# A boson-fermion correspondence in cohomological Donaldson–Thomas theory

Ben Davison  School of Mathematics
University of Edinburgh
Edinburgh EH9 3FD
UK<EMAIL_ADDRESS>
###### Abstract.

We introduce and study a fermionization procedure for the cohomological Hall
algebra $\mathcal{H}_{\Pi_{Q}}$ of representations of a preprojective algebra,
that selectively switches the cohomological parity of the BPS Lie algebra from
even to odd. We do so by determining the cohomological Donaldson–Thomas
invariants of central extensions of preprojective algebras studied in the work
of Etingof and Rains, via deformed dimensional reduction.

Via the same techniques, we determine the Borel–Moore homology of the stack of
representations of the $\upmu$-deformed preprojective algebra introduced by
Crawley–Boevey and Holland, for all dimension vectors. This provides a common
generalisation of the results of Crawley-Boevey and Van den Bergh on the
cohomology of smooth moduli schemes of representations of deformed
preprojective algebras, and my earlier results on the Borel–Moore homology of
the stack of representations of the undeformed preprojective algebra.

###### Contents

  1. 1 Introduction
  2. 2 Cohomological DT theory for quivers with potential
  3. 3 Deformed dimensional reduction and proofs of main results
  4. 4 Further directions

##  1\. Introduction

Given a quiver $Q$, in [KS11] Kontsevich and Soibelman define the
cohomological Hall algebra

(1)
$\operatorname{\mathcal{A}}_{Q}=\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\operatorname{\mathbf{H}}(\mathfrak{M}_{\mathbf{d}}(Q),\mathbb{Q})[-\upchi_{Q}(\mathbf{d},\mathbf{d})],$

which has as underlying vector space the singular cohomology of the stack of
finite-dimensional complex representations of $Q$, shifted in cohomological
degree by the Euler form (see (18) for the definition). As indicated by (1),
this algebra is graded by the dimension vectors of representations of $Q$. The
multiplication is defined by taking push-forward and pull-back of cohomology
in the usual correspondence diagram

(2)
$\textstyle{\mathfrak{M}(Q)\times\mathfrak{M}(Q)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathfrak{E}\mathrm{xact}(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\uppi\hskip
0.35002pt_{1}\times\uppi\hskip 0.35002pt_{3}}$$\scriptstyle{\uppi\hskip
0.35002pt_{2}}$$\textstyle{\mathfrak{M}(Q).}$

In the diagram (2),
$\mathfrak{M}(Q)=\coprod_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\mathfrak{M}_{\mathbf{d}}(Q)$
is the stack of finite dimensional $Q$-representations,
$\mathfrak{E}\mathrm{xact}(Q)$ is the stack of short exact sequences of
$Q$-representations, and $\uppi\hskip 0.50003pt_{i}$ is the morphism taking a
short exact sequence to its $i$th entry.

If $Q$ is moreover symmetric (i.e. for every pair of vertices $i,j$, there are
as many arrows from $i$ to $j$ as from $j$ to $i$), then the multiplication
respects the cohomological degree, and a theorem of Efimov [Efi12] states that
(a slight modification of) $\operatorname{\mathcal{A}}_{Q}$ is a free
supercommutative algebra; the “super” here means that elements in odd
cohomological degrees anti-commute with each other.

There are a couple of instances in which this result is possible to check by
hand, indeed it was observed in [KS11, Sec.2.5] that if $Q^{(l)}$ denotes the
quiver with one vertex and $l$ loops, then

(3) $\displaystyle\operatorname{\mathcal{A}}_{Q^{(0)}}\cong$
$\displaystyle\operatorname{\mathbf{Sym}}\left((\mathbb{Q}[u])[-1]\right)$ (4)
$\displaystyle\operatorname{\mathcal{A}}_{Q^{(1)}}\cong$
$\displaystyle\operatorname{\mathbf{Sym}}\left(\mathbb{Q}[u]\right)$

where $[-1]$ denotes a cohomological shift, and $u^{i}$ is placed in
cohomological degree $2i$. Both algebras are free supercommutative algebras
generated by a countable set of symbols $\upalpha_{i}$ for $i\geq 0$, with
$\upalpha_{i}$ placed in cohomological degree $1+2i$ in the zero-loop case,
and in cohomological degree $2i$ in the one-loop case. In other words, (3)
states that $\operatorname{\mathcal{A}}_{Q^{(0)}}$ is a free exterior algebra
with countably many generators, while (4) states that
$\operatorname{\mathcal{A}}_{Q^{(1)}}$ is a free commutative algebra with the
same generators. As observed in [KS11], the fact that the underlying vector
spaces of the two algebras are the same (since $\mathfrak{M}_{n}(Q^{(l)})$ is
homotopic to $\mathrm{BGl}_{n}$, regardless of $l$) can be seen as a
consequence of the boson-fermion correspondence in representation theory.

Now let $Q$ be an arbitrary finite quiver. We form the tripled quiver
$\tilde{Q}$, by adjoining to the quiver $Q$ an arrow $a^{*}$ with the opposite
orientation to $a$, for each $a$ an arrow of $Q$, and also adjoining a loop
$\omega_{i}$ at each vertex $i$ of $Q$. So for instance
$\widetilde{Q^{(1)}}\cong Q^{(3)}$. The quiver $\tilde{Q}$ carries a canonical
cubic potential

(5) $\tilde{W}=\sum_{a\in Q_{1}}[a,a^{*}]\sum_{i\in Q_{0}}\omega_{i},$

and one may define (again as in [KS11]) the critical cohomological Hall
algebra $\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$. Again, the
multiplication respects cohomological degree. The underlying vector space of
this algebra is the vanishing cycle cohomology of the function
$\operatorname{Tr}(\tilde{W})$ on the stack $\mathfrak{M}(\tilde{Q})$. The
multiplication is defined via pull-back and push-forward of vanishing cycle
cohomology along the same correspondence diagram (2). Via dimensional
reduction [Dav17a] the algebra
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$ is isomorphic [RS17, YZ16]
to the cohomological Hall algebra structure on the Borel–Moore homology of the
stack of representations of the preprojective algebra $\Pi_{Q}$ constructed by
Schiffmann and Vasserot in [SV13] and studied in [YZ18].

In general the algebra $\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$ is
not supercommutative; for example
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$ contains the universal
enveloping algebra of the Kac–Moody Lie algebra associated to the quiver
$Q^{\prime}$ obtained by removing all vertices from $Q$ that support 1-cycles
[Dav20]. On the other hand, via the cohomological integrality theorem [DM20],
the entire algebra $\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$
satisfies a Poincaré–Birkhoff–Witt theorem, meaning that we have an
$\mathbb{N}^{Q_{0}}$-graded isomorphism of cohomologically graded vector
spaces (but not of algebras):

(6) $\displaystyle\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}\cong$
$\displaystyle\operatorname{\mathbf{Sym}}\left(\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\left(\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}\otimes\mathbb{Q}[u]\right)[-1]\right).$

In the isomorphism (6), $\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}$
is a cohomologically graded vector space, the whole of
$\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}\otimes\mathbb{Q}[u]$ is
placed in $\mathbb{N}^{Q_{0}}$-degree $\mathbf{d}$, $u^{i}$ has cohomological
degree $2i$, and the symbol $[-1]$ denotes the cohomological shift as before.
Moreover by [Dav17b], the Poincaré polynomial of the cohomologically graded
vector space $\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}$ satisfies
the relation

(7)
$\operatorname{\mathtt{p}}(\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}},q^{1/2})\coloneqq\sum_{i\in\mathbb{Z}}\dim(\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}^{i})q^{i/2}=q^{-1/2}\mathtt{a}_{Q,\mathbf{d}}(q^{-1})$

where $\mathtt{a}_{Q,\mathbf{d}}(q)$ is the Kac polynomial [Kac80], counting
absolutely indecomposable $\mathbf{d}$-dimensional $Q$-representations over a
field of order $q$. For example, we observe that
$Q^{(1)}\cong\widetilde{Q^{(0)}}$ and calculate

$\mathtt{a}_{Q^{(0)},\mathbf{d}}(q)=\begin{cases}1&\textrm{if }\mathbf{d}=1\\\
0&\textrm{if }\mathbf{d}\geq 2\end{cases}$

recovering (4) from (6) and (7).

In particular, we see that for arbitrary finite $Q$, the algebra
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$ is bosonic, in the sense
that it is situated entirely in even cohomological degree. Having observed
that
$\operatorname{\mathcal{A}}_{Q^{(1)}}=\operatorname{\mathcal{A}}_{\widetilde{Q^{(0)}},\tilde{W}}$
has a fermionic counterpart $\operatorname{\mathcal{A}}_{Q^{(0)}}$, we may ask
the following

######  Question 1.1.

For a finite quiver $Q$, is there a “fermionic” version of
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$?

It is a consequence of the purity of the cohomological BPS invariants for the
quiver with potential $(\tilde{Q},\tilde{W})$ (again, see [Dav17b]) that there
is an equality

(8)
$\operatorname{\mathtt{p}}(\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}},q^{1/2})=\upchi_{q^{1/2}}(\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}})\
$

between the Poincaré polynomial and the virtual Poincaré polynomial111See
[Moz11] for the virtual Poincaré polynomial analogue of (7); any two of
purity, (7), and the virtual analogue of (7) imply the third. of
$\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}$. The polynomials
$\upchi_{q^{1/2}}(\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}})$ are
essentially by definition the refined BPS invariants for the quiver
$\tilde{Q}$ with potential $\tilde{W}$. Expressed in the language of refined
Donaldson–Thomas theory, the analogue of Question 1.1 is

######  Question 1.2.

Is there quiver $Q^{\prime}$ with potential $W^{\prime}$ such that the refined
BPS invariants satisfy

$\upchi_{q^{1/2}}(\operatorname{BPS}_{Q^{\prime},W^{\prime},\mathbf{d}})=q^{1/2}\upchi_{q^{1/2}}(\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}})?$

In this paper we will answer these two questions in the affirmative.

###  1.1. Counting rational curves

At least in the case in which $Q$ is an affine Dynkin quiver, there are strong
hints from the McKay correspondence that the answer to Question 1.2 should be
at least a partial “yes”. We recall some geometric background; see [KM92,
BKL01] for more details.

Fix a finite subgroup $G\subset\operatorname{SL}_{2}(\mathbb{C})$, and denote
by $X_{0}=\mathbb{C}^{2}/G$ the associated Kleinian singularity. We denote by
$\Upgamma$ the McKay graph of $G$, and by $\Upgamma^{\prime}$ the full ADE
type sub-graph of $\Upgamma$ obtained by removing the vertex corresponding to
the trivial representation. We denote by $p\colon Y_{0}\rightarrow X_{0}$ the
minimal resolution of $X_{0}$. The surface $X_{0}$ contains an isolated
singularity $x$, and the exceptional fibre $p^{-1}(x)$ consists of a chain of
rational curves, with incidence graph $\Upgamma^{\prime}$. The space $Y_{0}$
has a universal deformation $Y$ parametrised by $\mathfrak{h}$, the Cartan
subalgebra of the simple Lie algebra corresponding to $\Upgamma^{\prime}$, so
that we have a Cartesian diagram

$\textstyle{Y_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\uppi\hskip
0.35002pt}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{h}.}$

The generic fibre of $\uppi\hskip 0.50003pt$ contains no rational curves,
while if $h\in\mathfrak{h}$ lies in a root hyperplane corresponding to a
vertex $i$ of $\Upgamma^{\prime}$, the rational curve $C_{i}$ corresponding to
$i$ deforms along the line $t\cdot h$ for $t\in\mathbb{C}$ [BKL01, Prop.2.2].
We pick $h\in\mathfrak{h}$ and form the Cartesian diagram

(9)
$\textstyle{Y^{h}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\uppi\hskip
0.35002pt^{\prime}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\uppi\hskip
0.35002pt}$$\textstyle{\mathbb{A}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t\mapsto
t\cdot h}$$\textstyle{\mathfrak{h}.}$

For $\upbeta\in\operatorname{\mathbf{H}}_{2}(Y^{h},\mathbb{Z})$ and
$n\in\mathbb{N}$, let $\mathcal{M}_{\upbeta,n}(Y^{h})$ denote the moduli space
of semistable coherent sheaves $\mathcal{F}$ on $Y^{\prime}$, with fundamental
class of the support of $\mathcal{F}$ equal to $\upbeta$, and with
$\upchi(\mathcal{F})=n$.

Consider the case in which $\upbeta=[C_{i}]$ for some $i\in Q_{0}$, so that
stability is equivalent to semistability. Since a stable coherent sheaf
$\mathcal{F}$ cannot split as a direct sum, it is supported on a single fibre
of the morphism $\uppi\hskip 0.50003pt^{\prime}$. If $C_{i}$ deforms along
$\mathbb{A}^{1}$, with curve over $t\in\mathbb{A}_{1}$ labelled $C_{i,t}$,
then in each fibre $\uppi\hskip 0.50003pt^{\prime}(t)$ there is a unique
semistable coherent sheaf $\mathscr{O}_{C_{i,t}}(n-1)$ with Euler
characteristic $n$ and $[\mathscr{O}_{C_{i,t}}(n-1)]=\upbeta$. If $C_{i}$ does
not deform, then there is a unique semistable coherent sheaf on the whole of
$Y^{h}$ of class $(\upbeta,n)$, supported above $0\in\mathbb{A}^{1}$. So

$\mathcal{M}_{[C_{i}],n}(Y^{h})=\begin{cases}\operatorname{pt}&\textrm{if
}h\notin i^{\perp}\\\ \mathbb{A}^{1}&\textrm{if }h\in i^{\perp}.\end{cases}$

The cohomological BPS invariants $\operatorname{BPS}_{\upbeta,n}$ of the
3-fold $Y^{h}$ are in general hard to define rigorously, involving vanishing
cycle cohomology, d-critical structures [Joy15], orientation data, perverse
filtrations, etc. However, for simple classes like the ones we are considering
here, the definition/calculation boils down to something more straightforward:

$\displaystyle\operatorname{BPS}_{[C_{i}],n}(Y^{h})=$
$\displaystyle\operatorname{\mathbf{H}}(\mathcal{M}_{[C_{i}],n}(Y^{h}),\mathbb{Q})_{\operatorname{vir}}$
$\displaystyle\cong$ $\displaystyle\begin{cases}\mathbb{Q}&\textrm{if }h\notin
i^{\perp}\\\ \mathbb{Q}[1]&\textrm{if }h\in i^{\perp}.\end{cases}$

The subscript $\operatorname{vir}$ denotes the cohomological shift
$[\dim(\mathcal{M}_{[C_{i}],n})]$ by the dimension of the space we are taking
the singular cohomology of, and accounts for the shift in the $h\in i^{\perp}$
case. So in both cases, the cohomological BPS invariants for simple curve
classes are one-dimensional vector spaces, and are concentrated in even or odd
cohomological degree, depending on whether the choice of $h$ means that
$C_{i}$ is rigid or not.

