.1) $\varepsilon_{\lambda,\delta}=\begin{cases}1,\text{ if
}\frac{1}{2}n_{\lambda}+\langle\lambda,\delta\rangle\in\mathbb{Z},\\\ 0,\text{
otherwise}.\end{cases}$

For a partition $\textbf{d}=(d_{i})_{i=1}^{k}$ of $d$, let
$\varepsilon_{\textbf{d},\delta}=1$ if $\varepsilon_{\lambda,\delta}=1$ for
all cocharacters $\lambda$ with associated partition d and let
$\varepsilon_{\textbf{d},\delta}=0$ otherwise.

####  6.1.2. Sets of partitions

For a dimension vector $d=(d^{j})_{j\in I}\in\mathbb{N}^{I}$, recall that
$\underline{d}:=\sum_{j\in I}d^{j}$. Let $\delta\in
M(d)_{\mathbb{R}}^{W_{d}}$. Denote by $S^{d}_{\delta}$ the set of partitions
$\textbf{d}=(d_{i})_{i=1}^{k}$ of $d$ such that
$\varepsilon_{\textbf{d},\delta}=1$, where $\lambda$ is any antidominant
cocharacter with associated partition $(d_{i})_{i=1}^{k}$. If
$\delta=v\tau_{d}$, we use the notation $S^{d}_{v}$ instead of
$S^{d}_{v\tau_{d}}$.

Consider $(d_{i})_{i=1}^{k}\in S^{d}_{\delta}$, and an antidominant
cocharacter with associated partition $(d_{i})_{i=1}^{k}$. Define
$\theta_{i}\in\frac{1}{2}M(d_{i})$ with

$\sum_{i=1}^{k}\theta_{i}=-\frac{1}{2}R(d)^{\lambda>0}+\frac{1}{2}\mathfrak{g}(d)^{\lambda>0}.$

Let $\delta_{d_{i}}\in M(d_{i})_{\mathbb{R}}$ such that
$\sum_{i=1}^{k}\delta_{d_{i}}=\delta$. Then the Hall product induces a functor

$m=m_{\lambda}\colon\bigotimes_{i=1}^{k}\mathbb{M}(d_{i};\theta_{i}+\delta_{d_{i}})\to\mathbb{M}(d;\delta)$

and similarly for categories of matrix factorizations, see [Păda, Propositions
3.5 and 3.6] (in loc. cit. and using the notations used there, [Păda,
Proposition 3.6] is stated that $r>\frac{1}{2}$, but for $r=\frac{1}{2}$ it is
still true that $\chi-\sigma_{I}\in\frac{1}{2}\mathbb{W}$). If we assume that
$Q$ satisfies Assumption 2.1, then $\theta_{i}\in M(d_{i})^{W_{d_{i}}}$, and
so there are functors, see Remark 2.6:

(6.2)
$\bigotimes_{i=1}^{k}\mathbb{M}(d_{i};\delta_{d_{i}})\stackrel{{\scriptstyle\sim}}{{\to}}\bigotimes_{i=1}^{k}\mathbb{M}(d_{i};\theta_{i}+\delta_{d_{i}})\xrightarrow{m}\mathbb{M}(d;\delta_{d}).$

Assume that $\delta=v\tau_{d}$ and write $\delta_{i}=v_{i}\tau_{d_{i}}$ for
$1\leqslant i\leqslant k$. Then

(6.3) $\frac{v}{\underline{d}}=\frac{v_{i}}{\underline{d}_{i}}$

for any $1\leqslant i\leqslant k$. If we assume that $Q$ satisfies Assumption
2.1, the Hall product then induces functors, see (6.2):

$\bigotimes_{i=1}^{k}\mathbb{M}(d_{i})_{v_{i}}\stackrel{{\scriptstyle\sim}}{{\to}}\bigotimes_{i=1}^{k}\mathbb{M}(d_{i};\theta_{i}+\delta_{d_{i}})\xrightarrow{m}\mathbb{M}(d)_{v}.$

We end this subsection with the following computation:

######  Proposition 6.1.

Let $Q=(I,E)$ be a quiver satisfying Assumption 2.1. Let
$(d,v)\in\mathbb{N}^{I}\times\mathbb{Z}$. The set $S^{d}_{v}$ contains all
partitions $\textbf{d}=(d_{i})_{i=1}^{k}$ such that

$\underline{d}|v\cdot\gcd(\underline{d}_{1},\ldots,\underline{d}_{k}).$

In particular, if $\gcd(v,\underline{d})=1$, then $S^{d}_{v}$ contains only
the one term partition of $d$.

###### Proof.

Let $d=(d^{a})_{a\in I}\in\mathbb{N}^{I}$. Note that
$n_{\lambda}=\langle\lambda,\mathbb{L}^{\lambda>0}_{\mathscr{X}(d)}|_{0}\rangle\in
2\mathbb{Z}$ because $Q$ satisfies Assumption 2.1. Then
$\varepsilon_{\textbf{d},v\tau_{d}}=1$ if and only if
$\langle\lambda,v\tau_{d}\rangle\in\mathbb{Z}$ for all cocharacters $\lambda$
with associated partition d.

Write $\lambda=(\lambda^{a})_{a\in I}$, where
$\lambda^{a}\colon\mathbb{C}^{*}\to T(d^{a})$ is a cocharacter

$\displaystyle\lambda(t)=(t^{m_{1}},\ldots,t^{m_{1}},t^{m_{2}},\ldots,t^{m_{2}},\ldots,t^{m_{k}}),$

where $m_{i}$ appears $d^{a}_{i}$-times, and $m_{i}\neq m_{j}$ for $1\leqslant
i\neq j\leqslant k$. Then the condition
$\langle\lambda,v\tau_{d}\rangle\in\mathbb{Z}$ is equivalent to that

$v/\underline{d}\cdot\sum_{i=1}^{k}m_{i}\underline{d}_{i}\in\mathbb{Z}$

for all tuples of pairwise distinct integers
$(m_{i})_{i=1}^{k}\in\mathbb{Z}^{k}$, which implies the desired conclusion. ∎

####  6.1.3. BPS sheaves and cohomologies

Let $Q=(I,E)$ be a symmetric quiver, let $W$ be a potential of $Q$, and let
$d\in\mathbb{N}^{I}$. Consider the stack of dimension $d$ representations of
$Q$ and its good moduli space:

$\pi_{d}\colon\mathscr{X}(d):=R(d)/G(d)\to X(d):=R(d)/\\!\\!/G(d).$

We denote by
$\mathrm{IC}:=\mathrm{IC}_{\mathscr{X}(d)}=\mathbb{Q}_{\mathscr{X}(d)}[\dim\mathscr{X}(d)]$
and we may drop $\mathrm{Tr}\,W$ from the notation of the vanishing cycle
functor. Recall that
$\varphi:=\varphi_{\mathrm{Tr}\,W}\mathbb{Q}_{\mathscr{X}(d)}$. Following
[DM20], define the BPS sheaf

$\mathcal{BPS}_{d}:=\begin{cases}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{X(d)}[-1],\text{
if }X(d)^{\mathrm{st}}\neq\emptyset,\\\ 0,\text{ if
}X(d)^{\mathrm{st}}=\emptyset.\end{cases}$

Note that $\mathcal{BPS}_{d}\in\mathrm{Perv}(X(d))$.

Consider a partition $A=(d_{i})_{i=1}^{k}$ of $d$. Let $\ell(A):=k$. Assume
the set $\\{d_{1},\ldots,d_{k}\\}=\\{e_{1},\ldots,e_{s}\\}$ has cardinality
$s$ and that, for each $1\leqslant i\leqslant s$, there are $m_{i}$ elements
in $\\{d_{1},\ldots,d_{k}\\}$ equal to $e_{i}$. Define the addition maps
$\oplus_{i}\colon X(e_{i})^{\times m_{i}}\to X(m_{i}e_{i})$ for $1\leqslant
i\leqslant s$ and $\oplus^{\prime}\colon\times_{i=1}^{s}X(m_{i}e_{i})\to
X(d)$, which are finite. Define the sheaves:

$\displaystyle\mathrm{Sym}^{m_{i}}\big{(}\mathcal{BPS}_{e_{i}}\big{)}$
$\displaystyle:=\oplus_{i,\ast}\left(\mathcal{BPS}_{e_{i}}^{\boxtimes
m_{i}}\right)^{\mathfrak{S}_{m_{i}}}\in\mathrm{Perv}(X(m_{i}e_{i})),$ (6.4)
$\displaystyle\mathcal{BPS}_{A}$
$\displaystyle:=\oplus^{\prime}_{\ast}\left(\boxtimes_{i=1}^{s}\mathrm{Sym}^{m_{i}}(\mathcal{BPS}_{e_{i}})\right)\in\mathrm{Perv}(X(d)).$

Alternatively, by the Thom-Sebastiani theorem, the sheaf $\mathcal{BPS}_{A}$
has the following description. Let $\mathcal{BPS}^{0}_{A}$ be the sheaf
defined above for $W=0$. Then
$\mathcal{BPS}_{A}=\varphi_{\mathrm{Tr}\,W}\mathcal{BPS}^{0}_{A}[-1]$.

Define the complexes

(6.5) $\displaystyle\mathcal{BPS}_{d,\delta}$ $\displaystyle:=\bigoplus_{A\in
S^{d}_{\delta}}\mathcal{BPS}_{A}[-\ell(A)]\in D^{b}_{\mathrm{con}}(X(d)),$
$\displaystyle\mathcal{BPS}_{d,v}$
$\displaystyle:=\mathcal{BPS}_{d,v\tau_{d}}\in D^{b}_{\mathrm{con}}(X(d)).$

As we will see in (6.18), the complexes $\mathcal{BPS}_{A}$ and
$\mathcal{BPS}_{d,\delta}$ are direct summands of
$\pi_{*}\varphi\mathrm{IC}[-1]$ preserved by $1-\mathrm{T}$. Define
$\mathcal{BPS}_{A}^{\mathrm{inv}},\mathcal{BPS}_{d,\delta}^{\mathrm{inv}}\in
D^{b}_{\mathrm{con}}(X(d))$ by the exact triangles:

$\displaystyle\mathcal{BPS}_{A}^{\mathrm{inv}}[-1]\to\mathcal{BPS}_{A}\xrightarrow{1-\mathrm{T}}\mathcal{BPS}_{A}\to\mathcal{BPS}_{A}^{\mathrm{inv}},$
$\displaystyle\mathcal{BPS}_{d,\delta}^{\mathrm{inv}}[-1]\to\mathcal{BPS}_{d,\delta}\xrightarrow{1-\mathrm{T}}\mathcal{BPS}_{d,\delta}\to\mathcal{BPS}_{d,\delta}^{\mathrm{inv}}.$

####  6.1.4. Statement of the main theorem

Let $Q=(I,E)$ be a symmetric quiver and let $W$ be a quasi-homogeneous
potential. Consider the Chern character map (4.18):

(6.6) $\displaystyle\mathrm{ch}\colon
K^{\mathrm{top}}_{i}(\mathbb{S}(d)_{v})\hookrightarrow
K^{\mathrm{top}}_{i}(\mathrm{MF}(\mathscr{X}(d),\mathrm{Tr}\,W))\to\widetilde{H}^{i}(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}^{\mathrm{inv}}).$

Recall (4.19) and define the filtration:

$E_{\ell}K^{\mathrm{top}}_{i}(\mathbb{S}(d)_{v}):=K^{\mathrm{top}}_{i}(\mathbb{S}(d)_{v})\cap
E_{\ell}K^{\mathrm{top}}_{i}(\mathrm{MF}(\mathscr{X}(d),\mathrm{Tr}\,W))\subset
K^{\mathrm{top}}_{i}(\mathbb{S}(d)_{v}).$

There is an injective cycle map on the associated graded pieces:

(6.7)
$\displaystyle\mathrm{c}\colon\mathrm{gr}_{\ell}K^{\mathrm{top}}_{i}(\mathbb{S}(d)_{v})$
$\displaystyle\to
H^{2\dim\mathscr{X}(d)-2\ell-i}(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}^{\mathrm{inv}}[-2])$
$\displaystyle\xrightarrow{\sim}H^{\dim\mathscr{X}(d)-2\ell-i}(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}^{\mathrm{inv}}\,\mathrm{IC}_{\mathscr{X}(d)}[-2]),$

where we used
$\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}=\varphi_{\mathrm{Tr}\,W}[\dim\mathscr{X}(d)]$
for computing the cohomological degree. The following is the main result of
this section:

######  Theorem 6.2.

Assume the quiver $Q$ satisfies Assumption 2.1 and let $W$ be a quasi-
homogeneous potential of $Q$. Then the cycle map (6.7) induces an isomorphisms
for $i,\ell\in\mathbb{Z}$:

(6.8)
$\displaystyle\mathrm{c}\colon\mathrm{gr}_{\ell}K_{i}^{\mathrm{top}}\left(\mathbb{S}(d)_{v}\right)\xrightarrow{\sim}H^{\dim\mathscr{X}(d)-2\ell-i}(X(d),\mathcal{BPS}_{d,v}^{\mathrm{inv}}[-1]).$

The main part of proving Theorem 6.2 is the construction of a cycle map from
the topological K-theory of quasi-BPS categories to BPS cohomology, which
applies to all symmetric quivers $Q$.

######  Theorem 6.3.

Let $Q$ be an arbitrary symmetric quiver and let $W$ be a quasi-homogeneous
potential of $Q$. Let $d\in\mathbb{N}^{I}$, $\delta\in
M(d)^{W_{d}}_{\mathbb{R}}$, and $i,\ell\in\mathbb{Z}$. The cycle map (6.7)
induces a map:

(6.9)
$\mathrm{c}\colon\mathrm{gr}_{\ell}K_{i}^{\mathrm{top}}\left(\mathbb{S}(d;\delta)\right)\to
H^{\dim\mathscr{X}(d)-2\ell-i}\left(X(d),\mathcal{BPS}_{d,\delta}^{\mathrm{inv}}[-1]\right).$

We mention the following numerical corollary of Theorems 6.2 and 6.3.

######  Corollary 6.4.

Let $Q$ be an arbitrary symmetric quiver and let $W$ be a quasi-homogeneous
potential. Let $(d,v)\in\mathbb{N}^{I}\times\mathbb{Z}$ and let
$i\in\mathbb{Z}$. Then:

(6.10)
$\dim_{\mathbb{Q}}K^{\mathrm{top}}_{i}(\mathbb{S}(d)_{v})\leqslant\dim_{\mathbb{Q}}H^{\cdot}(X(d),\mathcal{BPS}_{d,v})^{\mathrm{inv}}.$

If $Q$ satisfies Assumption 2.1, then equality holds in (6.10).

When $\gcd(\underline{d},v)=1$ and $Q$ satisfies Assumption 2.1, we regard
$\mathbb{S}(d)_{v}$ as a categorification of the monodromy invariant BPS
cohomology of $(Q,W)$. Before we prove Corollary 6.4, note the following:

######  Proposition 6.5.

Let $Q$ be a quiver satisfying Assumption 2.1. The Chern character map (6.6)
is injective.

###### Proof.

It follows from Proposition 4.10 and Theorem 2.8. ∎

###### Proof of Corollary 6.4 from Theorem 6.3.

Note that there is a (non-canonical) isomorphism

(6.11) $\displaystyle
H^{\bullet}(X(d),\mathcal{BPS}_{d,v})^{\mathrm{inv}}\cong
H^{\bullet}(X(d),\mathcal{BPS}_{d,v})_{\mathrm{inv}}.$

The cycle map (6.9) is injective because (6.8) is injective. Then, by Theorem
6.3, we have that:

(6.12)
$\dim_{\mathbb{Q}}\mathrm{gr}_{\cdot}K^{\mathrm{top}}_{i}(\mathbb{S}(d)_{v})\leqslant\dim_{\mathbb{Q}}H^{\cdot}(X(d),\mathcal{BPS}_{d,v})^{\mathrm{inv}}.$

If $Q$ satisfies Assumption 2.1, then (6.12) is an equality. It suffices to
show that
$\dim_{\mathbb{Q}}K^{\mathrm{top}}_{i}(\mathbb{S}(d)_{v})=\dim_{\mathbb{Q}}\mathrm{gr}_{\cdot}K^{\mathrm{top}}_{i}(\mathbb{S}(d)_{v})$,
equivalently that (6.6) is injective, which is Proposition 6.5. ∎

Corollary 1.5 follows easily from Theorem 6.2.

###### Proof of Corollary 1.5.

Note that Proposition 6.1 implies that $\textbf{d}=(d_{i})_{i=1}^{k}\in
S^{d}_{v}$ if and only if $\underline{d}/\gcd(\underline{d},v)$ divides
$\underline{d}_{i}$ for $1\leqslant i\leqslant k$. Then
$S^{d}_{v}=S^{d}_{v^{\prime}}$ for $v,v^{\prime}\in\mathbb{Z}$ such that
$\gcd(\underline{d},v)=\gcd(\underline{d},v^{\prime})$. The statement then
follows from Theorem 6.2. ∎

In Section 7, we compute the topological K-theory of quasi-BPS categories of
preprojective algebras of quivers satisfying Assumption 2.2 using Theorem 6.2,
see Theorem 7.6. In [PTd], we further use Theorem 7.6 to compute the
topological K-theory of quasi-BPS categories of K3 surfaces. In particular, we
obtain categorifications of the BPS cohomology of a large class of
preprojective algebras and of K3 surfaces.

We end this subsection by discussing the zero potential case of Theorem 6.2.
Then $\mathcal{BPS}_{d}=\mathrm{IC}_{X(d)}$. Denote by
$\mathrm{IH}^{\bullet}(X(d)):=H^{\bullet}\big{(}X(d),\mathrm{IC}_{X(d)}\big{)}$.
Note that
$H^{\mathrm{odd}}(\mathscr{X}(d))=H^{\mathrm{odd}}(\mathscr{X}(d),\mathrm{IC}_{\mathscr{X}(d)})=0$.
We then have that $\mathrm{IH}^{\mathrm{even}}(X(d))=0$ because
$\mathrm{IC}_{X(d)}[-1]$ is a direct summand of
$R\pi_{*}\mathrm{IC}_{\mathscr{X}(d)}$, see (6.16); alternatively, the
vanishing $\mathrm{IH}^{\mathrm{even}}(X(d))=0$ follows from Kirwan
surjectivity. By Theorem 6.2 and Proposition 4.13, we obtain the following:

######  Theorem 6.6.

Let $Q$ be a quiver satisfying Assumption 2.1, let $d\in\mathbb{N}^{I}$, and
let $v\in\mathbb{Z}$ such that $\gcd\left(\underline{d},v\right)=1$. For
$\ell\in\mathbb{Z}$, the cycle map induces an isomorphism:

$\mathrm{c}\colon\mathrm{gr}_{\ell}K^{\mathrm{top}}_{0}(\mathbb{M}(d)_{v})\xrightarrow{\sim}\mathrm{IH}^{\dim\mathscr{X}(d)-2\ell-1}(X(d)).$

We note a consequence of Corollary 6.4, alternatively a numerical corollary of
Theorem 6.6.

######  Corollary 6.7.

Let $Q$ be a quiver satisfying Assumption 2.1, let $d\in\mathbb{N}^{I}$, and
let $v\in\mathbb{Z}$ such that $\gcd\left(\underline{d},v\right)=1$. Then

$\dim_{\mathbb{Q}}K^{\mathrm{top}}_{0}(\mathbb{M}(d)_{v})=\dim_{\mathbb{Q}}\mathrm{IH}^{\cdot}(X(d))$

for any $i\in\mathbb{Z}$.

