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 $s,t\,:\, Q_1\rightarrow Q_0$ 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

(2.1) \begin{align}\unicode{x03C7}_Q(\cdot,\cdot)\,:\, &\mathbb{Z}^{Q_0}\times\mathbb{Z}^{Q_0}\rightarrow\mathbb{Z}\end{align}

\begin{align}\nonumber &(\textbf{d},\textrm{e})\mapsto \sum_{a\in Q_1} \textbf{d}_{s(a)}\textbf{d}_{t(a)}-\sum_{i\in Q_0}\textbf{d}_i\textrm{e}_i.\end{align}

Where there is no possibility of confusion, we drop the quiver Q from the notation and just write $\unicode{x03C7}(\cdot,\cdot)$ . 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 $\unicode{x03BC}\in R$ . We recall the central extension of $\Pi_Q$ introduced by Etingof and Rains [Reference Etingof and Rains16]:

\begin{align*}\Pi_Q^{\unicode{x03BC}}=\mathbb{C}[\omega]\overline{Q}/\left\langle \sum_{a\in Q_1}[a,a^*]+\unicode{x03BC} \omega\right\rangle.\end{align*}

There is an obvious isomorphism

\begin{align*}\Pi_Q^0\cong\Pi_Q[\omega]\end{align*}

and natural isomorphisms

\begin{align*}\Pi_{Q,\unicode{x03BC}}\cong\ &\Pi_Q^{\unicode{x03BC}}/\langle \omega-1\rangle\\[4pt]\Pi_{Q}\cong\ &\Pi_Q^{\unicode{x03BC}}/\langle \omega\rangle,\end{align*}

where $\Pi_{Q,\unicode{x03BC}}$ is the deformed preprojective algebra recalled in the introduction. The algebra $\Pi_Q^{\unicode{x03BC}}$ provides an $\mathbb{A}^1$ -family of algebras interpolating between the preprojective algebra $\Pi_Q$ and the deformed preprojective algebra $\Pi_{Q,\unicode{x03BC}}$ .

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

\begin{align*}\partial W/\partial a=\sum_{a_m=a}a_{m+1}a_{m+2}\ldots a_na_1\ldots a_{m-1}\end{align*}

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

\begin{align*}\text{Jac}(Q,W)=\mathbb{C} Q/\langle \partial W/\partial a \;\lvert\; a\in Q_1\rangle.\end{align*}

In this paper, we will study Jacobi algebras obtained from the tripled QP $\left(\tilde{Q},\tilde{W}\right)$ defined in (1.5) by adding polynomials in the extra loops $\omega_i$ ; we refer the reader to [Reference Ginzburg18, Sect. 4] for general background on this construction, [Reference Quintero Vélez and Boer36, Reference Szendröi42] for the noncommutative geometry background in type ADE, and [Reference Cachazo, Katz and Vafa4] for the physics perspective.

Proposition 2.1. [Reference Ginzburg18, Ex. 4.3.5] Let $\unicode{x03BC}=\sum_i \unicode{x03BC}_ie_i$ . Set $\tilde{W}^{\unicode{x03BC}}=\tilde{W}+\frac{1}{2}\sum_{i\in Q_0} \unicode{x03BC}_i\omega_i^2$ . Then there is an isomorphism

\begin{align*}\Pi_Q^{\unicode{x03BC}}\cong \text{Jac}\!\left(\tilde{Q},\tilde{W}^{\unicode{x03BC}}\right).\end{align*}

In particular, there is a natural isomorphism

\begin{align*}\Pi_Q[\omega]\cong \text{Jac}\!\left(\tilde{Q},\tilde{W}\right).\end{align*}

Proof. This follows more or less from the definitions. The noncommutative derivatives of $\tilde{W}^{\unicode{x03BC}}$ 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 $\Pi^{\unicode{x03BC}}_Q$ as a quotient of $\mathbb{C}[\omega]\overline{Q}$ .

Example 2.2. Let Q be defined as in (1.10), and set $\unicode{x03BC}=e_0-e_1$ . Then

\begin{align*}\tilde{W}^{\unicode{x03BC}}=\omega_0\!\left(a^*a-bb^*\right)-\omega_1\!\left(aa^*-b^*b\right)+\frac{1}{2}\!\left(\omega_0^2-\omega_1^2\right).\end{align*}

After the noncommutative change of variables

\begin{align*}\omega_0&\mapsto \omega_0-a^*a+bb^*\\\omega_1&\mapsto \omega_1-aa^*+b^*b\end{align*}

the potential transforms to

\begin{align*}W=\frac{1}{2}\!\left(\omega_0^2-\omega_1^2\right)+b^*baa^*-bb^*a^*a.\end{align*}

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

\begin{align*}\text{Jac}\!\left(\tilde{Q},W\right)\cong \text{Jac}\!\left(Q_{\text{con}},W_{\texttt{KW}}\right),\end{align*}

giving the isomorphism (1.12). In particular, the noncommutative conifold is isomorphic to one of the central extensions of $\Pi_Q$ considered above, for Q as in (1.10).

Later we will use the following elementary result.

Proof. The first part follows from the fact that $\omega=\sum_{i\in Q_0}\omega_i$ is central in $\text{Jac}\!\left(\tilde{Q},\tilde{W}^{\unicode{x03BC}}\right)$ , so each eigenspace of $\rho(\omega)$ is preserved by the action of $\text{Jac}\!\left(\tilde{Q},\tilde{W}^{\unicode{x03BC}}\right)$ . For the second part, consider the relation in $\text{Jac}\!\left(\tilde{Q},\tilde{W}^{\unicode{x03BC}}\right)$

\begin{align*}0=\sum_{i\in Q_0}\partial \tilde{W}^{\unicode{x03BC}}/\partial \omega_i=\sum_{i\in Q_0}\unicode{x03BC}_i \omega_i+ \sum_{a\in Q_1}\left[a,a^*\right].\end{align*}

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

\begin{align*}0& = \sum_{i\in Q_0} \unicode{x03BC}_i \lambda \text{Tr}\!\left(\text{Id}_{\textbf{d}_i\times\textbf{d}_i}\right)\\& = \lambda \unicode{x03BC}\cdot \textbf{d}\end{align*}

as required.

2.2 Moduli spaces of quiver representations

Given a quiver Q and a dimension vector $\textbf{d}\in\mathbb{N}^{Q_0}$ we set

\begin{align*}\mathbb{A}_{\textbf{d}}(Q)\,:\!=\, &\prod_{a\in Q_1}\text{Hom}\!\left(\mathbb{C}^{\textbf{d}_{s(a)}},\mathbb{C}^{\textbf{d}_{t(a)}}\right)\\\text{GL}_{\textbf{d}}\,:\!=\, & \prod_{i\in Q_0}\text{GL}_{\textbf{d}_i}(\mathbb{C}).\end{align*}

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

\begin{align*}\mathfrak{M}_{\textbf{d}}(Q)\cong \mathbb{A}_{\textbf{d}}(Q)/\text{GL}_{\textbf{d}}.\end{align*}

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

\begin{align*}\mathcal{M}_{\textbf{d}}(Q)\cong \text{Spec}\!\left(\Gamma(\mathbb{A}_{\textbf{d}}(Q))^{\text{GL}_{\textbf{d}}}\right).\end{align*}

We denote by $\texttt{JH}_{\textbf{d}}\,:\, \mathfrak{M}_{\textbf{d}}(Q)\rightarrow \mathcal{M}_{\textbf{d}}(Q)$ the affinisation morphism. Although this morphism is not projective, it is approximated by projective maps in the sense of [Reference Davison and Meinhardt11], meaning that $\texttt{JH}_*$ and $\texttt{JH}_!$ commute with vanishing cycle functors (introduced in the next section).

