# Floer potentials, cluster algebras and quiver representations

Peter Albers, Maria Bertozzi, Markus Reineke  Peter Albers:
Universität Heidelberg, Institut für Mathematik, Im Neuenheimer Feld 205,
69120 Heidelberg, Germany<EMAIL_ADDRESS>Maria Bertozzi:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150,
44780 Bochum, Germany<EMAIL_ADDRESS>Markus Reineke:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150,
44780 Bochum, Germany<EMAIL_ADDRESS>
###### Abstract.

We use cluster algebras to interpret Floer potentials of monotone Lagrangian
tori in toric del Pezzo surfaces as cluster characters of quiver
representations (see Theorem 4.3 for the precise statement).

##  1\. Introduction

In the seminal work [11], R. Vianna constructed infinitely many pairwise non
Hamiltonian isotopic monotone Lagrangian tori in the complex projective plane.
These tori are naturally indexed by Markov triples (more precisely, by the
triple of squares), and were constructed inductively in [12] by a geometric
mutation procedure along the Markov tree. To distinguish these tori, Vianna
uses information on counts of Maslov index $2$ holomorphic disks contained in
them.
This construction was further conceptualized in [7], where it is put into a
context of cluster-like mutation operations originating in [3]. Vianna’s
mutation operation is extended to mutations of so-called Lagrangian seeds in
del Pezzo surfaces, the main result being a wall-crossing formula, stating
that the Floer potentials of mutated Lagrangian tori are related by an
explicit algebraic mutation rule for Laurent polynomials (see Sections 2.1,
2.2 for a summary).
A formally similar mutation invariance is known in the representation-
theoretic approach to cluster algebras [4], to be recalled in Sections 2.3,
2.4. Namely, the $F$-polynomials of representations of quivers with potentials
(or more precisely the cluster characters), encoding Euler characteristics of
Grassmannians of subrepresentations, are related by the mutation rule in
cluster algebras when the representations are mutated.
This formal similarity, and experiments for the first three Vianna tori, led
the authors to the prediction that, to any of the monotone Lagrangian tori in
toric del Pezzo constructed in [7], one can associate a representation of a
quiver with potential, whose cluster character corresponds naturally to the
Floer potential of the torus.
Indeed, this turns out to be feasible. Inspired by the setup of [6], we
construct (see Section 3) a comparison map from two-variable Laurent
polynomials to cluster algebras, which is compatible with the two algebraic
mutation rules mentioned above. We state and prove an appropriate version of
mutation invariance of cluster characters in Section 2.4.
After realizing the “initial” Landau-Ginzburg seeds of [7] as cluster
characters in Section 4 (under a mild additional hypothesis for the del Pezzos
of degree $6$ and $7$, see the proof of Lemma 4.1), the compatibilities of the
various mutations then almost directly imply the main result Theorem 4.3,
stating that indeed all relevant Floer potentials are cluster characters of
naturally constructed representations.
Acknowledgments: This work was initiated at the workshop “Interplay betweeen
symplectic geometry and cluster theory” at Heidelberg in January 2023. The
authors would like to thank all organizers, speakers and participants for the
inspiring atmosphere. M. R. would like to thank B. de Laporte, B. Keller, D.
Labardini-Fragoso and P. Plamondon for many clarifying discussions on mutation
and cluster characters. The work of all authors is supported by the DFG CRC-
TRR 191 “Symplectic structures in geometry, algebra and dynamics” (281071066).

##  2\. Recollections

###  2.1. Landau-Ginzburg potentials

Here we follow closely the presentation in [7, Section 4]. For real numbers
$a$ we define $[a]_{+}=\max(a,0)$, and make frequent use of the identity

$a=[a]_{+}-[-a]_{+}.$

We consider integral primitive vectors $v\in\mathbb{Z}^{2}$. For $v=(a,b)$ we
write

$v^{\perp}=(b,-a).$

Given another such vector $w=(c,d)$, we consider the standard scalar product

$(v,w)=ac+bd,$

as well as the symplectic form

$\\{v,w\\}=ad-bc$

and note that

$\\{v,w\\}=-(v^{\perp},w).$

The vector $v$ defines a mutation map $\mu_{v}$ on rational functions in
$\mathbb{C}(z_{1},z_{2})$ by

$\mu_{v}(F)(z_{1},z_{2})=F(z_{1}(1+z_{1}^{b}z_{2}^{-a})^{-a},z_{2}(1+z_{1}^{b}z_{2}^{-a})^{-b}).$

On monomials $z^{w}=z_{1}^{c}z_{2}^{d}$ we thus have

$\mu_{v}(z^{w})=z^{w}(1+z^{v^{\perp}})^{-(v,w)}.$

As a tropical version, we define a mutation map on vectors by

$\mu_{v}w=w+[\\{w,v\\}]_{+}v.$

######  Definition 2.1.

A seed is a collection

${\bf s}=(W,v_{1},\ldots,v_{n})$

consisting of a Laurent polynomial $W\in\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}]$,
called a potential, and pairwise different primitive vectors
$v_{1},\ldots,v_{n}\in\mathbb{Z}^{2}$, called directions. The seed ${\bf s}$
is called a Landau-Ginzburg (LG-)seed if $\mu_{v_{i}}W$ is again a Laurent
polynomial for $i=1,\ldots,n$. In this case, we define $\mu_{i}{\bf s}$, the
mutation of ${\bf s}$ in the direction $i=1,\ldots,n$, as the seed