Similarly, the definition and calculation of the simplest degree zero
cohomological BPS invariant $\operatorname{BPS}_{0,1}(Y^{h})$ is much easier
than the general case, and we have

$\operatorname{BPS}_{0,1}(Y^{h})\cong\operatorname{\mathbf{H}}(Y^{h},\mathbb{Q})_{\operatorname{vir}}.$

It is easy to verify that the singular cohomology of $Y^{h}$ does not depend
on the choice of $h$ at all. In general one expects
$\operatorname{BPS}_{0,n}\cong\operatorname{BPS}_{0,1}$ (compare with
[BBS13]), so it turns out that the degree zero222By which we mean the
Donaldson–Thomas theory of coherent sheaves with zero-dimensional support.
cohomological DT theory of $Y^{h}$ does not depend on $h$.

To put this geometric discussion in very leading language: for the most
degenerate case $h=0$, all of the cohomological DT theory is bosonic, since
the vanishing cycle cohomology ends up living in even cohomological degree
(taking into account the shift $[-1]$ in (6)). On the other hand, modifying
the deforming family defined by $h\in\mathfrak{h}$ to be more generic, a
portion of the cohomological DT theory is fermionized, depending on which root
hyperplanes $h$ avoids.

###  1.2. The noncommutative conifold and central extensions of the
preprojective algebra

Let $Q^{\prime}$ be the $A_{1}$ quiver, and let $Q$ be its affine extension,
which we label as follows:

(10)
$\textstyle{Q=}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{1.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$

Then as a special case of [KV00] there is a derived equivalence between the
category of finitely generated modules for the preprojective algebra $\Pi_{Q}$
of $Q$, and the category of coherent sheaves on $Y_{0}$, the minimal
resolution of the type $A_{1}$ singularity defined by the equation
$x^{2}+y^{2}=z^{2}$:

$\mathcal{D}^{\mathrm{b}}_{\mathrm{fg}}(\Pi_{Q}\operatorname{-Mod})\xrightarrow{\cong}\mathcal{D}^{\mathrm{b}}(\operatorname{Coh}(Y_{0})).$

We have that

$\textstyle{\tilde{Q}=}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{0}}$$\scriptstyle{b^{*}}$$\scriptstyle{a}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{a^{*}}$$\scriptstyle{\omega_{1}}$

The Kac polynomials of $Q$ are possible to calculate by hand, so that we can
calculate the cohomological BPS invariants for
$\operatorname{Jac}(\tilde{Q},\tilde{W})$ via (7)

(11)
$\operatorname{BPS}_{\tilde{Q},\tilde{W},(m,n)}\cong\begin{cases}\mathbb{Q}[3]\oplus\mathbb{Q}[1]&\textrm{if
}m=n\\\ \mathbb{Q}[1]&\textrm{if }m=n\pm 1\\\
0&\textrm{otherwise.}\end{cases}$

There is an isomorphism
$\operatorname{Jac}(\tilde{Q},\tilde{W})\cong\Pi_{Q}[\omega]=\Pi_{Q}\otimes\mathbb{C}[\omega]$,
so that we have in addition a derived equivalence (see e.g. [Sze08b])

$\mathcal{D}^{\mathrm{b}}_{\mathrm{fg}}(\operatorname{Jac}(\tilde{Q},\tilde{W})\operatorname{-Mod})\xrightarrow{\cong}\mathcal{D}^{\mathrm{b}}(\operatorname{Coh}(Y^{0}))$

where $Y^{0}$ is defined via the construction in the previous subsection by
setting $h=0\in\mathfrak{h}$, i.e. $Y^{0}=Y_{0}\times\mathbb{A}^{1}$.

For the $A_{1}$-singularity, the Cartan sub-algebra $\mathfrak{h}$ is one-
dimensional, so aside from $0$ there is an essentially unique choice of
$h\in\mathfrak{h}$. Defining $Y_{\operatorname{con}}=Y^{h}$ for a nonzero
choice of $h\in\mathfrak{h}$ in diagram (9), we obtain the resolved conifold.
As noted in the previous section, instead of having an $\mathbb{A}^{1}$-family
of rational curves giving rise to cohomological BPS invariants in odd degrees,
the resolved conifold contains a unique rigid curve. It follows that the
cohomological BPS invariants corresponding to sheaves supported on the curve
flip parity, and are supported in even cohomological degrees, so that they
contribute to the fermionic part of the cohomological DT theory of the
resolved conifold (as ever, taking into account the shift defined as in (6))

The resolved conifold also has a noncommutative model, studied in this context
by Szendrői, which we recall (see [Sze08a] for details on the noncommutative
Donaldson–Thomas theory of the conifold, and also [RSYZ20, GY20, LY20] for
more recent work on CoHAs related to toric 3-folds). We consider the double

$\textstyle{\overline{Q}=}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b^{*}}$$\scriptstyle{a}$$\textstyle{1.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{a^{*}}$

Set $W_{\operatorname{\mathtt{KW}}}=aa^{*}bb^{*}-a^{*}ab^{*}b$ to be the
Klebanov–Witten potential [KW98]. Then (e.g. as a special case of a result due
to Van den Bergh [vdB04]) there is a derived equivalence

$\mathcal{D}^{\mathrm{b}}_{\mathrm{fg}}(\operatorname{Jac}(\overline{Q},W_{\operatorname{\mathtt{KW}}})\operatorname{-Mod})\xrightarrow{\cong}\mathcal{D}^{\mathrm{b}}(\operatorname{Coh}(Y_{\operatorname{con}})).$

To interpolate between the two cases ($h=0,h\neq 0$) it turns out to be more
instructive to consider the quiver $\tilde{Q}$ with the potential

$\tilde{W}^{(1,-1)}=\tilde{W}+\omega_{0}^{2}-\omega_{1}^{2}.$

As we recall in §2.1 the resulting Jacobi algebra has already been studied: it
is a special case of the central extensions of $\Pi_{Q}$ introduced by Etingof
and Rains in [ER06]. There is an isomorphism (see Example 2.2)

(12)
$\operatorname{Jac}(\tilde{Q},\tilde{W}^{(1,-1)})\cong\operatorname{Jac}(\overline{Q},W_{\operatorname{\mathtt{KW}}})$

and the cohomological DT theory of the Jacobi algebras in (12) turns out to be
the same. The cohomological BPS invariants for the noncommutative conifold can
be deduced from [MMNS12] and purity (proved as in [DOS20, Thm.4.7]):

(13)
$\operatorname{BPS}_{\tilde{Q},\tilde{W}^{(1,-1)},m,n}\cong\begin{cases}\operatorname{\mathbf{H}}(Y_{\operatorname{con}},\mathbb{Q})[3]\cong\mathbb{Q}[3]\oplus\mathbb{Q}[1]&\textrm{if
}m=n\\\ \mathbb{Q}&\textrm{if }m=n\pm 1\\\ 0&\textrm{otherwise.}\end{cases}$

Comparing (11) with (13) we see exactly the same pattern as in the commutative
case of §1.1, with the passage from trivial $\mathbb{A}^{1}$-deformations of
$Y_{0}$ to nontrivial ones replaced in the context of noncommutative algebraic
geometry by the passage from the trivial central extension of the algebra
$\Pi_{Q}$ to the nontrivial ones constructed by Etingof and Rains; this change
provokes a flip in the parity of some, but not all, of the cohomological BPS
invariants.

###  1.3. Main results

In the rest of the paper we prove that the above discussion regarding the
(noncommutative) conifold forms part of a general procedure for (selectively)
“fermionizing” the cohomological Hall algebras of preprojective algebras. This
culminates in the following theorem:

######  Theorem A.

Let $\upmu\in\mathbb{C}^{Q_{0}}$. Set $\tilde{W}^{\upmu}=\sum_{a\in
Q_{1}}[a,a^{*}]\sum_{i\in Q_{0}}\omega_{i}+\frac{1}{2}\sum_{i\in
Q_{0}}\upmu_{i}\omega_{i}^{2}$. Then the cohomological BPS invariants for the
quiver $\tilde{Q}$ with potential $\tilde{W}^{\upmu}$ satisfy

(14)
$\operatorname{\mathtt{p}}(\operatorname{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}},q^{1/2})=\begin{cases}\mathtt{a}_{Q,\mathbf{d}}(q^{-1})&\textrm{if
}\mathbf{d}\cdot\upmu\neq 0\\\
q^{-1/2}\mathtt{a}_{Q,\mathbf{d}}(q^{-1})&\textrm{if
}\mathbf{d}\cdot\upmu=0\end{cases}$

where $\mathtt{a}_{Q,\mathbf{d}}(q)$ are the Kac polynomials for $Q$. In
addition, the natural mixed Hodge structure on the cohomological BPS
invariants $\operatorname{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}$ is
pure, of Tate type, so that we have isomorphisms of Hodge-theoretic BPS
invariants

(15)
$\operatorname{BPS}^{\operatorname{hdg}}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}\cong\begin{cases}\bigoplus_{i\in\mathbb{Z}}(\mathscr{L}^{i})^{\oplus\mathtt{a}_{Q,\mathbf{d},-i}}&\textrm{if
}\mathbf{d}\cdot\upmu\neq 0\\\
\bigoplus_{i\in\mathbb{Z}}(\mathscr{L}^{i-1/2})^{\oplus\mathtt{a}_{Q,\mathbf{d},-i}}&\textrm{if
}\mathbf{d}\cdot\upmu=0\end{cases}$

where
$\mathscr{L}\coloneqq\operatorname{\mathbf{H}}_{c}(\mathbb{A}^{1},\mathbb{Q})$.
By purity, the virtual Poincaré polynomials of the cohomological BPS
invariants agree with the above Poincaré polynomials, so that we have
equalities for the refined BPS invariants

(16)
$\upchi_{q^{1/2}}(\operatorname{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}})=\begin{cases}\mathtt{a}_{Q,\mathbf{d}}(q^{-1})&\textrm{if
}\mathbf{d}\cdot\upmu\neq 0\\\
q^{-1/2}\mathtt{a}_{Q,\mathbf{d}}(q^{-1})&\textrm{if
}\mathbf{d}\cdot\upmu=0.\end{cases}$

Comparing with (7) we see that for dimension vectors $\mathbf{d}$ satisfying
$\mathbf{d}\cdot\upmu\neq 0$ the cohomological BPS invariants have switched
parity. In particular, for generic choices of $\upmu$, the algebra
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}^{\upmu}}$ is a fermionized
version of $\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}}$, so (15) and
(16) answer Questions 1.1 and (1.2) respectively in the affirmative.

Still fixing $\upmu\in\mathbb{C}^{Q_{0}}$ the deformed preprojective algebra,
introduced by Crawley–Boevey and Holland in [CBH98], is defined by

$\Pi_{Q,\upmu}\coloneqq\mathbb{C}\overline{Q}/\langle\sum_{a\in
Q_{1}}[a,a^{*}]+\sum_{i\in Q_{0}}\upmu_{i}e_{i}\rangle.$

where $e_{i}$ is the path of length zero beginning and ending at the vertex
$i$. Via the methods used to prove Theorem A we are able to calculate the
Borel–Moore homology (along with its mixed Hodge structure) of all stacks of
representations of deformed preprojective algebras, simultaneously
generalising a result of Crawley–Boevey and Van den Bergh [CBdB04] from the
case of generic $\upmu$ and indivisible dimension vector $\mathbf{d}$, and the
result from [Dav17b] which deals with the case $\upmu=0$ and arbitrary
$\mathbf{d}$:

######  Theorem B.

For arbitrary $\upmu\in R$ and $\mathbf{d}\in\mathbb{N}^{Q_{0}}$ there is an
isomorphism of $\mathbb{N}^{Q_{0}}$-graded mixed Hodge structures

(17)
$\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\operatorname{\mathbf{H}}_{c}\\!\left(\mathfrak{M}_{\mathbf{d}}(\Pi_{Q,\upmu}),\mathbb{Q}\right)\otimes\mathscr{L}^{\upchi_{Q}(\mathbf{d},\mathbf{d})}\cong\operatorname{\mathbf{Sym}}\left(\bigoplus_{\begin{subarray}{c}0\neq\mathbf{d}\in\mathbb{N}^{Q_{0}}\\\
\mathbf{d}\cdot\upmu=0\end{subarray}}\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}^{\operatorname{hdg},\vee}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*},\mathbb{Q})_{\operatorname{vir}}\right)$

where

$\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}^{\operatorname{hdg},\vee}\cong\bigoplus_{i\in\mathbb{Z}}(\mathscr{L}^{i+1/2})^{\oplus\mathtt{a}_{Q,\mathbf{d},i}}$

and

$\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}=\bigoplus_{i\geq
0}\mathscr{L}^{-1/2-i}.$

In particular, the compactly supported cohomology of
$\mathfrak{M}(\Pi_{Q,\upmu})$, the stack of representations of the deformed
preprojective algebra, is pure, of Tate type.

See §3.3 for the proof of Theorems A and B.

###  1.4. Acknowledgments

The suggestion that partial fermionization of BPS Lie algebras should be
modelled by the passage from
$\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^{1}}(-2)\oplus\mathcal{O}_{\mathbb{P}^{1}})$
to
$\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^{1}}(-1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1))$
is based on conversations with, and unpublished work of Kevin Costello. I am
very grateful to him for being so generous with his time and ideas. I learnt
about connections between the various Jacobi algebras appearing in this paper
from Victor Ginzburg, and I am similarly grateful to him. Thanks also to Hans
Franzen, Tudor Pădurariu and Markus Reineke for stimulating conversations
about DT theory. My thanks also go to Tommaso Scognamiglio for questions that
led to the addition of §3.3 to an earlier draft.

This paper was written for the British Mathematical Colloquium, held
(virtually) in Glasgow in 2021. Many thanks are due to the organisers for
managing to run such an engaging online event. During the writing of the
paper, I was supported by the starter grant “Categorified Donaldson–Thomas
theory” No. 759967 of the European Research Council, and also supported by a
Royal Society university research fellowship.