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

There is a finite morphism [Reference Meinhardt and Reineke33]

\begin{align*}\oplus\,:\, \mathcal{M}(Q)\times\mathcal{M}(Q)\rightarrow \mathcal{M}(Q)\end{align*}

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 $\textbf{Perv}(\mathcal{M}(Q))$ , defined by

\begin{align*}\mathcal{F}^{\prime}\boxtimes_{\oplus}\mathcal{F}^{\prime\prime}\,:\!=\, \oplus_*\!\left(\mathcal{F}^{\prime}\boxtimes\mathcal{F}^{\prime\prime}\right).\end{align*}

Given a potential $W\in \mathbb{C}Q/[\mathbb{C} Q,\mathbb{C} Q],$ we form the function $\text{Tr}(W)_{\textbf{d}}$ on $\mathbb{A}_{\textbf{d}}(Q)$ . This is well defined and $\text{GL}_{\textbf{d}}$ -invariant by cyclic invariance of trace. As such, $\text{Tr}(W)_{\textbf{d}}$ induces functions on $\mathfrak{M}_{\textbf{d}}(Q)$ and $\mathcal{M}_{\textbf{d}}(Q)$ , which we continue to denote by $\text{Tr}(W)_{\textbf{d}}$ , or just 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

in which the square is Cartesian. Then, we define the nearby cycles functor $\psi_f\,:\, \mathcal{D}^{\textrm{b}}(\textbf{Perv}(X))\rightarrow \mathcal{D}^{\textrm{b}}(\textbf{Perv}(X))$ via

\begin{align*}\psi_f=\kappa_*\kappa^*p_*p^*.\end{align*}

The vanishing cycles functor $\unicode{x03D5}_f$ is defined so that for $\mathcal{F}\in \text{Ob}\!\left(\mathcal{D}^{\textrm{b}}\!\left(\textbf{Perv}(X)\right)\right)$ there is a distinguished triangle

\begin{align*}\kappa_*\kappa^*\mathcal{F}\rightarrow \psi_f\mathcal{F}\rightarrow \unicode{x03D5}_f\mathcal{F}.\end{align*}

Both ${{}^{\mathfrak{p}}\!}\psi_{f}\,:\!=\, \psi_f[{-}1]$ and ${{}^{\mathfrak{p}}\!}\unicode{x03D5}_{f}\,:\!=\,\unicode{x03D5}_f[{-}1]$ send perverse sheaves to perverse sheaves [Reference Kashiwara and Schapira23, Cor. 10.3.13] and (naturally) commute with Verdier duality [Reference Massey29].

We give a lightning account of the critical cohomological Hall algebra associated with a quiver with potential. More details can be found in [Reference Davison and Meinhardt11, Reference Kontsevich and Soibelman26]. For a stack $\mathfrak{M}$ for which each connected component is irreducible and generically smooth, we define the intersection complex

\begin{align*}\mathcal{IC}_{\mathfrak{M}}\,:\!=\,\coprod_{\mathfrak{N}\in\unicode{x03C0}_0(\mathfrak{M})}\mathcal{IC}_{\mathfrak{N}}\!\left(\mathbb{Q}_{\mathfrak{N}^{\textrm{sm}}}[{\dim}(\mathfrak{N})]\right),\end{align*}

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

\begin{align*}\mathcal{RA}_{Q,W}\,:\!=\, \texttt{JH}_*{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\mathcal{IC}_{\mathfrak{M}(Q)}.\end{align*}

The morphism $\unicode{x03C0}_2$ from (1.2) is proper, so that there is a natural integration map

\begin{align*}\unicode{x03B1}_{\textbf{d}^{\prime},\textbf{d}^{\prime\prime}}\,:\,\unicode{x03C0}_{2,*}\mathbb{Q}_{\mathfrak{E}\textrm{xact}_{\textbf{d}^{\prime},\textbf{d}^{\prime\prime}}(Q)}\rightarrow \mathbb{Q}_{\mathfrak{M}_{\textbf{d}^{\prime}+\textbf{d}^{\prime\prime}}(Q)}\left[{-}2\unicode{x03C7}\!\left(\textbf{d}^{\prime},\textbf{d}^{\prime\prime}\right)\right]\end{align*}

(the shift is given by the relative dimension of $\unicode{x03C0}_2$ ). Composing appropriate shifts of the morphisms

\begin{align*}&\oplus_*\texttt{JH}_*{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\!\left(\mathbb{Q}_{\mathfrak{M}(Q)\times\mathfrak{M}(Q)}\rightarrow (\unicode{x03C0}_1\times \unicode{x03C0}_3)_*\mathbb{Q}_{\mathfrak{E}\textrm{xact}} \right)\end{align*}

and the sum of $\texttt{JH}_*{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\unicode{x03B1}_{\textbf{d}^{\prime},\textbf{d}^{\prime\prime}}$ over pairs $(\textbf{d}^{\prime},\textbf{d}^{\prime\prime})$ , and using commutativity of vanishing cycle functors with proper and with smooth morphisms, we obtain the morphism

\begin{align*}\unicode{x03B2}\,:\, \oplus_*(\texttt{JH}\times\texttt{JH})_*{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\mathcal{IC}_{\mathfrak{M}_{\textbf{d}^{\prime}}(Q)\times\mathfrak{M}_{\textbf{d}^{\prime\prime}}(Q)}\rightarrow \texttt{JH}_*{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\mathcal{IC}_{\mathfrak{M}_{\textbf{d}}(Q)}.\end{align*}

Finally, composing $\unicode{x03B2}$ with $\oplus_*(\texttt{JH}\times\texttt{JH})_*\texttt{TS}$ , where $\texttt{TS}$ is (a shift of) the Thom–Sebastiani isomorphism [Reference Massey28]

\begin{align*}\texttt{TS}\,:\, {{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\mathbb{Q}_{\mathfrak{M}_{\textbf{d}^{\prime}}(Q)}\boxtimes{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\mathbb{Q}_{\mathfrak{M}_{\textbf{d}^{\prime\prime}}(Q)}\xrightarrow{\cong} {{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\mathbb{Q}_{\mathfrak{M}_{\textbf{d}^{\prime}}(Q)\times\mathfrak{M}_{\textbf{d}^{\prime\prime}}(Q)},\end{align*}

we define the (relative) Hall algebra multiplication

(2.2) \begin{equation}\mathcal{RA}_{Q,W}\boxtimes_{\oplus}\mathcal{RA}_{Q,W}\rightarrow \mathcal{RA}_{Q,W}.\end{equation}

The cohomology

\begin{align*}\mathcal{A}_{Q,W} & \,:\!=\, \textbf{H}\!\left(\mathfrak{M}(Q),{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\mathcal{IC}_{\mathfrak{M}(Q)}\right)\\[2pt]& \cong\textbf{H}\!\left(\mathcal{M}(Q),\mathcal{RA}_{Q,W}\right)\end{align*}

has a $\mathbb{N}^{Q_0}$ -grading by dimension vectors induced by the decomposition $\mathcal{M}(Q)=\coprod_{\textbf{d}\in\mathbb{N}^{Q_0}}\mathcal{M}_{\textbf{d}}(Q)$ , and the associative product induced by taking derived global sections of the morphism (2.2) respects this grading.

Assume that Q is symmetric^{Footnote 3}. For $\textbf{d}\in \mathbb{N}^{Q_0},$ we define

(2.3) \begin{equation}\mathcal{BPS}_{Q,W,\textbf{d}}\,:\!=\, \begin{cases}{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\mathcal{IC}_{\mathcal{M}_{\textbf{d}}(Q)}& \textrm{if there is a $\textbf{d}$-dimensional simple $\mathbb{C}Q$-module}\\ 0&\textrm{otherwise.}\end{cases}\end{equation}

According to our conventions, $\mathcal{IC}_{\mathcal{M}_{\textbf{d}}(Q)}$ is the intermediate extension of the constant perverse sheaf $\mathbb{Q}_{\mathcal{M}^{simp}_{\textbf{d}}(Q)}[1-\unicode{x03C7}(\textbf{d},\textbf{d})]$ on the (open, dense) subscheme of $\mathcal{M}_{\textbf{d}}(Q)$ corresponding to simple modules. Since $\mathcal{IC}_{\mathcal{M}_{\textbf{d}}(Q)}$ is Verdier self-dual, and vanishing cycle functors commute with Verdier duality [Reference Massey29], there are natural isomorphisms

(2.4) \begin{equation}\mathbb{D}\mathcal{BPS}_{Q,W,\textbf{d}}\cong\mathcal{BPS}_{Q,W,\textbf{d}}.\end{equation}

We recall the following version of the cohomological integrality theorem from [Reference Davison and Meinhardt11]

Theorem 2.4. There is an isomorphism of bounded above complexes of perverse sheaves

\begin{align*}\texttt{JH}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\mathcal{IC}_{\mathfrak{M}(Q)}\cong\textbf{Sym}_{\boxtimes_{\oplus}}\!\left(\bigoplus_{\mathbb{N}^{Q_0}\ni\textbf{d}\neq 0}\mathcal{BPS}_{Q,W,\textbf{d}}\otimes\textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}\right),\end{align*}

where

\begin{align*}\textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}\cong \bigoplus_{i\in\mathbb{Z}_{\geq 0}}\mathbb{Q}[1+2i].\end{align*}