$(\mu_{v_{i}}W,v_{1}^{\prime},\ldots,v_{n}^{\prime}),$

where

$v_{i}^{\prime}=-v_{i},\;v_{j}^{\prime}=\mu_{v_{i}}v_{j},\,j\not=i.$

Note that the operation of mutation of seeds is not involutive, since

$\mu_{-v}\mu_{v}w=w+\\{w,v\\}v$

defines a transvection, and similarly

$\mu_{-v}(\mu_{v}(z^{w}))=\mu_{-v}(z^{w}(1+z^{v^{\perp}})^{-(v,w)})=$
$=z^{w}(1+z^{-v^{\perp}})^{(v,w)}(1+z^{v^{\perp}})^{-(v,w)}=z^{w-(v^{\perp},w)v^{\perp}}=z^{w+\\{v,w\\}v^{\perp}}.$

######  Theorem 2.2.

[3] Any mutation of an LG seed is again an LG seed.

In other words, an LG seed stays an LG seed under arbitrary sequences of
mutations.
To see the concrete examples of LG seeds we will be interested in, we
reproduce the list of LG seeds corresponding to toric del Pezzo surfaces from
[7].

######  Theorem 2.3.

Any of the following combinations of toric del Pezzo surfaces, potentials and
directions constitutes an LG seed:

del Pezzo | potential, directions
---|---
$\mathbb{C}P^{2}$ |  $z_{1}+z_{2}+z_{1}^{-1}z_{2}^{-1}$,
| $(1,1),(-2,1),(1,-2)$
$\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ |  $z_{1}+z_{2}+z_{1}^{-1}+z_{2}^{-1}$,
| $(1,1),(1,-1),(-1,1),(-1,-1)$
${\rm Bl}_{1}\mathbb{C}P^{2}$ |  $z_{1}+z_{2}+z_{1}^{-1}z_{2}^{-1}+z_{1}z_{2}$,
| $(-2,1),(1,-2),(1,0),(0,1)$
${\rm Bl}_{2}\mathbb{C}P^{2}$ |  $z_{1}+z_{2}+z_{1}^{-1}+z_{2}^{-1}+z_{1}^{-1}z_{2}^{-1}$,
| $(1,-1),(-1,1),(-1,0),(0,-1),(1,1)$
${\rm Bl}_{3}\mathbb{C}P^{2}$ |  $z_{1}+z_{2}+z_{1}^{-1}+z_{2}^{-1}+z_{1}z_{2}+z_{1}^{-1}z_{2}^{-1}$,
| $\pm(1,-1),\pm(1,0),\pm(0,1)$

In the following, we take the above LG seeds as initial ones, and consider all
of their mutations:

######  Definition 2.4.

Let $X$ be a toric del Pezzo surface, and let

${\bf s}(X)=(W(X),v_{1},\ldots,v_{n}),$

be the corresponding LG seed in the above table. For ${\bf
i}=(i_{1},\ldots,i_{N})$ a sequence of indices in $\\{1,\ldots,n\\}$ without
repetitions, define

$s_{\bf i}(X)=\mu_{i_{N}}\cdots\mu_{i_{1}}{\bf s}(X)$

as the corresponding mutation of ${\bf s}(X)$, and denote by

$W_{\bf i}(X)=\mu_{i_{N}}\cdots\mu_{i_{1}}W(X)$

its potential.

###  2.2. Floer potentials

To define Floer potentials and their mutation, we continue to follow [7]. We
continue to denote by $X$ a toric del Pezzo surface viewed as Kaehler
manifold. For a monotone Lagrangian submanifold $L\subset X$, we define its
Floer potential (equivalently, disk or Landau-Ginzburg potential)

$W_{L}:{\rm Hom}(H_{1}(L,\mathbb{Z}),\mathbb{C}^{*})\rightarrow\mathbb{C}$

as the count of Maslov index $2$ holomorphic disks in $X$ with boundary on
$L$, weighted by the holonomy of a local system on $L$:

$W_{L}(\rho)=\sum_{\delta}n_{\delta}\rho(\partial\delta),$

where the sum ranges over classes $\delta\in H_{2}(X,L)$ of Maslov index $2$,
the number $n_{\delta}$ is the count of holomorphic disks in class $\delta$,
and $\rho(\partial\delta)$ is the holonomy along the boundary loop of the disk
of the local system determined by $\rho$. More precisely, $n_{\delta}$ is the
degree of the map “evaluation at the marked point”

$\mathcal{M}(L,\delta)\rightarrow L$

on the moduli space of holomorphic disks with boundary on $L$ of Maslov index
$2$ with a single marked point on the boundary of the disk representing the
class $\delta$ (with respect to a generic almost complex structure).
Identifying

$H_{1}(L,\mathbb{Z})/\mbox{torsion}\simeq\mathbb{Z}^{2}$

by chosing a basis, $\partial\delta$ becomes an element $(l_{1},l_{2})$ of
$\mathbb{Z}^{2}$. Monotonicity of $L$ and Gromov compactness then imply that
$W_{L}$, viewed as a function $(\mathbb{C}^{*})^{2}\rightarrow\mathbb{C}$
(using the previous identifications), is a finite sum of terms $n_{\delta}{\bf
z}^{\partial\delta}$ (where ${\bf
z}^{\partial\delta}=z_{1}^{l_{1}}z_{2}^{l_{2}}$) over the classes $\delta$ of
Maslov index $2$, and can thus be viewed as a Laurent polynomial in
$\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}]$,

