# A Thurston compactification of the space of stability conditions

Asilata Bapat and Anand Deopurkar and Anthony M. Licata

###### Abstract.

We propose a compactification of the moduli space of Bridgeland stability
conditions of a triangulated category. Under mild conditions on the
triangulated category, we conjecture that this compactification is a real
manifold with boundary, on which the action of the autoequivalence group
extends continuously. The key ingredient in the compactification is an
embedding of the stability space into an infinite-dimensional projective
space. We study this embedding in detail in the case of 2-Calabi–Yau
categories associated to quivers, and prove our conjectures in the $A_{2}$ and
$\widehat{A}_{1}$ cases. Central to our analysis is a detailed understanding
of Harder–Narasimhan multiplicities and how they transform under auto-
equivalences. We achieve this by introducing a structure called a
Harder–Narasimhan automaton and constructing examples for $A_{2}$ and
$\widehat{A}_{1}$.

##  1\. Introduction

A series of recent papers have established a fascinating analogy between the
Teichmüller space of a surface and the space of Bridgeland stability
conditions on a triangulated category [10, 8, 7]. In this analogy, a curve on
the surface corresponds to an object of the category, the topological
intersection number corresponds to the dimension of the hom spaces, a metric
corresponds to a stability condition, and the length of a curve prescribed by
a metric corresponds to the mass prescribed by the corresponding stability
condition.

Using this analogy, we can hope to transport powerful tools from geometry to
homological algebra. The goal of this paper is to outline one aspect of such a
program. We define a compactification of the space of Bridgeland stability
conditions on suitable triangulated categories, and propose a conjectural
description of its boundary. In the present paper, we work out this
compactification explicitly for the smallest non-trivial cases, namely the
case of the 2-Calabi–Yau categories associated to the $A_{2}$ quiver and the
$\widehat{A}_{1}$ quiver. We will extend our results to more general
triangulated categories in future work.

###  1.1. A projective embedding of the stability manifold

The key ingredient in Thurston’s compactification of the Teichmüller space is
its embedding in an infinite projective space. Roughly speaking, this
embedding sends a metric $\mu$ to the real-valued function on the set of
curves defined by the length with respect to $\mu$. By following the analogy,
we are led to the following construction.

Let $\mathbf{k}$ be a field, and $\mathcal{C}$ a $\mathbf{k}$-linear
triangulated category. Denote by $\mathbb{R}^{\mathcal{C}}$ the space of
functions from the set of objects of $\mathcal{C}$ to $\mathbb{R}$, endowed
with the product topology. Given a stability condition $\sigma$ on
$\mathcal{C}$ with central charge $Z$, the _mass_ of an object $x$ is defined
as

(1) $m_{\sigma}(x)=\sum_{i}|Z(z_{i})|,$

where $z_{i}$ are the semistable Harder–Narasimhan factors of $x$. The
association $\sigma\mapsto m_{\sigma}$ yields a continuous map

$m\colon\operatorname{Stab}(\mathcal{C})\to\mathbb{R}^{\mathcal{C}}.$

Recall that we have an action of $\mathbb{C}$ on
$\operatorname{Stab}(\mathcal{C})$. Let $\mathbb{P}^{\mathcal{C}}$ denote the
projective space
$(\mathbb{R}^{\mathcal{C}}\setminus\\{0\\})/\mathbb{R}^{\times}$. Then the
mass map descends to a continuous map

$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathcal{C}}.$

Guided by the analogy from Teichmüller theory, we may hope for the following.

######  Expectation 1.

The map
$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathcal{C}}$
is injective and a homeomorphism onto its image.

######  Expectation 2.

The closure of the image
$m\left(\operatorname{Stab}(\mathcal{C})/\mathbb{C}\right)\subset\mathbb{P}^{\mathcal{C}}$
is compact.

If these two expectations hold, then we obtain a compactification of
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$. They do seem to hold in a
number of cases of interest, such as for the Calabi–Yau categories associated
to connected quivers.

###  1.2. The boundary of the projective embedding

The next natural step is to describe the boundary. In Teichmüller theory, the
boundary can be described as follows. Each simple closed curve $\gamma$ on a
surface gives rise to a function on the set of all curves defined by the
topological (unsigned) intersection number with $\gamma$. The boundary is the
closure of the set of such functions.

These intersection functions have a natural categorical analogue. Assume that
the hom spaces in $\mathcal{C}$ are finite-dimensional. For each object
$x\in\mathcal{C}$, define $\hom(x)\in\mathbb{R}^{\mathcal{C}}$ as

$\hom(x)\colon y\mapsto\dim_{\mathbf{k}}\operatorname{Hom}^{*}(x,y).$

######  Expectation 3.

There is a suitable class of objects $\mathbf{S}\subset\mathcal{C}$ such that
the classes of functions $\hom(x)$ in $\mathbb{P}^{\mathcal{C}}$ form a dense
subset of the boundary of $m(\operatorname{Stab}(\mathcal{C})/\mathbb{C})$ in
$\mathbb{P}^{\mathcal{C}}$.

The class $\mathbf{S}$ should be regarded as the categorical analogue of the
set of simple closed curves on a surface. In the case of the 2-Calabi–Yau
categories associated to quivers, we expect $\mathbf{S}$ to be the set of
spherical objects of $\mathcal{C}$.