We define the cohomological BPS invariants

\begin{align*}\textrm{BPS}_{Q,W,\textbf{d}}\,:\!=\, \textbf{H}(\mathcal{M}_{\textbf{d}}(Q),\mathcal{BPS}_{Q,W,\textbf{d}}).\end{align*}

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

\begin{align*}\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0}}\textbf{H}_c(\mathfrak{M}_{\textbf{d}}(Q),{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(W)}\mathcal{IC}_{\mathfrak{M}_{\textbf{d}}(Q)})\cong \textbf{Sym}\!\left(\bigoplus_{\mathbb{N}^{Q_0}\ni\textbf{d}\neq 0}\textrm{BPS}^{\vee}_{Q,W,\textbf{d}}\otimes\textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}\right).\end{align*}

The BPS invariants of the Jacobi algebra Jac(Q,W) are defined via

\begin{align*}\omega_{Q,W,\textbf{d}}& = \unicode{x03C7}\!\left(\textbf{H}\!\left(\mathcal{M}_{\textbf{d}}(Q),\mathcal{BPS}_{Q,W,\textbf{d}}\right)\right)\\&\,:\!=\, \sum_{i\in\mathbb{Z}}(-1)^i\dim\!\left( \textbf{H}^i\!\left(\mathcal{M}_{\textbf{d}}(Q),\mathcal{BPS}_{Q,W,\textbf{d}}\right)\right)\\& = \unicode{x03C7}\!\left(\textbf{H}_c\!\left(\mathcal{M}_{\textbf{d}}(Q),\mathcal{BPS}_{Q,W,\textbf{d}}\right)\right),\end{align*}

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

Theorem 2.5. [Reference Davison and Meinhardt11] There is an isomorphism of unbounded complexes of perverse sheaves

\begin{align*}\mathcal{RA}_{Q,W}\cong\textbf{Sym}_{\boxtimes_{\oplus}}\!\left(\bigoplus_{\mathbb{N}^{Q_0}\ni\textbf{d}\neq 0}\mathcal{BPS}_{Q,W,\textbf{d}}\otimes\textbf{H}\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}\right),\end{align*}

where

\begin{align*}\textbf{H}\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}=\bigoplus_{i\in\mathbb{Z}_{\geq 0}}\mathbb{Q}[{-}1-2i],\end{align*}

so that

\begin{align*}{}^{\mathfrak{p}}\!\mathcal{H}^1\!\left(\mathcal{RA}_{Q,W}\right)=\left({}^{\mathfrak{p}}\unicode{x03C4}^{\leq 1}\mathcal{RA}_{Q,W}\right)[1]\cong \bigoplus_{\mathbb{N}^{Q_0}\ni\textbf{d}\neq 0}\mathcal{BPS}_{Q,W,\textbf{d}}.\end{align*}

Applying the natural transformation ${}^{\mathfrak{p}}\unicode{x03C4}^{\leq 1}\rightarrow \text{id}$ to $\mathcal{RA}_{Q,W}$ and taking hypercohomology, there is a natural inclusion

\begin{align*}\mathfrak{g}_{Q,W}\,:\!=\, \textrm{BPS}_{Q,W}[{-}1]= \textbf{H}\!\left(\mathcal{M}(Q),\mathcal{BPS}_{Q,W}\right)[{-}1]\hookrightarrow \mathcal{A}_{Q,W}.\end{align*}

The image of this inclusion is closed under the commutator^{Footnote 4} Lie bracket induced by the associative algebra structure on $\mathcal{A}_{Q,W}$ .

The resulting Lie algebra $\mathfrak{g}_{Q,W}$ is called the BPS Lie algebra for the pair (Q,W), see [Reference Davison and Meinhardt11] 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 [Reference Davison and Meinhardt11, Reference Kontsevich and Soibelman26], and for a comparison with the treatment of monodromic mixed Hodge modules in [Reference Saito39], we refer the reader to [Reference Davison and Meinhardt11, 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 $\text{Jac}\!\left(\tilde{Q},\tilde{W}^{\unicode{x03BC}}\right)$ with the Poincaré polynomials of the BPS cohomology.

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

(2.5) \begin{equation}\mathcal{H}^i\!\left(\left(\{x\}\times\mathbb{A}^1\rightarrow X\times\mathbb{A}^1\right)^*\mathcal{F}\right)\end{equation}

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 (2.5) is constant. Equivalently, we may define $\mathcal{C}_X$ as the essential image of $\unicode{x03C0}_X^*[1]$ , for $\unicode{x03C0}_X\,:\, X\times\mathbb{A}^1\rightarrow X$ the projection. We denote by $\textbf{MMHM}(X)$ the Serre quotient $\mathcal{B}_X/\mathcal{C}_X$ . There is an embedding of categories $\textbf{MHM}(X)\hookrightarrow \textbf{MMHM}(X)$ defined via

\begin{align*}\left(X\times\{0\}\hookrightarrow X\times\mathbb{A}^1\right)_*\,:\, \textbf{MHM}(X)\rightarrow \textbf{MHM}\!\left(X\times\mathbb{A}^1\right).\end{align*}

The direct image functor

\begin{align*}\Theta\,:\!=\, \left(X\times\mathbb{C}^*\hookrightarrow X\times\mathbb{A}^1\right)_*\end{align*}

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

\begin{align*}\textbf{rat}\circ \!\left(X\xrightarrow{x\mapsto (x,1)}X\times\mathbb{C}^*\right)^*\circ\Theta^{-1}[{-}1]\end{align*}

taking a monodromic mixed Hodge module to its underlying perverse sheaf on X. We abuse notation by denoting this functor also by $\textbf{rat}$ . 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

\begin{align*}\unicode{x03D5}^{\text{mon}}_{f}\,:\,&\textbf{MHM}(X)\rightarrow \textbf{MMHM}(X)\\&\mathcal{F}\mapsto \Theta \underline{\unicode{x03D5}}_{u\cdot f}\!\left(\mathcal{F}\boxtimes\underline{\mathbb{Q}}_{\mathbb{C}^*}\right)[1],\end{align*}

where u is the coordinate on $\mathbb{C}^*$ and $\underline{\unicode{x03D5}}_{u\cdot f}$ is the lift of ${{}^{\mathfrak{p}}\!}\unicode{x03D5}_{u\cdot f}$ to the categories of mixed Hodge modules. There is a natural isomorphism $\textbf{rat} \unicode{x03D5}^{\text{mon}}_{f}\cong {{}^{\mathfrak{p}}\!}\unicode{x03D5}_{f}\textbf{rat}$ .

An object $\mathcal{F}\in \text{Ob}(\textbf{MMHM}(X))$ inherits a weight filtration from the weight filtration on objects of $\textbf{MHM}(X\times\mathbb{A}^1)$ . 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 $\mathcal{F}\in \text{Ob}\!\left(\mathcal{D}^{\textrm{b}}(\textbf{MMHM}(X))\right)$ is pure if each $\mathcal{H}^i(\mathcal{F})$ is pure of weight i.

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

\begin{align*}\mathscr{L}^{\,\,1/2}\,:\!=\, \text{cone}\!\left(\underline{\mathbb{Q}}_{\mathbb{A}_1}\rightarrow d_*\underline{\mathbb{Q}}_{\mathbb{A}_1}\right),\end{align*}

where $d\,:\, \mathbb{A}^1\rightarrow \mathbb{A}^1$ 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

\begin{align*}\mathcal{IC}^{\text{hdg}}_{X}\,:\!=\, \mathcal{IC}_X\!\left(\underline{\mathbb{Q}}_{X^{\textrm{sm}}}[{\dim} X]\right)\otimes \mathscr{L}^{\,-\dim(X)/2}[{-}\dim X]\end{align*}

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

We define

(2.6) \begin{equation}\mathcal{BPS}^{\text{hdg}}_{Q,W,\textbf{d}}\,:\!=\, \begin{cases}\unicode{x03D5}^{\text{mon}}_{\text{Tr}(W)}\mathcal{IC}^{\text{hdg}}_{\mathcal{M}_{\textbf{d}}(Q)}& \textrm{if there is a $\textbf{d}$-dimensional simple $\mathbb{C}Q$-module}\\[4pt] 0&\textrm{otherwise.}\end{cases}\end{equation}

