# Geometric model for module categories of Dynkin quivers via hearts of total
stability conditions

Wen Chang  Wen Chang: School of Mathematics and Statistics, Shaanxi Normal
University, Xi’an 710062, China<EMAIL_ADDRESS>,  Yu Qiu  Yu Qiu: Yau
Mathematical Sciences Center and Department of Mathematical Sciences, Tsinghua
University, 100084 Beijing, China. & Beijing Institute of Mathematical
Sciences and Applications, Yanqi Lake, Beijing, China<EMAIL_ADDRESS>and
Xiaoting Zhang  Xiaoting Zhang: Beijing Advanced Innovation Center for Imaging
Theory and Technology, Academy for Multidisciplinary Studies, Capital Normal
University, Beijing 100048, China<EMAIL_ADDRESS>Dedicated to
Alastair King on the occasion of his sixtieth birthday

###### Abstract.

We derive a geometric model for the module category
$\operatorname{mod}\mathbf{k}Q$ of a Dynkin quiver $Q$ via the heart of a
total stability condition on the bounded derived category of
$\operatorname{mod}\mathbf{k}Q$. As an application, we prove Reineke’s
conjecture that there is a stability function on
$\operatorname{mod}\mathbf{k}Q$ making any indecomposables stable.

Key words: total stability condition, Dynkin diagram, module category,
geometric model, Reineke’s conjecture

###### Contents

  1. 1 Introduction
    1. 1.1 Stability structure on abelian and triangulated categories
    2. 1.2 Reineke’s conjecture and total stability
    3. 1.3 Geometric model for module category of a Dynkin diagram
  2. 2 Preliminaries
  3. 3 Hearts and triangles
    1. 3.1 Hearts of total stability conditions
    2. 3.2 Intersections as homomorphisms
  4. 4 Geometric model of hearts
    1. 4.1 Type $A_{n}$
    2. 4.2 Type $D_{n}$
    3. 4.3 Exceptional type $E_{6}$
    4. 4.4 Exceptional types $E_{n}$
  5. 5 Reineke’s conjecture
    1. 5.1 Type $A$ and $D$
    2. 5.2 Exceptional type
    3. 5.3 Summary
  6. A List of examples of stable $30$-gons in type $E_{8}$

##  1\. Introduction

The stability structures, e.g. King’s $\theta$-stability on abelian categories
and Bridgeland’s stability conditions on triangulated categories, play an
important role in many areas, such as, geometric invariant theory, Donaldson-
Thomas theory, cluster theory, etc.

In the prequel [QZ22], Qiu-Zhang give a geometric model for the root category
of the bounded derived category $\operatorname{\mathcal{D}}_{\infty}(Q)$
associated to a Dynkin diagram and describe the space $\operatorname{ToSt}(Q)$
of all total stability conditions on $\operatorname{\mathcal{D}}_{\infty}(Q)$.
In this paper, we describe the heart $\mathcal{H}_{\sigma}$ for
$\sigma\in\operatorname{ToSt}(Q)$ on the geometric model and hence obtain a
model for the module category of certain Dynkin quiver whose underlying
diagram is the same as $Q$. In fact, we show that such a heart can realize the
module category of a Dynkin quiver with any given orientation. As an
application, we prove Reineke’s conjecture that, for any Dynkin quiver $Q$,
there is a stability function on $\operatorname{mod}\mathbf{k}Q$ such that all
indecomposable objects are stable.

###  1.1. Stability structure on abelian and triangulated categories

An algebraic version of the notion of stability in geometric invariant theory
is introduced by King [K94] in the representation theory of finite-dimensional
algebras. The King’s $\theta$-stability is generalized by Rudakov [R97] to any
abelian category. A stability function on the module category
$\mathcal{H}(Q)\colon=\operatorname{mod}\mathbf{k}Q$ of a quiver $Q$ is a
group homomorphism $Z\colon K\mathcal{H}(Q)\to\mathbb{C}$. The associated
slope function is

$\mu_{Z}(\alpha)=\frac{\operatorname{Im}Z(\alpha)}{\operatorname{Re}Z(\alpha)}.$

An object $M$ is $Z$-stable (resp. $Z$-semistable) if and only if

$\mu_{Z}(L)<\mu_{Z}(M),\quad\text{(resp. $\mu_{Z}(L)\leq\mu_{Z}(M)$)}$

for any subobject $0\neq L\subsetneq M$.

Later, motivated by $\Pi$-stability from string theory, Bridgeland [B07]
introduce the notion of stability conditions on a triangulated category. To be
more precise, a stability condition $\sigma=(Z,\mathcal{P})$ on a triangulated
category $\operatorname{\mathcal{D}}$ consists of a group homomorphism
$Z\colon K\operatorname{\mathcal{D}}\to\mathbb{C}$, called the central charge,
and an $\mathbb{R}$-collection of abelian subcategories $\mathcal{P}(\phi)$ of
$\operatorname{\mathcal{D}}$, known as the slicing, which satisfy certain
conditions. The objects in each $\mathcal{P}(\phi)$ are called semistable and
the simple objects there are called stable. Such a notion is a kind of
triangulated generalization of the stability for an abelian category mentioned
above. Indeed, a stability condition $\sigma$ on $\operatorname{\mathcal{D}}$
induces a stability function $Z$ on the abelian category, i.e. its heart
$\mathcal{H}_{\sigma}$. Conversely, a stability function $Z$ on a heart
$\mathcal{H}$ of $\operatorname{\mathcal{D}}$, satisfying the so-called
Harder-Narasimhan property, is equivalent to a stability condition with heart
$\mathcal{H}$ on $\operatorname{\mathcal{D}}$.

###  1.2. Reineke’s conjecture and total stability

Denote by $\operatorname{Sim}\mathcal{H}(Q)$ the set of simples in
$\mathcal{H}(Q)$ and $\mathcal{A}_{Q}$ be the quantum affine space

$\displaystyle\mathbb{Q}(q^{1/2})\langle y^{S}\mid
S\in\operatorname{Sim}\mathcal{H}(Q)\rangle\big{/}(y^{S_{i}}y^{S_{j}}-q^{\lambda_{Q}(i,j)}y^{S_{j}}y^{S_{i}}),$

where $q^{1/2}$ is an indeterminate and $\lambda_{Q}(i,j)=\langle
S_{j},S_{i}\rangle-\langle S_{i},S_{j}\rangle$ with $\langle-,-\rangle$ given
by the Euler form associated to $Q$. One can obtain the completion
$\widehat{\mathcal{A}}_{Q}$ of $\mathcal{A}_{Q}$ with respect to the ideal
generated by $y^{S},\forall S\in\operatorname{Sim}\mathcal{H}(Q)$. Suppose
that a stability function $Z$ on $\mathcal{H}(Q)$ is discrete, i.e.
$\mu_{Z}(M)\neq\mu_{Z}(L)$ for any indecomposable $M\not\cong L$ in
$\mathcal{H}(Q)$. Then an invariant $\operatorname{DT}(\mathcal{H}(Q))$ (cf.
e.g. [Ke11]) can be calculated as

$\displaystyle\operatorname{DT}(\mathcal{H})\colon=\prod^{\longleftarrow}_{\text{$M$
stable}}\mathbb{E}(y^{\dim M})\quad\in\mathcal{A}_{Q}$ (1.1)

where the product is taking in phase-decreasing order and $\mathbb{E}$ is the
quantum dilogarithm defined as the formal series

$\mathbb{E}(X)=\sum_{j=0}^{\infty}\frac{q^{j^{2}/2}X^{j}}{\prod_{k=0}^{j-1}(q^{j}-q^{k})}.$

The formula 1.1 for calculating quantum dilogarithm identities is proved by
Reineke [R03], using Hall algebra and the integration map, for
$\mathcal{H}(Q)$ of a Dynkin quiver $Q$. An alternative proof can be found in
[Q15] which uses the fundamental group of exchange graph of hearts. In the
setting of Calabi-Yau-$3$ categories, this invariant produces the Donladson-
Thomas invariants, e.g. studied by Kontsevich-Sobliman [KS08]. In [R03],
Reineke conjectures the following.

######  Conjecture 1.1.

For any Dynkin quiver $Q$, there is a stability function $Z$ such that the
number of factors in the product (1.1) reaches maximal, i.e. equals $n\cdot
h_{Q}/2$, where $h_{Q}$ is the Coxeter number associated to $Q$. In other
words, any indecomposable object in $\mathcal{H}(Q)$ is $Z$-stable.

Such a stability function $Z$ is called a total stability function. This
conjecture has been confirmed for any quiver $Q$ of type $A_{n}$ (with
arbitrary orientation) in several papers via different approaches, such as
[BGMS19, AI20, HH20, Ki20], and also for any quiver $Q$ of type $D_{n}$ in
[A22].

In this paper, we prove Reineke’s conjecture from the point of view of
triangulated categories via geometric models. In the triangulated setting, one
also has the notion of total stability for a Bridgeland’s stability condition
$\sigma$, (cf. [Q15, Q18]), i.e. requiring all indecomposable objects in the
triangulated category $\sigma$-stable. The existence of such total stability
conditions is rare, if and only if under the circumstance that
$\operatorname{\mathcal{D}}=\operatorname{\mathcal{D}}_{\infty}(Q)$ for a
Dynkin diagram $Q$ (cf. [KOT19, Q20]). Clearly, a total stability condition
$\sigma$ on $\operatorname{\mathcal{D}}$ will induce a total stability
function on its heart $\mathcal{H}_{\sigma}$. The subtle point is to describe
the heart $\mathcal{H}_{\sigma}$ so that one can obtain the module category of
any Dynkin quiver (with arbitrary orientation). The geometric description (cf.
[QZ22]) of the space $\operatorname{ToSt}(Q)$ is the key here, that is we have
a geometric model for any module category of Dynkin type.

In type $A_{n}$, our approach is essentially the same as the one in [BGMS19].

######  Remark 1.2.

Note that in Reineke’s original paper [R03], he takes
$\operatorname{Re}Z(M)=\dim M$ to be the dimension function of $M$ for any
$M\in\mathcal{H}(Q)$, which is just for simplicity. Very recently, it is
proved in [M22] that, when restricted to $\operatorname{Re}Z(M)=\dim M$,
Reineke’s conjecture fails for the linear quiver $Q$ of type $E_{7}$. However,
such a constrain is unnecessary from the modern point of view. Our approach
proves the correct version of Reineke’s conjecture.

Another remark is that Hille-Juteau also have a proof of Reineke’s conjecture
(unpublished) according to [Ke11, § 1].

###  1.3. Geometric model for module category of a Dynkin diagram

There are many works on geometric models for various categories in
representation theory of algebras. Such models are in some sense, certain type
of Fukaya categories, cf. [HKK17, OPS18, HZZ20, BS21]. The geometric model
allows one to use surface combinatorics/ topology to produce many
applications, i.e. realizing arcs as indecomposables, intersections as
morphisms, (total) rotations of arcs as Auslander-Reiten translations, etc. In
particular, Qiu-Zhang [QZ22] construct a geometric model for the root category
$\operatorname{\mathcal{D}}_{\infty}(Q)/[2]$ of each Dynkin diagram $Q$. In
details, for a Dynkin diagram $Q$, they construct stable $h_{Q}$-gons
$\mathbf{V}$, which provide geometric models for
$\operatorname{\mathcal{D}}_{\infty}(Q)/[2]$, in the sense that each
indecomposable object in $\operatorname{\mathcal{D}}_{\infty}(Q)/[2]$ is
interpreted as an oriented admissible diagonal in $\mathbf{V}$, and naturally
induce a central charge of $\operatorname{\mathcal{D}}_{\infty}(Q)/[2]$. Note
that by upgrading orientation ($\mathbb{Z}_{2}$-grading) to
$\mathbb{Z}$-grading for admissible diagonals, the model also works for the
bounded derived category $\operatorname{\mathcal{D}}_{\infty}(Q)$.
Furthermore, by the construction, contractible triangles in the geometric
model imply triangles in the associated triangulated categories.