##  2\. Cohomological DT theory for quivers with potential

###  2.1. Some algebras from quivers

A quiver is determined by a set of vertices $Q_{0}$, a set of edges $Q_{1}$,
and two morphisms $s,t\colon Q_{1}\rightarrow Q_{0}$ taking an arrow to its
source and target respectively. We always assume that $Q_{0}$ and $Q_{1}$ are
finite. We define the Euler form

(18) $\displaystyle\upchi_{Q}(\cdot,\cdot)\colon$
$\displaystyle\mathbb{Z}^{Q_{0}}\times\mathbb{Z}^{Q_{0}}\rightarrow\mathbb{Z}$
$\displaystyle(\mathbf{d},\mathbf{e})\mapsto\sum_{a\in
Q_{1}}\mathbf{d}_{s(a)}\mathbf{d}_{t(a)}-\sum_{i\in
Q_{0}}\mathbf{d}_{i}\mathbf{e}_{i}.$

Where there is no possibility of confusion, we drop the quiver $Q$ from the
notation and just write $\upchi(\cdot,\cdot)$. We denote by $\overline{Q}$ the
doubled quiver of $Q$, obtained by adding an arrow $a^{*}$ for every arrow
$a\in Q_{1}$, where $a^{*}$ has the opposite orientation to $a$. We denote by
$\tilde{Q}$ the tripled quiver, obtained from $\overline{Q}$ by adding a loop
$\omega_{i}$ at each vertex $i\in Q_{0}$.

Given a ring $A$ and a quiver $Q$ we denote by $AQ$ the free path algebra of
$Q$ with coefficients in $A$. We denote by $R\subset\mathbb{C}Q$ the
semisimple subalgebra spanned by length zero paths, so we may identify
$R=\mathbb{C}^{Q_{0}}$. We denote by $\Pi_{Q}$ the preprojective algebra for
$Q$, defined to be the quotient of the free path algebra
$\mathbb{C}\overline{Q}$ by the two-sided ideal generated by the element
$\sum_{a\in Q_{1}}[a,a^{*}]$. As in the introduction, we denote by
$\Pi_{Q}[\omega]$ the trivial extension obtained by adjoining a central
element $\omega$ to the algebra $\Pi_{Q}$, i.e.
$\Pi_{Q}[\omega]=\Pi_{Q}\otimes\mathbb{C}[\omega]$.

Let $\upmu\in R$. We recall the central extension of $\Pi_{Q}$ introduced by
Etingof and Rains [ER06]:

$\Pi_{Q}^{\upmu}=\mathbb{C}[\omega]\overline{Q}/\langle\sum_{a\in
Q_{1}}[a,a^{*}]+\upmu\omega\rangle.$

There is an obvious isomorphism

$\Pi_{Q}^{0}\cong\Pi_{Q}[\omega]$

and natural isomorphisms

$\displaystyle\Pi_{Q,\upmu}\cong$
$\displaystyle\Pi_{Q}^{\upmu}/\langle\omega-1\rangle$
$\displaystyle\Pi_{Q}\cong$
$\displaystyle\Pi_{Q}^{\upmu}/\langle\omega\rangle,$

where $\Pi_{Q,\upmu}$ is the deformed preprojective algebra recalled in the
introduction. The algebra $\Pi_{Q}^{\upmu}$ provides an
$\mathbb{A}^{1}$-family of algebras interpolating between the preprojective
algebra $\Pi_{Q}$ and the deformed preprojective algebra $\Pi_{Q,\upmu}$.

Let $Q$ be a quiver and let $W\in\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]$ be a
potential, i.e. a linear combination of cyclic words, where cyclic words are
considered to be equivalent if they can be cyclically permuted to each other.
We will call the data of a quiver with potential $(Q,W)$ a QP. Given $a\in
Q_{1}$, if $W=a_{1}\ldots a_{n}$ is a single cyclic word we define

$\partial W/\partial a=\sum_{a_{m}=a}a_{m+1}a_{m+2}\ldots a_{n}a_{1}\ldots
a_{m-1}$

and define $\partial W/\partial a$ for general $W$ by extending linearly. We
define

$\operatorname{Jac}(Q,W)=\mathbb{C}Q/\langle\partial W/\partial
a\;\lvert\;a\in Q_{1}\rangle.$

In this paper we will study Jacobi algebras obtained from the tripled QP
$(\tilde{Q},\tilde{W})$ defined in (5) by adding polynomials in the extra
loops $\omega_{i}$; we refer the reader to [Gin06, Sec.4] for general
background on this construction, [Sze08b, VB10] for the noncommutative
geometry background in type ADE, and [CKV01] for the physics perspective.

######  Proposition 2.1.

[Gin06, Ex.4.3.5] Let $\upmu=\sum_{i}\upmu_{i}e_{i}$. Set
$\tilde{W}^{\upmu}=\tilde{W}+\frac{1}{2}\sum_{i\in
Q_{0}}\upmu_{i}\omega_{i}^{2}$. Then there is an isomorphism

$\Pi_{Q}^{\upmu}\cong\operatorname{Jac}(\tilde{Q},\tilde{W}^{\upmu}).$

In particular there is a natural isomorphism

$\Pi_{Q}[\omega]\cong\operatorname{Jac}(\tilde{Q},\tilde{W}).$

###### Proof.

This follows more or less from the definitions. The noncommutative derivatives
of $\tilde{W}^{\upmu}$ with respect to the arrows $a$ impose the relation that
$\omega$ commutes with the arrows $a^{*}$, and vice versa, while the
noncommutative derivatives with respect to the loops $\omega_{i}$ impose the
defining relations of $\Pi^{\upmu}_{Q}$ as a quotient of
$\mathbb{C}[\omega]\overline{Q}$. ∎

######  Example 2.2.

Let $Q$ be defined as in (10), and set $\upmu=e_{0}-e_{1}$. Then

$\tilde{W}^{\upmu}=\omega_{0}(a^{*}a-bb^{*})-\omega_{1}(aa^{*}-b^{*}b)+\frac{1}{2}(\omega_{0}^{2}-\omega_{1}^{2}).$

After the noncommutative change of variables

$\displaystyle\omega_{0}$ $\displaystyle\mapsto\omega_{0}-a^{*}a+bb^{*}$
$\displaystyle\omega_{1}$ $\displaystyle\mapsto\omega_{1}-aa^{*}+b^{*}b$

the potential transforms to

$W=\frac{1}{2}(\omega_{0}^{2}-\omega_{1}^{2})+b^{*}baa^{*}-bb^{*}a^{*}a$

Now the relations $\partial W/\partial\omega_{i}=\pm\omega_{i}$ mean that in
the Jacobi algebra we may simply remove the loops $\omega_{i}$. Thus there is
a natural isomorphism

$\operatorname{Jac}(\tilde{Q},W)\cong\operatorname{Jac}(Q_{\operatorname{con}},W_{\operatorname{\mathtt{KW}}}),$

giving the isomorphism (12). In particular, the noncommutative conifold is
isomorphic to one of the central extensions of $\Pi_{Q}$ considered above, for
$Q$ as in (10).

Later we will use the following elementary result.

######  Proposition 2.3.

Let $\rho$ be a $\mathbf{d}$-dimensional simple
$\operatorname{Jac}(\tilde{Q},\tilde{W}^{\upmu})$-module. Then the operator
$\rho(\omega)$ acts on the underlying vector space of $\rho$ by multiplication
by some scalar $\lambda\in\mathbb{C}$, and if $\upmu\cdot\mathbf{d}\neq 0$
then $\lambda=0$.

###### Proof.

The first part follows from the fact that $\omega=\sum_{i\in Q_{0}}\omega_{i}$
is central in $\operatorname{Jac}(\tilde{Q},\tilde{W}^{\upmu})$, so each
eigenspace of $\rho(\omega)$ is preserved by the action of
$\operatorname{Jac}(\tilde{Q},\tilde{W}^{\upmu})$. For the second part,
consider the relation in $\operatorname{Jac}(\tilde{Q},\tilde{W}^{\upmu})$

$0=\sum_{i\in Q_{0}}\partial\tilde{W}^{\upmu}/\partial\omega_{i}=\sum_{i\in
Q_{0}}\upmu_{i}\omega_{i}+\sum_{a\in Q_{1}}[a,a^{*}].$

Applying $\rho$ and taking the trace, the final sum vanishes, and we find

$\displaystyle 0=$ $\displaystyle\sum_{i\in
Q_{0}}\upmu_{i}\lambda\operatorname{Tr}(\operatorname{Id}_{\mathbf{d}_{i}\times\mathbf{d}_{i}})$
$\displaystyle=$ $\displaystyle\lambda\upmu\cdot\mathbf{d}$

as required. ∎

###  2.2. Moduli spaces of quiver representations

Given a quiver $Q$ and a dimension vector $\mathbf{d}\in\mathbb{N}^{Q_{0}}$ we
set

$\displaystyle\mathbb{A}_{\mathbf{d}}(Q)\coloneqq$ $\displaystyle\prod_{a\in
Q_{1}}\operatorname{Hom}(\mathbb{C}^{\mathbf{d}_{s(a)}},\mathbb{C}^{\mathbf{d}_{t(a)}})$
$\displaystyle\operatorname{GL}_{\mathbf{d}}\coloneqq$
$\displaystyle\prod_{i\in
Q_{0}}\operatorname{GL}_{\mathbf{d}_{i}}(\mathbb{C}).$

The group $\operatorname{GL}_{\mathbf{d}}$ acts on
$\mathbb{A}_{\mathbf{d}}(Q)$ via change of basis. We denote by
$\mathfrak{M}_{\mathbf{d}}(Q)$ the stack of $\mathbf{d}$-dimensional
$\mathbb{C}Q$-modules. There is an isomorphism of stacks

$\mathfrak{M}_{\mathbf{d}}(Q)\cong\mathbb{A}_{\mathbf{d}}(Q)/\operatorname{GL}_{\mathbf{d}}.$

We denote by $\mathcal{M}_{\mathbf{d}}(Q)$ the coarse moduli space of
$\mathbf{d}$-dimensional $\mathbb{C}Q$-modules. Geometric $K$-points of
$\mathcal{M}_{\mathbf{d}}(Q)$ are in natural bijection with semisimple
$KQ$-modules. There is an isomorphism

$\mathcal{M}_{\mathbf{d}}(Q)\cong\operatorname{Spec}(\Upgamma(\mathbb{A}_{\mathbf{d}}(Q))^{\operatorname{GL}_{\mathbf{d}}}).$

We denote by
$\mathtt{JH}_{\mathbf{d}}\colon\mathfrak{M}_{\mathbf{d}}(Q)\rightarrow\mathcal{M}_{\mathbf{d}}(Q)$
the affinization morphism. Although this morphism is not projective, it is
approximated by projective maps in the sense of [DM20], meaning that
$\mathtt{JH}_{*}$ and $\mathtt{JH}_{!}$ commute with vanishing cycle functors
(introduced in the next section).

For spaces and morphisms involving a subscript $\mathbf{d}$, if we omit the
subscript, the union over all dimension vectors is intended.

There is a finite morphism [MR19]

$\oplus\colon\mathcal{M}(Q)\times\mathcal{M}(Q)\rightarrow\mathcal{M}(Q)$

which at the level of geometric points, takes a pair of $KQ$-modules to their
direct sum. Since this morphism is invariant under swapping the two factors of
$\mathcal{M}(Q)$ in the domain, and finite morphisms are exact with respect to
the perverse t structure, we obtain an induced symmetric monoidal product on
$\operatorname{\mathbf{Perv}}(\mathcal{M}(Q))$, defined by

$\mathcal{F}^{\prime}\boxtimes_{\oplus}\mathcal{F}^{\prime\prime}\coloneqq\oplus_{*}\\!\left(\mathcal{F}^{\prime}\boxtimes\mathcal{F}^{\prime\prime}\right).$

Given a potential $W\in\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]$ we form the
function $\operatorname{Tr}(W)_{\mathbf{d}}$ on $\mathbb{A}_{\mathbf{d}}(Q)$.
This is well defined and $\operatorname{GL}_{\mathbf{d}}$-invariant by cyclic
invariance of trace. As such, $\operatorname{Tr}(W)_{\mathbf{d}}$ induces
functions on $\mathfrak{M}_{\mathbf{d}}(Q)$ and $\mathcal{M}_{\mathbf{d}}(Q)$,
which we continue to denote by $\operatorname{Tr}(W)_{\mathbf{d}}$, or just
$\operatorname{Tr}(W)$ if there is no risk of ambiguity.

###  2.3. Cohomological Donaldson–Thomas theory of quivers with potential

Given a function $f$ on a smooth complex variety $X$ we define
$X_{0}=f^{-1}(0)$, and consider the diagram

$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\mathbb{A}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{A}^{1}.}$

in which the square is Cartesian. Then we define the nearby cycles functor
$\psi_{f}\colon\mathcal{D}^{\mathrm{b}}(\operatorname{\mathbf{Perv}}(X))\rightarrow\mathcal{D}^{\mathrm{b}}(\operatorname{\mathbf{Perv}}(X))$
via

$\psi_{f}=\kappa_{*}\kappa^{*}p_{*}p^{*}.$

The vanishing cycles functor $\upphi_{f}$ is defined so that for
$\mathcal{F}\in\operatorname{Ob}(\mathcal{D}^{\mathrm{b}}(\operatorname{\mathbf{Perv}}(X)))$
there is a distinguished triangle

$\kappa_{*}\kappa^{*}\mathcal{F}\rightarrow\psi_{f}\mathcal{F}\rightarrow\upphi_{f}\mathcal{F}.$

Both ${}^{\mathfrak{p}}\\!\psi_{f}\coloneqq\psi_{f}[-1]$ and
${}^{\mathfrak{p}}\\!\upphi_{f}\coloneqq\upphi_{f}[-1]$ send perverse sheaves
to perverse sheaves [KS90, Cor.10.3.13] and (naturally) commute with Verdier
duality [Mas16].