For $(d,v)\in\mathbb{N}^{I}\times\mathbb{Z}$ and $Q$ as in the statement of
Corollary (6.7), we regard $\mathbb{M}(d)_{v}$ as a categorification of the
intersection cohomology of $X(d)$. Note that, in general, $X(d)$ is a singular
scheme.

###  6.2. The decomposition theorem

Let $\alpha\in\mathbb{N}$ and recall the construction of framed quivers
$Q^{\alpha f}$ from Subsection 2.14.

We review the explicit computation of summands in the BBDG decomposition
theorem [BBD82] for the pushforwards of the constant sheaves along the maps:

$\displaystyle\pi_{\alpha f,d}\colon\mathscr{X}^{\alpha f}(d)^{\text{ss}}\to
X(d),\ \pi_{d}\colon\mathscr{X}(d)\to X(d)$

due to Meinhardt–Reineke [MR19] and Davison–Meinhardt [DM20]. The maps
$\pi_{\alpha f,d}$ “approximate” the map $\pi_{d}$, see [DM20, Subsection
4.1]. The computation of $\pi_{d*}\mathrm{IC}_{\mathscr{X}(d)}$ is deduced
from the computation of $\pi_{\alpha f,d*}\mathbb{Q}_{\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}}}[\dim\mathscr{X}(d)]$.

We introduce some constructible sheaves on $X(d)$. Let $A$ be a tuplet
$(e_{i},m_{i,a})$ for $1\leqslant i\leqslant s$ and for $a\geqslant 0$, with
$(e_{i})_{i=1}^{s}\in\mathbb{Z}_{\geqslant 1}^{s}$ pairwise distinct and
$m_{i,a}\geqslant 0$ such that $\sum_{i=1}^{s}\sum_{a\geqslant
0}e_{i}m_{i,a}=d$. Let $\mathscr{P}$ be the set of all such tuplets $A$ and
let $\mathscr{P}_{\alpha}\subset\mathscr{P}$ be the subset of such tuplets
with $m_{i,a}=0$ for $a\geqslant\alpha e_{i}$. Note that each $A$ has a
corresponding partition with terms $e_{i}$ with multiplicity $\sum_{a\geqslant
0}m_{i,a}$ for $1\leqslant i\leqslant s$.

Consider the addition maps:

(6.13) $\displaystyle\oplus_{i,a}\colon X(e_{i})^{\times m_{i,a}}\to
X(m_{i,a}e_{i}),\ \oplus^{\prime}\colon\times_{i=1}^{s}\times_{a\geqslant
0}X(m_{i,a}e_{i})\to X(d).$

Define the constructible complexes:

$\displaystyle\mathrm{Sym}^{m_{i,a}}\left(\mathrm{IC}_{X(e_{i})}[-2a-1]\right)$
$\displaystyle:=\oplus_{i,a*}\left(\left(\mathrm{IC}_{X(e_{i})}[-2a-1]\right)^{\boxtimes
m_{i,a}}\right)^{\mathfrak{S}_{m_{i,a}}},$ $\displaystyle\mathrm{P}_{A}$
$\displaystyle:=\oplus^{\prime}_{*}\left(\boxtimes_{1\leqslant i\leqslant
s,a\geqslant
0}\mathrm{Sym}^{m_{i,a}}\left(\mathrm{IC}_{X(e_{i})}[-2a-1]\right)\right).$

Then $\mathrm{P}_{A}$ is supported on the image of $\oplus^{\prime}$ and is a
shifted perverse sheaf of degree

$p_{A}:=\sum_{i=1}^{k}\sum_{a\geqslant 0}m_{i,a}(2a+1),$

meaning that $\mathrm{P}_{A}[p_{A}]\in\mathrm{Perv}(X(d))$. Define analogously

(6.14) $\mathrm{Q}_{A}:=\oplus^{\prime}_{*}\left(\boxtimes_{1\leqslant
i\leqslant s,a\geqslant
0}\mathrm{Sym}^{m_{i,a}}\left(\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{X(e_{i})}[-2a-2]\right)\right).$

Then one can show, using the Thom-Sebastiani theorem, that

$\mathrm{Q}_{A}=\varphi_{\mathrm{Tr}\,W}\mathrm{P}_{A}[-1].$

Let $\alpha$ be an even positive natural number. The following explicit form
of the BBDG decomposition theorem for $\pi_{\alpha f,d}$ was determined by
Meinhardt–Reineke [MR19, Proposition 4.3]:

(6.15) $\pi_{\alpha f,d*}\left(\mathbb{Q}_{\mathscr{X}^{\alpha
f}(d)^{\text{ss}}}[\dim\mathscr{X}(d)]\right)=\bigoplus_{A\in\mathscr{P}_{\alpha}}P_{A}.$

The result in loc. cit. is stated as an equality in the Grothendieck group of
constructible sheaves, but the above stronger statement holds by the argument
in [DM20, Proof of Theorem 4.10]. Using the above, one can obtain, see [DM20,
Theorem C], the following decomposition:

(6.16)
$\pi_{d*}\mathrm{IC}_{\mathscr{X}(d)}=\bigoplus_{A\in\mathscr{P}}\mathrm{P}_{A}.$

The proper pushforward commutes with the vanishing cycle functor, so applying
the vanishing cycle functor to (6.15) one obtains the following decomposition,
which is also called the DT/ BPS wall-crossing:

(6.17) $\pi_{\alpha
f,d*}\varphi_{\mathrm{Tr}\,W}\left(\mathbb{Q}_{\mathscr{X}^{\alpha
f}(d)^{\text{ss}}}[\dim\mathscr{X}(d)-1]\right)=\bigoplus_{A\in\mathscr{P}_{\alpha}}Q_{A}.$

The map $\pi_{d}$ can be approximated by the proper maps $\pi_{\alpha f,d}$,
thus $\pi_{d*}$ also commutes with the vanishing cycle functor. From (6.16),
we obtain:

(6.18)
$\pi_{d*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1]=\bigoplus_{A\in\mathscr{P}}Q_{A}.$

The summands in all the above decompositions are induced via the Hall product.

We now state a corollary of (6.17).

######  Proposition 6.8.

Let $\alpha$ be an even positive integer and let $i\in\mathbb{Z}$. Then there
is an isomorphism of $\mathbb{N}^{I}\times\mathbb{N}$-graded vector spaces,
where the second grading is the cohomological grading:

(6.19) $\displaystyle\bigoplus_{d\in\mathbb{N}^{I}}H^{\bullet}$
$\displaystyle\left(\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}},\varphi\,[\dim\mathscr{X}(d)-1]\right)^{\mathrm{inv}}\cong$
$\displaystyle\left(\mathrm{Sym}\left(\bigoplus_{d\in\mathbb{N}^{I}}H^{\bullet}\left(X(d),\mathcal{BPS}_{d}[-1]\right)\otimes
H^{\bullet}(\mathbb{P}^{\alpha\underline{d}-1})\right)\right)^{\rm{inv}}.$

By forgetting the cohomological grading, there is an isomorphism of
$\mathbb{N}^{I}$-graded vector spaces:

$\bigoplus_{d\in\mathbb{N}^{I}}H^{\cdot}\left(\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}},\varphi\right)^{\mathrm{inv}}\cong\left(\mathrm{Sym}\left(\bigoplus_{d\in\mathbb{N}^{I}}H^{\cdot}\left(X(d),\mathcal{BPS}_{d}\right)^{\oplus\alpha\underline{d}}\right)\right)^{\rm{inv}}.$

###### Proof.

By taking global sections of the two sides of (6.17), we obtain an
isomorphism:

(6.20) $\displaystyle\bigoplus_{d\in\mathbb{N}^{I}}H^{\bullet}$
$\displaystyle\left(\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}},\varphi\,[\dim\mathscr{X}(d)-1]\right)\cong$
$\displaystyle\mathrm{Sym}\left(\bigoplus_{d\in\mathbb{N}^{I}}H^{\bullet}\left(X(d),\mathcal{BPS}_{d}[-1]\right)\otimes
H^{\bullet}(\mathbb{P}^{\alpha\underline{d}-1})\right).$

The isomorphism (6.19) follows by taking the monodromy invariant parts on the
two sides of the isomorphism (6.20). ∎

###  6.3. Semiorthogonal decompositions and the BBDG decomposition theorem

In this section, we prove Theorem 6.2 assuming Theorem 6.3. The proof follows
from a comparison of the pieces in the semiorthogonal decomposition from
Theorem 2.8 with the summands of the DT/ BPS wall-crossing (6.17). Actually,
the proof is based on a comparison of dimensions of certain vector spaces. In
the rest of this subsection, we will use certain non-canonical maps, but they
suffice for comparing dimensions of vector spaces.

We rewrite the Chern character isomorphism (4.21) for $\mathscr{X}$ a smooth
variety with a regular function $f\colon\mathscr{X}\to\mathbb{C}$. Observe
that there is a (non-canonical) isomorphism
$H^{i}(\mathscr{X},\varphi_{f})^{\mathrm{inv}}\cong
H^{i}(\mathscr{X},\varphi_{f})_{\mathrm{inv}}$ of $\mathbb{Q}$-vector spaces.
Rewrite (4.21) as the following (non-canonical) isomorphism of
$\mathbb{Q}$-vector spaces for every $i\in\mathbb{Z}$:

(6.21) $\mathrm{ch}\colon
K^{\mathrm{sg}}_{i}(\mathscr{X}_{0})\xrightarrow{\sim}H^{\cdot}(\mathscr{X},\varphi_{f})^{\mathrm{inv}}.$

Recall the notations
$\mathrm{gr}_{\cdot}K^{\mathrm{top}}_{i}:=\bigoplus_{a\in\mathbb{Z}}\mathrm{gr}_{a}K^{\mathrm{top}}_{i}$
and $H^{\cdot}:=\bigoplus_{a\in\mathbb{Z}}H^{a}.$ Given a vector space $V$
with a linear map $T\colon V\to V$, we denote by $V^{\rm{inv}}$ the kernel of
$(1-T)$. For a set $A$ of pairs $(V_{a},T_{a})_{a\in A}$ we denote by
$(\otimes_{a\in A}V_{a})^{\rm{inv}}$ the kernel of $1-\otimes_{a\in A}T_{a}$.
Note that $\otimes_{a\in A}V_{a}^{\rm{inv}}\subset(\otimes_{a\in
A}V_{a})^{\rm{inv}}$. The same notation is also used for symmetric products.
We will apply the above notation when $T_{a}$ are monodromy operators on
vanishing cycle cohomologies.

Note the following corollary of Theorem 6.3, which follows because the cycle
map (6.8) is injective:

######  Corollary 6.9.

Assume Theorem 6.3 holds. Then the cycle map (6.9) is injective.

###### Proof of Theorem 6.2 assuming Theorem 6.3.

Let $\alpha$ be an even positive integer and fix $i\in\mathbb{Z}$. By Theorem
2.8, there is a semiorthogonal decomposition:

$\mathrm{MF}\left(\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}},\mathrm{Tr}\,W\right)=\left\langle\bigotimes_{j=1}^{k}\mathbb{S}(d_{j})_{v_{j}}\right\rangle,$

where the right hand side is after all partitions $\sum_{j=1}^{k}d_{j}=d$ and
all weights $v_{j}\in\mathbb{Z}$ with $0\leqslant
v_{1}/\underline{d}_{1}<\ldots<v_{k}/\underline{d}_{k}<\alpha$. There is thus
an isomorphism of $\mathbb{N}^{I}$-graded vector spaces:

$\bigoplus_{d\in\mathbb{N}^{I}}\mathrm{gr}_{\cdot}K_{i}^{\mathrm{top}}\left(\mathrm{MF}\left(\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}},\mathrm{Tr}\,W\right)\right)\cong\bigoplus_{0\leqslant
v_{1}/\underline{d}_{1}<\cdots<v_{k}/\underline{d}_{k}<\alpha}\mathrm{gr}_{\cdot}K_{i}^{\rm{top}}\left(\bigotimes_{j=1}^{k}\mathbb{S}(d_{j})_{v_{j}}\right).$

Recall the isomorphism of $\mathbb{Q}$-vector spaces (6.11). By Corollary 6.9,
there is an injective (non-canonical) map:

(6.22)
$\displaystyle\mathrm{gr}_{\cdot}K_{i}^{\rm{top}}(\mathbb{S}(d)_{v})\hookrightarrow
H^{\cdot}(X(d),\mathcal{BPS}_{d,v})^{\rm{inv}}.$

Then we have injective maps

$\displaystyle\bigoplus_{0\leqslant
v_{1}/\underline{d}_{1}<\cdots<v_{k}/\underline{d}_{k}<\alpha}\mathrm{gr}_{\cdot}K_{i}^{\rm{top}}\left(\bigotimes_{j=1}^{k}\mathbb{S}(d_{j})_{v_{j}}\right)$
$\displaystyle\hookrightarrow\bigoplus_{0\leqslant
v_{1}/\underline{d}_{1}<\cdots<v_{k}/\underline{d}_{k}<\alpha}H^{\cdot}\left(\times_{j=1}^{k}X(d_{j}),\boxtimes_{j=1}^{k}\mathcal{BPS}_{d,v}\right)^{\mathrm{inv}}$
$\displaystyle\hookrightarrow\left(\bigotimes_{\begin{subarray}{c}\mu\in\mathbb{Q}\\\
0\leqslant\mu<\alpha\end{subarray}}\left(\mathrm{Sym}\left(\bigoplus_{\begin{subarray}{c}d\in\mathbb{N}^{I}\\\
\exists\,v\,\text{s.t.}\,\mu=v/\underline{d}\end{subarray}}H^{\cdot}(X(d),\mathcal{BPS}_{d})\right)\right)\right)^{\rm{inv}}$
$\displaystyle\xrightarrow{\sim}\left(\mathrm{Sym}\left(\bigoplus_{d\in\mathbb{N}^{I}}H^{\cdot}(X(d),\mathcal{BPS}_{d})^{\oplus\alpha\underline{d}}\right)\right)^{\mathrm{inv}}\xrightarrow{\sim}\bigoplus_{d\in\mathbb{N}^{I}}H^{\cdot}\big{(}\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}},\varphi\big{)}^{\mathrm{inv}}$

where the first inclusion follows from Corollary 6.9 (applied to disjoint
union of $k$-copies of $Q$, the $k=1$ case is (6.22)), the second inclusion
follows from the definition of $\mathcal{BPS}_{d,v}$, Proposition 6.1, and the
fact that the Thom-Sebastiani isomorphism is natural with respect to the
monodromy actions, the first isomorphism follows from a combinatorial
observation, and the second isomorphism follows from Proposition 6.8. We thus
obtain an injective map of $\mathbb{N}^{I}$-graded vector spaces:

$\bigoplus_{d\in\mathbb{N}^{I}}\mathrm{gr}_{\cdot}K_{i}^{\mathrm{top}}\left(\mathrm{MF}^{\mathrm{gr}}\left(\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}},\mathrm{Tr}\,W\right)\right)\hookrightarrow\bigoplus_{d\in\mathbb{N}^{I}}H^{\cdot}\big{(}\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}},\varphi\big{)}^{\mathrm{inv}}.$

By the isomorphism (6.21) together with the exact sequence (4.10), the
$\mathbb{N}^{I}$-graded piece of both sides of the above map has the same
(finite) dimension, hence the map above is an isomorphism. The map (6.22) is
then also an isomorphism, thus also the maps (6.8) are isomorphisms. ∎

It thus remains to prove Theorem 6.3. In Subsection 6.10, we reduce the proof
for a general symmetric quiver to that of a quiver with at least two loops at
every vertex. In Subsection 6.5, we prove a restriction statement of the image
under the cycle map of an object in a quasi-BPS category. In Subsection 6.6,
we combine the above restriction with the decomposition theorems (6.16) to
prove Theorem 6.3.

###  6.4. Reduction to quivers with enough edges

Consider an arbitrary symmetric quiver with potential $(Q,W)$. Let $Q=(I,E)$.
For $i\in I$, let $\omega_{i},\omega^{\prime}_{i}$ be two loops at $i$. Let
$E^{\gimel}:=E\sqcup\\{\omega_{i},\omega^{\prime}_{i}\mid i\in I\\}$ and
consider the quadratic potential $W^{q}:=\sum_{I\in
I}\omega_{i}\omega^{\prime}_{i}$. Define the quiver with potential:

$Q^{\gimel}:=(I,E^{\gimel}),\,W^{\gimel}:=W+W^{q}.$

######  Proposition 6.10.

Assume Theorem 6.3 holds for $(Q^{\gimel},W^{\gimel})$. Then Theorem 6.3 holds
for $(Q,W)$.

Recall the stack of representations $\mathscr{X}(d)=R(d)/G(d)$ of $Q$. For the
quiver $Q^{\gimel}$ and for $d\in\mathbb{N}^{I}$, we consider the following:
the stack of representations with its good moduli space

$\pi^{\gimel}_{d}\colon\mathscr{X}^{\gimel}(d):=\left(R(d)\oplus\mathfrak{g}(d)^{\oplus
2}\right)/G(d)\to X^{\gimel}(d),$

the BPS sheaves $\mathcal{BPS}^{\gimel}_{d,v}$ as defined in (6.5), the
polytope $\mathbb{W}^{\gimel}(d)$ as in (2.16), the integers
$n^{\gimel}_{\lambda}$ as in (2.20), the quasi-BPS categories
$\mathbb{M}^{\gimel}(d;\delta)$ from (2.19) and
$\mathbb{S}^{\gimel}(d;\delta)$ from (2.21). Let

$\mathscr{S}(d):=(R(d)\oplus\mathfrak{g}(d))/G(d)$

and consider the maps, where $v,t$ are the natural projections and $s$ is the
natural inclusion:

${\mathscr{X}(d)}$${\mathscr{S}(d)}$${\mathscr{X}^{\gimel}(d).}$$\scriptstyle{s}$$\scriptstyle{v}$$\scriptstyle{t}$

Let $G:=G(d)$ and $\mathfrak{g}:=\mathfrak{g}(d)$. We discuss two preliminary
propositions.