$W_{L}=\sum_{\delta}n_{\delta}\mathbf{z}^{\partial\delta}.$

For a monotone Lagrange torus $L\subset X$ and a Lagrangian disk $D\subset X$
with boundary on $L$ fulfilling suitable transversality conditions, there is
an operation of mutation of $L$ in direction $D$, producing a mutated
Lagrangian torus $\mu_{D}L$ and a mutated holomorphic disk $\mu_{D}D$. This
operation is involutive up to a Hamiltonian isotopy.
A Lagrangian seed $(L,\\{D_{i}\\}_{i})$ in $X$ consists of a monotone
Lagrangian torus $L$ and embedded Lagrangian holomorphic disks fulfilling
suitable transversality conditions. The above mutation procedure can be
extended to provide mutations

$\mu_{i}(L,\\{D_{j}\\}_{j})=(\mu_{D_{i}}L,\\{\mu_{D_{i}}D_{j}\\}_{j})$

of Lagrangian seeds in arbitrary directions.
To a Lagrangian seed $(L,\\{D_{i}\\}_{i})$, an LG seed

$(W_{L},([\partial D_{i}]^{\perp})_{i})$

can be assigned via the isomorphism
$H_{1}(L,\mathbb{Z})/\mbox{torsion}\simeq\mathbb{Z}^{2}$.

######  Theorem 2.5.

The assignment of an LG seed to a Lagrangian seed is compatible with the
mutation operations. Moreover, there exists an initial Lagrangian seed
$(L(X),\\{D_{i}(X)\\}_{i})$ realizing the LG seeds ${\bf s}(X)$ of Theorem
2.3.

We can then define iterated mutated Lagrangian seeds

$(L_{\bf i}(X),\\{D_{j}^{\bf
i}(X)\\}_{j})=\mu_{i_{N}}\cdots\mu_{i_{1}}(L(X),\\{D_{i}(X)\\}_{i})$

for any sequence ${\bf i}=(i_{1},\ldots,i_{N})$ without repetitions. By the
previous theorem, we thus have

######  Corollary 2.6.

For all sequences ${\bf i}$ as above, we have an equality of LG seeds

$(W_{L_{\bf i}(X)},\\{[\partial D_{j}^{\bf i}(X)]\\}_{j})={\bf s}_{\bf i}(X).$

###  2.3. Cluster algebras

We now turn to the purely algebraic part and recall some cluster algebra
terminology following [5]. Let $B=(b_{i,j})$ be a skew-symmetric $n\times
n$-matrix. We define the mutation $\mu_{i}B=B^{\prime}=(b_{j,k}^{\prime})$ in
direction $i=1,\ldots,n$ of $B$ by

$b_{j,k}^{\prime}=b_{j,k}\mbox{ if }j=i\mbox{ or }k=i,$
$b_{j,k}^{\prime}=b_{j,k}+[b_{j,i}]_{+}[b_{i,k}]_{+}-[-b_{j,i}]_{+}[-b_{i,k}]_{+}\mbox{
if }j\not=i\not=k.$

We define the mutation of $x$-variables as the automorphism of the rational
function field $\mathbb{C}(x_{1},\ldots,x_{n})$ given by
$\mu_{i}(x_{j})=x_{j}^{\prime}$ for

$x_{i}^{\prime}={x_{i}}^{-1}\prod_{j}{x_{j}}^{[b_{i,j}]_{+}}(1+\prod_{j}{x_{j}}^{-b_{i,j}})\mbox{
and }x_{j}^{\prime}=x_{j},\,j\not=i,$

a variant of the more common

$x_{i}^{\prime}=x_{i}^{-1}(\prod_{j}x_{j}^{[b_{i,j}]_{+}}+\prod_{j}x_{j}^{[b_{j,i}]_{+}}).$

We define the $y$-variables by

$y_{i}=\prod_{j}x_{j}^{b_{j,i}},$

and similarly for $y_{i}^{\prime}$, so that the previous mutation rule reads

$x_{i}^{\prime}=x_{i}^{-1}\prod_{j}{x_{j}}^{[b_{i,j}]_{+}}(1+{y_{i}}).$

The mutation of $y$-variables is then given by

$y_{i}^{\prime}={y_{i}}^{-1}\mbox{ and
}y_{j}^{\prime}={y_{j}}{y_{i}}^{[b_{i,j}]_{+}}(1+{y_{i}})^{-b_{i,j}},\,j\not=i.$

###  2.4. Representations of quivers with potentials

Turning to the quiver theoretic approach to cluster algebras and mutation, we
introduce representations of quivers with potential following [4]. Given a
matrix $B$ as in the previous subsection, we associate to it a quiver $Q=Q(B)$
with vertices $i=1,\ldots,n$, such that

$\mbox{ for all $i,j\in Q_{0}$, there are }[-b_{i,j}]_{+}\mbox{ arrows from
$i$ to $j$}.$

Note that $Q(B)$ has, by definition, no loops or two-cycles. We have

$Q(\mu_{i}B)=\mu_{i}Q(B),$

where the mutation $\mu_{i}Q$ of a quiver at a vertex $i$ is defined by
reversing all arrows incident with $i$, adding an arrow