In Teichmüller theory, the boundary also has a beautiful modular
interpretation as the space of so-called projective measured foliations. At
present, we do not know an appropriate categorical analogue of this notion.

Let $M$ be the image of $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ in
$\mathbb{P}^{\mathcal{C}}$. Let $\overline{M}$ be the closure of $M$ and set
$\partial M=\overline{M}\setminus M$.

######  Expectation 4.

The pair $(\overline{M},\partial M)$ is a manifold with boundary.

Furthermore, in the cases where $M$ is known to be an open ball (for example,
the 2-Calabi–Yau categories of $ADE$-quivers), we expect $\overline{M}$ to be
the closed ball and $\partial M$ to be the sphere.

###  1.3. Summary of results

We conjecture that all of the above expectations hold when $\mathcal{C}$ is
the 2-Calabi–Yau category associated to a finite connected quiver. We prove
parts of this conjecture in various levels of generality in this paper.

Let $\mathcal{C}$ be the 2-Calabi–Yau category associated to a finite
connected quiver $\Gamma$ (see § 2 for a detailed description), and
$\mathbf{S}$ a suitable subset of spherical objects in $\mathcal{C}$. We prove
the following statements.

  1. (1)

The map
$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathbf{S}}$
is injective (part of 1).

  2. (2)

The closure of the image of $m$ is compact (2).

  3. (3)

The reduced hom functionals corresponding to spherical objects are in the
boundary of the image of $m$ (part of 4).

  4. (4)

All four expectations hold when $\Gamma$ is the $A_{2}$ quiver, resulting in a
complete description of a compactification of
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ in this case. See Figure 1 for a
sketch of the resulting picture.

  5. (5)

The first three expectations hold when $\Gamma$ is the $\widehat{A}_{1}$
quiver.

We view these results as a proof-of-concept, suggesting that these
expectations are reasonable, at least when restricted to a suitable yet
interesting class of triangulated categories.

$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$$P_{1}$$P_{2}\to
P_{1}$$P_{2}$$P_{1}\to P_{2}$$P_{2}\to P_{1}\to P_{1}[1]$$P_{2}\to P_{2}\to
P_{1}[1]$ Figure 1. The compactification of
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$. The spherical objects appear as
a dense subset of the boundary.

###  1.4. Dynamics of equivalences

One of the applications of our compactification of $M$, at least when
$\overline{M}$ is homeomorphic to a Euclidean ball, is a dynamical
classification of autoequivalences which parallels the Nielsen–Thurston
classification of mapping classes as periodic, reducible, or pseudo-Anosov.

Let $\beta\colon\mathcal{C}\to\mathcal{C}$ be an auto-equivalence. Since the
construction of $\overline{M}$ is functorial, $\beta$ induces a homeomorphism
$\beta\colon\overline{M}\to\overline{M}$. By the Brouwer fixed point theorem,
this homeomorphism must have a fixed point. There are two basic cases: the
fixed point lies $M$ or it lies in $\partial M$.

In the first case, we say that $\beta$ is _periodic_. Indeed, under some
conditions on $\mathcal{C}$, we can prove that a power of $\beta$ acts simply
by a triangulated shift $[n]$ on $\mathcal{C}$.

In the second case, there are two further subcases. The first is when $\beta$
fixes the function $\operatorname{\overline{hom}}(x)$ (up to scaling) for some
object $x\in\mathcal{C}$. In this case we say that $\beta$ is _reducible_. The
rationale for this terminology is as follows. Under suitable assumptions on
$\mathcal{C}$ and $x$, this implies that $\beta(x)$ is isomorphic to a shift
of $x$. In that case, the orthogonal hom complement to $x$ will be preserved
by $\beta$, and the dynamical study of $\beta$ is reduced to the study of its
action on this subcategory.

If $\beta$ is neither periodic nor reducible, we call it _pseudo-Anosov_. In
this case, its fixed point(s) in $\partial M$ consist of functions not
represented by objects of $\mathcal{C}$. (Finding a moduli interpretation of
such functions, analogous to Thurston’s notion of projective measured
foliations, is an important problem we do not address). Under some reasonable
assumptions on $\mathcal{C}$, the auto-equivalence $\beta$ should have a dense
orbit on $\partial M$, and $\beta$ should have two fixed points which exhibit
sink/source dynamics. We conjecture that pseudo-Anosov auto-equivalences in
this sense are also pseudo-Anosov in the sense of [9].

In the $A_{2}$ case, the above classification agrees with the classical
Nielsen–Thurston classification of elements of the 3-strand braid group
$B_{3}$, as mapping classes of the 3-punctured disk.

###  1.5. Harder–Narasimhan automata

The 2-Calabi–Yau category $\mathcal{C}$ associated to a quiver $\Gamma$ admits
the action of a large symmetry group—the Artin–Tits braid group $B_{\Gamma}$.
Our analysis of the stability manifold and its compactification heavily uses
this symmetry. To do so, we must understand how the mass of an object, or
better, its HN filtration, transforms under an auto-equivalence.