We will derive a geometric model for any heart of a total stability condition
in $\operatorname{ToSt}(Q)$ from the geometric model of the root category. In
Proposition 3.2, we realize each admissible upward diagonal of a stable
$h_{Q}$-gon associated to a total stability condition $\sigma$ as an
indecomposable object in the heart $\mathcal{H}_{\sigma}$. Moreover, we locate
all the simples in the stable $h_{Q}$-gon and obtain the following.

######  Theorem 1.3 (Theorems 4.4, 4.9 and 4.15).

Let $\sigma\in\operatorname{ToSt}(Q)$ for a quiver $Q$ of Dynkin type and
$\mathbf{V}_{\sigma}$ be the stable $h_{Q}$-gon of type $Q$ associated to
$\sigma$. Then there is a bijection

$\mathfrak{M}\colon\mathrm{UD}(\mathbf{V}_{\sigma})\longrightarrow\operatorname{Ind}\mathcal{H}_{\sigma}$

sending an upward diagonal $\ell$ in $\mathbf{V}_{\sigma}$ to an
indecomposable object $\mathfrak{M}(\ell)$ in $\mathcal{H}_{\sigma}$.

Moreover, it restricts to a bijection

$\operatorname{Sim}\mathbf{V}_{\sigma}\to\operatorname{Sim}\mathcal{H}_{\sigma},$

between the set of $\operatorname{Im}Z$-Ind diagonals of $\mathbf{V}_{\sigma}$
and the set of simples of $\mathcal{H}_{\sigma}$. As a consequence,
$\mathcal{H}_{\sigma}$ is equivalent to the module category of the (ungraded)
intersection quiver of the $\operatorname{Im}Z$-Ind diagonals of
$\mathbf{V}_{\sigma}$.

Furthermore, we shall use such a model to prove Reineke’s conjecture.

######  Theorem 1.4 (Theorem 5.6).

Conjecture 1.1 holds.

In type $A_{n}$ and $D_{n}$, the proof bases on building stable $h_{Q}$-gon
directly. In type $E_{n\in\\{6,7,8\\}}$, we show that essentially we only need
to give $2^{n-5}$ concrete examples (of stable $h_{Q}$-gons) to verify the
conjecture.

### Acknowledgments

Qy would like to thank Alastair King for many inspiring discussions, Ryan
Kinser for the invitation of giving a talk in FD Seminar that pushed for this
paper and Jan Schröer for pointing out the reference [A22]. This work is
supported by National Key R&D Program of China (No.2020YFA0713000), Beijing
Natural Science Foundation (Grant No.Z180003), National Natural Science
Foundation of China (Grant No.12031007) and National Natural Science
Foundation of China (Grant No.12101422).

##  2\. Preliminaries

In this section, we will recall some conventions and constructions from
[QZ22].

#### Conventions

Consider the complex plane $\mathbb{C}=\mathbb{R}^{2}$. For any two points
$W_{1}=(x_{1},y_{1})$ and $W_{2}=(x_{2},y_{2})$ in $\mathbb{C}$, we define

$\displaystyle
W_{1}<W_{2}\Longleftrightarrow\begin{cases}y_{1}<y_{2}\text{\quad or}\\\
y_{1}=y_{2},\,x_{1}<x_{2}.\end{cases}$ (2.1)

We write a vector $\overrightarrow{VW}$ by $VW$ for simplicity and denote by
$\operatorname{arg}(VW)$ the angle of $VW$ which takes value in $[0,2\pi)$.

Let $\mathbf{V}$ be a (labelled) $h$-gon in $\mathbb{C}$ with vertices
$V_{0},V_{1},\cdots,V_{h}=V_{0}$ and (oriented) edges $0\neq
z_{j}=V_{j-1}V_{j}$, where $j\in\mathbb{Z}_{h}$. Then $\mathbf{V}$ is
_positively convex_ if all other $V_{i}$’s are on the left hand side of the
edge $z_{j}=V_{j-1}V_{j}$. Note that the positively convexity implies that the
vertices $V_{j}$’s are in anti-clockwise order. For any integer $0\leq s\leq
h/2$, the _length- $s$ diagonals_ of $\mathbf{V}$ are those $V_{j}V_{j+s}$.
For example, the length-$1$ diagonals are just the edges $z_{j}$. For a
positively convex $h$-gon $\mathbf{V}$, the _level- $s$ diagonal-gon_ is the
convex polygon bounded by its length-$s$ diagonals (i.e. on the left hand side
of).

######  Definition 2.1.

An _$h$ -gon $\mathbf{V}$ of type $Q$_ (a Dynkin quiver) is defined
respectively as:

$A_{n}\,(n\geq 1)$:

an $(n+1)$-gon.

$D_{n}\,(n\geq 4)$:

a (centrally) symmetric doubly punctured $2(n-1)$-gon with punctures
$B_{\pm}$.

$E_{6}$:

a $12$-gon satisfying:

$\displaystyle\begin{cases}z_{j}+z_{j+4}+z_{j+8}=0,\\\
z_{j}-z_{j-3}+z_{j-6}-z_{j-9}=0,\end{cases}\quad\forall j\in\mathbb{Z}_{12}.$
(2.2)

$E_{7}$:

a symmetric $18$-gon satisfying:

$\displaystyle z_{j}+z_{j+1}+z_{j+6}+z_{j+7}+z_{j+12}+z_{j+13}=0,\qquad\forall
j\in\mathbb{Z}_{18}.$ (2.3)

$E_{8}$:

a symmetric $30$-gon satisfying:

$\displaystyle\begin{cases}z_{j}+z_{j+10}+z_{j+20}=0,\\\
z_{j}+z_{j+6}+z_{j+12}+z_{j+18}+z_{j+24}=0,\end{cases}\quad\forall
j\in\mathbb{Z}_{30}.$ (2.4)

To summarize, an $h$-gon of type $Q$ is an $h_{Q}$-gon (satisfying above extra
conditions), where $h_{Q}$ is the Coxeter number associated to the underlying
Dynkin diagram of $Q$. In this case, we have $h=h_{Q}$. For simplicity, we
prefer to write $h$ instead of $h_{Q}$ if there is no confusion.

Let $\mathbf{V}$ be an $h$-gon of the exceptional type $E_{n}$ $(n=6,7,8)$. We
follow the notion in [QZ22] respectively:

$E_{6}$:

for each $j\in\mathbb{Z}_{12}$ we can draw an triangle
$\mathbf{T}_{j}\colon=V_{j-1}V_{j}W_{j}$ with edges

$V_{j-1}V_{j}=z_{j},\quad V_{j}W_{j}=z_{j+4},\quad W_{j}V_{j-1}=z_{j+8},$

cf. the triangle relations in (2.2).

$E_{7}$:

for each $j\in\mathbb{Z}_{18}$ we can draw a hexagon
$\mathbf{L}_{j}\colon=V_{j-1}V_{j}V_{j+1}W_{j+1}U_{j}W_{j-1}$ with edges

$\displaystyle V_{j-1}V_{j}=z_{j},\quad V_{j}V_{j+1}=z_{j+1},\quad
V_{j+1}W_{j+1}=z_{j+6},$ $\displaystyle W_{j+1}U_{j}=z_{j+7},\quad
U_{j}W_{j-1}=z_{j+12},\quad W_{j-1}V_{j-1}=z_{j+13},$

cf. the hexagon relations in (2.3).

$E_{8}$:

for each $j\in\mathbb{Z}_{30}$ we can draw a triangle
$\mathbf{T}_{j}\colon=V_{j-1}V_{j}W_{j-1}$ with edges

$V_{j-1}V_{j}=z_{j},\;V_{j}W_{j-1}=z_{j+10},\;W_{j-1}V_{j-1}=z_{j+20},$

cf. the triangle relations in (2.4).

The key feature in the exceptional cases is that there are two $h/2$-gons
inside $\mathbf{V}$, i.e. the ice/fire core with vertices $W_{j}$ of even/odd
indices.

######  Definition 2.2.

Let $\mathbf{V}$ be an $h$-gon of type $Q$. The _admissible diagonals_ of
$\mathbf{V}$ are defined respectively as:

$A_{n}$:

any usual (oriented) diagonals of $\mathbf{V}$.

$D_{n}$:

either the length-$s$ diagonals $\pm V_{j}V_{j+s}=\pm V_{j+s+h/2}V_{j+h/2}$
for $1\leq s<h/2$ and $j\in\mathbb{Z}_{h}$ or vectors $\pm V_{j}B_{\pm}=\pm
B_{\mp}V_{j+h/2}$ for $j\in\mathbb{Z}_{h}.$

$E_{n}$:

either the length-$s$ diagonals $\pm V_{j}V_{j+s}$ for $1\leq s\leq n-3$ and
$j\in\mathbb{Z}_{h}$ or vectors

$\begin{cases}\pm W_{j}W_{j+2},\\\ \pm V_{j-1}W_{j+2}=\pm W_{j+1}V_{j+n-3},\\\
\pm V_{j-1}W_{j+1}=\pm W_{j+2}V_{j+n-3},\end{cases}\quad\forall
j\in\mathbb{Z}_{h}.$

Here, the equalities above, together with the ones in the construction of the
exceptional case, i.e.

$\pm V_{j-1}V_{j}=\pm W_{j+4}V_{j+n-3}=\pm V_{j-2+n}W_{j-2+n},\quad\forall
j\in\mathbb{Z}_{h},$

and

$\begin{cases}\pm W_{j}W_{j+2}=\pm W_{j+8}V_{j+7}=\pm
V_{j-5}W_{j-6},&\text{for $E_{7}$}\\\ \pm W_{j}W_{j+2}=\pm
W_{j+14}V_{j+13}=\pm V_{j-10}W_{j-12},&\text{for
$E_{8}$}\end{cases}\quad\forall j\in\mathbb{Z}_{h}.$

as well as (central) symmetry in type $E_{7}/E_{8}$, are considered as
equivalent relations. We denote by $\mathrm{Ad}(\mathbf{V})$ the set of
(equivalent classes of) admissible diagonals in $\mathbf{V}$.

Let $\operatorname{\mathcal{D}}_{\infty}(Q)$ be the bounded derived category
of the module category of $\mathbf{k}Q$ and $\mathcal{R}(Q)$ the corresponding
root category $\operatorname{\mathcal{D}}_{\infty}(Q)/[2]$. Denote by
$\operatorname{Stab}(Q)$ the space of stability conditions on
$\operatorname{\mathcal{D}}_{\infty}(Q)$, cf [B07] for the detailed
definition, and $\operatorname{ToSt}(Q)$ the space of total stability
conditions on $\operatorname{\mathcal{D}}_{\infty}(Q)$. Now we recall the main
result from [QZ22] that the admissible diagonals give a central charge, which
is a combination of Proposition 3.9, Theorems 4.5 and 6.7 there. In fact, we
slightly generalize it to obtain a bijection between the set of (equivalent
classes of) admissible diagonals and the set of indecomposable objects in
$\mathcal{R}(Q)$.