$[\beta\alpha]:j\rightarrow k$

for every pair of arrows

$j\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}i\stackrel{{\scriptstyle\beta}}{{\rightarrow
k}},$

and deleting all resulting two-cycles.
We denote by $\widehat{\mathbb{C}Q}$ the completed (with respect to path
length) path algebra of $Q$, and by $\mathfrak{m}$ its maximal ideal. The
factor $\widehat{\mathbb{C}Q}/[\widehat{\mathbb{C}Q},\widehat{\mathbb{C}Q}]$
by (additive) commutators is spanned by cyclic equivalence classes of formal
sums of oriented cycles. An element $S$ in this space is called a potential
for $Q$, for which cyclic derivatives $\partial_{\alpha}S$ along the arrows
$\alpha$ in $Q$ are defined, as well as double cyclic derivatives
$\partial_{\beta\alpha}S$. We will never need their explicit definition in the
following and just refer to [4, (4.3),(4.4)].
For a sufficiently general (non-degenerate in the terminology of [4])
potential $S$ on $Q$, one can naturally define a mutation $\mu_{i}S$ as a
potential on $\mu_{i}Q$.
We consider representations $V$ of $Q$ which satisfy the Jacobian relations
that all $\partial_{\alpha}S$ are represented by zero in $V$, thus

$V_{\partial_{\alpha}S}=0,$

which we call representations of the quiver with potential $(Q,S)$. Moreover,
we require all such representations to be nilpotent, that is, every oriented
cycle $\omega$ is represented by a nilpotent endomorphism $V_{\omega}$.
Slightly more generally, we consider decorated representations $(V,V^{-})$,
consisting of a representation $V$ of $(Q,S)$, together with vector spaces
$V^{-}_{i}$ for $i=1,\ldots,n$. As a special case, we consider the negative
simple decorated representation $S_{i}^{-}$: it consists of the zero quiver
representation, and the additional vector spaces are given by
$(S_{i}^{-})_{j}^{-}=0$ for $j\not=i$ and $(S_{i}^{-})^{-}_{i}=\mathbb{C}$.
For every (decorated) representation $V$ of $(Q,S)$ (the potential $S$ still
assumed to be non-degenerate) and all $i\in Q_{0}$, a mutation $\mu_{i}V$ is
defined in [4], which is a (decorated) representation of
$(\mu_{i}Q,\mu_{i}S)$. The operation of mutation is involutive up to so-called
right equivalence, that is, $\mu_{i}\mu_{i}V$ is isomorphic to $\varphi^{*}V$
for some automorphism $\varphi$ of $\widehat{\mathbb{C}Q}$ fixing $S$.

###  2.5. $F$-polynomials and cluster characters

For a (decorated) representation $V$ of a quiver $Q$ and a dimension vector
${\bf e}$ with entries $e_{i}\leq\dim V_{i}$ for all $i$ we define the quiver
Grassmannian ${\rm Gr}_{\bf e}(V)$ as the complex projective variety of ${\bf
e}$-dimensional subrepresentations of $V$, viewed for example as a Zariski-
closed subset of the product $\prod_{i}{\rm Gr}_{e_{i}}(V_{i})$ of ordinary
Grassmannians. We define the $F$-polynomial of $V$ encoding the topological
Euler characteristic $\chi({\rm Gr}_{\bf e}(V))$ of all quiver Grassmannians:

$F_{V}(u)=\sum_{\bf e}\chi({\rm Gr}_{\bf
e}(V))\prod_{i}u_{i}^{e_{i}}\in\mathbb{C}[u_{1},\ldots,u_{n}].$

We define the ${\bf g}$\- and ${\bf h}$-vectors of $V$:
Fixing a vertex $i$ of $Q$, any decorated representation $V$ gives rise to a
triangle of linear maps

${V_{i}}$${V_{i}^{\rm in}}$${V_{i}^{\rm
out}}$$\scriptstyle{\beta_{i}}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{\gamma_{i}}$

where

$V_{i}^{\rm in}=\bigoplus_{\alpha:j\rightarrow i}V_{j},\;V_{i}^{\rm
out}=\bigoplus_{\beta:i\rightarrow j}V_{j},$
$\alpha_{i}=(V_{\alpha})_{\alpha:j\rightarrow
i},\;\beta_{i}=(V_{\beta})_{\beta:i\rightarrow j},$

and the $(\alpha,\beta)$-component of $\gamma_{i}$ is given by

$(\gamma_{i})_{(\alpha,\beta)}=V_{\partial_{\beta\alpha}S}.$

We then define

$h_{i}(V)=-\dim{\rm Ker}(\beta_{i})$

and

$g_{i}(V)=\dim{\rm Ker}(\gamma_{i})-\dim V_{i}+\dim V_{i}^{-}.$

We have the following relation between ${\bf g}$\- and ${\bf h}$-vectors:

$g_{i}(V)=h_{i}(V)-h_{i}(\mu_{i}V).$

The ${\bf g}$-vectors fulfill the mutation rule

$g_{i}(\mu_{i}V)=-g_{i}(V),\;g_{j}(\mu_{i}V)=g_{j}(V)+[b_{j,i}]_{+}g_{i}(V)-b_{j,i}h_{i}(V),\,i\not=j.$

We have the following mutation rule for $F$-polynomials:

######  Lemma 2.7.

[4, Lemma 5.2] We have

$(1+y_{i})^{h_{i}(V)}F_{V}(y)=(1+{y_{i}^{\prime}})^{h_{i}(\mu_{i}(V))}F_{\mu_{i}V}(y^{\prime}).$

