# Jacobi algebra presentations for fundamental group algebras

Vivek Mistry  School of Mathematics, University of Edinburgh, Edinburgh,
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We prove a special case of a conjecture of Davison which pertains to
superpotential descriptions of fundamental group algebras $k[\pi_{1}(X)]$. We
consider the case in which the manifold $X$ is the mapping torus
$M_{g,\varphi}$ of a genus $g$ Riemann surface $\Sigma_{g}$ and a finite order
automorphism $\varphi$, and the superpotential structure is given by the
Jacobi algebra of a quiver with potential.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Calabi-Yau algebras and Ginzburg differential graded algebras
    2. 2.2 Ginzburg differential graded algebras in the relative case
    3. 2.3 Quivers and Jacobi algebras
    4. 2.4 Brane tilings
    5. 2.5 Fundamental group algebras
  3. 3 Main results
    1. 3.1 A Jacobi algebra presentation for $\mathbb{C}[\pi_{1}(M_{g,\varphi})]$
  4. 4 Future directions
    1. 4.1 Calculating motivic DT invariants

##  1\. Introduction

Given a $d$-dimensional manifold $X$ we can define its fundamental group
algebra over a field $k$ as the group ring $k[\pi_{1}(X)].$ In [14] it is
stated that if $X$ is compact, orientable and has a contractible universal
cover then $k[\pi_{1}(X)]$ is a Calabi-Yau algebra of dimension $d$ (see [[14]
Corollary 6.1.4] and [[9] Proposition 5.2.6]). Many Calabi-Yau algebras turn
out to be what are known as superpotential algebras and so it was conjectured
in [14] that in the dimension 3 case these fundamental group algebras which
were Calabi-Yau were also superpotential algebras.

However in [9] Davison showed that this was not the case in general (for
$d\geq 2$), and that in order to have a superpotential structure the algebra
$k[\pi_{1}(X)]$ (and hence the manifold $X$) had to have certain specific
properties. In particular a superpotential structure implies a stronger notion
of an _exact_ Calabi-Yau structure which Davison showed required non-trivial
central units in $k[\pi_{1}(X)]$. Therefore Davison proposed an updated
conjecture [[9] Conjecture 7.1.1] regarding the possibility of a
superpotential structure on $k[\pi_{1}(X)]$ when $X$ is a circle bundle,
precisely because in such a manifold $X$ non-trivial central units in
$k[\pi_{1}(X)]$ are easily found. In this paper we focus on the 3-dimensional
case of this conjecture ie. when $X$ is a Seifert fibre bundle.

The ultimate goal of such a superpotential description of $k[\pi_{1}(X)]$ is
in regards to calculating the Donaldson-Thomas (DT) invariants of $X$. Given a
superpotential structure the DT invariants can then be though of in terms of
vanishing cycles and Milnor fibres of a globally defined function on a smooth
space which gives us a much stronger possibility of calculating them
concretely. We remark on how this may be accomplished towards the end of the
paper.

### Acknowledgements

I would like to thank my supervisor Ben Davison for proposing this project and
many helpful discussions and ideas while writing this paper. This research was
supported by the Royal Society studentship RGF\R1\180093.

##  2\. Background

Fix a field $k$ of characteristic 0 and let $A$ be a $k$-algebra. Henceforward
all unadorned tensor products will be taken over the base field $k$ while
tensor products over any other ring will be adorned with the respective ring,
unless otherwise stated.

######  Definition 2.1.

The _enveloping algebra_ of $A$ is the algebra

$A^{e}=A\otimes A^{\textnormal{op}}.$

Note that the category $A$-Bimod of $A$-bimodules is naturally isomorphic to
$A^{e}$-Mod the category of left $A^{e}$-modules and we will freely swap
between the two. From now on we shall work in the derived category and drop
all such extra notation on the relevant derived functors. Define a functor

$-^{\vee}:D(A\textnormal{-{Bimod}})\longrightarrow D(A\textnormal{-{Bimod}})$

by sending the bimodule $M$ to $\textnormal{Hom}_{A^{e}}(M,A\otimes A)$ where
$A\otimes A$ has the outer $A$-bimodule structure and $M^{\vee}$ has the
$A$-bimodule structure induced by the inner $A$-bimodule structure on
$A\otimes A$.

###  2.1. Calabi-Yau algebras and Ginzburg differential graded algebras

The Calabi-Yau structure is essential for defining DT invariants and suggests
a superpotential description so we give an overview here following [14] and
[9]. The subsequent definition for Calabi-Yau algebras is due to Ginzburg.

######  Definition 2.2.

An object $M^{\bullet}$ in the derived category $D(A\textnormal{-{Mod}})$ is
called _perfect_ if it is isomorphic to a bounded complex of projective
$A$-modules.

######  Definition 2.3.

An algebra $A$ is called _homologically finite_ if it is perfect as an
$A^{e}$-module.

######  Definition 2.4.

A _Calabi-Yau structure of dimension $d$_ on a homologically finite algebra
$A$ is an isomorphism in $D(A\textnormal{-{Bimod}})$

$f:A\xrightarrow{\,\,\sim\,\,}A^{\vee}[d]\quad\textnormal{such that}\quad
f=f^{\vee}[d].$

If $A$ is homologically finite we have an isomorphism

$\textnormal{HH}_{d}(A)\cong\textnormal{Ext}^{-d}_{A^{e}}(A^{\vee},A)=\textnormal{Hom}_{A^{e}}(A^{\vee}[d],A)$

where $\textnormal{HH}_{d}(A)$ is the $d$th Hochschild homology of $A.$ So an
isomorphism $A\cong A^{\vee}[d]$ corresponds to an element in
$\textnormal{HH}_{d}(A)$ (we call such elements _non-degenerate_), and the
self-duality $f=f^{\vee}[d]$ corresponds to the element being fixed by the
induced map on homology of the flip isomorphism
$\beta:A\otimes_{A^{e}}A\xrightarrow{\sim}A\otimes_{A^{e}}A$ that swaps the
copies of $A$ in the tensor product. However it turns out that the map on
Hochschild homology induced by $\beta$ is just the identity (see [[22]
Proposition C.1]) and so any non-degenerate element in
$\textnormal{HH}_{d}(A)$ or indeed any isomorphism $A\cong A^{\vee}[d]$ gives
rise to a $d$-dimensional Calabi-Yau structure on $A$.

For an algebra $A$ recall we have the following long exact sequence in cyclic
and Hochschild homology

$\cdots\rightarrow\textnormal{HC}_{n+1}(A)\rightarrow\textnormal{HC}_{n-1}(A)\xrightarrow{\partial}\textnormal{HH}_{n}(A)\rightarrow\textnormal{HC}_{n}(A)\rightarrow\cdots$

######  Definition 2.5.

An _exact Calabi-Yau structure of dimension $d$_ on an algebra $A$ is a non-
degenerate element $\nu\in\textnormal{HH}_{d}(A)$ that is in the image of the
boundary map $\partial$.

Now let $A$ be a differential graded algebra (dga). Its _bimodule of 1-forms_
is

$\Omega^{1}A=\textnormal{Ker}(A\otimes A\xrightarrow{\,\,\,m\,\,\,}A)$

where $m$ is the multiplication map. $\Omega^{1}A$ inherits a grading from $A$
making it a differential graded $A$-bimodule.

######  Definition 2.6.

A finitely generated dga $A$ is _smooth_ if $\Omega^{1}A$ is projective as an
$A$-bimodule.

Given a finitely generated negatively graded bimodule $V$ over a smooth
algebra $A$ we define $T_{A}(V)$ to be the tensor-algebra generated by $V$
over $A$. If $V$ is free as a bimodule then we call $T_{A}(V)$ a
_noncommutative vector bundle_ over $A.$

######  Lemma 2.1 ([7] Proposition 5.3 (3)).

Let $A$ be smooth and let $V$ be a finitely generated negatively graded
projective bimodule over $A$. Then the algebra $T_{A}(V)$ is smooth.

For a bimodule $M$ over a dga $A$ let $\textnormal{Der}(A,M)$ denote the
graded vector space of super-derivations from $A$ to $M$ and set
$\textnormal{Der}(A)=\textnormal{Der}(A,A)$. Giving $A\otimes A$ the outer
A-bimodule structure we let
$\mathbb{D}\textnormal{er}(A)=\textnormal{Der}(A,A\otimes A)$ be the bimodule
of _double derivations_ on $A$, where the $A$-bimodule structure on
$\mathbb{D}\textnormal{er}(A)$ is induced via the inner bimodule structure on
$A\otimes A$. Note there is a natural isomorphism
$\mathbb{D}\textnormal{er}(A)\cong(\Omega^{1}A)^{\vee}$. Let
$D:A\rightarrow\Omega^{1}A$ be the canonical derivation that sends $a\mapsto
a\otimes 1-1\otimes a.$ Then we can define the dga $(\Omega^{\bullet}A,D)$ of
_noncommutative differential forms_ of $A$ by

$\Omega^{\bullet}A=T_{A}(\Omega^{1}A)$

with differential induced by $D$. We next define the super-commutator quotient

$\textnormal{DR}(A)=\Omega^{\bullet}A/[\Omega^{\bullet}A,\Omega^{\bullet}A]$

called the _cyclic quotient_ of $\Omega^{\bullet}A$, which is a differential
graded vector space. Note that both $\Omega^{\bullet}A$ and
$\textnormal{DR}(A)$ are bigraded- they have the usual tensor grading as well
as a grading induced from $A.$ Denote by $\Omega^{i}A$ and
$\textnormal{DR}^{i}(A)$ the $i$th graded parts with respect to the tensor
grading.

######  Lemma 2.2.

Let $A$ be a finitely generated algebra over $k$ and let $V$ be a finitely
generated $A$-bimodule. Then $\Omega^{1}T_{A}(V)$ is generated by homogeneous
elements of degree 0 and 1 as a graded $T_{A}(V)$-bimodule.

###### Proof.

Because $T_{A}(V)$ is a free algebra $\Omega^{1}T_{A}(V)$ is generated as a
vector space by homogeneous elements of the form

$x=(x_{1}\otimes_{A}\ldots\otimes_{A}x_{i})\otimes(x_{i+1}\otimes_{A}\ldots\otimes_{A}x_{n})-(x_{1}\otimes_{A}\ldots\otimes_{A}x_{i-1})\otimes(x_{i}\otimes_{A}\ldots\otimes_{A}x_{n})\in(\Omega^{1}T_{A}(V))^{n}$

for $x_{m}\in V$ and $1\leq i\leq n$. Then

$(x_{1}\otimes_{A}\ldots\otimes_{A}x_{i-1})\cdot(x_{i}\otimes 1-1\otimes
x_{i})\cdot(x_{i+1}\otimes_{A}\ldots\otimes_{A}x_{n})=x$

and so because
$(x_{1}\otimes_{A}\ldots\otimes_{A}x_{i-1}),\,(x_{i+1}\otimes_{A}\ldots\otimes_{A}x_{n})\in
T_{A}(V)$ and $(a_{i}\otimes 1-1\otimes a_{i})\in(\Omega^{1}T_{A}(V))^{1}$ we
get the result. ∎

For $\theta\in\textnormal{Der}(A)$ define the degree 0 derivation
$L_{\theta}:\Omega^{\bullet}\rightarrow\Omega^{\bullet}$ by

$\displaystyle A\ni a$ $\displaystyle\mapsto\theta(a)$
$\displaystyle\Omega^{1}A\ni D(a)$ $\displaystyle\mapsto D(\theta(a))$

extended to the rest of $\Omega^{\bullet}A$ using the Leibniz rule and
linearity. We also define the _contraction mapping_
$i_{\theta}:\Omega^{\bullet}\rightarrow\Omega^{\bullet}$ as the super-
derivation given by

$\displaystyle A\ni a$ $\displaystyle\mapsto 0$ $\displaystyle\Omega^{1}A\ni
D(a)$ $\displaystyle\mapsto\theta(a).$

$i_{\theta}$ has degree $-1$ with respect to the tensor grading and degree 0
with respect to the induced grading by $A$. The derivations $L_{\theta}$ and
$i_{\theta}$ are related by the _Cartan identity_

$\displaystyle L_{\theta}=D\circ i_{\theta}+i_{\theta}\circ D\,.$ (2.1)

Both derivations descend to maps
$\textnormal{DR}(A)\rightarrow\textnormal{DR}(A)$; indeed if $x\in\Omega^{m}A$
and $y\in\Omega^{n}A$ then we have

$\displaystyle L_{\theta}([x,y])$
$\displaystyle=L_{\theta}\big{(}xy-(-1)^{mn}\,yx\big{)}$
$\displaystyle=L_{\theta}(x)y+x\,L_{\theta}(y)-(-1)^{mn}\,L_{\theta}(y)x-(-1)^{mn}\,y\,L_{\theta}(x)$
$\displaystyle=L_{\theta}(x)y-(-1)^{mn}\,y\,L_{\theta}(x)+x\,L_{\theta}(y)-(-1)^{mn}\,L_{\theta}(y)x$
$\displaystyle=[L_{\theta}(x),y]+[x,L_{\theta}(y)]$

and

$\displaystyle i_{\theta}([x,y])$
$\displaystyle=i_{\theta}\big{(}xy-(-1)^{mn}\,yx\big{)}$
$\displaystyle=i_{\theta}(x)y+(-1)^{m}\,x\,i_{\theta}(y)-(-1)^{mn}\,i_{\theta}(y)x-(-1)^{mn}(-1)^{n}\,y\,i_{\theta}(x)$
$\displaystyle=i_{\theta}(x)y-(-1)^{(m+1)n}\,y\,i_{\theta}(x)+(-1)^{m}\,x\,i_{\theta}(y)-(-1)^{mn}\,i_{\theta}(y)x$
$\displaystyle=i_{\theta}(x)y-(-1)^{(m-1)n}\,y\,i_{\theta}(x)+(-1)^{m}\Big{(}x\,i_{\theta}(y)-(-1)^{m(n-1)}\,i_{\theta}(y)x\Big{)}$
$\displaystyle=[i_{\theta}(x),y]+(-1)^{m}[x,i_{\theta}(y)].$

For $\lambda\in\mathbb{D}\textnormal{er}(A)$ we can also define a _contraction
mapping_
$i_{\lambda}:\Omega^{\bullet}A\rightarrow\Omega^{\bullet}A\otimes\Omega^{\bullet}A$
as the double super-derivation of degree $-1$ given by

$\displaystyle a\in\,A$ $\displaystyle\mapsto 0$ $\displaystyle
D(a)\in\,\Omega^{1}A$ $\displaystyle\mapsto\lambda(a).$

The _reduced contraction mapping_
$\iota_{\lambda}:\Omega^{\bullet}A\rightarrow\Omega^{\bullet}A$ is then
defined as

$\iota_{\lambda}=m(\beta\circ i_{\lambda})$

where
$m:\Omega^{\bullet}A\otimes\Omega^{\bullet}A\rightarrow\Omega^{\bullet}A$ is
the multiplication map and $\beta$ is the flip isomorphism on
$\Omega^{\bullet}A\otimes\Omega^{\bullet}A$ which sends homogeneous elements
$x\in\Omega^{m}A$ and $y\in\Omega^{n}A$ to

$x\otimes y\longmapsto(-1)^{mn}y\otimes x.$

The reduced contraction mapping $\iota_{\lambda}$ descends to a map

$\iota_{\lambda}:\textnormal{DR}(A)\rightarrow\Omega^{\bullet}A$

(see [[6] Lemma 2.8.6 (i)]).

######  Definition 2.7.

Let $\omega\in\textnormal{DR}^{2}(A)$ be closed with respect to the
differential $D$. Define a map
$i^{\omega}:\textnormal{Der}(A)\rightarrow\textnormal{DR}^{1}(A)$ by sending

$\textnormal{Der}(A)\ni\theta\mapsto i_{\theta}(\omega).$

We call $\omega$ _symplectic_ if $i^{\omega}$ is an isomorphism. Similarly we
can define a map
$\iota^{\omega}:\mathbb{D}\textnormal{er}(A)\rightarrow\Omega^{1}A$ and call
$\omega$ _bisymplectic_ if $\iota^{\omega}$ is an isomorphism.

From [[6] section 4.2] we have the following result.

######  Lemma 2.3.

Let $A$ be smooth and $\omega$ be a bisymplectic 2-form. Then $\omega$ is
symplectic.

For $\omega$ bisymplectic and any $a\in A$ we can define a derivation
$\\{a,-\\}_{\omega}:A\rightarrow A$ by

$b\mapsto m((\iota^{\omega})^{-1}(D(a))(b))$

which makes sense because, as $\omega$ is bisymplectic,
$(\iota^{\omega})^{-1}:\Omega^{1}A\rightarrow\mathbb{D}\textnormal{er}(A)$ is
a well-defined map. Similarly if $\omega$ is symplectic then for $W\in
A/[A,A]$ we can define a derivation in $\textnormal{Der}(A)$ by

$\\{W,-\\}_{\omega}:=(i^{\omega})^{-1}(D(W)).$

######  Definition 2.8.

An algebra $A$ is _connected_ if the sequence

$0\rightarrow
k\rightarrow\textnormal{DR}^{0}(A)\rightarrow\textnormal{DR}^{1}(A)$

is exact.

Let $\Delta:A\rightarrow A\otimes A$ be the distinguished double derivation
that sends $a\mapsto a\otimes 1-1\otimes a$. Then there exists a map

$\mu_{\textnormal{nc}}:\overline{\textnormal{DR}^{2}(A)}\rightarrow A/k$

where $\overline{\textnormal{DR}^{2}(A)}\subset\textnormal{DR}^{2}(A)$ are the
closed cyclic 2-forms, having the property that

$D(\mu_{\textnormal{nc}}(\omega))=\iota_{\Delta}(\omega)$

in $\Omega^{1}A$. $\mu_{\textnormal{nc}}$ is defined as the composition

$\overline{\textnormal{DR}^{2}(A)}\xrightarrow{\iota_{\Delta}}\overline{\Omega^{1}A}\xrightarrow{=}\widetilde{\Omega^{1}A}\xrightarrow{D^{-1}}A/k$

where $\widetilde{\Omega^{1}A}$ are the exact 1-forms (see [[6] section 4]).
Let $\textbf{w}\in A$ be a representative of $\mu_{\textnormal{nc}}(\omega)$.

######  Definition 2.9.

Let $(A,\omega,\xi)$ be a triple of a non-positively graded smooth algebra $A$
such that $\Omega^{1}A$ is generated by homogeneous elements of degree
$0,-1,\ldots,c$ as a graded $A$-bimodule, a bisymplectic 2-form
$\omega\in\textnormal{DR}^{2}(A)$ which is homogeneous of degree $c$ with
respect to the induced grading of $A$, and a super-derivation
$\xi\in\textnormal{Der}(A)$ of degree 1 such that $\xi^{2}=0$,
$L_{\xi}(\omega)=0$, and $\xi(\textbf{w})=0$. We define the _Ginzburg
differential graded algebra_ (Gdga) $\mathfrak{D}(A,\omega,\xi)$ of this
triple to be the following data:

  * •

The underlying algebra of $\mathfrak{D}(A,\omega,\xi)$ is the free product of
algebras $A*k[t]$ where $t$ has degree $c-1.$

  * •

The differential $d$ is given by

$\displaystyle A\ni a$ $\displaystyle\mapsto\xi(a)$ $\displaystyle t$
$\displaystyle\mapsto\textbf{w}.$

######  Remark 2.4.

  1.   2. (1)

As $\mu_{\textnormal{nc}}(\omega)\in A/k$ the representative w is only
determined up to some constant in $k$. Thus we choose w to be homogeneous (of
degree $\textnormal{deg}(\omega)=c$) so it is unique. This can cause a small
issue when $c=0$ but this will not be relevant to us; see [[9] Remark 4.3.1]
for more details.

  3. (2)

That $d^{2}=0$ follows from the assumptions $\xi^{2}=0$ and
$\xi(\textbf{w})=0$. One way of ensuring $\xi(\textnormal{{w}})=0$ is to
assume the additional condition $k\cap\xi(A)=0$. By [[6] Proposition 4.1.3] we
have for any $\theta\in\textnormal{Der}(A)$

$\mu_{\textnormal{nc}}(L_{\theta}(\omega))=L_{\theta}(\mu_{\textnormal{nc}}(\omega))$

hence

$\xi(\mu_{\textnormal{nc}}(\omega))=L_{\xi}(\mu_{\textnormal{nc}}(\omega))=\mu_{\textnormal{nc}}(L_{\xi}(\omega))=\mu_{\textnormal{nc}}(0)=0.$

It follows that $\xi(\textbf{w})\in k$ and so if $k\cap\xi(A)=0$ we get
$\xi(\textbf{w})=0$.

  4. (3)

Since $A$ is smooth $\omega$ is also symplectic by 2.3.

######  Definition 2.10.

A _superpotential algebra_ $B$ is an algebra (viewed as a dga concentrated in
degree 0) that is quasi-isomorphic to a Gdga $\mathfrak{D}(A,\omega,\xi)$ in
which $A$ is connected and $\xi=\\{W,-\\}_{\omega}$ for some $W\in A/[A,A].$ W
is called a _potential_.

We shall see some examples of Gdgas when we look at quivers and Jacobi
algebras.

######  Theorem 2.5 ([14] Theorem 3.6.4).

Let $\mathfrak{D}(A,\omega,\xi)$ be a Gdga with $\omega$ of degree $c$ and
suppose that $H^{i}(\mathfrak{D}(A,\omega,\xi))=0$ for all $i\neq 0.$ Then
$H^{0}(\mathfrak{D}(A,\omega,\xi))$ is Calabi-Yau of dimension $-c+2$.