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

\begin{align*}\textrm{BPS}^{\text{hdg}}_{Q,W,\textbf{d}}\,:\!=\, \textbf{H}\!\left(\mathcal{M}_{\textbf{d}}(Q),\mathcal{BPS}^{\text{hdg}}_{Q,W,\textbf{d}}\right)\end{align*}

satisfying $\textbf{rat}\!\left(\textrm{BPS}^{\text{hdg}}_{Q,W,\textbf{d}}\right)\cong \textrm{BPS}_{Q,W,\textbf{d}}$ . We define^{Footnote 5}

\begin{align*}\mathcal{RA}_{Q,W}^{\text{hdg}}=\texttt{JH}_*\unicode{x03D5}^{\text{mon}}_{\text{Tr}(W)}\mathcal{IC}_{\mathfrak{M}(Q)}^{\text{hdg}}.\end{align*}

Since all of the natural transformations defining the multiplication on the Hall algebra $\mathcal{A}_{Q,W}$ lift to categories of monodromic mixed Hodge modules [Reference Davison8, Reference Davison and Meinhardt11, Reference Kontsevich and Soibelman26], as does the Thom–Sebastiani theorem [Reference Saito39], we may define a multiplication on $\mathcal{RA}_{Q,W}^{\text{hdg}}$ that recovers the multiplication on $\mathcal{RA}_{Q,W}$ after applying the functor $\textbf{rat}$ . Likewise, taking derived direct image to a point we obtain the algebra object $\mathcal{A}^{\text{hdg}}_{Q,W}$ in monodromic mixed Hodge structures. Then by [Reference Davison and Meinhardt11], Theorems 2.3 and 2.4 lift to the categories of monodromic mixed Hodge modules and monodromic mixed Hodge structures. In particular, the BPS Lie algebra $\mathfrak{g}_{Q,W}=\textrm{BPS}_{Q,W}[{-}1]$ lifts to a Lie algebra object

\begin{align*}\mathfrak{g}^{\text{hdg}}_{Q,W}\,:\!=\, \textrm{BPS}^{\text{hdg}}_{Q,W}\otimes\mathscr{L}^{\,\,1/2}\end{align*}

inside $\mathcal{D}^{\textrm{b}}(\textbf{MMHM}(\text{pt}))$ . This is a Lie subalgebra of $\mathcal{A}_{Q,W}^{\text{hdg}}$ , 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 [Reference Davison8, Reference Davison and Meinhardt11] 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 $\left(\tilde{Q},\tilde{W}\right)$ , as in the introduction. Firstly, we recall the following purity result on the BPS cohomology of the Jacobi algebra $\text{Jac}\!\left(\tilde{Q},\tilde{W}\right)$ :

Theorem 2.6. [Reference Davison9] For an arbitrary quiver Q and dimension vector $\textbf{d}\in\mathbb{N}^{Q_0}$ , the mixed Hodge structure $\textrm{BPS}^{\text{hdg}}_{\tilde{Q},\tilde{W},\textbf{d}}$ is pure, of Tate type. In addition (or as a consequence of the cohomological integrality theorem), the mixed Hodge structure on

\begin{align*}\textbf{H}_c\!\left(\mathfrak{M}_{\textbf{d}}\!\left(\tilde{Q}\right),\unicode{x03D5}^{\text{mon}}_{\text{Tr}(\tilde{W})}\mathcal{IC}^{\text{hdg}}_{\mathfrak{M}_{\textbf{d}}\!\left(\tilde{Q}\right)}\right)\end{align*}

is pure, of Tate type.

We denote by

(2.7) \begin{equation}\tilde{e}\,:\, \mathcal{M}\!\left(\Pi_Q\right)\times\mathbb{A}^1\rightarrow \mathcal{M}\!\left(\tilde{Q}\right)\end{equation}

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. [Reference Davison9] For a quiver Q and dimension vector $\textbf{d}\in\mathbb{N}^{Q_0},$ there is a perverse sheaf

\begin{align*}\mathcal{BPS}_{\Pi_Q,\textbf{d}}\in \textbf{Perv}\!\left(\mathcal{M}_{\textbf{d}}\!\left(\Pi_Q\right)\right)\end{align*}

such that there is an isomorphism

\begin{align*}\mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}\cong \tilde{e}_!\left(\mathcal{BPS}_{\Pi_Q,\textbf{d}}\boxtimes \mathcal{IC}_{\mathbb{A}^1}\right).\end{align*}

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 [Reference Davison9] the description of the BPS cohomology of the Jacobi algebra $\text{Jac}\!\left(\tilde{Q},\tilde{W}\right)$ in terms of Kac polynomials:

Theorem 2.8. The Poincaré polynomials of the cohomological BPS invariants for the QP $\left(\tilde{Q},\tilde{W}\right)$ satisfy

(2.8) \begin{equation}\texttt{p}\!\left(\textrm{BPS}_{\tilde{Q},\tilde{W},\textbf{d}},q^{1/2}\right)=q^{-1/2}\texttt{a}_{Q,\textbf{d}}\!\left(q^{-1}\right),\end{equation}

where

\begin{align*}\texttt{a}_{Q,\textbf{d}}(q)=\sum_{i\in\mathbb{Z}_{\geq 0}} \texttt{a}_{Q,\textbf{d},i}\:q^i\end{align*}

is the Kac polynomial, counting the number of isomorphism classes of absolutely^{Footnote 6} indecomposable $\mathbb{F}_qQ$ -modules for $\mathbb{F}_q$ a field of order q. Furthermore, the natural mixed Hodge structure on $\textrm{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}$ is pure, of Tate type, so that we can write

(2.9) \begin{equation}\textrm{BPS}^{\text{hdg}}_{\tilde{Q},\tilde{W},\textbf{d}}\cong \bigoplus_{i\in\mathbb{Z}_{\geq 0}}\!\left(\mathscr{L}^{\,\,i-1/2}\right)^{\oplus \texttt{a}_{Q,\textbf{d},-i}}\end{equation}

and so

(2.10) \begin{equation}\mathfrak{g}^{\text{hdg}}_{\tilde{Q},\tilde{W},\textbf{d}}\cong \bigoplus_{i\in\mathbb{Z}_{\geq 0}}(\mathscr{L}^{\,\,i})^{\oplus \texttt{a}_{Q,\textbf{d},-i}}.\end{equation}

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 [Reference Davison and Pădurariu13]. 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. [Reference Davison and Pădurariu13, 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

\begin{align*}g=g_0+\sum_{1\leq j\leq m} g_jt_j,\end{align*}

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

\begin{align*}\unicode{x03C0}\,:\, &\overline{X}\rightarrow X\\q\,:\,&X\times\mathbb{A}^{n-m}\rightarrow X\end{align*}

the natural projections. Then, the natural transformation

(3.1) \begin{equation}\unicode{x03C0}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{g}\mathbb{Q}_{\overline{X}/G}\rightarrow \unicode{x03C0}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{g_0}\mathbb{Q}_{\overline{Z}/G}\cong q_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{g_0}\mathbb{Q}_{Z/G}[{-}2m]\end{equation}

is an isomorphism.

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

\begin{align*}\unicode{x03C0}_!\unicode{x03D5}^{\text{mon}}_{g}\underline{\mathbb{Q}}_{\overline{X}/G}\rightarrow \unicode{x03C0}_!\unicode{x03D5}^{\text{mon}}_{g_0}\underline{\mathbb{Q}}_{\overline{Z}/G}\cong q_!\unicode{x03D5}^{\text{mon}}_{g_0}\underline{\mathbb{Q}}_{Z/G}\otimes \mathscr{L}^{\,m}\end{align*}

is an isomorphism in^{Footnote 7} $\mathcal{D}^{\textrm{b}}(\textbf{MMHM}(X/G))$ .

3.2 Proof of Theorem A

We proceed by applying deformed dimensional reduction to the function $\text{Tr}\!\left(\tilde{W}^{\unicode{x03BC}}\right)$ .