We give a lightning account of the critical cohomological Hall algebra
associated to a quiver with potential. More details can be found in [KS11,
DM20]. For a stack $\mathfrak{M}$ for which each connected component is
irreducible and generically smooth, we define the intersection complex

$\mathcal{IC}_{\mathfrak{M}}\coloneqq\coprod_{\mathfrak{N}\in\uppi\hskip
0.35002pt_{0}(\mathfrak{M})}\mathcal{IC}_{\mathfrak{N}}(\mathbb{Q}_{\mathfrak{N}^{\mathrm{sm}}}[\dim(\mathfrak{N})]),$

i.e. it is the intermediate extension of the constant perverse sheaf from the
smooth locus. We define

$\operatorname{\mathcal{RA}}_{Q,W}\coloneqq\mathtt{JH}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathcal{IC}_{\mathfrak{M}(Q)}.$

The morphism $\uppi\hskip 0.50003pt_{2}$ from (2) is proper, so that there is
a natural integration map

$\upalpha_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}\colon\uppi\hskip
0.50003pt_{2,*}\mathbb{Q}_{\mathfrak{E}\mathrm{xact}_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}(Q)}\rightarrow\mathbb{Q}_{\mathfrak{M}_{\mathbf{d}^{\prime}+\mathbf{d}^{\prime\prime}}(Q)}[-2\upchi(\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime})]$

(the shift is given by the relative dimension of $\uppi\hskip 0.50003pt_{2}$).
Composing appropriate shifts of the morphisms

$\displaystyle\oplus_{*}\mathtt{JH}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\left(\mathbb{Q}_{\mathfrak{M}(Q)\times\mathfrak{M}(Q)}\rightarrow(\uppi\hskip
0.50003pt_{1}\times\uppi\hskip
0.50003pt_{3})_{*}\mathbb{Q}_{\mathfrak{E}\mathrm{xact}}\right)$

and the sum of
$\mathtt{JH}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\upalpha_{\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime}}$
over pairs $(\mathbf{d}^{\prime},\mathbf{d}^{\prime\prime})$, and using
commutativity of vanishing cycle functors with proper and with smooth
morphisms, we obtain the morphism

$\upbeta\colon\oplus_{*}(\mathtt{JH}\times\mathtt{JH})_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathcal{IC}_{\mathfrak{M}_{\mathbf{d}^{\prime}}(Q)\times\mathfrak{M}_{\mathbf{d}^{\prime\prime}}(Q)}\rightarrow\mathtt{JH}_{*}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathcal{IC}_{\mathfrak{M}_{\mathbf{d}}(Q)}.$

Finally, composing $\upbeta$ with
$\oplus_{*}(\mathtt{JH}\times\mathtt{JH})_{*}\operatorname{\mathtt{TS}}$,
where $\operatorname{\mathtt{TS}}$ is (a shift of) the Thom–Sebastiani
isomorphism [Mas01]

$\operatorname{\mathtt{TS}}\colon{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}_{\mathbf{d}^{\prime}}(Q)}\boxtimes{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}_{\mathbf{d}^{\prime\prime}}(Q)}\xrightarrow{\cong}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathbb{Q}_{\mathfrak{M}_{\mathbf{d}^{\prime}}(Q)\times\mathfrak{M}_{\mathbf{d}^{\prime\prime}}(Q)}$

we define the (relative) Hall algebra multiplication

(19)
$\operatorname{\mathcal{RA}}_{Q,W}\boxtimes_{\oplus}\operatorname{\mathcal{RA}}_{Q,W}\rightarrow\operatorname{\mathcal{RA}}_{Q,W}.$

The cohomology

$\displaystyle\operatorname{\mathcal{A}}_{Q,W}\coloneqq$
$\displaystyle\operatorname{\mathbf{H}}\\!\left(\mathfrak{M}(Q),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathcal{IC}_{\mathfrak{M}(Q)}\right)$
$\displaystyle\cong$
$\displaystyle\operatorname{\mathbf{H}}(\mathcal{M}(Q),\operatorname{\mathcal{RA}}_{Q,W})$

has a $\mathbb{N}^{Q_{0}}$-grading by dimension vectors induced by the
decomposition
$\mathcal{M}(Q)=\coprod_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\mathcal{M}_{\mathbf{d}}(Q)$,
and the associative product induced by taking derived global sections of the
morphism (19) respects this grading.

Assume that $Q$ is symmetric333There is a version of the integrality theorem
for non-symmetric quivers, concerning vanishing cycle cohomology of stacks of
semistable $\mathbb{C}Q$-modules, but we won’t need it in this paper. See
[DM20, Thm.A Thm.C]. For $\mathbf{d}\in\mathbb{N}^{Q_{0}}$ we define

(20)
$\mathcal{BPS}_{Q,W,\mathbf{d}}\coloneqq\begin{cases}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathcal{IC}_{\mathcal{M}_{\mathbf{d}}(Q)}&\textrm{if
there is a $\mathbf{d}$-dimensional simple $\mathbb{C}Q$-module}\\\
0&\textrm{otherwise.}\end{cases}$

According to our conventions, $\mathcal{IC}_{\mathcal{M}_{\mathbf{d}}(Q)}$ is
the intermediate extension of the constant perverse sheaf
$\mathbb{Q}_{\mathcal{M}^{\operatorname{simp}}_{\mathbf{d}}(Q)}[1-\upchi(\mathbf{d},\mathbf{d})]$
on the (open, dense) subscheme of $\mathcal{M}_{\mathbf{d}}(Q)$ corresponding
to simple modules. Since $\mathcal{IC}_{\mathcal{M}_{\mathbf{d}}(Q)}$ is
Verdier self-dual, and vanishing cycle functors commute with Verdier duality
[Mas16], there are natural isomorphisms

(21)
$\mathbb{D}\mathcal{BPS}_{Q,W,\mathbf{d}}\cong\mathcal{BPS}_{Q,W,\mathbf{d}}.$

We recall the following version of the cohomological integrality theorem from
[DM20]

######  Theorem 2.4.

There is an isomorphism of bounded above complexes of perverse sheaves

$\mathtt{JH}_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathcal{IC}_{\mathfrak{M}(Q)}\cong\operatorname{\mathbf{Sym}}_{\boxtimes_{\oplus}}\left(\bigoplus_{\mathbb{N}^{Q_{0}}\ni\mathbf{d}\neq
0}\mathcal{BPS}_{Q,W,\mathbf{d}}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right)$

where

$\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\cong\bigoplus_{i\in\mathbb{Z}_{\geq
0}}\mathbb{Q}[1+2i].$

We define the cohomological BPS invariants

$\operatorname{BPS}_{Q,W,\mathbf{d}}\coloneqq\operatorname{\mathbf{H}}(\mathcal{M}_{\mathbf{d}}(Q),\mathcal{BPS}_{Q,W,\mathbf{d}}).$

Applying the compactly supported cohomology functor to Theorem 2.4, and using
self-Verdier duality (21) of $\mathcal{BPS}_{Q,W,\mathbf{d}}$, yields

$\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\operatorname{\mathbf{H}}_{c}(\mathfrak{M}_{\mathbf{d}}(Q),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(W)}\mathcal{IC}_{\mathfrak{M}_{\mathbf{d}}(Q)})\cong\operatorname{\mathbf{Sym}}\left(\bigoplus_{\mathbb{N}^{Q_{0}}\ni\mathbf{d}\neq
0}\operatorname{BPS}^{\vee}_{Q,W,\mathbf{d}}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right).$

The BPS invariants of the Jacobi algebra $\operatorname{Jac}(Q,W)$ are defined
via

$\displaystyle\omega_{Q,W,\mathbf{d}}=$
$\displaystyle\upchi(\operatorname{\mathbf{H}}(\mathcal{M}_{\mathbf{d}}(Q),\mathcal{BPS}_{Q,W,\mathbf{d}}))$
$\displaystyle\coloneqq$
$\displaystyle\sum_{i\in\mathbb{Z}}(-1)^{i}\dim\\!\left(\operatorname{\mathbf{H}}^{i}\\!\left(\mathcal{M}_{\mathbf{d}}(Q),\mathcal{BPS}_{Q,W,\mathbf{d}}\right)\right)$
$\displaystyle=$
$\displaystyle\upchi(\operatorname{\mathbf{H}}_{c}(\mathcal{M}_{\mathbf{d}}(Q),\mathcal{BPS}_{Q,W,\mathbf{d}}))$

where the final identity again follows from Verdier self-duality of the BPS
sheaf. Turning to Verdier duals, we have instead

######  Theorem 2.5.

[DM20] There is an isomorphism of unbounded complexes of perverse sheaves

$\operatorname{\mathcal{RA}}_{Q,W}\cong\operatorname{\mathbf{Sym}}_{\boxtimes_{\oplus}}\left(\bigoplus_{\mathbb{N}^{Q_{0}}\ni\mathbf{d}\neq
0}\mathcal{BPS}_{Q,W,\mathbf{d}}\otimes\operatorname{\mathbf{H}}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right)$

where

$\operatorname{\mathbf{H}}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}=\bigoplus_{i\in\mathbb{Z}_{\geq
0}}\mathbb{Q}[-1-2i],$

so that

${}^{\mathfrak{p}}\\!\operatorname{\mathcal{H}}^{1}(\operatorname{\mathcal{RA}}_{Q,W})=({}^{\mathfrak{p}}\uptau^{\leq
1}\operatorname{\mathcal{RA}}_{Q,W})[1]\cong\bigoplus_{\mathbb{N}^{Q_{0}}\ni\mathbf{d}\neq
0}\mathcal{BPS}_{Q,W,\mathbf{d}}.$

Applying the natural transformation ${}^{\mathfrak{p}}\uptau^{\leq
1}\rightarrow\operatorname{id}$ to $\operatorname{\mathcal{RA}}_{Q,W}$ and
taking hypercohomology, there is a natural inclusion

$\mathfrak{g}_{Q,W}\coloneqq\operatorname{BPS}_{Q,W}[-1]=\operatorname{\mathbf{H}}(\mathcal{M}(Q),\mathcal{BPS}_{Q,W})[-1]\hookrightarrow\operatorname{\mathcal{A}}_{Q,W}.$

The image of this inclusion is closed under the commutator444Strictly
speaking, for this part of the theorem to be true, we need to twist the
symmetric monoidal structure on $\mathbb{N}^{Q_{0}}$-graded, cohomologically
graded vector spaces by a sign, over and above the Koszul sign rule (see
[DM20, Sec.1.6, Sec.6.1]). Thankfully for the quiver $\tilde{Q}$ this sign is
always $+$ (see [Dav20, Rem[2.3]). Lie bracket induced by the associative
algebra structure on $\operatorname{\mathcal{A}}_{Q,W}$.

The resulting Lie algebra $\mathfrak{g}_{Q,W}$ is called the BPS Lie algebra
for the pair $(Q,W)$, see [DM20] for more details.

###  2.4. Hodge theoretic BPS invariants

We give another lightning introduction, this time to Hodge theoretic DT
theory, via monodromic mixed Hodge modules. For more details, we refer the
reader to [KS11, DM20], and for a comparison with the treatment of monodromic
mixed Hodge modules in [Sai], we refer the reader to [DM20, Sec.2]. Mixed
Hodge structures are important in the subject of refined DT theory, since the
extra $q$-variable appearing in refined DT theory keeps track of the weight
filtration on certain mixed Hodge structures. On the other hand, our main
result states that all mixed Hodge structures appearing in this paper are
pure, so that weight polynomials can be replaced by Poincaré polynomials. The
takeaway is that this section can be skimmed by the reader that is happy to
use the purity part of Theorem A to identify the refined BPS invariants of
$\operatorname{Jac}(\tilde{Q},\tilde{W}^{\upmu})$ with the Poincaré
polynomials of the BPS cohomology.

For $X$ a variety we denote by $\operatorname{\mathbf{MHM}}(X)$ the category
of mixed Hodge modules on $X$. There is an equivalence of categories between
$\operatorname{\mathbf{MHM}}(\operatorname{pt})$ and the category of graded-
polarizable mixed Hodge structures. Let $\mathcal{B}_{X}$ denote the full
subcategory of $\operatorname{\mathbf{MHM}}(X\times\mathbb{A}^{1})$ containing
those mixed Hodge modules $\mathcal{F}$ such that for each $x\in X$ and
$i\in\mathbb{Z}$ the mixed Hodge modules

(22)
$\operatorname{\mathcal{H}}^{i}\\!\left((\\{x\\}\times\mathbb{A}^{1}\rightarrow
X\times\mathbb{A}^{1})^{*}\mathcal{F}\right)$

are locally constant away from $x\times\\{0\\}$. We denote by
$\mathcal{C}_{X}$ the full subcategory of $\mathcal{B}_{X}$ containing those
$\mathcal{F}$ such that each (22) is constant. Equivalently, we may define
$\mathcal{C}_{X}$ as the essential image of $\uppi\hskip
0.50003pt_{X}^{*}[1]$, for $\uppi\hskip 0.50003pt_{X}\colon
X\times\mathbb{A}^{1}\rightarrow X$ the projection. We denote by
$\operatorname{\mathbf{MMHM}}(X)$ the Serre quotient
$\mathcal{B}_{X}/\mathcal{C}_{X}$. There is an embedding of categories
$\operatorname{\mathbf{MHM}}(X)\hookrightarrow\operatorname{\mathbf{MMHM}}(X)$
defined via

$(X\times\\{0\\}\hookrightarrow
X\times\mathbb{A}^{1})_{*}\colon\operatorname{\mathbf{MHM}}(X)\rightarrow\operatorname{\mathbf{MHM}}(X\times\mathbb{A}^{1}).$

The direct image functor

$\Theta\coloneqq(X\times\mathbb{C}^{*}\hookrightarrow
X\times\mathbb{A}^{1})_{*}$

induces an equivalence of categories between the category of mixed Hodge
modules on $X\times\mathbb{C}^{*}$ with locally constant cohomology sheaves
after restriction to each $\\{x\\}\times\mathbb{C}^{*}$ and the category of
monodromic mixed Hodge modules on $X$. Denoting by $\Theta^{-1}$ an inverse
equivalence, there is a faithful forgetful functor

$\operatorname{\mathbf{rat}}\circ(X\xrightarrow{x\mapsto(x,1)}X\times\mathbb{C}^{*})^{*}\circ\Theta^{-1}[-1]$

taking a monodromic mixed Hodge module to its underlying perverse sheaf on
$X$. We abuse notation by denoting this functor also by
$\operatorname{\mathbf{rat}}$. For $X$ a variety, we denote by
$\underline{\mathbb{Q}}_{X}$ the lift of the constant sheaf $\mathbb{Q}_{X}$
to a complex of mixed Hodge modules on $X$. For $f$ a regular function on $X$
we define the vanishing cycles functor

$\displaystyle\upphi^{\operatorname{mon}}_{f}\colon$
$\displaystyle\operatorname{\mathbf{MHM}}(X)\rightarrow\operatorname{\mathbf{MMHM}}(X)$
$\displaystyle\mathcal{F}\mapsto\Theta\underline{\upphi}_{u\cdot
f}(\mathcal{F}\boxtimes\underline{\mathbb{Q}}_{\mathbb{C}^{*}})[1]$

where $u$ is the coordinate on $\mathbb{C}^{*}$ and
$\underline{\upphi}_{u\cdot f}$ is the lift of
${}^{\mathfrak{p}}\\!\upphi_{u\cdot f}$ to the categories of mixed Hodge
modules. There is a natural isomorphism
$\operatorname{\mathbf{rat}}\upphi^{\operatorname{mon}}_{f}\cong{}^{\mathfrak{p}}\\!\upphi_{f}\operatorname{\mathbf{rat}}$.