######  Theorem 2.6 ([9] Theorem 4.3.8).

Let $\mathfrak{D}(A,\omega,\xi)$ be as in 2.5 and also suppose that $A$ is
connected. Then $H^{0}(\mathfrak{D}(A,\omega,\xi))$ is exact Calabi-Yau of
dimension $-c+2$. In particular a superpotential algebra is exact Calabi-Yau.

###  2.2. Ginzburg differential graded algebras in the relative case

We can strengthen 2.5 by looking at a smaller class of Gdgas arising in
dimension 3. First we must extend the previous story of noncommutative
algebras over a field $k$ to noncommutative algebras over a finite dimensional
semisimple $k$-algebra $R$. Almost everything follows in a natural way to the
relative case. We introduce notation with ``$R$" in the subscript to
distinguish this change. Recall unadorned tensor products are to be taken over
the base field $k$. We shall explore examples of these relative Gdgas in the
next section on quivers and Jacobi algebras.

Let $A$ be an $R$-algebra, then

$\Omega^{1}_{R}A=\textnormal{Ker}(A\otimes_{R}A\xrightarrow{m}A)$

is the bimodule of _relative 1-forms_ , and similarly

$\Omega_{R}^{\bullet}A=T_{A}(\Omega_{R}^{1}A)$

is the dga of _relative noncommutative differential forms_ , with its cyclic
quotient

$\textnormal{DR}_{R}(A)=\Omega_{R}^{\bullet}A/[\Omega_{R}^{\bullet}A,\Omega_{R}^{\bullet}A].$

An important difference is in the bimodule of _relative double derivations_
$\mathbb{D}\textnormal{er}_{R}(A)$ which are derivations $\lambda:A\rightarrow
A\otimes A$ such that $\lambda\circ(R\rightarrow A)=0.$ This is defined so
that we again have a natural isomorphism
$\mathbb{D}\textnormal{er}_{R}(A)\cong(\Omega^{1}_{R}A)^{\vee}$. From now on
we let $R=kI$ be the algebra of $k$-valued functions on a finite indexing set
$I$. For each $i\in I$ let $1_{i}\in kI$ be the function that sends $i\mapsto
1$ and everything else to 0, and let $e=\sum_{i\in I}1_{i}\otimes 1_{i}\in
R\otimes R$. Define $\delta\in\mathbb{D}\textnormal{er}_{R}(A)$ as the
distinguished relative double derivation given by

$A\ni a\longmapsto a\cdot e-e\cdot a\in A\otimes A.$

Then $\delta$ is the appropriate relative counterpart to the distinguished
double derivation $\Delta$ we had in Section 2.1.

As in the non-relative case we have a map
$\mu_{\textnormal{nc}}:\overline{\textnormal{DR}_{R}^{2}(A)}\rightarrow(A/R)^{R}$,
where $(A/R)^{R}$ is the centraliser of $R$ in $A/R$, defined by the property
that

$D(\mu_{\textnormal{nc}}(\omega))=\iota_{\delta}(\omega)$

in $\Omega_{R}^{1}A$. $\mu_{\textnormal{nc}}$ is given by the composition

$\overline{\textnormal{DR}_{R}^{2}(A)}\xrightarrow{\iota_{\Delta}}(\overline{\Omega^{1}A})^{R}\xrightarrow{=}(\widetilde{\Omega^{1}A})^{R}\xrightarrow{D^{-1}}(A/R)^{R}$

(again see [[6] section 4]). Taking $\textbf{w}\in A$ to be a representative
of $\mu_{\textnormal{nc}}(\omega)$ we can define a relative Gdga analogously
to the non-relative case.

######  Definition 2.11.

Let $A$ be a non-positively graded smooth $R$-algebra such that
$\Omega_{R}^{1}A$ is generated by homogeneous elements of degree
$0,-1,\ldots,c$ as a graded $A$-bimodule,
$\omega\in\textnormal{DR}_{R}^{2}(A)$ a bisymplectic 2-form of degree $c$, and
$\xi\in\textnormal{Der}_{R}(A)$ a super-derivation of degree 1 such that
$\xi^{2}=0$, $L_{\xi}(\omega)=0$ and $\xi(\textbf{w})=0$. The _Ginzburg
differential graded algebra_ $\mathfrak{D}(A,\omega,\xi)$ is the free product
of $R$-algebras $A*_{R}R[t]$ with $t$ in degree $c-1$ and differential $d$
given by

$\displaystyle A\ni a$ $\displaystyle\mapsto\xi(a)$ $\displaystyle t$
$\displaystyle\mapsto\textbf{w}.$

Following [[14] section 5] let $F$ be a smooth algebra over $R$ and let
$\alpha\in\textnormal{DR}_{R}^{1}(F)$ be a cyclic 1-form such that

$\displaystyle D(\alpha)=0\quad\textnormal{and}\quad\iota_{\delta}(\alpha)=0.$
(2.2)

Set $A$ to be the tensor-algebra $T_{F}(\mathbb{D}\textnormal{er}_{R}(F))$
with non-positive grading given by negating the natural tensor grading. $A$ is
smooth by [[6] Theorem 5.1.1]. There is a canonical closed cyclic bisymplectic
2-form $\omega\in\textnormal{DR}_{R}^{2}(A)$ of degree -1 (see [[6] Theorem
5.1.1 and Proposition 5.4.1]) and by 2.2 $\Omega_{R}^{1}A$ is generated by
homogeneous elements of degree 0 and $-1$. Let $\xi_{\alpha}$ be the super-
derivation which is the image of $\alpha$ under the following composition

$\textnormal{DR}_{R}^{1}(F)\hookrightarrow\textnormal{DR}_{R}^{1}(A)\xrightarrow[\sim]{(i^{\omega})^{-1}}\textnormal{Der}_{R}(A)$

where the first inclusion is induced by the map
$F\xrightarrow{\sim}A^{0}\hookrightarrow A.$

######  Lemma 2.7.

With the algebra A, bisymplectic 2-form $\omega$ and super-derivation
$\xi_{\alpha}$ as above we have

$\xi_{\alpha}^{2}=0\quad\textnormal{and}\quad L_{\xi_{\alpha}}(\omega)=0.$

###### Proof.

In [14] in the proof of Proposition 5.2.4 we are given that $\xi_{\alpha}$
sends

$\displaystyle A^{0}=F\ni f$ $\displaystyle\mapsto 0$ (2.3) $\displaystyle
A^{1}=\mathbb{D}\textnormal{er}_{R}(F)\ni\theta$
$\displaystyle\mapsto\iota_{\theta}(\alpha)\in A^{0}.$

It follows that (omitting the $F$ subscript in the tensor product to reduce
clutter)

$A^{n}\ni\theta_{1}\otimes\ldots\otimes\theta_{n}\xmapsto{\,\,\,\xi_{\alpha}\,\,\,}\sum_{i=1}^{n}(-1)^{i-1}\theta_{1}\otimes\ldots\otimes\theta_{i-1}\otimes\iota_{\theta_{i}}(\alpha)\,\theta_{i+1}\otimes\ldots\otimes\theta_{n}\in
A^{n-1}$

hence

$\displaystyle\xi_{\alpha}^{2}(\theta_{1}\otimes\ldots\otimes\theta_{n})$
$\displaystyle=\sum_{i=1}^{n}(-1)^{i-1}\bigg{(}\sum_{j<i}(-1)^{j-1}\,\theta_{1}\otimes\ldots\iota_{\theta_{j}}(\alpha)\,\theta_{j+1}\ldots\iota_{\theta_{i}}(\alpha)\,\theta_{i+1}\ldots\otimes\theta_{n}$
$\displaystyle\qquad\qquad+\sum_{j>i}(-1)^{j}\,\theta_{1}\otimes\ldots\iota_{\theta_{i}}(\alpha)\,\theta_{i+1}\ldots\iota_{\theta_{j}}(\alpha)\,\theta_{j+1}\ldots\otimes\theta_{n}\bigg{)}$
$\displaystyle=\sum_{i=1}^{n-1}\bigg{(}\sum_{j>i}^{n}(-1)^{i+j-1}\,\theta_{1}\otimes\ldots\iota_{\theta_{i}}(\alpha)\,\theta_{i+1}\ldots\iota_{\theta_{j}}(\alpha)\,\theta_{j+1}\ldots\otimes\theta_{n}$
$\displaystyle\qquad\qquad+(-1)^{i+j-2}\,\theta_{1}\otimes\ldots\iota_{\theta_{i}}(\alpha)\,\theta_{i+1}\ldots\iota_{\theta_{j}}(\alpha)\,\theta_{j+1}\ldots\otimes\theta_{n}\bigg{)}$
$\displaystyle=0.$

Then because the elements $\theta_{1}\otimes\ldots\otimes\theta_{n}$ generate
$A^{n}$ over $F$ we have that $\xi_{\alpha}^{2}(x)=0$ for any $x\in A^{n}$ and
any $n\in\mathbb{N}$.

To show $L_{\xi_{\alpha}}(\omega)=0$, by definition we have
$\xi_{\alpha}=(i^{\omega})^{-1}(\alpha)$ and so
$i_{\xi_{\alpha}}(\omega)=\alpha$. Hence by the Cartan identity (2.1) we get

$L_{\xi_{\alpha}}(\omega)=D(i_{\xi_{\alpha}}(\omega))+i_{\xi_{\alpha}}(D(\omega))=D(\alpha)+0=0$

using the assumption in (2.2) that $D(\alpha)=0$ and the fact that $\omega$ is
closed. ∎

######  Definition 2.12.

Let $F$ be a smooth $R$-algebra and $\alpha\in\textnormal{DR}_{R}^{1}(F)$ a
cyclic 1-form satisfying the conditions in (2.2). Define the _Gdga associated
to the data $F,\alpha$_ to be

$\mathfrak{D}(F,\alpha)=\mathfrak{D}(A,\omega,\xi_{\alpha})$

where $A=T_{F}(\mathbb{D}\textnormal{er}_{R}(F))$, and $\omega$ and
$\xi_{\alpha}$ are as above.

######  Remark 2.8.

As per the definition of a Gdga we still need to establish the condition
$\xi_{\alpha}(\textbf{w})=0$. From [[14] Proposition 5.2.4 and Remark 5.2.5]
we do not necessarily have the additional condition $R\cap\xi_{\alpha}(A)=0$
analogous to $k\cap\xi(A)=0$ which we had in the non-relative setting in 2.4.
However $\xi_{\alpha}(\textbf{w})=0$ instead follows from the assumption
$\iota_{\delta}(\alpha)=0$ in (2.2). Indeed, from (2.3) we have that
$\xi_{\alpha}(\delta)=\iota_{\delta}(\alpha)=0$. Then by [[6] Theorem 5.1.1]
$\delta$ is a representative of $\mu_{\textnormal{nc}}(\omega)$ giving the
required result.

This smaller class of Gdgas have some nice properties which were studied by
Ginzburg in [[14] section 5.3]. In particular he showed that the zeroth
cohomology $H^{0}(\mathfrak{D}(F,\alpha))$ of the Gdga is Calabi-Yau of
dimension 3 if and only if the so-called _completed_ dga
$\widehat{\mathfrak{D}}(F,\alpha)$ is acyclic. Alternatively if
$\mathfrak{D}(F,\alpha)$ has an additional strictly positive grading (i.e.
$(\mathfrak{D}(F,\alpha))_{0}=k$) which is preserved by the differential $d$,
then $H^{0}(\mathfrak{D}(F,\alpha))$ is Calabi-Yau of dimension 3 if and only
if $\mathfrak{D}(F,\alpha)$ is acyclic. Later on it was shown by Keller and
Van den Bergh that $\mathfrak{D}(F,\alpha)$ is in fact itself Calabi-Yai of
dimension 3.

######  Theorem 2.9 ([18] Theorem A.12).

The Gdga $\mathfrak{D}(F,\alpha)$ is Calabi-Yau of dimension 3.

###  2.3. Quivers and Jacobi algebras

A quiver $Q$ is a pair $(Q_{0},Q_{1})$ consisting of a finite set of vertices
$Q_{0}$, and a finite set of directed edges or arrows $Q_{1}$ between those
vertices. We allow loops and cycles in our quivers. For each quiver we have
two maps, $s,t:Q_{1}\rightarrow Q_{0}$, the source and target maps that send
an arrow to its source vertex or target vertex respectively. For $i\in Q_{0}$
let $e_{i}$ denote the constant path in $Q$ at the vertex $i$.

######  Definition 2.13.

The path algebra $kQ$ of the quiver $Q$ is the algebra obtained by taking the
free $k$-algebra over all the constant paths $e_{i}$ and all the arrows in
$a\in Q_{1}$, modulo the relations that if two paths do not concatenate then
their product is 0.

######  Definition 2.14.

A representation of a quiver $Q$ is a vector space $V$ that comes with a
decomposition $V=\bigoplus_{i\in Q_{0}}V_{i}$ and linear maps
$f_{a}:V_{i}\rightarrow V_{j}$ for each arrow $a:i\rightarrow j\in Q_{1}$.

We shall focus on finite dimensional representations which can be grouped via
a dimension vector $n=(n_{i})\in\mathbb{N}^{Q_{0}}$ given by the
decomposition, i.e. $n_{i}=\textnormal{dim}(V_{i})$ for each $i$. Let
$\textnormal{Rep}_{n}(Q)$ denote the stack of $n$-dimensional representations
of $Q$. Explicitly we have the following global quotient stack description

$\textnormal{Rep}_{n}(Q)\cong\prod_{a:i\rightarrow j\in
Q_{1}}\textnormal{Mat}_{n_{i}\times n_{j}}(k)/\prod_{l\in
Q_{0}}\textnormal{GL}_{n_{l}}$

where $(g_{l})\in\prod_{l\in Q_{0}}\textnormal{GL}_{n_{l}}$ acts on
$(f_{a})\in\prod_{a\in Q_{1}}\textnormal{Mat}_{n_{i}\times n_{j}}(k)$ via
$(g_{l})\cdot(f_{a})=(g_{j}f_{a}g_{i}^{-1}).$

To get more interesting stacks of representations we may also consider quivers
with potential. Recall a potential is an element $W\in kQ/[kQ,kQ]$ and in this
case can be written as a sum of cycles in $Q$ up to cyclic permutation. Given
a potential $W$ we can consider its ``noncommutative derivatives" $\partial
W/\partial a$ which can be defined by taking all the cycles in $W$ that
contain the arrow $a$, cyclically permuting $a$ to the front of those cycles
and then deleting $a$. Concretely let $\mathfrak{d}:kQ/[kQ,kQ]\rightarrow kQ$
be the map that sends a cycle

$a_{n}a_{n-1}\ldots a_{1}\mapsto\sum_{j}a_{j}a_{j-1}\ldots a_{1}a_{n}\ldots
a_{j+1},$

and for $a\in Q_{1}$ let $a^{-1}:kQ\rightarrow kQ$ send the path

$p=b_{m}b_{m-1}\ldots b_{1}\mapsto\begin{cases}b_{m-1}\ldots
b_{1}&\textnormal{if}\,\,b_{m}=a\\\ 0&\textnormal{otherwise}\end{cases}$

then we define $\partial W/\partial a$ as $a^{-1}\circ\mathfrak{d}(W)$. These
derivatives give us an elements in the path algebra and so we obtain the ideal

$I_{W}=(\partial W/\partial a\,:\,a\in Q_{1}).$

######  Definition 2.15.

The Jacobi algebra of the quiver $Q$ with potential $W$ is the quotient
algebra

$\textnormal{Jac}(Q,W)=kQ/I_{W}.$

The potential $W$ also describes a natural map
$\textnormal{Tr}(W)_{n}:\textnormal{Rep}_{n}(Q)\rightarrow k$ that sends a
representation $(f_{a})$ to the trace of the matrix $W(f_{a})$.

######  Proposition 2.10 ([21] Proposition 3.8).

There is an isomorphism of stacks

$\textnormal{Rep}_{n}(\textnormal{Jac}(Q,W))\cong\textnormal{crit}(\textnormal{Tr}(W)_{n}).$

Unfortunately not all Jacobi algebras are superpotential algebras or even
Calabi-Yau algebras but there is a natural Gdga associated to them. The
following comes from [[14] section 4.2]. We work in the relative setting and
let $R=kQ_{0}$ be the $k$-algebra over the vertex set of the quiver $Q$ or
equivalently the $k$-algebra over the constant paths. The constant paths
$e_{i}$ then take the role of the functions $1_{i}$ mentioned in Section 2.2.

######  Lemma 2.11.

Let $V$ be the vector space $kQ_{1}$ viewed in the natural way as an
$R$-bimodule. Then we have an isomorphism of algebras

$\displaystyle kQ\cong T_{R}(V).$ (2.4)

In particular $kQ$ is a smooth $R$-algebra.

###### Proof.

The isomorphism is immediate from the definition of the multiplication on the
path algebra versus on the tensor algebra. For smoothness, clearly $R$ is
smooth and as $V$ is finite dimensional it is projective as an $R$-bimodule
hence by 2.1 $T_{R}(V)$ is smooth. ∎

Given a quiver $Q$ with potential $W$ consider the new quiver $\widehat{Q}$
which has vertex set $\widehat{Q}_{0}=Q_{0}$ and arrows

$\displaystyle a:i\rightarrow j\quad\,\,\textnormal{of degree 0 for
any}\,\,\,\,a:i\rightarrow j\in Q_{1}$ $\displaystyle a^{*}:j\rightarrow
i\quad\textnormal{of degree -1 for any}\,\,\,\,a:i\rightarrow j\in Q_{1}$
loops $\displaystyle t_{i}:i\rightarrow i\quad\,\,\textnormal{of degree -2 for
any}\,\,\,\,i\in Q_{0}.$

######  Definition 2.16.

The _Gdga associated to $(Q,W)$_, denoted by $\Gamma_{Q,W}$, is the path
algebra $k\widehat{Q}$ along with the differential $d$ that sends

$\displaystyle a\mapsto 0$ $\displaystyle a^{*}\mapsto\partial W/\partial a$
$\displaystyle t_{i}\mapsto e_{i}\Big{(}\sum_{a\in
Q_{1}}[a,a^{*}]\Big{)}e_{i}.$

######  Proposition 2.12.

$\Gamma_{Q,W}$ is a Gdga in the sense of 2.11.

###### Proof.

Let $\overline{Q}$ be the double quiver of $Q$, then the underlying algebra of
$\Gamma_{Q,W}$ is $k\overline{Q}*_{R}R[t]$ where $\sum_{i\in Q_{0}}t_{i}=t$.
As $k\overline{Q}$ is a path algebra of a quiver it is smooth by 2.11. If $p$
is a path in $k\overline{Q}$ and we write $p=p_{1}xp_{2}$ for some arrow
$x\in\overline{Q}_{1}$, then $\Omega^{1}_{R}k\overline{Q}$ is generated as a
$R$-bimodule by

$p_{1}x\otimes_{R}p_{2}-p_{1}\otimes_{R}xp_{2}$

for all paths $p$ and arrows $x$. Hence, because

$p_{1}\cdot(x\otimes_{R}1-1\otimes_{R}x)\cdot
p_{2}=p_{1}x\otimes_{R}p_{2}-p_{1}\otimes_{R}xp_{2}$

$\Omega^{1}_{R}k\overline{Q}$ is generated as a $k\overline{Q}$-bimodule by
the homogeneous elements of degree 0 and -1. From [[6] section 8.1 and
Proposition 8.1.1] we have that the 2-form $\omega$ is 111note this is
$-1\cdot$ the 2-form given in [6] due to a sign error in [[6] Lemma 3.1.1
(ii)]

$-\sum_{a\in Q_{1}}D(a)\cdot D(a^{*})=\sum_{a\in Q_{1}}D(a^{*})\cdot D(a)$

which has degree $-1$, and the super-derivation is $\xi=\\{W,-\\}_{\omega}$.

