# Brauer-Severi motives and Donaldson-Thomas invariants of quantized
threefolds

Lieven Le Bruyn  Department of Mathematics, University of Antwerp
Middelheimlaan 1, B-2020 Antwerp (Belgium)
<EMAIL_ADDRESS>
###### Abstract.

Motives of Brauer-Severi schemes of Cayley-smooth algebras associated to
homogeneous superpotentials are used to compute inductively the motivic
Donaldson-Thomas invariants of the corresponding Jacobian algebras. This
approach can be used to test the conjectural exponential expressions for these
invariants, proposed in [3]. As an example we confirm the second term of the
conjectured expression for the motivic series of the homogenized Weyl algebra.

##  1\. Introduction

We fix a homogeneous degree $d$ superpotential $W$ in $m$ non-commuting
variables $X_{1},\ldots,X_{m}$. For every dimension $n\geq 1$, $W$ defines a
regular functions, sometimes called the Chern-Simons functional

$Tr(W)~{}:~{}\mathbb{M}_{m,n}=\underbrace{M_{n}(\mathbb{C})\oplus\ldots\oplus
M_{n}(\mathbb{C})}_{m}\rTo\mathbb{C}$

obtained by replacing in $W$ each occurrence of $X_{i}$ by the $n\times n$
matrix n the $i$-th component, and taking traces.

We are interested in the (naive, equivariant) motives of the fibers of this
functional which we denote by

$\mathbb{M}_{m,n}^{W}(\lambda)=Tr(W)^{-1}(\lambda).$

Recall that to each isomorphism class of a complex variety $X$ (equipped with
a good action of a finite group of roots of unity) we associate its naive
equivariant motive $[X]$ which is an element in the ring
$K_{0}^{\hat{\mu}}(\mathrm{Var}_{\mathbb{C}})[\mathbb{L}^{-1/2}]$ (see [4] or
[3]) and is subject to the scissor- and product-relations

$[X]-[Z]=[X-Z]\quad\text{and}\quad[X].[Y]=[X\times Y]$

whenever $Z$ is a Zariski closed subvariety of $X$. A special element is the
Lefschetz motive $\mathbb{L}=[\mathbb{A}^{1}_{\mathbb{C}},id]$ and we recall
from [12, Lemma 4.1] that
$[GL_{n}]=\prod_{k=0}^{n-1}(\mathbb{L}^{n}-\mathbb{L}^{k})$ and from [3, 2.2]
that $[\mathbb{A}^{n},\mu_{k}]=\mathbb{L}^{n}$ for a linear action of
$\mu_{k}$ on $\mathbb{A}^{n}$. This ring is equipped with a plethystic
exponential  Exp, see for example [2] and [4].

The representation theoretic interest of the degeneracy locus
$Z=\\{dTr(W)=0\\}$ of the Chern-Simons functional is that it coincides with
the scheme of $n$-dimensional representations

$Z={\text{\em rep}}_{n}(R_{W})\quad\text{where}\quad
R_{W}=\frac{\mathbb{C}\langle
X_{1},\ldots,X_{m}\rangle}{(\partial_{X_{i}}(W):1\leq i\leq m)}$

of the corresponding Jacobi algebra $R_{W}$ where $\partial_{X_{i}}$ is the
cyclic derivative with respect to $X_{i}$. As $W$ is homogeneous it follows
from [4, Thm. 1.3] (or [1] if the superpotential allows ’a cut’) that its
virtual motive is equal to

$[{\text{\em
rep}}_{n}(R_{W})]_{virt}=\mathbb{L}^{-\frac{mn^{2}}{2}}([\mathbb{M}_{m,n}^{W}(0)]-[\mathbb{M}_{m,n}^{W}(1)])$

where $\hat{\mu}$ acts via $\mu_{d}$ on $\mathbb{M}^{W}_{m,n}(1)$ and
trivially on $\mathbb{M}^{W}_{m,n}(0)$. These virtual motives can be packaged
together into the motivic Donaldson-Thomas series

$U_{W}(t)=\sum_{n=0}^{\infty}\mathbb{L}^{-\frac{(m-1)n^{2}}{2}}\frac{[\mathbb{M}_{m,n}^{W}(0)]-[\mathbb{M}_{m,n}^{W}(1)]}{[GL_{n}]}t^{n}$

In [3] A. Cazzaniga, A. Morrison, B. Pym and B. Szendröi conjecture that this
generating series has an exponential expression involving simple rational
functions of virtual motives determined by representation theoretic
information of the Jacobi algebra $R_{W}$

$U_{W}(t)\overset{?}{=}{\text{\em
Exp}}(-\sum_{i=1}^{k}\frac{M_{i}}{\mathbb{L}^{1/2}-\mathbb{L}^{-1/2}}\frac{t^{m_{i}}}{1-t^{m_{i}}})$

where $m_{1}=1,\ldots,m_{k}$ are the dimensions of simple representations of
$R_{W}$ and $M_{i}\in\mathcal{M}_{\mathbb{C}}$ are motivic expressions without
denominators, with $M_{1}$ the virtual motive of the scheme parametrizing
(simple) $1$-dimensional representations. Evidence for this conjecture comes
from cases where the superpotential admits a cut and hence one can use
dimensional reduction, introduced by A. Morrison in [12], as in the case of
quantum affine three-space [3].

The purpose of this paper is to introduce an inductive procedure to test the
conjectural exponential expressions given in [3] in other interesting cases
such as the homogenized Weyl algebra and elliptic Sklyanin algebras. To this
end we introduce the following quotient of the free necklace algebra on $m$
variables

$\mathbb{T}_{m}^{W}(\lambda)=\frac{\mathbb{C}\langle
X_{1},\ldots,X_{m}\rangle\otimes{\text{\em
Sym}}(V_{m})}{(W-\lambda)},~{}\text{where}~{}V_{m}=\frac{\mathbb{C}\langle
X_{1},\ldots,X_{m}\rangle}{[\mathbb{C}\langle
X_{1},\ldots,X_{m}\rangle,\mathbb{C}\langle
X_{1},\ldots,X_{m}\rangle]_{vect}}$

is the vectorspace space having as a basis all cyclic words in
$X_{1},\ldots,X_{m}$. Note that any superpotential is an element of
${\text{\em Sym}}(V_{m})$. Substituting each $X_{k}$ by a generic $n\times n$
matrix and each cyclic word by the corresponding trace we obtain a quotient of
the trace ring of $m$ generic $n\times n$ matrices

$\mathbb{T}_{m,n}^{W}(\lambda)=\frac{\mathbb{T}_{m,n}}{(Tr(W)-\lambda)}\quad\text{with}\quad\mathbb{M}_{m,n}^{W}(\lambda)={\text{\em
trep}}_{n}(\mathbb{T}_{m,n}^{W})$

such that its scheme of trace preserving $n$-dimensional representations is
isomorphic to the fiber $\mathbb{M}_{m,n}^{W}(\lambda)$. We will see that if
$\lambda\not=0$ the algebra $\mathbb{T}_{m,n}^{W}(\lambda)$ shares many
ringtheoretic properties of trace rings of generic matrices, in particular it
is a Cayley-smooth algebra, see [10]. As such one might hope to describe
$\mathbb{M}_{m,n}^{W}(\lambda)$ using the Luna stratification of the quotient
and its fibers in terms of marked quiver settings given in [10]. However, all
this is with respect to the étale topology and hence useless in computing
motives.

For this reason we consider the Brauer-Severi scheme of
$\mathbb{T}_{m,n}^{W}(\lambda)$, as introduced by M. Van den Bergh in [17] and
further investigated by M. Reineke in [16], which are quotients of a principal
$GL_{n}$-bundles and hence behave well with respect to motives. More
precisely, the Brauer-Severi scheme of $\mathbb{T}_{m,n}^{W}(\lambda)$ is
defined as

${\text{\em
BS}}_{m,n}^{W}(\lambda)=\\{(v,\phi)\in\mathbb{C}^{n}\times{\text{\em
trep}}_{n}(\mathbb{T}_{m,n}^{W}(\lambda)~{}|~{}\phi(\mathbb{T}_{m,n}^{W}(\lambda))v=\mathbb{C}^{n}\\}/GL_{n}$

and their motives determine inductively the motives in the Donaldson-Thomas
series. In Proposition 5 we will show that

$(\mathbb{L}^{n}-1)\frac{[\mathbb{M}^{W}_{m,n}(0)]-[\mathbb{M}^{W}_{m,n}(1)]}{[GL_{n}]}$

is equal to

$[{\text{\em BS}}^{W}_{m,n}(0)]-[{\text{\em
BS}}_{m,n}^{W}(1)]+\sum_{k=1}^{n-1}\frac{\mathbb{L}^{(m-1)k(n-k)}}{[GL_{n-k}]}([{\text{\em
BS}}^{W}_{m,k}(0)]-[{\text{\em
BS}}^{W}_{m,k}(1)])([\mathbb{M}^{W}_{m,k}(0)]-[\mathbb{M}^{W}_{m,k}(1)])$

In section 4 we will compute the first two terms of $U_{W}(t)$ in the case of
the quantized $3$-space in a variety of ways. In the final section we repeat
the computation for the homogenized Weyl algebra and show that it coincides
with the conjectured expression of [3]. In a forthcoming paper [11] we will
compute the first two terms of the series for elliptic Sklyanin algebras both
in the generic case and the case of $2$-torsion points.