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 $f\in {\text{Hom}}_{\mathbb{C} Q}(\rho,\rho)$ . Then, we consider the commutative diagram

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

\begin{align*}\unicode{x03C0}&=qp\\\unicode{x03C0}^{\prime}&=q^{\prime}p^{\prime}.\end{align*}

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 $Q^{\text{op}}$ the quiver containing the same vertices as Q, and only the arrows $a^*$ for $a\in Q_1$ , i.e. $Q^{\text{op}}$ is the opposite quiver to Q. In the notation of Theorem 3.1, we set

(3.2) \begin{align}X&=\mathbb{A}_{\textbf{d}}(Q)\\[2pt]\mathbb{A}^m&=\mathbb{A}_{\textbf{d}}\!\left(Q^{\text{op}}\right)\nonumber\\[2pt]\nonumber \mathbb{A}^{n-m}&=\mathbb{A}_{\textbf{d}}(\Omega)\\[2pt] \nonumber G&= \text{GL}_{\textbf{d}}.\end{align}

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

We set

\begin{align*}L_{\unicode{x03BC}}=\frac{1}{2}\sum_{i\in Q_0}\unicode{x03BC}_i\omega_i^2\end{align*}

so $\tilde{W}^{\unicode{x03BC}}=\tilde{W}+L_{\unicode{x03BC}}$ . Then $\text{Tr}\!\left(\tilde{W}+L_{\unicode{x03BC}}\right)$ satisfies the conditions of Theorem 3.1; for example, it is $\mathbb{C}^*$ semi-invariant with weight two. In the notation of that theorem $Z\subset X\times \mathbb{A}^{n-m}=\mathbb{A}_{\textbf{d}}(Q^+)$ 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 isomorphism^{Footnote 8}

(3.3) \begin{align} \unicode{x03C0}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W}^{\unicode{x03BC}})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)}\cong\ & q_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}\mathbb{Q}_{\mathfrak{Z}(Q)}.\end{align}

If instead we set

\begin{align*}X & =\mathbb{A}_{\textbf{d}}(Q^+)\\\mathbb{A}^m & =\mathbb{A}_{\textbf{d}}(\Omega)\end{align*}

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

(3.4) \begin{align} p_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)}\cong& \mathbb{Q}_{\mathfrak{I}(Q)}.\end{align}

Since $\texttt{JH}^+$ is approximated by proper maps, there is a natural isomorphism (see [Reference Davison and Meinhardt11, Sec. 4.1])

(3.5) \begin{equation}\texttt{JH}^+_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}\mathbb{Q}_{\mathfrak{I}(Q)}\cong {{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}\texttt{JH}^+_!\mathbb{Q}_{\mathfrak{I}(Q)}.\end{equation}

Combining all of the above, we can write^{Footnote 9}

\begin{align*}\unicode{x03C0}^{\prime}_!\tilde{\texttt{JH}}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W}^{\unicode{x03BC}})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)} & \cong \texttt{JH}_!\unicode{x03C0}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W}^{\unicode{x03BC}})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)}&\\&\cong \texttt{JH}_!q_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}\mathbb{Q}_{\mathfrak{I}(Q)} &\textrm{by }(3.3)\\&\cong \texttt{JH}_!q_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}p_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)}&\textrm{by }(3.4)\\&\cong q^{\prime}_!\texttt{JH}^+_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}p_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)}\\&\cong q^{\prime}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}\texttt{JH}^+_!p_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)}&\textrm{by } (3.5)\\&\cong q^{\prime}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}p^{\prime}_!\tilde{\texttt{JH}}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)}\\&\cong q^{\prime}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}p^{\prime}_!\textbf{Sym}_{\oplus}\!\left(\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0}\setminus 0} \mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}\otimes\textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}\right)&\textrm{by Theorem }(2.4)\\&\cong q^{\prime}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}\textbf{Sym}_{\oplus}\!\left(\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0}\setminus 0} p^{\prime}_!\mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}\otimes\textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}\right)&\textrm{since }p^{\prime}\textrm{ is a monoid map}\\&\cong q^{\prime}_!\textbf{Sym}_{\oplus}\!\left(\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0}\setminus 0} {{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}p^{\prime}_!\mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}\otimes\textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}\right)&\textrm{by Thom\unicode{x2015}Sebastiani.}\end{align*}

By Lemma 2.7, we can write

(3.6) \begin{equation}p^{\prime}_!\mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}\cong e_!\left(\mathcal{F}\boxtimes \mathcal{IC}_{\mathbb{A}^1}\right),\end{equation}

where

\begin{align*}\mathcal{F}=(\mathcal{M}(\Pi_Q)\rightarrow \mathcal{M}(Q))_!\mathcal{BPS}_{\Pi_Q,\textbf{d}}.\end{align*}

Writing $f=\text{Tr}\!\left(L_{\unicode{x03BC}}\right)\lvert_{\mathcal{M}_{\textbf{d}}(Q)\times\mathbb{A}^1}$ , we have

\begin{align*}f\,:\, &\mathcal{M}_{\textbf{d}}(Q)\times\mathbb{A}^1\rightarrow\mathbb{C}\\&(\rho,t)\mapsto \frac{1}{2}(\unicode{x03BC}\cdot \textbf{d})t^2.\end{align*}

It follows from (3.6) that

\begin{align*}{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}p^{\prime}_!\mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}\cong\begin{cases}e_!(\mathcal{F}\boxtimes \mathbb{Q}_0)&\textrm{if }\unicode{x03BC}\cdot \textbf{d}\neq 0\\[5pt] e_!\left(\mathcal{F}\boxtimes \mathcal{IC}_{\mathbb{A}^1}\right)&\textrm{if }\unicode{x03BC}\cdot \textbf{d}= 0,\end{cases}\end{align*}

so that

\begin{align*}\textbf{H}_c(\mathcal{M}_{\textbf{d}}(Q^+),{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}\!\left(L_{\unicode{x03BC}}\right)}p^{\prime}_!\mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}})\cong \textbf{H}_c(\mathcal{M}_{\textbf{d}}(\Pi_Q),\mathcal{BPS}_{\Pi_Q,\textbf{d}})[{-}g(\textbf{d})],\end{align*}

where we define

\begin{align*}g(\textbf{d})=\begin{cases} 1&\textrm{if }\unicode{x03BC}\cdot\textbf{d}=0\\[5pt]0&\textrm{if }\unicode{x03BC}\cdot \textbf{d}\neq 0.\end{cases}\end{align*}

Finally, we deduce that

(3.7) \begin{align}\textbf{H}_c\!\left(\mathfrak{M}\!\left(\tilde{Q}\right),{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W}^{\unicode{x03BC}})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)}\right)& = \unicode{x03C4}_!\unicode{x03C0}^{\prime}_!\tilde{\texttt{JH}}_!{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W}^{\unicode{x03BC}})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)}\nonumber \\& \cong \textbf{Sym}\!\left(\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0}\setminus 0}\left(\textbf{H}_c(\mathcal{M}_{\textbf{d}}(\Pi_Q),\mathcal{BPS}_{\Pi_Q,\textbf{d}})\otimes \textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}} \right)[{-}g(\textbf{d})]\right).\end{align}

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

(3.8) \begin{equation}\textbf{H}_c(\mathfrak{M}\!\left(\tilde{Q}\right),{{}^{\mathfrak{p}}\!}\unicode{x03D5}_{\text{Tr}(\tilde{W}^{\unicode{x03BC}})}\mathcal{IC}_{\mathfrak{M}\!\left(\tilde{Q}\right)})\cong \textbf{Sym}\!\left(\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0}\setminus 0}\textrm{BPS}^{\vee}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}},\textbf{d}}\otimes \textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}\right).\end{equation}

Comparing (3.7) and (3.8), we find

\begin{align*}\textrm{BPS}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}},\textbf{d}}\cong\ &\left(\textbf{H}_c\!\left(\mathcal{M}_{\textbf{d}}\!\left(\Pi_Q\right),\mathcal{BPS}_{\Pi_Q,\textbf{d}}\right)\right)^{\vee}[g(\textbf{d})]\\\cong\ &\left(\textbf{H}_c\!\left(\mathcal{M}_{\textbf{d}}\!\left(\tilde{Q}\right),\mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}\right)[1]\right)^{\vee}[g(\textbf{d})]& \textrm{by Theorem 2.7}\\\cong\ &\textbf{H}_c\!\left(\mathcal{M}_{\textbf{d}}\!\left(\tilde{Q}\right),\mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}\right)^{\vee}[g(\textbf{d})-1]\\\cong\ &\textrm{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}[g(\textbf{d})-1]\end{align*}

and now (2.8) follows from Theorem 2.8.