######  Theorem 2.3.

[QZ22] A stable $h$-gon $\mathbf{V}$ of type $Q$ gives a geometric model for
$\mathcal{R}(Q)$ in the sense that we have a bijection

$\displaystyle\begin{array}[]{rcl}\mathfrak{M}:\mathrm{Ad}(\mathbf{V})&\longrightarrow&\operatorname{Ind}\mathcal{R}(Q),\\\
\ell&\mapsto&\mathfrak{M}(\ell)\end{array}$ (2.7)

which naturally induces a central charge
$Z_{\mathbf{V}}:K\operatorname{\mathcal{D}}_{\infty}(Q)\to\mathbb{C}$ with
$Z_{\mathbf{V}}(\mathfrak{M}(\ell))=\ell$.

There are two natural actions on $\operatorname{Stab}(Q)$, i.e. the
$\mathbb{C}$-action and the
$\operatorname{Aut}\operatorname{\mathcal{D}}_{\infty}(Q)$-action, which also
can be induced on $\operatorname{ToSt}(Q)$. In particular, the action of $[1]$
on $\operatorname{ToSt}(Q)$ coincide with the action of $1\in\mathbb{C}$ on
$\operatorname{ToSt}(Q)$.

######  Definition 2.4.

An $h$-gon $\mathbf{V}$ of type $Q$ is _stable_ if it is positively convex and
moreover:

$D_{n}$:

the two punctures $B_{\pm}$ are inside the level-$(n-2)$ diagonal-gon.

$E_{n}$:

the vertices $W_{j}$’s are inside the level-$(n-3)$ diagonal-gon, for
$n\in\\{6,7,8\\}$.

Note that a stable $h$-gon, without stating the types explicitly, just means a
positively convex $h$-gon (with above extra conditions).

Denote by $\operatorname{Stgon}(Q)$ the moduli space of stable $h$-gons of
type $Q$, where two stable $h$-gons are equivalent if and only if they are
parallel, i.e. related by a translation of $\mathbb{C}$.

######  Theorem 2.5.

[QZ22, Thm. 1] There is an isomorphism between complex manifolds

$\operatorname{ToSt}(Q)/[2]\cong\operatorname{Stgon}(Q),$

which sends a total stability condition $\sigma$ to a stable $h$-gon
$\mathbf{V}_{\sigma}$ of type $Q$ with central charge
$Z_{\mathbf{V}_{\sigma}}=Z_{\sigma}$.

##  3\. Hearts and triangles

In this section, we will describe the heart $\mathcal{H}_{\sigma}$ of
$\sigma\in\operatorname{ToSt}(Q)$ for a Dynkin quiver $Q$ and special short
exact sequence in $\mathcal{H}_{\sigma}$.

###  3.1. Hearts of total stability conditions

Recall that a stability condition $\sigma=(Z,\mathcal{P})$ consists of the
central charge $Z$ and the slicing $\mathcal{P}$, which is a
$\mathbb{R}$-collection of abelian subcategory $\mathcal{P}(\phi)$ for
$\phi\in\mathbb{R}$. Its heart $\mathcal{H}_{\sigma}\colon=\mathcal{P}[0,1)$
is defined to be the abelian category generated by $\mathcal{P}(\phi)$, where
$\phi\in[0,1)$. In particular, if $\sigma$ is totally stable, then the
indecomposables in $\mathcal{H}_{\sigma}$ are precisely those objects whose
phases are in $[0,1)$.

Define

$\displaystyle\upharpoonleft\hskip-3.61371pt{VW}\hskip-3.61371pt\upharpoonright\colon=\begin{cases}VW&\text{if~{}}~{}V<W,\\\
WV&\text{if~{}}~{}W<V.\end{cases}$ (3.1)

Then we obtain
$\operatorname{arg}(\upharpoonleft\hskip-3.61371pt{VW}\hskip-3.61371pt\upharpoonright)\in[0,\pi)$.

######  Definition 3.1.

An admissible diagonal $\ell$ of an $h$-gon $\mathbf{V}$ of type $Q$ is called
_upward_ if $\operatorname{arg}(\ell)\in[0,\pi)$. Denote by
$\mathrm{UD}(\mathbf{V})$ the set of upward (admissible) diagonals in
$\mathbf{V}$.

For any admissible diagonal $VW$, the upward version is exactly
$\upharpoonleft\hskip-3.61371pt{VW}\hskip-3.61371pt\upharpoonright$, where
$\upharpoonleft\hskip-3.61371pt{\cdot}\hskip-3.61371pt\upharpoonright$ is
defined in (3.1).

######  Proposition 3.2.

Let $\sigma\in\operatorname{ToSt}(Q)$ and $\mathbf{V}_{\sigma}$ be the
corresponding stable $h$-gon of type $Q$. Then $\mathfrak{M}$ in (2.7) induces
a bijection

$\mathfrak{M}\colon\mathrm{UD}(\mathbf{V}_{\sigma})\longrightarrow\operatorname{Ind}\mathcal{H}_{\sigma}.$
(3.2)

###### Proof.

We choose $\mathcal{H}_{\sigma}[1]\cup\mathcal{H}_{\sigma}$ as a fundamental
domain for $\mathcal{R}(Q)$. For an upward diagonal $\ell$, the phase of
$\mathfrak{M}(\ell)$ is $\operatorname{arg}(\ell)/\pi$, which belongs to
$[0,1)$. Hence the map $\mathfrak{M}$ in (2.7) restricts to the required
bijection. ∎

Recall that the heart $\mathcal{H}$ in
$\operatorname{\mathcal{D}}_{\infty}(Q)$ is _standard_ if it is equivalent to
the module category of some quiver with the same underlying graph of $Q$. When
$\sigma$ is totally stable, we have
$\operatorname{Ind}\operatorname{\mathcal{D}}_{\infty}(Q)=\operatorname{Ind}\mathcal{P}_{\sigma}\cup\operatorname{Ind}\mathcal{P}_{\sigma}^{\perp}$
for $\mathcal{P}_{\sigma}=\mathcal{P}[0,+\infty)$. Thus, by [Q15, Proposition
2.7] we have the following statement.

######  Lemma 3.3.

The heart $\mathcal{H}_{\sigma}$ is standard if
$\sigma\in\operatorname{ToSt}(Q)$.

###  3.2. Intersections as homomorphisms

In this section, we let $\mathbf{V}$ be an $h$-gon of type $Q$ and will
establish a correspondence between certain intersections of admissible
diagonals in $\mathrm{Ad}(\mathbf{V})$ and the homomorphisms between the
associated objects in the root category $\mathcal{R}(Q)$.

######  Definition 3.4.

An _oriented intersection_ from an upward diagonal $\ell_{1}$ to another
upward diagonal $\ell_{2}$ is an anticlockwise angle $\alpha$ from $\ell_{1}$
to $\ell_{2}$ locally at an intersection $P\in\ell_{1}\cap\ell_{2}$ such that
$\alpha$ is in the interior of $\mathbf{V}$.

The opposite angles are considered equivalent if $P$ is an interior
intersection, cf. Figure 1.

For an oriented intersection $\alpha$ from $\ell_{1}$ to $\ell_{2}$, we set

$\displaystyle\rho(\alpha)\colon=\begin{cases}0&\text{if~{}}arg(\ell_{1})<arg(\ell_{2}),\\\
1&\text{if~{}}arg(\ell_{2})<arg(\ell_{1}),\end{cases}$ (3.3)

called the _intersection index_ of $\alpha$. Note that $arg(\ell_{1})\neq
arg(\ell_{2})$ if $\ell_{1}$ and $\ell_{2}$ intersect. For instance, in Figure
1, $\alpha$ is an oriented intersection from $\ell_{1}$ to $\ell_{2}$ with
index $\rho(\alpha)=0$, while $\beta$ is an oriented intersection from
$\ell_{2}$ to $\ell_{1}$ with index $\rho(\beta)=1$.

$\alpha$$\alpha$$\beta$$\beta$$\ell_{1}$$\ell_{2}$$\bullet$ Figure 1. Two
oriented intersections $\alpha$ and $\beta$

######  Lemma 3.5.

Assume that there is a contractible triangle formed by upward diagonals
$\ell_{1}$, $\ell_{2}$ and $\ell_{3}$ in $\mathbf{V}$ such that
$\ell_{3}=\ell_{1}+\ell_{2}$ and $arg(\ell_{1})<arg(\ell_{2})$. Then we have
the following triangle in $\mathcal{R}(Q)$:

$\displaystyle\mathfrak{M}(\ell_{1})\xrightarrow{\alpha}\mathfrak{M}(\ell_{3})\xrightarrow{\beta}\mathfrak{M}(\ell_{2})\xrightarrow{\gamma}\mathfrak{M}(\ell_{1})[1],$
(3.4)

where morphisms correspond to oriented intersections.

Moreover, such a triangle is a short exact sequence when restricted to
$\mathcal{H}_{\sigma}$:

$0\xrightarrow{}\mathfrak{M}(\ell_{1})\xrightarrow{\alpha}\mathfrak{M}(\ell_{3})\xrightarrow{\beta}\mathfrak{M}(\ell_{2})\xrightarrow{}0.$

$\gamma$$\alpha$$\beta$$\ell_{1}$$\ell_{2}$$\ell_{3}$$P$$\bullet$$\bullet$$\bullet$

$\gamma$$\beta$$\alpha$$\ell_{2}$$\ell_{1}$$\ell_{3}$$P$$\bullet$$\bullet$$\bullet$

Figure 2. Contractible triangle of upward diagonals

###### Proof.

There are two cases of the contractible triangle depending on the positions of
$\ell_{1}$ and $\ell_{2}$, see Figure 2, but the triangle (3.4) will always be
as above (where the order of objects depends on $\operatorname{arg}$ of
$\ell_{i}$’s).

The statement follows from [HKK17, OPS18] and [QZZ] for type $A$ and type $D$,
respectively. As for the exceptional cases, it can be checked
straightforwardly, cf. [QZ22, § 5] for couple forms of the triangles. ∎

######  Remark 3.6.

In general, oriented intersections between admissible diagonals (arcs) induce
(a basis of) morphisms between the corresponding objects and the index of an
intersection corresponds to the degree of the morphism. More precisely, one
has (cf. [HKK17, OPS18, QZZ] or [IQZ20, (6.7)])

$\displaystyle\operatorname{Int}^{d}(\ell_{1},\ell_{2})=\dim\operatorname{Hom}^{d}(\mathfrak{M}(\ell_{1}),\mathfrak{M}(\ell_{2}))$
(3.5)

in the setting of (graded skew) gentle algebras.

Moreover, the mapping cone of such a morphism usually (e.g. when the
intersection is at the boundary) corresponds to the diagonals obtained from
the union of the original two diagonals by smoothing out the oriented
intersection.

######  Definition 3.7.

For two upward diagonals $l_{1}$ and $l_{2}$, there is an _essential oriented
intersection_ from $l_{2}$ to $l_{1}$ if there is a contractible triangle with
edges (equivalent to) $l_{1}$ and $l_{2}$, as shown in one of the pictures in
Figure 2.