We first show that $\omega$ is bisymplectic. Let
$\lambda\in\mathbb{D}\textnormal{er}_{R}(k\overline{Q})$ be a double
derivation defined on arrows $x\in\overline{Q}_{1}$ by

$\lambda(x)=\sum_{r}c_{r}^{x}p_{r}^{x}\otimes q_{r}^{x}$

where $c_{r}^{x}\in k$ and $p_{r}^{x},\,q_{r}^{x}$ are paths in
$k\overline{Q}$. Note that for each term in $\lambda(x)$ we have
$s(x)=s(q_{r}^{x})$ and $t(x)=t(p_{r}^{x})$ for the paths
$p_{r}^{x},q_{r}^{x}$. We consider the map $\iota^{\omega}$ that sends
$\lambda\mapsto\iota_{\lambda}(\omega).$ Then

$\displaystyle\iota_{\lambda}(\omega)$ $\displaystyle=m(\beta\circ
i_{\lambda}(\omega))$ $\displaystyle=m\Big{(}\beta\Big{(}\sum_{a\in
Q_{1}}-\lambda(a)\cdot D(a^{*})+D(a)\cdot\lambda(a^{*})\Big{)}\Big{)}$
$\displaystyle=m\Big{(}\beta\Big{(}\sum_{a\in
Q_{1}}-\Big{(}\sum_{r}c_{r}^{a}p_{r}^{a}\otimes q_{r}^{a}\Big{)}\cdot
D(a^{*})+D(a)\cdot\Big{(}\sum_{s}c_{s}^{a^{*}}p_{s}^{a^{*}}\otimes
q_{s}^{a^{*}}\Big{)}\Big{)}\Big{)}$
$\displaystyle=m\Big{(}\beta\Big{(}\sum_{a\in
Q_{1}}-\Big{(}\sum_{r}c_{r}^{a}p_{r}^{a}\otimes(q_{r}^{a}D(a^{*}))+\sum_{s}c_{s}^{a^{*}}(D(a)\,p_{s}^{a^{*}})\otimes
q_{s}^{a^{*}}\Big{)}\Big{)}\Big{)}$ $\displaystyle=\sum_{a\in
Q_{1}}-\Big{(}\sum_{r}c_{r}^{a}q_{r}^{a}D(a^{*})\,p_{r}^{a}-\sum_{s}c_{s}^{a^{*}}q_{s}^{a^{*}}D(a)\,p_{s}^{a^{*}}\Big{)}$
$\displaystyle=\sum_{a\in
Q_{1}}-\Big{(}\sum_{r}c_{r}^{a}(q_{r}^{a}a^{*}\otimes_{R}p_{r}^{a}-q_{r}^{a}\otimes_{R}a^{*}p_{r}^{a})-\sum_{s}c_{s}^{a^{*}}(q_{s}^{a^{*}}a\otimes_{R}p_{s}^{a^{*}}-q_{s}^{a^{*}}\otimes_{R}ap_{s}^{a^{*}})\Big{)}.$
(2.5)

For surjectivity of $\iota^{\omega}$, as $\Omega^{1}_{R}k\overline{Q}$ is
generated by $p_{1}x\otimes_{R}p_{2}-p_{1}\otimes_{R}xp_{2}$ for all paths
$p=p_{1}xp_{2}$ and arrows $x\in k\overline{Q}_{1}$, if we define
$\lambda_{p,x}\in\mathbb{D}\textnormal{er}_{R}(k\overline{Q})$ by
$\lambda_{p,x}(x^{*})=p_{2}\otimes p_{1}$ and $\lambda(y)=0$ for all other
$y\in\overline{Q}_{1}$ (notation: $(x^{*})^{*}=x$) then we can see that
$\iota_{\lambda_{p,x}}(\omega)=p_{1}x\otimes_{R}p_{2}-p_{1}\otimes_{R}xp_{2}$
using equation (2.5). For injectivity suppose that
$\iota_{\lambda}(\omega)=0$. Each simple tensor element in the sum given in
equation (2.5) is non-zero since $s(q_{r}^{x})=s(x)=t(x^{*})$ and
$t(p_{r}^{x})=t(x)=s(x^{*})$. Then because simple tensor elements give a basis
of $k\overline{Q}\otimes_{R}k\overline{Q}$, for each
$x_{0}\in\overline{Q}_{1}$ and each $r_{0}$ with paths
$p_{r_{0}}^{x_{0}},q_{r_{0}}^{x_{0}}$ in $k\overline{Q}$ given by
$\lambda(x_{0})$ there must exist at least one $x_{1}\in\overline{Q}_{1}$ and
some $r_{1}$ with paths $p_{r_{1}}^{x_{1}},q_{r_{1}}^{x_{1}}$ in
$k\overline{Q}$ given by $\lambda(x_{1})$ such that

$q_{r_{0}}^{x_{0}}x_{0}^{*}\otimes_{R}p_{r_{0}}^{x_{0}}=q_{r_{1}}^{x_{1}}\otimes_{R}x_{1}^{*}p_{r_{1}}^{x_{1}}$

in order for the overall coefficient of the term
$q_{r_{0}}^{x_{0}}x_{0}^{*}\otimes_{R}p_{r_{0}}^{x_{0}}$ in equation (2.5) to
be 0. Let the length of the path $p_{r_{0}}^{x_{0}}$ be $l_{1}$ and the length
of $q_{r_{0}}^{x_{0}}$ be $l_{2}$. Then the length of
$p_{r_{1}}^{x_{1}}=l_{1}-1$ and the length of $q_{r_{1}}^{x_{1}}=l_{2}+1$. Now
equation (2.5) also contains the non-zero term
$c_{r_{1}}^{x_{1}}q_{r_{1}}^{x_{1}}x_{1}^{*}\otimes_{R}p_{r_{1}}^{x_{1}}$.
Hence we repeat this for $x_{1}$ to find a $x_{2}\in\overline{Q}_{1}$ and
$r_{2}$ such that
$q_{r_{1}}^{x_{1}}x_{1}^{*}\otimes_{R}p_{r_{1}}^{x_{1}}=q_{r_{2}}^{x_{2}}\otimes_{R}x_{2}^{*}p_{r_{2}}^{x_{2}}$
where the length of $p_{r_{2}}^{x_{2}}=l_{1}-2$ and the length of
$q_{r_{2}}^{x_{2}}=l_{2}+2$. We continue to repeat this argument and after
$l_{1}$ successive iterations we find an arrow $x_{l_{1}}\in\overline{Q}_{1}$
and some $r_{l_{1}}$ such that the length of $p_{r_{l_{1}}}^{x_{l_{1}}}=0$
i.e. $p_{r_{l_{1}}}^{x_{l_{1}}}$ is just a constant path in $\overline{Q}$.
However now we have the non-zero term
$c_{r_{l_{1}}}^{x_{l_{1}}}q_{r_{l_{1}}}^{x_{l_{1}}}x_{l_{1}}^{*}\otimes_{R}p_{r_{l_{1}}}^{x_{l_{1}}}$
in equation (2.5) but there cannot exist some $y\in\overline{Q}_{1}$ some $s$
and paths $p_{s}^{y},q_{s}^{y}$ in $k\overline{Q}$ such that

$q_{r_{l_{1}}}^{x_{l_{1}}}x_{l_{1}}^{*}\otimes_{R}p_{r_{l_{1}}}^{x_{l_{1}}}=q_{s}^{y}\otimes_{R}y^{*}p_{s}^{y}$

because $y^{*}p_{s}^{y}$ cannot be a constant path. It follows that the
coefficient of the term
$q_{r_{l_{1}}}^{x_{l_{1}}}x_{l_{1}}^{*}\otimes_{R}p_{r_{l_{1}}}^{x_{l_{1}}}$
in equation (2.5) is $c_{r_{l_{1}}}^{x_{l_{1}}}\neq 0$ which contradicts our
initial assumption that $\iota_{\lambda}(\omega)=0$.

Next we check the differential $d$. In the definition of the Gdga the
differential is given on $A$ by the super-derivation $\xi$, so we want to
explicitly verify that for all $x\in\overline{Q}_{1}$ we have
$d(x)=\xi(x)=\\{W,x\\}_{\omega}$. $\xi$ is defined as the unique super-
derivation such that $i_{\xi}(\omega)=D(W)$ so write
$W=\sum_{t=1}^{m}a_{t,n_{t}}\ldots a_{t,1}$ for arrows
$a_{t,j}\in\overline{Q}_{1}$. Then

$\frac{\partial W}{\partial a}=\sum_{\begin{subarray}{c}t,j|\\\
a_{t,j}=a\end{subarray}}a_{t,j-1}\ldots a_{t,1}\,a_{t,n_{t}}\ldots a_{t,j+1}.$

Now because we are in the quotient $\textnormal{DR}_{R}^{1}(k\overline{Q})$ we
can cyclically permute terms (up to a sign, which in this case is always $+1$
since we are dealing with terms of degree 0 and a term of degree 1 in
$\Omega_{R}^{\bullet}k\overline{Q}$), hence we get that

$\displaystyle D(W)$
$\displaystyle=\sum_{t=1}^{m}\bigg{(}\sum_{j=1}^{n_{t}}a_{t,n_{t}}\ldots
a_{t,j+1}\,D(a_{t,j})\,a_{t,j-1}\ldots a_{t,1}\bigg{)}$
$\displaystyle=\sum_{t=1}^{m}\bigg{(}\sum_{j=1}^{n_{t}}D(a_{t,j})\,a_{t,j-1}\ldots
a_{t,1}\,a_{t,n_{t}}\ldots a_{t,j+1}\bigg{)}$ $\displaystyle=\sum_{a\in
Q_{1}}\bigg{(}\sum_{\begin{subarray}{c}t,j|\\\
a_{t,j}=a\end{subarray}}D(a_{t,j})\,a_{t,j-1}\ldots a_{t,1}\,a_{t,n_{t}}\ldots
a_{t,j+1}\bigg{)}$ $\displaystyle=\sum_{a\in Q_{1}}D(a)\frac{\partial
W}{\partial a}.$

Then as

$i_{\xi}(\omega)=\sum_{a\in Q_{1}}-\xi(a)D(a^{*})+D(a)\xi(a^{*})$

by comparing coefficients between these two expressions we must have that
$\xi(a)=0$ and $\xi(a^{*})=\partial W/\partial a$ for any $a\in Q_{1}$, as
required.

We must also check the differential on $t$. To do this we calculate a
representative w of $\mu_{\textnormal{nc}}(\omega)$. Recall
$\mu_{\textnormal{nc}}(\omega)$ was defined such that
$D(\mu_{\textnormal{nc}}(\omega))=\iota_{\delta}(\omega)$ where
$\delta:k\overline{Q}\rightarrow k\overline{Q}\otimes k\overline{Q}$ was the
distinguished double derivation introduced in Section 2.2. So

$\displaystyle\iota_{\delta}(\omega)$ $\displaystyle=m(\beta\circ
i_{\delta}(\omega))$ $\displaystyle=m\Big{(}\beta\Big{(}-\sum_{a\in
Q_{1}}\delta(a)\cdot D(a^{*})-D(a)\cdot\delta(a^{*})\Big{)}\Big{)}$
$\displaystyle=m\Big{(}\beta\Big{(}-\sum_{a\in Q_{1}}(a\otimes
e_{s(a)}-e_{t(a)}\otimes a)\cdot(a^{*}\otimes_{R}1-1\otimes_{R}a^{*})$
$\displaystyle\qquad\qquad\quad-(a\otimes_{R}1-1\otimes_{R}a)\cdot(a^{*}\otimes
e_{s(a^{*})}-e_{t(a^{*})}\otimes a^{*})\Big{)}\Big{)}$
$\displaystyle=m\Big{(}\beta\Big{(}-\sum_{a\in
Q_{1}}a\otimes(a^{*}\otimes_{R}1-e_{s(a)}\otimes_{R}a^{*})-e_{t(a)}\otimes(aa^{*}\otimes_{R}1-a\otimes_{R}a^{*})$
$\displaystyle\qquad\qquad\quad-(a\otimes_{R}a^{*}-1\otimes_{R}aa^{*})\otimes
e_{t(a)}+(a\otimes_{R}e_{t(a^{*})}-1\otimes_{R}a)\otimes a^{*}\Big{)}\Big{)}$
$\displaystyle=-\sum_{a\in
Q_{1}}a^{*}\otimes_{R}a-e_{s(a)}\otimes_{R}a^{*}a-aa^{*}\otimes_{R}e_{t(a)}+a\otimes_{R}a^{*}$
$\displaystyle\qquad\qquad-a\otimes_{R}a^{*}+e_{t(a)}\otimes_{R}aa^{*}+a^{*}a\otimes_{R}e_{s(a)}-a^{*}\otimes_{R}a$
$\displaystyle=\sum_{a\in
Q_{1}}e_{s(a)}\otimes_{R}a^{*}a-a^{*}a\otimes_{R}e_{s(a)}+aa^{*}\otimes_{R}e_{t(a)}-e_{t(a)}\otimes_{R}aa^{*}$

while on the other hand

$\displaystyle D\big{(}[a,a^{*}]\big{)}$
$\displaystyle=D(a)a^{*}+aD(a^{*})-D(a^{*})a-a^{*}D(a)$
$\displaystyle=(a\otimes_{R}e_{s(a)}-e_{t(a)}\otimes_{R}a)a^{*}+a(a^{*}\otimes_{R}e_{s(a^{*})}-e_{t(a^{*})}\otimes_{R}a^{*})$
$\displaystyle\quad\,\,-(a^{*}\otimes_{R}e_{s(a^{*})}-e_{t(a^{*})}\otimes_{R}a^{*})a-a^{*}(a\otimes_{R}e_{s(a)}-e_{t(a)}\otimes_{R}a)$
$\displaystyle=a\otimes_{R}a^{*}-e_{t(a)}\otimes_{R}aa^{*}+aa^{*}\otimes_{R}e_{t(a)}-a\otimes_{R}a^{*}$
$\displaystyle\quad\,\,-a^{*}\otimes_{R}a+e_{s(a)}\otimes_{R}a^{*}a-a^{*}a\otimes_{R}e_{s(a)}+a^{*}\otimes_{R}a$
$\displaystyle=e_{s(a)}\otimes_{R}a^{*}a-a^{*}a\otimes_{R}e_{s(a)}+aa^{*}\otimes_{R}e_{t(a)}-e_{t(a)}\otimes_{R}aa^{*}.$

It follows that a homogeneous representative of degree -1 of
$\mu_{\textnormal{nc}}(\omega)$ is $\sum_{a\in Q_{1}}[a,a^{*}]$ hence

$d(t)=\textbf{w}=\sum_{a\in Q_{1}}[a,a^{*}]=\sum_{i\in
Q_{0}}d(t_{i})=d\Big{(}\sum_{i\in Q_{0}}t_{i}\Big{)}.$

Finally we must check the conditions $\xi^{2}=0$, $L_{\xi}(\omega)=0$ and
$\xi(\textbf{w})=0$. The first equality is clear from the explicit description
given above of how $\xi$ acts on arrows. For the second equality we have

$\displaystyle L_{\xi}(\omega)$ $\displaystyle=\sum_{a\in
Q_{1}}L_{\xi}(D(a))\cdot D(a^{*})+D(a)\cdot L_{\xi}(D(a^{*}))$
$\displaystyle=\sum_{a\in Q_{1}}D(\xi(a))\cdot D(a^{*})+D(a)\cdot
D(\xi(a^{*}))$ $\displaystyle=\sum_{a\in Q_{1}}D(a)\cdot D\left(\frac{\partial
W}{\partial a}\right)$ $\displaystyle=\sum_{a\in
Q_{1}}D^{2}(a)\cdot\frac{\partial W}{\partial a}+D(a)\cdot
D\left(\frac{\partial W}{\partial a}\right)$ $\displaystyle=D^{2}(W)=0.$

For the third equality again write $W=\sum_{t=1}^{m}a_{t,n_{t}}\ldots
a_{t,1}$. Then

$\displaystyle\xi(\textbf{w})$ $\displaystyle=\xi\bigg{(}\sum_{a\in
Q_{1}}[a,a^{*}]\bigg{)}=\sum_{a\in Q_{1}}\xi(aa^{*})-\xi(a^{*}a)$
$\displaystyle=\sum_{a\in
Q_{1}}\xi(a)a^{*}+a\,\xi(a^{*})-\xi(a^{*})a-(-1)a^{*}\,\xi(a)$
$\displaystyle=\sum_{a\in Q_{1}}a\,\frac{\partial W}{\partial
a}-\frac{\partial W}{\partial a}\,a$ $\displaystyle=\sum_{a\in
Q_{1}}\bigg{(}\sum_{\begin{subarray}{c}t,j|\\\
a_{t,j}=a\end{subarray}}a_{t,j}\,a_{t,j-1}\ldots a_{t,1}\,a_{t,n_{t}}\ldots
a_{t,j+1}$ $\displaystyle\qquad\quad-\sum_{\begin{subarray}{c}s,i|\\\
a_{s,i}=a\end{subarray}}a_{s,i-1}\ldots a_{s,1}\,a_{s,n_{s}}\ldots
a_{s,i+1}\,a_{s,i}\bigg{)}$ $\displaystyle=\sum_{t,j}a_{t,j}\,a_{t,j-1}\ldots
a_{t,1}\,a_{t,n_{t}}\ldots a_{t,j+1}$
$\displaystyle\quad\,-\sum_{s,i}a_{s,i-1}\ldots a_{s,1}\,a_{s,n_{s}}\ldots
a_{s,i+1}\,a_{s,i}$ $\displaystyle=0.$

∎

######  Proposition 2.13.

$\textnormal{Jac}(Q,W)=H^{0}(\Gamma_{Q,W})$.

###### Proof.

Since $(\Gamma_{Q,W})_{0}=kQ$ and $(\Gamma_{Q,W})_{-1}=kQ\\{a^{*}:a\in
Q_{1}\\}$ we get

$\displaystyle H^{0}(\Gamma_{Q,W})$
$\displaystyle=\frac{\textnormal{Ker}\,\,d_{0}}{\textnormal{Im}\,\,d_{-1}}$
$\displaystyle=\frac{kQ}{(\partial W/\partial a:a\in Q_{1})}$
$\displaystyle=\textnormal{Jac}(Q,W).$

∎

There is a natural identification of graded algebras $k\overline{Q}\cong
T_{kQ}(\mathbb{D}\textnormal{er}_{R}(kQ))$ that sends the arrow $a^{*}$ to the
double derivation $\lambda_{a^{*}}$ given on arrows by

$Q_{1}\ni x\longmapsto\begin{cases}1\otimes 1&\textnormal{if}\,\,x=a\\\
0&\textnormal{otherwise}.\end{cases}$

It is easy to see that under this identification the distinguished relative
double derivation $\delta\in\mathbb{D}\textnormal{er}_{R}(kQ)$ is sent to
$\sum_{a\in Q_{1}}[a,a^{*}]\in k\overline{Q}$. Setting the cyclic 1-form
$\alpha\in\textnormal{DR}_{R}^{1}(kQ)$ to be $D(W)$ we have that the super-
derivation $\\{W,-\\}_{\omega}\in\textnormal{Der}_{R}(k\overline{Q})$
corresponds to the super-derivation
$\xi_{\alpha}\in\textnormal{Der}_{R}(T_{kQ}(\mathbb{D}\textnormal{er}_{R}(kQ)))$.
Indeed it suffices to check this for degree 0 and -1 elements. The degree 0
parts of $k\overline{Q}$ and $T_{kQ}(\mathbb{D}\textnormal{er}_{R}(kQ))$ are
both isomorphic to $kQ$ and both super-derivations send degree 0 elements to
0. Then in degree -1 we have $\\{W,a^{*}\\}_{\omega}=\partial W/\partial a$
which under the identification above corresponds to sending
$\lambda_{a^{*}}\mapsto\partial W/\partial a$. Hence we must show that
$\xi_{\alpha}(\lambda_{a^{*}})=\iota_{\lambda_{a^{*}}}(D(W))$ is equal to
$\partial W/\partial a$. Then

$\displaystyle\iota_{\lambda_{a^{*}}}(D(W))$
$\displaystyle=m\circ\beta\big{(}i_{\lambda_{a^{*}}}(D(W)\big{)}$
$\displaystyle=m\circ\beta\bigg{(}i_{\lambda_{a^{*}}}\bigg{(}\sum_{b\in
Q_{1}}D(b)\frac{\partial W}{\partial b}\bigg{)}\bigg{)}$
$\displaystyle=m\circ\beta\bigg{(}\sum_{b\in
Q_{1}}\lambda_{a^{*}}(b)\frac{\partial W}{\partial b}\bigg{)}$
$\displaystyle=m\circ\beta\bigg{(}1\otimes_{R}1\cdot\frac{\partial W}{\partial
a}\bigg{)}$ $\displaystyle=\frac{\partial W}{\partial a}.$

We also have $\iota_{\delta}(\alpha)=0$ from 2.8. It follows that the Gdga
$\Gamma_{Q,W}$ is naturally isomorphic to the Gdga $\mathfrak{D}(kQ,D(W))$
described in 2.12. The proceeding result then follows immediately from 2.9.

######  Proposition 2.14.

The Gdga $\Gamma_{Q,W}$ is Calabi-Yau of dimension 3.

######  Proposition 2.15 ([1] Proposition 2.3).

There exists a natural $t$-structure for the derived category
$D(\Gamma_{Q,W})$ whose heart is equivalent to $\textnormal{Jac}(Q,W)$-Mod.

2.15 allows us to view the moduli stack of representations
$\textnormal{Rep}_{n}(\textnormal{Jac}(Q,W))$ as a moduli stack in a 3 Calabi-
Yau category regardless of whether the algebra $\textnormal{Jac}(Q,W)$ is 3
Calabi-Yau or not. This is important for us to be able to define DT
invariants. If however $\textnormal{Jac}(Q,W)$ is in fact Calabi-Yau and $W$
is homogeneous we get something stronger. There is a natural grading on the
path algebra $kQ$ of a quiver given by the path length, which makes the
isomorphism (2.4) an isomorphism of graded algebras. If $W$ is homogeneous of
degree $m$ then this descends to a grading on $\textnormal{Jac}(Q,W)$ too.
This gives us a second grading on $\Gamma_{Q,W}$, which is strictly positive
and preserved by the differential $d$ if we set the degree of the dual arrows
$a^{*}$ to be $m-1$ and degree of the loops $t_{i}$ to be $m$. By [[6]
Proposition 8.1.1] and its proof we have that the path algebra $k\overline{Q}$
of the doubled quiver is connected. Then by 2.15, and [[14] Proposition 3.7.7
and Theorem 5.3.1] we get the following:

######  Theorem 2.16 (cf. [14] Corollary 5.4.3).

Let $W$ be a homogeneous potential with respect to the path grading on $kQ$.
Then $\textnormal{Jac}(Q,W)$ is Calabi-Yau of dimension 3 if and only if it is
a superpotential algebra.

###  2.4. Brane tilings

######  Definition 2.17.

A brane tiling $\Delta$ of a genus $g$ Riemann surface $\Sigma_{g}$ is an
embedding $\Gamma\hookrightarrow\Sigma_{g}$ of a bipartite graph $\Gamma$ such
that each connected component of $\Sigma_{g}\setminus\Gamma$ is simply
connected. We choose a partition of the vertex set of $\Gamma$ into 2 disjoint
subsets of black and white vertices, such that every edge in $\Gamma$ goes
between a black vertex and a white vertex.

From a brane tiling we can obtain a quiver with potential
$(Q_{\Delta},W_{\Delta})$. The underlying graph of $Q_{\Delta}$ is the dual
graph to $\Gamma$ in $\Sigma_{g}$ and it is directed so that arrows in
$Q_{\Delta}$ go clockwise around a white vertex and anticlockwise around a
black vertex. For a vertex $v\in\Gamma$ let $c_{v}$ denote the minimal cycle
in $Q_{\Delta}$ that goes around it, i.e. $c_{v}$ consists of all the arrows
that are dual to the edges that come out of $v$. Then we take the potential to
be

$W_{\Delta}=\sum_{v\,\textnormal{white}}c_{v}-\sum_{u\,\textnormal{black}}c_{u}.$

######  Example 2.17.

The following genus 2 example will be used throughout the following sections
and will be useful to keep in mind to picture what's going on. It can also
easily be extended to higher genus surfaces. A brane tiling $\Delta$ of
$\Sigma_{2}$ and its dual quiver $Q_{\Delta}$ are given in the following
picture.