######  Proposition 6.11.

Let $\delta\in M(d)^{W_{d}}_{\mathbb{R}}$ and let $i\in\mathbb{Z}$. There is
an isomorphism:

$\displaystyle s_{*}v^{*}\colon
H^{i}(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1])$
$\displaystyle\xrightarrow{\sim}H^{i}(\mathscr{X}^{\gimel}(d),\varphi_{\mathrm{Tr}\,W^{\gimel}}\mathrm{IC}_{\mathscr{X}^{\gimel}(d)}[-1]),$
$\displaystyle H^{i}(X(d),\mathcal{BPS}_{d,\delta})$
$\displaystyle\xrightarrow{\sim}H^{i}(X^{\gimel}(d),\mathcal{BPS}^{\gimel}_{d,\delta}).$

###### Proof.

Consider the diagram:

(6.23)
${\mathscr{X}(d)}$${\mathscr{S}(d)}$${\mathscr{X}^{\gimel}(d)}$${X(d)}$${X^{\gimel}(d).}$$\scriptstyle{\pi_{d}}$$\scriptstyle{s}$$\scriptstyle{v}$$\scriptstyle{\pi^{\gimel}_{d}}$$\scriptstyle{u}$

We first show there is an isomorphism of sheaves on
$D^{b}_{\mathrm{con}}(X^{\gimel}(d))$:

(6.24) $s_{*}v^{*}\colon
u_{*}\pi_{d*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1]\xrightarrow{\sim}\pi^{\gimel}_{d*}\varphi_{\mathrm{Tr}\,W^{\gimel}}\mathrm{IC}_{\mathscr{X}^{\gimel}(d)}[-1].$

First, there is an isomorphism of sheaves on
$D^{b}_{\mathrm{con}}(X^{\gimel}(d))$:

(6.25) $s_{*}v^{*}\colon
u_{*}\pi_{d*}\mathrm{IC}_{\mathscr{X}(d)}\xrightarrow{\sim}\pi^{\gimel}_{d*}\varphi_{\mathrm{Tr}\,W^{q}}\mathrm{IC}_{\mathscr{X}^{\gimel}(d)}[-1].$

The above map is obtained from base-change from the the map for
$\mathscr{X}(d)=\mathrm{pt}$, that is, from the map

(6.26)
$s_{0*}v_{0}^{*}\colon\pi_{d,0*}u_{0*}\mathrm{IC}_{BG}\xrightarrow{\sim}\pi_{d,0*}\varphi_{\mathrm{Tr}\,W^{q}}\mathrm{IC}_{\mathfrak{g}^{\oplus
2}/G}[-1],$

where $s_{0},v_{0},u_{0}$ are the maps as in (6.23) for $\mathscr{X}(d)$
replaced by $\mathrm{Spec}\,\mathbb{C}=\mathrm{pt}$, and where
$\pi_{d,0}\colon\mathfrak{g}^{\oplus 2}/G\to\mathfrak{g}^{\oplus 2}/\\!\\!/G$.
By a direct computation, we have that

$\varphi_{\mathrm{Tr}\,W^{q}}\mathrm{IC}_{\mathfrak{g}^{\oplus
2}/G}[-1]=\mathrm{IC}_{BG}$

because $W^{q}$ is a Morse function with critical locus $BG$, the origin in
$\mathfrak{g}^{\oplus 2}/G$. Further, (6.26) is an isomorphism for global
sections by dimensional reduction, see Subsection 5.1, so (6.26) is an
isomorphism. Then (6.25) is also an isomorphism.

Abuse notation and write
$\mathrm{Tr}\,W\colon\mathscr{X}^{\gimel}(d)\xrightarrow{\mathrm{proj}}\mathscr{X}(d)\xrightarrow{\mathrm{Tr}\,W}\mathbb{C}.$
Note that $\pi_{d*}$ commutes with $\varphi_{\mathrm{Tr}\,W}$ because
$\pi_{d}$ can be approximated with the proper maps $\pi_{\alpha f,d}$, see
Subsection 6.2. Further, $\varphi_{\mathrm{Tr}\,W}$ commutes with proper
pushforward and smooth pullback. Apply $\varphi_{\mathrm{Tr}\,W}$ to both
sides of (6.25) and use the Thom-Sebastiani theorem for vanishing cycles to
obtain:

$\displaystyle s_{*}v^{*}\colon
u_{*}\pi_{d*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1]$
$\displaystyle\xrightarrow{\sim}\pi^{\gimel}_{d*}\varphi_{\mathrm{Tr}\,W}\left(\varphi_{\mathrm{Tr}\,W^{q}}\mathrm{IC}_{\mathscr{X}^{\gimel}(d)}[-1]\right)$
$\displaystyle\cong\pi^{\gimel}_{d*}\varphi_{\mathrm{Tr}\,W^{\gimel}}\mathrm{IC}_{\mathscr{X}^{\gimel}(d)}[-1].$

We now explain that the isomorphism (6.24) induces an isomorphism of sheaves
in $D^{b}_{\mathrm{con}}(X^{\gimel}(d))$:

$u_{*}\mathcal{BPS}_{d,\delta}\xrightarrow{\sim}\mathcal{BPS}^{\gimel}_{d,\delta}.$

First, we explain that $S^{d}_{\delta}(Q)=S^{d}_{\delta}(Q^{\gimel})$. Let
$\lambda$ be a cocharacter of $T(d)$. Let $n_{\lambda}$ and
$n^{\gimel}_{\lambda}$ be the integers (2.20) for $Q$ and $Q^{\gimel}$,
respectively. Let $\varepsilon_{\lambda,\delta}$ and
$\varepsilon_{\lambda,\delta}^{\gimel}$ be the integers (6.1) for $Q$ and
$Q^{\gimel}$, respectively. Then

$n^{\gimel}_{\lambda}-n_{\lambda}=2\langle\lambda,\mathfrak{g}(d)^{\lambda>0}\rangle,$

thus $\varepsilon_{\lambda,\delta}=\varepsilon_{\lambda,\delta}^{\gimel}$, so
indeed $S^{d}_{\delta}(Q)=S^{d}_{\delta}(Q^{\gimel})=:S^{d}_{\delta}$.

It suffices to check that (6.24) induces isomorphisms:

(6.27) $u_{*}\mathcal{BPS}_{A}\xrightarrow{\sim}\mathcal{BPS}^{\gimel}_{A}$

for any $A\in S^{d}_{\delta}$. The isomorphism (6.24) is obtained by applying
the functor $\varphi_{\mathrm{Tr}\,W}$ to the isomorphism (6.24) for $W=0$,
that is, from the isomorphism (6.25). Therefore it suffices to check (6.27)
when $W=0$, so we assume that $W=0$ in the rest of the proof. Assume $A$ has a
corresponding partition $(d_{i})_{i=1}^{k}$ of $d$. Let $X_{A}$ be the image
of $\oplus\colon\times_{i=1}^{k}X(d_{i})\to X(d)$. There is an isomorphism:

$u_{*}{}^{p}\mathcal{H}^{k}(\pi_{d*}\mathrm{IC}_{\mathscr{X}(d)})\xrightarrow{\sim}{}^{p}\mathcal{H}^{k}(\pi^{\gimel}_{d*}\varphi_{\mathrm{Tr}\,W^{q}}\mathrm{IC}_{\mathscr{X}^{\gimel}(d)}[-1]).$

There are either no summands of support $X_{A}$ on both sides, case in which
both $u_{*}\mathcal{BPS}_{A}$ and $\mathcal{BPS}^{\gimel}_{A}$ are zero, or
there are unique summands of support $X_{A}$ on both sides, namely
$u_{*}\mathcal{BPS}_{A}$ and $\mathcal{BPS}^{\gimel}_{A}$, and thus (6.27)
follows. ∎

We note the following corollary of Proposition 6.11.

######  Corollary 6.12.

Let $\delta\in M(d)^{W_{d}}_{\mathbb{R}}$ and let $i\in\mathbb{Z}$. There is
an isomorphism:

$\displaystyle s_{*}v^{*}\colon
H^{i}(\mathscr{X}(d),\varphi^{\mathrm{inv}}_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1])$
$\displaystyle\xrightarrow{\sim}H^{i}(\mathscr{X}^{\gimel}(d),\varphi^{\mathrm{inv}}_{\mathrm{Tr}\,W^{\gimel}}\mathrm{IC}_{\mathscr{X}^{\gimel}(d)}[-1]),$
$\displaystyle H^{i}(X(d),\mathcal{BPS}^{\mathrm{inv}}_{d,\delta})$
$\displaystyle\xrightarrow{\sim}H^{i}(X^{\gimel}(d),\mathcal{BPS}^{\gimel,\mathrm{inv}}_{d,\delta}).$

We also relate quasi-BPS categories under Knorrer periodicity:

######  Proposition 6.13.

There is an equivalence:

$\displaystyle s_{*}v^{*}\colon\mathrm{MF}(\mathscr{X}(d),\mathrm{Tr}\,W)$
$\displaystyle\xrightarrow{\sim}\mathrm{MF}(\mathscr{X}^{\gimel}(d),\mathrm{Tr}\,W^{\gimel}),$
$\displaystyle\mathbb{S}(d;\delta)$
$\displaystyle\xrightarrow{\sim}\mathbb{S}^{\gimel}(d;\delta).$

###### Proof.

(cf. [PTe, Proposition 2.14]) Consider the Koszul complex

$\mathscr{K}:=s_{*}v^{*}\mathcal{O}_{\mathscr{X}}\in\mathrm{MF}(\mathscr{X}^{\gimel}(d),\mathrm{Tr}\,W^{q}),$

where
$s_{*}v^{*}\colon\mathrm{MF}(\mathscr{X}(d),0)\xrightarrow{\sim}\mathrm{MF}(\mathscr{X}^{\gimel}(d),\mathrm{Tr}\,W^{q})$
is an equivalence by Knorrer periodicity. By the Thom-Sebastiani theorem for
matrix factorizations [Pre], there is then an equivalence:

$t^{*}(-)\otimes\mathscr{K}\colon\mathrm{MF}(\mathscr{X}(d),\mathrm{Tr}\,W)\xrightarrow{\sim}\mathrm{MF}(\mathscr{X}^{\gimel}(d),\mathrm{Tr}\,W^{\gimel}).$

Note that $t^{*}(-)\otimes\mathscr{K}=s_{*}v^{*}(-)$. It remains to show that

$t^{*}(-)\otimes\mathscr{K}\colon\mathbb{S}(d;\delta)\xrightarrow{\sim}\mathbb{S}^{\gimel}(d;\delta).$

It suffices to show that, for $F\in D^{b}(\mathscr{X}(d))$, we have that
$F\in\mathbb{M}(d;\delta)$ if and only if
$t^{*}(F)\otimes\mathscr{K}\in\mathbb{M}^{\gimel}(d;\delta)$.

We use Lemma 2.3. Let $\nu\colon B\mathbb{C}^{*}\to\mathscr{X}(d)$, let $F\in
D^{b}(\mathscr{X}(d))$, and let $A_{F}\subset\mathbb{Z}$ be the set of weights
of $\nu^{*}(F)$. Note that for any $\nu^{\gimel}\colon
B\mathbb{C}^{*}\to\mathscr{X}^{\gimel}(d)$ such that $t\circ\nu^{\gimel}=\nu$,
we have that the weights of $(\nu^{\gimel})^{*}(t^{*}(F)\otimes\mathscr{K})$
are the Minkowski sum
$A_{F}+\left[-\langle\nu,\mathfrak{g}\rangle,\langle\nu,\mathfrak{g}\rangle\right]$.
We have that

$A_{F}\subset\left[-\frac{1}{2}n_{\lambda}+\langle\lambda,\delta_{d}\rangle,\frac{1}{2}n_{\lambda}+\langle\lambda,\delta_{d}\rangle\right]$

if and only if

$A_{F}+\left[-\langle\nu,\mathfrak{g}\rangle,\langle\nu,\mathfrak{g}\rangle\right]\subset\left[-\frac{1}{2}n^{\gimel}_{\lambda}+\langle\lambda,\delta_{d}\rangle,\frac{1}{2}n^{\gimel}_{\lambda}+\langle\lambda,\delta_{d}\rangle\right].$

The conclusion then follows. ∎

###### Proof of Proposition 6.10.

Let $i,\ell\in\mathbb{Z}$. By Corollary 4.9, there is a commutative diagram,
where
$b=\dim\mathscr{X}(d)-i-2\ell=\dim\mathscr{X}^{\gimel}(d)-i-2(\ell+\dim\mathfrak{g})$:

${\mathrm{gr}_{\ell}K^{\mathrm{top}}_{i}(\mathrm{MF}(\mathscr{X}(d),\mathrm{Tr}\,W))}$${\mathrm{gr}_{\ell+\dim\mathfrak{g}}K^{\mathrm{top}}_{i}(\mathrm{MF}(\mathscr{X}^{\gimel}(d),\mathrm{Tr}\,W^{\gimel}))}$${H^{b}(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}^{\mathrm{inv}}\mathrm{IC}_{\mathscr{X}(d)}[-2])}$${H^{b}(\mathscr{X}^{\gimel}(d),\varphi_{\mathrm{Tr}\,W^{\gimel}}^{\mathrm{inv}}\mathrm{IC}_{\mathscr{X}^{\gimel}(d)}[-2]).}$$\scriptstyle{\sim}$$\scriptstyle{s_{*}v^{*}}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{\sim}$$\scriptstyle{s_{*}v^{*}}$

The conclusion follows from Corollary 6.12 and Proposition 6.13. ∎

It will be convenient to make the following assumption on a quiver:

######  Assumption 6.1.

Assume the quiver $Q$ is symmetric and has at least two loops at any vertex.

We introduce some notation. For any cocharacter $\lambda$ with associated
partition d, define
$c_{\textbf{d}}:=c_{\lambda}:=\dim\mathscr{X}(d)-\dim\mathscr{X}(d)^{\lambda\geqslant
0}$.

######  Lemma 6.14.

Let $Q=(I,E)$ be a quiver which satisfies Assumption 6.1 and let
$d\in\mathbb{N}^{I}$ be non-zero.

(a) For any cocharacter $\lambda$ of $T(d)$, we have $c_{\lambda}\geqslant 0$,
and the inequality is strict if $\lambda$ has an associated partition with at
least two terms. Moreover, we have

$\dim\mathscr{X}(d)^{\lambda\geqslant
0}-\dim\mathscr{X}(d)^{\lambda}=c_{\lambda}.$

(b) The map $\pi_{d}\colon\mathscr{X}(d)\to X(d)$ is generically a
$\mathbb{C}^{*}$-gerbe, in particular there exists a stable representation of
dimension $d$.

###### Proof.

We only discuss the first claim of part (a), the second claim and part (b) are
similar. Let $S(d)$ be the affine space of dimension $d$ representations of
the quiver obtained from $Q$ by deleting one loop at every vertex in $I$. Then

$\mathscr{X}(d)=\left(S(d)\oplus\mathfrak{g}(d)\right)/G(d).$

We have that

$\dim\mathscr{X}(d)=\dim S(d)\text{ and }\dim\mathscr{X}(d)^{\lambda\geqslant
0}=\dim\left(S(d)\right)^{\lambda\geqslant 0},$

so $c_{\lambda}=\dim\left(S(d)\right)^{\lambda<0}$, and the first claim
follows. ∎

###  6.5. Coproduct-like maps in K-theory

In this subsection,we assume that $Q$ satisfies Assumption 6.1. Consider an
antidominant cocharacter $\lambda$ of $T(d)$ and let
$a_{\lambda}\colon\mathscr{X}(d)^{\lambda}\to\mathscr{X}(d)$ be the natural
morphism inducing pullback maps for any $i,\ell\in\mathbb{Z}$:

$\displaystyle a_{\lambda}^{*}$ $\displaystyle\colon
K^{\mathrm{top}}_{i}\left(\mathrm{MF}\left(\mathscr{X}(d),\mathrm{Tr}\,W\right)\right)\to
K^{\mathrm{top}}_{i}\left(\mathrm{MF}\left(\mathscr{X}(d)^{\lambda},\mathrm{Tr}\,W\right)\right),$
$\displaystyle a^{*}_{\lambda}$
$\displaystyle\colon\mathrm{gr}_{\ell}K_{i}^{\mathrm{top}}(\mathrm{MF}\left(\mathscr{X}(d),\mathrm{Tr}\,W\right))\to\mathrm{gr}_{\ell+2d_{\lambda}}K_{i}^{\mathrm{top}}\left(\mathrm{MF}\left(\mathscr{X}(d)^{\lambda},\mathrm{Tr}\,W\right)\right),$

where $d_{\lambda}:=\dim\mathscr{X}(d)^{\lambda}-\dim\mathscr{X}(d)$ is the
relative dimension of $a_{\lambda}$. Consider the quotient
$G(d)^{\prime}:=G(d)^{\lambda}/\text{image}(\lambda)$ and the stack
$\mathscr{X}(d)^{\prime\lambda}:=R(d)^{\lambda}/G(d)^{\prime}$. There is an
isomorphism:

$\displaystyle
K^{\mathrm{top}}_{i}\left(\mathrm{MF}\left(\mathscr{X}(d)^{\lambda},\mathrm{Tr}\,W\right)\right)\cong
K^{\mathrm{top}}_{i}\left(\mathrm{MF}\left(\mathscr{X}(d)^{\prime\lambda},\mathrm{Tr}\,W\right)\right)[q^{\pm
1}].$

There is an analogous isomorphism for graded K-theory. There are also maps in
cohomology:

$\displaystyle a_{\lambda}^{*}$ $\displaystyle\colon
H^{\cdot}\left(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}\right)\to
H^{\cdot}\left(\mathscr{X}(d)^{\prime\lambda},\varphi_{\mathrm{Tr}\,W}\right)[h],$
$\displaystyle a_{\lambda}^{*}$ $\displaystyle\colon
H^{\cdot}\left(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}^{\mathrm{inv}}\right)\to
H^{\cdot}\left(\mathscr{X}(d)^{\prime\lambda},\varphi_{\mathrm{Tr}\,W}^{\mathrm{inv}}\right)[h].$

Assume the associated partition of $\lambda$ is
$\textbf{d}=(d_{i})_{i=1}^{k}$. Recall that
$c_{\lambda}:=c_{\textbf{d}}:=\dim\mathscr{X}(d)-\dim\mathscr{X}(d)^{\lambda\geqslant
0}$. We define the following integers (which we call widths of magic or quasi-
BPS categories in this paper):

(6.28)
$c_{\lambda,\delta}:=c_{\lambda}+\varepsilon_{\lambda,\delta},\,c_{\textbf{d},\delta}:=c_{\textbf{d}}+\varepsilon_{\textbf{d},\delta}.$

######  Proposition 6.15.

Let $\lambda$ be an antidominant cocharacter of $G(d)$ and let
$i,\ell\in\mathbb{Z}$. Consider the diagram:

${\mathrm{gr}_{\ell}K^{\mathrm{top}}_{i}(\mathbb{S}(d;\delta))}$${\mathrm{gr}_{\ell+2d_{\lambda}}K^{\mathrm{top}}_{i}\left(\mathrm{MF}\left(\mathscr{X}(d)^{\lambda},\mathrm{Tr}\,W\right)\right)}$${H^{2\dim\mathscr{X}(d)-2\ell-i}\left(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}^{\mathrm{inv}}[-2]\right)}$${H^{\cdot}\left(\mathscr{X}(d)^{\prime\lambda},\varphi_{\mathrm{Tr}\,W}^{\mathrm{inv}}[-2]\right)[h].}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{a^{*}_{\lambda}}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{a^{*}_{\lambda}}$

Then the image of
$\mathrm{c}\,a^{*}_{\lambda}\mathrm{gr}_{\bullet}K^{\mathrm{top}}_{\bullet}\left(\mathbb{S}(d;\delta)\right)$
lies in the subspace

$\bigoplus_{j=0}^{c_{\lambda,\delta}-1}H^{2\dim\mathscr{X}(d)-2\ell-i-2j}\left(\mathscr{X}(d)^{\prime\lambda},\varphi^{\mathrm{inv}}[-2]\right)h^{j}\subset
H^{\cdot}\left(\mathscr{X}(d)^{\prime\lambda},\varphi^{\mathrm{inv}}[-2]\right)[h].$

Note that $a_{\lambda}$ only depends on the partition d, so we obtain that the
image of
$\mathrm{c}\,a^{*}_{\lambda}\mathrm{gr}_{\ell}K^{\mathrm{top}}_{i}\left(\mathbb{S}(d;\delta)\right)$
lies in the subspace

$\bigoplus_{j=0}^{c_{\textbf{d},\delta}-1}H^{2\dim\mathscr{X}(d)-2\ell-i-2j}\left(\mathscr{X}(d)^{\prime\lambda},\varphi^{\mathrm{inv}}[-2]\right)h^{j}\subset
H^{\cdot}\left(\mathscr{X}(d)^{\prime\lambda},\varphi^{\mathrm{inv}}[-2]\right)[h].$