Acknowledgement :  I would like to thank Brent Pym for stimulating
conversations concerning the results of [3] and Balazs Szendröi for explaining
the importance of the monodromy action (which was lacking in a previous
version) and for sharing his calculations on the Exp-expressions of [3]. I am
grateful to Ben Davison for pointing out a computational error in summing up
the terms in the homogenized Weyl algebra case and explaining the equality
with the conjectured motive.

##  2\. Brauer-Severi motives

With $\mathbb{T}_{m,n}$ we will denote the trace ring of $m$ generic $n\times
n$ matrices. That is, $\mathbb{T}_{m,n}$ is the $\mathbb{C}$-subalgebra of the
full matrix-algebra $M_{n}(\mathbb{C}[x_{ij}(k)~{}|~{}1\leq i,j\leq n,1\leq
k\leq m])$ generated by the $m$ generic matrices

$X_{k}=\begin{bmatrix}x_{11}(k)&\ldots&x_{1n}(k)\\\ \vdots&&\vdots\\\
x_{n1}(k)&\ldots&x_{nn}(k)\end{bmatrix}$

together with all elements of the form $Tr(M)1_{n}$ where $M$ runs over all
monomials in the $X_{i}$. These algebras have been studied extensively by
ringtheorists in the 80ties and some of the results are summarized in the
following result

######  Proposition 1.

Let $\mathbb{T}_{m,n}$ be the trace ring of $m$ generic $n\times n$ matrices,
then

  1. (1)

$\mathbb{T}_{m,n}$ is an affine Noetherian domain with center
$Z(\mathbb{T}_{m,n})$ of dimension $(m-1)n^{2}+1$ and generated as
$\mathbb{C}$-algebra by the $Tr(M)$ where $M$ runs over all monomials in the
generic matrices $X_{k}$.

  2. (2)

$\mathbb{T}_{m,n}$ is a maximal order and a noncommutative UFD, that is all
twosided prime ideals of height one are generated by a central element and
$Z(\mathbb{T}_{m,n})$ is a commutative UFD which is a complete intersection if
and only if $n=1$ or $(m,n)=(2,2),(2,3)$ or $(3,2)$.

  3. (3)

$\mathbb{T}_{m,n}$ is a reflexive Azumaya algebra unless $(m,n)=(2,2)$, that
is, every localization at a central height one prime ideal is an Azumaya
algebra.

###### Proof.

For (1) see for example [13] or [15]. For (2) see for example [8], for (3) for
example [7]. ∎

A Cayley-Hamilton algebra of degree $n$ is a $\mathbb{C}$-algebra $A$ ,
equipped with a linear trace map $tr:A\rTo A$ satisfying the following
properties:

  1. (1)

$tr(a).b=b.tr(a)$

  2. (2)

$tr(a.b)=tr(b.a)$

  3. (3)

$tr(tr(a).b)=tr(a).tr(b)$

  4. (4)

$tr(a)=n$

  5. (5)

$\chi_{a}^{(n)}(a)=0$ where $\chi_{a}^{(n)}(t)$ is the formal Cayley-Hamilton
polynomial of degree $n$, see [14]

For a Cayley-Hamilton algebra $A$ of degree $n$ it is natural to look at the
scheme ${\text{\em trep}}_{n}(A)$ of all trace preserving $n$-dimensional
representations of $A$, that is, all trace preserving algebra maps $A\rTo
M_{n}(\mathbb{C})$. A Cayley-Hamilton algebra $A$ of degree $n$ is said to be
a smooth Cayley-Hamilton algebra if ${\text{\em trep}}_{n}(A)$ is a smooth
variety. Procesi has shown that these are precisely the algebras having the
smoothness property of allowing lifts modulo nilpotent ideals in the category
of all Cayley-Hamilton algebras of degree $n$, see [14]. The étale local
structure of smooth Cayley-Hamilton algebras and their centers have been
extensively studied in [10].

######  Proposition 2.

Let $W$ be a homogeneous superpotential in $m$ variables and define the
algebra

$\mathbb{T}_{m,n}^{W}(\lambda)=\frac{\mathbb{T}_{m,n}}{(Tr(W)-\lambda)}\quad\text{then}\quad\mathbb{M}_{m,n}^{W}(\lambda)={\text{\em
trep}}_{n}(\mathbb{T}_{m,n}^{W}(\lambda))$

If $Tr(W)-\lambda$ is irreducible in the UFD $Z(\mathbb{T}_{m,n})$, then for
$\lambda\not=0$

  1. (1)

$\mathbb{T}_{m,n}^{W}(\lambda)$ is a reflexive Azumaya algebra.

  2. (2)

$\mathbb{T}_{m,n}^{W}(\lambda)$ is a smooth Cayley-Hamilton algebra of degree
$n$ and of Krull dimension $(m-1)n^{2}$.

  3. (3)

$\mathbb{T}_{m,n}^{W}(\lambda)$ is a domain.

  4. (4)

The central singular locus is the the non-Azumaya locus of
$\mathbb{T}_{m,n}^{W}(\lambda)$ unless $(m,n)=(2,2)$.

###### Proof.

(1) : As $\mathbb{M}_{m,n}^{W}(\lambda)={\text{\em
trep}}_{n}(\mathbb{T}_{m,n}^{W}(\lambda))$ is a smooth affine variety for
$\lambda\not=0$ (due to homogeneity of $W$) on which $GL_{n}$ acts by
automorphisms, we know that the ring of invariants,

$\mathbb{C}[{\text{\em
trep}}_{n}(\mathbb{T}_{m,n}^{W}(\lambda))]^{GL_{n}}=Z(\mathbb{T}_{m,n}^{W}(\lambda))$

which coincides with the center of $\mathbb{T}_{m,n}^{W}(\lambda)$ by e.g.
[10, Prop. 2.12], is a normal domain. Because the non-Azumaya locus of
$\mathbb{T}_{m,n}$ has codimension at least $3$ (if $(m,n)\not=(2,2)$) by [7],
it follows that all localizations of $\mathbb{T}_{m,n}^{W}(\lambda)$ at height
one prime ideals are Azumaya algebras. Alternatively, using (2) one can use
the theory of local quivers as in [10].

(2) : That the Cayley-Hamilton degree of the quotient
$\mathbb{T}_{m,n}^{W}(\lambda)$ remains $n$ follows from the fact that
$\mathbb{T}_{m,n}$ is a reflexive Azumaya algebra and irreducibility of
$Tr(W)-\lambda$. Because $\mathbb{M}_{m,n}^{W}(\lambda)={\text{\em
trep}}_{n}(\mathbb{T}_{m,n}^{W}(\lambda))$ is a smooth affine variety,
$\mathbb{T}_{m,n}^{W}(\lambda)$ is a smooth Cayley-Hamilton algebra. The
statement on Krull dimension follows from the fact that the Krull dimension of
$\mathbb{T}_{m,n}$ is known to be $(m-1)n^{2}+1$.

(3) : After taking determinants, this follows from factoriality of
$Z(\mathbb{T}_{m,n})$ and irreducibility of $Tr(W)-\lambda$.

(4) : This follows from the theory of local quivers as in [10]. The most
general non-simple representations are of representation type $(1,a;1,b)$ with
the dimensions of the two simple representations $a,b$ adding up to $n$. The
corresponding local quiver is

$\textstyle{\scriptscriptstyle
1}$$\scriptstyle{(m-1)ab}$$\scriptstyle{(m-1)a^{2}+1}$$\textstyle{\scriptscriptstyle
1}$$\scriptstyle{(m-1)ab}$$\scriptstyle{(m-1)b^{2}}$

and as $(m-1)ab\geq 2$ under the assumptions, it follows that the
corresponding singular point is singular. ∎

Let us define for all $k\leq n$ and all $\lambda\in\mathbb{C}$ the locally
closed subscheme of $\mathbb{C}^{n}\times{\text{\em
trep}}_{n}(\mathbb{T}_{m,n}^{W}(\lambda))$

${\text{\em X}}_{k,n,\lambda}=\\{(v,\phi)\in\mathbb{C}^{n}\times{\text{\em
trep}}_{n}(\mathbb{T}_{m,n}^{W}(\lambda))~{}|~{}dim_{\mathbb{C}}(\phi(\mathbb{T}_{m,n}^{W}(\lambda)).v)=k\\}$

Sending a point $(v,\phi)$ to the point in the Grassmannian ${\text{\em
Gr}}(k,n)$ determined by the $k$-dimensional subspace
$V=\phi(\mathbb{T}_{m,n}^{W}(\lambda)).v\subset\mathbb{C}^{n}$ we get a
Zariskian fibration as in [12]

${\text{\em X}}_{k,n,\lambda}\rOnto{\text{\em Gr}}(k,n)$

To compute the fiber over $V$ we choose a basis of $\mathbb{C}^{n}$ such that
the first $k$ base vectors span $V=\phi(\mathbb{T}_{m,n}^{W}(\lambda)).v$.
With respect to this basis, the images of the generic matrices $X_{i}$ all are
of the following block-form

$\phi(X_{i})=\begin{bmatrix}\phi_{k}(X_{i})&\sigma(X_{i})\\\
0&\phi_{n-k}(X_{i})\end{bmatrix}\quad\text{with}\quad\begin{cases}\phi_{k}(X_{i})\in
M_{k}(\mathbb{C})\\\ \phi_{n-k}(X_{i})\in M_{n-k}(\mathbb{C})\\\
\sigma(X_{i})\in M_{n-k\times k}(\mathbb{C})\end{cases}$