Such a notion will be used to define the intersection quiver of an $h$-gon of
type $Q$ in Section 3.1. Note that there is at most one essential oriented
intersection between two upward diagonals. In type $A_{n}$, any oriented
intersection at the boundary between two upward diagonals is essential.
However, this is not true in other types. For instance, in type $D_{n}$, the
oriented intersection of upward diagonals $V_{j}B_{-}$ and $V_{j}B_{+}$ is not
essential since $B_{-}B_{+}$ is not an admissible diagonal.

##  4\. Geometric model of hearts

In this section, we give algorithms to find admissible diagonals corresponding
to simples in the hearts of total stability conditions and hence obtain all
the information (that we need) of the hearts in the geometric model.

###  4.1. Type $A_{n}$

Let $\mathbf{V}$ be a stable $(n+1)$-gon in $\mathbb{C}$. We reorder the
vertices $V_{j}$ of $\mathbf{V}$ as

$Y_{1}<Y_{2}<\cdots<Y_{n+1}$

with respect to the order (2.1), where $Y_{i}=(x_{i},y_{i})$. Note that there
are at most two $y_{i}$’s coincide. Then an upward diagonal in $\mathbf{V}$ is
of the form $Y_{i}Y_{j}$ with $1\leq i<j\leq n+1$.

######  Definition 4.1.

The diagonals $s_{i}:=Y_{i}Y_{i+1}$ are called _${\operatorname{Im}}Z$ -ind
diagonals_, for $1\leq i\leq n$. We denote by

$\operatorname{Sim}\mathbf{V}\colon=\\{s_{i}\mid 1\leq i\leq n\\}.$

Note that any upward diagonal $Y_{i}Y_{j}$ ($1\leq i<j\leq n+1$) uniquely
decomposes into the sum:

$\displaystyle
Y_{i}Y_{j}=Y_{i}Y_{i+1}+\cdots+Y_{j-1}Y_{j}=s_{i}+s_{i+1}\cdots+s_{j-1},$
(4.1)

with respect to $y$-coordinate (i.e. the imaginary part of the central
charge).

######  Lemma 4.2.

Let $\sigma\in\operatorname{ToSt}(A_{n})$ and
$Y_{i}Y_{j}\in\mathrm{UD}(\mathbf{V}_{\sigma})$ with $\mathbf{V}_{\sigma}$
being the associated stable $(n+1)$-gon. The decomposition (4.1) induces a
filtration of short exact sequences for $\mathfrak{M}(Y_{i}Y_{j})$ in the
heart $\mathcal{H}_{\sigma}$ with factors $\mathfrak{M}(s_{k})$ for $i\leq
k\leq j-1$.

###### Proof.

Note that $Y_{i}Y_{j}$, $Y_{i}Y_{j-1}$ and $s_{j-1}=Y_{j-1}Y_{j}$ form a
contractible triangle. By Lemma 3.5, there exists a short exact sequence (in
$\mathcal{H}_{\sigma}$) either of the form

$0\xrightarrow{}\mathfrak{M}(Y_{i}Y_{j-1})\xrightarrow{}\mathfrak{M}(Y_{i}Y_{j})\xrightarrow{}\mathfrak{M}(s_{j-1})\xrightarrow{}0,$

if $arg(Y_{i}Y_{j-1})<arg(s_{j-1})$, or

$0\xrightarrow{}\mathfrak{M}(s_{j-1})\xrightarrow{}\mathfrak{M}(Y_{i}Y_{j})\xrightarrow{}\mathfrak{M}(Y_{i}Y_{j-1})\xrightarrow{}0,$

if $arg(s_{j-1})<arg(Y_{i}Y_{j-1})$. Thus the required filtration can be
constructed inductively. ∎

######  Definition 4.3.

The _intersection quiver_ $\mathcal{Q}(\mathbf{V})$ of $\mathbf{V}$ is the
graded quiver whose

  * •

vertices are ${\operatorname{Im}}Z$-ind diagonals and

  * •

arrows are essential oriented intersections with gradings given by
intersection indices.

For example, one can draw $\mathcal{Q}(\mathbf{V})$ for a stable $6$-gon
$\mathbf{V}$ of type $A_{5}$ as in Figure 3.

$Y_{1}$$Y_{2}$$Y_{3}$$Y_{5}$$Y_{6}$$Y_{4}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{5}$

$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{5}$

Figure 3. The intersection quiver of a $6$-gon of type $A_{5}$

Note that any oriented intersection at the boundary is essential in a stable
$(n+1)$-gon. Denote by $\mathcal{Q}^{1-?}(\mathbf{V})$ the quiver obtained
from $\mathcal{Q}(\mathbf{V})$ by grading change $?\mapsto(1-?)$. We will show
that the quiver $\mathcal{Q}^{1-?}(\mathbf{V})$ will have trivial grading for
all Dynkin case.

Recall from [KQ15] that the _Ext-quiver_ $\mathcal{Q}(\mathcal{H})$ of a
(finite length) heart $\mathcal{H}$ is the graded quiver whose vertices are
the simples of $\mathcal{H}$ and whose graded arrows from $S$ to $T$ are given
by a basis of $\operatorname{Ext}^{\bullet}(S,T)$.

Let $\operatorname{Sim}\mathcal{A}$ denote the set of simples of an abelian
category $\mathcal{A}$. As a consequence, we have the following.

######  Theorem 4.4.

Let $\sigma\in\operatorname{ToSt}(Q)$ with heart $\mathcal{H}_{\sigma}$ and
the corresponding stable $h$-gon $\mathbf{V}_{\sigma}$ of type $Q$. If $Q$ is
of type $A_{n}$, then the bijection (3.2) restricts to a bijection

$\mathfrak{M}\colon\operatorname{Sim}\mathbf{V}_{\sigma}\to\operatorname{Sim}\mathcal{H}_{\sigma}.$

Moreover, the Ext-quiver $\mathcal{Q}(\mathcal{H}_{\sigma})$ coincides with
the intersection quiver $\mathcal{Q}(\mathbf{V}_{\sigma})$. Thus, we have

$\displaystyle\mathcal{H}_{\sigma}\cong\mathcal{H}(\mathcal{Q}^{1-?}(\mathbf{V}_{\sigma})).$
(4.2)

Recall that
$\mathcal{H}(\mathcal{Q}^{1-?}(\mathbf{V}_{\sigma}))=\operatorname{mod}\mathbf{k}\mathcal{Q}^{1-?}(\mathbf{V}_{\sigma})$.

###### Proof.

Any object in $\mathcal{H}_{\sigma}$ admits a filtration with factors in
$\\{\mathfrak{M}(s)\mid s\in\operatorname{Sim}\mathbf{V}_{\sigma}\\}$. Hence
we have

$\operatorname{Sim}\mathcal{H}_{\sigma}\subset\\{\mathfrak{M}(s)\mid
s\in\operatorname{Sim}\mathbf{V}_{\sigma}\\}.$

Noticing that both sets have $n$ elements, they must coincide. The
intersection quiver matches the $\operatorname{Ext}$-quiver follows from the
general formula (3.5).

The final statement, i.e. (4.2), follows from the standard simple-projective
duality:
$\operatorname{Ext}^{1}(S_{i},S_{j})\cong\operatorname{Irr}(P_{j},P_{i})^{*}$.
In particular, since the heart $\mathcal{H}_{\sigma}$ is hereditary by Lemma
3.3, the grading of each arrow in $\mathcal{Q}(\mathcal{H}_{\sigma})$ is one.
Thus $\mathcal{Q}^{1-?}(\mathbf{V}_{\sigma})$ has trivial grading. ∎

######  Remark 4.5.

By Theorem 4.4, the filtration in Lemma 4.2 is exactly the Jordan-Hölder
filtration of $\mathfrak{M}(Y_{i}Y_{j})$ in $\mathcal{H}_{\sigma}$.

If we forget the central charges in the theorem above and treat $\mathbf{V}$
as a topological polygon, then it becomes the result of [BGMS19].

###  4.2. Type $D_{n}$

Let $\mathbf{V}$ be a stable $2(n-1)$-gon of type $D_{n}$ with center
$O=(0,0)$ in $\mathbb{C}$ and $B_{\pm}=(\pm x_{B},\pm y_{B})$ the two
punctures. Without loss of generality, we assume that $B_{-}\leq B_{+}$, i.e.
either $B_{-}<B_{+}$ or $B_{-}=O=B_{+}$.

Reorder the points in $\\{V_{j},B_{\pm}\mid j\in\mathbb{Z}_{2(n-1)}\\}$ as

$\displaystyle Y_{-n}<\cdots
Y_{-1}\leq{\color[rgb]{0.75,0.75,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0.75,0.75,0.75}\pgfsys@color@gray@stroke{0.75}\pgfsys@color<EMAIL_ADDRESS>Y_{1}<\cdots Y_{n}$ (4.3)

with respect to (2.1), where $Y_{i}=(x_{i},y_{i})\in\mathbb{C}$. The symmetry
of $\mathbf{V}$ implies that $Y_{-i}=-Y_{i}$. Also note that the two
equalities in (4.3) can only hold simultaneously and, in which case, we have
$Y_{\pm 1}=O=B_{\pm}$. The stability of $\mathbf{V}$ implies that there are at
least $(n-2)$ $V_{j}$’s, which are less then $B_{-}$. Hence $B_{\pm}$ must be
either $Y_{\pm 1}$ or $Y_{\pm 2}$. In such notation, an upward diagonal in
$\mathbf{V}$ is of the form $Y_{i}Y_{j}$ such that $-n\leq i<j\leq n$ and
$i+j\neq 0$.

Denote by

  * •

$\mathbf{V}_{n}^{-}$ the convex hull of vertices in
$\\{Y_{-n},\cdots,Y_{-2},Y_{-1}\\}$ and

  * •

$\mathbf{V}_{n}^{+}$ the convex hull of vertices in
$(\\{Y_{-n},\cdots,Y_{-2},Y_{-1}\\}\setminus\\{B_{-}\\})\cup\\{B_{+}\\}$.

######  Lemma 4.6.

Both $\mathbf{V}_{n}^{-}$ and $\mathbf{V}_{n}^{+}$ are stable $n$-gons of type
$A_{n-1}$.

###### Proof.

Consider the polygon with $(n-1)$ consecutive vertices $V_{j+1},\ldots
V_{j+(n-1)}$, which inherit the positive convexity of $\mathbf{V}$. By the
stability of $\mathbf{V}$, $B_{\pm}$ is bounded in the parallelogram with
vertices $V_{j},V_{j+1},V_{j+(n-1)},V_{j+n}$. Thus, the polygon with vertices
$V_{j+1},\ldots V_{j+(n-1)}$ and $B_{\epsilon}$ is positively convex, for
$\epsilon\in\\{+,-\\}$.