###### Proof.

Consider a complex $A$ in $\mathbb{S}(d;\delta)$. Then $a_{\lambda}^{*}(A)$ is
in the subcategory of $\mathrm{MF}(\mathscr{X}(d)^{\lambda},\mathrm{Tr}\,W)$
generated by $\mathrm{MF}(\mathscr{X}(d)^{\prime\lambda},\mathrm{Tr}\,W)_{v}$
for

$v\in
S_{\lambda,\delta}:=\left[-\frac{1}{2}\langle\lambda,\mathbb{L}_{\mathscr{X}(d)}^{\lambda>0}\rangle+\langle\lambda,\delta\rangle,\frac{1}{2}\langle\lambda,\mathbb{L}_{\mathscr{X}(d)}^{\lambda>0}\rangle+\langle\lambda,\delta\rangle\right]\cap\mathbb{Z}.$

Thus

(6.29)
$a^{*}_{\lambda}K^{\mathrm{top}}_{i}\left(\mathbb{S}(d;\delta)\right)\subset
K^{\mathrm{top}}_{i}\left(\mathrm{MF}\left(\mathscr{X}(d)^{\prime\lambda},\mathrm{Tr}\,W\right)\right)\otimes\mathscr{A},$

where $\mathscr{A}:=\bigoplus_{j\in S_{\lambda,\delta}}\mathbb{Q}\cdot q^{j}$.
There are filtrations pulled back from cohomology by the Chern character for
both
$K_{i}^{\mathrm{top}}\left(\mathrm{MF}(\mathscr{X}(d)^{\prime\lambda},\mathrm{Tr}\,W)\right)$
and $K_{0}^{\mathrm{top}}\left(B\mathbb{C}^{*}\right)$, and there is an
isomorphism obtained by the Kunneth formula:

(6.30)
$\mathrm{gr}_{\ell}K_{i}^{\mathrm{top}}\left(\mathrm{MF}(\mathscr{X}(d)^{\lambda},\mathrm{Tr}\,W)\right)\cong\bigoplus_{a+b=\ell}\mathrm{gr}_{a}K_{i}^{\mathrm{top}}\left(\mathrm{MF}(\mathscr{X}(d)^{\prime\lambda},\mathrm{Tr}\,W)\right)\otimes\mathrm{gr}_{b}K_{0}^{\mathrm{top}}\left(B\mathbb{C}^{*}\right).$

The filtration $E_{b}K^{\mathrm{top}}_{0}(B\mathbb{C}^{*})$ on
$K^{\mathrm{top}}_{0}(B\mathbb{C}^{*})$ induces a filtration

$E_{b}\mathscr{A}:=\mathscr{A}\cap E_{b}K^{\mathrm{top}}_{0}(B\mathbb{C}^{*})$

on $\mathscr{A}$. There are natural inclusions
$\mathrm{gr}_{b}\mathscr{A}\hookrightarrow\mathrm{gr}_{b}K^{\mathrm{top}}_{0}(B\mathbb{C}^{*})$.
We obtain a Kunneth formula:

(6.31)
$\displaystyle\mathrm{gr}_{\ell}\left(K_{i}^{\mathrm{top}}\left(\mathrm{MF}(\mathscr{X}(d)^{\prime\lambda},\mathrm{Tr}\,W)\right)\otimes\mathscr{A}\right)\cong\bigoplus_{a+b=\ell}\mathrm{gr}_{a}K_{i}^{\mathrm{top}}\left(\mathrm{MF}(\mathscr{X}(d)^{\prime\lambda},\mathrm{Tr}\,W)\right)\otimes\mathrm{gr}_{b}\mathscr{A}.$

By (6.29) and (6.31), we have

$a^{*}_{\lambda}\mathrm{gr}_{\ell}K^{\mathrm{top}}_{i}\left(\mathbb{S}(d;\delta)\right)\subset\mathrm{gr}_{\cdot}K^{\mathrm{top}}_{i}\left(\mathrm{MF}\left(\mathscr{X}(d)^{\prime\lambda},\mathrm{Tr}\,W\right)\right)\otimes\mathrm{gr}_{\cdot}\mathscr{A}.$

It suffices to show that:

(6.32)
$\mathrm{c}\left(\mathrm{gr}_{\cdot}\mathscr{A}\right)\subset\bigoplus_{i=0}^{c_{\lambda,\delta}-1}\mathbb{Q}\cdot
h^{i}.$

For any $1\leqslant i\leqslant k$, let $F_{i}$ be a stable representation of
dimension $d_{i}$ (which exists by Lemma 6.14, and note that this is the only
place where we use that $Q$ satisfies Assumption 6.1) and let
$F:=\bigoplus_{i=1}^{k}F_{i}$. Let $V/(\mathbb{C}^{*})^{k}$ be the moduli
stack of representations of the Ext quiver of $F$ and dimension vector
$(1,\ldots,1)\in\mathbb{N}^{k}$. Note that there is an étale map

$V/(\mathbb{C}^{*})^{k}\to\mathscr{X}(d),\,0\mapsto F.$

We have the equality of sets

$S_{\lambda,\delta}=\Big{[}-\frac{1}{2}\langle\lambda,V^{\lambda>0}\rangle+\langle\lambda,\delta\rangle,\frac{1}{2}\langle\lambda,V^{\lambda>0}\rangle+\langle\lambda,\delta\rangle\Big{]}\cap\mathbb{Z}.$

We denote the image of $\lambda$ in $(\mathbb{C}^{*})^{k}$ by
$\mathbb{C}^{*}$. Consider the maps:

$V^{\lambda}\xleftarrow{q^{\prime}}V^{\lambda\geqslant
0}\xrightarrow{p^{\prime}}V.$

Let $\ell$ be a generic linearization of $\mathbb{C}^{*}$. By [HL12, Theorem
2.10], see also [HLS16, Equation (3)], the subcategory of
$D^{b}(V/\mathbb{C}^{*})$ generated by $\mathcal{O}_{V}(v)$ for weights $v\in
S_{\lambda,\delta}$ is equivalent to
$D^{b}(V^{\ell\text{-ss}}/\mathbb{C}^{*})$ if
$\varepsilon_{\lambda,\delta}=0$, and has a semiorthogonal decomposition with
pieces $D^{b}(V^{\ell\text{-ss}}/\mathbb{C}^{*})$ and
$p^{\prime}_{*}q^{\prime*}D^{b}(V^{\lambda})$ if
$\varepsilon_{\lambda,\delta}=1$. Define the map

$\mathrm{s}\colon K^{\mathrm{top}}_{0}(B\mathbb{C}^{*})\cong
K^{\mathrm{top}}_{0}(V/\mathbb{C}^{*})\to
K^{\mathrm{top}}_{0}(V^{\ell\text{-ss}}/\mathbb{C}^{*})\oplus
p^{\prime}_{*}q^{\prime*}K^{\mathrm{top}}_{0}(V^{\lambda})^{\oplus\varepsilon_{\lambda,\delta}}$

as the direct sum of the restriction onto $V^{\ell\text{-ss}}/\mathbb{C}^{*}$
and the inverse of the inclusion:

$p^{\prime}_{*}q^{\prime*}\colon
K^{\mathrm{top}}_{0}\left(D^{b}(V^{\lambda})_{a}\right)^{\oplus\varepsilon_{\lambda,\delta}}\cong
K^{\mathrm{top}}_{0}(V^{\lambda})^{\oplus\varepsilon_{\lambda,\delta}}\to
K^{\mathrm{top}}_{0}(V/\mathbb{C}^{*})$

for a weight
$a=\big{\lfloor}\frac{1}{2}\langle\lambda,V^{\lambda>0}\rangle+\langle\lambda,\delta\rangle\big{\rfloor}\in\mathbb{Z}$
of $\lambda$, constructed by the semiorthogonal decomposition [HL12]. The
following composition is an isomorphism:

$\mathscr{A}\hookrightarrow K^{\mathrm{top}}_{0}(B\mathbb{C}^{*})\cong
K^{\mathrm{top}}_{0}(V/\mathbb{C}^{*})\xrightarrow{\mathrm{s}}K^{\mathrm{top}}_{0}(V^{\ell\text{-ss}}/\mathbb{C}^{*})\oplus
p^{\prime}_{*}q^{\prime*}K^{\mathrm{top}}_{0}(V^{\lambda})^{\oplus\varepsilon_{\lambda,\delta}}.$

Note that the Hall product $p^{\prime}_{*}q^{\prime*}\colon
H^{\cdot}(V^{\lambda})\to H^{\cdot}(V/\mathbb{C}^{*})$ has image
$\mathbb{Q}\cdot h^{c_{\lambda}}$, and thus it has a natural inverse
$H^{\cdot}(V/\mathbb{C}^{*})\to
p^{\prime}_{*}q^{\prime*}H^{\cdot}(V^{\lambda})$. Let $\mathrm{t}$ be the
direct sum of this inverse and the restriction map:

$\mathrm{t}\colon H^{\mathrm{BM}}_{\cdot}(V/\mathbb{C}^{*})\to
H^{\mathrm{BM}}_{\cdot}(V^{\ell\text{-ss}}/\mathbb{C}^{*})\oplus
p^{\prime}_{*}q^{\prime*}H^{\mathrm{BM}}_{\cdot}(V^{\lambda})^{\oplus\varepsilon_{\lambda,\delta}}.$

Recall that the Hall products in K-theory and cohomology are compatible via
the cycle map, see Proposition 3.6. There is a commutative diagram:

(6.33)
${\mathrm{gr}_{\cdot}\mathscr{A}}$${\mathrm{gr}_{\cdot}K^{\mathrm{top}}_{0}(V/\mathbb{C}^{*})}$${\mathrm{gr}_{\cdot}K^{\mathrm{top}}_{0}(V^{\ell\text{-ss}}/\mathbb{C}^{*})\oplus
p^{\prime}_{*}q^{\prime*}\mathrm{gr}_{\cdot}K^{\mathrm{top}}(V^{\lambda})^{\oplus\varepsilon_{\lambda,\delta}}}$${H^{\mathrm{BM}}_{\cdot}(V/\mathbb{C}^{*})}$${H^{\mathrm{BM}}_{\cdot}(V^{\ell\text{-ss}}/\mathbb{C}^{*})\oplus
p^{\prime}_{*}q^{\prime*}H^{\mathrm{BM}}_{\cdot}(V^{\lambda})^{\oplus\varepsilon_{\lambda,\delta}}.}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{\mathrm{s}}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{\mathrm{t}}$

Note that $V^{\ell\text{-ss}}/\mathbb{C}^{*}=V^{\lambda}\times
V^{\prime\ell\text{-ss}}/\mathbb{C}^{*}$, where $V^{\prime}\subset V$ is the
subspace spanned by non-zero $\lambda$-weights, and thus

$H^{\cdot}(V^{\ell\text{-ss}}/\mathbb{C}^{*})\cong
H^{\cdot}(V^{\prime\ell\text{-ss}}/\mathbb{C}^{*})\cong\bigoplus_{i=0}^{c_{\lambda}-1}\mathbb{Q}\cdot
h^{i}.$

Then the map $\mathrm{t}$ is the truncation of polynomials:

(6.34)
${H^{\mathrm{BM}}_{\cdot}(V/\mathbb{C}^{*})}$${H^{\mathrm{BM}}_{\cdot}(V^{\ell\text{-ss}}/\mathbb{C}^{*})\oplus
p^{\prime}_{*}q^{\prime*}H^{\mathrm{BM}}_{\cdot}(V^{\lambda})^{\oplus\varepsilon_{\lambda,\delta}}}$${\mathbb{Q}[h]}$${\bigoplus_{i=0}^{c_{\lambda,\delta}-1}\mathbb{Q}\cdot
h^{i}.}$$\scriptstyle{\mathrm{t}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

The conclusion then follows. ∎

###  6.6. Coproduct-like maps in cohomology

In this section, we assume that $Q=(I,E)$ satisfies Assumption 6.1. By
Proposition 6.10, we obtain Theorem 6.3 for general symmetric quivers $Q$.

For $\lambda$ an antidominant cocharacter with associated partition
$(d_{i})_{i=1}^{k}$, consider the good moduli space map

$\pi_{\lambda}:=\times_{i=1}^{k}\pi_{d_{i}}\colon\mathscr{X}(d)^{\lambda}\to
X(d)^{\lambda}:=\times_{i=1}^{k}X(d_{i}).$

Consider the projection map
$t_{\lambda}\colon\mathscr{X}(d)^{\lambda}\to\mathscr{X}(d)^{\prime\lambda}$.
The maps introduced fit in the following diagram:

Consider the following perverse truncation

(6.35) $\displaystyle\mathrm{S}\colon
t_{\lambda*}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}[c_{\lambda}]$
$\displaystyle\cong\mathrm{IC}_{\mathscr{X}(d)^{\prime\lambda}}[c_{\lambda}-1]\otimes\mathbb{Q}[h]$
$\displaystyle\to{}^{p}\tau^{\geqslant
c_{\lambda}+1}\left(\mathrm{IC}_{\mathscr{X}(d)^{\prime\lambda}}[c_{\lambda}-1]\otimes\mathbb{Q}[h]\right)$
$\displaystyle\cong\bigoplus_{j\geqslant
0}\mathrm{IC}_{\mathscr{X}(d)^{\prime\lambda}}[-c_{\lambda}-1-2j]\cong
t_{\lambda*}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}[-c_{\lambda}].$

Define the map $\Delta_{\lambda}$ as the composition:

(6.36)
$\Delta_{\lambda}\colon\pi_{d\ast}\mathrm{IC}_{\mathscr{X}(d)}\to\pi_{d\ast}a_{\lambda\ast}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}[2c_{\lambda}]=i_{\lambda\ast}\pi_{\lambda\ast}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}[2c_{\lambda}]\stackrel{{\scriptstyle\mathrm{S}}}{{\to}}i_{\lambda\ast}\pi_{\lambda\ast}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}.$

Recall the notations from Subsection 6.2 and the decomposition theorem (6.16).
Consider the total perverse cohomology

$\mathcal{H}\left(\pi_{d*}\mathrm{IC}_{\mathscr{X}(d)}\right):=\bigoplus_{i\in\mathbb{Z}}{}^{p}\mathcal{H}^{i}\left(\pi_{d*}\mathrm{IC}_{\mathscr{X}(d)}\right)[-i].$

For $A\in\mathscr{P}$ as in Subsection 6.2, there are then natural maps

$\displaystyle\mathrm{P}_{A}\to\mathcal{H}\left(\pi_{d*}\mathrm{IC}_{\mathscr{X}(d)}\right)\to\mathrm{P}_{A}.$

######  Proposition 6.16.

Let $A,B\in\mathscr{P}$ with corresponding sheaves $\mathrm{P}_{A}$ and
$\mathrm{P}_{B}$ of different support. Assume that $p_{B}\leqslant p_{A}$.

(a) The map (6.36) induces an isomorphism

(6.37) $\Delta_{\lambda}\colon\mathrm{P}_{A}\xrightarrow{\sim}\mathrm{P}_{A}.$

(b) The map $\Delta_{\lambda}\colon\mathrm{P}_{B}\to\mathrm{P}_{A}$ is zero.

###### Proof.

(a) Assume $\lambda$ has associated partition $(d_{i})_{i=1}^{k}$. Assume
further that the set $\\{d_{1},\ldots,d_{k}\\}=\\{e_{1},\ldots,e_{s}\\}$ has
cardinality $s$ and that each $e_{i}$ appears $m_{i}$ times among the $d_{j}$
for $1\leqslant j\leqslant k$. Let $A^{\circ}\in\mathscr{P}$ be the tuplet
$(e_{i},m_{i,a})$ with $m_{i,0}=m_{i}$ and $m_{i,a}=0$ for $a\geqslant 1$.

For $d\in\mathbb{N}^{I}$, let $\hbar_{d}:=c_{1}(\mathcal{O}(\sigma_{d}))\in
H^{2}(\mathscr{X}(d))$. By [DM20, Theorem C], the summand $\mathrm{P}_{A}$ of
$\mathcal{H}\left(\pi_{d*}\mathrm{IC}_{\mathscr{X}(d)}\right)$ is obtained
from $\boxtimes_{i=1}^{k}(\mathrm{IC}_{X(d_{i})}[-1])$ by multiplication with
the equivariant parameters $\hbar_{d_{i}}$ for $1\leqslant i\leqslant k$. The
map $\Delta_{\lambda}\colon\mathrm{P}_{A}\to\mathrm{P}_{A}$ is thus obtained
by multiplication by equivariant parameters $\hbar_{d_{i}}$ for $1\leqslant
i\leqslant k$ from the map

$\displaystyle\Delta_{\lambda}m_{\lambda}\colon
i_{\lambda\ast}\boxtimes_{i=1}^{k}(\mathrm{IC}_{X(d_{i})}[-1])\to
i_{\lambda\ast}\boxtimes_{i=1}^{k}(\mathrm{IC}_{X(d_{i})}[-1]).$

By [DM20, Theorem C], the image of
$m_{\lambda}\left(i_{\lambda\ast}\boxtimes_{i=1}^{k}(\mathrm{IC}_{X(d_{i})}[-1])\right)$
in $\mathcal{H}\left(\pi_{d*}\mathrm{IC}_{\mathscr{X}(d)}\right)$ is
$\mathrm{P}_{A^{\circ}}$. Thus the map (6.37) is obtained by multiplication by
equivariant parameters from the map

$\Delta_{\lambda}\colon\mathrm{P}_{A^{\circ}}\to\mathrm{P}_{A^{\circ}}.$

We may thus assume that $A=A^{\circ}$. The Hall product is induced by a map

$m_{\lambda}\colon
i_{\lambda\ast}\pi_{\lambda*}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}\to\pi_{d*}\mathrm{IC}_{\mathscr{X}(d)}.$

The lowest non-zero piece of the perverse filtration on
$i_{\lambda\ast}\pi_{\lambda*}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}$ is given
by

${}^{p}\tau^{\leqslant
k}i_{\lambda\ast}\pi_{\lambda*}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}=i_{\lambda\ast}\boxtimes_{i=1}^{s}\left(\mathrm{IC}_{X(e_{i})}[-1]\right)^{\boxtimes
m_{i}}.$

The (shifted) perverse sheaf
$i_{\lambda\ast}\boxtimes_{i=1}^{s}\left(\mathrm{IC}_{X(e_{i})}[-1]\right)^{\boxtimes
m_{i}}$ splits as a direct sum of simple sheaves, and one such sheaf is
$\mathrm{P}_{A}$. There is thus a natural inclusion $\mathrm{P}_{A}\subset
i_{\lambda\ast}\pi_{\lambda*}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}$. The map

(6.38) $\Delta_{\lambda}m_{\lambda}\colon\mathrm{P}_{A}\to
i_{\lambda\ast}\pi_{\lambda*}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}$

has image in the lowest non-zero perverse truncation of
$i_{\lambda\ast}\pi_{\lambda*}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}$, and
thus (6.38) induces a map:

(6.39)
$\Delta_{\lambda}m_{\lambda}\colon\mathrm{P}_{A}\to{}^{p}\tau^{\leqslant
s}i_{\lambda\ast}\pi_{\lambda*}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}=\boxtimes_{i=1}^{s}\left(\mathrm{IC}_{X(e_{i})}[-1]\right)^{\boxtimes
m_{i}}.$

The (shifted) perverse sheaf
$i_{\lambda\ast}\boxtimes_{i=1}^{s}\left(\mathrm{IC}_{X(e_{i})}[-1]\right)^{\boxtimes
m_{i}}$ has only one summand isomorphic to $\mathrm{P}_{A}$, which is a simple
(shifted) perverse sheaf. Thus the map (6.39) restricts to a map

(6.40) $\mathrm{P}_{A}\to\mathrm{P_{A}}$

All such maps are given by multiplication by scalars. It is thus an
isomorphism if it is not the zero map. It suffices to show that the maps
(6.38) or (6.39) are not zero. We will show this after passing to a convenient
open set of $X(d)^{\lambda}$.