Using these matrix-form it is easy to see that

$Tr(\phi(W(X_{1},\ldots,X_{m})))=Tr(\phi_{k}(W(X_{1},\ldots,X_{m})))+Tr(\phi_{n-k}(W(X_{1},\ldots,X_{m})))$

That is, if $\phi_{k}\in{\text{\em trep}}_{k}(\mathbb{T}_{m,k}^{W}(\mu))$ then
$\phi_{n-k}\in{\text{\em trep}}(\mathbb{T}_{m,n-k}^{W}(\lambda-\mu))$ and
moreover we have that $(v,\phi_{k})\in{\text{\em X}}_{k,k,\mu}$. Further, the
$m$ matrices $\sigma(X_{i})\in M_{n-k\times k}(\mathbb{C})$ can be taken
arbitrary. Rephrasing this in motives we get

$[{\text{\em X}}_{k,n,\lambda}]=\mathbb{L}^{mk(n-k)}[{\text{\em
Gr}}(k,n)]\sum_{\mu\in\mathbb{C}}[{\text{\em X}}_{k,k,\mu}][{\text{\em
trep}}_{n-k}(\mathbb{T}_{m,n-k}(\lambda-\mu))]$

Here the summation $\sum_{\mu\in\mathbb{C}}$ is shorthand for distinguishing
between zero and non-zero values of $\mu$ and $\lambda-\mu$. For example, with
$\sum_{\mu\in\mathbb{C}}[{\text{\em X}}_{k,k,\mu}][{\text{\em
trep}}_{n-k}(\mathbb{T}_{m,n-k}(\lambda-\mu))]$ we mean for $\lambda\not=0$

$(\mathbb{L}-2)[{\text{\em X}}_{k,k,1}][{\text{\em
trep}}_{n-k}(\mathbb{T}_{m,n-k}(1))]+[{\text{\em X}}_{k,k,0}][{\text{\em
trep}}_{n-k}(\mathbb{T}_{m,n-k}(\lambda))]+[{\text{\em
X}}_{k,k,\lambda}][{\text{\em trep}}_{n-k}(\mathbb{T}_{m,n-k}(0))]$

and when $\lambda=0$

$(\mathbb{L}-1)[{\text{\em X}}_{k,k,1}][{\text{\em
trep}}_{n-k}(\mathbb{T}_{m,n-k}(1))]+[{\text{\em X}}_{k,k,0}][{\text{\em
trep}}_{n-k}(\mathbb{T}_{m,n-k}(0))].$

Further, we have

$[{\text{\em
Gr}}(k,n)]=\frac{[GL_{n}]}{[GL_{k}][GL_{n-k}]\mathbb{L}^{k(n-k)}}\quad\text{and}\quad[{\text{\em
X}}_{k,k,\mu}]=[GL_{k}][{\text{\em BS}}_{m,k}^{W}(\mu)]$

and substituting this in the above, and recalling that
$\mathbb{M}_{m,l}^{W}(\alpha)={\text{\em
trep}}_{l}(\mathbb{T}_{m,l}^{W}(\alpha))$, we get

######  Proposition 3.

With notations as before we have for all $0<k<n$ and all
$\lambda\in\mathbb{C}$ that

$[{\text{\em
X}}_{k,n,\lambda}]=[GL_{n}]\mathbb{L}^{(m-1)k(n-k)}\sum_{\mu\in\mathbb{C}}[{\text{\em
BS}}_{m,k}^{W}(\mu)]\frac{[\mathbb{M}_{m,n-k}^{W}(\lambda-\mu)]}{[GL_{n-k}]}$

Further, we have

$[{\text{\em
X}}_{0,n,\lambda}]=[\mathbb{M}_{m,n}^{W}(\lambda)]\quad\text{and}\quad[{\text{\em
X}}_{n,n,\lambda}]=[GL_{n}][{\text{\em BS}}_{m,n}^{W}(\lambda)]$

We can also express this in terms of generating series. Equip the commutative
ring $\mathcal{M}_{\mathbb{C}}[[t]]$ with the modified product

$t^{a}\ast t^{b}=\mathbb{L}^{(m-1)ab}t^{a+b}$

and consider the following two generating series for all
$\frac{1}{2}\not=\lambda\in\mathbb{C}$

${\text{\em B}}_{\lambda}(t)=\sum_{n=1}^{\infty}[{\text{\em
BS}}_{m,n}^{W}(\lambda)]t^{n}\quad\text{and}\quad{\text{\em
R}}_{\lambda}(t)=\sum_{n=1}^{\infty}\frac{[\mathbb{M}_{m,n}^{W}(\lambda)]}{[GL_{n}]}t^{n}$
${\text{\em B}}_{\frac{1}{2}}(t)=\sum_{n=0}^{\infty}[{\text{\em
BS}}_{m,n}^{W}(\frac{1}{2})]t^{n}\quad\text{and}\quad{\text{\em
R}}_{\frac{1}{2}}(t)=\sum_{n=0}^{\infty}\frac{[\mathbb{M}_{m,n}^{W}(\frac{1}{2})]}{[GL_{n}]}t^{n}$

######  Proposition 4.

With notations as before we have the functional equation

$1+{\text{\em R}}_{1}(\mathbb{L}t)=\sum_{\mu}{\text{\em
B}}_{\mu}(t)\ast{\text{\em R}}_{1-\mu}(t)$

###### Proof.

The disjoint union of the strata of the dimension function on
$\mathbb{C}^{n}\times{\text{\em trep}}_{n}(\mathbb{T}_{m,n}^{W}(\lambda))$
gives

$\mathbb{C}^{n}\times\mathbb{M}_{m,n}^{W}(\lambda)={\text{\em
X}}_{0,n,\lambda}\sqcup{\text{\em
X}}_{1,n,\lambda}\sqcup\ldots\sqcup{\text{\em X}}_{n,n,\lambda}$

Rephrasing this in terms of motives gives

$\mathbb{L}^{n}[\mathbb{M}_{m,n}^{W}(\lambda)]=[\mathbb{M}_{m,n}^{W}(\lambda)]+\sum_{k=1}^{n-1}[{\text{\em
X}}_{k,n,\lambda}]+[GL_{n}][{\text{\em BS}}_{m,n}^{W}(\lambda)]$

and substituting the formula of proposition 3 into this we get

$\frac{[\mathbb{M}_{m,n}^{W}(\lambda)]}{[GL_{n}]}\mathbb{L}^{n}t^{n}=\frac{[\mathbb{M}_{m,n}^{W}(\lambda)]}{[GL_{n}]}t^{n}+$
$\sum_{k=1}^{n-1}\sum_{\mu\in\mathbb{C}}([{\text{\em
BS}}_{m,k}^{W}(\mu)]t^{k})\ast(\frac{[\mathbb{M}_{m,n-k}^{W}(\lambda-\mu)]}{[GL_{n-k}]}t^{n-k})+[{\text{\em
BS}}_{m,n}^{W}(\lambda)]t^{n}$

Now, take $\lambda=1$ then on the left hand side we have the $n$-th term of
the series $1+{\text{\em R}}_{1}(\mathbb{L}t)$ and on the right hand side we
have the $n$-th factor of the series $\sum_{\mu}{\text{\em
B}}_{\mu}(t)\ast{\text{\em R}}_{1-\mu}(t)$. The outer two terms arise from the
product ${\text{\em B}}_{\frac{1}{2}}(t)\ast{\text{\em R}}_{\frac{1}{2}}(t)$,
using that $W$ is homogeneous whence for all $\lambda\not=0$

${\text{\em BS}}_{m,n}^{W}(\lambda)\simeq{\text{\em
BS}}_{m,n}^{W}(1)\quad\text{and}\quad\mathbb{M}_{m,n}^{W}(\lambda)\simeq\mathbb{M}_{m,n}^{W}(1)$

This finishes the proof. ∎

These formulas allow us to determine the motive
$[\mathbb{M}_{m,n}^{W}(\lambda)]$ inductively from lower dimensional
contributions and from the knowledge of the motive of the Brauer-Severi scheme
$[{\text{\em BS}}_{m,n}^{W}(\lambda)]$.