In particular, $\mathbf{V}_{n}^{\pm}$ are both positively convex since the
vertices of $\mathbf{V}^{\pm}_{n}$ except $B_{\pm}$ are $(n-1)$ consecutive
vertices of $\mathbf{V}$. ∎

Set $s_{i}\colon=Y_{i-n-1}Y_{i-n}$ for $1\leq i\leq n-1$. Then, by Definition
4.1, we have

$\displaystyle\operatorname{Sim}\mathbf{V}_{n}^{-}=\\{s_{i}\mid 1\leq i\leq
n-1\\},\quad\text{and}$ (4.4)
$\displaystyle\operatorname{Sim}\mathbf{V}_{n}^{+}=\begin{cases}\\{s_{i}\mid
1\leq i\leq n-2\\}\cup\\{Y_{-2}B_{+}\\}&\text{if $B_{-}=Y_{-1}$},\\\
\\{s_{i}\mid 1\leq i\leq n-3\\}\cup\\{Y_{-3}Y_{-1},Y_{-1}B_{+}\\}&\text{if
$B_{-}=Y_{-2}$}.\end{cases}$ (4.5)

######  Definition 4.7.

For type $D_{n}$, the set $\operatorname{Sim}\mathbf{V}$ of
_${\operatorname{Im}}Z$ -ind diagonals_ is defined to be $\\{s_{i}\mid 1\leq
i\leq n-1\\}$ together with a diagonal

$\displaystyle
s_{n}=\begin{cases}Y_{-2}B_{+}=B_{-}Y_{2}&\text{if~{}}~{}B_{\pm}=Y_{\pm 1},\\\
Y_{-1}B_{+}=B_{-}Y_{1}&\text{if~{}}~{}B_{\pm}=Y_{\pm 2}.\end{cases}$ (4.6)

######  Lemma 4.8.

Let $\sigma\in\operatorname{ToSt}(D_{n})$ and
$Y_{i}Y_{j}\in\mathrm{UD}(\mathbf{V}_{\sigma})$ with
$\mathbf{V}:=\mathbf{V}_{\sigma}$ being the associated stable $2(n-1)$-gon.
Then $Y_{i}Y_{j}$ decomposes into a sum of ${\operatorname{Im}}Z$-ind
diagonals in $\operatorname{Sim}\mathbf{V}$, which induces a filtration of
short exact sequences for $\mathfrak{M}(Y_{i}Y_{j})$ in the heart
$\mathcal{H}_{\sigma}$ with factors
$\mathfrak{M}(s),s\in\operatorname{Sim}\mathbf{V}$.

###### Proof.

We only prove the case that $B_{\pm}=Y_{\pm 1}$ as the argument for
$B_{\pm}=Y_{\pm 2}$ is similar.

Note that there are the following four possibilities of the indices $i,j$:

  * $($a$)$

$i<j\leq-1$;

  * $($b$)$

$i<-1<1\leq j$;

  * $($a${}^{\prime})$

$j>i\geq 1$;

  * $($b${}^{\prime})$

$j>1>-1\geq i$,

where the latter two cases are the dual of the former two since
$Y_{i}Y_{j}=Y_{-j}Y_{-i}$. It suffices to discuss the former two cases. To be
more precise,

  * $($a$)$

if $i<j\leq-1$, then $Y_{i}Y_{j}$ is an upward diagonal in
$\mathbf{V}_{n}^{-}$ and thus a sum of elements in
$\operatorname{Sim}\mathbf{V}_{n}^{-}(\subset\operatorname{Sim}\mathbf{V})$ by
Lemma 4.2;

  * $($b1$)$

if $i<-1$ and $j=1$, then $Y_{i}Y_{j}=Y_{i}B_{+}$ is an upward diagonal in
$\mathbf{V}_{n}^{+}$ and thus a sum of elements in
$\operatorname{Sim}\mathbf{V}_{n}^{+}(\subset\operatorname{Sim}\mathbf{V})$ by
Lemma 4.2;

  * $($b2$)$

if $i<-1<1<j$, then we have

$Y_{i}Y_{j}=Y_{i}B_{-}+B_{-}Y_{j}=Y_{i}B_{-}+Y_{-j}B_{+}.$

Note that $Y_{i}Y_{-1}\in\mathrm{UD}(\mathbf{V}_{n}^{-})$ and
$Y_{-j}B_{+}\in\mathrm{UD}(\mathbf{V}_{n}^{+})$. Hence, combining the
decompositions in the two situations above, we obtain a required
decomposition.

Furthermore, similar to the proof of Lemma 4.2, we can iteratively use Lemma
3.5 to show that the sum gives rise to a filtration of short exact sequences
whose factors are ${\operatorname{Im}}Z$-ind diagonals. ∎

Following the Definition 4.3, we can define the _intersection quiver_
$\mathcal{Q}(\mathbf{V})$ for a stable $2(n-1)$-gon $\mathbf{V}$ of type
$D_{n}$. Note that the arrows in $\mathcal{Q}(\mathbf{V})$ arise from
essential oriented intersections. See Figure 4 for an example, that there is
no arrow from $Y_{-2}Y_{1}$ to $Y_{-2}Y_{-1}$ as the oriented intersection at
$Y_{-2}$ is not essential.

$Y_{-5}$$Y_{-4}$$Y_{-2}$$Y_{3}$$Y_{5}$$Y_{4}$$Y_{2}$$Y_{-3}$$Y_{-5}$$Y_{-4}$$Y_{-2}$$Y_{3}$$Y_{5}$$Y_{4}$$Y_{2}$$Y_{-3}$$Y_{-5}$$Y_{-4}$$Y_{-2}$$Y_{3}$$Y_{5}$$Y_{4}$$Y_{2}$$Y_{-3}$$Y_{-5}$$Y_{-4}$$Y_{-2}$$Y_{3}$$Y_{5}$$Y_{4}$$Y_{2}$$Y_{-3}$$Y_{-5}$$Y_{-4}$$Y_{-2}$$Y_{3}$$Y_{5}$$Y_{4}$$Y_{2}$$Y_{-3}$$Y_{-5}$$Y_{-4}$$Y_{-2}$$Y_{3}$$Y_{5}$$Y_{4}$$Y_{2}$$Y_{-3}$$Y_{-5}$$Y_{-4}$$Y_{-2}$$Y_{3}$$Y_{5}$$Y_{4}$$Y_{2}$$Y_{-3}$$Y_{-5}$$Y_{-4}$$Y_{-2}$$Y_{3}$$Y_{5}$$Y_{4}$$Y_{2}$$Y_{-3}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$Y_{1}$$\bullet$$Y_{-1}$$s_{1}$$s_{2}$$s_{3}$$s_{5}$$s_{4}$

$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{5}$

Figure 4. The intersection quiver of an $8$-gon of type $D_{5}$.

As a consequence, we have the following theorem.

######  Theorem 4.9.

Theorem 4.4 holds for any quiver $Q$ of type $D_{n}$. In particular, the
underlying diagram of the Ext-quiver/intersection quiver is of the form:

$s_{1}$$s_{2}$$s_{3}$$\cdots$$s_{n-2}$$s_{n-1}$$s_{n}$

###### Proof.

Mutatis mutandis the proof of Theorem 4.4. ∎

###  4.3. Exceptional type $E_{6}$

Let $\mathbf{V}$ be a stable $12$-gon of type $E_{6}$ in $\mathbb{C}$. Reorder
the points in $\\{V_{j},W_{j}\mid j\in\mathbb{Z}_{12}\\}$ with respect to
(2.1) as $\\{Y_{i}\mid 1\leq i\leq 24\\}$ such that $Y_{i}<Y_{i+1}$ for all
$i$. The stability of $\mathbf{V}$ implies that there are at least three
$V_{j}$’s that are less than all $W_{j}$’s. In other words,
$Y_{1},Y_{2},Y_{3}$ must all be $V_{j}$’s and thus are consecutive vertices of
$\mathbf{V}$. Without loss of generality, we can assume that
$\\{Y_{1},Y_{2},Y_{3}\\}=\\{V_{1},V_{2},V_{3}\\}$. Denote by
$\bigtriangleup_{-}$ the triangle with vertices $V_{1}$, $V_{2}$ and $V_{3}$.

######  Remark 4.10.

Similarly, one can deduce that $Y_{22},Y_{23},Y_{24}$ are consecutive vertices
of $\mathbf{V}$ and they are vertices in $\\{V_{7},V_{8},V_{9}\\}$. We leave
it as an exercise to the reader. Denote by $\bigtriangleup_{0}$ the triangle
with vertices $V_{7}$, $V_{8}$ and $V_{9}$.

Consider the two triangles (cf. Figure 6) respectively:

  * •

$\bigtriangleup_{L}$ with vertices $W_{3}$, $W_{1}$ and $V_{0}$,

  * •

$\bigtriangleup_{R}$ with vertices $V_{4}$, $W_{4}$ and $W_{2}$.

$V_{0}$$W_{1}$$V_{1}$$V_{2}$$W_{2}$$W_{3}$$V_{3}$$V_{4}$$W_{4}$$W_{5}$$V_{5}$$W_{6}$$V_{6}$$V_{7}$$W_{7}$$W_{8}$$V_{8}$$V_{9}$$W_{9}$$W_{10}$$V_{10}$$V_{11}$$W_{11}$$W_{0}$$\bigtriangleup_{0}$$\bigtriangleup_{L}$$\bigtriangleup_{R}$$\bigtriangleup_{-}$$s_{3,L}$$s_{3}$

Figure 5. The triangles $\bigtriangleup_{?}$ in type $E_{6}$ with
$?=\\{0,-,L,R\\}$.

$V_{0}$$W_{1}$$V_{1}$$V_{2}$$W_{2}$$W_{3}$$V_{3}$$V_{4}$$W_{4}$$W_{5}$$V_{5}$$W_{6}$$V_{6}$$V_{7}$$W_{7}$$W_{8}$$V_{8}$$V_{9}$$W_{9}$$W_{10}$$V_{10}$$V_{11}$$W_{11}$$W_{0}$$s_{6}$$s_{5}$$s_{4}$$s_{2}$$s_{3}$$s_{1}$

$s_{1}$$s_{2}$$s_{4}$$s_{5}$$s_{6}$$s_{3}$

Figure 6. The intersection quiver of a $12$-gon of type $E_{6}$

######  Lemma 4.11.

The triangles $\bigtriangleup_{L}$ and $\bigtriangleup_{R}$ (as well as
$\bigtriangleup_{0}$) are parallel (i.e. related by translations).

###### Proof.

The statement follows from the equalities of diagonals/vectors:

$V_{7}V_{8}=V_{4}W_{4}=W_{3}W_{1}\quad\text{and}\quad
V_{8}V_{9}=W_{4}W_{2}=W_{1}V_{0},$

cf. [QZ22, Construction 6.3 or § 6.3]. ∎

Comparing the vertices in $\bigtriangleup_{L}$ and $\bigtriangleup_{R}$ with
respect to (2.1), we set $\bigtriangleup_{+}$ to be the smaller one among them
in the sense that

$\displaystyle\bigtriangleup_{+}\colon=\begin{cases}\bigtriangleup_{L}&\text{if~{}}~{}W_{1}<W_{4},\\\
\bigtriangleup_{R}&\text{if~{}}~{}W_{4}<W_{1}.\end{cases}$ (4.7)

######  Lemma 4.12.

Let $\mathbf{J}_{6}$ be the convex hull of
$\bigtriangleup_{-}\cup\bigtriangleup_{+}$. Then it is a positively
convex/stable hexagon (of type $A_{5}$).

###### Proof.

By [QZ22, § 6.3], we know that $\mathbf{J}_{6}$ is in fact a narrow hexagon
either $\mathbf{H}_{1}$ or $\mathbf{H}_{2}$, where

$\mathbf{H}_{j}=V_{j-1}V_{j}V_{j+1}V_{j+2}W_{j+2}W_{j}.$

Note that those $\mathbf{H}_{j}$’s inherit the positive convexity from
$\mathbf{V}$. ∎

Denote the ${\operatorname{Im}}Z$-ind diagonals of $\mathbf{J}_{6}$ from top
to bottom by $s_{1},s_{2},s_{4},s_{5},$ and $s_{6}$ respectively. Set
$s_{3}\colon=\upharpoonleft\hskip-3.61371pt{V_{0}W_{2}}\hskip-3.61371pt\upharpoonright=\upharpoonleft\hskip-3.61371pt{W_{3}V_{4}}\hskip-3.61371pt\upharpoonright$,
i.e.