An object $\mathcal{F}\in\operatorname{Ob}(\operatorname{\mathbf{MMHM}}(X))$
inherits a weight filtration from the weight filtration on objects of
$\operatorname{\mathbf{MHM}}(X\times\mathbb{A}^{1})$. We say that
$\mathcal{F}$ is pure of weight $n$ if the associated graded object with
respect to this filtration is concentrated in degree $n$. We say that an
object
$\mathcal{F}\in\operatorname{Ob}(\mathcal{D}^{\mathrm{b}}(\operatorname{\mathbf{MMHM}}(X)))$
is pure if each $\operatorname{\mathcal{H}}^{i}(\mathcal{F})$ is pure of
weight $i$.

The cohomologically graded mixed Hodge structure
$\mathscr{L}=\operatorname{\mathbf{H}}_{c}(\mathbb{A}^{1},\mathbb{Q})$ is
pure: it is concentrated in cohomological degree two, and is pure of weight
two. This object has a tensor square root in
$\mathcal{D}^{\mathrm{b}}(\operatorname{\mathbf{MMHM}}(\operatorname{pt}))$
provided by

$\mathscr{L}^{1/2}\coloneqq\operatorname{cone}(\underline{\mathbb{Q}}_{\mathbb{A}_{1}}\rightarrow
d_{*}\underline{\mathbb{Q}}_{\mathbb{A}_{1}})$

where $d\colon\mathbb{A}^{1}\rightarrow\mathbb{A}^{1}$ is the morphism
$z\mapsto z^{2}$. We say a monodromic mixed Hodge structure is of Tate type if
it is a direct sum of (possibly negative) tensor powers of the monodromic
mixed Hodge structure $\mathscr{L}^{1/2}[1]$.

For $X$ an irreducible variety we denote by

$\mathcal{IC}^{\operatorname{hdg}}_{X}\coloneqq\mathcal{IC}_{X}\left(\underline{\mathbb{Q}}_{X^{\mathrm{sm}}}[\dim
X]\right)\otimes\mathscr{L}^{-\dim(X)/2}[-\dim X]$

the natural lift of $\mathcal{IC}_{X}$ to a pure weight zero monodromic mixed
Hodge module.

We define

(23)
$\mathcal{BPS}^{\operatorname{hdg}}_{Q,W,\mathbf{d}}\coloneqq\begin{cases}\upphi^{\operatorname{mon}}_{\operatorname{Tr}(W)}\mathcal{IC}^{\operatorname{hdg}}_{\mathcal{M}_{\mathbf{d}}(Q)}&\textrm{if
there is a $\mathbf{d}$-dimensional simple $\mathbb{C}Q$-module}\\\
0&\textrm{otherwise.}\end{cases}$

This is the natural lift of the BPS sheaf to a monodromic mixed Hodge module,
i.e.
$\operatorname{\mathbf{rat}}(\mathcal{BPS}^{\operatorname{hdg}}_{Q,W,\mathbf{d}})\cong\mathcal{BPS}_{Q,W,\mathbf{d}}$.
Similarly we define the monodromic mixed Hodge structure

$\operatorname{BPS}^{\operatorname{hdg}}_{Q,W,\mathbf{d}}\coloneqq\operatorname{\mathbf{H}}(\mathcal{M}_{\mathbf{d}}(Q),\mathcal{BPS}^{\operatorname{hdg}}_{Q,W,\mathbf{d}})$

satisfying
$\operatorname{\mathbf{rat}}(\operatorname{BPS}^{\operatorname{hdg}}_{Q,W,\mathbf{d}})\cong\operatorname{BPS}_{Q,W,\mathbf{d}}$.
We define555Since there is not a fully developed theory of mixed Hodge modules
for stacks, some care has to be taken care with this definition. See [DM20]
for the details.

$\operatorname{\mathcal{RA}}_{Q,W}^{\operatorname{hdg}}=\mathtt{JH}_{*}\upphi^{\operatorname{mon}}_{\operatorname{Tr}(W)}\mathcal{IC}_{\mathfrak{M}(Q)}^{\operatorname{hdg}}.$

Since all of the natural transformations defining the multiplication on the
Hall algebra $\operatorname{\mathcal{A}}_{Q,W}$ lift to categories of
monodromic mixed Hodge modules [KS11, Dav17a, DM20], as does the
Thom–Sebastiani theorem [Sai], we may define a multiplication on
$\operatorname{\mathcal{RA}}_{Q,W}^{\operatorname{hdg}}$ that recovers the
multiplication on $\operatorname{\mathcal{RA}}_{Q,W}$ after applying the
functor $\operatorname{\mathbf{rat}}$. Likewise, taking derived direct image
to a point we obtain the algebra object
$\operatorname{\mathcal{A}}^{\operatorname{hdg}}_{Q,W}$ in monodromic mixed
Hodge structures. Then by [DM20], Theorems 2.5 and 2.4 lift to the categories
of monodromic mixed Hodge modules and monodromic mixed Hodge structures. In
particular, the BPS Lie algebra
$\mathfrak{g}_{Q,W}=\operatorname{BPS}_{Q,W}[-1]$ lifts to a Lie algebra
object

$\mathfrak{g}^{\operatorname{hdg}}_{Q,W}\coloneqq\operatorname{BPS}^{\operatorname{hdg}}_{Q,W}\otimes\mathscr{L}^{1/2}$

inside
$\mathcal{D}^{\mathrm{b}}(\operatorname{\mathbf{MMHM}}(\operatorname{pt}))$.
This is a Lie subalgebra of
$\operatorname{\mathcal{A}}_{Q,W}^{\operatorname{hdg}}$, considered as a Lie
algebra in the category of $\mathbb{N}^{Q_{0}}$-graded, cohomologically graded
monodromic mixed Hodge structures, via the commutator Lie bracket. See
[Dav17a, DM20] for full details.

###  2.5. Cohomological Donaldson–Thomas theory for preprojective algebras

In this section we restrict our attention to “tripled” QPs of the form
$(\tilde{Q},\tilde{W})$, as in the introduction. Firstly, we recall the
following purity result on the BPS cohomology of the Jacobi algebra
$\operatorname{Jac}(\tilde{Q},\tilde{W})$:

######  Theorem 2.6.

[Dav17b] For an arbitrary quiver $Q$ and dimension vector
$\mathbf{d}\in\mathbb{N}^{Q_{0}}$, the mixed Hodge structure
$\operatorname{BPS}^{\operatorname{hdg}}_{\tilde{Q},\tilde{W},\mathbf{d}}$ is
pure, of Tate type. In addition (or as a consequence of the cohomological
integrality theorem) the mixed Hodge structure on

$\operatorname{\mathbf{H}}_{c}(\mathfrak{M}_{\mathbf{d}}(\tilde{Q}),\upphi^{\operatorname{mon}}_{\operatorname{Tr}(\tilde{W})}\mathcal{IC}^{\operatorname{hdg}}_{\mathfrak{M}_{\mathbf{d}}(\tilde{Q})})$

is pure, of Tate type.

We denote by

(24)
$\tilde{e}\colon\mathcal{M}(\Pi_{Q})\times\mathbb{A}^{1}\rightarrow\mathcal{M}(\tilde{Q})$

the closed embedding that sends a pair $(\rho,t)$ to the
$\mathbb{C}\tilde{Q}$-module $\rho^{\prime}$ for which the action of the
arrows $a,a^{*}\in\overline{Q}$ are the same as for $\rho$, and the action of
each $\rho^{\prime}(\omega_{i})$ is given by multiplication by $t$.

We will need the following result on the support and equivariance of the BPS
sheaf itself:

######  Lemma 2.7.

[Dav17b] For a quiver $Q$ and dimension vector
$\mathbf{d}\in\mathbb{N}^{Q_{0}}$ there is a perverse sheaf

$\mathcal{BPS}_{\Pi_{Q},\mathbf{d}}\in\operatorname{\mathbf{Perv}}(\mathcal{M}_{\mathbf{d}}(\Pi_{Q}))$

such that there is an isomorphism

$\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}\cong\tilde{e}_{!}\left(\mathcal{BPS}_{\Pi_{Q},\mathbf{d}}\boxtimes\mathcal{IC}_{\mathbb{A}^{1}}\right).$

The same result holds at the level of monodromic mixed Hodge modules.

In words, the theorem says that the BPS sheaf is supported on the subspace of
$\mathbb{C}\tilde{Q}$-modules for which all of the generalised eigenvalues of
the operators $\rho(\omega_{i})$ are the same, and the sheaf is moreover
equivariant for the $\mathbb{A}^{1}$-action that acts by adding scalar
multiples of the identity to all of the operators $\rho(\omega_{i})$
simultaneously.

We recall from [Dav17b] the description of the BPS cohomology of the Jacobi
algebra $\operatorname{Jac}(\tilde{Q},\tilde{W})$ in terms of Kac polynomials:

######  Theorem 2.8.

The Poincaré polynomials of the cohomological BPS invariants for the QP
$(\tilde{Q},\tilde{W})$ satisfy

(25)
$\operatorname{\mathtt{p}}(\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}},q^{1/2})=q^{-1/2}\mathtt{a}_{Q,\mathbf{d}}(q^{-1})$

where

$\mathtt{a}_{Q,\mathbf{d}}(q)=\sum_{i\in\mathbb{Z}_{\geq
0}}\mathtt{a}_{Q,\mathbf{d},i}\>q^{i}$

is the Kac polynomial, counting the number of isomorphism classes of
absolutely666A module is called absolutely indecomposable if it remains
indecomposable after extending scalars to the algebraic closure
$\overline{\mathbb{F}}_{q}$. indecomposable $\mathbb{F}_{q}Q$-modules for
$\mathbb{F}_{q}$ a field of order $q$. Furthermore the natural mixed Hodge
structure on $\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}$ is pure, of
Tate type, so that we can write

(26)
$\operatorname{BPS}^{\operatorname{hdg}}_{\tilde{Q},\tilde{W},\mathbf{d}}\cong\bigoplus_{i\in\mathbb{Z}_{\geq
0}}(\mathscr{L}^{i-1/2})^{\oplus\mathtt{a}_{Q,\mathbf{d},-i}}$

and so

(27)
$\mathfrak{g}^{\operatorname{hdg}}_{\tilde{Q},\tilde{W},\mathbf{d}}\cong\bigoplus_{i\in\mathbb{Z}_{\geq
0}}(\mathscr{L}^{i})^{\oplus\mathtt{a}_{Q,\mathbf{d},-i}}.$

##  3\. Deformed dimensional reduction and proofs of main results

###  3.1. Deformed dimensional reduction

The main tool in proving Theorem A will be deformed dimensional reduction, as
introduced in joint work with Tudor Pădurariu [DP21]. This is a geometric
result about vanishing cycle functors for functions satisfying certain
$\mathbb{C}^{*}$-equivariance properties. We state the version that we need
below.

######  Theorem 3.1.

[DP21, Thm.1.3] Let the algebraic group $G$ act on a variety $X$ and affine
space $\mathbb{A}^{n}$. Assume that $\mathbb{A}^{n}$ is also given a
$\mathbb{C}^{*}$-action, with non-negative weights, which commutes with the
$G$-action. Let $\mathbb{C}^{*}$ act on $\overline{X}=X\times\mathbb{A}^{n}$
via the product of the given action on $\mathbb{A}^{n}$ with the trivial
action on $X$. Let $g$ be a function on $\overline{X}$ that is $G$-invariant
and $\mathbb{C}^{*}$-semi-invariant, with strictly positive weight. Assume
that we are given a $G\times\mathbb{C}^{*}$-equivariant decomposition
$\mathbb{A}^{n}=\mathbb{A}^{m}\times\mathbb{A}^{n-m}$, and that we can write

$g=g_{0}+\sum_{1\leq j\leq m}g_{j}t_{j}$

where the functions $g_{0},\ldots g_{m}$ are pulled back from
$X\times\mathbb{A}^{n-m}$ and $t_{1},\ldots,t_{m}$ are a system of coordinates
for $\mathbb{A}^{m}$. Let $Z\subset X\times\mathbb{A}^{n-m}$ be the vanishing
locus of the functions $g_{1},\ldots,g_{m}$. Then $Z$ is $G$-invariant. Set
$\overline{Z}=Z\times\mathbb{A}^{m}\subset\overline{X}$. We denote by

$\displaystyle\uppi\hskip 0.50003pt\colon$
$\displaystyle\overline{X}\rightarrow X$ $\displaystyle q\colon$
$\displaystyle X\times\mathbb{A}^{n-m}\rightarrow X$

the natural projections. Then the natural transformation

(28) $\uppi\hskip
0.50003pt_{!}{}^{\mathfrak{p}}\\!\upphi_{g}\mathbb{Q}_{\overline{X}/G}\rightarrow\uppi\hskip
0.50003pt_{!}{}^{\mathfrak{p}}\\!\upphi_{g_{0}}\mathbb{Q}_{\overline{Z}/G}\cong
q_{!}{}^{\mathfrak{p}}\\!\upphi_{g_{0}}\mathbb{Q}_{Z/G}[-2m]$

is an isomorphism.

Since the functor $\operatorname{\mathbf{rat}}$ is faithful, the same
statement is true at the level of (monodromic) mixed Hodge modules: the
natural transformation

$\uppi\hskip
0.50003pt_{!}\upphi^{\operatorname{mon}}_{g}\underline{\mathbb{Q}}_{\overline{X}/G}\rightarrow\uppi\hskip
0.50003pt_{!}\upphi^{\operatorname{mon}}_{g_{0}}\underline{\mathbb{Q}}_{\overline{Z}/G}\cong
q_{!}\upphi^{\operatorname{mon}}_{g_{0}}\underline{\mathbb{Q}}_{Z/G}\otimes\mathscr{L}^{m}$

is an isomorphism in777The derived category of monodromic mixed Hodge modules
on this global quotient stack is defined following e.g. [Ach].
$\mathcal{D}^{\mathrm{b}}(\operatorname{\mathbf{MMHM}}(X/G))$.

###  3.2. Proof of Theorem A

We proceed by applying deformed dimensional reduction to the function
$\operatorname{Tr}(\tilde{W}^{\upmu})$.