######  Proposition 5.

For all $n$ we have the following inductive description of the motives in the
Donalson-Thomas series

$(\mathbb{L}^{n}-1)\frac{[\mathbb{M}^{W}_{m,n}(0)]-[\mathbb{M}^{W}_{m,n}(1)]}{[GL_{n}]}$

is equal to

$[{\text{\em BS}}^{W}_{m,n}(0)]-[{\text{\em
BS}}_{m,n}^{W}(1)]+\sum_{k=1}^{n-1}\frac{\mathbb{L}^{(m-1)k(n-k)}}{[GL_{n-k}]}([{\text{\em
BS}}^{W}_{m,k}(0)]-[{\text{\em
BS}}^{W}_{m,k}(1)])([\mathbb{M}^{W}_{m,k}(0)]-[\mathbb{M}^{W}_{m,k}(1)])$

###### Proof.

Follows from Proposition 3 and the fact that for all $\mu\not=0$ we have that
$[\mathbb{M}_{m,k}^{W}(\mu)]=[\mathbb{M}_{m,k}^{W}(1)]$ and $[{\text{\em
BS}}_{m,k}^{W}(\mu)]=[{\text{\em BS}}_{m,k}^{W}(1)]$. ∎

##  3\. Deformations of affine $3$-space

The commutative polynomial ring $\mathbb{C}[x,y,z]$ is the Jacobi algebra
associated with the superpotential $W=XYZ-XZY$. For this reason we restrict in
the rest of this paper to cases where the superpotential $W$ is a cubic
necklace in three non-commuting variables $X,Y$ and $Z$, that is $m=3$ from
now on. As even in this case the calculations become quickly unmanageable we
restrict to $n\leq 2$, that is we only will compute the coefficients of $t$
and $t^{2}$ in $U_{W}(t)$. We will have to compute the motives of fibers of
the Chern-Simons functional

$M_{2}(\mathbb{C})\oplus M_{2}(\mathbb{C})\oplus
M_{2}(\mathbb{C})\rTo^{Tr(W)}\mathbb{C}$

so we want to express $Tr(W)$ as a function in the variables of the three
generic $2\times 2$ matrices

$X=\begin{bmatrix}n&p\\\ q&r\end{bmatrix},~{}Y=\begin{bmatrix}s&t\\\
u&v\end{bmatrix},~{}Z=\begin{bmatrix}w&x\\\ y&z\end{bmatrix}.$

We will call $\\{n,r,s,v,w,x\\}$ (resp. $\\{p,t,x\\}$ and $\\{q,u,y\\}$) the
diagonal- (resp. upper- and lower-) variables. We claim that

$Tr(W)=C+Q_{q}.q+Q_{u}.u+Q_{y}.y$

where $C$ is a cubic in the diagonal variables and $Q_{q},Q_{u}$ and $Q_{y}$
are bilinear in the diagonal and upper variables, that is, there are linear
terms $L_{ab}$ in the diagonal variables such that

$\begin{cases}Q_{q}=L_{qp}.p+L_{qt}.t+L_{qx}.x\\\
Q_{u}=L_{up}.p+L_{ut}.t+L_{ux}.x\\\
Q_{y}=L_{yp}.p+L_{yt}.t+L_{yx}.x\end{cases}$

This follows from considering the two diagonal entries of a $2\times 2$ matrix
as the vertices of a quiver and the variables as arrows connecting these
vertices as follows

$\scriptstyle{n}$$\scriptstyle{s}$$\scriptstyle{w}$$\scriptstyle{q}$$\scriptstyle{u}$$\scriptstyle{y}$$\scriptstyle{p}$$\scriptstyle{t}$$\scriptstyle{x}$$\scriptstyle{r}$$\scriptstyle{v}$$\scriptstyle{z}$

and observing that only an oriented path of length $3$ starting and ending in
the same vertex can contribute something non-zero to $Tr(W)$. Clearly these
linear and cubic terms are fully determined by $W$. If we take

$W=\alpha X^{3}+\beta Y^{3}+\gamma Z^{3}+\delta XYZ+\epsilon XZY$

then we have $C=W(n,s,w)+W(r,v,z)$ and

$\begin{cases}L_{qp}&=3\alpha(n+r)\\\ L_{qt}&=\epsilon w+\delta z\\\
L_{qx}&=\delta s+\epsilon v\end{cases}\quad\begin{cases}L_{up}&=\delta
w+\epsilon z\\\ L_{ut}&=3\beta(s+v)\\\ L_{ux}&=\epsilon n+\delta r\\\
\end{cases}\quad\begin{cases}L_{yp}&=\epsilon s+\delta v\\\ L_{yt}&=\delta
n+\epsilon r\\\ L_{yx}&=3\gamma(w+z)\\\ \end{cases}$

By using the cellular decomposition of the Brauer-Severi scheme of
$\mathbb{T}_{3,2}$ one can simplify the computations further by specializing
certain variables. From [16] we deduce that ${\text{\em
BS}}_{2}(\mathbb{T}_{3,2})$ has a cellular decomposition as
$\mathbb{A}^{10}\sqcup\mathbb{A}^{8}\sqcup\mathbb{A}^{8}$ where the three
cells have representatives

$\begin{cases}{\text{\em cell}}_{1}~{}:~{}v=\begin{bmatrix}1\\\
0\end{bmatrix},\quad X=\begin{bmatrix}0&p\\\ 1&r\end{bmatrix},\quad
Y=\begin{bmatrix}s&t\\\ u&v\end{bmatrix},\quad Z=\begin{bmatrix}w&x\\\
y&z\end{bmatrix}\\\ \\\ {\text{\em cell}}_{2}~{}:~{}v=\begin{bmatrix}1\\\
0\end{bmatrix},\quad X=\begin{bmatrix}n&p\\\ 0&r\end{bmatrix},\quad
Y=\begin{bmatrix}0&t\\\ 1&v\end{bmatrix},\quad Z=\begin{bmatrix}w&x\\\
y&z\end{bmatrix}\\\ \\\ {\text{\em cell}}_{3}~{}:~{}v=\begin{bmatrix}1\\\
0\end{bmatrix},\quad X=\begin{bmatrix}n&p\\\ 0&r\end{bmatrix},\quad
Y=\begin{bmatrix}s&t\\\ 0&v\end{bmatrix},\quad Z=\begin{bmatrix}0&x\\\
1&z\end{bmatrix}\end{cases}$

It follows that ${\text{\em BS}}_{3,2}^{W}(1)$ decomposes as
$\mathbf{S_{1}}\sqcup\mathbf{S_{2}}\sqcup\mathbf{S_{3}}$ where the subschemes
$\mathbf{S_{i}}$ of $\mathbb{A}^{11-i}$ have defining equations

$\begin{cases}\mathbf{S_{1}}~{}:~{}(C+Q_{u}.u+Q_{y}.y+Q_{q})|_{n=0}=1\\\
\mathbf{S_{2}}~{}:~{}(C+Q_{y}.y+Q_{u})|_{s=0}=1\\\
\mathbf{S_{3}}~{}:~{}(C+Q_{y})|_{w=0}=1\end{cases}$

Note that in using the cellular decomposition, we set a variable equal to $1$.
So, in order to retain a homogeneous form we let $\mathbb{G}_{m}$ act on
$n,s,w,r,v,z$ with weight one, on $q,u,y$ with weight two and on $x,t,p$ with
weight zero. Thus, we need a slight extension of [4, Thm. 1.3] as to allow
$\mathbb{G}_{m}$ to act with weight two on certain variables.

From now on we will assume that $W$ is as above with $\delta=1$ and
$\epsilon\not=0$. In this generality we can prove:

######  Proposition 6.

With assumptions as above

$[\mathbf{S_{3}}]=\begin{cases}\mathbb{L}^{7}-\mathbb{L}^{4}+\mathbb{L}^{3}[W(n,s,0)+W(-\epsilon^{-1}n,-\epsilon
s,0)=1]_{\mathbb{A}^{2}}&\text{if $\gamma\not=0$}\\\
\mathbb{L}^{7}-\mathbb{L}^{5}+\mathbb{L}^{3}[W(n,s,0)+W(-\epsilon^{-1}n,-\epsilon
s,z)=1]_{\mathbb{A}^{3}}&\text{if $\gamma=0$}\end{cases}$

###### Proof.

$\mathbf{S_{3}}$ : The defining equation in $\mathbb{A}^{8}$ is equal to

$W(n,s,0)+W(r,v,z)+(\epsilon s+v)p+(n+\epsilon r)t+3\gamma(z)x=1$

If $\epsilon s+v\not=0$ we can eliminate $p$ and get a contribution
$\mathbb{L}^{5}(\mathbb{L}^{2}-\mathbb{L})$. If $v=-\epsilon s$ but
$n+\epsilon r\not=0$ we can eliminate $t$ and get a term
$\mathbb{L}^{4}(\mathbb{L}^{2}-\mathbb{L})$. From now on we may assume that
$v=-\epsilon s$ and $r=-\epsilon^{-1}n$.

$\gamma\not=0$ : Assume first that $z\not=0$ then we can eliminate $x$ and get
a contribution $\mathbb{L}^{4}(\mathbb{L}-1)$. If $z=0$ then we get a term

$\mathbb{L}^{3}[W(n,s,0)+W(-\epsilon^{-1}n,-\epsilon s,0)=1]_{\mathbb{A}^{2}}$

$\gamma=0$ : Then we have a remaining contribution

$\mathbb{L}^{3}[W(n,s,0)+W(-\epsilon^{-1}n,-\epsilon s,z)=1]_{\mathbb{A}^{3}}$

Summing up all contributions gives the result. ∎

Calculating the motives of $\mathbf{S_{2}}$ and $\mathbf{S_{1}}$ in this
generality quickly leads to a myriad of subcases to consider. For this reason
we will defer the calculations in the cases of interest to the next sections.
Specializing Proposition 5 to the case of $n=2$ we get

######  Proposition 7.

For $n=2$ we have that

$(\mathbb{L}^{2}-1)\frac{[\mathbb{M}^{W}_{3,2}(0)]-[\mathbb{M}^{W}_{3,2}(1)]}{[GL_{2}]}$

is equal to

$[{\text{\em BS}}^{W}_{3,2}(0)]-[{\text{\em
BS}}^{W}_{3,2}(1)]+\frac{\mathbb{L}^{2}}{(\mathbb{L}-1)}([\mathbb{M}^{W}_{3,1}(0)]-[\mathbb{M}^{W}_{3,1}(1)])^{2}$

###### Proof.

The result follows from Proposition 5 and from the fact that
$\mathbf{BS}_{3,1}^{W}(1)=\mathbb{M}_{3,1}^{W}(1)$ and
$\mathbf{BS}_{3,1}^{W}(0)=\mathbb{M}_{3,1}^{W}(0)$. ∎