$\displaystyle
s_{3}=\begin{cases}V_{0}W_{2}=W_{3}V_{4}&\text{if~{}}~{}V_{0}<W_{2}(\Longleftrightarrow
W_{3}<V_{4}),\\\
W_{2}V_{0}=V_{4}W_{3}&\text{if~{}}~{}W_{2}<V_{0}(\Longleftrightarrow
V_{4}<W_{3})\end{cases}$ (4.8)

and define $\operatorname{Sim}\mathbf{V}\colon=\\{s_{i}\mid 1\leq i\leq 6\\}$.
Then, following Definition 4.3, we also have the _intersection quiver_
$\mathcal{Q}(\mathbf{V})$. See Figure 6 for an example, which corresponds to
Figure 6.

$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$000$$M_{1}$$000$$M_{2}$$000$$M_{3}$$000$$M_{4}$$000$$M_{5}$$000$$M_{6}$$V_{0}V_{3}$$V_{1}V_{4}$$V_{2}V_{5}$$V_{3}V_{6}$$V_{4}V_{7}$$V_{5}V_{8}$$V_{11}V_{2}$$V_{0}V_{2}$$V_{1}V_{3}$$V_{2}V_{4}$$V_{3}V_{5}$$V_{4}V_{6}$$V_{5}V_{7}$$V_{11}W_{2}$$V_{0}W_{3}$$V_{1}W_{4}$$V_{2}W_{5}$$V_{3}W_{6}$$V_{4}W_{7}$$000$$V_{0}V_{1}$$000$$V_{1}V_{2}$$000$$V_{2}V_{3}$$000$$V_{3}V_{4}$$000$$V_{4}V_{5}$$000$$V_{5}V_{6}$$000$$V_{1}V_{2}$$000$$V_{2}V_{3}$$000$$W_{0}W_{2}$$000$$W_{1}W_{3}$$000$$W_{2}W_{4}$$000$$W_{3}W_{5}$$000$$W_{4}W_{6}$$000$$W_{5}W_{7}$$000$$W_{6}W_{8}$$000$$W_{1}W_{3}$$000$$W_{2}W_{4}$
Figure 7. Part of the AR quiver of
$\operatorname{\mathcal{D}}_{\infty}(E_{6})$

######  Proposition 4.13.

Theorem 4.4 holds for any quiver $Q$ of type $E_{6}$. In particular, the
underlying diagram of the Ext-quiver/intersection quiver is of the form (of
homogenous degree one):

$s_{1}$$s_{2}$$s_{4}$$s_{5}$$s_{6}$$s_{3}$ (4.9)

###### Proof.

Recall the assumption that $\sigma\in\operatorname{ToSt}(Q)$ and
$\mathbf{V}:=\mathbf{V}_{\sigma}$ be the corresponding stable $12$-gon of type
$E_{6}$. Following the construction above, we have the set of upward diagonals

$\operatorname{Sim}\mathbf{V}=\operatorname{Sim}\mathbf{J}_{6}\cup\\{s_{3}\\}=\\{s_{i}\mid
1\leq i\leq 6\\},$

where $\mathbf{J}_{6}$ is the narrow hexagon (either $\mathbf{H}_{1}$ or
$\mathbf{H}_{2}$) in Lemma 4.12 and
$\operatorname{Sim}\mathbf{J}_{6}=\\{s_{1},s_{2},s_{4},s_{5},s_{6}\\}$.

Denote by $\operatorname{thick}\mathbb{L}$ the thick subcategory of
$\operatorname{\mathcal{D}}_{\infty}(Q)$ generated by objects in
$\\{\mathfrak{M}(\ell)\mid\ell\in\mathbb{L}\\}$ for some set $\mathbb{L}$
consisting of upward diagonals.

We claim that $\operatorname{Sim}_{0}\colon=\\{\mathfrak{M}(s_{i})\mid 1\leq
i\leq 6\\}\subset\mathcal{H}_{\sigma}$ is a simple minded collection of
$\operatorname{\mathcal{D}}_{\infty}(Q)$, in the sense that

  * •

$\operatorname{Hom}^{\leq 0}(S,T)=0$ for any $S,T\in\operatorname{Sim}_{0}$
and

  * •

$\operatorname{thick}\operatorname{Sim}\mathbf{V}=\operatorname{\mathcal{D}}_{\infty}(Q)$.

If so, $\operatorname{Sim}_{0}$ generates a heart that is contained in
$\mathcal{H}_{\sigma}$, which must coincide with $\mathcal{H}_{\sigma}$. In
other words, $\operatorname{Sim}_{0}=\operatorname{Sim}\mathcal{H}_{\sigma}$
and then the proposition follows easily.

For the first condition, we actually have

$\displaystyle\operatorname{Hom}^{\bullet}(S,T)=\operatorname{Hom}^{1}(S,T).$
(4.10)

To verify this, we first list several full subcategories of
$\operatorname{\mathcal{D}}_{\infty}(Q)$:

  * $($a$)$

$\operatorname{thick}\operatorname{Sim}\mathbf{J}_{6}\cong\operatorname{\mathcal{D}}_{\infty}(A_{5})$
since $\mathbf{J}_{6}$ is a stable hexagon (of type $A_{5}$).

  * $($b$)$

$\operatorname{thick}\\{s_{3},s_{4},s_{5},s_{6}\\}\cong\operatorname{\mathcal{D}}_{\infty}(A_{4})$
with the corresponding stable pentagon $\mathbf{J}_{5}$ (of type $A_{4}$)
formed by the vertices of $s_{3,?}$ and those of $\bigtriangleup_{-}$, where

$s_{3,?}\colon=\begin{cases}s_{3,L}=V_{0}W_{2}&\text{if
$\bigtriangleup_{+}=\bigtriangleup_{L}$,}\\\ s_{3,R}=V_{4}W_{3}&\text{if
$\bigtriangleup_{+}=\bigtriangleup_{R}$.}\end{cases}$

Indeed, we have
$\operatorname{Sim}\mathbf{J}_{5}=\\{s_{3},s_{4},s_{5},s_{6}\\}$.

  * $($c$)$

$\operatorname{thick}\operatorname{Sim}\bigtriangleup_{\pm}\cong\operatorname{\mathcal{D}}_{\infty}(A_{2})$
with the corresponding stable triangle $\bigtriangleup_{\pm}$ of type $A_{2}$.
Its indecomposable objects (up to shift) are the ones in green (for
$\bigtriangleup_{+}$) and in blue (for $\bigtriangleup_{-}$) in Figure 7.

By Theorem 4.4 and $($a$)$, any objects $S,T$ in $\\{\mathfrak{M}(s)\mid
s\in\operatorname{Sim}\mathbf{J}_{6}\\}$ satisfies (4.10). Note that the
object with labeling $XY\in\mathrm{Ad}(\mathbf{V})$ in Figure 7 is precisely
$\mathfrak{M}(XY)\in\mathcal{R}(Q)$ and the one in yellow corresponds to some
shift of $\mathfrak{M}(s_{3})$. Using $($c$)$, it is straightforward to see
that $\operatorname{Hom}^{\bullet}$ vanishes between (i.e. from and to)
$\mathfrak{M}(s_{3})$ and
$\mathfrak{M}(s_{i})\in\operatorname{thick}\operatorname{Sim}\bigtriangleup_{\pm}$
for $i\in\\{1,2,5,6\\}$. By $($b$)$, there exists exactly one
$\operatorname{Hom}^{1}$ between $\mathfrak{M}(s_{3})$ and
$\mathfrak{M}(s_{4})$ as they are consecutive $\operatorname{Im}Z$-Ind
diagonals of $\mathbf{J}_{5}$. In all, the equality (4.10) holds for any
$S,T\in\operatorname{Sim}_{0}$, which implies the first condition. In fact,
from the discussion above, we have actually shown that the Ext-quiver of
$\operatorname{Sim}_{0}$ is of the form (4.9).

For any $j\in\mathbb{Z}_{12}$, define

$\mathrm{UD}_{j}\colon=\mathrm{UD}(\mathbf{H}_{j})\cup\\{\upharpoonleft\hskip-3.61371pt{V_{j-1}W_{j+1}}\hskip-3.61371pt\upharpoonright,\upharpoonleft\hskip-3.61371pt{W_{j+1}V_{j+2}}\hskip-3.61371pt\upharpoonright\\}.$

The second condition can be proved as follows:

  * $($i$)$

By Lemma 3.5, there is a triangle with objects
$\upharpoonleft\hskip-3.61371pt{\ell}\hskip-3.61371pt\upharpoonright$, where
$\ell$ runs through the edges of a contractible triangle.

  * $($ii$)$

By Lemma 4.2, we know that
$\operatorname{thick}\operatorname{Sim}\mathbf{J}_{6}=\operatorname{thick}\mathrm{UD}(\mathbf{J}_{6})$.
Note that

$\operatorname{Sim}\mathbf{V}=\operatorname{Sim}\mathbf{J}_{6}\cup\\{\upharpoonleft\hskip-3.61371pt{V_{0}W_{2}}\hskip-3.61371pt\upharpoonright=\upharpoonleft\hskip-3.61371pt{W_{3}V_{4}}\hskip-3.61371pt\upharpoonright\\},$

where $\mathbf{J}_{6}=\mathbf{H}_{j}$ and
$\upharpoonleft\hskip-3.61371pt{V_{0}W_{2}}\hskip-3.61371pt\upharpoonright=\upharpoonleft\hskip-3.61371pt{W_{3}V_{4}}\hskip-3.61371pt\upharpoonright$
is one element in $\mathrm{UD}_{j}\setminus\mathrm{UD}(\mathbf{H}_{j})$ for
some $j\in\\{1,2\\}$. It immediately follows from $($i$)$ that
$\operatorname{thick}\operatorname{Sim}\mathbf{V}=\operatorname{thick}\mathrm{UD}_{1}$
or $\operatorname{thick}\mathrm{UD}_{2}$.

  * $($iii$)$

For any $\ell\in\mathrm{UD}_{j+1}$, one checks that $\ell$ is an edge of a
contractible triangle whose other two edges are in $\mathrm{UD}_{j}$. By
$($i$)$, we have
$\operatorname{thick}\mathrm{UD}_{j+1}\subset\operatorname{thick}\mathrm{UD}_{j}$.
Hence all $\mathrm{UD}_{j}$’s coincide for $j\in\mathbb{Z}_{12}$.