For any non-zero $e\in\mathbb{N}^{I}$, by the same argument used to prove that
(6.14), there exists a stable point in $R(e)$, equivalently the map
$\pi_{e}\colon\mathscr{X}(e)\to X(e)$ is generically a $\mathbb{C}^{*}$-gerbe.
For $1\leqslant i\leqslant k$, let $R_{i}$ be a simple representation of $Q$
of dimension $d_{i}$ such that $R_{i}$ and $R_{j}$ are not isomorphic for
$1\leqslant i<j\leqslant k$. Let $R:=\bigoplus_{i=1}^{k}R_{i}$. Note that the
stabilizer of $R$ is $T=(\mathbb{C}^{*})^{k}$. By the etale slice theorem,
there is an analytic smooth open substack
$R\in\mathscr{U}/T\subset\mathscr{X}(d)$ such that

$\mathscr{U}/\\!\\!/T\to X(d)\text{ and }\mathscr{U}^{\lambda}\to
X(d)^{\lambda}$

are an analytic neighborhoods of $\pi_{d}(R)$ and
$\times_{i=1}^{k}\pi_{d_{i}}(R_{i})=\pi_{\lambda}(R)$, respectively. After
possibly shrinking $\mathscr{U}$, we may assume that $\mathscr{U}$ and
$\mathscr{U}^{\lambda}$ are contractible. The maps

${\mathscr{X}(d)^{\lambda}}$${\mathscr{X}(d)^{\lambda\geqslant
0}}$${\mathscr{X}(d)}$$\scriptstyle{a_{\lambda}}$$\scriptstyle{q_{\lambda}}$$\scriptstyle{p_{\lambda}}$

are, analytically locally over $\pi_{d}(R)\in X(d)$, isomorphic to the
following:

(6.41) ${\mathscr{U}^{\lambda}/T}$${\mathscr{U}^{\lambda\geqslant
0}/T}$${\mathscr{U}/T.}$$\scriptstyle{a_{\lambda}}$$\scriptstyle{q_{\lambda}}$$\scriptstyle{p_{\lambda}}$

Note that the maps $p_{\lambda}$ and $a_{\lambda}$ in (6.41) are closed
immersions. To show that the map (6.40) is non-zero, it suffices to check that
the map

(6.42)
$\Delta_{\lambda}m_{\lambda}|_{\mathscr{U}^{\lambda}}\colon\mathrm{P}_{A}|_{\mathscr{U}^{\lambda}}\to\mathrm{P}_{A}|_{\mathscr{U}^{\lambda}}$

is non-zero. It suffices to check that the map is non-zero after passing to
global sections. We drop the restriction to $\mathscr{U}^{\lambda}$ from the
notation from now on. The element $1\in H^{0}(\mathscr{U}^{\lambda}/T)$ is in
$P^{\leqslant s}H^{\cdot}(\mathscr{U}^{\lambda}/T)$. We check by a direct
computation that

(6.43) $\Delta_{\lambda}m_{\lambda}(1)=1\in H^{0}(\mathscr{U}^{\lambda}/T).$

Note that the computation (6.43) shows that the map (6.42) is non-zero, and
thus the conclusion follows. It suffices to check the computation in (6.43)
for $\mathscr{U}^{\lambda}/\mathbb{C}^{*}$, where by $\mathbb{C}^{*}$ we
denote the image of $\lambda$, because
$H^{0}(\mathscr{U}^{\lambda}/\mathbb{C}^{*})\cong
H^{0}(\mathscr{U}^{T}/T)\cong\mathbb{Q}$. Observe that
$H^{\cdot}(\mathscr{U}^{\lambda}/\mathbb{C}^{*})\cong\mathbb{Q}[h]$ and that

$m_{\lambda}(1)=p_{\lambda*}q_{\lambda}^{*}(1)=h^{c_{\lambda}}$

because $p_{\lambda}$ has relative dimension $-c_{\lambda}$. Note that
$\Delta_{\lambda}(h^{c_{\lambda}})=1$ from the construction of
$\Delta_{\lambda}$, and thus the conclusion follows.

(b) If $p_{B}<p_{A}$, the map
$\Delta_{\lambda}\colon\mathrm{P}_{B}\to\mathrm{P}_{A}$ is zero by considering
the perverse degree. If $p_{B}=p_{A}$, then the map is zero because, after a
shift, it is a map of simple perverse sheaves with different support. ∎

We next prove the analogue of Proposition 6.16 for a non-zero potential. Let
$W$ be an arbitrary potential of $Q$. Recall the sheaves $\mathrm{Q}_{A}$ from
Subsection 6.2. Let
$\mathcal{H}(\pi_{*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1])$
be the total perverse cohomology of
$\pi_{*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1]$. There are
natural maps:

$\mathrm{Q}_{A}\to\mathcal{H}(\pi_{*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1])\to\mathrm{Q}_{A}.$

Apply the vanishing cycle functor to the maps (6.36) to obtain:

(6.44)
$\displaystyle\Delta_{\lambda}\colon\pi_{d*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1]$
$\displaystyle\to
i_{\lambda*}\pi_{\lambda*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}[-1]$
$\displaystyle=i_{\lambda*}\boxtimes_{i=1}^{k}\left(\pi_{d_{i}*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d_{i})}[-1]\right).$

Let $\mathrm{Q}_{A}^{\mathrm{inv}}$ be defined by the exact triangle

$\mathrm{Q}_{A}^{\mathrm{inv}}[-1]\to\mathrm{Q}_{A}\xrightarrow{1-\mathrm{T}}\mathrm{Q}_{A}\to\mathrm{Q}_{A}^{\mathrm{inv}}.$

######  Proposition 6.17.

Let $A,B\in\mathscr{P}$ with corresponding sheaves $\mathrm{Q}_{A}$ and
$\mathrm{Q}_{B}$ of different support. Assume that $p_{B}\leqslant p_{A}$.

(a) The map (6.44) induces isomorphisms

(6.45)
$\Delta_{\lambda}\colon\mathrm{Q}_{A}\xrightarrow{\sim}\mathrm{Q}_{A},\,\Delta_{\lambda}\colon\mathrm{Q}^{\mathrm{inv}}_{A}\xrightarrow{\sim}\mathrm{Q}^{\mathrm{inv}}_{A}.$

(b) The maps $\Delta_{\lambda}\colon\mathrm{Q}_{B}\to\mathrm{Q}_{A}$ and
$\Delta_{\lambda}\colon\mathrm{Q}^{\mathrm{inv}}_{B}\to\mathrm{Q}^{\mathrm{inv}}_{A}$
are zero.

###### Proof.

The maps above are induced from the map (6.36), thus the conclusion follows
from Proposition 6.16. ∎

We now record corollaries to be used in the proof of Theorem 6.3. Fix a
splitting

(6.46)
$H^{\bullet}\left(\mathscr{X}(d),\varphi^{\mathrm{inv}}_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1]\right)=\bigoplus_{A\in\mathscr{P}}H^{\bullet}(X(d),\mathrm{Q}^{\mathrm{inv}}_{A}).$

Let $x\in
H^{\bullet}\left(\mathscr{X}(d),\varphi^{\mathrm{inv}}_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1]\right)$.
Use the decomposition above to write

(6.47) $x=\sum_{A\in\mathscr{P}}x_{A}$

with $x_{A}\in H^{\bullet}(X(d),\mathrm{Q}^{\mathrm{inv}}_{A})$.

######  Corollary 6.18.

Let $\delta\in M(d)_{\mathbb{R}}^{W_{d}}$. Let $\lambda$ be an antidominant
cocharacter of $T(d)$ with associated partition d such that
$\varepsilon_{\lambda,\delta}=\varepsilon_{\textbf{d},\delta}$. Let $x\in
H^{i}(\mathscr{X}(d),\varphi^{\mathrm{inv}}_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1])$
and assume that

$a^{*}_{\lambda}(x)\in\bigoplus_{j=0}^{c_{\textbf{d},\delta}-1}H^{i-2j}(\mathscr{X}(d)^{\prime\lambda},\varphi^{\mathrm{inv}}_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)^{\prime\lambda}}[-2])h^{j}.$

(a) If $\varepsilon_{\textbf{d},\delta}=1$, then $\Delta_{\lambda}(x)=0$.

(b) If $\varepsilon_{\textbf{d},\delta}=0$, then $\Delta_{\lambda}(x)$ is in
the image of

$H^{i}(\mathscr{X}(d)^{\prime\lambda},\varphi^{\mathrm{inv}}_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)^{\prime\lambda}}[-2])\hookrightarrow
H^{i}(\mathscr{X}(d)^{\lambda},\varphi^{\mathrm{inv}}_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)^{\lambda}}[-1]).$

###### Proof.

Recall the definition of $\mathrm{S}$ from (6.35).

(a) If $\varepsilon_{\textbf{d},\delta}=1$, then
$\mathrm{S}p_{\lambda}^{*}(x)=0$, so $\Delta_{\lambda}(x)=0$.

(b) If $\varepsilon_{\textbf{d},\delta}=0$, then
$\mathrm{S}p_{\lambda}^{*}(x)\in
H^{\cdot}\left(X(d)^{\lambda},\varphi_{\mathrm{Tr}\,W}^{\mathrm{inv}}\mathrm{IC}_{\mathscr{X}(d)^{\prime\lambda}}[-c_{\lambda}-2]\right)$.
The conclusion follows from the definition of $\Delta_{\lambda}$ in (6.44). ∎

######  Corollary 6.19.

Let $\delta\in M(d)_{\mathbb{R}}^{W_{d}}$. Let $\lambda$ be an antidominant
cocharacter of $T(d)$ with associated partition d such that
$\varepsilon_{\lambda,\delta}=\varepsilon_{\textbf{d},\delta}$. Let $x\in
H^{i}(\mathscr{X}(d),\varphi^{\mathrm{inv}}_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1])$
and assume that

$a^{*}_{\lambda}(x)\in\bigoplus_{j=0}^{c_{\textbf{d},\delta}-1}H^{i-2j}(\mathscr{X}(d)^{\prime\lambda},\varphi^{\mathrm{inv}}_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)^{\prime\lambda}}[-2])h^{j}.$

Recall the decomposition (6.47).

(a) If $\varepsilon_{\textbf{d},\delta}=1$, then $x_{A}=0$ for all tuples
$A\in\mathscr{P}$ with corresponding cocharacter $\lambda$.

(b) If $\varepsilon_{\textbf{d},\delta}=0$, then $x_{A}=0$ for all tuples
$A\in\mathscr{P}$ with corresponding cocharacter $\lambda$ and different from
$A^{\circ}$.

###### Proof.

Both claims follow from Proposition 6.17 and Corollary 6.18. ∎

###### Proof of Theorem 6.3.

Recall the cycle map in (6.7)

$\mathrm{c}\colon\mathrm{gr}_{a}K_{i}^{\mathrm{top}}(\mathbb{S}(d;\delta))\to
H^{\dim\mathscr{X}(d)-2a-i}(\mathscr{X}(d),\varphi_{\mathop{\rm
Tr}W}^{\rm{inv}}\mathrm{IC}_{\mathscr{X}(d)}[-2]).$

By Proposition 6.10, we may assume that $Q$ has at least two loops at every
vertex. Let $y$ be in the image of the above map. By Proposition 6.15 and
Corollary 6.19, we have that $y_{A}=0$ unless $A=A^{\circ}$ for some partition
$\textbf{d}=(d_{i},m_{i})_{i=1}^{k}$ of $d$ with $m_{i}\geqslant 1$ and
$d_{i}$ pairwise distinct with $\varepsilon_{\textbf{d},\delta}=0$. The
statement thus follows. ∎

##  7\. Topological K-theory of quasi-BPS categories for preprojective
algebras

In this section, we use the results of Sections 5 and 6 to compute the
topological K-theory of preprojective algebras of quivers satisfying
Assumption 2.1 in terms of BPS cohomology, see Theorem 7.6.

###  7.1. The preprojective BPS sheaf

Let $Q^{\circ}=(I,E^{\circ})$ be a quiver. Recall the moduli stack of
dimension $d$ representations of the tripled quiver $Q$ of $Q^{\circ}$ of
dimension $d$ and its good moduli space:

$\pi_{X,d}:=\pi_{d}\colon\mathscr{X}(d)\to X(d).$

Recall also the moduli stack of dimension $d$ representation of the
preprojective algebra of $Q^{\circ}$ and its good moduli space:

$\pi_{P,d}\colon\mathscr{P}(d)^{\rm{cl}}\to P(d).$

Consider the moduli stack of dimension $d$ representations of the double
quiver of $Q^{\circ}$ and its good moduli space:

$\pi_{Y,d}\colon\mathscr{Y}(d):=(R^{\circ}(d)\oplus
R^{\circ}(d)^{\vee})/G(d)\to Y(d).$

Consider the diagram:

${\mathscr{P}(d)^{\mathrm{cl}}}$${\mathscr{Y}(d)}$${\mathscr{X}(d)}$${P(d)}$${Y(d)}$${X(d).}$$\scriptstyle{\pi_{P,d}}$$\scriptstyle{j}$$\scriptstyle{\pi_{Y,d}}$$\scriptstyle{\eta}$$\scriptstyle{\pi_{X,d}}$$\scriptstyle{j}$$\scriptstyle{\eta}$

Here $\eta\colon\mathscr{X}(d)\to\mathscr{Y}(d)$ is the projection which
forgets the $\mathfrak{g}(d)$-component and the bottom horizontal arrows are
induced maps on good moduli spaces. Let
$\mathbb{C}\hookrightarrow\mathfrak{g}(d)$ be the diagonal embedding, which
induces the closed immersion

$\displaystyle\gamma\colon X^{\prime}(d):=(R^{\circ}(d)\oplus
R^{\circ}(d)^{\vee}\oplus\mathbb{C})/\\!\\!/G(d)\hookrightarrow X(d).$

Let $\eta^{\prime}:=\eta|_{X^{\prime}(d)}$. By [Dava, Theorem/ Definition
4.1], there exists a preprojective BPS sheaf

$\mathcal{BPS}^{p}_{d}\in\mathrm{Perv}(P(d))$

such that the BPS sheaf of the tripled quiver with potential $(Q,W)$
associated to $Q^{\circ}$ is

(7.1)
$\mathcal{BPS}_{d}=\gamma_{\ast}\eta^{\prime\ast}j_{*}(\mathcal{BPS}^{p}_{d})[1]\in\mathrm{Perv}(X(d)).$

For a partition $A=(d_{i})_{i=1}^{k}$ of $d$, define
$\mathcal{BPS}^{p}_{A}\in\mathrm{Perv}(P(d))$ as in (6.1.3). For $\delta\in
M(d)^{W_{d}}_{\mathbb{R}}$, define the following perverse sheaves on $P(d)$:

(7.2) $\mathcal{BPS}^{p}_{\delta}:=\bigoplus_{A\in
S^{d}_{\delta}}\mathcal{BPS}^{p}_{A},\,\mathcal{BPS}^{p}_{d,v}:=\mathcal{BPS}^{p}_{v\tau_{d}},$

where the set of partitions $S^{d}_{\delta}$ is defined from the tripled
quiver $Q$, see Subsection 6.1.2. Then $\mathcal{BPS}^{p}_{d,v}$ is a direct
summand of $\pi_{P,d*}\omega_{\mathscr{P}(d)^{\mathrm{cl}}}$, see [Dava,
Theorem A], and so $H^{-a}(P(d),\mathcal{BPS}^{p}_{d,v})$ is a direct summand
of

$H^{\mathrm{BM}}_{a}(\mathscr{P}(d)^{\mathrm{cl}})=H^{-a}\big{(}P(d),\pi_{P,d*}\omega_{\mathscr{P}(d)^{\mathrm{cl}}}\big{)}.$

Recall the maps

$\mathscr{P}(d)\xleftarrow{\eta^{\prime}}\eta^{-1}(\mathscr{P}(d))\xrightarrow{j^{\prime}}\mathscr{X}(d).$

The dimension of $\mathscr{P}(d)$ as a quasi-smooth stack is
$\dim\mathscr{P}(d):=\dim\mathscr{Y}(d)-\dim\mathfrak{g}(d)$. Recall the
dimensional reduction isomorphism from Subsection 5.1:

$\displaystyle j^{\prime}_{*}\eta^{\prime*}\colon
H^{\mathrm{BM}}_{a}(\mathscr{P}(d)^{\mathrm{cl}})\cong
H^{\mathrm{BM}}_{a}(\mathscr{P}(d))$
$\displaystyle\xrightarrow{\sim}H^{2\dim\mathscr{Y}(d)-a}(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}\mathbb{Q}_{\mathscr{X}(d)}[-1])$
$\displaystyle=H^{\dim\mathscr{P}(d)-a}(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}[-1]).$

By the construction of the PBW isomorphism for preprojective Hall algebras
[Dava, Equation (31)], the above isomorphism preserves the BPS cohomologies:

(7.3) $j^{\prime}_{*}\eta^{\prime*}\colon
H^{-a}(P(d),\mathcal{BPS}^{p}_{d,v})\xrightarrow{\sim}H^{\dim\mathscr{P}(d)-a}(X(d),\mathcal{BPS}_{d,v}).$

###  7.2. Computations

Recall the categories

$\mathbb{T}(d)_{v}\subset D^{b}(\mathscr{P}(d))\text{ and
}\mathbb{T}(d)_{v}^{\mathrm{red}}\subset D^{b}(\mathscr{P}(d)^{\mathrm{red}})$

from Subsection 2.13. Consider the natural closed immersion
$l^{\prime}\colon\mathscr{P}(d)^{\mathrm{red}}\hookrightarrow\mathscr{P}(d)$.
The closed immersion
$l\colon\mathscr{P}(d)^{\mathrm{cl}}\hookrightarrow\mathscr{P}(d)$ factors
through
$\mathscr{P}(d)^{\mathrm{cl}}\hookrightarrow\mathscr{P}(d)^{\mathrm{red}}\stackrel{{\scriptstyle
l^{\prime}}}{{\hookrightarrow}}\mathscr{P}(d)$.

######  Proposition 7.1.

Let $Q$ be a symmetric quiver. Then there is a weak equivalence of spectra
$l^{\prime}_{*}\colon K^{\mathrm{top}}(\mathbb{T}(d)^{\mathrm{red}}_{v})\to
K^{\mathrm{top}}(\mathbb{T}(d)_{v})$.

###### Proof.

There is a weak equivalence of spectra $l^{\prime}_{*}\colon
G^{\mathrm{top}}(\mathscr{P}(d)^{\mathrm{red}})\xrightarrow{\sim}G^{\mathrm{top}}(\mathscr{P}(d))$.
The claim then follows from Theorem 2.9. ∎

For $i\in\mathbb{Z}$, consider the Chern character map (3.8) for the quasi-
smooth stack $\mathscr{P}(d)$:

(7.4) $\mathrm{ch}\colon
G^{\mathrm{top}}_{i}\left(\mathscr{P}(d)\right)\to\widetilde{H}^{\mathrm{BM}}_{i}(\mathscr{P}(d)).$

It induces a Chern character map

(7.5) $\mathrm{ch}\colon
K^{\mathrm{top}}_{i}(\mathbb{T}(d)_{v})\hookrightarrow
G^{\mathrm{top}}_{i}\left(\mathscr{P}(d)\right)\to\widetilde{H}^{\mathrm{BM}}_{i}(\mathscr{P}(d)).$

######  Corollary 7.2.

The maps (7.4) and (7.5) are injective.