Fix a stability condition on $\mathcal{C}$, and let $\Sigma$ be the set of
semi-stable objects. For an object $x\in\mathcal{C}$, let
$HN(x)\in\mathbb{Z}^{\Sigma}$ denote the HN multiplicity vector of $x$. Let a
group $G$ act on $\mathcal{C}$ by auto-equivalences. The action of $G$ on
$\mathcal{C}$ does not usually induce a compatible linear action of $G$ on
$\mathbb{Z}^{\Sigma}$. Nevertheless, we prove in the $A_{2}$ and
$\widehat{A}_{1}$ cases that the action is “piecewise linear”. Roughly
speaking, the objects of $\mathcal{C}$ may be partitioned into subsets, called
_states_. For each pair of states, there are certain allowable elements of $G$
that send objects of the first state to objects of the second, and induce
linear transformations on the HN multiplicities. We formalise this phenomenon
using structures called HN automata, in § 3.

As an application of this construction, we recover a theorem of
Rouquier–Zimmermann [16, Proposition 4.8] in the type $A_{2}$ case and prove a
new analogue of this theorem in the type $\widehat{A}_{1}$ case. More
generally, we anticipate that automata should play an important role in
analysing group actions on categories.

###  1.6. $q$-analogue

The entire story above admits a $q$-analogue for any positive real number
$q>0$. Let $\sigma$ be a stability condition with central charge $Z$ and phase
function $\phi$. Define the _$q$ -mass_ of an object $x$ as follows (deforming
(1)):

$m_{q,\sigma}(x)=\sum_{i}|q^{\phi(z_{i})}Z(z_{i})|.$

We get a continuous map

$m_{q}\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathcal{C}},$

which agrees with our original map $m$ when $q=1$. We expect that for all
$q>0$, the map
$m_{q}\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathcal{C}}$
is injective and a homeomorphism onto its image. We thus obtain a family of
compactifications of $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ given by
the closures of $m_{q}(\operatorname{Stab}(\mathcal{C})/\mathbb{C})$ in
$\mathbb{P}^{\mathcal{C}}$. We conjecture that the closures
$\\{\overline{M}_{q}\\}_{q>0}$ are all homeomorphic to each other.

There are some interesting differences between the $q=1$ and $q\neq 1$ cases,
even in the case of the $A_{2}$ quiver. For example, at $q=1$, the functions
$\operatorname{\overline{hom}}(x)$ for spherical $x$ are dense in the
boundary: under the identifications $M_{q=1}\cong\mathbb{H}$, $\partial
M_{q=1}=\mathbb{R}\cup\infty$, these functions are identified with the
rational points $\mathbb{Q}\cup\infty$ of the boundary. When $q\neq 1$,
however, this is no longer the case. Instead, the $\hom(x)$ functions are
identified with a non-dense (in fact, fractal) subset of
$\mathbb{R}\cup\infty$, arising as an orbit of the Burau matrices of
$\operatorname{SL}_{2}(\mathbb{R})$. Equally compelling, and even less
understood at present, is the behaviour of $\overline{M}_{q}$ as $q\to\infty$
or $q\to 0$.

###  1.7. Organisation

In § 2 we give precise definitions of the categories we work with, and recall
some background on Bridgeland stability conditions. In § 3 we introduce the
definition of Harder–Narasimhan automata. In § 4 we describe an embedding of
the space of stability conditions into an infinite-dimensional projective
space, via mass functionals. We prove several general results about this
embedding: that it is injective and pre-compact, and that reduced hom
functionals lie in the boundary of this embedding. We also prove that this
embedding is a homeomorphism onto its image if we have access to a suitable HN
automaton.

In § 5 and § 6 we explicitly construct suitable HN automata for the cases of
the $A_{2}$ and $\widehat{A}_{1}$ quivers. Using these automata, we prove the
results mentioned in the summary. Along the way we also obtain new proofs of
existing results in the literature. For instance, we construct an explicit
isomorphism of $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ with an open disk
in the case of the $A_{2}$ quiver. Another isomorphism has been constructed in
[6]. We recover a result of Rouquier–Zimmermann [16, Proposition 4.8] for the
$A_{2}$ case, and prove an analogue of this theorem in the $\widehat{A}_{1}$
case.

In Appendix A we establish general results about rearrangements of
Harder–Narasimhan filtrations in a stability condition.

### Acknowledgements

We thank Louis Becker and Edmund Heng for carefully reading multiple drafts of
this paper, finding errors, and suggesting improvements. We thank Tom
Bridgeland for enlightening conversations and pointing out connections to
other work. We benefited from interactions with Fabian Haiden, Amnon Neeman,
Alexander Polishchuk, and Catharina Stroppel. A.M.L. is supported by the
Australian Research Council Award FT180100069 and DP180103150. A.D. is
supported by the Australian Research Council Award DE180101360.

##  2\. CY categories associated to quivers

###  2.1. The zigzag algebra and its homotopy category

Let $\Gamma$ be a connected graph without loops and multiple edges. Let
$B_{\Gamma}$ be the Artin–Tits braid group associated to $\Gamma$. This group
is generated by $\sigma_{i}$, where $i$ ranges over the set of vertices of
$\Gamma$, modulo the following relations:

$\displaystyle\sigma_{i}\sigma_{j}\sigma_{i}$
$\displaystyle=\sigma_{j}\sigma_{i}\sigma_{j}\text{ if there is an edge
between $i$ and $j$,}$ $\displaystyle\sigma_{i}\sigma_{j}$
$\displaystyle=\sigma_{j}\sigma_{i}\text{ otherwise.}$

Following [11], we describe a category $\mathcal{C}_{\Gamma}$ with a (weak)
action of $B_{\Gamma}$ that categorifies the standard representation of the
associated Coxeter group.