Figure 1. A brane tiling $\Delta$ of $\Sigma_{2}$ in green with its dual
quiver $Q_{\Delta}$ in red.

Explicitly the dual quiver $Q_{\Delta}$ is

${1}$${2}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{c}$$\scriptstyle{d}$$\scriptstyle{e}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{f}$

with potential

$W_{\Delta}=abfjie+gc+hd-agic-bhjd-fe.$

The idea behind using brane tilings in the context of fundamental groups of
Riemann surfaces is as follows (from [10]). If the arrow $a\in Q_{\Delta,1}$
is dual to the edge between the white vertex $v$ and black vertex $u$ in
$\Delta$ then $a\cdot\partial W_{\Delta}/\partial a=c_{v}-c_{u}.$ This tells
us that in $\textnormal{Jac}(Q_{\Delta},W_{\Delta})$ we can identify the paths
$c_{v}$ and $c_{u}$, which are homotopic in $\Sigma_{g}$ via a homotopy which
is spanned by the edge from the brane tiling. This allows us to link the
algebra over the fundamental group of the Riemann surface $\Sigma_{g}$ with a
Jacobi algebra. However not all homotopic paths in $\mathbb{C}Q_{\Delta}$ are
identified in $\textnormal{Jac}(Q_{\Delta},W_{\Delta})$, for example the
minimal cycles $c_{v}$ are null-homotopic but are not equal to any constant
paths $e_{i}$ in $\textnormal{Jac}(Q_{\Delta},W_{\Delta})$.

######  Definition 2.18.

A dimer $D$ for a brane tiling $\Delta$ is a choice of edges in $\Delta$ such
that every vertex in $\Delta$ is an endpoint of exactly one edge in $D$.

Dimers are useful because after dualising they give us a collection of arrows
in $Q_{\Delta}$ with the property that every cycle in $W_{\Delta}$ contains
exactly one arrow from this set. This is straightforward from the definition
of the dimer and the potential $W_{\Delta}$; if $a$ is dual to the edge in $D$
that goes between the vertices $v$ and $u$ then $a$ will appear in the minimal
cycles $c_{v}$ and $c_{u}$ in $W_{\Delta}$, and since vertices in $\Delta$ are
endpoints to exactly one edge in $D$ no other arrow in the dual of $D$ will
appear in $c_{v}$ or $c_{u}$ and every term in $W$ contains one of these
arrows.

###  2.5. Fundamental group algebras

This final preliminary section gives an overview of existing results relating
algebras over fundamental groups to the aforementioned Calabi-Yau and
superpotential properties, motivating the conjecture we aim to partially
solve.

######  Definition 2.19.

A manifold $X$ is called _acyclic_ if its universal cover is contractible.

For the rest of this section let $X$ be a compact, acyclic, orientable
manifold of dimension $n$ and let $k[\pi_{1}(X)]$ be its fundamental group
algebra.

######  Theorem 2.18 ([9] Theorem 5.2.2).

The fundamental group algebra $k[\pi_{1}(X)]$ is homologically finite.

Let $LX$ denote the _free loop-space_ of $X$ i.e. the space of continuous maps
$S^{1}\rightarrow X.$ As path components of $LX$ are in one-to-one
correspondence with conjugacy classes $c\in\pi_{1}(X)$, let $(LX)_{c}$ denote
the path component corresponding to $c.$ Denote by $const:X\rightarrow LX$ the
map that sends a point to the constant loop at that point, let
$bp:LX\rightarrow X$ denote the map that sends a loop to its basepoint, and
let $const_{*}$ and $bp_{*}$ denote the respective pushforwards in homology.

######  Theorem 2.19 ([16] Theorem 6.2).

For all $d\in\mathbb{N}$ we have a natural isomorphism

$\textnormal{HH}_{d}(k[\pi_{1}(X)])\cong H_{d}(LX).$

Recall for an algebra $A$ that an element $\nu\in\textnormal{HH}_{d}(A)$ is
called non-degenerate if induces a (quasi-) isomorphism $A\cong A^{\vee}[d]$.
There is an action of the centre $Z(A)$ on $\textnormal{HH}_{*}(A)$ induced by
$Z(A)$ acting on the first copy of $A$ in $A\otimes_{A^{e}}A$.

######  Proposition 2.20 ([9] Proposition 5.2.6).

An element $\nu\in\textnormal{HH}_{n}(k[\pi_{1}(X)])$ is non-degenerate if and
only if $\nu=z\cdot const_{*}([X])$ for some central unit $z\in
k[\pi_{1}(X)].$ In particular $k[\pi_{1}(X)]$ is Calabi-Yau of dimension $n.$

######  Theorem 2.21 ([9] Theorem 6.1.3).

$k[\pi_{1}(X)]$ is exact Calabi-Yau of dimension $n$ only if it contains a
non-trivial central unit $z.$

The idea behind the proof of 2.21 is as follows. First, from [16], we have an
isomorphism of long exact sequences (where $A=k[\pi_{1}(X)]$)

${\cdots}$${\textnormal{HC}_{n}(A)}$${\textnormal{HC}_{n-2}(A)}$${\textnormal{HH}_{n-1}(A)}$${\textnormal{HC}_{n-1}(A)}$${\cdots}$${\cdots}$${H^{S^{1}}_{n}(LX)}$${H^{S^{1}}_{n-2}(LX)}$${H_{n-1}(LX)}$${H^{S^{1}}_{n-1}(LX)}$${\cdots}$$\scriptstyle{\partial}$$\scriptstyle{\partial^{\prime}}$

Hence an exact Calabi-Yau structure of dimension $d$ on $A$ is equivalent to a
non-degenerate element $\eta\in H_{d}(LX)$ which is in the image of
$\partial^{\prime}.$ By 2.20 a non-degenerate $\eta$ must be of the form
$\eta=z\cdot const_{*}([X])$ for a central unit $z\in A$. Now there exists a
natural grading of both $H^{S^{1}}_{d-1}(LX)$ and $H_{d}(LX)$ by conjugacy
classes in $\pi_{1}(X)$ (the grading exists at the level of chains) which is
preserved by $\partial^{\prime}$, so the existence of some $\lambda\in
H^{S^{1}}_{d-1}(LX)$ such that $\partial^{\prime}(\lambda)=\eta$ is equivalent
to equalities $\partial(\lambda_{c})=a_{c}s_{c}\cdot const_{*}([X])$ where $c$
is a conjugacy class in $\pi_{1}(X)$, $a_{c}\in k$, $s_{c}=\sum_{g\in c}g$,
and $\lambda_{c}\in H^{S^{1}}_{d-1}((LX)_{c})$ such that
$\eta=\sum_{c}a_{c}s_{c}\cdot const_{*}([X])$. If we were to take the central
unit $z=1$ then we would need to look for some $\lambda_{0}\in
H^{S^{1}}_{d-1}((LX)_{0})$ such that
$\partial^{\prime}(\lambda_{0})=const_{*}([X]).$ But, from [[9] Theorem 6.1.3]
and its proof, it turns out the map $H^{S^{1}}_{d-1}((LX)_{0})\rightarrow
H_{d}(LX)$ is 0 hence 1 does not give an exact Calabi-Yau structure.

######  Example 2.22 ([9] Corollary 6.2.3 and Corollary 6.2.4).

Let $X$ be a compact hyperbolic manifold of dimension $>1.$ Then
$k[\pi_{1}(X)]$ is not a superpotential algebra. Indeed one can show that
$k[\pi_{1}(X)]$ has trivial centre, hence by 2.21 it is not exact Calabi-Yau.
But 2.6 says that every superpotential algebra is exact Calabi-Yau.

We saw earlier that when $A$ is connected a Gdga with cohomology concentrated
in degree 0 will have an exact Calabi-Yau cohomology algebra. 2.21 describes
the manifolds which cannot have an exact Calabi-Yau fundamental group algebra.
This gives a rudimentary justification for looking for superpotential
descriptions for the fundamental group algebras of circle bundles; often
circle bundles have a natural non-trivial central element in their fundamental
groups. In particular for the dimension 3 case we will be looking for
isomorphisms between the fundamental group algebra and the Jacobi algebra of a
quiver with potential. Then because our fundamental group algebras are always
Calabi-Yau (by 2.20) we get using 2.16 that the Jacobi algebra presentation is
a superpotential algebra provided that the potential is homogeneous. As a
starting point of this conjecture we have:

######  Theorem 2.23 ([10] Proposition 4.2).

Let $\Delta$ be a brane tiling of a Riemann surface $\Sigma_{g}$ and let
$\mathbb{C}\widetilde{Q}_{\Delta}$ denote the localisation of the path algebra
$\mathbb{C}Q_{\Delta}$ with respect to all the arrows in $Q_{\Delta,1}$. Then
we have an isomorphism of algebras

$\textnormal{Jac}(\mathbb{C}\widetilde{Q}_{\Delta},W_{\Delta})\cong\textnormal{Mat}_{r\times
r}(\mathbb{C}[\pi_{1}(\Sigma_{g}\times S^{1})])$

where $r$ is the number of vertices in $Q_{\Delta}.$

As remarked earlier there is a natural way to think about how the relations in
the Jacobi algebra of the brane tiling identify homotopic paths in the surface
$\Sigma_{g}$, but also not all relations in the fundamental group can be
obtained from the potential. Adding in this extra $S^{1}$ direction fully
rectifies this issue whereby paths in the Jacobi algebra that would be null-
homotopic in $\Sigma_{g}$ are instead sent to loops around the circle in
$\Sigma_{g}\times S^{1}$. Explicitly we grade the arrows in $Q_{\Delta}$ such
that the potential $W_{\Delta}$ is homogeneous of degree 1 (note this is
entirely separate to the path grading we already have on the path algebra) and
then define a new embedding $Q_{\Delta}\hookrightarrow\Sigma_{g}\times S^{1}$
where this grading determines how far the arrows go around the $S^{1}$
direction. In particular, for any genus $g$ surface it is possible to find a
tiling $\Delta$ such that the potential $W_{\Delta}$ is homogeneous with
respect to this grading (e.g. see [[9] Figure 2]).

If in addition $W_{\Delta}$ is homogeneous with respect to the path grading,
since 2.23 holds for any brane tiling, we get a superpotential description for
the fundamental group algebra $\mathbb{C}[\pi_{1}(\Sigma_{g}\times S^{1})]$.

######  Conjecture 2.24.

Let $X$ be a compact, acyclic, orientable circle bundle of dimension $n$. Then
$k[\pi_{1}(X)]$ is a superpotential algebra.

Guided by 2.23 we shall provide a Jacobi algebra description for the case of a
mapping torus of a Riemann surface by a finite-order, orientation-preserving
automorphism. Thereby we provide more evidence for 2.24 if a path-graded
homogeneous potential can be found.

##  3\. Main results

###  3.1. A Jacobi algebra presentation for
$\mathbb{C}[\pi_{1}(M_{g,\varphi})]$

From now on we work over the field $\mathbb{C}$.

Let $\Sigma_{g}$ be a genus $g$ Riemann surface and let $\varphi$ be an
orientation-preserving automorphism of $\Sigma_{g}$ of order $n$. Let
$M_{g,\varphi}=\Sigma_{g}\times[0,1]/\sim_{\varphi}$ denote the mapping torus
of $\varphi$.

Consider a brane tiling $\Delta$ of $\Sigma_{g}$ giving rise to a dual quiver
$Q=Q_{\Delta}$ and potential $W=W_{\Delta}$ as described in Section 2.4. We
assume that the brane tiling and the chosen colouring of the vertices in the
tiling are preserved by $\varphi.$ This implies that $\varphi$ induces an
automorphism on $Q$ and hence also on the path algebra $\mathbb{C}Q$ (we
denote this automorphism by $\varphi$ as well). We focus on the case in which
the size of the orbit of every vertex in $Q$ under $\varphi$ is $n$.

######  Lemma 3.1 ([5] Proposition 3.3 1.).

Let $f:D^{2}\xrightarrow{\sim}D^{2}$ be an orientation-preserving automorphism
of the disc $D^{2}$ of order $n$. Then

  1. a)

the fixed point set of $f$ is a single point in the interior of $D^{2}$.

  2. b)

the fixed point set of the composition $f^{i}$ is equal to the fixed point set
of $f$ for all $i\neq 0\,\,\textnormal{mod}\,\,n$.

######  Lemma 3.2.

Let $\varphi$ be an orientation-preserving automorphism of $\Sigma_{g}$ of
order $n$. Then there exists a brane tiling $\Delta$ which is preserved by
$\varphi$ such that every vertex in the dual quiver $Q=Q_{\Delta}$ has an
orbit of size $n$ under the induced action of $\varphi$ on $Q$.

###### Proof.

A point in $p\in\Sigma_{g}$ will have orbit size strictly less than $n$ under
$\varphi$ if and only if $p$ is a fixed point of $\varphi^{d}$ for some
$0<d<n$ that divides $n$. Let $\Delta_{0}$ be a brane tiling of $\Sigma_{g}$
with colouring that is preserved by $\varphi$. Then a vertex $i$ in the dual
quiver $Q_{\Delta_{0}}$ will have orbit size strictly less than $n$ if and
only if the tile $S_{i}$ in $\Delta_{0}$ that is dual to $i$ is such that
$\varphi^{d}(S_{i})=S_{i}$ for some $0<d<n$ that divides $n$. Suppose that
there exists a vertex $i\in Q_{\Delta_{0},0}$ with orbit size less than $n$
and suppose that $d$ is the smallest positive integer such that
$\varphi^{d}(S_{i})=S_{i}$. Then
$\varphi^{d}|_{S_{i}}:S_{i}\xrightarrow{\sim}S_{i}$ is an (orientation-
preserving) automorphism of $S_{i}$. Therefore by 3.1 a) it has a single fixed
point $p$ in the interior of $S_{i}$, and because by 3.1 b) the fixed point
set of $\varphi^{dj}$ is the same point for all $j=1,\ldots,n/d-1$ every other
point in $S_{i}$ has orbit size $n$ under $\varphi$.

We add new brane tiling vertices of the same colour at
$p,\varphi(p),\ldots,\varphi^{d-1}(p)$, then choose an opposite colour vertex
$v$ on the boundary of $S_{i}$ and add an edge between the new vertex at $p$
and $v$. We then add the images of this edge under $\varphi^{j}$ for all
$j=1,\ldots,n-1$, which go between the vertices $\varphi^{j}(p)$ and
$\varphi^{j}(v)$, to the brane tiling. This gives us a new brane tiling
$\Delta_{1}$ that is also preserved by $\varphi$, and in which the tile
$S_{i}$ in $\Delta_{0}$ has been subdivided into tiles
$S_{i_{1}},\ldots,S_{i_{n/d}}$ (see Fig. 2). Indeed because the vertex $v$ has
orbit size $n$ we have added $n/d$ distinct edges into the tile $S_{i}$ that
only intersect at the point $p$ and so we do in fact end up with a brane
tiling (each of the distinct tiles
$S_{i},\varphi(S_{i}),\ldots,\varphi^{d-1}(S_{i})$ each have $n/d$ edges
added, giving us a total of $n$ new edges added to $\Delta_{0}$ as expected).

We claim that $\varphi^{k}(S_{i_{j}})\neq S_{i_{j}}$ for all $i_{j}$ and all
$0<k<n$, and hence the vertex $i\in Q_{\Delta_{0},0}$ which had orbit size
less than $n$ has been replaced by the vertices $i_{j}\in Q_{\Delta_{1},0}$
all of orbit size $n$. So suppose not i.e. there exists some $j$ and $k$ such
that $\varphi^{k}(S_{i_{j}})=S_{i_{j}}$. Note that $k$ must be multiple of $d$
since otherwise

$\varphi^{md+l}(S_{i_{j}})\subset\varphi^{l}(S_{i})\neq S_{i}$

for $0<l<d$. If $k=md$ then we have that
$\varphi^{md}|_{S_{i_{j}}}:S_{i_{j}}\xrightarrow{\sim}S_{i_{j}}$ is an
automorphism of $S_{i_{j}}$ and so again by 3.1 a) it has a fixed point in the
interior of $S_{i_{j}}$. But this would also be a fixed point of
$\varphi^{md}|_{S_{i}}$ and it would be distinct from $p$ contradicting 3.1
b).

We repeat this procedure for all such vertices in the dual quiver with orbit
size less than $n$ giving us the required brane tiling $\Delta$. ∎

$p$

$\widehat{a}$$\varphi^{d}(\widehat{a})$$\varphi^{2d}(\widehat{a})$

Figure 2. An example of a tile in $\Delta_{0}$ for which we suppose the
induced action of $\varphi^{d}$ is clockwise rotation by $120^{\circ}$ and the
tile has orbit size $d=n/3<n$. The fixed point of this rotation is the point
$p$. We add a vertex at this point (in this case a white vertex), and then
attach this to the brane tiling via an edge $\widehat{a}$. We then add all the
images under $\varphi,\ldots,\varphi^{n-1}$ of both the new vertex and edge to
$\Delta_{0}$. In the resulting brane tiling the three new tiles are no longer
preserved by $\varphi^{d}$.

######  Remark 3.3.

Take a brane tiling $\Delta$ such that every vertex in the dual quiver has an
orbit of size $n$ under the induced action of $\varphi$ on $Q=Q_{\Delta}$.
Then we may freely add additional vertices and edges to $\Delta$ along with
their images under $\varphi,\ldots,\varphi^{n-1}$ and still maintain a brane
tiling which is preserved by $\varphi$ such that the orbit of every vertex in
the dual quiver has size $n$. Indeed, given such a $\Delta$ adding extra
vertices and edges will amount to subdividing existing tiles in $\Delta$. From
the proof of 3.2 we see that if one of these new tiles $S_{i_{j}}$, which
subdivides the old tile $S_{i}$, gives a dual vertex with orbit size less than
$n$ then we must have that $\varphi^{d}(S_{i_{j}})=S_{i_{j}}$ for some $d<n$,
and hence $\varphi^{d}|_{S_{i_{j}}}$ has a fixed point in $S_{i_{j}}$. Then as
$\varphi$ and hence $\varphi^{d}$ preserves $\Delta$ we know that
$\varphi^{d}(S_{i})$ is also a tile in $\Delta$. But $\varphi^{d}$ has a fixed
point inside $S_{i_{j}}\subset S_{i}$ and so $\varphi^{d}(S_{i})$ must equal
$S_{i}$ which contradicts $\Delta$ not having any dual vertices of orbit size
less than $n$.

Consider the semidirect product
$S:=\mathbb{C}Q\rtimes_{\varphi}\mathbb{Z}/n\mathbb{Z}$, which has the
following description

$\mathbb{C}Q\rtimes_{\varphi}\mathbb{Z}/n\mathbb{Z}=\bigoplus_{l=0}^{n-1}\mathbb{C}Q\times\\{l\\}$

with multiplication given by

$(a,l)\cdot(b,m)=(a\,\varphi^{l}(b),\,\,l+m)$

where $a,b$ are paths in $\mathbb{C}Q$ and $l,m$ and $l+m$ are elements in
$\mathbb{Z}/n\mathbb{Z}.$

######  Lemma 3.4.

$S$ is Morita equivalent to a localised path algebra of a quiver.

###### Proof.

A representation of $S$ is a representation of $Q$ with the extra data of the
action of $\mathbbm{1}\in\mathbb{Z}/n\mathbb{Z}$, subject to the
multiplication relations in $S$. Let $V=\bigoplus_{i\in Q_{0}}V_{i}$ and
$(f_{a})_{a\in Q_{1}}$ be the data of a representation of $Q$. The action of
$\mathbbm{1}$ is then given by isomorphisms
$V_{i}\xrightarrow{r_{i}}V_{\varphi(i)}$ for each $i\in Q_{0}$ such that if we
let $r:V\xrightarrow{\sim}V$ be the isomorphism given component-wise on
$V_{i}$ by $r_{i}$, we have $r^{n}=\textnormal{id}_{V}$. The data
$(V,(f_{a}),(r_{i}))$ then gives a representation of $S$ if we also have that
for all $a:i\rightarrow j\in Q_{1},\,\,\,f_{\varphi(a)}=r_{j}\circ f_{a}\circ
r_{i}^{-1}.$ Equivalently if we partition the orbits of $\varphi$ in $Q$ as
follows; for the vertices we have orbits
$O_{1}=\\{i_{1},\varphi(i_{1}),\ldots,\varphi^{n-1}(i_{1})\\},\,\ldots,\,O_{y}=\\{i_{y},\ldots,\varphi^{n-1}(i_{y})\\}$
and for the arrows we have orbits
$P_{1}=\\{a_{1},\ldots,\varphi^{n-1}(a_{1})\\},\,\ldots,\,P_{z}=\\{a_{z},\ldots,\varphi^{n-1}(a_{z})\\}$,
then the data of a representation of $S$ is given by the vector space
$V=\bigoplus_{i\in Q_{0}}V_{i}$, linear maps $f_{a_{t}}$ for
$t\in\\{1,\ldots,z\\}$ and isomorphisms
$r_{i_{u},0}:V_{i_{u}}\xrightarrow{\sim}V_{\varphi(i_{u})},\,\,r_{i_{u},1}:V_{\varphi(i_{u})}\xrightarrow{\sim}V_{\varphi^{2}(i_{u})},\,\ldots,\,\,r_{i_{u},n-2}:V_{\varphi^{n-2}(i_{u})}\xrightarrow{\sim}V_{\varphi^{n-1}(i_{u})}$
for $u\in\\{1,\ldots,y\\}$.