###### Proof.

It suffices to check that (7.4) is injective. This follows from Proposition
4.10, Theorem 2.8 (applied to a fixed $\mu$ and all
$\alpha\in\mathbb{Z}_{\geqslant 1}$), and the Koszul equivalence (2.14). ∎

######  Corollary 7.3.

We have that $G^{\mathrm{top}}_{1}(\mathscr{P}(d))=0$. Thus also
$K^{\mathrm{top}}_{1}(\mathbb{T}(d)_{v})=0$.

###### Proof.

We have that $H^{\mathrm{BM}}_{\mathrm{odd}}(\mathscr{P}(d)^{\mathrm{cl}})=0$
by [Davb, Theorem A]. The conclusion follows by Proposition 7.2. ∎

Recall the filtration $E_{\ell}G^{\mathrm{top}}_{0}(\mathscr{P}(d))$ of
$G^{\mathrm{top}}_{0}(\mathscr{P}(d))$ from Subsection 3.3. Define the
filtration:

$E_{\ell}K^{\mathrm{top}}_{0}(\mathbb{T}(d)_{v}):=E_{\ell}G^{\mathrm{top}}_{0}(\mathscr{P}(d))\cap
K^{\mathrm{top}}_{0}(\mathbb{T}(d)_{v})\subset
K^{\mathrm{top}}_{0}(\mathbb{T}(d)_{v}).$

We denote by $\mathrm{gr}_{\ell}K^{\mathrm{top}}_{0}(\mathbb{T}(d)_{v})$ the
associated graded piece, and note that it is a direct summand of
$\mathrm{gr}_{\ell}G^{\mathrm{top}}_{0}(\mathscr{P}(d))$ by Theorem 2.9.
Define similarly a filtration
$E_{\ell}G^{\mathrm{top}}_{0}(\mathscr{P}(d)^{\mathrm{red}})\subset
G^{\mathrm{top}}_{0}(\mathscr{P}(d)^{\mathrm{red}})$ and a filtration
$E_{\ell}K^{\mathrm{top}}_{0}(\mathbb{T}(d)^{\mathrm{red}}_{v})\subset
K^{\mathrm{top}}_{0}(\mathbb{T}(d)^{\mathrm{red}}_{v})$.

######  Corollary 7.4.

The forget-the-potential functor $\Theta$ induces an isomorphism:

(7.6)
$\mathrm{gr}_{\ell}K^{\mathrm{top}}_{0}\left(\mathrm{MF}^{\mathrm{gr}}(\mathscr{X}(d),\mathrm{Tr}\,W)\right)\xrightarrow{\sim}\mathrm{gr}_{\ell}K^{\mathrm{top}}_{0}\left(\mathrm{MF}(\mathscr{X}(d),\mathrm{Tr}\,W)\right).$

There are thus also isomorphisms:

$\mathrm{gr}_{\ell}K^{\mathrm{top}}_{0}\left(\mathbb{T}(d)_{v}\right)\xrightarrow{\sim}\mathrm{gr}_{\ell+\dim\mathfrak{g}(d)}K^{\mathrm{top}}_{0}\left(\mathbb{S}^{\mathrm{gr}}(d)_{v}\right)\xrightarrow{\sim}\mathrm{gr}_{\ell+\dim\mathfrak{g}(d)}K^{\mathrm{top}}_{0}\left(\mathbb{S}(d)_{v}\right).$

###### Proof.

The isomorphism (7.6) follows from Corollaries 5.2 and 7.3. The other
isomorphism follow from the Koszul equivalence, see Proposition 5.2 for an
explanation of the degree of the graded pieces. ∎

######  Corollary 7.5.

There is a commutative diagram, where the vertical maps are cycle maps and the
left horizontal maps are the dimensional reduction maps
$i^{\prime}_{*}p^{\prime*}$.

${\mathrm{gr}_{\cdot}G^{\mathrm{top}}_{0}(\mathscr{P}(d))}$${\mathrm{gr}_{\cdot}K^{\mathrm{top}}_{0}(\mathrm{MF}^{\mathrm{gr}}(\mathscr{X}(d),\mathrm{Tr}\,W))}$${\mathrm{gr}_{\cdot}K^{\mathrm{top}}_{0}(\mathrm{MF}(\mathscr{X}(d),\mathrm{Tr}\,W))}$${\widetilde{H}^{\mathrm{BM}}_{0}(\mathscr{P}(d))}$${\widetilde{H}^{0}(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}[-1])}$${\widetilde{H}^{0}(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}^{\mathrm{inv}}[-2]).}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

Here we have suppressed the cohomological degrees to make the diagram simpler.

###### Proof.

The claim follows from Proposition 5.2 and Corollary 7.4. ∎

######  Theorem 7.6.

For an arbitrary quiver $Q^{\circ}$, the cycle map (7.4) for $\mathscr{P}(d)$
induces a cycle map

(7.7)
$\mathrm{c}\colon\mathrm{gr}_{\ell}K^{\mathrm{top}}_{0}(\mathbb{T}(d)_{v})\cong\mathrm{gr}_{\ell}K^{\mathrm{top}}_{0}(\mathbb{T}(d)^{\mathrm{red}}_{v})\to
H^{-2\ell}(P(d),\mathcal{BPS}^{p}_{d,v}).$

If $Q^{\circ}$ satisfies Assumption 2.2, then (7.7) is an isomorphism.

###### Proof.

The isomorphism
$\mathrm{gr}_{\ell}K^{\mathrm{top}}_{0}(\mathbb{T}(d)_{v})\cong\mathrm{gr}_{\ell}K^{\mathrm{top}}_{0}(\mathbb{T}(d)^{\mathrm{red}}_{v})$
follows from Proposition 7.1. Consider the diagram, whose lower square
commutes from Corollary 7.5 and the top horizontal map is an isomorphism by
Corollary 7.4:

${\mathrm{gr}_{\ell}K^{\mathrm{top}}_{0}(\mathbb{T}(d)_{v})}$${\mathrm{gr}_{\ell+\dim\mathfrak{g}(d)}K^{\mathrm{top}}_{0}(\mathbb{S}(d)_{v})}$${\mathrm{gr}_{\ell}G^{\mathrm{top}}_{0}(\mathscr{P}(d))}$${\mathrm{gr}_{\ell+\dim\mathfrak{g}(d)}K^{\mathrm{top}}_{0}(\mathrm{MF}(\mathscr{X}(d),\mathrm{Tr}\,W))}$${H^{\mathrm{BM}}_{2\ell}(\mathscr{P}(d))}$${H^{2\dim\mathscr{Y}(d)-2\ell}(\mathscr{X}(d),\varphi_{f}[-1]).}$$\scriptstyle{\sim}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\mathrm{c}}$$\scriptstyle{\sim}$

By Theorem 6.3, the map $\beta$ has image in

$H^{\dim\mathscr{P}(d)-2\ell}(\mathscr{X}(d),\mathcal{BPS}_{d,v})\subset
H^{2\dim\mathscr{Y}(d)-2\ell}(\mathscr{X}(d),\varphi_{\mathrm{Tr}\,W}[-1]).$

If $Q^{\circ}$ satisfies Assumption 2.2, it is an isomorphism onto
$H^{\dim\mathscr{P}(d)-2\ell}(\mathscr{X}(d),\mathcal{BPS}_{d,v})$ by Theorem
6.2. By (7.3), the map $\alpha$ has image in
$H^{-2\ell}(P(d),\mathcal{BPS}^{p}_{d,v})$, and, if $Q^{\circ}$ satisfies
Assumption 2.2, it is an isomorphism onto
$H^{-2\ell}(P(d),\mathcal{BPS}^{p}_{d,v})$. ∎

######  Remark 7.7.

There are two perverse filtrations on
$H^{\mathrm{BM}}_{\cdot}(\mathscr{P}(d))$ for any quiver $Q^{\circ}$. One of
them is induced from the tripled quiver with potential $(Q,W)$ and studied in
[DM20]; the first non-zero piece in the perverse filtration is
${}^{p}\tau^{\leqslant
1}\pi_{d*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}=\mathcal{BPS}_{d}$.
Another filtration is induced from the map $\pi_{P,d}$ and studied in [Dava],
where it is called the “less perverse filtration”; the first non-zero piece in
the perverse filtration is ${}^{p}\tau^{\leqslant
0}\pi_{P,d*}\omega_{\mathscr{P}(d)^{\mathrm{cl}}}$. Note that, for any
$v\in\mathbb{Z}$, ${}^{p}\tau^{\leqslant
1}\pi_{d*}\varphi_{\mathrm{Tr}\,W}\mathrm{IC}_{\mathscr{X}(d)}$ is a direct
summand of $\mathcal{BPS}_{d,v}$, which itself is a direct summand of
${}^{p}\tau^{\leqslant 0}\pi_{P,d*}\omega_{\mathscr{P}(d)^{\mathrm{cl}}}$.
Thus the topological K-theory of quasi-BPS categories (for $Q^{\circ}$
satisfying Assumption 2.2, and for any $v\in\mathbb{Z}$) lies between the
first non-zero pieces of these two perverse filtrations.

######  Remark 7.8.

Davison–Hennecart–Schlegel Mejia [DHSMb, DHSMa] computed the preprojective BPS
sheaves in terms of the intersection complexes of the varieties $P(d)$.

We note the following numerical corollary of Theorem 7.6.

######  Corollary 7.9.

Let $Q^{\circ}$ be a quiver satisfying Assumption 2.2 and let
$(d,v)\in\mathbb{N}^{I}\times\mathbb{Z}$. Then

$\dim_{\mathbb{Q}}K^{\mathrm{top}}_{0}(\mathbb{T}(d)_{v})=\dim_{\mathbb{Q}}H^{\cdot}(P(d),\mathcal{BPS}_{d,v}).$

###### Proof.

The map (7.5) is injective by Proposition 7.2. The conclusion then follows
from Theorem 7.6. ∎

##  8\. Examples

In this section, we discuss some explicit examples of computations of the
topological K-theory of quasi-BPS categories. All vector spaces considered in
this section are $\mathbb{Q}$-vector spaces. We first note a preliminary
proposition.

######  Proposition 8.1.

Let $Q=(I,E)$ be a symmetric quiver, let $d\in\mathbb{N}^{I}$, and let
$v\in\mathbb{Z}$. Then

$\dim
K^{\mathrm{top}}_{0}(\mathbb{M}(d)_{v})=\\#\left(M(d)^{+}\cap(\mathbf{W}(d)+v\tau_{d}-\rho)\right).$

###### Proof.

There is a natural isomorphism

$K_{0}(\mathscr{X}(d))\xrightarrow{\sim}K_{0}^{\mathrm{top}}(\mathscr{X}(d))\cong
K_{0}(BG(d)).$

The category $\mathbb{M}(d)_{v}$ is admissible in $D^{b}(\mathscr{X}(d))$, so
the above isomorphism restricts to the isomorphism

$K_{0}(\mathbb{M}(d)_{v})\xrightarrow{\sim}K_{0}^{\mathrm{top}}(\mathbb{M}(d)_{v}).$

The generators of $K_{0}(\mathscr{X}(d))$ are the classes of the vector
bundles $\mathcal{O}_{\mathscr{X}(d)}\otimes\Gamma_{G(d)}(\chi)$, where $\chi$
is a dominant weight of $G(d)$ and $\Gamma_{G(d)}(\chi)$ is the irreducible
representation of $G(d)$ of highest weight $\chi$. The computation

$\dim
K_{0}(\mathbb{M}(d)_{v})=\\#\left(M(d)^{+}\cap(\textbf{W}(d)+v\tau_{d}-\rho)\right)$

follows then from the definition of $\mathbb{M}(d)_{v}$. ∎

######  Remark 8.2.

In view of Proposition 8.1 and Theorem 6.6, the total intersection cohomology
of the spaces $X(d)$ can be determined by counting lattice points inside the
polytope $(\textbf{W}(d)+v\tau_{d}-\rho)$.

###  8.1. Toric examples

Let $g\in\mathbb{N}$. Consider the quiver $Q=(I,E)$, where $I=\\{1,2\\}$ and
$E$ has one loop at $1$, one loop at $2$, $2g+1$ edges
{$e_{1},\ldots,e_{2g+1}\\}$ from $0$ to $1$ and $2g+1$ edges
$\\{\overline{e}_{1},\ldots,\overline{e}_{2g+1}\\}$ from $1$ to $0$. The
following is a figure for $g=1$.

${1}$${2}$$\scriptstyle{e_{2}}$$\scriptstyle{e_{3}}$$\scriptstyle{e_{1}}$$\scriptstyle{\overline{e}_{1}}$$\scriptstyle{\overline{e}_{2}}$$\scriptstyle{\overline{e}_{3}}$

Fix $d=(1,1)\in\mathbb{N}^{I}$. Then

$\mathscr{X}(d)=\left(\mathbb{C}^{2}\oplus\mathbb{C}^{2(2g+1)}\right)\big{/}(\mathbb{C}^{*})^{2}.$

The diagonal $\mathbb{C}^{*}\hookrightarrow(\mathbb{C}^{*})^{2}$ acts
trivially on $\mathbb{C}^{2}\oplus\mathbb{C}^{2(2g+1)}$. The factor
$\mathbb{C}^{*}$ corresponding to the vertex $1$ acts with weight $0$ on
$\mathbb{C}^{2}$, weight $1$ on $\mathbb{C}^{2g+1}$, and weight $-1$ on
$\mathbb{C}^{2g+1}$. We consider the stack, which is the
$\mathbb{C}^{*}$-rigidification of $\mathscr{X}(d)$:

$\mathscr{X}^{\prime}(d)=\left(\mathbb{C}^{2}_{0}\oplus\mathbb{C}^{2g+1}_{1}\oplus\mathbb{C}^{2g+1}_{-1}\right)\big{/}\mathbb{C}^{*}.$

The GIT quotient for any non-trivial stability condition provides a small
resolution of singularities:

$\tau\colon
Y:=\left(\mathbb{C}^{2}_{0}\oplus\mathbb{C}^{2g+1}_{1}\oplus\mathbb{C}^{2g+1}_{-1}\right)^{\mathrm{ss}}\big{/}\mathbb{C}^{*}=\mathbb{C}^{2}\times\mathrm{Tot}_{\mathbb{P}^{2g}}\left(\mathcal{O}(-1)^{2g+1}\right)\to
X(d).$

Here, small means that $\dim Y\times_{X(d)}Y=\dim X(d)$ and $Y\times_{X(d)}Y$
has a unique irreducible component of maximal dimension. Then, by the BBDG
decomposition theorem, we have that
$\tau_{*}\mathrm{IC}_{Y}=\mathrm{IC}_{X(d)}$. We decorate the BPS sheaves with
a superscript zero to indicate that the potential is zero. We obtain that:

(8.1) $\mathcal{BPS}^{0}_{d}=\tau_{*}\mathrm{IC}_{Y}=\mathrm{IC}_{X(d)}\text{
and
}\mathcal{BPS}^{0}_{(1,0)}=\mathcal{BPS}^{0}_{(0,1)}=\mathrm{IC}_{\mathbb{C}}.$

######  Proposition 8.3.

If $v$ is odd, then $\mathbb{M}(d)_{v}\cong D^{b}(Y)$ and
$\mathcal{BPS}^{0}_{d,v}=\mathcal{BPS}^{0}_{d}$.

If $v$ is even, then $\mathbb{M}(d)_{v}$ has a semiorthogonal decomposition
with summands equivalent to $D^{b}(Y)$ and $D^{b}(\mathbb{C}^{2})$, and
$\mathcal{BPS}^{0}_{d,v}=\mathcal{BPS}^{0}_{d}\oplus\mathcal{BPS}^{0}_{(1,0)}\boxtimes\mathcal{BPS}^{0}_{(0,1)}$.

###### Proof.

The category $\mathbb{M}(d)_{v}$ is the subcategory of $D^{b}(\mathscr{X}(d))$
generated by the line bundles
$\mathcal{O}_{\mathscr{X}(d)}(w\beta_{2}+(v-w)\beta_{1})$ for $w\in\mathbb{Z}$
such that

(8.2) $\frac{v}{2}\leqslant w\leqslant 2g+1+\frac{v}{2}.$

One can show that $\mathbb{M}(d)_{v}$ is equivalent to the “window
subcategory” (in the sense of [HL15]) of $D^{b}(\mathscr{X}^{\prime}(d))$
containing objects $F$ such that the weights of $\mathbb{C}^{*}$ on $F|_{0}$
are in $\left[\frac{v}{2},\frac{v}{2}+2g+1\right]\cap\mathbb{Z}$.

If $v$ is odd, then $\mathbb{M}(d)_{v}\cong D^{b}(Y)$ by [HL15, Theorem 2.10].
The boundary points $\frac{v}{2}$ and $\frac{v}{2}+2g+1$ are not integers, so
$\mathcal{BPS}^{0}_{d,v}=\mathcal{BPS}^{0}_{d}$.

If $v$ is even, then
$\mathcal{BPS}^{0}_{d,v}=\mathcal{BPS}^{0}_{d}\oplus\mathcal{BPS}^{0}_{(1,0)}\boxtimes\mathcal{BPS}^{0}_{(0,1)}$.
The fixed locus of the unique Kempf-Ness locus in the construction of $Y$ is
$(\mathbb{C}^{2}_{0}\oplus\mathbb{C}^{2g+1}_{1}\oplus\mathbb{C}^{2g+1}_{-1})^{\mathbb{C}^{*}}=\mathbb{C}^{2}$.
As a corollary of [HL15, Theorem 2.10], see the remark in [HLS16, Equation
(3)], the category $\mathbb{M}(d)_{v}$ has a semiorthogonal decomposition with
summands $D^{b}(Y)$ and $D^{b}(\mathbb{C}^{2})$. ∎

As a corollary of the above proposition and of the computations (8.1), we
obtain the following:

$\dim
K^{\mathrm{top}}_{0}(\mathbb{M}(d)_{v})\stackrel{{\scriptstyle(*)}}{{=}}\dim
H^{\cdot}(X(d),\mathcal{BPS}^{0}_{d,v})=\begin{cases}2g+1,\text{ if }v\text{
is odd},\\\ 2g+2,\text{ if }v\text{ is even}.\end{cases}$

The equality $(*)$ is also the consequence (6.4) of Theorem 6.2. Note that the
dimensions of $K^{\mathrm{top}}(\mathbb{M}(d)_{v})$ can be computed
immediately using Proposition 8.1 and (8.2), and then $(*)$ can be checked
without using window categories. However, by Proposition 8.3, the equality
$(*)$ is obtained as a corollary of the Atiyah-Hirzebruch theorem for the
smooth varieties $Y$ and $\mathbb{C}^{2}$.