##  4\. Quantum affine three-space

For $q\in\mathbb{C}^{*}$ consider the superpotential $W_{q}=XYZ-qXZY$, then
the associated algebra $R_{W_{q}}$ is the quantum affine $3$-space

$R_{W_{q}}=\frac{\mathbb{C}\langle X,Y,Z\rangle}{(XY-qYX,ZX-qXZ,YZ-qZY)}$

It is well-known that $R_{W_{q}}$ has finite dimensional simple
representations of dimension $n$ if and only if $q$ is a primitive $n$-th root
of unity. For other values of $q$ the only finite dimensional simples are
$1$-dimensional and parametrized by $XYZ=0$ in $\mathbb{A}^{3}$. In this case
we have

$\begin{cases}[\mathbb{M}_{3,1}^{W_{q}}(1)]=[(q-1)XYZ=1]_{\mathbb{A}^{3}}=(\mathbb{L}-1)^{2}\\\
[\mathbb{M}_{3,1}^{W_{q}}(0)]=[(1-q)XYZ=0]_{\mathbb{A}^{3}}=3\mathbb{L}^{2}-3\mathbb{L}+1\end{cases}$

That is, the coefficient of $t$ in $U_{W_{q}}(t)$ is equal to

$\mathbb{L}^{-1}\frac{[\mathbb{M}_{3,1}^{W_{q}}(0)-[\mathbb{M}_{3,1}^{W_{q}}(1)]}{[GL_{1}]}=\mathbb{L}^{-1}\frac{2\mathbb{L}^{2}-\mathbb{L}}{\mathbb{L}-1}=\frac{2\mathbb{L}-1}{\mathbb{L}-1}$

In [3, Thm. 3.1] it is shown that in case $q$ is not a root of unity, then

$U_{W_{q}}(t)={\text{\em
Exp}}(\frac{2\mathbb{L}-1}{\mathbb{L}-1}\frac{t}{1-t})$

and if $q$ is a primitive $n$-th root of unity then

$U_{W_{q}}(t)={\text{\em
Exp}}(\frac{2\mathbb{L}-1}{\mathbb{L}-1}\frac{t}{1-t}+(\mathbb{L}-1)\frac{t^{n}}{1-t^{n}})$

In [3, 3.4.1] a rather complicated attempt is made to explain the term
$\mathbb{L}-1$ in case $q$ is an $n$-th root of unity in terms of certain
simple $n$-dimensional representations of $R_{W_{q}}$. Note that the geometry
of finite dimensional representations of the algebra $R_{W_{q}}$ is studied
extensively in [5] and note that there are additional simple $n$-dimensional
representations not taken into account in [3, 3.4.1].

Perhaps a more conceptual explanation of the two terms in the exponential
expression of $U_{W_{q}}(t)$ in case $q$ is an $n$-th root of unity is as
follows. As $W_{q}$ admits a cut $W_{q}=X(YZ-qZY)$ it follows from [12] that
for all dimensions $m$ we have

$[\mathbb{M}_{3,m}^{W_{q}}(0)]-[\mathbb{M}_{3,m}^{W_{q}}(1)]=\mathbb{L}^{m^{2}}[{\text{\em
rep}}_{m}(\mathbb{C}_{q}[Y,Z])]$

where $\mathbb{C}_{q}[Y,Z]=\mathbb{C}\langle Y,Z\rangle/(YZ-qZY)$ is the
quantum plane. If $q$ is an $n$-th root of unity the only finite dimensional
simple representations of $\mathbb{C}_{q}[Y,Z]$ are of dimension $1$ or $n$.
The $1$-dimensional simples are parametrized by $YZ=0$ in $\mathbb{A}^{2}$
having as motive $2\mathbb{L}-1$ and as all have $GL_{1}$ as stabilizer group,
this explains the term $(2\mathbb{L}-1)/(\mathbb{L}-1)$. The center of
$\mathbb{C}_{q}[Y,Z]$ is equal to $\mathbb{C}[Y^{n},Z^{n}]$ and the
corresponding variety $\mathbb{A}^{2}={\text{\em
Max}}(\mathbb{C}[Y^{n},Z^{n}])$ parametrizes $n$-dimensional semi-simple
representations.The $n$-dimensional simples correspond to the Zariski open set
$\mathbb{A}^{2}-(Y^{n}Z^{n}=0)$ which has as motive $(\mathbb{L}-1)^{2}$.
Again, as all these have as $GL_{2}$-stabilizer subgroup $GL_{1}$, this
explains the term

$\mathbb{L}-1=\frac{(\mathbb{L}-1)^{2}}{[GL_{1}]}$

As the superpotential allows a cut in this case we can use the full strength
of [1]and can obtain $[\mathbb{M}^{W}_{3,2}(0)]$ from
$[\mathbb{M}^{W}_{3,2}(1)]$ from the equality

$\mathbb{L}^{12}=[\mathbb{M}^{W}_{3,2}(0)]+(\mathbb{L}-1)[\mathbb{M}^{W}_{3,2}(1)]$

To illustrate the inductive procedure using Brauer-Severi motives we will
consider the case $n=2$, that is $q=-1$ with superpotential $W=XYZ+XZY$. In
this case we have from [3, Thm. 3.1] that

$U_{W}(t)={\text{\em
Exp}}(\frac{2\mathbb{L}-1}{\mathbb{L}-1}\frac{t}{1-t}+(\mathbb{L}-1)\frac{t^{2}}{1-t^{2}}$

The basic rules of the plethystic exponential on
$\mathcal{M}_{\mathbb{C}}[[t]]$ are

${\text{\em Exp}}(\sum_{n\geq 1}[A_{n}]t^{n})=\prod_{n\geq
1}(1-t^{n})^{-[A_{n}]}\quad\text{where}\quad(1-t)^{-\mathbb{L}^{m}}=(1-\mathbb{L}^{m}t)^{-1}$

and one has to extend all infinite products in $t$ and $\mathbb{L}^{-1}$. One
starts by rewriting $U_{W}(t)$ as a product

$U_{W}(t)={\text{\em Exp}}(\frac{t}{1-t}){\text{\em
Exp}}(\frac{\mathbb{L}}{\mathbb{L}-1}\frac{t}{1-t}){\text{\em
Exp}}(\frac{\mathbb{L}t^{2}}{1-t^{2}}){\text{\em
Exp}}(\frac{t^{2}}{1-t^{2}})^{-1}$

where each of the four terms is an infinite product

${\text{\em Exp}}(\frac{t}{1-t})=\prod_{m\geq
1}(1-t^{m})^{-1},\qquad{\text{\em
Exp}}(\frac{\mathbb{L}}{\mathbb{L}-1}\frac{t}{1-t})=\prod_{m\geq
1}\prod_{j\geq 0}(1-\mathbb{L}^{-j}t^{m})^{-1}$ ${\text{\em
Exp}}(\frac{\mathbb{L}t^{2}}{1-t^{2}})=\prod_{m\geq
1}(1-\mathbb{L}t^{2m})^{-1},\qquad{\text{\em
Exp}}(\frac{t^{2}}{1-t^{2}})^{-1}=\prod_{m\geq 1}(1-t^{2m}$

That is, we have to work out the infinite product

$\prod_{m\geq 1}((1-t^{2m-1})^{-1}(1-\mathbb{L}t^{2m})^{-1})\prod_{m\geq
1}\prod_{j\geq 0}(1-\mathbb{L}^{-j}t^{m})^{-1}$

as a power series in $t$, at least up to quadratic terms. One obtains

$U_{W}(t)=1+\frac{2\mathbb{L}-1}{\mathbb{L}-1}t+\frac{\mathbb{L}^{4}+3\mathbb{L}^{3}-2\mathbb{L}^{2}-2\mathbb{L}+1}{(\mathbb{L}^{2}-1)(\mathbb{L}-1)}t^{2}+\ldots$

That is, if $W=XYZ+XZY$ one must have the relation:

$[\mathbb{M}_{3,2}^{W}(0)]-[\mathbb{M}_{3,2}^{W}(1)]=\mathbb{L}^{5}(\mathbb{L}^{4}+3\mathbb{L}^{3}-2\mathbb{L}^{2}-2\mathbb{L}+1)$

###  4.1. Dimensional reduction

It follows from the dimensional reduction argument of [12] that

$[\mathbb{M}_{3,2}^{W}(0)]-[\mathbb{M}_{3,2}^{W}(1)]=\mathbb{L}^{4}[{\text{\em
rep}}_{2}~{}\mathbb{C}_{-1}[X,Y]]$

where $\mathbb{C}_{-1}[X,Y]$ is the quantum plane at $q=-1$, that is,
$\mathbb{C}\langle X,Y\rangle/(XY+YX)$. The matrix equation

$\begin{bmatrix}a&b\\\ c&d\end{bmatrix}\begin{bmatrix}e&f\\\
g&h\end{bmatrix}+\begin{bmatrix}e&f\\\ g&h\end{bmatrix}\begin{bmatrix}a&b\\\
c&d\end{bmatrix}=\begin{bmatrix}0&0\\\ 0&0\end{bmatrix}$

gives us the following system of equations

$\begin{cases}2ae+bg+fc=0\\\ 2hd+bg+fc=0\\\ f(a+d)+b(e+h)=0\\\
c(h+e)+g(a+d)=0\end{cases}$

where the two first are equivalent to $ae=hd$ and $2ae+bg+fc=0$. Changing
variables