Performing the same calculations in the category of monodromic mixed Hodge structures, isomorphisms (3.7) and (3.8) yield the isomorphism of $\mathbb{N}^{Q_0}$ -graded complexes of monodromic mixed Hodge structures

(3.9) \begin{equation}\textbf{Sym}\!\left(\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0}\setminus 0}\textrm{BPS}^{\text{hdg},\vee}_{\tilde{Q},\tilde{W},\textbf{d}}\otimes \mathscr{L}^{\,\,(g(\textbf{d})-1)/2}\otimes \textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}} \right)\cong \textbf{Sym}\!\left(\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0}\setminus 0}\textrm{BPS}^{\text{hdg},\vee}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}},\textbf{d}}\otimes \textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}\right),\end{equation}

where

\begin{align*}\textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}=\bigoplus_{i\geq 0}\mathscr{L}^{\,-1/2-i}.\end{align*}

On the other hand, by Theorem 2.6 the mixed Hodge structure $\textrm{BPS}^{\text{hdg}}_{\Pi_Q,\textbf{d}}$ is pure, so that $\textrm{BPS}^{\text{hdg},\vee}_{\Pi_Q,\textbf{d}}\otimes\mathscr{L}^{\,n/2}$ is pure for all $n\in\mathbb{Z}$ . It follows that both sides of (3.9) are pure. The isomorphism

\begin{align*}\textrm{BPS}^{\text{hdg}}_{\tilde{Q},\tilde{W},\textbf{d}}\otimes \mathscr{L}^{\,\,(g(\textbf{d})-1)/2}\cong \textrm{BPS}^{\text{hdg}}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}},\textbf{d}}\end{align*}

then follows from (3.9) and semisimplicity of the category of pure monodromic mixed Hodge structures as in [Reference Davison and Pădurariu13, 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 $\unicode{x03BC}=\unicode{x03BC}_0e_0$ to be generic, we just have to pick $\unicode{x03BC}_0\neq 0$ . Then, we find

\begin{align*}\text{Jac}\left(Q^{(1)},\tilde{W}^{\unicode{x03BC}}\right) & = \text{Jac}\left(Q^{(1)},\omega_0^2\right)\\& \cong \mathbb{C} Q^{(0)}\end{align*}

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

Remark 3.4. Along the course of the proof of Theorem A, we have shown that

\begin{align*}\unicode{x03C0}^{\prime}_*\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}},\textbf{d}}\cong \begin{cases} \unicode{x03C0}^{\prime}_*\mathcal{BPS}_{\tilde{Q},\tilde{W}}[{-}1] &\textrm{if }\unicode{x03BC}\cdot \textbf{d}\neq 0\\[4pt]\unicode{x03C0}^{\prime}_*\mathcal{BPS}_{\tilde{Q},\tilde{W}}&\textrm{otherwise.}\end{cases}\end{align*}

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 $\unicode{x03BC}\in R$ , and a number $n\in\mathbb{Z}_{\geq 1}$ , define

\begin{align*}\tilde{W}^{\unicode{x03BC}}_n=\tilde{W}+\frac{1}{n}\unicode{x03BC}\omega^n.\end{align*}

Then in the decomposition (3.2), we let $\mathbb{C}^*$ act with weight 1 on $\mathbb{A}_{\textbf{d}}(\Omega)$ , weight $n-1$ on $\mathbb{A}_{\textbf{d}}\!\left(Q^{\text{op}}\right)$ , and trivially on $\mathbb{A}_{\textbf{d}}(Q)$ , so $\text{Tr}(\tilde{W}^{\unicode{x03BC}}_{\textbf{d}})$ satisfies the conditions of Theorem 3.1 (now it is a weight n function). Then, the argument of Section 3.2 gives that

(3.10) \begin{equation}\textrm{BPS}^{\text{hdg}}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}_n,\textbf{d}}\cong \textrm{BPS}^{\text{hdg}}_{\tilde{Q},\tilde{W},\textbf{d}}\otimes \mathscr{L}^{\,\,1/2}\otimes \textbf{H}\!\left(\mathbb{A}^1,\unicode{x03D5}^{\text{mon}}_{(\unicode{x03BC}\cdot\textbf{d}) x^n}\mathcal{IC}_{\mathbb{A}^1}\right).\end{equation}

Theorem A follows from the special case $n=2$ and Theorem 2.8, observing that $\textbf{H}\!\left(\mathbb{A}^1,\unicode{x03D5}^{\text{mon}}_{x^2}\underline{\mathbb{Q}}_{\mathbb{A}^1}\right)=\mathscr{L}^{\,\,1/2}$ .

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

\begin{align*}\tilde{W}_1^{\unicode{x03BC}}=\sum_{a\in Q_1}[a,a^*]+\unicode{x03BC}\omega.\end{align*}

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

\begin{align*}\unicode{x03C0}_!\unicode{x03D5}^{\text{mon}}_{\text{Tr}(\tilde{W}_1^{\unicode{x03BC}})}\underline{\mathbb{Q}}_{\mathfrak{M}_{\textbf{d}}\!\left(\tilde{Q}\right)}\rightarrow \underline{\mathbb{Q}}_{\mathfrak{M}_{\textbf{d}}\left(\Pi_{Q,\unicode{x03BC}}\right)}\otimes\mathscr{L}^{\,\,\textbf{d}\cdot\textbf{d}}.\end{align*}

Proof. In the setup of Theorem 3.1, we put $X=\mathbb{A}_{\textbf{d}}(\overline{Q})$ , $n=m$ , $\mathbb{A}^m=\mathbb{A}_{\textbf{d}}(\Omega)$ and $G=\text{GL}_{\textbf{d}}$ . Then, $g_0$ is the zero function, so that there is a natural isomorphism $\unicode{x03D5}^{\text{mon}}_{g_0}\rightarrow \text{id}$ , and the result follows from Theorem 3.1 and $\dim(\mathbb{A}_{\textbf{d}}(\Omega))=\textbf{d}\cdot\textbf{d}$ .

Proof of Theorem B. We have isomorphisms of $\mathbb{N}^{Q_0}$ -graded mixed Hodge structures

\begin{align*}\bigoplus_{\textbf{d}\in \mathbb{N}^{Q_0}}\textbf{H}_c\!\left(\mathfrak{M}_{\textbf{d}}(\Pi_{Q,\unicode{x03BC}}),\mathbb{Q}\right)\otimes \mathscr{L}^{\,\,\unicode{x03C7}_Q(\textbf{d},\textbf{d})}\cong &\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0}}\textbf{H}_c\!\left(\mathfrak{M}_{\textbf{d}}\!\left(\tilde{Q}\right),\unicode{x03D5}^{\text{mon}}_{\text{Tr}(W_1^{\unicode{x03BC}})}\mathcal{IC}_{\mathfrak{M}_{\textbf{d}}\!\left(\tilde{Q}\right)}\right)\\\cong&\textbf{Sym}\!\left(\bigoplus_{\mathbb{N}^{Q_0}\ni \textbf{d}\neq 0}\textrm{BPS}^{\text{hdg},\vee}_{\tilde{Q},W_1^{\unicode{x03BC}},\textbf{d}}\otimes\textbf{H}_c\!\left({\text{pt}}/\mathbb{C}^*\right)_{\text{vir}}\right),\end{align*}

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 (2.9) and the claim that

\begin{align*}\textrm{BPS}^{\text{hdg}}_{\tilde{Q},W_1^{\unicode{x03BC}},\textbf{d}}\cong\begin{cases} \textrm{BPS}^{\text{hdg}}_{\tilde{Q},\tilde{W},\textbf{d}} &\textrm{if }\textbf{d}\cdot\unicode{x03BC}=0\\0& \textrm{otherwise.}\end{cases}\end{align*}

This follows from (3.10) and the observation that for $\lambda\in\mathbb{C}$

\begin{align*}\textbf{H}\!\left(\mathbb{A}^1,\unicode{x03D5}^{\text{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}\tag*{$\square$}\end{align*}

Assume that the dimension vector $\textbf{d}$ is indivisible, meaning that there is no $n\in \mathbb{Z}_{\geq 2}$ such that $\frac{1}{n}\textbf{d}\in\mathbb{N}^{Q_0}$ . Assume also that $\unicode{x03BC}$ is chosen to be generic, subject to the constraint that $\textbf{d}\cdot\unicode{x03BC}=0$ . Equivalently, $\textbf{d}$ and $\unicode{x03BC}$ are chosen so that $\textbf{d}^{\prime}\cdot \unicode{x03BC}=0$ implies that $\textbf{d}^{\prime}$ is an integer multiple of $\textbf{d}$ . Then (1.17) simplifies to

(3.11) \begin{equation}\bigoplus_{n\in\mathbb{Z}_{\geq 0}}\textbf{H}_c\!\left(\mathfrak{M}_{n \textbf{d}}(\Pi_{Q,\unicode{x03BC}}),\mathbb{Q}\right)\otimes \mathscr{L}^{\,n^2\unicode{x03C7}_Q(\textbf{d},\textbf{d})}\cong \textbf{Sym}\!\left(\bigoplus_{n\in\mathbb{Z}_{>0}}\textrm{BPS}_{\tilde{Q},\tilde{W},n\textbf{d}}^{\text{hdg},\vee}\otimes\textbf{H}_c({\text{pt}}/\mathbb{C}^*,\mathbb{Q})_{\text{vir}}\right).\end{equation}