  * $($iv$)$

By definition, we have

$\mathrm{UD}(\mathbf{V})=\bigcup_{j\in\mathbb{Z}_{12}}\mathrm{UD}_{j}.$

Then $($iii$)$ implies that
$\operatorname{thick}\mathrm{UD}_{j}=\operatorname{thick}\mathrm{UD}(\mathbf{V})=\operatorname{\mathcal{D}}_{\infty}(Q).$
Together with $($ii$)$, we obtain the required claim.∎

###  4.4. Exceptional types $E_{n}$

$V_{2}$$V_{3}$$V_{4}$$W_{4}$$W_{2}$$V_{5}$$V_{0}$$V_{1}$$W_{1}$$W_{3}$$\bigtriangleup_{L}$$\bigtriangleup_{R}$$\square_{4}$$s_{3,R}$$s_{3}$

Figure 8. The triangles $\bigtriangleup_{L},\bigtriangleup_{R}$ and the square
$\square_{-}$ in type $E_{7}$

$V_{2}$$V_{3}$$V_{4}$$W_{4}$$W_{2}$$V_{5}$$V_{0}$$V_{1}$$W_{1}$$W_{3}$$s_{7}$$s_{6}$$s_{5}$$s_{4}$$s_{2}$$s_{1}$$s_{3}$
$s_{1}$$s_{2}$$s_{4}$$s_{5}$$s_{6}$$s_{7}$$s_{3}$

Figure 9. The intersection quiver of an $18$-gon of type $E_{7}$

Let $\sigma\in\operatorname{ToSt}(E_{n})$, where $n\in\\{6,7,8\\}$, and
$\mathbf{V}\colon=\mathbf{V}_{\sigma}$ the associated stable $h$-gon of type
$E_{n}$. Reorder the points in $\\{V_{j},W_{j}\mid j\in\mathbb{Z}_{h_{Q}}\\}$
increasingly with respect to (2.1) as $\\{Y_{i}\mid 1\leq i\leq 2h_{Q}\\}$. As
in § 4.3 for the case $n=6$, we have the following in all exceptional cases:

  * •

the points in $\\{Y_{i}\mid 1\leq i\leq n-3\\}$ are consecutive vertices of
$\mathbf{V}$. Thus we can assume that they are $\\{V_{j}\mid 1\leq j\leq
n-3\\}$. Those vertices $V_{j}$’s form a positively convex $(n-3)$-gon
$\mathbf{P}_{n-3}$. More precisely, $\mathbf{P}_{3}$ is a triangle
$\bigtriangleup_{-}$, $\mathbf{P}_{4}$ is a square $\square_{-}$ and
$\mathbf{P}_{5}$ is a pentagon $\pentagon_{-}$.

  * •

there are two distinct parallel triangles $\bigtriangleup_{L}$ and
$\bigtriangleup_{R}$ (since $W_{1}V_{0}=W_{4}W_{2}$ and
$W_{3}W_{1}=V_{n-2}W_{4}$), where

    * –

$\bigtriangleup_{L}$ is the triangle with vertices $V_{0},W_{1}$ and $W_{3}$,
and

    * –

$\bigtriangleup_{R}$ is the triangle with vertices $W_{2},W_{4}$ and
$V_{n-2}$.

  * •

define $\bigtriangleup_{+}$ to be the smaller one among the above two
triangles as in (4.7) and $\mathbf{J}_{n}$ the convex hull of
$\mathbf{P}_{n-3}\cup\bigtriangleup_{+}$, which is a positively convex $n$-gon
due to the stability of $\mathbf{V}$.

  * •

denote the ${\operatorname{Im}}Z$-ind diagonals of $\mathbf{J}_{n}$ from top
to bottom by $s_{1},s_{2},s_{4},\ldots,s_{n}$ respectively and set
$s_{3}\colon=\upharpoonleft\hskip-3.61371pt{V_{0}W_{2}}\hskip-3.61371pt\upharpoonright=\upharpoonleft\hskip-3.61371pt{W_{3}V_{n-2}}\hskip-3.61371pt\upharpoonright$.

######  Definition 4.14.

For type $E_{n}$, the set $\operatorname{Sim}\mathbf{V}$ of
_${\operatorname{Im}}Z$ -ind diagonals_ of $\mathbf{V}$ is defined to be
$\\{s_{i}\mid 1\leq i\leq n\\}$. Thus we have the _intersection quiver_
$\mathcal{Q}(\mathbf{V})$ as in Definition 4.3. See Figures 9-9 and Figures
14-14 for examples of the associated intersection quivers in types $E_{7}$ and
$E_{8}$ respectively.

Now, we can upgrade Proposition 4.13 to all exceptional cases.

######  Theorem 4.15.

Theorem 4.4 holds for any quiver $Q$ of type $E_{n}$, where $n\in\\{6,7,8\\}$.
In particular, the underlying diagram of the Ext-quiver/intersection quiver is
of the form

$s_{1}$$s_{2}$$s_{4}$$s_{5}$$\cdots$$s_{n}.$$s_{3}$ (4.11)

##  5\. Reineke’s conjecture

In this section, we will use the geometric model constructed in § 4 to prove
Reineke’s conjecture about the existence of total stability functions on the
module category of any Dynkin quiver. The conjecture for type $A$ case is
proved in [BGMS19], where the construction is more or less the same. However,
we include the details as well for self-containedness.

###  5.1. Type $A$ and $D$

######  Proposition 5.1.

For any quiver $Q$ of type $A_{n}$, there is a stable $(n+1)$-gon $\mathbf{V}$
such that $\mathcal{Q}^{1-?}(\mathbf{V})\cong Q$.

$Y_{1}$$Y_{n+1}$$Y_{i}$$\epsilon_{i}=-$$Y_{j}$$\epsilon_{j}=+$$s_{i}$$s_{i-1}$$s_{j-1}$$s_{j}$
Figure 10. Stable $(n+1)$-gon of type $A_{n}$

###### Proof.

Assume that the underlying diagram of $Q$ is as follows:

$1$$2$$3$$\cdots$$n.$

We define a sign function $\epsilon$ on $Q_{1}=\\{a_{i}\mid 2\leq i\leq n\\}$,
where $a_{i}$ is the arrow between $i-1$ and $i$, as:

$\displaystyle\epsilon_{i}\colon=\begin{cases}+&\textrm{if
$i-1\xrightarrow{a_{i}}i$},\\\ -&\textrm{if
$i-1\xleftarrow{a_{i}}i$}.\end{cases}$ (5.1)

Set $Y_{1}=(0,-1)$ and $Y_{n+1}=(1,0)$ in $\mathbb{C}$. For any $2\leq i\leq
n$, let $Y_{i}=(x_{i},y_{i})$ be the point on $S^{1}=\\{x^{2}+y^{2}=1\\}$ with

$y_{i}=-1+\frac{2(i-1)}{n},$

and $\operatorname{sign}(x_{i})=-\epsilon_{i}$. Clearly, we have

$Y_{1}<Y_{2}<\cdots<Y_{n+1}.$

Denote by $\mathbf{V}$ the convex $(n+1)$-gon with vertices $\\{Y_{i}\mid
1\leq i\leq n+1\\}$. By Definition 4.3, the intersection quiver of
$\mathbf{V}$ is isomorphic to $Q$, cf. Figure 10. ∎

######  Proposition 5.2.

For any quiver $Q$ of type $D_{n}$, there is a stable $2(n-1)$-gon
$\mathbf{V}$ of type $D_{n}$ such that $\mathcal{Q}^{1-?}(\mathbf{V})\cong Q$.

###### Proof.

Assume that the underlying diagram of $Q$ is as follows:

$1$$2$$3$$\cdots$$n-2$$n-1$$n.$

Denote by $Q_{1}=\\{a_{i}\mid 2\leq i\leq n\\}$, where $a_{i}$ is the arrow
between $i-1$ and $i$ for $2\leq i\leq n-1$ and $a_{n}$ is the arrow between
$n-2$ and $n$. We define a sign function $\epsilon$ on $Q_{1}$ with
$\epsilon_{i}$ given by (5.1) for $2\leq i\leq n-1$ and

$\displaystyle\epsilon_{n}\colon=\begin{cases}+&\textrm{if
$n-2\xrightarrow{a_{n}}n$},\\\ -&\textrm{if
$n-2\xleftarrow{a_{n}}n$}.\end{cases}$ (5.2)

Set $Y_{\pm n}=(0,\pm 1)$ in $\mathbb{C}$. For any $3\leq i\leq n-1$, let
$Y_{i}=(x_{i},y_{i})$ be the point on $S^{1}=\\{x^{2}+y^{2}=1\\}$ with

$y_{\pm i}=\frac{\pm i}{n},$

and $\operatorname{sign}(x_{\pm i})=\pm\epsilon_{n+1-|i|}$. To choose vertices
$Y_{\pm 1}$ and $Y_{\pm 2}$, there are three cases.

  * $($a$)$

If $\epsilon_{n-1}=\epsilon_{n}$, then we set $Y_{\pm 2}=(x_{\pm 2},\pm 2/n)$
be the points on $S^{1}$ with $\operatorname{sign}(x_{\pm
2})=\pm\epsilon_{n-1}$ and $Y_{\pm 1}=(0,0)$. Let $\mathbf{V}$ be the convex
$2(n-1)$-gon with vertices $\\{Y_{\pm i}\mid 2\leq i\leq n\\}$ and two
punctures $B_{\pm}=Y_{\pm 1}$, which is a stable $2(n-1)$-gon of type $D_{n}$
by construction. Moreover, the ordering (4.3) holds. By Definitions 4.3 and
4.7, the intersection quiver of $\mathbf{V}$ is isomorphic to $Q$. The
pictures in Figure 11 give two examples for the case $n=6$.

$Y_{-6}$$Y_{-5}$$Y_{-4}$$Y_{-3}$$Y_{-2}$$Y_{6}$$Y_{5}$$Y_{4}$$Y_{3}$$Y_{2}$$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{5}=s_{6}$$Y_{-6}$$Y_{-5}$$Y_{-4}$$Y_{-3}$$Y_{-2}$$Y_{6}$$Y_{5}$$Y_{4}$$Y_{3}$$Y_{2}$$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{5}=s_{6}$$Y_{\pm
1}=O=B_{\pm 1}$$Y_{\pm 1}=O=B_{\pm 1}$mirror

$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{6}$$s_{5}$

$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{6}$$s_{5}$

Figure 11. Two mirror stable $10$-gons of type $D_{6}$ in Case $($a$)$

  * $($b$)$

If $\epsilon_{n-1}=-\epsilon_{n}$ and $(\epsilon_{n-1},\epsilon_{n})=(-,+)$,
then we set $Y_{\pm 1}=(\pm 1,0)$. Let $\mathbf{V}$ be the convex $2(n-1)$-gon
with vertices $\\{Y_{\pm i}\mid 1\leq i\leq n,i\neq 2\\}$. As $\mathbf{V}$ is
inscribed in $S^{1}$, the level-$(n-2)$ diagonal-gon is the intersection of
$(n-1)$ symmetric rectangles containing the origin $O$. Hence, such
level-$(n-2)$ diagonal-gon contains a disk with center at $O$ whose diameter
$d$ is the shortest edge of those rectangles. Take $B_{\pm}=Y_{\pm 2}=(0,\pm
d/3)$ inside such a disk. Then the convex $2(n-1)$-gon $\mathbf{V}$ with
punctures $B_{\pm}$ is a stable $2(n-1)$-gon of type $D_{n}$. Moreover, the
ordering (4.3) holds. By Definitions 4.3 and 4.7, the intersection quiver of
$\mathbf{V}$ is isomorphic to $Q$. The left picture in Figure 12 gives an
example for the case $n=6$.