Let $Q^{\\#}$ denote the quiver with vertices $Q^{\\#}_{0}=Q_{0}$ and arrows
$Q^{\\#}_{1}=\\{a_{1},\ldots,a_{z},\,r_{i_{u},t}\,:\,u=1,\ldots,y\,\,\textnormal{and}\,\,t=0,\ldots,n-2\\}$,
and let $\mathbb{C}Q^{\prime}$ denote the localisation of the path algebra
$\mathbb{C}Q^{\\#}$ at the arrows $\\{r_{i_{u},t}\\}$. Then it is clear that
the data of a representation of $S$ is then exactly the same as the data of a
representation of $\mathbb{C}Q^{\prime}$. ∎

We shall use the notation $Q^{\prime}$ for the ``localised" quiver associated
to the partially localised path algebra $\mathbb{C}Q^{\prime}$. We call the
arrows in $Q^{\\#}\subset Q^{\prime}$ _generating arrows_ and the arrows
$r_{i_{u},t}\in Q^{\prime}_{1}$ _isomorphism arrows_.

######  Example 3.5.

Consider a Riemann surface of genus 2 and let the automorphism $\varphi$ be
rotation by $180^{\circ}$ around the $z$-axis through the centre of the
surface. We take the brane tiling from 2.17.

Figure 3. Let $\varphi$ be the rotation by $180^{\circ}$ around the blue axis.

Recall this gives us the following dual quiver $Q$

${1}$${2}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{c}$$\scriptstyle{d}$$\scriptstyle{e}$$\scriptstyle{i}$$\scriptstyle{j}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{f}$

and potential $W=abfjie+gc+hd-agic-bhjd-fe$. $\varphi$ then swaps the two
vertices 1 and 2 and also the arrows $a$ and $j$, $b$ and $i$, $c$ and $h$,
$d$ and $g$, $e$ and $f$. The quiver $Q^{\prime}$ corresponding to the algebra
$\mathbb{C}Q\rtimes\mathbb{Z}/2\mathbb{Z}$ is

${1}$${2}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{r^{-1}}$$\scriptstyle{c}$$\scriptstyle{d}$$\scriptstyle{e}$$\scriptstyle{r}$

as the arrows $a,b,c,d,e$ generate $Q$ under $\varphi$ and the arrows
$r,r^{-1}$ give the isomorphism between the vector spaces at the two vertices.

######  Lemma 3.6.

There is an inclusion of algebras
$\xi:\mathbb{C}Q\hookrightarrow\mathbb{C}Q^{\prime}$.

###### Proof.

The map $\xi$ is induced by the natural inclusion
$\mathbb{C}Q\times\\{0\\}\hookrightarrow S$. As per the proof of 3.4 we make a
choice of a generating set of arrows $\\{a_{1},\ldots,a_{z}\\}$ of $Q$ under
$\varphi$ as well as a choice of isomorphism arrows
$r_{i_{u},t}:\varphi^{t}(i_{u})\rightarrow\varphi^{t+1}(i_{u})$ for
$u=1,\ldots,y$ and $t=0,\ldots,n-2$. Then define the map $\xi$ as follows;
since $Q_{0}=Q_{0}^{\prime}$ send the constant paths $e_{i}\mapsto e_{i}$, and
then map the arrow $a\in Q_{1}$ to the path $p_{a}a_{j}q_{a}$ where $a$ is in
the orbit under $\varphi$ of the generating arrow $a_{j}$ and
$p_{a}:t(a_{j})\rightarrow t(a),\,q_{a}:s(a)\rightarrow s(a_{j})$ are paths
comprised solely of the isomorphism arrows or their inverses (see Fig. 4 for
an example).

$a$$\varphi(a)$$\varphi^{2}(a)$$\varphi^{3}(a)$$r_{1}$$r_{2}$$s_{2}$$r_{3}$$s_{3}$$s_{4}$
Figure 4. An example of how $Q$ fits inside $Q^{\prime}$. Let the red vertices
and (dashed) arrows be in $Q$ with $\varphi$ being clockwise rotation by
$90^{\circ}$. If we take the generating arrow of the orbit of $a$ to be
$\varphi(a)$ and let the blue arrows denote the chosen isomorphism arrows,
then $\xi(a)=s_{4}s_{3}s_{2}\varphi(a)r_{1}$, $\xi(\varphi(a))=\varphi(a)$,
$\xi(\varphi^{2}(a))=s_{2}\varphi(a)r_{2}^{-1}$, and
$\xi(\varphi^{3}(a))=s_{3}s_{2}\varphi(a)r_{2}^{-1}r_{3}^{-1}$.

$\xi$ is well-defined because by construction of $Q^{\prime}$ all such paths
$a_{j},\,p_{a},\,q_{a}$ are unique and because both $a$ and $\xi(a)$ are paths
$s(a)\rightarrow t(a)$ all the trivial relations in $\mathbb{C}Q$ are
satisfied.

To see that $\xi$ is an inclusion it suffices to show that $\xi$ is injective
on paths and that

$\\{\xi(p):p\,\,\textnormal{is a path in}\,\,\mathbb{C}Q\\}$

is linearly independent. So first suppose we have paths $p,\,q\in\mathbb{C}Q$
such that $\xi(p)=\xi(q).$ Write

$\displaystyle p$ $\displaystyle=b_{n}b_{n-1}\ldots b_{2}b_{1}$ $\displaystyle
q$ $\displaystyle=c_{m}c_{m-1}\ldots c_{2}c_{1}$

for arrows $b_{i},c_{j}\in Q_{1}$. Then the equality $\xi(p)=\xi(q)$ implies
that both $b_{1}$ and $c_{1}$ lie in the same orbit of some generating arrow
$a_{1}$ i.e.

$b_{1}=\varphi^{i_{1}}(a_{1})\quad\textnormal{and}\quad
c_{1}=\varphi^{j_{1}}(a_{1})$

for some $i_{1},j_{1}$. But as paths in $\mathbb{C}Q^{\prime}$ they must have
the same source, and so by the assumption that the size of the orbit of each
vertex in $Q$ under $\varphi$ has size $n$ we know that the source of
$\varphi^{i_{1}}(a_{1})$ and $\varphi^{j_{1}}(a_{1})$ are equal only when
$i_{1}=j_{1}$ and so $b_{1}=c_{1}$. We can then repeat this argument for the
rest of the arrows in $p$ and $q$ giving us that $m=n$ and $b_{i}=c_{i}$ for
all $i$, hence $p=q$ and $\xi$ is injective on paths. Then certainly we have

$\\{\xi(p):p\,\,\textnormal{is a path
in}\,\,\mathbb{C}Q\\}\subseteq\\{p^{\prime}:p^{\prime}\,\,\textnormal{is a
path in}\,\,\mathbb{C}Q^{\prime}\\}$

and since $\\{p^{\prime}:p^{\prime}\,\,\textnormal{is a path
in}\,\,\mathbb{C}Q^{\prime}\\}$ is a $\mathbb{C}$-basis of
$\mathbb{C}Q^{\prime}$ we get the required result. ∎

We grade the isomorphism arrows $r_{i_{u},t}$ in $Q^{\prime}$ with degree 1,
their inverses $r_{i_{u},t}^{-1}$ with degree $-1$, and all other arrows in
$Q^{\prime}$ with degree 0.

######  Lemma 3.7.

For all arrows $a\in Q_{1}$ the element $\xi(a)\in\mathbb{C}Q^{\prime}$ either
has degree 0, degree $n$ or degree $-n$.

###### Proof.

This is clear from the assumption that each orbit of vertices/arrows under
$\varphi$ has size $n.$ Indeed if the arrow $a$ is in the orbit of the
generating arrow $b$ then we can write $a=\varphi^{k}(b)$ for some $k.$
Therefore, depending upon the choice of the isomorphism arrows $r_{i_{u},t}$
for each orbit of vertices, between the source vertices of $a$ and $b$ there
will either be $k$ or $n-k$ isomorphism arrows and similarly between the
target vertices of $a$ and $b$ there will either be $k$ or $n-k$ isomorphism
arrows. Hence $\xi(a)$ is either going to be the composition of $k$ (resp.
$n-k$) isomorphism arrows followed by $b$ followed by $k$ (resp. $n-k$)
inverses and hence it will have degree 0, or it will be the composition of $k$
(resp. $n-k$) isomorphism arrows followed by $b$ followed by $n-k$ (resp. $k$)
isomorphism arrows and hence it will have degree $n$ or it will be the
composition of $k$ (resp. $n-k$) inverses arrows followed by $b$ followed by
$n-k$ (resp. $k$) more inverse arrows and hence it will have degree $-n$ (see
Fig. 5 for an illustration of each case).

$a$$\varphi(a)$$\varphi^{2}(a)$$\varphi^{3}(a)$$r_{1}$$r_{2}$$s_{2}$$r_{3}$$s_{3}$$s_{4}$

$a$$\varphi(a)$$\varphi^{2}(a)$$\varphi^{3}(a)$$r_{1}$$r_{2}$$s_{2}$$r_{3}$$s_{3}$$s_{4}$

Figure 5. Let $\varphi$ be clockwise rotation by $90^{\circ}$ on the quiver.
The left-hand side is the same setup as we had in Fig. 4 with generating arrow
for the orbit $\varphi(a)$. As we saw $\xi(a)=s_{4}s_{3}s_{2}\varphi(a)r_{1}$
has degree 4 whilst $\xi(\varphi^{2}(a))=s_{2}\varphi(a)r_{2}^{-1}$ has degree
0 due to the choices of isomorphism arrows between the sources and targets of
the arrows in this orbit. On the right-hand side we instead take the
generating arrow of this orbit to be $a$ and choose the same isomorphism
arrows. In this case we would have, for example,
$\xi(\varphi(a))=s_{2}^{-1}s_{3}^{-1}s_{4}^{-1}ar_{1}^{-1}$ which has degree
$-4$.

∎

From now on when we say a path $p\in\mathbb{C}Q^{\prime}$ is made from
isomorphism arrows we mean that the only arrows that appear in $p$ are the
isomorphism arrows $r_{i_{u},t}$ or their inverses $r_{i_{u},t}^{-1}$.

######  Lemma 3.8.

For all paths $p^{\prime}\in\mathbb{C}Q^{\prime}$ we can write

$p^{\prime}=q\,\xi(p)$

for some path $p\in\mathbb{C}Q$ and a path $q\in\mathbb{C}Q^{\prime}$ made
from isomorphism arrows. In particular if $\textnormal{deg}(p^{\prime})\equiv
0\,\textnormal{mod}\,n$ then $p^{\prime}\in\,\textnormal{Im}(\xi)$.

######  Remark 3.9.

Having the path of isomorphism arrows $q$ at the end is somewhat arbitrary, it
is also possible to write $p^{\prime}=\xi(p)q^{\prime}$ or
$p^{\prime}=q^{\prime\prime}\xi(p)q^{\prime\prime\prime}$.

###### Proof.

Write

$p^{\prime}=r_{m+1}a_{m}\ldots a_{2}r_{2}a_{1}r_{1}$

for arrows $a_{i}\in Q^{\prime}_{1}$ coming from the generating set of $Q_{1}$
under $\varphi$, and paths of isomorphism arrows $r_{i}$. Let
$\widetilde{a_{1}}$ be the arrow in $Q_{1}$ such that $r_{1}$ is the unique
path of isomorphism arrows in $\mathbb{C}Q^{\prime}$ between $s(a_{1})$ and
$s(\widetilde{a_{1}})$ (in particular
$\widetilde{a_{1}}=\varphi^{k_{1}}(a_{1})$ for some $k_{1}$). Then
$\xi(\widetilde{a_{1}})=s_{1}a_{1}r_{1}$ where $s_{1}$ is the unique path of
isomorphism arrows in $\mathbb{C}Q^{\prime}$ between $t(a_{1})$ and
$t(\widetilde{a_{1}})$. Then let $\widetilde{a_{2}}$ be the arrow in $Q_{1}$
such that $r_{2}s_{1}^{-1}$ is the unique path of isomorphism arrows in
$\mathbb{C}Q^{\prime}$ between $s(a_{2})$ and $s(\widetilde{a_{2}})$, and
write $\xi(\widetilde{a_{2}})=s_{2}a_{2}r_{2}s_{1}^{-1}$. Repeating this
procedure for all the generating arrows $a_{i}$ in $p^{\prime}$ it is clear
that

$p^{\prime}=r_{m+1}s_{m}^{-1}\xi(\widetilde{a_{m}}\ldots\widetilde{a_{1}}).$

If $\textnormal{deg}(p^{\prime})\equiv 0\,\,\textnormal{mod}\,\,n$ this
implies that

$\textnormal{deg}(r_{m+1}s_{m}^{-1})+\textnormal{deg}(\xi(\widetilde{a_{m}}\ldots\widetilde{a_{1}}))\equiv
0\,\,\textnormal{mod}\,\,n.$

By 3.7 $\textnormal{deg}(\xi(\widetilde{a_{m}}\ldots\widetilde{a_{1}}))\equiv
0\,\,\textnormal{mod}\,\,n$ and so the path of isomorphism arrows
$r_{m+1}s_{m}^{-1}$ has degree a multiple of $n$. Now a path made of
isomorphism arrows must go between vertices in $Q^{\prime}$ in the same orbit,
but in any orbit of a vertex there are only $n-1$ isomorphism arrows that
connect the whole orbit. Hence the degree of $r_{m+1}s_{m}^{-1}$ is 0 and so
it is just a constant path. ∎

It is clear that $\xi$ descends to a map
$\mathbb{C}Q/[\mathbb{C}Q,\mathbb{C}Q]\rightarrow\mathbb{C}Q^{\prime}/[\mathbb{C}Q^{\prime},\mathbb{C}Q^{\prime}]$
so let $W^{\prime}$ be the image of $W$ under this morphism, giving a
potential on $Q^{\prime}$. Therefore we can construct the Jacobi algebra
$\textnormal{Jac}(Q^{\prime},W^{\prime})$. In our running example 3.5 $\xi$
sends

$\displaystyle a$ $\displaystyle\longmapsto a,\qquad\quad f\longmapsto rer$
$\displaystyle b$ $\displaystyle\longmapsto b,\qquad\quad g\longmapsto rdr$
$\displaystyle c$ $\displaystyle\longmapsto c,\qquad\quad h\longmapsto rcr$
$\displaystyle d$ $\displaystyle\longmapsto d,\qquad\quad i\longmapsto
r^{-1}br$ $\displaystyle e$ $\displaystyle\longmapsto e,\qquad\quad
j\longmapsto r^{-1}ar$

and

$\displaystyle W^{\prime}=abreabre+2rdrc-2ardbrc-rere.$ (3.1)

There is a choice involved in taking generators of $Q$ under $\varphi$ and the
isomorphism arrows $r_{i_{u},t}$ (or equivalently a choice of $\xi$) and hence
the algebra $\mathbb{C}Q^{\prime}$ is not unique (but it is unique up to
canonical isomorphism). We choose $\xi$ so that $W^{\prime}$ does not contain
both $r_{i_{u},t}$ and $r_{i_{u},t}^{-1}$, so that we may view it as a
potential on the un-localised quiver algebra and hence the noncommutative
derivative description of $\textnormal{Jac}(Q^{\prime},W^{\prime})$ makes
sense.

######  Lemma 3.10.

There exists a choice of $\xi$ such that for each isomorphism arrow
$r_{i_{u},t}\in Q^{\prime}_{1}$ the potential $W^{\prime}$ does not contain
both $r_{i_{u},t}$ and $r_{i_{u},t}^{-1}$.

###### Proof.

Consider the orbit of vertices
$\\{i_{u},\varphi(i_{u}),\ldots,\varphi^{n-1}(i_{u})\\}$ in $Q^{\prime}$,
where $r_{i_{u},t}:\varphi^{t}(i_{u})\rightarrow\varphi^{t+1}(i_{u})$. If
$W^{\prime}$ were to contain both $r_{i_{u},t}$ and $r_{i_{u},t}^{-1}$ then
there would need to be arrows $a,\,b,\,c,\,d\in Q_{1}$ such that $ba$ and $dc$
are part of cycles in $W$ and we would need to choose $\xi$ so that if we let
$a_{0},\,b_{0},\,c_{0},\,d_{0}$ be the corresponding generators of the orbits
of the arrows then $t(a_{0})=\varphi^{k_{1}}(i_{u})$,
$s(b_{0})=\varphi^{k_{2}}(i_{u})$, and $t(c_{0})=\varphi^{l_{1}}(i_{u})$,
$s(d_{0})=\varphi^{l_{2}}(i_{u})$ with $k_{1},l_{2}\leq t$, $k_{2},l_{1}\geq
t+1$. Indeed if this was the case then

$\xi(ba)=p_{4}\,b_{0}\,p_{3}\,r_{i_{u},t}\,p_{2}\,a_{0}\,p_{1}$

and

$\xi(dc)=q_{4}\,d_{0}\,q_{3}\,r_{i_{u},t}^{-1}\,q_{2}\,c_{0}\,q_{1}$

for paths of isomorphism arrows $p_{i},\,q_{i}$ (see Fig. 6 for an
illustration of this). Hence if we simply choose $\xi$ so that the generators
of arrows of two different paths in the cycles in $W$ that cross in the same
orbit of vertices do not overlap in this way we remove instances of both
$r_{i_{u},t}$ and $r_{i_{u},t}^{-1}$ appearing in $W^{\prime}$. One way of
ensuring this is for each orbit of vertices in $Q$, when choosing generators
of the orbits of arrows whose sources lie in this orbit of vertices we have
that the sources of the each of the generators are equal. This would amount to
taking $s(b_{0})=s(d_{0})$ and hence at most either $r_{i_{u},t}$ or
$r_{i_{u},t}^{-1}$ can appear in $W^{\prime}$. ∎

Figure 6. Two parts $ba$ and $dc$ of cycles that appear in $W$ that cross at
the orbit of the vertex $i_{u}$. The chosen generators for the orbits are
$a_{0},\,b_{0},\,c_{0},\,d_{0}$ giving $\xi(ba)$ and $\xi(dc)$ as the paths
highlighted in yellow. We can see that, due to these choices of generators,
$r_{i_{u},t}$ appears in $\xi(ba)$ and $r_{i_{u},t}^{-1}$ appears in $\xi(dc)$
and so both appear in $W^{\prime}=\xi(W)$. A choice of $\xi$ where this will
not occur would be for example to take $b$ and $d$ to be the generators of
their orbits. As the sources of $b$ and $d$ are equal this means no matter
what choice we take for the generators of the orbits for $a$ and $c$, we can
never see both $r_{i_{u},t}$ and $r_{i_{u},t}^{-1}$ in $W^{\prime}$ (assuming
no other parts of cycles in $W$ go through the orbit of $i_{u}$).

######  Lemma 3.11.

  1.   2. a)

Let $a\in Q^{\prime}_{1}$ be a generating arrow which is dual to the edge in
$\Delta$ that goes between the white vertex $v$ and the black vertex $u$. Then

$\displaystyle a\cdot\frac{\partial W^{\prime}}{\partial
a}=n\,\big{(}\xi(c_{v})-\xi(c_{u})\big{)}$ (3.2)

for the minimal cycles $c_{v}$ and $c_{u}$ in $W$.

  3. b)

The relations $\partial W^{\prime}/\partial r_{i_{u},t}$ (or if relevant
$\partial W^{\prime}/\partial r_{i_{u},t}^{-1}$) can be derived from the
relations $\partial W^{\prime}/\partial a$ for generating arrows $a\in
Q^{\prime}_{1}$.

###### Proof.

a) Write $\varphi^{j}(a)=a_{j}\in Q_{1}$ (so $a_{0}=a$). Then
$\xi(a_{j})=p_{j}a\,q_{j}$ where $p_{j},q_{j}$ are paths in
$\mathbb{C}Q^{\prime}$ made from isomorphism arrows. For each $j$ write
$c_{v_{j}}$ and $c_{u_{j}}$ for the minimal cycles in $W$ such that $a_{j}$ is
dual to the edge in $\Delta$ between the white vertex $v_{j}$ and black vertex
$u_{j}$; as these cycles can be written up to cyclic permutation we write them
so that $a_{j}$ appears at the end of these cycles. As $\varphi$ preserves the
tiling $\Delta$ we have for any vertex $v\in\Delta$ that
$\varphi(c_{v})=c_{\varphi(v)}$. Hence $c_{v_{j}}=\varphi^{j}(c_{v_{0}})$ and
$c_{u_{j}}=\varphi^{j}(c_{u_{0}})$. Then because we wrote the cycles so that
$a_{j}$ appears at the end it follows that
$\xi(c_{v_{j}})=p_{j}\xi(c_{v_{0}})p_{j}^{-1}$ and
$\xi(c_{u_{j}})=p_{j}\xi(c_{u_{0}})p_{j}^{-1}$, and hence up to cyclic
permutation $\xi(c_{v_{j}})=\xi(c_{v_{0}})$ and
$\xi(c_{u_{j}})=\xi(c_{u_{0}})$. This is because $\xi(c_{v_{0}})$ (resp.
$\xi(c_{u_{0}})$) ends with $a$ so is a cycle at the vertex $t(a)$ whilst
$c_{v_{j}}$ (resp. $c_{u_{j}}$) is a cycle at the vertex $t(a_{j})$, and
$p_{j}$ is the unique path of isomorphism arrows between $t(a)$ and
$t(a_{j})$.