$x=\frac{1}{2}(a+d),\quad y=\frac{1}{2}(a-d),\quad u=\frac{1}{2}(e+h),\quad
v=\frac{1}{2}(e-h)$

the equivalent system then becomes (in the variables $b,c,f,g,u,v,x,y$)

$\begin{cases}xv+yu=0\\\ xu+yv+bg+fc=0\\\ fx+bu=0\\\ cu+gx=0\end{cases}$

######  Proposition 8.

The motive of $R_{2}={\text{\em rep}}_{2}~{}\mathbb{C}_{-1}[x,y]$ is equal to

$[R_{2}]=\mathbb{L}^{5}+3\mathbb{L}^{4}-2\mathbb{L}^{3}-2\mathbb{L}^{2}+\mathbb{L}$

###### Proof.

If $x\not=0$ we obtain

$v=-\frac{yu}{x},\quad f=-\frac{bu}{x},\quad g=-\frac{cu}{x}$

and substituting these in the remaining second equation we get the equation(s)

$u(y^{2}-x^{2}+2bc)=0\quad\text{and}\quad x\not=0$

If $u\not=0$ then $y^{2}-x^{2}+2bc=0$. If in addition $b\not=0$ then
$c=\tfrac{x^{2}-y^{2}}{2b}$ and $y$ is free. As $x,u$ and $b$ are non-zero
this gives a contribution $(\mathbb{L}-1)^{3}\mathbb{L}$. If $b=0$ then $c$ is
free and $x^{2}-y^{2}=0$, so $y=\pm x$. This together with $x\not=0\not=u$
leads to a contribution of $2\mathbb{L}(\mathbb{L}-1)^{2}$. If $u=0$ then
$y,b$ and $c$ are free variables, and together with $x\not=0$ this gives
$(\mathbb{L}-1)\mathbb{L}^{3}$.

Remains the case that $x=0$. Then the system reduces to

$\begin{cases}yu=0\\\ yv+bg+fc=0\\\ bu=0\\\ cu=0\end{cases}$

If $u\not=0$ then $y=0,b=0$ and $c=0$ leaving $c,g,v$ free. This gives
$(\mathbb{L}-1)\mathbb{L}^{3}$. If $u=0$ then the only remaining equation is
$yv+bg+fc=0$. That is, we get the cone in $\mathbb{A}^{6}$ of the Grassmannian
$Gr(2,4)$ in $\mathbb{P}^{5}$. As the motive of $Gr(2,4)$ is

$[Gr(2,4)]=(\mathbb{L}^{2}+1)(\mathbb{L}^{2}+\mathbb{L}+1)$

we get a contribution of

$(\mathbb{L}-1)(\mathbb{L}^{2}+1)(\mathbb{L}^{2}+\mathbb{L}+1)+1$

Summing up all contributions gives the desired result. ∎

###  4.2. Brauer-Severi motives

In the three cells of the Brauer-Severi scheme of $\mathbb{T}_{3,2}$ of
dimensions resp. $10,9$ and $8$ the superpotential $Tr(XYZ+XZY)$ induces the
equations:

$\begin{cases}\mathbf{S_{1}}~{}:~{}2rvz+puz+pvy+rty+psy+rux+puw+tz+vx+sx+tw=1\\\
\mathbf{S_{2}}~{}:~{}2rvz+pvy+rty+nty+pz+rx+nx+pw=1\\\
\mathbf{S_{3}}~{}:~{}2rvz+pv+rt+nt+ps=1\end{cases}$

######  Proposition 9.

With notations as above, the Brauer-Severi scheme of $\mathbb{T}_{3,2}^{W}(1)$
has a decomposition

$\mathbf{BS}_{3,2}^{W}(1)=\mathbf{S_{1}}\sqcup\mathbf{S_{2}}\sqcup\mathbf{S_{3}}$

where the schemes $\mathbf{S_{i}}$ have motives

$\begin{cases}[\mathbf{S_{1}}]=\mathbb{L}^{9}-\mathbb{L}^{6}-2\mathbb{L}^{5}+3\mathbb{L}^{4}-\mathbb{L}^{3}\\\
[\mathbf{S_{2}}]=\mathbb{L}^{8}-2\mathbb{L}^{5}+\mathbb{L}^{4}\\\
[\mathbf{S_{3}}]=\mathbb{L}^{7}-2\mathbb{L}^{4}+\mathbb{L}^{3}\\\ \end{cases}$

Therefore, the Brauer-Severi scheme has motive

$[\mathbf{BS}_{3,2}^{W}(1)]=\mathbb{L}^{9}+\mathbb{L}^{8}+\mathbb{L}^{7}-\mathbb{L}^{6}-4\mathbb{L}^{5}+2\mathbb{L}^{4}$

###### Proof.

$\mathbf{S_{1}}$ : From Proposition 6 we obtain

$[\mathbf{S_{3}}]=\mathbb{L}^{7}-\mathbb{L}^{5}+\mathbb{L}^{3}[W(n,s,0)+W(-n,-s,z)=1]_{\mathbb{A}^{3}}$

and as $W(n,s,0)+W(-n,-s,z)=2nsz$ we get
$\mathbb{L}^{7}-\mathbb{L}^{5}+\mathbb{L}^{3}(\mathbb{L}-1)^{2}$.

$\mathbf{S_{2}}$ : The defining equation is

$2rvz+y(pv+(r+n)t)+p(z+w)+x(r+n)=1$

If $r+n\not=0$ we can eliminate $x$ and have a contribution
$\mathbb{L}^{6}(\mathbb{L}^{2}-\mathbb{L})$. If $r+n=0$ we get the equation

$2rvz+p(yv+z+w)=1$

If $yv+z+w\not=0$ we can eliminate $p$ and get a term
$\mathbb{L}^{3}(\mathbb{L}^{4}-\mathbb{L}^{3})$. If $r+n=0$ and $yv+z+w=0$ we
have $2rvz=1$ so a term $\mathbb{L}^{4}(\mathbb{L}-1)^{2}$. Summing up gives
us

$[\mathbf{S}_{2}]=\mathbb{L}^{4}(\mathbb{L}-1)(\mathbb{L}^{3}+\mathbb{L}^{2}+\mathbb{L}-1)=\mathbb{L}^{8}-2\mathbb{L}^{5}+\mathbb{L}^{4}$

$\mathbf{S_{1}}$ : The defining equation is

$2rvz+p(u(z+w)+y(v+s))+t(z+w+ry)+x(v+s+ru)=1$

If $v+s+ru\not=0$ we can eliminate $x$ and get
$\mathbb{L}^{5}(\mathbb{L}^{4}-\mathbb{L}^{3})$. If $v+s+ru=0$ and
$z+w+ry\not=0$ we can eliminate $t$ and have a term
$\mathbb{L}^{4}(\mathbb{L}^{4}-\mathbb{L}^{3})$. If $v+s+ru=0$ and $z+w+ry=0$,
the equation becomes (in $\mathbb{A}^{8}$, with $t,x$ free variables)

$2r(vz-puy)=1$

giving a term $\mathbb{L}^{2}(\mathbb{L}^{5}-[vz=puy])$. To compute
$[vz=puy]_{\mathbb{A}^{5}}$ assume first that $v\not=0$, then this gives
$\mathbb{L}^{3}(\mathbb{L}-1)$ and if $v=0$ we get
$\mathbb{L}(3\mathbb{L}^{2}-3\mathbb{L}+1)$. That is,
$[vz=puy]_{\mathbb{A}^{5}}=\mathbb{L}^{4}+2\mathbb{L}^{3}-3\mathbb{L}^{2}+\mathbb{L}$.
In total this gives us

$[\mathbf{S}_{1}]=\mathbb{L}^{3}(\mathbb{L}-1)(\mathbb{L}^{5}+\mathbb{L}^{4}+\mathbb{L}^{3}-2\mathbb{L}+1)=\mathbb{L}^{9}-\mathbb{L}^{6}-2\mathbb{L}^{5}+3\mathbb{L}^{4}-\mathbb{L}^{3}$

finishing the proof. ∎

######  Proposition 10.

From the Brauer-Severi motive we obtain

$\begin{cases}[\mathbb{M}_{3,2}^{W}(1)]&=\mathbb{L}^{11}-\mathbb{L}^{8}-3\mathbb{L}^{7}+2\mathbb{L}^{6}+2\mathbb{L}^{5}-\mathbb{L}^{4}\\\
[\mathbb{M}_{3,2}^{W}(0)]&=\mathbb{L}^{11}+\mathbb{L}^{9}+2\mathbb{L}^{8}-5\mathbb{L}^{7}+3\mathbb{L}^{5}-\mathbb{L}^{4}\end{cases}$

As a consequence we have,

$[\mathbb{M}_{3,2}^{W}(0)]-[\mathbb{M}_{3,2}^{W}(1)]=\mathbb{L}^{4}(\mathbb{L}^{5}+3\mathbb{L}^{4}-2\mathbb{L}^{3}-2\mathbb{L}^{2}+\mathbb{L})$

###### Proof.

We have already seen that $\mathbb{M}_{3,1}^{W}(1)=\\{(x,y,z)~{}|~{}2xyz=1\\}$
and $\mathbb{M}_{3,1}^{W}(0)=\\{(x,y,z)~{}|~{}xyz=0\\}$ whence

$[\mathbb{M}_{3,1}^{W}(1)]=(\mathbb{L}-1)^{2}\quad\text{and}\quad[\mathbb{M}_{3,1}^{W}(0)]=3\mathbb{L}^{2}-3\mathbb{L}+1$