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

\begin{align*}\textbf{H}_c\!\left(\mathfrak{M}_{\textbf{d}}(\Pi_{Q,\unicode{x03BC}}),\mathbb{Q}\right)\cong \textbf{H}_c(\mathcal{M}_{\textbf{d}}(\Pi_Q),\mathbb{Q})\otimes \textbf{H}_c({\text{pt}}/\mathbb{C}^*,\mathbb{Q}).\end{align*}

Recall that there is an isomorphism

\begin{align*}\textbf{H}_c({\text{pt}}/\mathbb{C}^*,\mathbb{Q})\cong\bigoplus_{i\geq 0}\mathscr{L}^{\,-i-1}.\end{align*}

From the $n=1$ piece of (3.11), we deduce the following slightly stronger version of a result of Crawley–Boevey and Van den Bergh [Reference Crawley-Boevey and Van den Bergh7]:

Corollary 3.6. Let $\textbf{d}$ be indivisible, and $\unicode{x03BC}$ be generic. Then, there is an isomorphism of mixed Hodge structures

\begin{align*}\textbf{H}_c(\mathcal{M}_{\textbf{d}}(\Pi_{Q,\unicode{x03BC}}),\mathbb{Q})\cong \bigoplus_{i\geq 0}(\mathscr{L}^{\,\,1+i- \unicode{x03C7}_Q(\textbf{d},\textbf{d})})^{\oplus \texttt{a}_{Q,\textbf{d},i}}.\end{align*}

In particular, $\textbf{H}_c(\mathcal{M}_{\textbf{d}}(\Pi_{Q,\unicode{x03BC}}),\mathbb{Q})$ is pure of Tate type.

4.1 Calculating the BPS sheaves

While our main theorem gives a way to calculate the BPS cohomology $\textrm{BPS}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}},\textbf{d}}$ for arbitrary Q, $\textbf{d}\in \mathbb{N}^{Q_0}$ and $\unicode{x03BC}\in R$ , the actual BPS sheaf $\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}},\textbf{d}}$ remains a little mysterious. We can at least generalise (part of) the support lemma (Lemma 2.7) from the case $\unicode{x03BC}=0$ .

Lemma 4.1. For a quiver Q, and dimension vector $\textbf{d}\in\mathbb{N}^{Q_0}$ , we have

\begin{align*}\text{supp}(\mathcal{BPS}_{\Pi^{\unicode{x03BC}}_Q,\textbf{d}})\subset\begin{cases}\tilde{e}(\mathcal{M}_{\textbf{d}}(\Pi_Q)\times\mathbb{A}^1)& \textrm{if } \unicode{x03BC}\cdot \textbf{d}=0\\[5pt]\tilde{e}(\mathcal{M}_{\textbf{d}}(\Pi_Q)\times\{0\})&\textrm{if }\unicode{x03BC}\cdot\textbf{d}\neq 0,\end{cases}\end{align*}

where $\tilde{e}$ is defined in (2.7).

Proof. The proof is very similar to [Reference Davison9, Lem. 4.1]. Let $\rho$ be a $\text{Jac}\!\left(\tilde{Q},\tilde{W}^{\unicode{x03BC}}\right)$ -module lying in the support of $\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}},\textbf{d}}$ . By definition, $\rho$ is semisimple, and so since $\rho\!\left(\sum_{i\in Q_0}\omega_i\right)$ is central, it acts via a diagonal matrix. Arguing as in [Reference Davison9, Lem. 4.1] and using centrality of $\rho\!\left(\sum_{i\in Q_0}\omega_i\right)$ 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 $\unicode{x03BC}\cdot\textbf{d}\neq 0$ . Arguing as in Proposition 2.3, if $\unicode{x03BC}\cdot\textbf{d}\neq 0$ , we must have $\lambda=0$ . This establishes the lemma in the case $\unicode{x03BC}\cdot\textbf{d}\neq 0$ .

Even in the case $\unicode{x03BC}\cdot \textbf{d}=0,$ this is a slightly weaker statement than Lemma 2.7; it is harder to show that $\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}},\textbf{d}}$ 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 $\text{Tr}(\tilde{W}^{\unicode{x03BC}})$ is not $\mathbb{A}^1$ -invariant unless $\unicode{x03BC}=0$ .

It would be interesting to compare $\mathcal{BPS}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}},\textbf{d}}$ with $\mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}\cong \tilde{e}_!\left(\mathcal{BPS}_{\Pi_Q,\textbf{d}}\boxtimes \mathcal{IC}_{\mathbb{A}^1}\right)$ in the case $\unicode{x03BC}\cdot\textbf{d}=0$ , and with $\tilde{e}_!\left(\mathcal{BPS}_{\Pi_Q,\textbf{d}}\boxtimes \mathbb{Q}_0\right)$ in the case $\unicode{x03BC}\cdot \textbf{d}\neq 0$ . In particular, we may ask the following question:

By [Reference Davison10, Thm. A], we know that $\mathcal{BPS}_{\tilde{Q},\tilde{W},\textbf{d}}$ is a pure monodromic mixed Hodge module, i.e. we know that for $\unicode{x03BC}=0$ the answer to Question 4.2 is yes.

4.2 The BPS algebra

Identifying the BPS cohomology of the Jacobi algebra $\mathbb{C}\!\left(\tilde{Q},\tilde{W}^{\unicode{x03BC}}\right)$ is only part of understanding its cohomological DT theory; it tells us the size of the graded pieces of $\mathcal{A}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}$ , but nothing about the algebra structure above what we already know from [Reference Davison and Meinhardt11] regarding general quivers with potential (e.g. the PBW theorem). At least for $\unicode{x03BC}=0$ , the algebra $\mathcal{A}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}$ has been studied from various points of view (see e.g. [Reference Diaconescu, Porta and Sala14, Reference Kapranov and Vasserot22, Reference Maulik and Okounkov31, Reference Nakajima35, Reference Varagnolo44]) while for the $\unicode{x03BC}\neq 0$ 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 [Reference Galakhov and Yamazaki17, Reference Li and Yamazaki27].

For general Q and $\unicode{x03BC}$ , the shifted BPS cohomology $\mathfrak{g}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}\,:\!=\, \textrm{BPS}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}[{-}1]$ carries a Lie algebra structure (by Theorem 2.5). Even in the case $\unicode{x03BC}=0,$ we do not yet fully understand this Lie algebra for general Q. We can at least try to relate the case of general $\unicode{x03BC}$ to the case $\unicode{x03BC}=0$ , via the following construction: Fix Q and $\unicode{x03BC}$ , and define

\begin{align*}\mathfrak{g}_{\text{even}}\,:\!=\, &\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0} \;\lvert\; \textbf{d}\cdot\unicode{x03BC}=0}\mathfrak{g}_{\tilde{Q},\tilde{W},\textbf{d}}\\\mathfrak{g}_{\text{odd}}\,:\!=\, &\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0} \;\lvert\; \textbf{d}\cdot\unicode{x03BC}\neq 0}\mathfrak{g}_{\tilde{Q},\tilde{W},\textbf{d}}.\end{align*}

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

\begin{align*}\mathfrak{g}^{\unicode{x03BC}}=\mathfrak{g}_{\textrm{even}}\oplus \mathfrak{g}_{\textrm{odd}}[{-}1],\end{align*}

with the Lie bracket

\begin{align*}[(\unicode{x03B1}, \unicode{x03B2}),(\unicode{x03B1}^{\prime}, \unicode{x03B2}^{\prime})]=([\unicode{x03B1},\unicode{x03B1}^{\prime}], [\unicode{x03B1},\unicode{x03B2}^{\prime}]-[\unicode{x03B1}^{\prime},\unicode{x03B2}]).\end{align*}

Question 4.3. Is there is an isomorphism of Lie algebras

\begin{align*}\mathfrak{g}^{\unicode{x03BC}}\cong\mathfrak{g}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}?\end{align*}

Proof. The case $\unicode{x03BC}=0$ is trivial, since then

\begin{align*}\mathfrak{g}^{\unicode{x03BC}}& = \mathfrak{g}_{\textrm{even}}\\& = \mathfrak{g}_{\tilde{Q},\tilde{W}}\end{align*}

and $\tilde{W}^{\unicode{x03BC}}=\tilde{W}$ . In the case of generic $\unicode{x03BC}$ , Theorem A tells us that there is an isomorphism