$v$$1$$2$$3$$4$$\varphi(v)$$\varphi(1)$$\varphi(2)$$\varphi(3)$$\varphi(4)$$a$$b$$c$$d$$\varphi(a)$$\varphi(b)$$\varphi(c)$$\varphi(d)$$\varphi$
Figure 7. An example of part of a brane tiling with the action of $\varphi$.
Let $\varphi$ send the part of the brane tiling on the left to the part on the
right. Take the generators of the orbits to be $a,\,b,\,c,\,d$ and write
$p_{i}$ for the path in $\mathbb{C}Q^{\prime}$ made of isomorphism arrows
between the vertices $i$ and $\varphi(i)$. Hence
$\xi(\varphi(a))=p_{2}ap_{1}^{-1}$, $\xi(\varphi(b))=p_{3}bp_{2}^{-1}$,
$\xi(\varphi(c))=p_{4}cp_{3}^{-1}$, and $\xi(\varphi(d))=p_{1}dp_{4}^{-1}$.
Therefore
$\xi(c_{\varphi(v)})=\xi(\varphi(d))\xi(\varphi(c))\xi(\varphi(b))\xi(\varphi(a))=p_{1}dcbap_{1}^{-1}=p_{1}\xi(c_{v})p_{1}^{-1}$
and all the intermediary paths of isomorphism arrows cancel, as required.

To calculate $\partial W^{\prime}/\partial a$, the relevant terms in
$W^{\prime}$ are

$\xi(c_{v_{j}})-\xi(c_{u_{j}})=\xi(c_{v_{0}})-\xi(c_{u_{0}})\,\,\textnormal{(up
to cyclic permutation)}$

for $j\in\\{0,\ldots,n-1\\}$. Writing the cycles $\xi(c_{v_{0}})$ and
$\xi(c_{u_{0}})$ so that $a$ is at the end, it follows that

$a\cdot\frac{\partial W^{\prime}}{\partial
a}=n\,\big{(}\xi(c_{v_{0}})-\xi(c_{u_{0}})\big{)}.$

b) Morally this is true because $r_{i_{u},t}\cdot\partial W^{\prime}/\partial
r_{i_{u},t}$ is a sum whose terms are just the cycles in $W^{\prime}$ that
contain $r_{i_{u},t}$ with $r_{i_{u},t}$ cyclically permuted to the front. But
the cycles in $W^{\prime}$ can be made from the relations $\partial
W^{\prime}/\partial a$ for generating arrows $a$ and so because $r_{i_{u},t}$
is invertible in $\mathbb{C}Q^{\prime}$, $\partial W^{\prime}/\partial
r_{i_{u},t}$ is $r_{i-u,t}^{-1}$ multiplied by terms that can be obtained from
the relations $\partial W^{\prime}/\partial a$. We just need to check which
generating arrows $a$ are needed to obtain the relevant cycles in $W^{\prime}$
which contain $r_{i_{u},t}$.

So consider the relation $\partial W^{\prime}/\partial r_{i_{u},t}$ and
without loss of generality we assume that $W^{\prime}$ does not contain
$r_{i_{u},t}^{-1}$. For clarity write $r=r_{i_{u},t}$. If $a\in Q_{1}$ has $r$
in its image under $\xi$ and if $b\in Q_{1}$ has $r^{-1}$ in its image under
$\xi$ then we can write

$\xi(a)=p^{\prime}_{a}\,r^{x_{a}}\,p^{\prime\prime}_{a}\,a_{0}\,q^{\prime}_{a}\,r^{y_{a}}\,q^{\prime\prime}_{a}$

and

$\xi(b)=p^{\prime}_{b}\,r^{x_{b}}\,p^{\prime\prime}_{b}\,b_{0}\,q^{\prime}_{b}\,r^{y_{b}}\,q^{\prime\prime}_{b}$

for $a_{0}$ and $b_{0}$ the chosen generators of the orbits,
$p^{\prime}_{a},\,p^{\prime\prime}_{a},\,q^{\prime}_{a},\,q^{\prime\prime}_{a}$
and
$p^{\prime}_{b},\,p^{\prime\prime}_{b},\,q^{\prime}_{b},\,q^{\prime\prime}_{b}$
paths of isomorphism arrows that do not contain $r$ or $r^{-1}$, and
$x_{a},\,y_{a}\in\\{0,1\\}$ and $x_{b},\,y_{b}\in\\{-1,0\\}$. If $a$ is dual
to the edge in $\Delta$ between the white vertex $v_{a}$ and black vertex
$u_{a}$ and we write the minimal cycles $c_{v_{a}}$ and $c_{u_{a}}$ so that
$a$ is at the end, then $r\cdot\partial W^{\prime}/\partial r$ will contain
terms like

$\displaystyle
r\,p^{\prime\prime}_{a}\,a_{0}\,q^{\prime}_{a}\,r^{y_{a}}\,q^{\prime\prime}_{a}\Big{(}\xi(a^{-1}c_{v_{a}})-\xi(a^{-1}c_{u_{a}})\Big{)}p^{\prime}_{a}$
(3.3)

and

$\displaystyle
r\,q^{\prime\prime}_{a}\Big{(}\xi(a^{-1}c_{v_{a}})-\xi(a^{-1}c_{u_{a}})\Big{)}p^{\prime}_{a}\,r^{x_{a}}\,p^{\prime\prime}_{a}\,a_{0}\,q^{\prime}_{a}.$
(3.4)

However since $W^{\prime}$ does not contain $r^{-1}$ whenever an arrow $b$
appears in $W$ the $r^{-1}$ it gives under $\xi$ must be cancelled out by an
$r$ given by some $a$ under $\xi$. Hence we must remove those terms from
$r\cdot\partial W^{\prime}/\partial r$ that are cancelled. What is removed is
exactly of the form

$\displaystyle
r\,r^{-1}\,p^{\prime\prime}_{b}\,b_{0}\,q^{\prime}_{b}\,r^{y_{b}}\,q^{\prime\prime}_{b}\Big{(}\xi(b^{-1}c_{v_{b}})-\xi(b^{-1}c_{u_{b}})\Big{)}p^{\prime}_{b}$
(3.5)

or

$\displaystyle
r\,r^{-1}\,q^{\prime\prime}_{b}\Big{(}\xi(b^{-1}c_{v_{b}})-\xi(b^{-1}c_{u_{b}})\Big{)}p^{\prime}_{b}\,r^{x_{b}}\,p^{\prime\prime}_{b}\,b_{0}\,q^{\prime}_{b}$
(3.6)

and so $r\cdot\partial W^{\prime}/\partial r$ is a sum of terms like (3.3) and
(3.4) minus terms like (3.5) and (3.6). Now (3.3) can be written as

$\displaystyle
r\,p^{\prime\prime}_{a}\,a_{0}\,q^{\prime}_{a}\,r^{y_{a}}\,q^{\prime\prime}_{a}\Big{(}\xi(a^{-1}c_{v_{a}})-\xi(a^{-1}c_{u_{a}})\Big{)}p^{\prime}_{a}$
$\displaystyle=p_{a}^{\prime-1}\Big{(}p_{a}^{\prime}\,r\,p^{\prime\prime}_{a}\,a_{0}\,q^{\prime}_{a}\,r^{y_{a}}\,q^{\prime\prime}_{a}\Big{(}\xi(a^{-1}c_{v_{a}})-\xi(a^{-1}c_{u_{a}})\Big{)}\Big{)}p^{\prime}_{a}$
$\displaystyle=p_{a}^{\prime-1}\Big{(}\xi(a)\xi(a^{-1})\Big{(}\xi(c_{v_{a}})-\xi(c_{u_{a}})\Big{)}\Big{)}p^{\prime}_{a}$
$\displaystyle=\frac{1}{n}\,p_{a}^{\prime-1}\Big{(}p_{a}\,a_{0}\cdot\partial
W^{\prime}/\partial a_{0}\,p_{a}^{-1}\Big{)}p^{\prime}_{a}$

where $p_{a}=p^{\prime}_{a}rp^{\prime\prime}_{a}$ and the last line follows
from part a). Similarly (3.4), (3.5), (3.6) can all be written in terms of the
derivatives of $W^{\prime}$ with respect to the generating arrows $a_{0}$ or
$b_{0}$. This gives the result. ∎

######  Lemma 3.12.

$\xi$ induces an inclusion
$\widetilde{\xi}:\textnormal{Jac}(Q,W)\hookrightarrow\textnormal{Jac}(Q^{\prime},W^{\prime})$.

###### Proof.

We first show the induced map is well-defined. Let $b\in Q_{1}$ and suppose
$b=\varphi^{k}(a)$ for some arrow $a$ in the chosen generating set of $Q_{1}$.
As in the proof of 3.11 we write $\xi(b)=p_{k}a\,q_{k}$ where $p_{k},\,q_{k}$
are paths of isomorphism arrows in $\mathbb{C}Q^{\prime}$. Then we see that

$\displaystyle\xi\left(b\cdot\frac{\partial W}{\partial b}\right)$
$\displaystyle=\xi(c_{v_{b}})-\xi(c_{u_{b}})$
$\displaystyle=p_{k}\big{(}\xi(c_{v_{a}})-\xi(c_{u_{a}})\big{)}p_{k}^{-1}.$
(3.7)

Combining (3.7) with (3.2) gives

$\xi\left(b\cdot\frac{\partial W}{\partial
b}\right)=\frac{1}{n}\,p_{k}\,a\cdot\frac{\partial W^{\prime}}{\partial
a}p_{k}^{-1}$

and so from the equation $\xi(b)=p_{k}a\,q_{k}$ we get that

$\displaystyle\xi\left(\frac{\partial W}{\partial
b}\right)=\frac{1}{n}q_{k}^{-1}\frac{\partial W^{\prime}}{\partial
a}\,p_{k}^{-1}.$ (3.8)

As for injectivity, we must show that given some $x\in\mathbb{C}Q$ such that
$\xi(x)\in I_{W^{\prime}}$ then $x\in I_{W}$. Write

$\xi(x)=\sum_{a}p^{\prime}_{a}\frac{\partial W^{\prime}}{\partial
a}\,q^{\prime}_{a}$

for paths $p^{\prime}_{a},\,q^{\prime}_{a}\in\mathbb{C}Q^{\prime}$. Using 3.8
we can write this as

$\xi(x)=\sum_{a}p^{\prime}_{a}\frac{\partial W^{\prime}}{\partial
a}\,r_{a}\,\xi(q_{a})$

for paths $q_{a}\in\mathbb{C}Q$ and paths of isomorphism arrows
$r_{a}\in\mathbb{C}Q^{\prime}$. Let $\varphi^{k_{a}}(a)$ be the arrow in
$Q_{1}$ such that $r_{a}^{-1}$ is the unique path of isomorphism arrows in
$\mathbb{C}Q^{\prime}$ between $t(a)$ and $t(\varphi^{k_{a}}(a))$, then using
(3.8) we can write

$\xi(x)=\sum_{a}p^{\prime\prime}_{a}\,\xi\left(\frac{\partial
W}{\partial\varphi^{k_{a}}(a)}\,q_{a}\right)$

where $p^{\prime\prime}_{a}=p^{\prime}_{a}s_{a}$ and $s_{a}$ is the unique
path of isomorphism arrows between $s(a)$ and $s(\varphi^{k_{a}}(a))$. Taking
degrees we see from 3.7 that $\textnormal{deg}(p^{\prime\prime}_{a})\equiv
0\,\textnormal{mod}\,n$ and so 3.8 gives us a path $p_{a}\in\mathbb{C}Q$ such
that $\xi(p_{a})=p^{\prime\prime}_{a}$. As $\xi$ is injective we get

$x=\sum_{a}p_{a}\,\frac{\partial W}{\partial\varphi^{k_{a}}(a)}\,q_{a}$

giving the result. ∎

We make the further assumption that we can choose generators and isomorphism
arrows such that additionally $W^{\prime}$ will be homogeneous of degree
$n=\textnormal{deg}(\varphi)$. We can see that in the case of the running
example 3.5 that $W^{\prime}=abreabre+2rdrc-2ardbrc-rere$ indeed has degree 2
in the isomorphism arrow $r$. We give some justification about why this
assumption can be made.

######  Lemma 3.13.

Let $\varphi$ be an automorphism of order $n$ of the Riemann surface
$\Sigma_{g}$ and let $\Delta_{0}$ be a brane tiling of $\Sigma_{g}$ which is
preserved by $\varphi$. Then we can extend $\Delta_{0}$ into a brane tiling
$\Delta$ which is also preserved by $\varphi$ such that there exists a dimer
for $\Delta$.

###### Proof.

In a brane tiling define the _distance_ between two distinct vertices $u$ and
$v$, denoted by $\textnormal{dist}(u,v)$, to be 1 if there exists a tile in
the brane tiling for which both $u$ and $v$ are in the perimeter of it. Then
recursively define the distance of two vertices to be $x+1$ if there exists
another vertex $u^{\prime}$ such that $\textnormal{dist}(u,u^{\prime})=1$ and
$\textnormal{dist}(u^{\prime},v)=x$ or vice versa.

For our brane tiling $\Delta_{0}$ we first add vertices and their images under
$\varphi,\ldots,\varphi^{n-1}$ until the number of black vertices equals the
number of white vertices. Next we add edges and their images under
$\varphi,\ldots,\varphi^{n-1}$ until we end up with a brane tiling which we
call $\Delta_{0}^{\prime}$. We then begin to construct a dimer $D$ for
$\Delta_{0}^{\prime}$ by choosing pairs of vertices in $\Delta_{0}^{\prime}$
(one black and one white) which are connected by an edge. Once a vertex has
been chosen in a pair it is then removed from further consideration for future
pairings.

In this way we can see that if we are able to have all vertices in
$\Delta_{0}^{\prime}$ in one of these pairs then we will indeed have a dimer
for $\Delta_{0}^{\prime}$. However it might turn out that due to our choices
there exists a (w.l.o.g) black vertex $u$ such that all the white vertices
adjacent to it are already paired off.

To remedy such an issue with constructing our dimer, we note that because the
number of black and white vertices is equal there must exist at least one
white vertex $v$ that has not been paired off. If $\textnormal{dist}(u,v)=1$
we add an edge to $\Delta_{0}^{\prime}$ (along with all its images under
$\varphi,\ldots,\varphi^{n-1}$) connecting $u$ to $v$ and then take this to be
in $D$. If $\textnormal{dist}(u,v)>1$ we consider all the white vertices
$v_{1}$ such that $\textnormal{dist}(u,v_{1})=1$. We then choose a white
vertex $v_{1}$ such that $v_{1}$ is paired with the black vertex $u_{1}$ and
$\textnormal{dist}(u_{1},v)<\textnormal{dist}(u,v)$. Such a vertex $u_{1}$
must exist due to our definition of distance in the brane tiling. We then add
an edge between $u$ and $v_{1}$ to the brane tiling (along with its images
under $\varphi,\ldots,\varphi^{n-1}$) unless one already exists in
$\Delta_{0}^{\prime}$. Then we replace the pair $(u_{1},v_{1})$ in $D$ with
$(u,v_{1})$, thereby shifting the issue we are having with constructing our
dimer onto $u_{1}$. We continue to repeat this, each time noting that the
distance between the black vertex $u_{i}$ and our white vertex $v$ is
decreasing. Hence after a finite number of steps we obtain a black vertex
$u_{y}$ such that $\textnormal{dist}(u_{y},v)=1$ and so we can add an edge
between $u_{y}$ and $v$ into the brane tiling and then add the pair
$(u_{y},v)$ into $D$. See Fig. 8 for an illustrative example.

This will fully rectify the issue we had of not being able to include the
vertex $u$ as a pair into $D$ and also not introduce any further issues since
all vertices that had been paired up in $D$ beforehand still remain in $D$
(albeit some will be paired with different vertices now). Continuing to do
this for each vertex will ensure that all vertices end up in $D$ and hence $D$
will be a dimer for the brane tiling $\Delta$ we end up with.

$v$$u$

$v$$v^{\prime}$$u$

Figure 8. The top picture shows an example of part of a brane tiling with the
partial dimer $D$ in green. There are two vertices $v$ and $u$ which are not
contained by an edge in $D$ with $\textnormal{dist}(u,v)=2$. With the current
setup it is not possible to include $v$ in $D$ to get a dimer. We rectify this
in the second picture by changing the edges we include in $D$ (with the new
edge in $D$ shown in blue). This shifts the problem onto the vertex
$v^{\prime}$ which is such that $\textnormal{dist}(u,v^{\prime})=1$. We can
then add an edge that connects $v^{\prime}$ and $u$ into $\Delta^{\prime}_{0}$
(along with all its images under $\varphi,\ldots,\varphi^{n-1}$), and then
include this new edge into $D$.

∎

We must always ensure that our brane tilings are preserved by $\varphi$ and
that the orbits of the vertices in the dual quiver $Q$ have size $n$. By 3.3
we can add to a brane tiling with this property and retain the property, so if
we take a brane tiling $\Delta_{0}$ that is preserved by $\varphi$ then apply
3.2 and then 3.13 to it we end up with a brane tiling $\Delta$ for which both
the orbit size of every vertex in $Q=Q_{\Delta}$ is $n$ and for which there
exists a dimer $D$ for $\Delta$. To try to make $W^{\prime}$ homogeneous of
degree $n$ in the isomorphism arrows we consider the set $E$ of arrows in $Q$
that are dual to the edges in $D$. It is then hoped that there is a choice of
$\xi$ for which the conditions explained in the proof of in 3.10 hold and for
which $\xi(e)$ has degree $n$ for each $e\in E$ and $\xi(a)$ has degree 0 for
every other arrow $a\in Q_{1}\setminus E$. This will then give a potential
$W^{\prime}$ which has the properties of not containing both $r_{i_{u},t}$ and
$r_{i_{u},t}^{-1}$, and having degree $n$ in the isomorphism arrows since $W$
is homogeneous of degree 1 in the arrows in $E$.

Given these assumptions on $W^{\prime}$ hold, similarly to [10] Proposition
4.2 and the preceding discussion, we define a homomorphism

$\displaystyle\textnormal{Jac}(\widetilde{Q^{\prime}},W^{\prime})\xrightarrow{\,\,\Psi\,\,}\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(M_{g,\varphi})])$ (3.9)

where $m$ is the number of vertices in $Q^{\prime}$ (which is also the number
of vertices in $Q$) and the tilde denotes that we are localising the path
algebra $\mathbb{C}Q^{\prime}$ with respect to every arrow $a\in
Q^{\prime}_{1}$ before taking the quotient. To construct $\Psi$ explicitly
consider a maximal tree $T\subset Q$ such that $\xi(t)$ has degree 0 for all
$t\in T$. With regards to the dimer $D$ discussed above, this means that the
intersection of $T$ and the set of arrows dual to edges in $D$ is empty. Fix a
basepoint $\widetilde{\textnormal{bp}}\in\Sigma_{g}$ that is invariant under
$\varphi$ (note such a basepoint will be a vertex in the brane tiling $\Delta$
as per 3.2), fix a vertex $\textnormal{bp}\in Q_{0}$, and fix a path
$\delta:\widetilde{\textnormal{bp}}\rightarrow\textnormal{bp}$ in
$\Sigma_{g}$. Then we can view

$\pi_{1}\big{(}M_{g,\varphi},\,[\widetilde{\textnormal{bp}},0]\big{)}\cong\pi_{1}\big{(}\Sigma_{g},\,\widetilde{\textnormal{bp}}\big{)}\rtimes_{\varphi}\mathbb{Z}.$

From now on, although technically all loops in $M_{g,\varphi}$ and
$\Sigma_{g}$ have basepoint $\widetilde{\textnormal{bp}}$, we work with loops
at the basepoint $\textnormal{bp}\in Q_{0}\subset\Sigma_{g}$ and implicitly
use $\delta$ to formally view them as loops at $\widetilde{\textnormal{bp}}$.
For each $i\in Q_{0}$ let $i_{s}=\varphi^{s}(i)$ and let $t_{i}$ be the unique
path $\textnormal{bp}\rightarrow i$ in $\mathbb{C}\widetilde{Q}^{\prime}$
comprised solely from arrows in $T$ or their inverses. Let $E_{x,y}(z)$ denote
the matrix with entries
$z\in\mathbb{C}[\pi_{1}(\Sigma_{g})\rtimes_{\varphi}\mathbb{Z}]$ in the the
$(x,y)$-th position and zeroes everywhere else. Then define
$\Psi:\textnormal{Jac}(\widetilde{Q^{\prime}},W^{\prime})\rightarrow\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(\Sigma_{g})\rtimes_{\varphi}\mathbb{Z}])$ by sending

$\displaystyle a:i\rightarrow j$ $\displaystyle\longmapsto
E_{j,i}\big{(}([t_{j}^{-1}a\,t_{i}],0)\big{)}$ $\displaystyle
r_{i_{u},s}:i_{u,s}\rightarrow i_{u,s+1}$ $\displaystyle\longmapsto
E_{i_{u,s+1},i_{u,s}}\big{(}([\varphi^{-1}(t_{i_{u,s+1}}^{-1})\,t_{i_{u,s}}],-1)\big{)}$

where $a\in Q^{\prime}_{1}$ is a generating arrow, $r_{i_{u},s}\in
Q^{\prime}_{1}$ is an isomorphism arrow,
$[l]\in\pi_{1}(\Sigma_{g},\widetilde{\textnormal{bp}})$ denotes the class of
the loop $l\in\Sigma_{g}$ and we view paths $p\in\mathbb{C}Q$ as paths in
$\Sigma_{g}$ via the natural inclusion $Q\hookrightarrow\Sigma_{g}$.

######  Remark 3.14.

It is not immediate that $\Psi$ as given above is well-defined.

######  Remark 3.15.

The assignment of an arrow $a:i\rightarrow j$ to $t_{j}^{-1}a\,t_{i}$ in the
above homomorphism is equivalent to the contraction of the tree $T$ seen in
[[10] Proposition 4.2].