###### Proof of Theorem A.

Given a quiver $Q$ we denote by $Q^{+}$ the quiver obtained by adding a loop
$\omega_{i}$ at each vertex $i\in Q_{0}$, or equivalently the quiver obtained
by removing all of the arrows $a^{*}$ from $\tilde{Q}$. We denote by
$\mathfrak{I}(Q)$ the stack of pairs $(\rho,f)$, consisting of a
$\mathbb{C}Q$-module $\rho$ and an endomorphism
$f\in\operatorname{Hom}_{\mathbb{C}Q}(\rho,\rho)$. Then we consider the
commutative diagram

$\textstyle{\mathfrak{I}(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathfrak{M}(\tilde{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\mathtt{JH}}}$$\scriptstyle{p}$$\textstyle{\mathfrak{M}(Q^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathtt{JH}^{+}}$$\scriptstyle{q}$$\textstyle{\mathfrak{M}(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathtt{JH}}$$\textstyle{\mathcal{M}(\tilde{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{\mathcal{M}(Q^{+})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\prime}}$$\textstyle{\mathcal{M}(Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\uptau}$$\textstyle{\mathcal{M}(\Pi_{Q})\times\mathbb{A}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{e}}$$\scriptstyle{h}$$\textstyle{\mathcal{M}(Q)\times\mathbb{A}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\textstyle{\operatorname{pt}.}$

In the above diagram, $i$ is the natural embedding of stacks, sending a pair
$(\rho,f)$ to the $\mathbb{C}Q^{+}$-module $\rho^{\prime}$ for which the
underlying $\mathbb{C}Q$-module is $\rho$, and the action of the loops
$\omega_{i}$ is given by $f$. All of the horizontal arrows are the natural
forgetful maps. We set

$\displaystyle\uppi\hskip 0.50003pt=$ $\displaystyle qp$
$\displaystyle\uppi\hskip 0.50003pt^{\prime}=$ $\displaystyle
q^{\prime}p^{\prime}.$

We denote by $\Omega$ the quiver containing the same vertices as $Q$, and for
which the only arrows are the loops $\omega_{i}$. We denote by
$Q^{\operatorname{op}}$ the quiver containing the same vertices as $Q$, and
only the arrows $a^{*}$ for $a\in Q_{1}$, i.e. $Q^{\operatorname{op}}$ is the
opposite quiver to $Q$. In the notation of Theorem 3.1 we set

(29) $\displaystyle X=$ $\displaystyle\mathbb{A}_{\mathbf{d}}(Q)$
$\displaystyle\mathbb{A}^{m}=$
$\displaystyle\mathbb{A}_{\mathbf{d}}(Q^{\operatorname{op}})$
$\displaystyle\mathbb{A}^{n-m}=$
$\displaystyle\mathbb{A}_{\mathbf{d}}(\Omega)$ $\displaystyle G=$
$\displaystyle\operatorname{GL}_{\mathbf{d}}.$

We let $\mathbb{C}^{*}$ act on $\mathbb{A}^{m}$ and $\mathbb{A}^{n-m}$ with
weight one.

We set

$L_{\upmu}=\frac{1}{2}\sum_{i\in Q_{0}}\upmu_{i}\omega_{i}^{2}$

so $\tilde{W}^{\upmu}=\tilde{W}+L_{\upmu}$. Then
$\operatorname{Tr}(\tilde{W}+L_{\upmu})$ satisfies the conditions of Theorem
3.1, for example it is $\mathbb{C}^{*}$ semi-invariant with weight two. In the
notation of that theorem $Z\subset
X\times\mathbb{A}^{n-m}=\mathbb{A}_{\mathbf{d}}(Q^{+})$ is the locus
containing those $\mathbb{C}Q^{+}$-modules $\rho$ such that the endomorphisms
$\rho(\omega_{i})$ determine an endomorphism of the underlying
$\mathbb{C}Q$-module of $\rho$. It follows that $Z/G=\mathfrak{Z}(G)$. By
Theorem 3.1 there is an isomorphism888Note that the shift $2\sum_{a\in
Q_{1}}\mathbf{d}_{s(a)}\mathbf{d}_{t(a)}$ appearing in the definition of the
intersection complex is equal to the $2m$ appearing in (28).

(30) $\displaystyle\uppi\hskip
0.50003pt_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W}^{\upmu})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})}\cong$
$\displaystyle
q_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}\mathbb{Q}_{\mathfrak{Z}(Q)}.$

If instead we set

$\displaystyle X=$ $\displaystyle\mathbb{A}_{\mathbf{d}}(Q^{+})$
$\displaystyle\mathbb{A}^{m}=$ $\displaystyle\mathbb{A}_{\mathbf{d}}(\Omega)$

set $n=m$, and again let $\mathbb{C}^{*}$ act on $\mathbb{A}^{m}$ with weight
one, then the function $\operatorname{Tr}(\tilde{W})$ still satisfies the
conditions of Theorem 3.1 (now it is semi-invariant with weight one). So
instead we arrive at the isomorphism

(31) $\displaystyle
p_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})}\cong$
$\displaystyle\mathbb{Q}_{\mathfrak{I}(Q)}.$

Since $\mathtt{JH}^{+}$ is approximated by proper maps, there is a natural
isomorphism (see [DM20, Sec.4.1])

(32)
$\mathtt{JH}^{+}_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}\mathbb{Q}_{\mathfrak{I}(Q)}\cong{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}\mathtt{JH}^{+}_{!}\mathbb{Q}_{\mathfrak{I}(Q)}.$

Combining all of the above we can write999For the final isomorphism, see
[DM20, Prop.3.11] for the required compatibility with the symmetrizing
morphism.

$\displaystyle\uppi\hskip
0.50003pt^{\prime}_{!}\tilde{\mathtt{JH}}_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W}^{\upmu})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})}\cong$
$\displaystyle\mathtt{JH}_{!}\uppi\hskip
0.50003pt_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W}^{\upmu})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})}$
$\displaystyle\cong$
$\displaystyle\mathtt{JH}_{!}q_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}\mathbb{Q}_{\mathfrak{I}(Q)}$
$\displaystyle\textrm{by }\eqref{DR1}$ $\displaystyle\cong$
$\displaystyle\mathtt{JH}_{!}q_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}p_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})}$
$\displaystyle\textrm{by }\eqref{DR2}$ $\displaystyle\cong$ $\displaystyle
q^{\prime}_{!}\mathtt{JH}^{+}_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}p_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})}$
$\displaystyle\cong$ $\displaystyle
q^{\prime}_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}\mathtt{JH}^{+}_{!}p_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})}$
$\displaystyle\textrm{by }\eqref{ncomm}$ $\displaystyle\cong$ $\displaystyle
q^{\prime}_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}p^{\prime}_{!}\tilde{\mathtt{JH}}_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})}$
$\displaystyle\cong$ $\displaystyle
q^{\prime}_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}p^{\prime}_{!}\operatorname{\mathbf{Sym}}_{\oplus}\left(\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}\setminus
0}\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right)$
by Theorem 2.4 $\displaystyle\cong$ $\displaystyle
q^{\prime}_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}\operatorname{\mathbf{Sym}}_{\oplus}\left(\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}\setminus
0}p^{\prime}_{!}\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right)$
$\displaystyle\textrm{since }p^{\prime}\textrm{ is a monoid map}$
$\displaystyle\cong$ $\displaystyle
q^{\prime}_{!}\operatorname{\mathbf{Sym}}_{\oplus}\left(\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}\setminus
0}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}p^{\prime}_{!}\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right)$
by Thom–Sebastiani.

By Lemma 2.7 we can write

(33) $p^{\prime}_{!}\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}\cong
e_{!}(\mathcal{F}\boxtimes\mathcal{IC}_{\mathbb{A}^{1}})$

where

$\mathcal{F}=(\mathcal{M}(\Pi_{Q})\rightarrow\mathcal{M}(Q))_{!}\mathcal{BPS}_{\Pi_{Q},\mathbf{d}}.$

Writing
$f=\operatorname{Tr}(L_{\upmu})\lvert_{\mathcal{M}_{\mathbf{d}}(Q)\times\mathbb{A}^{1}}$,
we have

$\displaystyle f\colon$
$\displaystyle\mathcal{M}_{\mathbf{d}}(Q)\times\mathbb{A}^{1}\rightarrow\mathbb{C}$
$\displaystyle(\rho,t)\mapsto\frac{1}{2}(\upmu\cdot\mathbf{d})t^{2}.$

It follows from (33) that

${}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}p^{\prime}_{!}\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}\cong\begin{cases}e_{!}(\mathcal{F}\boxtimes\mathbb{Q}_{0})&\textrm{if
}\upmu\cdot\mathbf{d}\neq 0\\\
e_{!}(\mathcal{F}\boxtimes\mathcal{IC}_{\mathbb{A}^{1}})&\textrm{if
}\upmu\cdot\mathbf{d}=0,\end{cases}$

so that

$\operatorname{\mathbf{H}}_{c}(\mathcal{M}_{\mathbf{d}}(Q^{+}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(L_{\upmu})}p^{\prime}_{!}\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}})\cong\operatorname{\mathbf{H}}_{c}(\mathcal{M}_{\mathbf{d}}(\Pi_{Q}),\mathcal{BPS}_{\Pi_{Q},\mathbf{d}})[-g(\mathbf{d})]$

where we define

$g(\mathbf{d})=\begin{cases}1&\textrm{if }\upmu\cdot\mathbf{d}=0\\\
0&\textrm{if }\upmu\cdot\mathbf{d}\neq 0.\end{cases}$

Finally we deduce that

$\displaystyle\operatorname{\mathbf{H}}_{c}(\mathfrak{M}(\tilde{Q}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W}^{\upmu})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})})=$
$\displaystyle\uptau_{!}\uppi\hskip
0.50003pt^{\prime}_{!}\tilde{\mathtt{JH}}_{!}{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W}^{\upmu})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})}$
(34) $\displaystyle\cong$
$\displaystyle\operatorname{\mathbf{Sym}}\left(\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}\setminus
0}\left(\operatorname{\mathbf{H}}_{c}(\mathcal{M}_{\mathbf{d}}(\Pi_{Q}),\mathcal{BPS}_{\Pi_{Q},\mathbf{d}})\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right)[-g(\mathbf{d})]\right).$

On the other hand, by Theorem 2.4 again, we have

(35)
$\operatorname{\mathbf{H}}_{c}(\mathfrak{M}(\tilde{Q}),{}^{\mathfrak{p}}\\!\upphi_{\operatorname{Tr}(\tilde{W}^{\upmu})}\mathcal{IC}_{\mathfrak{M}(\tilde{Q})})\cong\operatorname{\mathbf{Sym}}\left(\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}\setminus
0}\operatorname{BPS}^{\vee}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right).$

Comparing (34) and (35) we find

$\displaystyle\operatorname{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}\cong$
$\displaystyle(\operatorname{\mathbf{H}}_{c}(\mathcal{M}_{\mathbf{d}}(\Pi_{Q}),\mathcal{BPS}_{\Pi_{Q},\mathbf{d}}))^{\vee}[g(\mathbf{d})]$
$\displaystyle\cong$
$\displaystyle(\operatorname{\mathbf{H}}_{c}(\mathcal{M}_{\mathbf{d}}(\tilde{Q}),\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}})[1])^{\vee}[g(\mathbf{d})]$
by Theorem 2.7 $\displaystyle\cong$
$\displaystyle\operatorname{\mathbf{H}}_{c}(\mathcal{M}_{\mathbf{d}}(\tilde{Q}),\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}})^{\vee}[g(\mathbf{d})-1]$
$\displaystyle\cong$
$\displaystyle\operatorname{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}[g(\mathbf{d})-1]$

and now (25) follows from Theorem 2.8.

Performing the same calculations in the category of monodromic mixed Hodge
structures, isomorphisms (34) and (35) yield the isomorphism of
$\mathbb{N}^{Q_{0}}$-graded complexes of monodromic mixed Hodge structures

(36)
$\operatorname{\mathbf{Sym}}\left(\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}\setminus
0}\operatorname{BPS}^{\operatorname{hdg},\vee}_{\tilde{Q},\tilde{W},\mathbf{d}}\otimes\mathscr{L}^{(g(\mathbf{d})-1)/2}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right)\cong\operatorname{\mathbf{Sym}}\left(\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}\setminus
0}\operatorname{BPS}^{\operatorname{hdg},\vee}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right)$

where

$\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}=\bigoplus_{i\geq
0}\mathscr{L}^{-1/2-i}.$

On the other hand, by Theorem 2.6 the mixed Hodge structure
$\operatorname{BPS}^{\operatorname{hdg}}_{\Pi_{Q},\mathbf{d}}$ is pure, so
that
$\operatorname{BPS}^{\operatorname{hdg},\vee}_{\Pi_{Q},\mathbf{d}}\otimes\mathscr{L}^{n/2}$
is pure for all $n\in\mathbb{Z}$. It follows that both sides of (36) are pure.
The isomorphism

$\operatorname{BPS}^{\operatorname{hdg}}_{\tilde{Q},\tilde{W},\mathbf{d}}\otimes\mathscr{L}^{(g(\mathbf{d})-1)/2}\cong\operatorname{BPS}^{\operatorname{hdg}}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}$

then follows from (36) and semisimplicity of the category of pure monodromic
mixed Hodge structures as in [DP21, Cor.7.1]. ∎

######  Example 3.2.

To recover the example that we started the paper with, consider the quiver
$Q^{(0)}$ with one vertex and no arrows. We label the unique arrow of the one-
loop quiver $Q^{(1)}=\widetilde{Q^{(0)}}$ by $\omega_{0}$. For
$\upmu=\upmu_{0}e_{0}$ to be generic we just have to pick $\upmu_{0}\neq 0$.
Then we find

$\displaystyle\operatorname{Jac}(Q^{(1)},\tilde{W}^{\upmu})=$
$\displaystyle\operatorname{Jac}(Q^{(1)},\omega_{0}^{2})$ $\displaystyle\cong$
$\displaystyle\mathbb{C}Q^{(0)}$

and so the fermionic version of
$\operatorname{\mathcal{A}}_{\widetilde{Q^{(0)}},\tilde{W}}=\operatorname{\mathcal{A}}_{Q^{(1)}}$
is indeed $\operatorname{\mathcal{A}}_{Q^{(0)}}$.

######  Example 3.3.

We return once more to the noncommutative conifold. As observed in Example
2.2, this algebra is obtained by setting $\upmu=(1,-1)$ for the quiver (10)
and considering $\operatorname{Jac}(\tilde{Q},\tilde{W}^{\upmu})$. The
cohomological BPS invariants of the QP $(\tilde{Q},\tilde{W})$ are given in
(11), while the cohomological BPS invariants of the noncommutative conifold
are given in (13). Comparing the two, there is a cohomological shift between
the respective $(m,n)$th cohomological BPS invariants precisely if $m\neq n$,
i.e. precisely if $\upmu\cdot(m,n)\neq 0$.