(4.1) \begin{equation}\mathfrak{g}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}\cong \mathfrak{g}_{\tilde{Q},\tilde{W}}[{-}1]\end{equation}

as cohomologically graded vector spaces, and so it is sufficient to show that the Lie bracket on $\mathfrak{g}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}$ vanishes, since the Lie bracket on $\mathfrak{g}^{\unicode{x03BC}}\,:\!=\, \mathfrak{g}_{\textrm{odd}}[{-}1]$ does by definition. Combining (4.1) and (2.10), we deduce that $\mathfrak{g}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}$ 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 $\mathfrak{g}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}$ . Despite Proposition 4.4 the expected answer to Question 4.3, once extended to the double of $\mathfrak{g}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}$ is no; one of the motivations for this paper is work of Kevin Costello [Reference Costello5], in which he conjectures that in the case of, for example, the resolved conifold, the Lie bracket does not vanish on the fermionic part of the (doubled) BPS Lie algebra. As we saw in Section 1.2, and Example 3.3 the (noncommutative) resolved conifold is a partial fermionisation of the QP $\left(\tilde{Q},\tilde{W}\right)$ 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 $\mathcal{A}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}$ .

Fix a quiver Q, and a framing dimension vector $\textbf{f}\in \mathbb{N}^{Q_0}$ . We form the quiver $Q_{\textbf{f}}$ by adding one extra vertex (labelled $\infty$ ) to $Q_0$ , and $\textbf{f}_i$ arrows from $\infty$ to i for each $i\in Q_0$ . Set $\mathscr{Q}=\widetilde{Q_{\textbf{f}}}$ . 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 $\unicode{x03BC}=\sum_{i\in Q_0} \unicode{x03BC}_i e_i$ we define

\begin{align*}\tilde{W}_{\textbf{f}}^{\unicode{x03BC}}=\left(\sum_{i\in \mathscr{Q}_0}\omega_i\right) \left(\sum_{a\in(Q_{\textbf{f}})_{1}}[a,a^*]\right)+\sum_{i\in Q_0}\unicode{x03BC}_i\omega_i^2.\end{align*}

We write dimension vectors for $\mathscr{Q}$ as $\textrm{e}=(\textbf{d},n)$ where $\textbf{d}\in\mathbb{N}^{Q_0}$ and $n=\textrm{e}_{\infty}\in\mathbb{N}$ . We define

\begin{align*}\mathbb{A}_{(\textbf{d},1)}^{st}(\mathscr{Q})\subset \mathbb{A}_{(\textbf{d},1)}(\mathscr{Q})\end{align*}

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

\begin{align*}\mathscr{N}_{\textbf{f},\textbf{d}}(Q)\,:\!=\, \mathbb{A}_{(\textbf{d},1)}^{st}(\mathscr{Q})/\text{GL}_{\textbf{d}}.\end{align*}

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

\begin{align*}\mathcal{F}l_{(\textbf{d},1),\textbf{d}^{\prime}}(\mathscr{Q})\end{align*}

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

the morphism $\unicode{x03C0}_2$ is proper, and so via push-forward and pull-back in vanishing cycle cohomology, we obtain an action of $\mathcal{A}_{\tilde{Q},\tilde{W}^{\unicode{x03BC}}}$ on

\begin{align*}\textbf{M}^{\unicode{x03BC}}_{\textbf{f}}(Q)\,:\!=\,\bigoplus_{\textbf{d}\in\mathbb{N}^{Q_0}}\textbf{H}\!\left(\mathscr{N}_{\textbf{f},\textbf{d}^{\prime\prime}}(Q),\unicode{x03D5}^{\text{mon}}_{\text{Tr}(\tilde{W}_{\textbf{f}}^{\unicode{x03BC}})}\underline{\mathbb{Q}}_{\mathscr{N}_{\textbf{f},\textbf{d}^{\prime\prime}}(Q)}\right)\otimes \mathscr{L}^{\,\,\unicode{x03C7}_{\mathscr{Q}}\!\left((1,\textbf{d}),(1,\textbf{d})\right)/2}.\end{align*}

Given a quiver Q, we denote by

\begin{align*}\unicode{x03BC}_{Q,\textbf{d}}\,:\, \mathbb{A}_{\textbf{d}}(\overline{Q})&\rightarrow \mathfrak{{gl}}_{\textbf{d}}\\\rho&\mapsto \sum_{a\in Q_1}[\rho(a),\rho(a^*)].\end{align*}

the usual moment map. We define

\begin{align*}Z(\textbf{f},\textbf{d})\subset \mathbb{A}^{st}_{(\textbf{d},1)}\left(\overline{Q_{\textbf{f}}}\right)\end{align*}

the vanishing locus of the composition of $\unicode{x03BC}_{Q_{\textbf{f}},(\textbf{d},1)}$ with the natural projection $\mathfrak{{gl}}_{(\textbf{d},1)}\rightarrow \mathfrak{{gl}}_{\textbf{d}}$ . 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

\begin{align*}X(\textbf{f},\textbf{d})=Z(\textbf{f},\textbf{d})/\text{GL}_{\textbf{d}}.\end{align*}

Proposition 4.5. Denote by g the restriction of the function $\text{Tr}\!\left(\tilde{W}_{\textbf{f}}^{\unicode{x03BC}}\right)$ to $\mathscr{N}_{\textbf{f},\textbf{d}}(Q)$ , then

\begin{align*}\text{crit}(g)\cong \begin{cases}X(\textbf{f},\textbf{d})&\textrm{if }\unicode{x03BC}\cdot\textbf{d}\neq 0\\X(\textbf{f},\textbf{d})\times\mathbb{A}^1&\textrm{if }\unicode{x03BC}\cdot\textbf{d}=0.\end{cases}\end{align*}

Moreover, the monodromic mixed Hodge module

\begin{align*}\unicode{x03D5}^{\text{mon}}_{g}\underline{\mathbb{Q}}_{\mathscr{N}_{\textbf{f},\textbf{d}}(Q)}\otimes \mathscr{L}^{\,\,\unicode{x03C7}_{\mathscr{Q}}\!\left((\textbf{d},1),(\textbf{d},1)\right)/2}\end{align*}

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 crit(g) in the form

\begin{align*}g=x_1^2+\ldots+x_c^2,\end{align*}

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

In the special case $\unicode{x03BC}=0,$ it is possible to show that the monodromy of the rank one local system ${{}^{\mathfrak{p}}\!}\unicode{x03D5}_{g}\underline{\mathbb{Q}}_{\mathscr{N}_{\textbf{f},\textbf{d}^{\prime\prime}}(Q)}$ is trivial [Reference Davison10, Prop. 6.3], and we conjecture that this is always true. Assuming the conjecture, Proposition 4.5 implies that there are isomorphisms

(4.2) \begin{equation}\textbf{M}^{\unicode{x03BC}}_{\textbf{f}}(Q)\cong \!\left(\bigoplus_{\substack{\textbf{d}\in\mathbb{N}^{Q_0}\\ \unicode{x03BC}\cdot\textbf{d}\neq 0}}\textbf{H}(X(\textbf{f},\textbf{d}),\mathbb{Q})_{\text{vir}}\right)\oplus \!\left(\bigoplus_{\substack{\textbf{d}\in\mathbb{N}^{Q_0}\\ \unicode{x03BC}\cdot\textbf{d}= 0}}\textbf{H}(X(\textbf{f},\textbf{d}),\mathbb{Q})_{\text{vir}}\otimes \mathscr{L}^{\,-1/2}\right).\end{equation}

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

Let Q ^′ 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 with Q ^′. By [Reference Davison10, Thm. 6.6], there is an inclusion of Lie algebras

\begin{align*}\mathfrak{n}^-_{Q^{\prime}}\hookrightarrow \mathfrak{g}_{\tilde{Q},\tilde{W}}\end{align*}

as the part of the BPS Lie algebra lying in cohomological degree zero, so that (4.2) in case $\unicode{x03BC}=0$ allows us to reconstruct (one half of) Nakajima’s action of $\mathfrak{g}_{Q^{\prime}}$ on the cohomology of quiver varieties [Reference Nakajima34]. The conjecture suggests that the cohomology of Nakajima quiver varieties should be as crucial to the representation theory of partially fermionised BPS Lie algebras as they are to the representation theory of their bosonic counterparts.