######  Remark 3.16.

$\varphi^{-1}(t_{i_{q,p+1}}^{-1})\,t_{i_{q,p}}$ is not a loop in $\Sigma_{g}$
but in fact is a path
$\textnormal{bp}\rightarrow\varphi^{-1}(\textnormal{bp})$. So, first fixing
some path $\gamma:\textnormal{bp}\rightarrow\varphi(\textnormal{bp})$ in
$\Sigma_{g}$, what we actually mean by
$[\varphi^{-1}(t_{i_{q,p+1}}^{-1})t_{i_{q,p}}]\in\pi_{1}(\Sigma_{g},\textnormal{bp})$
is the class of the loop
$\varphi^{-1}(\gamma)\circ\varphi^{-1}(t_{i_{q,p+1}}^{-1})t_{i_{q,p}}$. Since
$\textnormal{bp}\in Q_{0}$ we take $\gamma=t_{\varphi(\textnormal{bp})}$. In a
similar vein when multiplying in the semi-direct product we technically should
write

$([\beta],m)\cdot([\alpha],k)=([\gamma^{-1}\circ\varphi(\gamma^{-1})\circ\cdots\circ\varphi^{k-1}(\gamma^{-1})\circ\varphi^{k}(\beta)\circ\varphi^{k-1}(\gamma)\circ\cdots\circ\varphi(\gamma)\circ\gamma\circ\alpha],k+m)$

for $k>0$ and

$([\beta],m)\cdot([\alpha],k)=([\varphi^{-1}(\gamma)\circ\cdots\circ\varphi^{k+1}(\gamma^{-1})\circ\varphi^{k}(\gamma)\circ\varphi^{k}(\beta)\circ\varphi^{k}(\gamma^{-1})\circ\varphi^{k+1}(\gamma^{-1})\circ\cdots\circ\varphi^{-1}(\gamma^{-1})\circ\alpha],k+m)$

for $k<0$ in order to ensure we are multiplying loops at bp. However it will
turn out that in the multiplications we present later these intermediary
paths, that are needed to correct for the basepoints of the loops involved,
will mostly all cancel and so we omit them from the proceeding discussion to
aid in notational clarity. This would not be necessary if we worked with the
$\varphi$-invariant basepoint $\widetilde{\textnormal{bp}}$, but it must be
done to allow us to work within the quiver $Q^{\prime}$.

Note that $\Psi$ sends

$r_{i_{u},s}^{-1}\longmapsto
E_{i_{u,s},i_{u,s+1}}\big{(}([\varphi(t_{i_{u,s}}^{-1})\,t_{i_{u,s+1}}],1)\big{)}.$

######  Theorem 3.17.

Let $\Sigma_{g}$ be a Riemann surface of genus $g$ and let $\varphi$ be an
orientation-preserving automorphism of $\Sigma_{g}$ of order $n$. Let $\Delta$
be a brane tiling of $\Sigma_{g}$ which is preserved under $\varphi$ such that
the size of the order of each vertex in $Q=Q_{\Delta}$ is $n$. Choose
generating arrows from $Q$ and isomorphism arrows to construct the quiver
$Q^{\prime}$ as before, such that the potential $W^{\prime}$ is homogeneous of
degree $n$ and does not contain both an isomorphism arrow $r_{i_{u},t}$ and
its inverse $r_{i_{u},t}^{-1}$. Then the homomorphism of algebras (3.9)

$\textnormal{Jac}(\widetilde{Q^{\prime}},W^{\prime})\xrightarrow[\,\,\,\,\sim\,\,\,\,]{\Psi}\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(M_{g,\varphi})])$

is well-defined and an isomorphism.

###### Proof.

Using the definition of $\Psi$ we have that

$\displaystyle\Psi\circ\widetilde{\xi}(p)=E_{y,x}\big{(}([t_{y}^{-1}p\,t_{x}],-\textnormal{deg}(\widetilde{\xi}(p)))\big{)}$
(3.10)

for any path $p:x\rightarrow y\in\textnormal{Jac}(\widetilde{Q},W)$. Indeed
for an arrow $b\in Q_{1}$ we have that $b=\varphi^{l}(a)$ for some generating
arrow $a\in Q_{1}$ and $l\in\\{0,\ldots,n-1\\}$, and hence we can write
$\widetilde{\xi}(b)$ as equal to

$\displaystyle s_{1}^{-1}\ldots s_{k}^{-1}\,a\,r_{k}\ldots r_{1}$ or
$\displaystyle s_{l-k}\ldots s_{1}\,a\,r_{k}\ldots r_{1}$ or $\displaystyle
s_{k}\ldots s_{1}\,a\,r^{-1}_{1}\ldots r^{-1}_{k}$ or $\displaystyle
s_{1}^{-1}\ldots s_{l-k}^{-1}\,a\,r^{-1}_{1}\ldots r^{-1}_{k}$

for isomorphism arrows $r_{i}:x_{i-1}\rightarrow
x_{i},\,s_{i}:y_{i-1}\rightarrow y_{i}$, and some $0\leq k\leq l$. In the
first of these cases we get that $a:x_{k}\rightarrow y_{k}$ and
$b:x_{0}\rightarrow y_{0}$, and therefore

$\displaystyle\Psi(\widetilde{\xi}(b))$ $\displaystyle=\Psi(s_{1}^{-1}\ldots
s_{k}^{-1}\,a\,r_{k}\ldots r_{1})$
$\displaystyle=E_{y_{0},y_{1}}\big{(}([\varphi(t_{y_{0}}^{-1})t_{y_{1}}],1)\big{)}\,\cdots\,E_{y_{k-1},y_{k}}\big{(}([\varphi(t_{y_{k-1}}^{-1})t_{y_{k}}],1)\big{)}\,\,E_{y_{k},x_{k}}\big{(}([t_{y_{k}}^{-1}a\,t_{x_{k}}],0)\big{)}$
$\displaystyle\quad\,\,E_{x_{k},x_{k-1}}\big{(}([\varphi^{-1}(t_{x_{k}}^{-1})t_{x_{k-1}}],-1)\big{)}\,\cdots\,E_{x_{1},x_{0}}\big{(}([\varphi^{-1}(t_{x_{1}}^{-1})t_{x_{0}}],-1)\big{)}$
$\displaystyle=E_{y_{0},x_{0}}\big{(}([\varphi(t_{y_{0}}^{-1})t_{y_{1}}],1)\cdot\ldots\cdot([\varphi(t_{y_{k-1}}^{-1})t_{y_{k}}],1)\cdot([t_{y_{k}}^{-1}a\,t_{x_{k}}],0)\cdot$
$\displaystyle\qquad\qquad\cdot([\varphi^{-1}(t_{x_{k}}^{-1})t_{x_{k-1}}],-1)\cdot\ldots\cdot([\varphi^{-1}(t_{x_{1}}^{-1})t_{x_{0}}],-1)\big{)}$
$\displaystyle=E_{y_{0},x_{0}}\big{(}([t_{y_{0}}^{-1}\varphi^{k}(a)\,t_{x_{0}}],0)\big{)}$
$\displaystyle=E_{y_{0},x_{0}}\big{(}([t_{y_{0}}^{-1}b\,t_{x_{0}}],-\textnormal{deg}(\widetilde{\xi}(b)))\big{)}$

as required. The other cases follow in a similar manner and since we have
shown the statement for all arrows in $Q$ it readily extends to all paths in
$\mathbb{C}\widetilde{Q}$.

By 3.11, to show that $\Psi$ is well-defined it suffices to check that

$\Psi\left(\frac{\partial W^{\prime}}{\partial a}\right)=0$

for all generating arrows $a\in Q_{1}$. But by 3.12 we have that

$\frac{\partial W^{\prime}}{\partial a}=n\,\xi\left(\frac{\partial W}{\partial
a}\right)$

and hence we just need to show that $\Psi(\widetilde{\xi}(\partial W/\partial
a))=0$. Writing $\partial W/\partial a=p-q$ for $a:i\rightarrow j$, (3.10)
tells us that

$\displaystyle\Psi\left(\widetilde{\xi}\left(\frac{\partial W}{\partial
a}\right)\right)$
$\displaystyle=E_{i,j}\big{(}([t_{i}^{-1}pt_{j}^{-1}],-\textnormal{deg}(\widetilde{\xi}(p))\big{)}-E_{i,j}\big{(}([t_{i}^{-1}qt_{j}^{-1}],-\textnormal{deg}(\widetilde{\xi}(q))\big{)}$
$\displaystyle=E_{i,j}\big{(}([t_{i}^{-1}pt_{j}^{-1}],-\textnormal{deg}(\widetilde{\xi}(p))-([t_{i}^{-1}qt_{j}^{-1}],-\textnormal{deg}(\widetilde{\xi}(q))\big{)}.$

But [[10] Proposition 4.2 and Proposition 5.4] implies that
$[p]=[q]\in\pi_{1}(\Sigma_{g})$ and since $W^{\prime}$ is homogeneous
$\textnormal{deg}(\widetilde{\xi}(p))=\textnormal{deg}(\widetilde{\xi}(q))$,
hence

$E_{i,j}\big{(}([t_{i}^{-1}pt_{j}^{-1}],-\textnormal{deg}(\widetilde{\xi}(p))-([t_{i}^{-1}qt_{j}^{-1}],-\textnormal{deg}(\widetilde{\xi}(q))\big{)}=E_{i,j}\big{(}0\big{)}=0$

as required.

We now move onto showing that $\Psi$ is an isomorphism. For surjectivity we
choose $\textnormal{bp}\in Q^{\prime}_{0}$ such that there exists an
isomorphism arrow $r:\textnormal{bp}\rightarrow\varphi(\textnormal{bp})$ and
recall we take $\gamma=t_{\varphi(\textnormal{bp})}$. Then $r$ has image under
$\Psi$ given by

$\displaystyle
E_{\varphi(\textnormal{bp}),\textnormal{bp}}\big{(}([\varphi^{-1}(\gamma)\circ\varphi^{-1}(t_{\varphi(\textnormal{bp})}^{-1})],-1)\big{)}$
$\displaystyle=E_{\varphi(\textnormal{bp}),\textnormal{bp}}\big{(}([\varphi^{-1}(t_{\varphi(\textnormal{bp})})\circ\varphi^{-1}(t_{\varphi(\textnormal{bp})}^{-1})],-1)$
$\displaystyle=E_{\varphi(\textnormal{bp}),\textnormal{bp}}\big{(}([1],-1)\big{)}$

where $[1]\in\pi_{1}(\Sigma_{g})$ is the class of the constant path. For
$i,j\in Q^{\prime}_{0}$ consider the path

$p_{ij}=\widetilde{\xi}(t_{i}\,t_{\varphi(\textnormal{bp})}^{-1})\,r\,\widetilde{\xi}(t_{j}^{-1})$

in $\widetilde{Q}^{\prime}$. Since $\textnormal{deg}(t)=0$ for all $t\in T$ we
have that $\textnormal{deg}(t_{i})=0$ for all $i\in Q^{\prime}_{0}$, and so by
(3.10) applying $\Psi$ to $p_{ij}$ gives

$\displaystyle
E_{i,\varphi(\textnormal{bp})}\big{(}([t_{i}^{-1}t_{i}t_{\varphi(\textnormal{bp})}^{-1}t_{\varphi(\textnormal{bp})}],0)\big{)}\cdot
E_{\varphi(\textnormal{bp}),\textnormal{bp}}\big{(}([1],-1)\big{)}\cdot
E_{\textnormal{bp},j}\big{(}([t_{\textnormal{bp}}^{-1}t_{j}^{-1}t_{j}],0)\big{)}$
$\displaystyle=E_{i,j}\big{(}([\varphi^{-1}(\gamma)\cdot\varphi^{-1}(t_{i}^{-1}t_{i}t_{\varphi(\textnormal{bp})}^{-1}t_{\varphi(\textnormal{bp})})\cdot\varphi^{-1}(\gamma^{-1})\cdot
t_{\textnormal{bp}}^{-1}t_{j}^{-1}t_{j}],-1)\big{)}$
$\displaystyle=E_{i,j}\big{(}([1],-1)\big{)}$

because by definition $t_{\textnormal{bp}}=1$. In a similar way replacing the
$r$ with $r^{-1}$ gives the matrices $E_{i,j}\big{(}([1],1)\big{)}$ in the
image of $\Psi$ too. Hence these paths, varying over all vertices $i,j$ in
$Q^{\prime}$, will generate the $\mathbb{Z}$ summand in the semi-direct
product $\pi_{1}(\Sigma_{g})\rtimes_{\varphi}\mathbb{Z}$ for all coordinates
$(i,j)$. Next let

$\pi:\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(\Sigma_{g})\rtimes_{\varphi}\mathbb{Z}])\rightarrow\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(\Sigma_{g})])$

be the projection map, and note that $\pi$ is not an algebra homomorphism but
becomes one when we restrict to the subalgebra $\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(\Sigma_{g})\rtimes_{\varphi}n\mathbb{Z}])=\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(\Sigma_{g})\times n\mathbb{Z}])$, since
$\varphi^{n}=\textnormal{id}_{\Sigma_{g}}$. Then (3.10) tells us that the
composition

$\textnormal{Jac}(\widetilde{Q},W)\xrightarrow{\,\,\widetilde{\xi}\,\,}\textnormal{Jac}(\widetilde{Q}^{\prime},W^{\prime})\xrightarrow{\,\,\Psi\,\,}\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(\Sigma_{g})\rtimes_{\varphi}\mathbb{Z}])\xrightarrow{\,\,\pi\,\,}\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(\Sigma_{g})])$

is the homomorphism that sends a path $p:x\rightarrow
y\in\textnormal{Jac}(\widetilde{Q},W)$ to

$E_{y,x}([t_{y}^{-1}p\,t_{x}])\in\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(\Sigma_{g})])$

since
$\textnormal{Im}(\Psi\circ\widetilde{\xi})\subset\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(\Sigma_{g})\rtimes_{\varphi}n\mathbb{Z}])$. This
homomorphism is surjective by 2.23. Therefore $\Psi$ is surjective.

To show injectivity suppose $\Psi(\alpha)=\Psi(\beta)$ for paths
$\alpha,\beta:i\rightarrow j$ in $\mathbb{C}Q^{\prime}$. Then

$\Psi(\beta^{-1}\alpha)=E_{i,i}\big{(}([1],0)\big{)}.$

Let $\pi_{\mathbb{Z}}:\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(\Sigma_{g})\rtimes_{\varphi}\mathbb{Z}])\rightarrow\textnormal{Mat}_{m\times
m}(\mathbb{C}[\mathbb{Z}])$ then from the definition of $\Psi$ it is clear
that the composition $\pi_{\mathbb{Z}}\circ\Psi$ sends a path $p:i\rightarrow
j$ in $\mathbb{C}\widetilde{Q}^{\prime}$ to

$E_{j,i}(-\textnormal{deg}(p)).$

Hence $\textnormal{deg}(\beta^{-1}\alpha)=0$ and so by 3.8 $\beta^{-1}\alpha$
lies in the image of $\widetilde{\xi}$. So we can view $\beta^{-1}\alpha$ as a
path in $\textnormal{Jac}(\widetilde{Q},W)$ and so by [[8] Lemma 2.7] we have
that $\beta^{-1}\alpha=c_{v}^{d}$ for some minimal cycle $c_{v}$ around a
vertex $v\in\Delta$ and some $d\in\mathbb{Z}.$ Since
$\textnormal{deg}(\widetilde{\xi}(c_{v}))=n$ from our assumption that the
potential $W^{\prime}$ is homogeneous of degree $n$, this implies that
$\pi_{\mathbb{Z}}(\Psi(\widetilde{\xi}(c_{v}))\neq 0$. Therefore $d$ must be
equal to 0 and hence $\beta^{-1}\alpha=c_{v}^{0}=e_{i}$ the constant path at
the vertex $i\in Q^{\prime}_{0}$, and so $\alpha=\beta$ in
$\mathbb{C}\widetilde{Q}^{\prime}$. ∎

######  Example 3.18.

We apply the homomorphism $\Psi$ to our running example of the genus 2 surface
with $\varphi$ the rotation by $180^{\circ}$ from 3.5, using the brane tiling
given in 2.17.

Note that a finite presentation for the algebra
$\mathbb{C}[\pi_{1}(\Sigma_{2})\rtimes_{\varphi}\mathbb{Z}]$ is given by the
generators $\\{x,y,z\\}$ and the relations

$\displaystyle xyx^{-1}y^{-1}z^{-1}xyx^{-1}y^{-1}z=1$ (3.11) $\displaystyle
xz^{2}=z^{2}x$ $\displaystyle yz^{2}=z^{2}y.$

We saw from (3.1) that for our brane tiling and that particular choice of
generators and isomorphism arrows we get the potential

$W^{\prime}=abreabre+2rdrc-2ardbrc-rere.$

This satisfies the conditions that $W^{\prime}$ does not contain both $r$ and
$r^{-1}$ and that $W^{\prime}$ is homogeneous of degree
$2=\textnormal{deg}(\varphi)$ in the isomorphism arrow $r$. We choose the
maximal tree $T=\\{e\\}\subset Q$ since $\xi(e)=e$ has degree $0$. We have the
following relations in $\textnormal{Jac}(\widetilde{Q}^{\prime},W^{\prime})$

$\displaystyle\frac{\partial W^{\prime}}{\partial a}$
$\displaystyle=2breabre-2rdbrc$ $\displaystyle\frac{\partial
W^{\prime}}{\partial b}=2reabrea-2rcard$ $\displaystyle\frac{\partial
W^{\prime}}{\partial c}$ $\displaystyle=2rdr-2ardbr$
$\displaystyle\frac{\partial W^{\prime}}{\partial d}=2rcr-2brcar$
$\displaystyle\frac{\partial W^{\prime}}{\partial e}$
$\displaystyle=2abreabr-2rer$

and recall from 3.11 b) that we do not need to consider $\partial
W^{\prime}/\partial r$.

Since the arrow $e$ is in our maximal tree $T$, we can think of $e$ as a means
to go between the vertices in $\widetilde{Q}^{\prime}$ and therefore as a
means to go between the different coordinates in $\textnormal{Mat}_{2\times
2}(\mathbb{C}[\pi_{1}(\Sigma_{2})\rtimes_{\varphi}\mathbb{Z}])$. Equivalently
we can simply contract the arrow $e$ in $\widetilde{Q}^{\prime}$ giving a
quiver with 1 vertex and with all the arrows becoming loops at this vertex. So
to simplify things, in the above relations we set $e=1$.

Then we have

$\displaystyle\frac{\partial W^{\prime}}{\partial a}=0$
$\displaystyle\Longrightarrow\,\,\,$ $\displaystyle brabr=rdbrc$

which combined with $\partial W^{\prime}/\partial e=0$ gives

$ardbrc=rr$

and then combining with $\partial W^{\prime}/\partial c=0$ gives

$\displaystyle rdrc=rr$ $\displaystyle\Longrightarrow\,\,\,$ $\displaystyle
d=rc^{-1}r^{-1}.$ (3.12)

Additionally

$\displaystyle\frac{\partial W^{\prime}}{\partial b}=0$
$\displaystyle\Longrightarrow\,\,\,$ $\displaystyle b=a^{-1}carda^{-1}r^{-1}$
$\displaystyle\Longrightarrow\,\,\,$ $\displaystyle
b=a^{-1}carrc^{-1}r^{-1}a^{-1}r^{-1}.$ (3.13)

Using these substitutions we get

$\displaystyle\frac{\partial W^{\prime}}{\partial e}=0$
$\displaystyle\Longrightarrow\,\,\,$ $\displaystyle
carrc^{-1}r^{-1}a^{-1}carrc^{-1}r^{-1}a^{-1}=rr$

and

$\displaystyle\frac{\partial W^{\prime}}{\partial a}=0$
$\displaystyle\Longrightarrow\,\,\,$ $\displaystyle
a^{-1}carrc^{-1}r^{-1}a^{-1}carrc^{-1}r^{-1}a^{-1}=rrc^{-1}r^{-1}a^{-1}carrc^{-1}r^{-1}a^{-1}c$

which combined give

$\displaystyle a^{-1}rr=a^{-1}c^{-1}rrc$ $\displaystyle\Longrightarrow\,\,\,$
$\displaystyle crr=rrc.$ (3.14)

Then combining $\partial W^{\prime}/\partial b=0$ and $\partial
W^{\prime}/\partial d=0$ gives

$\displaystyle rabra=b^{-1}rcrd$ $\displaystyle\Longrightarrow\,\,\,$
$\displaystyle brabra=rcrd$ $\displaystyle\Longrightarrow\,\,\,$
$\displaystyle a^{-1}carrrc^{-1}r^{-1}a^{-1}carrc^{-1}r^{-1}=rcrrc^{-1}r^{-1}$

and then using $rrc=crr$ implies that

$a^{-1}carrrc^{-1}r^{-1}a^{-1}carrc^{-1}r^{-1}=rr.$

Combining this with $\partial W^{\prime}/\partial a=0$ and $\partial
W^{\prime}/\partial e=0$ gives

$rra^{-1}=a^{-1}c^{-1}rrc$

so again using $rrc=crr$ implies that

$\displaystyle arr=rra.$ (3.15)

Finally we have

$\displaystyle\frac{\partial W^{\prime}}{\partial d}=0$
$\displaystyle\Longrightarrow\,\,\,$ $\displaystyle
rcr=a^{-1}carrrc^{-1}r^{-1}a^{-1}r^{-1}rcar$
$\displaystyle\Longrightarrow\,\,\,$ $\displaystyle
rc=a^{-1}carrrc^{-1}r^{-1}a^{-1}ca$ $\displaystyle\Longrightarrow\,\,\,$
$\displaystyle 1=a^{-1}carc^{-1}ra^{-1}cac^{-1}r^{-1}.$ (3.16)

again using $rrc=crr$ in the last step.