Recall from the introduction that the BPS invariants for the resolved conifold
are only a partially fermionized version of the BPS invariants of
$Y_{0}\times\mathbb{A}^{1}$. In view of the main result, we see that the
generic deformation in the commutative algebraic geometry context corresponds
to the deformation $\upmu=(1,-1)$. To fully fermionize the DT theory we are
obliged to work in the fully noncommutative101010I.e. with a Jacobi algebra
that is not derived equivalent to a threefold. context provided by generic
$\upmu$. Note that deformations within algebraic geometry of the Kleinian
singularity were parameterised by $\mathfrak{h}$, the Cartan subalgebra of the
(reduced) McKay graph $\Upgamma^{\prime}$, whereas noncommutative deformations
are parameterised by the Cartan subalgebra of the full McKay graph.

######  Remark 3.4.

Along the course of the proof of Theorem A we have shown that

$\uppi\hskip
0.50003pt^{\prime}_{*}\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}\cong\begin{cases}\uppi\hskip
0.50003pt^{\prime}_{*}\mathcal{BPS}_{\tilde{Q},\tilde{W}}[-1]&\textrm{if
}\upmu\cdot\mathbf{d}\neq 0\\\ \uppi\hskip
0.50003pt^{\prime}_{*}\mathcal{BPS}_{\tilde{Q},\tilde{W}}&\textrm{otherwise.}\end{cases}$

With a little effort, one may lift this statement to the level of monodromic
mixed Hodge module complexes.

###  3.3. Proof of Theorem B

Given a quiver $Q$, an element $\upmu\in R$, and a number
$n\in\mathbb{Z}_{\geq 1}$, define

$\tilde{W}^{\upmu}_{n}=\tilde{W}+\frac{1}{n}\upmu\omega^{n}.$

Then in the decomposition (29), we let $\mathbb{C}^{*}$ act with weight $1$ on
$\mathbb{A}_{\mathbf{d}}(\Omega)$, weight $n-1$ on
$\mathbb{A}_{\mathbf{d}}(Q^{\operatorname{op}})$, and trivially on
$\mathbb{A}_{\mathbf{d}}(Q)$, so
$\operatorname{Tr}(\tilde{W}^{\upmu}_{\mathbf{d}})$ satisfies the conditions
of Theorem 3.1 (now it is a weight $n$ function). Then the argument of §3.2
gives that

(37)
$\operatorname{BPS}^{\operatorname{hdg}}_{\tilde{Q},\tilde{W}^{\upmu}_{n},\mathbf{d}}\cong\operatorname{BPS}^{\operatorname{hdg}}_{\tilde{Q},\tilde{W},\mathbf{d}}\otimes\mathscr{L}^{1/2}\otimes\operatorname{\mathbf{H}}(\mathbb{A}^{1},\upphi^{\operatorname{mon}}_{(\upmu\cdot\mathbf{d})x^{n}}\mathcal{IC}_{\mathbb{A}^{1}}).$

Theorem A follows from the special case $n=2$ and Theorem 2.8, observing that
$\operatorname{\mathbf{H}}(\mathbb{A}^{1},\upphi^{\operatorname{mon}}_{x^{2}}\underline{\mathbb{Q}}_{\mathbb{A}^{1}})=\mathscr{L}^{1/2}$.

In this section we consider instead the special case $n=1$, i.e. we consider
the quiver $\tilde{Q}$ with the potential

$\tilde{W}_{1}^{\upmu}=\sum_{a\in Q_{1}}[a,a^{*}]+\upmu\omega.$

This potential is linear in the loops $\omega_{i}$, so that the following
proposition is a straightforward application of (undeformed) dimensional
reduction:

######  Proposition 3.5.

There is an isomorphism in the derived category of complexes of mixed Hodge
modules

$\uppi\hskip
0.50003pt_{!}\upphi^{\operatorname{mon}}_{\operatorname{Tr}(\tilde{W}_{1}^{\upmu})}\underline{\mathbb{Q}}_{\mathfrak{M}_{\mathbf{d}}(\tilde{Q})}\rightarrow\underline{\mathbb{Q}}_{\mathfrak{M}_{\mathbf{d}}(\Pi_{Q,\upmu})}\otimes\mathscr{L}^{\mathbf{d}\cdot\mathbf{d}}.$

###### Proof.

In the setup of Theorem 3.1, we put $X=\mathbb{A}_{\mathbf{d}}(\overline{Q})$,
$n=m$, $\mathbb{A}^{m}=\mathbb{A}_{\mathbf{d}}(\Omega)$ and
$G=\operatorname{GL}_{\mathbf{d}}$. Then $g_{0}$ is the zero function, so that
there is a natural isomorphism
$\upphi^{\operatorname{mon}}_{g_{0}}\rightarrow\operatorname{id}$, and the
result follows from Theorem 3.1 and
$\dim(\mathbb{A}_{\mathbf{d}}(\Omega))=\mathbf{d}\cdot\mathbf{d}$. ∎

###### Proof of Theorem B.

We have isomorphisms of $\mathbb{N}^{Q_{0}}$-graded mixed Hodge structures

$\displaystyle\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\operatorname{\mathbf{H}}_{c}\\!\left(\mathfrak{M}_{\mathbf{d}}(\Pi_{Q,\upmu}),\mathbb{Q}\right)\otimes\mathscr{L}^{\upchi_{Q}(\mathbf{d},\mathbf{d})}\cong$
$\displaystyle\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\operatorname{\mathbf{H}}_{c}\left(\mathfrak{M}_{\mathbf{d}}(\tilde{Q}),\upphi^{\operatorname{mon}}_{\operatorname{Tr}(W_{1}^{\upmu})}\mathcal{IC}_{\mathfrak{M}_{\mathbf{d}}(\tilde{Q})}\right)$
$\displaystyle\cong$
$\displaystyle\operatorname{\mathbf{Sym}}\left(\bigoplus_{\mathbb{N}^{Q_{0}}\ni\mathbf{d}\neq
0}\operatorname{BPS}^{\operatorname{hdg},\vee}_{\tilde{Q},W_{1}^{\upmu},\mathbf{d}}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*})_{\operatorname{vir}}\right)$

where the first isomorphism is dimensional reduction (as in Proposition 3.5)
and the second is the cohomological integrality theorem. So the theorem
follows from (26) and the claim that

$\operatorname{BPS}^{\operatorname{hdg}}_{\tilde{Q},W_{1}^{\upmu},\mathbf{d}}\cong\begin{cases}\operatorname{BPS}^{\operatorname{hdg}}_{\tilde{Q},\tilde{W},\mathbf{d}}&\textrm{if
}\mathbf{d}\cdot\upmu=0\\\ 0&\textrm{otherwise.}\end{cases}$

This follows from (37) and the observation that for $\lambda\in\mathbb{C}$

$\operatorname{\mathbf{H}}\\!\left(\mathbb{A}^{1},\upphi^{\operatorname{mon}}_{\lambda
x}\mathcal{IC}_{\mathbb{A}^{1}}\right)\cong\begin{cases}\mathscr{L}^{-1/2}&\textrm{if
}\lambda=0\\\ 0&\textrm{if }\lambda\neq 0.\end{cases}$

∎

Assume that the dimension vector $\mathbf{d}$ is indivisible, meaning that
there is no $n\in\mathbb{Z}_{\geq 2}$ such that
$\frac{1}{n}\mathbf{d}\in\mathbb{N}^{Q_{0}}$. Assume also that $\upmu$ is
chosen to be generic, subject to the constraint that $\mathbf{d}\cdot\upmu=0$.
Equivalently, $\mathbf{d}$ and $\upmu$ are chosen so that
$\mathbf{d}^{\prime}\cdot\upmu=0$ implies that $\mathbf{d}^{\prime}$ is an
integer multiple of $\mathbf{d}$. Then (17) simplifies to

(38) $\bigoplus_{n\in\mathbb{Z}_{\geq
0}}\operatorname{\mathbf{H}}_{c}\\!\left(\mathfrak{M}_{n\mathbf{d}}(\Pi_{Q,\upmu}),\mathbb{Q}\right)\otimes\mathscr{L}^{n^{2}\upchi_{Q}(\mathbf{d},\mathbf{d})}\cong\operatorname{\mathbf{Sym}}\left(\bigoplus_{n\in\mathbb{Z}_{>0}}\operatorname{BPS}_{\tilde{Q},\tilde{W},n\mathbf{d}}^{\operatorname{hdg},\vee}\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*},\mathbb{Q})_{\operatorname{vir}}\right).$

Our assumptions imply that $\mathcal{M}_{\mathbf{d}}(\Pi_{Q})$ is a fine
moduli scheme, and $\mathfrak{M}_{\mathbf{d}}(\Pi_{Q})$ is a trivial
$\mathrm{B}\mathbb{C}^{*}$-gerbe over it, meaning that

$\operatorname{\mathbf{H}}_{c}\\!\left(\mathfrak{M}_{\mathbf{d}}(\Pi_{Q,\upmu}),\mathbb{Q}\right)\cong\operatorname{\mathbf{H}}_{c}(\mathcal{M}_{\mathbf{d}}(\Pi_{Q}),\mathbb{Q})\otimes\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*},\mathbb{Q}).$

Recall that there is an isomorphism

$\operatorname{\mathbf{H}}_{c}(\operatorname{pt}/\mathbb{C}^{*},\mathbb{Q})\cong\bigoplus_{i\geq
0}\mathscr{L}^{-i-1}.$

From the $n=1$ piece of (38) we deduce the following slightly stronger version
of a result of Crawley–Boevey and Van den Bergh [CBdB04]:

######  Corollary 3.6.

Let $\mathbf{d}$ be indivisible, and $\upmu$ be generic. Then there is an
isomorphism of mixed Hodge structures

$\operatorname{\mathbf{H}}_{c}(\mathcal{M}_{\mathbf{d}}(\Pi_{Q,\upmu}),\mathbb{Q})\cong\bigoplus_{i\geq
0}(\mathscr{L}^{1+i-\upchi_{Q}(\mathbf{d},\mathbf{d})})^{\oplus\mathtt{a}_{Q,\mathbf{d},i}}.$

In particular,
$\operatorname{\mathbf{H}}_{c}(\mathcal{M}_{\mathbf{d}}(\Pi_{Q,\upmu}),\mathbb{Q})$
is pure of Tate type.

##  4\. Further directions

###  4.1. Calculating the BPS sheaves

While our main theorem gives a way to calculate the BPS cohomology
$\operatorname{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}$ for arbitrary
$Q$, $\mathbf{d}\in\mathbb{N}^{Q_{0}}$ and $\upmu\in R$, the actual BPS sheaf
$\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}$ remains a little
mysterious. We can at least generalise (part of) the support lemma (Lemma 2.7)
from the case $\upmu=0$.

######  Lemma 4.1.

For a quiver $Q$, and dimension vector $\mathbf{d}\in\mathbb{N}^{Q_{0}}$, we
have

$\displaystyle\operatorname{supp}(\mathcal{BPS}_{\Pi^{\upmu}_{Q},\mathbf{d}})\subset\begin{cases}\tilde{e}(\mathcal{M}_{\mathbf{d}}(\Pi_{Q})\times\mathbb{A}^{1})&\textrm{if
}\upmu\cdot\mathbf{d}=0\\\
\tilde{e}(\mathcal{M}_{\mathbf{d}}(\Pi_{Q})\times\\{0\\})&\textrm{if
}\upmu\cdot\mathbf{d}\neq 0\end{cases}$

where $\tilde{e}$ is defined in (24).

###### Proof.

The proof is very similar to [Dav17b, Lem.4.1]. Let $\rho$ be a
$\operatorname{Jac}(\tilde{Q},\tilde{W}^{\upmu})$-module lying in the support
of $\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}$. By definition,
$\rho$ is semisimple, and so since $\rho(\sum_{i\in Q_{0}}\omega_{i})$ is
central, it acts via a diagonal matrix. Arguing as in [Dav17b, Lem.4.1] and
using centrality of $\rho(\sum_{i\in Q_{0}}\omega_{i})$ along with the
cohomological integrality theorem, we deduce that the generalised eigenvalues
of $\rho(\omega)$ are all the same, i.e. $\rho(\omega)$ acts via scalar
multiplication, establishing the lemma in the case $\upmu\cdot\mathbf{d}\neq
0$. Arguing as in Proposition 2.3, if $\upmu\cdot\mathbf{d}\neq 0$, we must
have $\lambda=0$. This establishes the lemma in the case
$\upmu\cdot\mathbf{d}\neq 0$. ∎

Even in the case $\upmu\cdot\mathbf{d}=0$ this is a slightly weaker statement
than Lemma 2.7; it is harder to show that
$\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}$ is
$\mathbb{A}^{1}$-equivariant (where $\mathbb{A}^{1}$ acts by adding a scalar
multiple of the identity matrix to $\rho(\omega)$) since the function
$\operatorname{Tr}(\tilde{W}^{\upmu})$ is not $\mathbb{A}^{1}$-invariant
unless $\upmu=0$.

It would be interesting to compare
$\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}$ with
$\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}\cong\tilde{e}_{!}\left(\mathcal{BPS}_{\Pi_{Q},\mathbf{d}}\boxtimes\mathcal{IC}_{\mathbb{A}^{1}}\right)$
in the case $\upmu\cdot\mathbf{d}=0$, and with
$\tilde{e}_{!}\left(\mathcal{BPS}_{\Pi_{Q},\mathbf{d}}\boxtimes\mathbb{Q}_{0}\right)$
in the case $\upmu\cdot\mathbf{d}\neq 0$. In particular, we may ask the
following question:

######  Question 4.2.

Is the monodromic mixed Hodge module
$\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\upmu},\mathbf{d}}$ pure?

By [Dav20, Thm.A], we know that
$\mathcal{BPS}_{\tilde{Q},\tilde{W},\mathbf{d}}$ is a pure monodromic mixed
Hodge module, i.e. we know that for $\upmu=0$ the answer to Question 4.2 is
yes.

###  4.2. The BPS algebra

Identifying the BPS cohomology of the Jacobi algebra
$\mathbb{C}(\tilde{Q},\tilde{W}^{\upmu})$ is only part of understanding its
cohomological DT theory; it tells us the size of the graded pieces of
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}^{\upmu}}$, but nothing about
the algebra structure above what we already know from [DM20] regarding general
quivers with potential (e.g. the PBW theorem). At least for $\upmu=0$, the
algebra $\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}^{\upmu}}$ has been
studied from various points of view (see e.g. [Nak01, Var00, MO19, KV19,
DPS20]) while for the $\upmu\neq 0$ case, it would be interesting (at least
for noncommutative resolutions of toric CY3s) to relate these algebras to the
Yangians defined in terms of (generalised) McMahon modules and crystal melting
in [LY20, GY20].