Further, Proposition 8.3 is useful when considering a non-zero potential for
$Q$. For example, consider the potential

$W:=\sum_{i=1}^{2g+1}e_{i}\overline{e}_{i}.$

Note that $W\colon Y\to\mathbb{C}$ is smooth. The computation (8.1) implies
that:

(8.3)
$\mathcal{BPS}_{d}=\varphi_{W}\mathrm{IC}_{X(d)}=\tau_{*}\varphi_{W}\mathrm{IC}_{Y}=0\text{
and
}\mathcal{BPS}^{0}_{(1,0)}=\mathcal{BPS}^{0}_{(0,1)}=\mathrm{IC}_{\mathbb{C}}.$

The BPS sheaves have trivial monodromy. Further, Proposition 8.3 implies that:

$\displaystyle\mathbb{S}(d)_{v}$ $\displaystyle\simeq\mathrm{MF}(Y,W)=0\text{
if }v\text{ is odd},$ $\displaystyle\mathbb{S}(d)_{v}$
$\displaystyle=\langle\mathrm{MF}(Y,W),\mathrm{MF}(\mathbb{C}^{2},0)\rangle\simeq\mathrm{MF}(\mathbb{C}^{2},0)\text{
if }v\text{ is even}.$

Let $i\in\mathbb{Z}$. The following equality (which also follows by (6.4))
holds by a direct computation:

$\dim K^{\mathrm{top}}_{i}(\mathbb{S}(d)_{v})=\dim
H^{\cdot}(X(d),\mathcal{BPS}_{d,v})=\begin{cases}0,\text{ if }v\text{ is
odd},\\\ 1,\text{ if }v\text{ is even}.\end{cases}$

######  Remark 8.4.

A similar analysis can be done for any symmetric quiver $Q=(I,E)$ (not
necessarily satisfying Assumption 2.1) and a dimension vector $d=(d^{i})_{i\in
I}\in\mathbb{N}^{I}$ such that $d^{i}\in\\{0,1\\}$ for every $i\in I$. We do
not give the details for the proofs. Let $v\in\mathbb{Z}$ such that
$\gcd(\underline{d},v)=1$. Assume $W=0$.

One can show that, for a generic GIT stability $\ell\in
M(d)^{W_{d}}_{\mathbb{R}}\cong M(d)_{\mathbb{R}}$, the GIT quotient
$Y:=R(d)^{\ell\text{-ss}}/G(d)\cong R(d)^{\ell\text{-ss}}/\\!\\!/G(d)$ is a
small resolution

$\tau\colon Y\to X(d).$

Then $\mathcal{BPS}^{0}_{d}=\tau_{*}\mathrm{IC}_{Y}.$ By [HLS20], there is an
equivalence:

$\displaystyle\mathbb{M}(d)_{v}\cong D^{b}(Y).$

The following equality (which is a corollary of Theorem 6.2) follows then by
the Atiyah-Hirzebruch theorem for the smooth variety $Y$:

$\dim K^{\mathrm{top}}_{0}(\mathbb{M}(d)_{v})=\dim
K^{\mathrm{top}}_{0}(Y)=\dim H^{\cdot}(Y)=\dim
H^{\cdot}(X(d),\mathcal{BPS}^{0}_{d}).$

Similar computations can be done also for a general $v\in\mathbb{Z}$.

###  8.2. Quivers with one vertex and an odd number of loops

Let $g\in\mathbb{N}$. Consider $Q$ the quiver with one vertex and $2g+1$
loops. The following is a picture for $g=1$.

(8.4) ${\bullet}$

For $d\in\mathbb{N}$, recall the good moduli space map:

$\mathscr{X}(d):=\mathfrak{gl}(d)^{\oplus(2g+1)}/GL(d)\to
X(d):=\mathfrak{gl}(d)^{\oplus(2g+1)}/\\!\\!/GL(d).$

For $g>0$, the variety $X(d)$ is singular. For every stability condition
$\ell\in M(d)^{W_{d}}_{\mathbb{R}}$, we have that
$\mathscr{X}(d)^{\ell\text{-ss}}=\mathscr{X}(d)$, so we do not obtain
resolutions of singularities of $X(d)$ as in the previous example. There are
no known crepant geometric resolutions (in particular, small resolutions) of
$X(d)$. For $\gcd(d,v)=1$, Špenko–Van den Bergh [ŠVdB17] proved that
$\mathbb{M}(d)_{v}$ is a twisted noncommutative crepant resolution of $X(d)$.
In view of Theorem 6.6, we regard $\mathbb{M}(d)_{v}$ as the categorical
analogue of a small resolution of $X(d)$.

Reineke [Rei12] and Meinhardt–Reineke [MR19] provided multiple combinatorial
formulas for the dimensions of the individual intersection cohomology vector
spaces $\mathrm{IH}^{\bullet}(X(d))$. As noted in Remark 8.2, Theorem 6.6 also
provides combinatorial formulas for the total intersection cohomology of
$X(d)$. We explain that our formula recovers a formula already appearing in
the work of Reineke [Rei12, Theorem 7.1].

Fix $v\in\mathbb{Z}$. By Proposition 8.1, we need to determine the number of
(integral, dominant) weights $\chi=\sum_{i=1}^{d}c_{i}\beta_{i}\in M(d)^{+}$
with $\sum_{i=1}^{d}c_{i}=v$ and $c_{i}\geqslant c_{i-1}$ for every
$2\leqslant i\leqslant d$, such that

(8.5)
$\chi+\rho-v\tau_{d}\in\frac{2g+1}{2}\mathrm{sum}[0,\beta_{i}-\beta_{j}],$

where the Minkowski sum is after all $1\leqslant i,j\leqslant d$. Define
$\widetilde{\chi}\in M(d)$ and $\widetilde{c}_{i}\in\mathbb{Z}$ for
$1\leqslant i\leqslant d$ as follows:

$\widetilde{\chi}:=\chi-g\cdot(2\rho)=\sum_{i=1}^{d}\widetilde{c}_{i}\beta_{i}.$

Note that, for every $2\leqslant i\leqslant d$, the inequality $c_{i}\geqslant
c_{i-1}$ becomes:

(8.6) $\widetilde{c}_{i}-\widetilde{c}_{i-1}+2g\geqslant 0.$

A dominant weight $\chi$ satisfies (8.5) if and only if, for all dominant
cocharacters $\lambda$ of $T(d)\subset GL(d)$, we have:

(8.7)
$\langle\lambda,\chi+\rho-v\tau_{d}\rangle\leqslant\frac{2g+1}{2}\langle\lambda,\mathfrak{g}^{\lambda>0}\rangle=\frac{2g+1}{2}\langle\lambda,\rho\rangle.$

######  Proposition 8.5.

The inequalities (8.7) hold for all dominant cocharacters $\lambda$ if and
only they hold for the cocharacters
$\lambda_{k}(z)=(\overbrace{1\ldots,1}^{d-k},\overbrace{z,\ldots,z}^{k})\in
T(d)$ for $1\leqslant k\leqslant d$.

###### Proof.

In the cocharacter lattice, any dominant cocharacter $\lambda$ is a linear
combination with nonnegative coefficients of $\lambda_{k}$ for $1\leqslant
k\leqslant d$. Then, if (8.7) holds for all $\lambda_{k}$, it also holds for
all dominant $\lambda$. ∎

We rewrite the conditions (8.7) for $\lambda_{k}$ using the weight
$\widetilde{\chi}$:

$\langle\lambda_{k},\widetilde{\chi}\rangle\leqslant\langle\lambda_{k},v\tau_{d}\rangle.$

Alternatively, the condition above can be written as:

(8.8) $\sum_{i=d-k+1}^{d}\widetilde{c}_{i}\leqslant\frac{vk}{d}.$

######  Definition 8.6.

Let $\mathscr{H}^{2g+1}_{d,v}$ be the set of tuplets of integers
$(\widetilde{c}_{i})_{i=1}^{d}\in\mathbb{Z}^{d}$ satisfying the inequality
(8.6) and (8.8) for every $2\leqslant k\leqslant d$ and such that
$\sum_{i=1}^{d}\widetilde{c}_{i}=v$. Let
$H^{2g+1}_{d,v}:=\\#\mathscr{H}^{2g+1}_{d,v}$.

######  Remark 8.7.

The numbers $H^{2g+1}_{d,0}$ appear in combinatorics as “score sequences of
complete tournaments”, and in the study of certain $\mathbb{C}^{*}$-fixed
points in the moduli of $SL(n)$-Higgs bundles, see [Rei12, Section 7]. By
Proposition 8.1, we have that:

$\dim K^{\mathrm{top}}_{0}(\mathbb{M}(d)_{v})=H^{2g+1}_{d,v}.$

By Theorem 6.6, for any $v\in\mathbb{Z}$ such that $\gcd(d,v)=1$, we obtain
that:

(8.9) $\dim\mathrm{IH}^{\cdot}(X(d))=H^{2g+1}_{d,v}.$

The above statement was already proved (by different methods) by Reineke and
Meinhardt–Reineke by combining [Rei12, Theorem 7.1] and [MR19, Theorem 4.6],
see also [MR19, Section 4.3]. Note that we assume that the number of loops is
odd in order to apply Theorem 6.2. In loc. cit., Reineke also provided
combinatorial formulas for $m$-loop quivers for $m$ even.

######  Remark 8.8.

Note that, as a corollary of (8.9), we obtain that
$H^{2g+1}_{d,v}=H^{2g+1}_{d,v^{\prime}}$ if $\gcd(d,v)=\gcd(d,v^{\prime})=1$.
There are natural bijections
$\mathscr{H}^{2g+1}_{d,v}\xrightarrow{\sim}\mathscr{H}^{2g+1}_{d,v^{\prime}}$
for $d|v-v^{\prime}$ or for $v^{\prime}=-v$, but we do not know such natural
bijections for general $v,v^{\prime}$ coprime with $d$.

For $\gcd(d,v)=1$ and $n\in\mathbb{Z}_{\geqslant 1}$, the topological K-theory
$K_{i}^{\rm{top}}(\mathbb{M}(nd)_{nv})$ is computed from the intersection
cohomology of $X(e)$ for $e\in\mathbb{N}$, and the set $S_{nv}^{nd}$. The
following is a corollary of Proposition 6.1:

######  Corollary 8.9.

For $\gcd(d,v)=1$ and $n\in\mathbb{Z}_{\geqslant 1}$, the set $S_{nv}^{nd}$
consists of partitions $(d_{i})_{i=1}^{k}$ of $nd$ such that $d_{i}=n_{i}d$
for $(n_{i})_{i=1}^{k}\in\mathbb{N}^{k}$ a partition of $n$.

######  Example 8.10.

Suppose that $g=0$. In this case, the variety $X(d)$ is smooth:

$X(d)=\mathfrak{gl}(d)/\\!\\!/GL(d)\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{Sym}^{d}(\mathbb{C})\cong\mathbb{C}^{d}.$

The above isomorphism is given by sending an element of $\mathfrak{gl}(d)$ to
the set of generalized eigenvalues. However $X(d)^{\mathrm{st}}=\emptyset$ if
$d>1$, thus $\mathcal{BPS}_{d}=\mathrm{IC}_{\mathbb{C}}$ if $d=1$, and
$\mathcal{BPS}_{d}=0$ for $d>1$. Then by Corollary 8.9, we have
$\mathcal{BPS}_{d,v}=0$ unless $d|v$, case in which
$\mathcal{BPS}_{d,v}=\mathrm{Sym}^{d}(\mathcal{BPS}_{1})=\mathrm{IC}_{X(d)}$.
Thus for $g=0$, we have

(8.10) $\displaystyle\dim
H^{\cdot}(X(d),\mathcal{BPS}_{d,v})=\begin{cases}1,\text{ if }d|v,\\\ 0,\text{
otherwise.}\end{cases}$

On the other hand, by [Tod23b, Lemma 3.2] we have that $\mathbb{M}(d)_{v}=0$
unless $d|v$, case in which it is the subcategory of $D^{b}(\mathscr{X}(d))$
generated by $\mathcal{O}_{\mathscr{X}(d)}(v\tau_{d})$, and thus equivalent to
$D^{b}(X(d))$, see [Tod23b, Lemma 3.3]. Then:

(8.11) $\displaystyle\dim
K^{\mathrm{top}}_{0}(\mathbb{M}(d)_{v})=\begin{cases}1,\text{ if }d|v,\\\
0,\text{ otherwise.}\end{cases}$

For $g=0$, we can thus verify (6.4) by the direct computations (8.10), (8.11).

###  8.3. The three loop quiver

In this subsection, we make explicit the corollary of Theorem 6.2 for the
three loop quiver (8.4) with loops $\\{x,y,z\\}$ and with potential
$W=x[y,z]$. The quasi-BPS categories $\mathbb{S}(d)_{v}$ are the quasi-BPS
categories of $\mathbb{C}^{3}$ and are admissible in the DT category
$\mathcal{DT}(d)$ studied in [PTa]. The quasi-BPS categories
$\mathbb{T}(d)_{v}$ are the quasi-BPS categories of $\mathbb{C}^{2}$ and are
admissible in $D^{b}(\mathscr{C}(d))$, where $\mathscr{C}(d)$ is the commuting
stack of matrices of size $d$. For $n\in\mathbb{N}$, we denote by $p_{2}(n)$
the number of partitions of $n$.

######  Proposition 8.11.

Let $(d,v)\in\mathbb{N}\times\mathbb{Z}$ be coprime, let $n\in\mathbb{N}$ and
$i\in\mathbb{Z}$. Then:

$\dim K^{\mathrm{top}}_{i}(\mathbb{S}(nd)_{nv})=\dim
K^{\mathrm{top}}_{0}(\mathbb{T}(nd)_{nv})=p_{2}(n).$

###### Proof.

By a theorem of Davison [Davb, Theorem 5.1], we have that

$\mathcal{BPS}_{e}=\mathrm{IC}_{\mathbb{C}^{3}}$

for every $e\in\mathbb{N}$, where $\mathbb{C}^{3}\hookrightarrow X(e)$ is the
subvariety parameterizing three diagonal matrices. Then $\dim
H^{\cdot}(X(e),\mathcal{BPS}_{e})=1$, and so
$\mathrm{Sym}^{k}\left(H^{\cdot}(X(e),\mathcal{BPS}_{e})\right)=1$ for every
positive integers $e$ and $k$. Then $H^{\cdot}(X(nd),\mathcal{BPS}_{A})$ is
also one dimensional for every $A\in S^{nd}_{nv}$. Note that
$\\#S^{nd}_{nv}=p_{2}(n)$ by Corollary 8.9. Then

$H^{\cdot}(X(nd),\mathcal{BPS}_{nd,nv})=\bigoplus_{A\in
S^{nv}_{nd}}H^{\cdot}(X(nd),\mathcal{BPS}_{A})=\mathbb{Q}^{\oplus p_{2}(n)}.$

The monodromy is trivial on $H^{\cdot}(X(nd),\mathcal{BPS}_{nd,nv})$. By
Theorems 6.2 and 7.6, we obtain the desired computations. ∎

######  Remark 8.12.

By Theorem 6.2, the topological K-theory of quasi-BPS categories may be
determined whenever one can compute the BPS cohomology and the set
$S^{v}_{d}$. Proposition 8.11 is an example of such a computation. We mention
two other computations for the three loop quiver with potentials
$W^{\prime}:=x[y,z]+z^{a}$ (for $a\geqslant 2$) and
$W^{\prime\prime}:=x[y,z]+yz^{2}$.

Let $\mathcal{BPS}^{\prime}_{d}$ and $\mathbb{S}^{\prime}(d)_{v}$ be the BPS
sheaves and the quasi-BPS categories of $(Q,W^{\prime})$. Denote similarly the
BPS sheaves and the quasi-BPS categories of $(Q,W^{\prime\prime})$. By [DP22,
Theorem 1.5], we have that
$H^{\cdot}(X(d),\mathcal{BPS}^{\prime}_{d})^{\mathrm{inv}}=0$ because
$H^{\cdot}(\mathbb{C},\varphi_{t^{a}})^{\mathrm{inv}}=0$. Then Theorem 6.2
implies that, for every $i,v\in\mathbb{Z}$ with $\gcd(d,v)=1$:

$K^{\mathrm{top}}_{i}(\mathbb{S}^{\prime}(d)_{v})=0.$

By [DP22, Corollary 7.2], we have that
$H^{\cdot}(X(d),\mathcal{BPS}^{\prime\prime}_{d})^{\mathrm{inv}}=H^{\cdot}(X(d),\mathcal{BPS}^{\prime\prime}_{d})$
is one dimensional. As in Proposition 8.11, we have that, for every
$i,v\in\mathbb{Z}$:

$\dim
K^{\mathrm{top}}_{i}(\mathbb{S}^{\prime\prime}(d)_{v})=p_{2}(\gcd(d,v)).$

##  9\. Étale covers of preprojective algebras

In this section, we prove an extension of Theorem 7.6 to étale covers of
preprojective stacks which we use to compute the topological K-theory of
quasi-BPS categories of K3 surfaces in [PTd].

We first define quasi-BPS categories and BPS cohomology for étale covers of
preprojective stacks. BPS sheaves are defined by base-change from the BPS
sheaves of preprojective algebras. Quasi-BPS categories are defined via graded
matrix factorizations and the Koszul equivalence, see Subsection 9.2.

Recall that Theorem 7.6 follows, via dimensional reduction and the Koszul
equivalence, from Theorem 6.2. The two main ingredients in the proof of
Theorem 6.2 are the semiorthogonal decomposition from Theorem 2.5 and the
construction of the cycle map (6.9) from Theorem 6.3. The analogous statements
for étale covers are Propositions 9.5 and 9.6, respectively.

We will use the notations and constructions from Subsection 7.1. Throughout
this section, we fix a quiver $Q^{\circ}$ satisfying Assumption 2.2 and a
dimension vector $d\in\mathbb{N}^{I}$.

We begin by discussing the setting and by stating Theorem 9.2, the main result
of this section.

###  9.1. Preliminaries

Let $E$ be an affine variety with an action of $G:=G(d)$ and with a
$G$-equivariant étale map

$e\colon E\to R^{\circ}(d)\oplus R^{\circ}(d)^{\vee}.$

Consider the quotient stacks with good moduli spaces

$\displaystyle\pi_{F}\colon\mathscr{F}:=(E\oplus\mathfrak{g})/G\to
F:=(E\oplus\mathfrak{g})/\\!\\!/G.$

Consider the moment map $\mu\colon E\to\mathfrak{g}^{\vee}$ and the induced
function $f\colon\mathscr{F}\to\mathbb{C}$, where
$f(x,v)=\langle\mu(x),v\rangle$ for $x\in E$ and $v\in\mathfrak{g}$. Consider
the quotient stack with good moduli space

$\pi_{L}\colon\mathscr{L}:=\mu^{-1}(0)/G\to L:=\mu^{-1}(0)/\\!\\!/G.$

There are maps:

${\mathscr{L}^{\mathrm{cl}}}$${\mathscr{P}(d)^{\mathrm{cl}}}$${L}$${P(d).}$$\scriptstyle{e}$$\scriptstyle{\pi_{L}}$$\scriptstyle{\pi_{P,d}}$$\scriptstyle{e}$

Throughout this section, we assume that both horizontal maps in the above
diagram are étale.

Note that the moment map $\mu$ has image in the traceless subalgebra
$\mathfrak{g}^{\vee}_{0}\cong\mathfrak{g}_{0}\subset\mathfrak{g}$. Let
$\mu_{0}\colon E\to\mathfrak{g}^{\vee}_{0}$ and let
$\mathscr{L}^{\mathrm{red}}:=\mu_{0}^{-1}(0)/G$.

######  Definition 9.1.

Let $\mathcal{BPS}^{p}_{d}\in\mathrm{Perv}(P(d))$ be the preprojective BPS
sheaf and let

(9.1) $\mathcal{BPS}^{L}:=e^{*}(\mathcal{BPS}^{p}_{d})\in\mathrm{Perv}(L).$

One defines $\mathcal{BPS}^{L}_{\delta}$ for $\delta\in
M(d)^{W_{d}}_{\mathbb{R}}$ and
$\mathcal{BPS}^{L}_{d,v}:=\mathcal{BPS}^{L}_{v\tau_{d}}$ as in (7.2).