Let $D\Gamma$ denote the doubled quiver, which has the same vertices as
$\Gamma$ and two edges for each edge of $\Gamma$. More precisely, each edge f
of $\Gamma$ is replaced in $D\Gamma$ by two oriented edges connecting the same
vertices but with opposite orientations. Let $\mbox{Path}(D\Gamma)$ denote the
path algebra of $D\Gamma$. As a vector space over $\mathbf{k}$,
$\mbox{Path}(D\Gamma)$ is spanned by all oriented paths in $D\Gamma$, and the
multiplication is given by concatenation. The natural length function on
paths, which declares edges to have length one, induces a non-negative grading
on $\mbox{Path}(D\Gamma)$.

Since $\Gamma$ does not have multiple edges, we may express a path
unambiguously by listing the vertices it travels through in order. So, for
example, if $i$ and $j$ are neighbouring vertices, the element $(i|j|i)$
denotes the length two path which begins at $i$, travels along the unique
oriented edge from $i$ to $j$, and then travels back to $i$ along the unique
oriented edge from $j$ to $i$. Suppose $\Gamma$ has at least two vertices. We
let $A(\Gamma)$ denote the quotient algebra of the path algebra by the two-
sided ideal generated by the following elements:

  1. (1)

all length three paths,

  2. (2)

length two paths $(i|j|k)$ with $i\neq k$, and

  3. (3)

$(i|j|i)-(i|k|i)$ when $i$ is connected to both $j$ and $k$.

These relations are all homogeneous (of degree 2), so that the natural grading
on the path algebra descends to a grading on $A(\Gamma)$. If $\Gamma$ has a
single vertex, (type $A_{1}$), we set $A(\Gamma)=\mathbb{C}[x]/x^{2}$, graded
by setting $\deg(x)=2$. Following [11], we call $A(\Gamma)$ the zigzag algebra
of $\Gamma$.

The minimal idempotents of $A(\Gamma)$ are the length zero paths $(i)$ for
$i\in\Gamma$. The modules $P_{i}=A(\Gamma)(i)$ are indecomposable projective
left $A(\Gamma)$ modules. Any finitely-generated graded projective left
$A(\Gamma)$ module is isomorphic to a direct sum of grading shifts of the
modules $P_{i}$.

The category considered in [11] is the homotopy category of graded projective
left $A(\Gamma)$-modules. This admits two shift functors, namely the
homological shift and the grading shift. Following [17, § 4] and [18, § 3], we
work in a simpler setting where the two shifts are identified. We consider
$A(\Gamma)$ as a differential graded algebra (dga), where all the
differentials are zero. Let ${\rm DGM}_{\Gamma}$ be the category of
differential graded modules (dgm) over the dga $A(\Gamma)$. Objects of ${\rm
DGM}_{\Gamma}$ are dgms, and morphisms are chain maps, compatible with the
action of $A(\Gamma)$. Let ${\mathcal{D}}_{\Gamma}$ be the category obtained
from ${\rm DGM}_{\Gamma}$ by inverting the quasi-isomorphisms. Note that each
graded $A(\Gamma)$-module can be considered as an element of ${\rm
DGM}_{\Gamma}$ (and hence of $\mathcal{D}_{\Gamma}$); we simply take all
differentials to be zero. We let $\mathcal{C}_{\Gamma}\subset D_{\Gamma}$ be
the full triangulated subcategory generated by the objects $P_{i}$.

Having given the construction, we state the salient properties of
$\mathcal{C}_{\Gamma}$.

  1. (1)

$\mathcal{C}_{\Gamma}$ is a $\mathbf{k}$-linear triangulated category

  2. (2)

$\mathcal{C}_{\Gamma}$ is 2-Calabi–Yau: that is, for a pair of objects
$x,y\in\mathcal{C}$, we have natural isomorphisms

$\operatorname{Hom}(x,y)\cong\operatorname{Hom}(y,x[2])^{*}.$

  3. (3)

$\mathcal{C}$ is classically generated by the objects $P_{i}$, satisfying

$\displaystyle\operatorname{Hom}(P_{i},P_{i}[n])$
$\displaystyle=\begin{cases}\mathbf{k}&\text{ if $n=0$ or $n=2$,}\\\ 0&\text{
otherwise};\end{cases}$ $\displaystyle\operatorname{Hom}(P_{i},P_{j}[n])$
$\displaystyle=\begin{cases}\mathbf{k}&\text{ if $n=1$ and $i$ and $j$ are
neighbours,}\\\ 0&\text{ otherwise, for $i\neq j$}.\end{cases}$

The Grothendieck group $K_{\Gamma}$ of $\mathcal{C}_{\Gamma}$ is a free
$\operatorname{\mathbb{Z}}$ module generated by the classes $\\{[P_{i}]\\}$.