The relations (3.12) and (3.13) tell us that the generators $b$ and $d$ depend
upon $a,c,r$, and the relations (3.14), (3.15), and (3.16) are exactly the
relations given in (3.11). Hence the map $a^{-1}\mapsto x$, $c\mapsto y$,
$r\mapsto z$ is an isomorphism. It is easy to check that the map $\Psi$
described in 3.17 is this isomorphism; for example at the $(1,1)$-coordinate
$\Psi(a)=([a],0)$, $\Psi(c)=([e^{-1}c],0)$ and $\Psi(r)=([1],-1)$ and the
loops $[a],\,[e^{-1}c]\in\Sigma_{2}$ can be seen using Fig. 2 as the first
pair of the standard generators of $\pi_{1}(\Sigma_{2})$, i.e. $x_{1}^{-1}$
and $y_{1}$, as required.

##  4\. Future directions

One of the benefits of a superpotential description of the fundamental group
algebra $k[\pi_{1}(X)]$ is a more approachable way of calculating the
(motivic) DT invariants associated to $\pi_{1}(X)$ (see [11] and [19] for good
introductions to the Grothendieck ring of varieties and motivic DT
invariants). [[10] Proposition 5.5] says that for a finitely generated algebra
$A$ there is an equivalence of stacks

$\textnormal{Rep}_{d}(A)\cong\textnormal{Rep}_{dm}(\textnormal{Mat}_{m\times
m}(A))$

and so it follows from 3.17 and 2.10 that we get a description of
$\textnormal{Rep}_{d}(\mathbb{C}[\pi_{1}(M_{g,\varphi})])$ as a critical
locus. Thus it is straightforward to define its motivic DT partition function.
For an Artin stack $S$ locally of finite type over $\mathbb{C}$ let
$\mathcal{M}_{S}^{G}=\textnormal{K}_{0}^{G}(\textnormal{Var}/S)[\mathbb{L}^{-\frac{1}{2}}]$
denote the Grothendieck ring of naive $G$-equivariant motives over $S$
adjoined $\mathbb{L}^{-\frac{1}{2}}$, where $\mathbb{L}$ is the motive of
$\mathbb{A}^{1}$. Let
$\mathcal{M}^{\widehat{\mu}}_{S}=\lim_{\underset{r}{\to}}\mathcal{M}^{\mu_{r}}_{S}$
where $\mu_{r}$ is the group of $r$-th roots of unity. Then explicitly this
partition function is

$\Phi_{\pi_{1}(M_{g,\varphi})}(t)=\sum_{m=0}^{\infty}\Big{[}\textnormal{Rep}_{d}(\mathbb{C}[\pi_{1}(M_{g,\varphi})])\Big{]}_{\textnormal{vir}}t^{m}$

where for a finitely-generated algebra $A$ whose stack of representations can
be written as a critical locus
$\textnormal{Rep}_{d}(A)\cong\textnormal{crit}(f_{d})$ for
$f_{d}:X_{d}\rightarrow\mathbb{C}$ and $X_{d}$ smooth we write

$\Big{[}\textnormal{Rep}_{d}(A)\Big{]}_{\textnormal{vir}}=\int_{\textnormal{Rep}_{d}(A)}[\phi_{f_{d}}]\in\mathcal{M}^{\widehat{\mu}}_{\mathbb{C}}$

where $[\phi_{f_{d}}]\in\mathcal{M}^{\widehat{\mu}}_{\textnormal{Rep}_{d}(A)}$
is the (normalised) motivic vanishing cycle of $f_{d}$.

For the fundamental group algebras of the mapping tori of Riemann surfaces we
are considering the isomorphism from 3.17 to the Jacobi algebra gives us the
following critical locus structure

$\textnormal{Rep}_{d}(\mathbb{C}[\pi_{1}(M_{g,\varphi})])\cong\textnormal{crit}(\widetilde{f}_{d})$

where
$\widetilde{f}_{d}=\textnormal{Tr}(W^{\prime})_{d}:\textnormal{Rep}_{d}(\widetilde{Q}^{\prime})\rightarrow\mathbb{C}$,
for $d=(d,\ldots,d)\in\mathbb{N}^{m}$. Let $M_{d}(\widetilde{Q}^{\prime})$ be
the space of representations of $\mathbb{C}\widetilde{Q}^{\prime}$ which gives
a smooth atlas of $\textnormal{Rep}_{d}(\widetilde{Q}^{\prime})$ since

$\textnormal{Rep}_{d}(\widetilde{Q}^{\prime})\cong
M_{d}(\widetilde{Q}^{\prime})/G$

for $G=\prod_{i\in Q^{\prime}_{0}}\textnormal{GL}_{d}$, and let
$f_{d}:M_{d}(\widetilde{Q}^{\prime})\rightarrow\mathbb{C}$ be the lift of
$\widetilde{f}_{d}$ to this atlas. It follows that

$\Big{[}\textnormal{Rep}_{d}(\mathbb{C}[\pi_{1}(M_{g,\varphi})])\Big{]}_{\textnormal{vir}}=\mathbb{L}^{\frac{r}{2}}[G]^{-1}\int_{\textnormal{crit}(f_{d})}[\phi_{f_{d}}]$

where $r=\textnormal{dim}(G)$. Hence the problem of calculating motivic DT
partition function for $\mathbb{C}[\pi_{1}(M_{g,\varphi})]$ comes down to
calculating the motivic vanishing cycles $\int[\phi_{f_{d}}]$. We now give
some ideas on how one could go about doing this using motivic dimensional
reduction [2], [11], [20] and power structures [15], [4].

###  4.1. Calculating motivic DT invariants

In order to calculate the motivic vanishing cycle $\int[\phi_{f_{d}}]$ we
would like to use [[11] Theorem 5.9] which says that

$\int_{\textnormal{crit}(f_{d})}[\phi_{f_{d}}]=\int_{\textnormal{crit}(f_{d})}[\phi_{f_{d}}^{\textnormal{eq}}]:=\mathbb{L}^{-\frac{\textnormal{dim}\,X_{d}}{2}}\Big{(}\big{[}f_{d}^{-1}(0)\big{]}-\big{[}f_{d}^{-1}(1)\big{]}\Big{)}$

for the equivariant vanishing cycle $\int[\phi_{f_{d}}^{\textnormal{eq}}]$ of
$f_{d}$. But we can't apply [[11] Theorem 5.9] directly as the variety
$M_{d}(\widetilde{Q}^{\prime})$ cannot locally be written in the form
$\mathbb{A}^{r}\times Z$ with a $\mathbb{G}_{d}$-action such that $f_{d}$ is
equivariant and the induced action on $Z$ is trivial. To remedy this we
present a conjecture that utilises power structures to overcome this (see
[15], [4] for an explanation of power structures on the Grothendieck ring of
varieties).

Let $B$ be a finitely generated algebra with potential $W$ and let
$A=\textnormal{Jac}(B,W)$ be its Jacobi algebra. Let $\widetilde{\omega}\in B$
be such that its image $\omega\in A$ is central. Then define two new partition
functions $\Phi_{A}^{\omega-\textnormal{nilp}}(t)$ and
$\Phi_{A}^{\omega-\textnormal{inv}}(t)$ for the substacks
$\textnormal{Rep}^{\omega-\textnormal{nilp}}(A)$ of representations of $A$ for
which $\omega$ acts nilpotently and
$\textnormal{Rep}^{\omega-\textnormal{inv}}(A)$ of representations of $A$ for
which $\omega$ acts invertibly, by

$\displaystyle\Phi_{A}^{\omega-\textnormal{nilp}}(t)=\sum_{d=0}^{\infty}\Big{[}\textnormal{Rep}_{d}(A)\Big{]}_{\textnormal{vir}}^{\omega-\textnormal{nilp}}t^{d}$
$\displaystyle\Phi_{A}^{\omega-\textnormal{inv}}(t)=\sum_{d=0}^{\infty}\Big{[}\textnormal{Rep}_{d}(A)\Big{]}_{\textnormal{vir}}^{\omega-\textnormal{inv}}t^{d}$

where
$\big{[}\textnormal{Rep}_{d}(A)\big{]}_{\textnormal{vir}}^{\omega-\textnormal{nilp}}$
(resp.
$\big{[}\textnormal{Rep}_{d}(A)\big{]}_{\textnormal{vir}}^{\omega-\textnormal{inv}}$)
denotes the pushforward to $\mathcal{M}^{\hat{\mu}}_{\mathbb{C}}$ of the
pullback of $\big{[}\textnormal{Rep}_{d}(A)\big{]}_{\textnormal{relvir}}$ to
$\textnormal{Rep}_{d}^{\omega-\textnormal{nilp}}(A)$ (resp.
$\textnormal{Rep}_{d}^{\omega-\textnormal{inv}}(A)$).

######  Remark 4.1.

If we have a function $f:X\rightarrow\mathbb{C}$ with critical locus
$Z=\textnormal{crit}(f)$ and an open subset $U\subset X$ then

$[\phi_{f}]|_{Z\cap U}=[\phi_{f|_{U}}].$

Then $\widetilde{\omega}$ acting invertibly on representations of $B$ is an
open condition and

$\textnormal{Rep}_{d}^{\omega-\textnormal{inv}}(A)=\textnormal{crit}(\textnormal{Tr}(W)_{d})\cap\textnormal{Rep}_{d}^{\widetilde{\omega}-\textnormal{inv}}(B).$

Hence defining the virtual motive

$\Big{[}\textnormal{Rep}_{d}^{\omega-\textnormal{inv}}(A)\Big{]}_{\textnormal{vir}}:=\int_{\textnormal{crit}(\textnormal{Tr}(W)_{d})}[\phi_{f_{d}^{\textnormal{inv}}}]$

where
$f_{d}^{\textnormal{inv}}=\textnormal{Tr}(W)_{d}|_{\textnormal{Rep}_{d}^{\widetilde{\omega}-\textnormal{inv}}(B)}$,
we get that

$\Big{[}\textnormal{Rep}_{d}(A)\Big{]}_{\textnormal{vir}}^{\omega-\textnormal{inv}}=\Big{[}\textnormal{Rep}_{d}^{\omega-\textnormal{inv}}(A)\Big{]}_{\textnormal{vir}}.$

If we let $A_{\omega}$ denote the localisation of $A$ with respect to $\omega$
then it is clear that
$\textnormal{Rep}_{d}^{\omega-\textnormal{inv}}(A)\cong\textnormal{Rep}_{d}(A_{\omega})$
hence

$\Big{[}\textnormal{Rep}_{d}(A)\Big{]}_{\textnormal{vir}}^{\omega-\textnormal{inv}}=\Big{[}\textnormal{Rep}_{d}^{\omega-\textnormal{inv}}(A)\Big{]}_{\textnormal{vir}}=\Big{[}\textnormal{Rep}_{d}(A_{\omega})\Big{]}_{\textnormal{vir}}$

and so

$\displaystyle\Phi_{A}^{\omega-\textnormal{inv}}(t)=\Phi_{A_{\omega}}(t).$
(4.1)

We construct a new space of representations for our fundamental group algebras
which will arise from a partially localised quiver algebra. As before consider
a brane tiling $\Delta$ for a Riemann surface $\Sigma_{g}$ with automorphism
$\varphi:\Sigma_{g}\xrightarrow{\sim}\Sigma_{g}$ of order $n$, giving dual
quiver $Q_{\Delta}=Q$ and potential $W_{\Delta}=W$. Choose generating arrows
$a\in Q_{1}$ for the action of $\varphi$ and choose isomorphism arrows
$r_{i_{u},t}:\varphi^{t}(i_{u})\rightarrow\varphi^{t+1}(i_{u})$ for all orbits
of vertices and $t=0,\ldots,n-2$ giving the quiver $Q^{\\#}$ as explained in
the proof of 3.4. Then define the algebra $\mathbb{C}\widehat{Q}^{\prime}$ to
be the localisation of $\mathbb{C}Q^{\\#}$ with respect to the generating
arrows $\\{a\\}$ (recall that $\mathbb{C}Q^{\prime}$ was the localisation of
$\mathbb{C}Q^{\\#}$ with respect to the isomorphism arrows $r_{i_{u},t}$).
Since we assumed the potential $W^{\prime}$ does not contain any inverse
arrows (more correctly it did not contain both $r_{i_{u},t}$ and
$r_{i_{u},t}^{-1}$ so if it contained an inverse then we simply change the
arrow $r_{i_{u},t}$ in $Q^{\\#}$ to its inverse $r_{i_{u},t}^{-1}$) then
$W^{\prime}$ also gives a potential on $\mathbb{C}\widehat{Q}^{\prime}$.

######  Conjecture 4.2.

Let $B=\mathbb{C}\widehat{Q^{\prime}}$ be the path algebra of
$\widehat{Q}^{\prime}$ with potential $W^{\prime}$ and let
$A=\textnormal{Jac}(B,W^{\prime})$ be the Jacobi algebra of $(B,W^{\prime})$.
Let $\omega\in A$ be a central element and consider the three partition
functions

$\displaystyle\Phi_{A}(t)=\sum_{d=0}^{\infty}\Big{[}\textnormal{Rep}_{d}(A)\Big{]}_{\textnormal{vir}}t^{d}$
$\displaystyle\Phi_{A}^{\omega-\textnormal{nilp}}(t)=\sum_{d=0}^{\infty}\Big{[}\textnormal{Rep}_{d}(A)\Big{]}_{\textnormal{vir}}^{\omega-\textnormal{nilp}}t^{d}$
$\displaystyle\Phi_{A}^{\omega-\textnormal{inv}}(t)=\sum_{d=0}^{\infty}\Big{[}\textnormal{Rep}_{d}(A)\Big{]}_{\textnormal{vir}}^{\omega-\textnormal{inv}}t^{d}\,.$

Then

$\Phi_{A}(t)=\big{(}\Phi_{A}^{\omega-\textnormal{nilp}}(t)\big{)}^{\mathbb{L}}$

and

$\Phi_{A}^{\omega-\textnormal{inv}}(t)=\big{(}\Phi_{A}^{\omega-\textnormal{nilp}}(t)\big{)}^{\mathbb{L}-1}.$

Since $\textnormal{Jac}(\widetilde{Q}^{\prime},W^{\prime})$ is a localisation
of $A$ (with respect to the isomorphism arrows $\\{r_{i_{u},t}\\}$) we try
find a central element $\omega\in A$ such that
$A_{\omega}=\textnormal{Jac}(\widetilde{Q}^{\prime},W^{\prime})$. We expect to
be able to do this because
$\mathbb{C}[\pi_{1}(M_{g,\varphi})]\cong\mathbb{C}[\pi_{1}(\Sigma_{g})\rtimes_{\varphi}\mathbb{Z}]$
and so $\omega=([1],n)\in\pi_{1}(\Sigma_{g})\rtimes_{\varphi}\mathbb{Z}$ is
central, then using 3.17 we have
$\textnormal{Jac}(\widetilde{Q}^{\prime},W^{\prime})\cong\textnormal{Mat}_{m\times
m}(\mathbb{C}[\pi_{1}(M_{g,\varphi})])$ so by construction $A$ should have
such elements. We can then apply 4.2 to write
$\Phi_{\textnormal{Jac}(\widetilde{Q}^{\prime},W^{\prime})}(t)=\Phi_{A_{\omega}}(t)=\Phi_{A}^{\omega-\textnormal{inv}}(t)$
in terms of $\Phi_{A}(t)$. We get that

$\Phi_{\textnormal{Jac}(\widetilde{Q}^{\prime},W^{\prime})}(t)=\big{(}\Phi_{A}(t)\big{)}^{\frac{\mathbb{L}-1}{\mathbb{L}}}.$

Then because 3.17 implies that

$\Phi_{\mathbb{C}[\pi_{1}(M_{g,\varphi})]}(t^{m})=\Phi_{\textnormal{Jac}(\widetilde{Q}^{\prime},W^{\prime})}(t)$

where $m=|Q_{0}|$, it follows that

$\displaystyle\Phi_{\mathbb{C}[\pi_{1}(M_{g,\varphi})]}(t^{m})=\big{(}\Phi_{A}(t)\big{)}^{\frac{\mathbb{L}-1}{\mathbb{L}}}\,.$
(4.2)

Writing $d=(d,\ldots,d)\in\mathbb{N}^{m}$ we have that
$\textnormal{Rep}_{d}(B)\cong M_{d}(B)/G$ with

$M_{d}(B)=\prod_{r_{i_{u},t}}\textnormal{Mat}_{d\times
d}(\mathbb{C})\times\prod_{a\in Q^{\\#}_{1}}\textnormal{GL}_{d}(\mathbb{C})$

which is globally of the form $\mathbb{A}^{r}\times Z$. Let
$f_{d}=\textnormal{Tr}(W^{\prime})_{d}:M_{d}(B)\rightarrow\mathbb{C}$ and
define a $\mathbb{G}_{m}$-action on $M_{d}(B)$ by scaling the non-invertible
matrices only. This gives us a $\mathbb{G}_{m}$-action such that the induced
action on $Z$ is trivial, and because $W^{\prime}$ is homogeneous of degree
$n$ in the isomorphism arrows $\\{r_{i_{u},t}\\}$ it follows that $f_{d}$ is
equivariant of degree $n$. Hence we can apply [[11] Theorem 5.9] to the
virtual motives in the partition function $\Phi_{A}(t)$ reducing the problem
to calculating the motives $[f_{d}^{-1}(0)]$ and $[f_{d}^{-1}(1)]$ for each
$d$.

To see this in action let us consider our running example, namely $\Sigma_{2}$
with $\varphi$ being the rotation by $180^{\circ}$ and with the brane tiling
given in 2.17 and quivers $Q,Q^{\prime}$ and potentials $W,W^{\prime}$ given
in 3.5 and 3.18. Let $B=\mathbb{C}\widehat{Q}^{\prime}$ and
$A=\textnormal{Jac}(\widehat{Q}^{\prime},W^{\prime})$ be the algebras
described above. In particular the generators in $B$ are $\\{a^{\pm 1},b^{\pm
1},c^{\pm 1},d^{\pm 1},e^{\pm 1},r\\}$.

######  Lemma 4.3.

The element $\omega=rere+erer\in A$ is central and
$A_{\omega}=\textnormal{Jac}(\widetilde{Q^{\prime}},W^{\prime})$.

###### Proof.

We first show that
$A_{\omega}=\textnormal{Jac}(\widetilde{Q^{\prime}},W^{\prime}).$ It suffices
to show that $r^{-1}$ exists in $A_{\omega}$. And so because

$er\omega^{-1}e=er(e^{-1}r^{-1}e^{-1}r^{-1}+r^{-1}e^{-1}r^{-1}e^{-1})e=err^{-1}e^{-1}r^{-1}e^{-1}e=r^{-1}$

and $e,r,\omega^{-1}$ are all elements of $A_{\omega}$, then $r^{-1}\in
A_{\omega}$.

Then to show $\omega\in A$ is central, since $\omega$ is not a 0-divisor it
suffices to show that $\omega\in A_{\omega}$ is central. Now

$A_{\omega}=\textnormal{Jac}(\widetilde{Q^{\prime}},W^{\prime})\xrightarrow[\sim]{\Psi}\textnormal{Mat}_{2\times
2}(\mathbb{C}[\pi_{1}(M_{2,\varphi})])$

where, viewing
$\pi_{1}(M_{2,\varphi})\cong\pi_{1}(\Sigma_{2})\rtimes_{\varphi}\mathbb{Z}$,
$\omega$ is sent under this isomorphism to the matrix

$\omega^{\prime}=\begin{pmatrix}([1],-2)&0\\\ 0&([1],-2)\end{pmatrix}.$

Since $\varphi^{2}=\varphi^{-2}=\textnormal{id}_{\Sigma_{2}}$ we have for
$([\alpha],m)\in\pi_{1}(\Sigma_{2})\rtimes_{\varphi}\mathbb{Z}$

$([\alpha],m)\cdot([1],-2)=([\varphi^{-2}(\alpha)],m-2)=([\alpha],-2+m)=([1],-2)\cdot([\alpha],m).$

Hence $\omega^{\prime}$ is central and therefore so is $\omega$. ∎

It follows that we can apply the conjecture in this case for the particular
choice of $\omega=rere+erer$. We have that
$\textnormal{Rep}_{(d,d)}(B)\cong\big{(}\textnormal{Mat}_{d\times
d}(\mathbb{C})\times\textnormal{GL}_{d}^{5}\big{)}/\textnormal{GL}_{d}^{2}$
and $\textnormal{Rep}_{(d,d)}(A)\cong\textnormal{crit}(\widetilde{f_{d}})$
where
$\widetilde{f_{d}}=\textnormal{Tr}(W^{\prime})_{d}:\textnormal{Rep}_{(d,d)}(B)\rightarrow\mathbb{C}$.
Therefore

$\displaystyle\Phi_{\mathbb{C}[\pi_{1}(M_{2,\varphi})]}(t^{2})$
$\displaystyle=\sum_{d=0}^{\infty}\Big{[}\textnormal{Rep}_{d}(\mathbb{C}[\pi_{1}(M_{2,\varphi})])\Big{]}_{\textnormal{vir}}t^{2d}$
$\displaystyle=\sum_{d=0}^{\infty}\Big{[}\textnormal{Rep}_{2d}(A_{\omega})\Big{]}_{\textnormal{vir}}t^{2d}$
$\displaystyle=\sum_{d=0}^{\infty}\Big{[}\textnormal{Rep}_{(d,d)}(A_{\omega})\Big{]}_{\textnormal{vir}}t^{2d}$
$\displaystyle=\sum_{d=0}^{\infty}\Big{[}\textnormal{Rep}_{(d,d)}(A)\Big{]}_{\textnormal{vir}}^{\omega-\textnormal{inv}}t^{2d}$
$\displaystyle=\Phi_{A}^{\omega-\textnormal{inv}}(t)$

where we take the series in the variable $t^{2}$ because a $d$-dimensional
representation of $\mathbb{C}[\pi_{1}(M_{2,\varphi})]$ corresponds to a
$(d,d)$-dimensional representation of the Jacobi algebra