For general $Q$ and $\upmu$, the shifted BPS cohomology
$\mathfrak{g}_{\tilde{Q},\tilde{W}^{\upmu}}\coloneqq\operatorname{BPS}_{\tilde{Q},\tilde{W}^{\upmu}}[-1]$
carries a Lie algebra structure (by Theorem 2.5). Even in the case $\upmu=0$
we do not yet fully understand this Lie algebra for general $Q$. We can at
least try to relate the case of general $\upmu$ to the case $\upmu=0$, via the
following construction:

Fix $Q$ and $\upmu$, and define

$\displaystyle\mathfrak{g}_{\operatorname{even}}\coloneqq$
$\displaystyle\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}\;\lvert\;\mathbf{d}\cdot\upmu=0}\mathfrak{g}_{\tilde{Q},\tilde{W},\mathbf{d}}$
$\displaystyle\mathfrak{g}_{\operatorname{odd}}\coloneqq$
$\displaystyle\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}\;\lvert\;\mathbf{d}\cdot\upmu\neq
0}\mathfrak{g}_{\tilde{Q},\tilde{W},\mathbf{d}}.$

Then $\mathfrak{g}_{\operatorname{even}}$ is a Lie subalgebra of
$\mathfrak{g}_{\tilde{Q},\tilde{W}}$, and $\mathfrak{g}_{\operatorname{odd}}$
is a Lie module for it, and we consider the extension

$\mathfrak{g}^{\upmu}=\mathfrak{g}_{\operatorname{even}}\oplus\mathfrak{g}_{\operatorname{odd}}[-1],$

with the Lie bracket

$[(\upalpha,\upbeta),(\upalpha^{\prime},\upbeta^{\prime})]=([\upalpha,\upalpha^{\prime}],[\upalpha,\upbeta^{\prime}]-[\upalpha^{\prime},\upbeta]).$

######  Question 4.3.

Is there is an isomorphism of Lie algebras

$\mathfrak{g}^{\upmu}\cong\mathfrak{g}_{\tilde{Q},\tilde{W}^{\upmu}}?$

######  Proposition 4.4.

The answer to Question 4.3 is yes in the case $\upmu=0$, and for generic
$\upmu$.

###### Proof.

The case $\upmu=0$ is trivial, since then

$\displaystyle\mathfrak{g}^{\upmu}=$
$\displaystyle\mathfrak{g}_{\operatorname{even}}$ $\displaystyle=$
$\displaystyle\mathfrak{g}_{\tilde{Q},\tilde{W}}$

and $\tilde{W}^{\upmu}=\tilde{W}$. In the case of generic $\upmu$, Theorem A
tells us that there is an isomorphism

(39)
$\mathfrak{g}_{\tilde{Q},\tilde{W}^{\upmu}}\cong\mathfrak{g}_{\tilde{Q},\tilde{W}}[-1]$

as cohomologically graded vector spaces, and so it is sufficient to show that
the Lie bracket on $\mathfrak{g}_{\tilde{Q},\tilde{W}^{\upmu}}$ vanishes,
since the Lie bracket on
$\mathfrak{g}^{\upmu}\coloneqq\mathfrak{g}_{\operatorname{odd}}[-1]$ does by
definition. Combining (39) and (27), we deduce that
$\mathfrak{g}_{\tilde{Q},\tilde{W}^{\upmu}}$ lies entirely in odd
cohomological degree. Since the Hall algebra multiplication, and hence the
commutator Lie bracket, preserves cohomological degree, the Lie bracket
vanishes as required. ∎

In this paper, we have ignored completely the question of whether there is a
natural double to the Lie algebra
$\mathfrak{g}_{\tilde{Q},\tilde{W}^{\upmu}}$. Despite Proposition 4.4 the
expected answer to Question 4.3, once extended to the double of
$\mathfrak{g}_{\tilde{Q},\tilde{W}^{\upmu}}$ is no; one of the motivations for
this paper is work of Kevin Costello [Cos18], in which he conjectures that in
the case of e.g. the resolved conifold, the Lie bracket does not vanish on the
fermionic part of the (doubled) BPS Lie algebra. As we saw in §1.2, and
Example 3.3 the (noncommutative) resolved conifold is a partial fermionization
of the QP $(\tilde{Q},\tilde{W})$ where $\tilde{Q}$ is the affine $A_{1}$
quiver; so the conifold is precisely the kind of case that Proposition 4.4
does not cover.

###  4.3. Representation theory

The boson-fermion correspondence is typically considered (by mathematicians)
to be a part of representation theory, and so it would be remiss to finish the
paper without saying anything about representations of
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}^{\upmu}}$.

Fix a quiver $Q$, and a framing dimension vector
$\mathbf{f}\in\mathbb{N}^{Q_{0}}$. We form the quiver $Q_{\mathbf{f}}$ by
adding one extra vertex (labelled $\infty$) to $Q_{0}$, and $\mathbf{f}_{i}$
arrows from $\infty$ to $i$ for each $i\in Q_{0}$. Set
$\mathscr{Q}=\widetilde{Q_{\mathbf{f}}}$. In other words, this is the usual
doubled framed quiver that one uses to define Nakajima quiver varieties, but
with an extra loop at every vertex (including a loop $\omega_{\infty}$ at the
framing vertex).

For $\upmu=\sum_{i\in Q_{0}}\upmu_{i}e_{i}$ we define

$\tilde{W}_{\mathbf{f}}^{\upmu}=(\sum_{i\in\mathscr{Q}_{0}}\omega_{i})(\sum_{a\in
Q_{\mathbf{f}}}[a,a^{*}])+\sum_{i\in Q_{0}}\upmu_{i}\omega_{i}^{2}.$

We write dimension vectors for $\mathscr{Q}$ as $\mathbf{e}=(\mathbf{d},n)$
where $\mathbf{d}\in\mathbb{N}^{Q_{0}}$ and
$n=\mathbf{e}_{\infty}\in\mathbb{N}$. We define

$\mathbb{A}_{(\mathbf{d},1)}^{\operatorname{st}}(\mathscr{Q})\subset\mathbb{A}_{(\mathbf{d},1)}(\mathscr{Q})$

to be the subvariety containing those $\rho$ such that $\rho_{\infty}$
generates $\rho$ as a $\mathbb{C}\mathscr{Q}$-module. We call such
$\mathbb{C}\mathscr{Q}$-modules stable. Then we define

$\mathscr{N}_{\mathbf{f},\mathbf{d}}(Q)\coloneqq\mathbb{A}_{(\mathbf{d},1)}^{\operatorname{st}}(\mathscr{Q})/\operatorname{GL}_{\mathbf{d}}.$

This is a smooth variety, and is naturally isomorphic to the stack of stable
$(\mathbf{d},1)$-dimensional $\mathbb{C}\mathscr{Q}$-modules along with a
trivialisation $\rho_{\infty}\cong\mathbb{C}$. We define

$\operatorname{\mathcal{F}l}_{(\mathbf{d},1),\mathbf{d}^{\prime}}(\mathscr{Q})$

to be the stack of pairs of a stable $(\mathbf{d},1)$-dimensional
$\mathbb{C}\mathscr{Q}$-module $\rho$, along with a
$(\mathbf{d}^{\prime},0)$-dimensional submodule $\rho^{\prime}\subset\rho$ and
a trivialisation $\rho_{\infty}\cong\mathbb{C}$. Set
$\mathbf{d}^{\prime\prime}=\mathbf{d}-\mathbf{d}^{\prime}$. In the
correspondence diagram

$\textstyle{\mathfrak{M}_{\mathbf{d}^{\prime}}(\widetilde{Q})\times\mathscr{N}_{\mathbf{f},\mathbf{d}^{\prime\prime}}(Q)}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathcal{F}l}_{(\mathbf{d},1),\mathbf{d}^{\prime}}(\mathscr{Q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\uppi\hskip
0.35002pt_{1}\times\uppi\hskip 0.35002pt_{3}}$$\scriptstyle{\uppi\hskip
0.35002pt_{2}}$$\textstyle{\mathscr{N}_{\mathbf{f},\mathbf{d}}(Q)}$

the morphism $\uppi\hskip 0.50003pt_{2}$ is proper, and so via push-forward
and pull-back in vanishing cycle cohomology, we obtain an action of
$\operatorname{\mathcal{A}}_{\tilde{Q},\tilde{W}^{\upmu}}$ on

$\mathbf{M}^{\upmu}_{\mathbf{f}}(Q)\coloneqq\bigoplus_{\mathbf{d}\in\mathbb{N}^{Q_{0}}}\operatorname{\mathbf{H}}(\mathscr{N}_{\mathbf{f},\mathbf{d}^{\prime\prime}}(Q),\upphi^{\operatorname{mon}}_{\operatorname{Tr}(\tilde{W}_{\mathbf{f}}^{\upmu})}\underline{\mathbb{Q}}_{\mathscr{N}_{\mathbf{f},\mathbf{d}^{\prime\prime}}(Q)})\otimes\mathscr{L}^{\upchi_{\mathscr{Q}}\left((1,\mathbf{d}),(1,\mathbf{d})\right)/2}.$

Given a quiver $Q$ we denote by

$\displaystyle\upmu_{Q,\mathbf{d}}\colon\mathbb{A}_{\mathbf{d}}(\overline{Q})$
$\displaystyle\rightarrow\mathfrak{\operatorname{\mathfrak{gl}}}_{\mathbf{d}}$
$\displaystyle\rho$ $\displaystyle\mapsto\sum_{a\in
Q_{1}}[\rho(a),\rho(a^{*})].$

the usual moment map. We define

$Z(\mathbf{f},\mathbf{d})\subset\mathbb{A}^{\operatorname{st}}_{(\mathbf{d},1)}(\overline{Q_{\mathbf{f}}})$

the vanishing locus of the composition of
$\upmu_{Q_{\mathbf{f}},(\mathbf{d},1)}$ with the natural projection
$\mathfrak{\operatorname{\mathfrak{gl}}}_{(\mathbf{d},1)}\rightarrow\mathfrak{\operatorname{\mathfrak{gl}}}_{\mathbf{d}}$.
Here the stability condition is the same as above: we restrict to the open
locus of those $\rho$ such that $\rho_{\infty}$ generates. Then the Nakajima
quiver variety is defined to be the smooth variety

$X(\mathbf{f},\mathbf{d})=Z(\mathbf{f},\mathbf{d})/\operatorname{GL}_{\mathbf{d}}.$

######  Proposition 4.5.

Denote by $g$ the restriction of the function
$\operatorname{Tr}(\tilde{W}_{\mathbf{f}}^{\upmu})$ to
$\mathscr{N}_{\mathbf{f},\mathbf{d}}(Q)$, then

$\operatorname{crit}(g)\cong\begin{cases}X(\mathbf{f},\mathbf{d})&\textrm{if
}\upmu\cdot\mathbf{d}\neq 0\\\
X(\mathbf{f},\mathbf{d})\times\mathbb{A}^{1}&\textrm{if
}\upmu\cdot\mathbf{d}=0.\end{cases}$

Moreover the monodromic mixed Hodge module

$\upphi^{\operatorname{mon}}_{g}\underline{\mathbb{Q}}_{\mathscr{N}_{\mathbf{f},\mathbf{d}}(Q)}\otimes\mathscr{L}^{\upchi_{\mathscr{Q}}\left((\mathbf{d},1),(\mathbf{d},1)\right)/2}$

is analytically locally isomorphic to the constant mixed Hodge module on the
critical locus of $g$.

###### Proof.

The second part follows from the first: the holomorphic Bott–Morse lemma tells
us that since the critical locus of $g$ is scheme-theoretically smooth, $g$
can be written analytically locally around $\operatorname{crit}(g)$ in the
form

$g=x_{1}^{2}+\ldots+x_{c}^{2}$

where $c$ is the codimension of $\operatorname{crit}(g)$ inside
$\mathcal{N}_{\mathbf{f},\mathbf{d}}(Q)$. The first part follows by the same
argument as Proposition 2.3. ∎

In the special case $\upmu=0$ it is possible to show that the monodromy of the
rank one local system
${}^{\mathfrak{p}}\\!\upphi_{g}\underline{\mathbb{Q}}_{\mathscr{N}_{\mathbf{f},\mathbf{d}^{\prime\prime}}(Q)}$
is trivial [Dav20, Prop.6.3], and we conjecture that this is always true.
Assuming the conjecture, Proposition 4.5 implies that there are isomorphisms

(40)
$\mathbf{M}^{\upmu}_{\mathbf{f}}(Q)\cong\left(\bigoplus_{\begin{subarray}{c}\mathbf{d}\in\mathbb{N}^{Q_{0}}\\\
\upmu\cdot\mathbf{d}\neq
0\end{subarray}}\operatorname{\mathbf{H}}(X(\mathbf{f},\mathbf{d}),\mathbb{Q})_{\operatorname{vir}}\right)\oplus\left(\bigoplus_{\begin{subarray}{c}\mathbf{d}\in\mathbb{N}^{Q_{0}}\\\
\upmu\cdot\mathbf{d}=0\end{subarray}}\operatorname{\mathbf{H}}(X(\mathbf{f},\mathbf{d}),\mathbb{Q})_{\operatorname{vir}}\otimes\mathscr{L}^{-1/2}\right).$

Note that the vacuum vector, spanning
$\operatorname{\mathbf{H}}(X(\mathbf{f},0),\mathbb{Q})$, lies in the second
summand.

Let $Q^{\prime}$ be the full subquiver of $Q$ obtained by removing all
vertices that support loops, as well as arrows to or from them, and let
$\mathfrak{n}^{-}_{Q^{\prime}}$ be the negative piece of the Kac–Moody Lie
algebra associated to $Q^{\prime}$. By [Dav20, Thm.6.6] there is an inclusion
of Lie algebras

$\mathfrak{n}^{-}_{Q^{\prime}}\hookrightarrow\mathfrak{g}_{\tilde{Q},\tilde{W}}$

as the part of the BPS Lie algebra lying in cohomological degree zero, so that
(40) in case $\upmu=0$ allows us to reconstruct (one half of) Nakajima’s
action of $\mathfrak{g}_{Q^{\prime}}$ on the cohomology of quiver varieties
[Nak98]. The conjecture suggests that the cohomology of Nakajima quiver
varieties should be as crucial to the representation theory of partially
fermionized BPS Lie algebras as they are to the representation theory of their
bosonic counterparts.

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