###  2.2. Spherical objects and twists

The indecomposable projective modules $P_{i}$ are _spherical_ in the sense of
[17, Definition 2.9]. In particular,
$\operatorname{Hom}^{*}(x,x)\cong\mathbf{k}[t]/t^{2}$ as a graded
$\mathbf{k}$-algebra. Any spherical object $x$ gives rise to an
autoequivalence $\sigma_{x}\colon\mathcal{C}_{\Gamma}\to\mathcal{C}_{\Gamma}$
called the _spherical twist_ in $x$ (see [17]). The twists in the $P_{i}$
satisfy the defining relations of the Artin–Tits braid group $B_{\Gamma}$
associated to $\Gamma$. That is,

$\displaystyle\sigma_{P_{i}}\sigma_{P_{j}}\sigma_{P_{i}}$
$\displaystyle\cong\sigma_{P_{j}}\sigma_{P_{i}}\sigma_{P_{j}}\text{ when $i$
and $j$ are neighbours},$ $\displaystyle\sigma_{P_{i}}\sigma_{P_{j}}$
$\displaystyle\cong\sigma_{P_{j}}\sigma_{P_{i}}\text{ when $i\neq j$ are not
connected by an edge of $\Gamma$}.$

Via the homomorphism $B_{\Gamma}\to\operatorname{Aut}(\mathcal{C}_{\Gamma}$
defined by $\sigma_{i}\mapsto\sigma_{P_{i}}$, we have a (weak) action of
$B_{\Gamma}$ on $\mathcal{C}_{\Gamma}$. This action is known to be faithful in
some cases—for example when $\Gamma$ is an ADE Dynkin diagram [4, 14]—and is
conjecturally faithful for all $\Gamma$. The induced action on $K_{\Gamma}$ is
the standard representation of the Coxeter group associated to $\Gamma$.

###  2.3. Stability conditions

We assume some familiarity with the theory of Bridgeland stability conditions
(see [5]). We will denote the space of stability conditions on $\mathcal{C}$
by $\operatorname{Stab}(\mathcal{C})$. A point
$\tau=(\mathcal{P},Z)\in\operatorname{Stab}(\mathcal{C})$ is specified by two
compatible structures: a slicing $\mathcal{P}$ and a central change $Z$.

There is an action of the complex numbers $\operatorname{\mathbb{C}}$ on
$\operatorname{Stab}(\mathcal{C})$ lifting the scaling action of
$\operatorname{\mathbb{C}}^{*}$ on the space of central charges. Explicitly,
if $\tau=(\mathcal{P},Z)$ and $\omega=a+i\pi b$, then

$(\omega\cdot\mathcal{P})(\phi)=\mathcal{P}(\phi-b)\text{ and }\omega\cdot
Z=e^{\omega}Z.$

####  2.3.1. Harder–Narasimhan filtrations

The slicing $\mathcal{P}$ of a stability condition $\tau$ associates to every
object $X\in\mathcal{C}$ a Harder-Narasimhan filtration

$X_{0}\to X_{1}\to\dots\to X_{n}=X,$

uniquely determined by the fact that they objects
$Y_{k}=\operatorname{Cone}(X_{k-1}\rightarrow X_{k})$ are semi-stable of
decreasing phase

$\phi(Y_{1})>\phi(Y_{2})>\dots>\phi(Y_{n}).$

####  2.3.2. Mass functionals

For a $\tau$ \- semi-stable object $Y$, we define the mass of $Y$ to be the
modulus of the value of the central charge at $Y$, $m_{\tau}(Y)=|Z(Y)|$. We
extend this definition to arbitrary objects of $\mathcal{C}$ by defining the
$\tau$-mass of $X$ to be the sum of the $\tau$ masses of the semi-stable terms
in the Harder-Narasimhan filtration of $X$:

$m_{\tau}(X)=\sum_{k}m_{\tau}(Y_{k}).$

##  3\. Harder–Narasimhan (HN) automata

A Harder–Narasimhan automaton (or HN automaton) is a device that allows for an
effective computation of HN multiplicities. This is an example of a more
general structure, which we call automata, that we now introduce.

Fix a group $G$. We think of $G$ as a category with one object and invertible
arrows, and denote this category by $\underline{G}$. A $G$-_automaton_ is a
category $\Theta$ with a functor $\sigma\colon\Theta\to\underline{G}$. We
refer to objects of $\Theta$ as _states_ and arrows of $\Theta$ as
_transitions_. To ease notation, we often drop $\sigma$.

Note that each arrow $e$ in $\Theta$ is labelled with an element of $G$,
namely $\sigma(e)$. We can thus represent $\Theta$ by a $G$-labelled graph (we
usually show only a subset of arrows that generates the category). Conversely,
a $G$-labelled graph represents an automaton, whose states are the vertices of
the graph, whose transitions are (free) compositions of the edges in the
graph, and where the functor to $\underline{G}$ is defined by the edge labels.

We say that an element $g$ of $G$ is _recognised by $\Theta$_ if $g=\sigma(e)$
for some arrow $e$ of $\Theta$. Likewise, we say that a writing $g=g_{1}\cdots
g_{n}$ is _recognised by $\Theta$_ if $g_{i}=\sigma(e_{i})$ where
$e_{1},\dots,e_{n}$ is a sequence of composable arrows in $\Theta$.

Just as a $G$-set is a functor $\underline{G}\to{\rm\bf Set}$, we define an
$\Theta$-set to be a functor $\Theta\to{\rm\bf Set}$. Likewise, an
$\Theta$-representation is a functor $\Theta\to{\rm\bf Ab}$ or
$\Theta\to{\rm\bf Vect}_{k}$ for some field $k$. Morphisms of sets and
representations are natural transformations between the respective functors,
as expected. Keeping with the usual terminology, we refer to these as
_$\Theta$ -equivariant maps_.