The following is the central definition for our purposes, originating in [2],
and written in this form in [9, Remark 5.2]:

######  Definition 2.8.

We define the cluster character of $V$ by

${\rm CC}_{V}(x)=x^{{\bf
g}(V)}F_{V}(y)\in\mathbb{C}[x_{1}^{\pm},\ldots,x_{n}^{\pm}].$

For two decorated representations $V$ and $W$ of $Q$, we have

$F_{V\oplus W}(x)=F_{V}(x)\cdot F_{W}(x)$

and

${\bf g}(V\oplus W)={\bf g}(V)+{\bf g}(W),$

so that

${\rm CC}_{V\oplus W}(x)={\rm CC}_{V}(x)\cdot{\rm CC}_{W}(x).$

It is thus natural to extend cluster characters to virtual representations,
that is, elements of the split Grothendieck group $K_{0}^{\oplus}$ of the
category of decorated representations. Namely, every element of this group can
be written as a difference $[V]-[W]$ of isomorphism classes of
representations, and we define

${\rm CC}_{[V]-[W]}(x)=\frac{{\rm CC}_{V}(x)}{{\rm
CC}_{W}(x)}\in\mathbb{C}(x_{1},\ldots,x_{n}).$

Note also that the mutation operation $\mu_{i}$ naturally extends to virtual
representations.

######  Lemma 2.9.

The cluster character of virtual representations is invariant under mutation,
that is,

${\rm CC}_{V}(x)={\rm CC}_{\mu_{i}V}(x^{\prime}).$

This lemma is essentially [8, Corollary 4.14]. Due to different conventions in
notation, we give a complete proof here. We can assume $V$ to be a decorated
representation. Using the mutation rule for $F$-polynomials, the relation
between ${\bf g}$\- and ${\bf h}$-vectors and the definition of $y$-variables,
we have

$x^{{\bf g}(V)}F_{V}(y)=x^{{\bf
g}(V)}(1+y_{i})^{-h_{i}(V)}(1+y_{i})^{h_{i}(V)}F_{V}(y)=$ $=x^{{\bf
g}(V)}(1+y_{i})^{-h_{i}(V)}(1+y_{i}^{-1})^{h_{i}(\mu_{i}V)}F_{\mu_{i}V}(y^{\prime})=$
$=x^{{\bf
g}(V)}(1+y_{i})^{-h_{i}(V)+h_{i}(\mu_{i}V)}y_{i}^{-h_{i}(\mu_{i}V)}F_{\mu_{i}V}(y^{\prime})=$
$=x^{{\bf
g}(V)}(1+y_{i})^{-g_{i}(V)}\prod_{j\not=i}x_{j}^{-b_{j,i}h_{i}(\mu_{i}V)}F_{\mu_{i}V}(y^{\prime})=$
$=x_{i}^{g_{i}(V)}\prod_{j\not=i}x_{j}^{g_{j}(V)-b_{j,i}h_{i}(\mu_{i}V)}(1+y_{i})^{-g_{i}(V)}F_{\mu_{i}V}(y^{\prime}).$

We rewrite the exponent of $x_{j}$ using the mutation rule for ${\bf
g}$-vectors as

$g_{j}(V)-b_{j,i}h_{i}(\mu_{i}V)=g_{j}(V)-b_{j,i}h_{i}(V)+b_{j,i}g_{i}(V)=$
$=g_{j}(\mu_{i}V)-[b_{j,i}]_{+}g_{i}(V)+b_{j,i}g_{i}(V)=$
$=g_{j}(\mu_{i}V)-[b_{i,j}]_{+}g_{i}(V)$

and thus $x^{{\bf g}(V)}F_{V}(y)=$

$=x_{i}^{g_{i}(V)}\prod_{j\not=i}x_{j}^{g_{j}(\mu_{i}V)-[b_{i,j}]_{+}g_{i}(V)}(1+y_{i})^{-g_{i}(V)}F_{\mu_{i}V}(y^{\prime})=$
$=(x_{i}^{-1}\prod_{j\not=i}x_{j}^{[b_{i,j}]_{+}}(1+y_{i}))^{g_{i}(\mu_{i}V)}\prod_{j\not=i}x_{j}^{g_{j}(\mu_{i}V)}F_{\mu_{i}V}(y^{\prime})=$
$=(x^{\prime})^{{\bf g}(\mu_{i}V)}F_{\mu_{i}V}(y^{\prime}),$

proving the lemma.

##  3\. LG seeds and cluster algebras

To an LG seed ${\bf s}=(W,v_{1},\ldots,v_{n})$ we associate the skew-symmetric
matrix $B=B({\bf s})=(b_{i,j})$ given by

$b_{i,j}=\\{v_{i},v_{j}\\}.$

Note that it has rank at most two. This definition is compatible with
mutation, that is,

$B(\mu_{i}({\bf s}))=\mu_{i}(B({\bf s})).$

Namely, we have, in the notation of Definition 2.1:

$\\{v_{i}^{\prime},v_{k}^{\prime}\\}=\\{-v_{i},v_{k}+[\\{v_{k},v_{i}\\}]_{+}v_{i}\\}=-\\{v_{i},v_{k}\\},$

for $i\not=k$, and similarly for $\\{v_{j}^{\prime},v_{i}^{\prime}\\}$ and
$j\not=i$, whereas, for $j\not=i\not=k$, we have