  * $($b${}^{\prime})$

If If $\epsilon_{n-1}=-\epsilon_{n}$ and
$(\epsilon_{n-1},\epsilon_{n})=(+,-)$, then we take the stable $2(n-1)$-gon
$\mathbf{V}^{\prime}$ of type $D_{n}$ obtained from $\mathbf{V}$ in Case
$($b$)$ by clockwise rotating $R_{-\theta}=e^{-\mathbf{i}\pi\theta}$ of a
small angle $0<\theta\ll 1$. Such a rotation does not change the order between
almost all $Y_{i}$’s, except the one between $Y_{\pm 1}$. Set
$Y^{\prime}_{i}=R_{-\theta}(Y_{i})$ for $i\neq\pm 1$ and $Y^{\prime}_{\pm
1}=R_{-\theta}(Y_{\mp 1})$. Then the ordering (4.3) holds for all
$Y^{\prime}_{i}$’s. By Definitions 4.3 and 4.7, the intersection quiver of
$\mathbf{V}^{\prime}$ is isomorphic to $Q$. The right picture in Figure 12
gives an example for the case $n=6$.∎

$x$$y$$Y_{-6}$$Y_{-5}$$Y_{-4}$$Y_{-3}$$Y_{1}$$Y_{6}$$Y_{5}$$Y_{4}$$Y_{3}$$Y_{-1}$$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{5}$$s_{6}$$O$$Y_{2}=B_{+}$$Y_{-2}=B_{-}$

$x$$y$$Y^{\prime}_{-6}$$Y^{\prime}_{-5}$$Y^{\prime}_{-4}$$Y^{\prime}_{-3}$$Y^{\prime}_{-1}$$Y^{\prime}_{6}$$Y^{\prime}_{5}$$Y^{\prime}_{4}$$Y^{\prime}_{3}$$Y^{\prime}_{1}$$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{6}$$s_{5}$$O$$Y^{\prime}_{2}=B_{+}$$Y^{\prime}_{-2}=B_{-}$

$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{6}$$s_{5}$

$s_{1}$$s_{2}$$s_{3}$$s_{4}$$s_{6}$$s_{5}$

Figure 12. Two stable $10$-gons of type $D_{6}$ in Cases $($b$)$ and
$($b${}^{\prime})$

Note that any quiver in Case $($b${}^{\prime})$ is isomorphic to some quiver
in Case $($b$)$ by swapping the vertices $n-1$ and $n$. Thus, one can omit the
proof in Case $($b${}^{\prime})$. However we keep it to demonstrate the
$\mathbb{C}$-action effect.

###  5.2. Exceptional type

Firstly, note that if we prove the conjecture for $E_{8}$, then $E_{6}$ and
$E_{7}$ follows directly by restricted to full subcategories. Secondly, we
have the following easy observation, cf. Lemma 5.3 and an illustration in
Figure 11.

Consider the mirror operation $\mathbf{V}\mapsto\mathbf{V}^{\vee}$ with
respect to $y$-axis, which also mirror on the order (2.1) to be

$\displaystyle
W_{1}<W_{2}\Longleftrightarrow\begin{cases}y_{1}<y_{2}\text{\quad or}\\\
y_{1}=y_{2},\,x_{1}>x_{2},\end{cases}$ (5.3)

for any two points $W_{1}=(x_{1},y_{1})$ and $W_{2}=(x_{2},y_{2})$ in
$\mathbb{C}$. Let $\sigma^{\vee}$ be the total stability condition to
$\mathbf{V}^{\vee}$. The order (5.3) effects the definition of upward
diagonals, in the sense that we need to take $\operatorname{arg}$ to be in
$(0,\pi]$. Consequently, we need choose
$\mathcal{H}^{\vee}\colon=\mathcal{P}^{\vee}(0,1]$ as the heart of
$\sigma^{\vee}$ so that everything in the paper still hold, e.g. Proposition
3.2.

######  Lemma 5.3.

The intersection quiver $\mathcal{Q}(\mathbf{V}^{\vee})$ is the opposite
quiver of $\mathcal{Q}(\mathbf{V})$ and it is the Ext-quiver of the heart
$\mathcal{H}^{\vee}$.

Note that by this lemma, one can also easily reduce Case $($b${}^{\prime})$ to
Case $($b$)$ using the mirror operation.

######  Remark 5.4.

As there are only finitely many indecomposable objects in
$\operatorname{\mathcal{D}}_{\infty}(Q)/[2]$, then
$\mathcal{P}(\theta)=\emptyset$ for $0<\theta\ll 1$. This implies that

$\mathcal{P}(0,1]=\mathcal{P}[\theta,1+\theta),$

which equals the canonical heart of $\theta\cdot\sigma$ (the
$\mathbb{C}$-action). Thus, in the setting of Lemma 5.3, if one combines the
mirror operation with a rotation smaller enough, the effect on the Ext-quiver
of the canonical heart would be precisely taking opposite quiver.

Next, we have another important observation.

Observation: Suppose that we have constructed a stable $h$-gon $\mathbf{V}$ of
type $E_{n}$, where $n\in\\{6,7,8\\}$, with the intersection quiver

$s_{1}$$s_{2}$$s_{4}$$s_{5}$$\cdots$$s_{n}$$s_{3}$ (5.4)

such that

$\displaystyle\operatorname{arg}Z_{\mathbf{V}}(S_{1})=\operatorname{arg}Z_{\mathbf{V}}(S_{3})=\operatorname{arg}Z_{\mathbf{V}}(S_{n})=0,$
(5.5)

where $S_{j}=\mathfrak{M}(s_{j})$ for $1\leq j\leq 8$. Then, by a small
deformation, $\operatorname{arg}Z_{\mathbf{V}}(S_{i})$ can become near $\pi$
for any leaf vertex $i\in\\{1,3,n\\}$ and thus reverse the arrow at $S_{i}$.
Note that since the space $\operatorname{Stgon}(Q)$ of stable $h$-gons of type
$Q$ is open, such small deformation is permitted.

######  Proposition 5.5.

For any quiver $Q$ of type $E_{n}$, where $n\in\\{6,7,8\\}$, there is a stable
$h$-gon $\mathbf{V}$ of type $E_{n}$ such that
$\mathcal{Q}^{1-?}(\mathbf{V})\cong Q$.

###### Proof.

By Lemma 5.3, we can fix the orientation of a chosen arrow (say between vertex
2 and 4). Further, by the observation above, we only need to construct
$2^{3}=8$ stable $30$-gons $\mathbf{V}$ with the following intersection
quivers:

$s_{1}$$s_{2}$$s_{4}$$s_{5}$$s_{6}$$s_{7}$$s_{8}$$s_{3}$ (5.6)

satisfying (5.5). All possibilities of those $8$ stable $30$-gons are shown in
Appendix A. ∎

###  5.3. Summary

Combining Propositions 5.1 5.2 and 5.5, we verify Reineke’s conjecture:

######  Theorem 5.6.

For any Dynkin quiver $Q$, there is a stability function on $\mathcal{H}(Q)$
such that any indecomposable is stable.

Final remark is that all the examples in the exceptional cases are produce by
GeoGebra where the precise coordinate information is in the tex/tikz code.

##  Appendix A List of examples of stable $30$-gons in type $E_{8}$

######  Example A.1.

Figures 15 and 16 give the eight examples of stable 30-gon $\mathbf{V}$, in
which we only draw the partial part containing $\operatorname{Sim}\mathbf{V}$
such that the associated intersection quiver $\mathcal{Q}(\mathbf{V})$ can be
read. More precisely, the intersection quiver is of the form

$s_{1}$$s_{2}$$s_{4}$$s_{5}$$s_{6}$$s_{7}$$s_{8}$$s_{3}$$a_{5}$$a_{6}$$a_{7}$
(A.1)

with $\epsilon_{i}$ denoting the sign of the orientation of $a_{i}$ for $5\leq
i\leq 7$ as in (5.1). And we have the sign labeling
$\epsilon_{5}\epsilon_{6}\epsilon_{7}$ with the following eight possibilities:

  * •

$---,+--,-+-,--+$ for the pictures in Figure 15, respectively, and

  * •

$++-,+-+,-++,+++$ for the pictures in Figure 16, respectively.

$V_{1}$$V_{2}$$W_{1}$$V_{3}$$W_{2}$$V_{4}$$W_{3}$$V_{5}$$V_{0}$$V_{6}$$W_{4}$$s_{3,L}$$\bigtriangleup_{L}$$\bigtriangleup_{R}$$\pentagon_{-}$$s_{3}$$B$

Figure 13. The triangles $\bigtriangleup_{L},\bigtriangleup_{R}$ and the
pentagon $\pentagon_{-}$ in type $E_{8}$

$V_{1}$$V_{2}$$W_{1}$$V_{3}$$W_{2}$$V_{4}$$W_{3}$$V_{5}$$V_{0}$$s_{8}$$s_{1}$$s_{3}$$s_{7}$$s_{6}$$s_{5}$$s_{4}$$s_{2}$

$s_{1}$$s_{2}$$s_{4}$$s_{5}$$s_{6}$$s_{7}$$s_{8}$$s_{3}$

Figure 14. The intersection quiver of a $30$-gon of type $E_{8}$

$s_{4}$$s_{3}$$s_{1}$$s_{2}$$s_{8}$$s_{5}$$s_{7}$$s_{6}$$V_{1}$$V_{2}$$W_{1}$$V_{3}$$W_{2}$$V_{4}$$W_{3}$$V_{5}$$V_{0}$$N$

$s_{1}$$s_{3}$$s_{4}$$s_{2}$$s_{8}$$s_{7}$$s_{6}$$s_{5}$$V_{1}$$V_{2}$$W_{1}$$V_{3}$$W_{2}$$V_{4}$$W_{3}$$V_{5}$$V_{0}$$N$$I^{\prime}_{1}$

$s_{4}$$s_{3}$$s_{1}$$s_{2}$$s_{8}$$s_{5}$$s_{7}$$s_{6}$$V_{1}$$V_{2}$$W_{1}$$V_{3}$$W_{2}$$V_{4}$$W_{3}$$V_{5}$$V_{0}$

$s_{4}$$s_{3}$$s_{1}$$s_{2}$$s_{7}$$s_{8}$$s_{6}$$s_{5}$$V_{1}$$V_{2}$$W_{1}$$V_{3}$$W_{2}$$V_{4}$$W_{3}$$V_{5}$$V_{0}$

Figure 15. Four stable 30-gons of type $E_{8}$

$s_{8}$$s_{1}$$s_{3}$$s_{4}$$s_{2}$$s_{7}$$s_{6}$$s_{5}$$V_{1}$$V_{2}$$W_{1}$$V_{3}$$W_{2}$$V_{4}$$W_{3}$$V_{5}$$V_{0}$

$s_{8}$$s_{1}$$s_{3}$$s_{7}$$s_{6}$$s_{5}$$s_{4}$$s_{2}$$V_{1}$$V_{2}$$W_{1}$$V_{3}$$W_{2}$$V_{4}$$W_{3}$$V_{5}$$V_{0}$

$s_{4}$$s_{3}$$s_{8}$$s_{1}$$s_{2}$$s_{6}$$s_{7}$$s_{5}$$V_{1}$$V_{2}$$W_{1}$$V_{3}$$W_{2}$$V_{4}$$W_{3}$$V_{5}$$V_{0}$$N$$I^{\prime}_{1}$

$s_{1}$$s_{3}$$s_{4}$$s_{2}$$s_{5}$$s_{8}$$s_{7}$$s_{6}$$V_{1}$$V_{2}$$W_{1}$$V_{3}$$W_{2}$$V_{4}$$W_{3}$$V_{5}$$V_{0}$

Figure 16. Another four stable 30-gons of type $E_{8}$

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