Note that a $G$-set gives a $\Theta$-set, obtained by pre-composing with
$\Theta\to\underline{G}$. Thus, the idea of an $\Theta$-set is weaker than
that of a $G$-set.

Let $S$ be a $\Theta$-set. Suppose we have a subset $T(v)\subset S(v)$ for
every state $v$. The $\Theta$-set _generated by_ $T$, denoted $\langle
T\rangle$, is the smallest $\Theta$-subset of $S$ such that $\langle
T\rangle(v)$ contains $T(v)$ for all $v$. Explicitly, $\langle T\rangle(v)$ is
the set consisting of $e(x)$ for all arrows $e\colon w\to v$ and $x\in T(w)$.

Let $\mathcal{C}$ be a triangulated category and $G$ a group acting on
$\mathcal{C}$ by triangulated auto-equivalences. Let
$\operatorname{ob}\mathcal{C}$ be the set of isomorphism classes of objects of
$\mathcal{C}$. Then $\operatorname{ob}\mathcal{C}$ is a $G$-set. Fix a
stability condition $\tau$ on $\mathcal{C}$. Let
$\Sigma\subset\operatorname{ob}\mathcal{C}$ be the set of $\tau$-semistable
objects. For $x\in\operatorname{ob}\mathcal{C}$, let
$HN_{\tau}(x)\in\mathbb{Z}^{\Sigma}$ be the $\tau$-HN mutiplicity vector of
$x$ and $m_{\tau}(x)$ the $\tau$-mass of $x$.

######  Definition 3.1.

A _$\tau$ -HN automaton_ consists of a $G$-automaton $\Theta$ along with the
following additional data:

  1. (1)

a sub $\Theta$-set $S\subset\operatorname{ob}\mathcal{C}$,

  2. (2)

a $\Theta$-representation $M$ and an $\Theta$-equivariant map $i\colon S\to
M$,

  3. (3)

for every state $v$, a linear map $HN_{v}\colon M(v)\to\mathbb{Z}^{\Sigma}$.

The data is required to satisfy the following for every $s\in S(v)$:

$HN_{\tau}(s)=HN_{v}(i(s)).$

A _$\tau$ -mass automaton_ is analogous, with $HN_{v}$ replaced by
$m_{v}\colon M(v)\to\mathbb{R}$ that satisfies

$m_{\tau}(s)=m_{v}(i(s)).$

######  Remark 3.2.

A $\tau$-HN automaton yields a $\tau$-mass automaton. Indeed, we take $m_{v}$
to be the composite of $HN_{v}$ and the linear function
$\mathbb{Z}^{\Sigma}\to\mathbb{R}$ that sends a semi-stable object to its
mass.

Abusing notation, we denote an HN (or mass) automaton by the same name as the
underlying automaton. We also leave out $i\colon S\to M$ from the notation,
and instead write $[x]$ for $i(x)$. We refer to an element of
$\operatorname{ob}\mathcal{C}$ contained in $S(v)$ as an object _supported at_
the state $v$. Note that an object may be supported at multiple states. We say
that an object is _recognised by $\Theta$_ if it is supported at some state.

HN automata give a partial linearisation of the action of $G$ on HN
multiplicities. To see how, suppose for simplicity that the maps $HN_{v}\colon
M(v)\to\mathbb{Z}^{\Sigma}$ are injective. Consider a state $v$ and an arrow
$e\colon v\to w$. We have a linear transformation

$M(e)\colon M(v)\to M(w)$

that agrees with the transformation

$HN_{\tau}(s)\mapsto HN_{\tau}(e\cdot s)$

for every $s\in S(v)$. This linearisation is only partial for two reasons:
first, it only applies to objects recognised by the automaton; and second,
even for an object recognised by the automaton, say for an object supported at
$v$, it only applies to the arrows originating from $v$.

A useful automaton would recognise a rich class of objects and
transformations. In the $A_{2}$ and $\widehat{A}_{1}$ cases, we construct
automata that recognise all braid actions and all spherical objects that are
in the braid group orbit of a standard spherical object. It also has
additional nice properties—for example, it is “mass degenerating”, as
explained in § 4.2. These properties allow us to prove structural results
about stability conditions on $\mathcal{C}$.

##  4\. The projective embedding and its boundary

For this section, let $\mathcal{C}$ be the 2-CY category associated to any
finite connected quiver, as defined in § 2. Fix $\mathbf{S}$ to be a set of
objects of $\mathcal{C}$ satisfying the following properties.

  1. (1)

The set $\mathbf{S}$ contains the objects $\\{P_{i}\mid i\in\Gamma\\}$.

  2. (2)

The set $\mathbf{S}$ is closed under the action of the Artin–Tits braid group
$B_{\Gamma}$.

Given a stability condition $\tau$ and an object $x$ of $\mathcal{C}$, the
$\tau$-mass of $x$ is a non-negative real number. Acting on $\tau$ be the
natural action of $\mathbb{C}$ on $\operatorname{Stab}(\mathcal{C})$ rescales
the $\tau$-mass of every object $x$ by the same constant. Therefore we can
define a _mass map_

$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathbf{S}}=\mathbb{P}(\mathbb{R}^{\mathbf{S}}),$

sending the class of a stability condition $\tau$ to the projectivised
infinite vector with coordinates $m_{\tau}(x)$ for each $x\in\mathbf{S}$.