$\\{v_{j}^{\prime},v_{k}^{\prime}\\}=\\{v_{j}+[\\{v_{j},v_{i}\\}]_{+}v_{i},v_{k}+[\\{v_{k},v_{i}\\}]_{+}v_{i}\\}=$
$=\\{v_{j},v_{k}\\}+[\\{v_{k},v_{i}\\}]_{+}\\{v_{j},v_{i}\\}+[\\{v_{j},v_{i}\\}]_{+}\\{v_{i},v_{k}\\}=$
$=b_{j,k}+[b_{j,i}]_{+}b_{i,k}+b_{j,i}[b_{k,i}]_{+}=b_{j,k}+[b_{j,i}]_{+}[b_{i,k}]_{+}-[-b_{j,i}]_{+}[-b_{i,k}]_{+}.$

Inspired by [6], we define

######  Definition 3.1.

The comparison map

$\Phi=\Phi_{\bf
s}:\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}]\rightarrow\mathbb{C}[x_{1}^{\pm},\ldots,x_{n}^{\pm}]$

is defined by

$\Phi_{\bf s}(z^{v})=\prod_{i}x_{i}^{-(v,v_{i})}.$

######  Lemma 3.2.

The comparison map $\Phi_{\bf s}$ is compatible with the mutations of Sections
2.1 and 2.3, respectively, that is, we have

$\mu_{i}\circ\Phi_{\bf s}=\Phi_{\mu_{i}{\bf s}}\circ\mu_{i}.$

Namely, we have

$\mu_{i}(\Phi_{\bf s}(z^{v}))=\mu_{i}(\prod_{j}x_{j}^{-(v,v_{j})})=$
$=({x_{i}}^{-1}\prod_{j}{x_{j}}^{[b_{i,j}]_{+}}(1+y_{i}))^{-(v,v_{i})}\prod_{j\not=i}{x_{j}}^{-(v,v_{j})}=$
$={x_{i}}^{(v,v_{i})}\prod_{j\not=i}{x_{j}}^{-(v,v_{j})-[b_{i,j}]_{+}(v,v_{i})}(1+y_{i})^{-(v,v_{i})}.$

On the other hand, we have

$\Phi_{\mu_{i}{\bf s}}(\mu_{i}(z^{v}))=\Phi_{\mu_{i}{\bf
s}}(z^{v}(1+z^{v_{i}^{\perp}})^{-(v_{i},v)})=$
$=\prod_{j}{x_{j}}^{-(v,v_{j}^{\prime})}(1+\prod_{j}{x_{j}}^{-(v_{i}^{\perp},v_{j}^{\prime})})^{-(v_{i},v)}.$

Now

$\prod_{j}{x_{j}}^{-(v_{i}^{\perp},v_{j}^{\prime})}=\prod_{j}{x_{j}}^{\\{v_{i},v_{j}^{\prime}\\}}=\prod_{j}{x_{j}}^{b_{i,j}}={y_{i}^{-1}},$

so that

$(1+\prod_{j}{x_{j}}^{-(v_{i}^{\perp},v_{j}^{\prime})})^{-(v_{i},v)}=(1+{y_{i}}^{-1})^{-(v_{i},v)}={y_{i}}^{(v_{i},v)}(1+{y_{i}})^{-(v_{i},v)},$

and

$\prod_{j}{x_{j}}^{-(v,v_{j}^{\prime})}={x_{i}}^{(v,v_{i})}\prod_{j\not=i}{x_{j}}^{-(v,v_{j})-[b_{j,i}]_{+}(v,v_{i})},$

which combines to $\Phi_{\mu_{i}{\bf s}}(\mu_{i}(z^{v}))=$

$={x_{i}}^{(v,v_{i})}\prod_{j\not=i}{x_{j}}^{-(v,v_{j})-[b_{j,i}]_{+}(v,v_{i})}{y_{i}}^{(v_{i},v)}(1+{y_{i}})^{-(v_{i},v)}=$
$={x_{i}}^{(v,v_{i})}\prod_{j\not=i}{x_{j}}^{-(v,v_{j})-[b_{j,i}]_{+}(v,v_{i})-b_{i,j}(v_{i},v)}(1+{y_{i}})^{-(v_{i},v)}=$
$={x_{i}}^{(v,v_{i})}\prod_{j\not=i}{x_{j}}^{-(v,v_{j})-[b_{i,j}]_{+}(v,v_{i})}(1+{y_{i}})^{-(v_{i},v)}=$
$=\mu_{i}(\Phi_{\bf s}(z^{v}))$

as claimed.

##  4\. Floer potentials as $F$-polynomials

For each of the LG seeds of Theorem 2.3, we will now describe the quiver
corresponding to the $B$-matrix induced by the seed:

$\left[\begin{array}[]{rrr}0&3&-3\\\ -3&0&3\\\
3&-3&0\end{array}\right],\;\left[\begin{array}[]{rrrr}0&-2&2&0\\\ 2&0&0&-2\\\
-2&0&0&2\\\
0&2&-2&0\end{array}\right],\;\left[\begin{array}[]{rrrr}0&3&-1&-2\\\
-3&0&2&1\\\ 1&-2&0&1\\\ 2&-1&-1&0\end{array}\right],$
$\left[\begin{array}[]{rrrrr}0&0&-1&-1&2\\\ 0&0&1&1&-2\\\ 1&-1&0&1&-1\\\
1&-1&-1&0&1\\\
-2&2&1&-1&0\end{array}\right],\;\left[\begin{array}[]{rrrrrr}0&0&1&-1&1&-1\\\
0&0&-1&1&-1&1\\\ -1&1&0&0&1&-1\\\ 1&-1&0&0&-1&1\\\ -1&1&-1&1&0&0\\\
1&-1&1&-1&0&0\end{array}\right].$
$3$$2$$1$$2$$3$$1$$4$$4$$2$$1$$3$$4$$5$$1$$3$$2$$4$$5$$6$$3$$1$$2$