Plugging this and the obtained Brauer-Severi motive into Proposition 5 gives
$[\mathbb{M}_{3,2}^{W}(1)]$. From this $[\mathbb{M}_{3,2}^{W}(0)]$ follows
from the equation
$\mathbb{L}^{12}=(\mathbb{L}-1)[\mathbb{M}_{3,2}^{W}(1)]+[\mathbb{M}_{3,2}^{W}(0)]$.
∎

##  5\. The homogenized Weyl algebra

If we consider the superpotential $W=XYZ-XZY-\frac{1}{3}X^{3}$ then the
associated algebra $R_{W}$ is the homogenized Weyl algebra

$R_{W}=\frac{\mathbb{C}\langle X,Y,Z\rangle}{(XZ-ZX,XY-YX,YZ-ZY-X^{2})}$

In this case we have $\mathbb{M}_{3,1}^{W}(1)=\\{x^{3}=-3\\}$ and
$\mathbb{M}_{3,1}^{W}(0)=\\{x^{3}=0\\}$, whence

$[\mathbb{M}_{3,1}^{W}(1)]=\mathbb{L}^{2}[\mu_{3}],\quad\text{and}\quad[\mathbb{M}_{3,1}^{W}(0)]=\mathbb{L}^{2}$

where, as in [3, 3.1.3] we denote by $[\mu_{3}]$ the equivariant motivic class
of $\\{x^{3}=1\\}\subset\mathbb{A}^{1}$ carrying the canonical action of
$\mu_{3}$. Therefore, the coefficient of $t$ in $U_{W}(t)$ is equal to

$\mathbb{L}^{-1}\frac{[\mathbb{M}_{3,1}^{W}(0)]-[\mathbb{M}_{3,1}^{W}(0)]}{[GL_{1}]}=\frac{\mathbb{L}(1-[\mu_{3}])}{\mathbb{L}-1}$

As all finite dimensional simple representations of $R_{W}$ are of dimension
one, this leads to the conjectural expression [3, Conjecture 3.3]

$U_{W}(t)\overset{?}{=}{\text{\em
Exp}}(\frac{\mathbb{L}(1-[\mu_{3}])}{\mathbb{L}-1}\frac{t}{1-t})$

Balazs Szendröi kindly provided the calculation of the first two terms of this
series. Denote with $\tilde{\mathbf{M}}=1-[\mu_{3}]$, then

$U_{W}(t)\overset{?}{=}1+\frac{\mathbb{L}\tilde{\mathbf{M}}}{\mathbb{L}-1}t+\frac{\mathbb{L}^{2}\tilde{\mathbf{M}}^{2}+\mathbb{L}(\mathbb{L}^{2}-1)\tilde{\mathbf{M}}+\mathbb{L}^{2}(\mathbb{L}-1)\sigma_{2}(\tilde{\mathbf{M}})}{(\mathbb{L}^{2}-1)(\mathbb{L}-1)}t^{2}+\ldots$

As was pointed out by B. Pym and B. Davison it follows from [4, Defn 4.4 and
Prop 4.5 (4)] that $\sigma_{2}(\tilde{\mathbf{M}})=\mathbb{L}$, so the second
term is equal to

$\frac{\mathbb{L}^{3}(\mathbb{L}-1)+\tilde{\mathbf{M}}\mathbb{L}(\mathbb{L}^{2}-1)+\tilde{\mathbf{M}}^{2}\mathbb{L}^{2}}{(\mathbb{L}^{2}-1)(\mathbb{L}-1)}$

We will now compute the this second term using Brauer-Severi motives.

Recall that ${\text{\em BS}}_{3,2}^{W}(i)$, for $i=0,1$, decomposes as
$\mathbf{S_{1}}\sqcup\mathbf{S_{2}}\sqcup\mathbf{S_{3}}$ where the subschemes
$\mathbf{S_{i}}$ of $\mathbb{A}^{11-i}$ have defining equations

$\begin{cases}\mathbf{S_{1}}~{}:~{}-\frac{1}{3}r^{3}+((w-z)p+rx)u+((v-s)p-rt)y-rp+(z-w)t+(s-v)x=\delta_{i1}\\\
\mathbf{S_{2}}~{}:~{}-\frac{1}{3}n^{3}-\frac{1}{3}r^{3}+(vp+(n-r)t)y+(w-z)p+(r-n)x=\delta_{i1}\\\
\mathbf{S_{3}}~{}:~{}-\frac{1}{3}n^{3}-\frac{1}{3}r^{3}+(v-s)p+(n-r)t=\delta_{i1}\end{cases}$

If we let the generator of $\mu_{3}$ act with weight one on the variables
$n,s,w,r,v,z$, with weight two on $x,t,p$ and with weight zero on $q,u,y$ we
see that the schemes $S_{j}$ for $i=1$ are indeed $\mu_{3}$-varieties. We will
now compute their equivariant motives:

######  Proposition 11.

With notations as above, the Brauer-Severi scheme of $\mathbb{T}_{3,2}^{W}(1)$
has a decomposition

$\mathbf{BS}_{3,2}^{W}(1)=\mathbf{S_{1}}\sqcup\mathbf{S_{2}}\sqcup\mathbf{S_{3}}$

where the schemes $\mathbf{S_{i}}$ have equivariant motives

$\begin{cases}[\mathbf{S_{1}}]=\mathbb{L}^{9}-\mathbb{L}^{6}\\\
[\mathbf{S_{2}}]=\mathbb{L}^{8}+([\mu_{3}]-1)\mathbb{L}^{6}=\mathbb{L}^{8}-\tilde{\mathbf{M}}\mathbb{L}^{6}\\\
[\mathbf{S_{3}}]=\mathbb{L}^{7}+([\mu_{3}]-1)\mathbb{L}^{5}=\mathbb{L}^{7}-\tilde{\mathbf{M}}\mathbb{L}^{5}\\\
\end{cases}$

Therefore, the Brauer-Severi scheme $\mathbf{BS}^{W}_{3,2}(1)$ has equivariant
motive

$[\mathbf{BS}_{3,2}^{W}(1)]=\mathbb{L}^{9}+\mathbb{L}^{8}+\mathbb{L}^{7}+([\mu_{3}]-2)\mathbb{L}^{6}+([\mu_{3}]-1)\mathbb{L}^{5}$

###### Proof.

$\mathbf{S_{3}}$ : If $v-s\not=0$ we can eliminate $p$ and obtain a
contribution $\mathbb{L}^{5}(\mathbb{L}^{2}-\mathbb{L})$. If $v=s$ and
$n-r\not=0$ we can eliminate $t$ and obtain a term
$\mathbb{L}^{4}(\mathbb{L}^{2}-\mathbb{L})$. Finally, if $v=s$ and $n=r$ we
have the identity $-\frac{2}{3}n^{3}=1$ and a contribution
$\mathbb{L}^{5}[\mu_{3}]$.

$\mathbf{S_{2}}$ : If $r-n\not=0$ we can eliminate $x$ and get a term
$\mathbb{L}^{6}(\mathbb{L}^{2}-\mathbb{L})$. If $r-n=0$ we get the equation in
$\mathbb{A}^{8}$

$-\frac{2}{3}n^{3}+p(vy+w-z)=1$

If $vy+w-z\not=0$ we can eliminate $p$ and get a contribution
$\mathbb{L}^{3}(\mathbb{L}^{4}-\mathbb{L}^{3})$. Finally, if $vy+w-z=0$ we get
the equation $-\frac{2}{3}n^{3}=1$ and hence a term
$\mathbb{L}^{3}.\mathbb{L}^{3}[\mu_{3}]$.

$\mathbf{S_{1}}$ : If $(w-z)p+rx\not=0$ then we can eliminate $u$ and get a
contribution

$\mathbb{L}^{4}(\mathbb{L}^{5}-[(w-z)p+rx=0]_{\mathbb{A}^{5}})=\mathbb{L}^{6}(\mathbb{L}-1)(\mathbb{L}^{2}-1)$

If $(w-z)p+rx=0$ but $(v-s)p-rt\not=0$ we can eliminate $y$ and get a term

$\mathbb{L}.[(w-z)p+rx=0,(v-s)p-rt\not=0]_{\mathbb{A}^{8}}$

To compute the equivariant motive in $\mathbb{A}^{8}$ assume first that
$r\not=0$ then we can eliminate $x$ from the equation and obtain

$\mathbb{L}^{2}[r\not=0,(v-s)p-rt\not=0]_{\mathbb{A}^{5}}=\mathbb{L}^{2}(\mathbb{L}^{4}(\mathbb{L}-1)-[r\not=0,(v-s)p-rt=0]_{\mathbb{A}^{5}})=\mathbb{L}^{5}(\mathbb{L}-1)^{2}$

If $r=0$ we have to compute
$[(w-z)p=0,(v-s)p\not=0]_{\mathbb{A}^{7}}=\mathbb{L}^{2}(\mathbb{L}-1)(\mathbb{L}^{2}-\mathbb{L})\mathbb{L}=\mathbb{L}^{4}(\mathbb{L}-1)^{2}$.
So, in total this case gives a contribution

$\mathbb{L}.[(w-z)p+rx=0,(v-s)p-rt\not=0]_{\mathbb{A}^{8}}=\mathbb{L}^{5}(\mathbb{L}-1)(\mathbb{L}^{2}-1)$

If $(w-z)p+rx=0$, $(v-s)p-rt=0$ and $r\not=0$ we can eliminate
$x=\tfrac{z-w}{r}p$ and $t=\tfrac{v-s}{r}p$ and substituting in the defining
equation of $\mathbf{S_{1}}$ we get