The aim of this section is to prove some general properties about the mass
map.

###  4.1. Injectivity

We first show that the map
$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathbf{S}}$
is injective. Let $\heartsuit_{\operatorname{std}}$ be the heart of the
standard $t$-structure on $\mathcal{C}$. Say that a stability condition
$\tau=(Z,P)$ is standard if there is some $\alpha\in\mathbb{R}$ such that the
abelian category $P([\alpha,\alpha+1))$ is equal to
$\heartsuit_{\operatorname{std}}$. Note that the property of being standard is
invariant under the action of $\mathbb{C}$ on
$\operatorname{Stab}(\mathcal{C})$.

Any standard stability condition can be continuously deformed to any other
standard stability condition: we can first match up the standard hearts by
rotation, and then deform the central charge without changing the standard
heart from one stability condition to the other. Let
$\operatorname{Stab}^{\circ}(\mathcal{C})/\mathbb{C}$ be the connected
component of $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ containing the
standard stability conditions.

######  Proposition 4.1 (Injectivity).

Let $\mathcal{C}$ be the 2CY category associated to a finite connected quiver.
Then the mass map
$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathbf{S}}$
is injective on $\operatorname{Stab}^{\circ}(\mathcal{C})/\mathbb{C}$.

###### Proof.

Let
$\widetilde{m}\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{R}^{\mathbf{S}}\backslash\\{0\\}$
be the lift of the map $m$ characterised by

$\sum\widetilde{m}(P_{i})=1.$

We show that $\widetilde{m}$ is injective on
$\operatorname{Stab}^{\circ}(\mathcal{C})/\mathbb{C}$, which is equivalent to
showing that $m$ is injective.

Let $\tau$ be a stability condition. By [12, Proposition 4.13], $\tau$ is in
the braid group orbit of a standard stability condition. (Although [12] treats
the case of preprojective algebras, the same proof works in our setting.) So
by applying a braid if necessary, suppose that $\tau$ is standard. By applying
a phase shift if necessary, suppose that
$P_{\tau}([0,1))=\heartsuit_{\operatorname{std}}$. In particular, the objects
$\\{P_{i}\mid i\in Q_{0}\\}$ are stable objects in $P_{\tau}([0,1))$.

Assume that the central charge $Z_{\tau}$ is scaled so that for any semistable
object $x$, we have

$\widetilde{m}_{\tau}(x)=|Z_{\tau}(x)|.$

In fact, the semistable objects of $\tau$ are characterised by this property.

Let $\tau$ and $\tau^{\prime}$ be two stability conditions such that
$\widetilde{m}_{\tau}=\widetilde{m}_{\tau^{\prime}}$, with central charges
rescaled as above. Let $x$ be a stable object of $\tau$, which means in
particular that

$\widetilde{m}_{\tau}(x)=|Z_{\tau}(x)|.$

If $x$ is not $\tau^{\prime}$-stable, then it has a filtration (a refinement
of the Harder–Narasimhan filtration) with stable pieces $y_{1},\ldots,y_{k}$,
such that

$\widetilde{m}_{\tau^{\prime}}(x)=\sum_{i=1}^{k}\widetilde{m}_{\tau^{\prime}}(y_{i}).$

Since $\widetilde{m}_{\tau}=\widetilde{m}_{\tau^{\prime}}$, the same equality
holds for $\tau$:

$|Z_{\tau}(x)|=\widetilde{m}_{\tau}(x)=\sum_{i=1}^{k}\widetilde{m}_{\tau}(y_{i}).$

By the triangle inequality for the vectors $Z_{\tau}(x)$ and $Z_{\tau}(y_{i})$
for each $i$, we see that all these vectors lie on the same real ray in the
complex plane. A non-trivial equality as above cannot hold if $x$ is
$\tau$-stable, because a stable object of phase $\phi$ is simple in the
abelian category $P_{\tau}(\phi)$. Therefore $x$ is also
$\tau^{\prime}$-stable.

We see that $\tau$ and $\tau^{\prime}$ have the same collections of stable
objects. We now show that up to the action of $\mathbb{C}$, they have
identical central charges and slicings.

Let us show inductively that up to a simultaneous $\mathbb{C}$-action, we have
$Z_{\tau^{\prime}}(P_{i})=Z_{\tau}(P_{i})$ and
$\phi_{\tau^{\prime}}(P_{i})=\phi_{\tau}(P_{i})$ for each $i\in Q_{0}$. Order
$Q_{0}$ such that for each $i>1$ there is some $j<i$ such that the sub-quiver
$\\{i,j\\}$ forms a doubled quiver of type $A_{2}$.

For the base case, by rotating and scaling $\tau^{\prime}$ if necessary,
assume that $Z_{\tau^{\prime}}(P_{1})=Z_{\tau}(P_{1})$, and
$\phi_{\tau^{\prime}}(P_{1})=\phi_{\tau}(P_{1})$. For the induction step,
consider some $i>1$. Fix $j<i$ such that the sub-quiver $\\{i,j\\}$ is a
doubled quiver of type $A_{2}$. By the induction hypothesis, we have
$Z_{\tau^{\prime}}(P_{j})=Z_{\tau}(P_{j})$ and
$\phi_{\tau^{\prime}}(P_{j})=\phi_{\tau}(P_{j})$.