For each of these quivers $Q(X)$, associated to a toric del Pezzo surface $X$,
we choose a non-degenerate potential $S(X)$. For any sequence ${\bf
i}=(i_{1},\ldots,i_{N})$ of vertices in $Q(X)$ without repetitions, we define

$(Q_{\bf i}(X),S_{\bf i}(X))=\mu_{i_{N}}\cdots\mu_{i_{1}}(Q(X),S(X)).$

Since the mutation operations on LG seeds, skew-symmetric matrices and
quivers, respectively, are compatible, we have

$Q(B({\bf s}_{\bf i}(X)))=Q_{\bf i}(X).$

For example, for the quiver associated to $X=\mathbb{C}P^{2}$, the quivers
$Q_{\bf i}(X)$ are (all) Markov quivers, that is, triangles with $3a$, $3b$,
$3c$ parallel arrows, respectively, where $(a,b,c)$ is a Markov triple.
We now claim:

######  Lemma 4.1.

For every toric del Pezzo surface $X$, there is a virtual representation
$P(X)$ of $(Q(X),S(X))$ of the form $P(X)=[V]-[(S_{i}^{-})^{c}]$ for
nonnegative $c$, such that

${\rm CC}_{P(X)}(x)=\Phi_{{\bf s}(X)}(W(X)).$

In other words, the LG potential $W(X)\in\mathbb{C}[z_{1}^{\pm},z_{2}^{\pm}]$
associated to $X$ in Theorem 2.3 corresponds to the cluster character ${\rm
CC}_{P(X)}(x)$ of $P(X)$, which belongs to
$\mathbb{C}[x_{1}^{\pm},\ldots,x_{n}^{\pm}]$, under the map $\Phi_{{\bf
s}(X)}$.

Note that the way they are depicted, for every of the above quivers there is a
canonical outer cycle containing every vertex once. We fix a vertex $i_{0}$ of
$Q$ arbitrarily.
We then define a representation $V$ as follows: we have $V_{i_{0}}$=0 and
$V_{i}=\mathbb{C}$ for all $i\not=i_{0}$. Moreover, we have $V^{-}_{i}=0$ for
all $i$. We have $V_{\alpha}=0$ whenever $\alpha$ does not belong to the outer
cycle, and $V_{\alpha}\not=0$ whenever $\alpha$ belongs to the outer cycle and
is not incident with $i_{0}$. Recalling the maps $\gamma_{i}$ used in the
definition of ${\bf g}$-vectors, we moreover define $c=\dim{\rm
Ker}(\gamma_{i_{0}})+1$.
We claim that (under mild hypotheses on the non-degenerate potential in the
cases $X={\rm Bl}_{2}\mathbb{C}P^{2}$ and ${\rm Bl}_{3}\mathbb{C}P^{2}$) the
above $V$ defines a representation of the quiver with potential $(Q(X),S(X))$,
and that $P(X)=[V]-[(S_{1}^{-})^{c}]$ fulfills the assumption of the lemma. We
fix, for example, $i_{0}=1$.
In the case $X=\mathbb{C}P^{2}$, the representation $V$ clearly fulfills the
Jacobian relations since there are no consecutive arrows supporting non-zero
maps in $V$. We have

$y_{1}=x_{2}^{-3}x_{3}^{3},\;y_{2}=x_{1}^{3}x_{3}^{-3},\;y_{3}=x_{1}^{-3}x_{2}^{3},$

and thus

$F_{P}(y)=F_{V}(y)=1+y_{2}+y_{2}y_{3}=1+x_{1}^{3}x_{3}^{-3}+x_{2}^{3}x_{3}^{-3}.$

The $\gamma$-maps are of the form $0=\gamma_{2}:\mathbb{C}^{3}\rightarrow 0$
and $0=\gamma_{3}:0\rightarrow\mathbb{C}^{3}$, so that

${\bf g}_{V}=(\dim{\rm Ker}(\gamma_{1}),2,-1),$

and thus

${\bf g}_{P}=(-1,2,-1),$

yielding

${\rm
CC}_{P}(x)=x_{1}^{-1}x_{2}^{2}x_{3}^{-1}(1+x_{1}^{3}x_{3}^{-3}+x_{2}^{3}x_{3}^{-3})=x_{1}^{-1}x_{2}^{2}x_{3}^{-1}+x_{1}^{-1}x_{2}^{-1}x_{3}^{2}+x_{1}^{2}x_{2}^{-1}x_{3}^{-1}.$

This indeed coincides with $\Phi(z_{1}+z_{2}+z_{1}^{-1}z_{2}^{-1})$ since, by
definition,

$\Phi(z_{1})=x_{1}^{-1}x_{2}^{2}x_{3}^{-1},\;\Phi(z_{2})=x_{1}^{-1}x_{2}^{-1}x_{3}^{2}.$

The verification in the cases $X=\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ and
$X={\rm Bl}_{1}\mathbb{C}P^{2}$ is entirely similar.
For the remaining two cases, we first depict representations of the relevant
quivers:

$V_{4}$$V_{5}$$V_{1}$$V_{3}$$V_{2}$$\scriptstyle{M^{1}_{1}}$$\scriptstyle{M^{1}_{2}}$$\scriptstyle{M^{4}_{1}}$$\scriptstyle{M^{4}_{2}}$$\scriptstyle{M^{3}_{1}}$$\scriptstyle{M^{2}_{1}}$$\scriptstyle{M^{2}_{2}}$$\scriptstyle{M^{5}_{2}}$$\scriptstyle{M^{5}_{3}}$$\scriptstyle{M^{5}_{1}}$$V_{4}$$V_{5}$$V_{6}$$V_{3}$$V_{1}$$V_{2}$$\scriptstyle{M^{1}_{1}}$$\scriptstyle{M^{1}_{2}}$$\scriptstyle{M^{3}_{1}}$$\scriptstyle{M^{5}_{2}}$$\scriptstyle{M^{4}_{1}}$$\scriptstyle{M^{6}_{2}}$$\scriptstyle{M^{2}_{1}}$$\scriptstyle{M^{2}_{2}}$$\scriptstyle{M^{6}_{1}}$$\scriptstyle{M^{5}_{1}}$$\scriptstyle{M^{3}_{2}}$$\scriptstyle{M^{4}_{2}}$

In the case $X={\rm Bl}_{2}\mathbb{C}P^{2}$, there are two new phenomena to be
discussed. First of all, we assume that the potential $S$ does not involve the
outer $5$-cycle with non-zero coefficient (otherwise $V$ had to fulfill a
Jacobian relation which might force the compositions
$M_{1}^{2}M_{1}^{3}M_{1}^{4}$ and $M_{2}^{2}M_{1}^{3}M_{1}^{4}$ to be
proportional). Moreover, in computing the ${\bf g}$-vector, we will see why it
is necessary to represent the arrows outside the outer cycle by zero. Namely,
this forces the $\gamma$-maps
$\gamma_{2}:\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}$ and
$\gamma_{3}:\mathbb{C}^{2}\rightarrow\mathbb{C}$ to be zero, whereas
$\gamma_{4}:\mathbb{C}^{2}\rightarrow\mathbb{C}$ is non-zero since it involves
a linear combination of $M_{1}^{2}$ and $M_{2}^{2}$ which we can, in general,
assume to be non-zero. The resulting ${\bf g}$-vector is thus

${\bf g}_{V}=(\dim{\rm Ker}(\gamma_{1}),1,1,0,-1),$

and thus

${\bf g}_{P}=(-1,1,1,0,-1).$

The rest of the verification is again straight-forward, so that both ${\rm
CC}_{P}(x)$ and $\Phi(W(x))$ coincide with

$x_{1}^{-1}x_{2}x_{3}x_{5}^{-1}+x_{1}x_{2}^{-1}x_{4}x_{5}^{-1}+x_{1}x_{2}^{-1}x_{3}^{-1}x_{5}+x_{1}^{-1}x_{2}x_{4}^{-1}x_{5}+x_{3}^{-1}x_{4}^{-1}x_{5}^{2}.$

The verification in the final case $X={\rm Bl}_{3}\mathbb{C}P^{2}$ is then
again similar: we assume that the potential $S$ does not involve the $5$-cycle
through all vertices except $1$ with non-zero coefficient (which could
otherwise force the composition $M_{1}^{5}M_{1}^{2}M_{1}^{4}M_{1}^{6}$ to be
zero, resulting in the wrong $F$-polynomial of $V$). This finishes the proof
of the lemma.
We can now mutate the virtual representation $P(X)$ to all mutated quivers
with potentials:

######  Definition 4.2.

For every sequence of vertices ${\bf i}=(i_{1},\ldots,i_{N})$ without
repetition, define $P_{\bf i}(X)$ as the representation of $(Q_{\bf
i}(X),S_{\bf i}(X))$ given by

$P_{\bf i}(X)=\mu_{i_{N}}\cdots\mu_{i_{1}}P(X).$

The main result of this paper now follows immediately:

######  Theorem 4.3.

Every Floer potential of a monotone Lagrangian torus $L_{\bf i}(X)$ appearing
in a Lagrangian seed with associated LG seed $s_{\bf i}(X)$ corresponds to the
cluster character ${\rm CC}_{P_{\bf i}(X)}(x)$ of the virtual representation
$P_{\bf i}(X)$ of $(Q_{\bf i}(X),S_{\bf i}(X))$ under the comparison map
$\Phi_{{\bf s}_{\bf i}(X)}$, that is,

$\Phi_{{\bf s}_{\bf i}(X)}(W_{L_{\bf i}(X)})={\rm CC}_{P_{\bf i}(X)}(x).$

Indeed, the claimed equality holds for the empty sequence ${\bf i}$, the right
hand side is mutation invariant by Lemma 2.9 and the definition of the $P_{\bf
i}(X)$, and the left hand side is mutation invariant by Lemma 3.2, the
definition of the $W_{\bf i}(X)$, and the mutation compatibility of Floer
potentials of Theorem 2.5.
Remark: The authors expect that, using [10], explicit (graded) potentials for
the above five quivers $Q(X)$ can be constructed, so that the representations
$P_{\bf i}(X)$, and conjecturally also their cluster characters, can be
determined more explicitly as graded representations of the $Q_{\bf i}(X)$.

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