$-\frac{1}{3}r^{3}-rp=1$

so we can eliminate $p$ and obtain a contribution
$\mathbb{L}^{6}(\mathbb{L}-1)$. Finally, if $(w-z)p+rx=0$, $(v-s)p-rt=0$ and
$r=0$ we get the system of equations

$\begin{cases}(w-z)p=0\\\ (v-s)p=0\\\ (z-w)t+(s-v)x=1\end{cases}$

If $p\not=0$ we must have $w-z=0$ and $v-s=0$ which is impossible, so we must
have $p=0$ and the remaining equation is $(z-w)t+(s-v)x=1$ giving a
contribution $\mathbb{L}^{5}(\mathbb{L}^{2}-1)$. Summing up these
contributions gives the claimed motive. ∎

######  Proposition 12.

With notations as above, the Brauer-Severi scheme of $\mathbb{T}_{3,2}^{W}(0)$
has a decomposition

$\mathbf{BS}_{3,2}^{W}(0)=\mathbf{S_{1}}\sqcup\mathbf{S_{2}}\sqcup\mathbf{S_{3}}$

where the schemes $\mathbf{S_{i}}$ have (equivariant) motives

$\begin{cases}[\mathbf{S_{1}}]=\mathbb{L}^{9}+\mathbb{L}^{7}-\mathbb{L}^{6}\\\
[\mathbf{S_{2}}]=\mathbb{L}^{8}\\\ [\mathbf{S_{3}}]=\mathbb{L}^{7}\\\
\end{cases}$

Therefore, the Brauer-Severi scheme $\mathbf{BS}^{W}_{3,2}(0)$ has
(equivariant) motive

$[\mathbf{BS}_{3,2}^{W}(0)]=\mathbb{L}^{9}+\mathbb{L}^{8}+2\mathbb{L}^{7}-\mathbb{L}^{6}$

###### Proof.

$\mathbf{S_{3}}$ : If $v-s\not=0$ we can eliminate $p$ and obtain a
contribution $\mathbb{L}^{5}(\mathbb{L}^{2}-\mathbb{L})$. If $v=s$ and
$n-r\not=0$ we can eliminate $t$ and obtain a term
$\mathbb{L}^{4}(\mathbb{L}^{2}-\mathbb{L})$. Finally, if $v=s$ and $n=r$ we
have the identity $n^{3}=0$ and a contribution $\mathbb{L}^{5}$.

$\mathbf{S_{2}}$ : If $r-n\not=0$ we can eliminate $x$ and get a term
$\mathbb{L}^{6}(\mathbb{L}^{2}-\mathbb{L})$. If $r-n=0$ we get the equation in
$\mathbb{A}^{8}$

$-\frac{2}{3}n^{3}+p(vy+w-z)=1$

If $vy+w-z\not=0$ we can eliminate $p$ and get a contribution
$\mathbb{L}^{3}(\mathbb{L}^{4}-\mathbb{L}^{3})$. Finally, if $vy+w-z=0$ we get
the equation $n^{3}=0$ and hence a term $\mathbb{L}^{6}$.

$\mathbf{S_{1}}$ : If $(w-z)p+rx\not=0$ we can eliminate $u$ and obtain a term

$\mathbb{L}^{4}(\mathbb{L}^{5}-[(w-z)p+rx=0]_{\mathbb{A}^{5}})=\mathbb{L}^{6}(\mathbb{L}-1)(\mathbb{L}^{2}-1)$

If $(w-z)p+rx=0$ but $(v-s)p-rt\not=0$ then we can eliminate $y$ and obtain a
contribution

$\mathbb{L}[(w-z)p+rx=0,(v-s)p-rt\not=0]_{\mathbb{A}^{8}}=\mathbb{L}^{5}(\mathbb{L}-1)(\mathbb{L}^{2}-1)$

Now, assume that $(w-z)p+rx=0$ and $(v-s)p-rt=0$. If $r\not=0$ then we can
eliminate $p,t$ as before and substituting them in the defining equation of
$\mathbf{S_{1}}$ we get

$-\frac{1}{3}r^{3}-rp=0$

and we can eliminate $p$ giving a contribution $\mathbb{L}^{6}(\mathbb{L}-1)$.
Finally, if $(w-z)p+rx=0$ and $(v-s)p-rt=0$ and $r=0$ we have the system of
equations

$\begin{cases}(w-z)p=0\\\ (v-s)p=0\\\ (z-w)t+(s-v)x=0\end{cases}$

If $p\not=0$ we get $w-z=0$ and $v-s=0$ giving a contribution
$\mathbb{L}^{6}(\mathbb{L}-1)$. If $p=0$ the only remaining equation is
$(z-w)t+(s-v)x=0$ which gives a contribution
$\mathbb{L}^{5}(\mathbb{L}^{2}+\mathbb{L}-1)$. Summing up all terms gives the
claimed motive. ∎

Now, we have all the information to compute the second term of the motivic
Donaldson-Thomas series. We have

$\begin{cases}[\mathbf{BS}_{3,2}^{W}(0)]-[\mathbf{BS}_{3,2}^{W}(1)]=\mathbb{L}^{7}+\tilde{\mathbf{M}}\mathbb{L}^{6}+\tilde{\mathbf{M}}\mathbb{L}^{5}\\\
[\mathbb{M}^{W}_{3,1}(0)]-[\mathbb{M}^{W}_{3,1}(1)]=\tilde{\mathbf{M}}\mathbb{L}^{2}\end{cases}$

By Proposition 7 this implies that

$(\mathbb{L}^{2}-1)\frac{[\mathbb{M}^{W}_{3,2}(0)]-[\mathbb{M}^{W}_{3,2}(1)]}{[GL_{2}]}=\mathbb{L}^{7}+\tilde{\mathbf{M}}\mathbb{L}^{6}+\tilde{\mathbf{M}}\mathbb{L}^{5}+\tilde{\mathbf{M}}^{2}\frac{\mathbb{L}^{6}}{(\mathbb{L}-1)}$

Therefore the virtual motive is equal to

$\mathbb{L}^{-4}\frac{[\mathbb{M}^{W}_{3,2}(0)]-[\mathbb{M}^{W}_{3,2}(1)]}{[GL_{2}]}=\frac{\mathbb{L}^{3}(\mathbb{L}-1)+\tilde{\mathbf{M}}\mathbb{L}(\mathbb{L}^{2}-1)+\tilde{\mathbf{M}}^{2}\mathbb{L}^{2}}{(\mathbb{L}^{2}-1)(\mathbb{L}-1)}$

which coincides with the conjectured term in [3, Conjecture 3.3].

## References

  * [1] K. Behrend, J. Bruan and B. Szendroi, Motivic degree zero Donaldson-Thomas invariants, arXiv:0909.5088 (2009)
  * [2] J. Bryan and A. Morrison, Motivic classes of commuting varieties via power structures, J. Algebraic Geom. 24 (2015) 183-199
  * [3] Alberto Cazzaniga, Andrew Morrison, Brent Pym and Balazs Szendroi, Motivic Donaldson-Thomas invariants for some quantized threefolds, arXiv:1510.08116 (2015)
  * [4] Ben Davison and Sven Meinhardt, Motivic DT-invariants for the one loop quiver with potential, arXiv:1108.5956 (2011)
  * [5] Kevin De Laet, Geometry of representations of quantum spaces, arXiv: 1405.1938 (2014)
  * [6] Kevin De Laet and Lieven Le Bruyn, The geometry of representations of $3$-dimensional Sklyanin algebras, Algebras and Representation Theory, 18 , 761-776 (2015)
  * [7] Lieven Le Bruyn, Quiver concomitants are often reflexive Azumaya, Proc. AMS, 105 (1989) 10-16
  * [8] Lieven Le Bruyn, Trace rings of generic matrices are unique factorization domains, Glasgow Math. J. 28 (1986) 11-13
  * [9] Lieven Le Bruyn and Michel Van den Bergh, An explicit description of $\mathbb{T}(3,2)$. In ”Ring Theory, Proceedings Antwerp 1985”, 109-113, Lecture Notes in Mathematics 1197, (1986).
  * [10] Lieven Le Bruyn, Noncommutative geometry and Cayley-smooth orders, Pure and Appl. Math. 290, Chapman & Hall (2008)
  * [11] Lieven Le Bruyn, The superpotential $XYZ+XZY-\tfrac{c}{3}(X^{3}+Y^{3}+Z^{3})$, to appear.
  * [12] Andrew Morrison, Motivic invariants of quivers via dimensional reduction, arXiv:1103.3819 (2011)
  * [13] Claudio Procesi, The invariant theory of $n\times n$ matrices, Adv. in Math. 19 (1976) 306-381
  * [14] Claudio Procesi, A formal inverse to the Cayley-Hamilton theorem, J. Alg. 107 (1987) 63-74
  * [15] Y. P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR-Izv. 8 (1974) 727-760
  * [16] Markus Reineke, Cohomology of non-commutative Hilbert schemes, Alg. Repr. Theory 8 (2005) 541-561
  * [17] Michel Van den Bergh, The Brauer-Severi scheme of the trace ring of generic matrices, Perspectives in Ring Theory (Antwerp 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol 233, Kluwer (1988)
  * [18] Chelsea Walton, Representation theory of three-dimensional Sklyanin algebras, Nuclear Phys. B, 860, 167-185 (2012)