By Theorem 2.9, there is a semiorthogonal decomposition:

$D^{b}\left(\mathscr{P}(d)\right)_{v}=\big{\langle}\mathbb{A}(d)_{v},\mathbb{T}(d)_{v}\big{\rangle}$

for any $v\in\mathbb{Z}$. The purpose of this section is to prove the
following:

######  Theorem 9.2.

Let $v\in\mathbb{Z}$. There are subcategories $\mathbb{T}=\mathbb{T}(L)_{v}$
and $\mathbb{A}=\mathbb{A}(L)_{v}$ of $D^{b}(\mathscr{L})_{v}$ such that:

  1. (1)

there is a semiorthogonal decomposition
$D^{b}(\mathscr{L})_{v}=\langle\mathbb{A},\mathbb{T}\rangle$,

  2. (2)

if $e$ is the identity, then $\mathbb{T}=\mathbb{T}(d)_{v}$ and
$\mathbb{A}=\mathbb{A}(d)_{v}$,

  3. (3)

if $h\colon E^{\prime}\to E$ is an étale map inducing $e^{\prime}:=e\circ
h\colon E^{\prime}\to R^{\circ}(d)\oplus R^{\circ}(d)^{\vee}$, and if we
consider $\pi_{L^{\prime}}\colon\mathscr{L^{\prime}}\to L^{\prime}$ and the
categories $\mathbb{A}(L^{\prime}),\mathbb{T}(L^{\prime})\subset
D^{b}(\mathscr{L}^{\prime})$ for $E^{\prime}$, then $h$ induces functors
$h^{*}\colon\mathbb{T}(L)_{v}\to\mathbb{T}(L^{\prime})_{v}$ and
$h^{*}\colon\mathbb{A}(L)_{v}\to\mathbb{A}(L^{\prime})_{v}$,

  4. (4)

for any $i,\ell\in\mathbb{Z}$, the cycle map (3.6) for $\mathscr{L}$ induces
isomorphisms

$\mathrm{c}\colon\mathrm{gr}_{\ell}K^{\mathrm{top}}_{i}(\mathbb{T})\xrightarrow{\sim}H^{-2\ell-i}(L,\mathcal{BPS}^{L}_{d,v}).$

Further, one can also define categories
$\mathbb{T}^{\mathrm{red}},\mathbb{A}^{\mathrm{red}}\subset
D^{b}(\mathscr{L}^{\mathrm{red}})_{v}$ which satisfy the analogous conditions
to (1)-(4) above. In particular, the map
$l^{\prime}\colon\mathscr{L}^{\mathrm{red}}\to\mathscr{L}$ induces an
isomorphism

(9.2) $\mathrm{c}\circ
l^{\prime}_{*}\colon\mathrm{gr}_{\ell}K^{\mathrm{top}}_{i}(\mathbb{T}^{\mathrm{red}})\xrightarrow{\sim}\mathrm{gr}_{\ell}K^{\mathrm{top}}_{i}(\mathbb{T})\xrightarrow{\sim}H^{-2\ell-i}(L,\mathcal{BPS}^{L}_{d,v}).$

We will only explain the constructions for $\mathscr{L}$, the case of
$\mathscr{L}^{\mathrm{red}}$ is similar. In Subsection 9.2, we define the
categories $\mathbb{T}$ and $\mathbb{A}$ using graded matrix factorizations
and the Koszul equivalence. In Subsection 9.3, we prove the third claim in
Theorem 9.2.

###  9.2. Quasi-BPS categories for étale covers

We will use the setting from Subsection 9.1. There is a Cartesian diagram,
where the maps $e$ are étale maps:

${\mathscr{F}}$${\mathscr{X}(d)}$${F}$${X(d).}$${\square}$$\scriptstyle{\pi_{F}}$$\scriptstyle{e}$$\scriptstyle{\pi_{X,d}}$$\scriptstyle{e}$

By Theorem 2.9, there is a semiorthogonal decomposition

(9.3)
$D^{b}\big{(}\mathscr{X}(d)\big{)}_{v}=\big{\langle}\mathbb{B}(d)_{v},\mathbb{M}(d)_{v}\big{\rangle}.$

Define subcategories $\mathbb{B}=\mathbb{B}(F)$, $\mathbb{M}=\mathbb{M}(F)$ of
$D^{b}(\mathscr{F})_{v}$ to be classically generated (see Subsection 2.15) by
$e^{\ast}\mathbb{B}(d)_{v}$, $e^{\ast}\mathbb{M}(d)_{v}$ respectively. Note
that, for $\delta\in M(d)^{W_{d}}_{\mathbb{R}}$, we can define analogously

(9.4) $\mathbb{M}(\delta)=\mathbb{M}(F;\delta)\subset D^{b}(\mathscr{F}).$

Lemma 2.11 implies that:

######  Corollary 9.3.

There is a semiorthogonal decomposition

$D^{b}(\mathscr{F})_{v}=\langle\mathbb{B},\mathbb{M}\rangle.$

If $e$ is the identity, then $\mathbb{M}(d)=\mathbb{M}(d)_{v}$ and
$\mathbb{B}(d)=\mathbb{B}(d)_{v}$.

Consider the category of graded matrix factorizations
$\mathrm{MF}^{\mathrm{gr}}(\mathscr{F},f)$, where the grading is of weight $2$
for the summand $\mathfrak{g}$ and is of weight $0$ on $E$. By the Koszul
equivalence, we have that:

(9.5) $\kappa_{L}\colon
D^{b}(\mathscr{L})\xrightarrow{\sim}\mathrm{MF}^{\mathrm{gr}}(\mathscr{F},f).$

Define the subcategories of $D^{b}(\mathscr{L})$:

$\mathbb{T}=\mathbb{T}(L):=\kappa_{L}^{-1}\left(\mathrm{MF}^{\mathrm{gr}}(\mathbb{M},f)\right),\
\mathbb{A}=\mathbb{A}(L):=\kappa_{L}^{-1}\left(\mathrm{MF}^{\mathrm{gr}}(\mathbb{B},f)\right).$

By [PTa, Proposition 2.5], we obtain:

######  Corollary 9.4.

The properties (1), (2), and (3) in the statement of Theorem 9.2 hold for the
categories $\mathbb{A}$ and $\mathbb{T}$ of $D^{b}(\mathscr{L})$.

We also need a version of Theorem 2.8 for étale covers. Consider the forget-
the-framing map $\tau_{\alpha}\colon\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}}\to\mathscr{X}(d)$. Let $\mathscr{F}^{\alpha f}$ be such
that the following diagram is Cartesian:

(9.6) ${\mathscr{F}^{\alpha f}}$${\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}}}$${\mathscr{F}}$${\mathscr{X}(d)}$${F}$${X(d).}$${\square}$$\scriptstyle{\tau_{\alpha,F}}$$\scriptstyle{e}$$\scriptstyle{\tau_{\alpha}}$$\scriptstyle{e}$$\scriptstyle{\pi_{F}}$$\scriptstyle{\pi_{X,d}}$$\scriptstyle{e}$

Recall the semiorthogonal decomposition of Theorem 2.5:

$D^{b}\left(\mathscr{X}^{\alpha
f}(d)^{\mathrm{ss}}\right)=\Big{\langle}\tau_{\alpha}^{*}\left(\otimes_{i=1}^{k}\mathbb{M}(d_{i})_{v_{i}}\right)\Big{\rangle}.$

For a partition $\underline{d}:=(d_{i})_{i=1}^{k}$ of $d\in\mathbb{N}^{I}$ and
for integer weights $\underline{v}:=(v_{i})_{i=1}^{k}$, define
$\mathbb{M}(\underline{d},\underline{v})\subset D^{b}\left(\mathscr{F}^{\alpha
f}\right)$ to be classically generated by
$e^{\ast}\tau_{\alpha}^{\ast}\left(\otimes_{i=1}^{k}\mathbb{M}(d_{i})_{v_{i}}\right)$.
By Lemma 2.11, we obtain that:

######  Proposition 9.5.

There is a semiorthogonal decomposition

$D^{b}\left(\mathscr{F}^{\alpha
f}\right)=\Big{\langle}\mathbb{M}(\underline{d},\underline{v})\Big{\rangle},$

where the left hand side is as in Theorem 2.5.

The category $\mathbb{M}(\underline{d},\underline{v})$ can be described via
the Hall product, same as in Theorem 2.5. Let $\lambda$ be an antidominant
cocharacter associated to $(d_{i})_{i=1}^{k}$. Consider the diagram:

$\mathscr{F}^{\lambda}\xleftarrow{q_{F}}\mathscr{F}^{\lambda\geqslant
0}\xrightarrow{p_{F}}\mathscr{F}.$

There is an étale map
$e\colon\mathscr{F}^{\lambda}\to\mathscr{X}(d)^{\lambda}\cong\times_{i=1}^{k}\mathscr{X}(d_{i})$.
Then the Hall product

$\ast_{F}=p_{F*}q_{F}^{*}\colon D^{b}(\mathscr{F}^{\lambda})\to
D^{b}(\mathscr{F})$

is base-change of the categorical Hall product
$D^{b}\big{(}\times_{i=1}^{k}\mathscr{X}(d_{i})\big{)}\cong\otimes_{i=1}^{k}D^{b}(\mathscr{X}(d_{i}))\to
D^{b}(\mathscr{X}(d))$. Let
$\widetilde{\mathbb{M}}(\underline{d},\underline{v})\subset
D^{b}\left(\mathscr{F}(\underline{d})\right)$ be the subcategory classically
generated by
$e^{\ast}\left(\otimes_{i=1}^{k}\mathbb{M}(d_{i})_{v_{i}}\right)$. There is
then an equivalence:

(9.7)
$\tau^{*}_{\alpha,F}\circ\ast_{F}\colon\widetilde{\mathbb{M}}(\underline{d},\underline{v})\xrightarrow{\sim}\mathbb{M}(\underline{d},\underline{v}).$

###  9.3. Comparison with BPS cohomology

Recall the notation from Subsection 7.1. Consider the commutative diagram:

${\mathscr{L}^{\mathrm{cl}}}$${\mathscr{P}(d)^{\mathrm{cl}}}$${\mathscr{F}}$${E/G}$${\mathscr{Y}(d)}$${L}$${P(d)}$${F}$${E/\\!\\!/G}$${Y(d)}$$\scriptstyle{\eta}$$\scriptstyle{\pi_{F}}$$\scriptstyle{\pi_{E}}$$\scriptstyle{\eta}$

Recall the sheaf $\mathcal{BPS}^{L}\in\mathrm{Perv}(L)$ defined in (9.1) and
consider the BPS sheaf $\mathcal{BPS}_{d}\in\mathrm{Perv}(X(d))$ for the
tripled quiver with potential $(Q,W)$ associated to $Q^{\circ}$. Define the
BPS sheaf:

$\mathcal{BPS}^{F}=e^{*}(\mathcal{BPS}_{d})\in\mathrm{Perv}(F).$

For $\delta\in M(d)^{W_{d}}_{\mathbb{R}}$, define

$\mathcal{BPS}^{F}_{\delta}\in
D^{b}_{\mathrm{con}}(F),\,\mathcal{BPS}^{F}_{d,v}:=\mathcal{BPS}^{F}_{v\tau_{d}}$

as in (6.5). Note that, by base-change of the decomposition (6.18), we obtain
the analogous decomposition for $\pi_{F}\colon\mathscr{F}\to F$:

(9.8)
$\pi_{F*}\varphi_{f}\mathrm{IC}_{\mathscr{F}}[-1]=\bigoplus_{A\in\mathscr{P}}e^{*}(\mathrm{Q}_{A}).$

By base-change of (6.17), we obtain the following decomposition for the map
$\pi_{\alpha,F}:=\pi_{F}\circ\tau_{\alpha,F}\colon\mathscr{F}^{\alpha f}\to
F$:

(9.9) $\pi_{\alpha,F*}\varphi_{f}\mathbb{Q}_{\mathscr{F}^{\alpha
f}}[\dim\mathscr{F}-1]=\bigoplus_{A\in\mathscr{P}_{\alpha}}e^{*}(\mathrm{Q}_{A}).$

The monodromy on $H^{\bullet}(\mathscr{F},\varphi_{f})$ is trivial, so there
is a cycle map:

(9.10)
$\mathrm{c}\colon\mathrm{gr}_{a}K^{\mathrm{top}}_{i}\left(\mathrm{MF}(\mathscr{F},f)\right)\xrightarrow{\sim}H^{2\dim\mathscr{F}-i-2a}(\mathscr{F},\varphi_{f}\mathbb{Q}_{\mathscr{F}}[-1])\oplus
H^{2\dim\mathscr{F}-i-2a}(\mathscr{F},\varphi_{f}\mathbb{Q}_{\mathscr{F}}[-2]).$

We now define a cycle map from topological K-theory of quasi-BPS categories to
BPS cohomology, which is the analogue of Theorem 6.2.

######  Proposition 9.6.

Let $\delta\in M(d)^{W_{d}}_{\mathbb{R}}$ and recall the categories
$\mathbb{M}(\delta)$ from (9.4). The cycle map (9.10) has image in

(9.11)
$\mathrm{c}\colon\mathrm{gr}_{a}K^{\mathrm{top}}_{i}\left(\mathrm{MF}(\mathbb{M}(\delta),f)\right)\to
H^{\dim\mathscr{F}-i-2a}(F,\mathcal{BPS}^{F}_{\delta})\oplus
H^{\dim\mathscr{F}-i-2a}(F,\mathcal{BPS}^{F}_{\delta}[-1]).$

Thus, for $\delta=v\tau_{d}$ and $\mathbb{M}=\mathbb{M}(v\tau_{d})$, the cycle
map (9.10) has image in

(9.12)
$\mathrm{c}\colon\mathrm{gr}_{a}K^{\mathrm{top}}_{i}\left(\mathrm{MF}(\mathbb{M},f)\right)\to
H^{\dim\mathscr{F}-i-2a}(F,\mathcal{BPS}^{F}_{d,v})\oplus
H^{\dim\mathscr{F}-i-2a}(F,\mathcal{BPS}^{F}_{d,v}[-1]).$

###### Proof.

The same argument used in the proof of Theorem 6.3 applies here. The
$\lambda$-widths (see (6.28)) of the category $\mathbb{M}(\delta)$ are equal
to the $\lambda$-widths of the category $\mathbb{M}(d;\delta)$ for all
cocharacters $\lambda$. The analogue of Proposition 6.15 then holds for
$\mathrm{MF}(\mathbb{M}(\delta),f)$.

There is an explicit decomposition of $\pi_{F*}\mathrm{IC}_{\mathscr{F}}$
obtained by base-change from (6.16), and where the summands are in the image
of (the base-change of the) Hall product. In Subsection 6.6, we constructed
the map (6.36), proved Proposition 6.16, and noted corollaries of Proposition
6.16. There are versions of the map (6.36) and of Proposition 6.16 by $F$ by
base-change, and the results in Subsection 6.6 also apply for
$\pi_{F*}\mathrm{IC}_{\mathscr{F}}$, and thus for
$\pi_{F*}\varphi_{f}\mathrm{IC}_{\mathscr{F}}[-1]$. ∎

We next prove the analogue of Theorem 6.2.

######  Proposition 9.7.

The cycle map (9.12) is an isomorphism.

###### Proof.

The same argument used to prove Theorem 6.2 applies here, see Subsection 6.3,
that is, the statement follows from comparing summands in the semiorthogonal
decomposition (9.5) with summands in the decomposition (9.9). The cycle map
(9.12) is injective by (9.10) and the admissibility of
$\mathrm{MF}^{\mathrm{gr}}(\mathbb{M},f)$ in
$\mathrm{MF}^{\mathrm{gr}}(\mathscr{F},f)$.

Consider a partition $\underline{d}=(d_{i})_{i=1}^{k}$ of $d$ and weights
$\underline{v}=(v_{i})_{i=1}^{k}\in\mathbb{Z}^{k}$. Consider the perverse
sheaf

$\mathcal{BPS}_{\underline{d},\underline{v}}:=\oplus_{*}\left(\boxtimes_{i=1}^{k}\mathcal{BPS}_{d_{i},v_{i}}\right)\in\mathrm{Perv}(X(d)),$

where $\oplus\colon\times_{i=1}^{k}X(d_{i})\to X(d)$ is the direct sum map. By
Proposition 9.6 for the disjoint union of $k$ copies of $Q$, dimension vector
$(d_{i})_{i=1}^{k}\in\left(\mathbb{N}^{I}\right)^{k}$, and
$\delta=\sum_{i=1}^{k}v_{i}\tau_{d_{i}}$, there is an injective map for any
$i\in\mathbb{Z}$:

$\mathrm{gr}_{\cdot}K^{\mathrm{top}}_{i}\left(\mathbb{M}(\underline{d},\underline{v})\right)\hookrightarrow
H^{\cdot}\left(F,e^{*}\mathcal{BPS}_{\underline{d},\underline{v}}\right).$

The claim now follows as in the proof of Theorem 6.2. ∎

We prove the analogue of Theorem 7.6.

######  Proposition 9.8.

The cycle map obtained by composing (9.10) with the forget-the-potential map
is an isomorphism:

(9.13)
$\mathrm{c}\colon\mathrm{gr}_{a}K^{\mathrm{top}}_{i}\left(\mathrm{MF}^{\mathrm{gr}}(\mathscr{F},f)\right)\xrightarrow{\sim}H^{2\dim\mathscr{F}-i-2a}(\mathscr{F},\varphi_{f}[-1]).$

Thus there is an isomorphism:

(9.14)
$\mathrm{c}\colon\mathrm{gr}_{a}K^{\mathrm{top}}_{i}\left(\mathrm{MF}^{\mathrm{gr}}(\mathbb{M},f)\right)\to
H^{\dim\mathscr{F}-i-2a}(F,\mathcal{BPS}^{F}_{d,v}).$

Further, there is an isomorphism:

$\mathrm{gr}_{a}K^{\mathrm{top}}_{i}(\mathbb{T})\xrightarrow{\sim}H^{-2a-i}(L,\mathcal{BPS}^{L}_{d,v})$

###### Proof.

The isomorphism (9.13) follows from Proposition 5.2. The isomorphism (9.14)
follows then from Proposition 9.6. The last isomorphism follows from (9.14)
and the compatibility between dimensional reduction and the Koszul equivalence
from Proposition 5.2. ∎

###### Proof of Theorem 9.2.

The first three properties hold by Corollary 9.4. The fourth property follows
from Proposition 9.8. The statement for reduced stacks follows similarly. The
isomorphism (9.2) also follows directly from Proposition 3.5. ∎

We also note the following analogue of Corollary 7.2.

######  Proposition 9.9.

The Chern character map

$\mathrm{ch}\colon K^{\mathrm{top}}_{i}(\mathbb{T})\hookrightarrow
G^{\mathrm{top}}_{i}(\mathscr{L})\to\bigoplus_{j\in\mathbb{Z}}H^{\mathrm{BM}}_{i+2j}(\mathscr{L})$

is injective.

###### Proof.

The proof is analogous to that of Corollary 7.2. The claim follows from
Proposition 4.10, a version of Proposition 9.5 involving potential, and the
Koszul equivalence. ∎

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Tudor Pădurariu: Max Planck Institute for Mathematics, Vivatsgasse 7 Bonn
53111, Germany.
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>