First suppose that $\phi_{\tau}(P_{i})=\phi_{\tau}(P_{j})$. In this case, the
objects $P_{j}\to P_{i}$ and $P_{i}\to P_{j}$ are both $\tau$-semistable, and
moreover

$\widetilde{m}_{\tau}(P_{j}\to P_{i})=\widetilde{m}_{\tau}(P_{i}\to
P_{j})=\widetilde{m}_{\tau}(P_{i})+\widetilde{m}_{\tau}(P_{j}).$

Note that the same equalities hold for $\widetilde{m}_{\tau}$. Let us show
that $\phi_{\tau^{\prime}}(P_{i})=\phi_{\tau^{\prime}}(P_{j})$. If not, then
one of them is smaller than the other; suppose for instance that
$\phi_{\tau^{\prime}}(P_{i})<\phi_{\tau^{\prime}}(P_{j})$. In this case,
$P_{j}\to P_{i}$ is $\tau^{\prime}$-semistable, and so

$\widetilde{m}_{\tau^{\prime}}(P_{j}\to P_{i})=|Z_{\tau^{\prime}}(P_{j}\to
P_{i})|.$

By the triangle inequality, we have

$\widetilde{m}_{\tau^{\prime}}(P_{j}\to
P_{i})<\widetilde{m}_{\tau^{\prime}}(P_{i})+\widetilde{m}_{\tau^{\prime}}(P_{j}),$

which is a contradiction. So
$\phi_{\tau^{\prime}}(P_{i})=\phi_{\tau^{\prime}}(P_{j})$, and by the
induction hypothesis, we conclude that
$Z_{\tau^{\prime}}(P_{i})=Z_{\tau}(P_{i})$.

Next suppose that $\phi_{\tau}(P_{i})<\phi_{\tau}(P_{j})$; the other direction
is analogous. In this case, the object $P_{j}\to P_{i}$ is $\tau$-stable, and

$\phi_{\tau}(P_{i})<\phi_{\tau}(P_{j}\to
P_{i})<\phi_{\tau}(P_{j})<\phi_{\tau}(P_{i})+1.$

Since the stable objects in $\tau$ and $\tau^{\prime}$ agree, $P_{j}\to P_{i}$
is also $\tau^{\prime}$-stable. Since we know that

$\operatorname{Hom}(P_{i},P_{j}\to P_{i})\neq
0,\quad\operatorname{Hom}(P_{j}\to P_{i},P_{j})\neq 0,\quad\text{and
}\operatorname{Hom}(P_{j},P_{i}[1])\neq 0,$

the same inequalities hold for $\phi_{\tau^{\prime}}$:

(2) $\phi_{\tau}^{\prime}(P_{i})<\phi_{\tau}^{\prime}(P_{j}\to
P_{i})<\phi_{\tau}^{\prime}(P_{j})<\phi_{\tau}^{\prime}(P_{i})+1.$

We also know that

$m_{\tau^{\prime}}(P_{i})=m_{\tau}(P_{i}),\quad m_{\tau^{\prime}}(P_{j}\to
P_{i})=m_{\tau}(P_{j}\to P_{i}).$

The three numbers $m_{\tau^{\prime}}(P_{i})$, $m_{\tau^{\prime}}(P_{j}\to
P_{i})$, and $m_{\tau^{\prime}}(P_{j})$ form the three side lengths of a
triangle. Using this together with (2) above, we can uniquely reconstruct the
central charge $Z_{\tau^{\prime}}(P_{i})$. Therefore we see that
$Z_{\tau^{\prime}}(P_{i})=Z_{\tau}(P_{i})$. The phase
$\phi_{\tau^{\prime}}(P_{i})$ is now uniquely determined up to an integer
shift. Once again, (2) fixes this integer shift with respect to
$\phi_{\tau^{\prime}}(P_{j})$, which is known to equal $\phi_{\tau}(P_{j})$.
Therefore we see that $\phi_{\tau^{\prime}}(P_{i})=\phi_{\tau}(P_{i})$, and
the proof is complete.

The Grothendieck group of $\mathcal{C}$ is spanned by the classes of
$\\{P_{i}\mid i\in Q_{0}\\}$. So for any object $x$ of $\mathcal{C}$, we now
have $Z_{\tau^{\prime}}(x)=Z_{\tau}(x)$. Consequently,
$\phi_{\tau^{\prime}}(x)$ is an integer shift of $\phi_{\tau}(x)$ for each
stable $x$. It remains to show that $\phi_{\tau^{\prime}}(x)=\phi_{\tau}(x)$
for each stable $x$.

Let $x$ be stable in $\tau$ (equivalently, in $\tau^{\prime}$). Since $\tau$
is standard, $x$ belongs to $\heartsuit_{\operatorname{std}}[n]$ for some
integer $n$. Then $x$ is an iterated extension of $\\{P_{i}[n]\mid i\in
Q_{0}\\}$ in the abelian category $\heartsuit_{\operatorname{std}}[n]$.
Suppose that we have nonzero maps $P_{i}[n]\to x$ and $x\to P_{j}[n]$ in the
abelian category $\heartsuit_{\operatorname{std}}[n]$, for some $i,j\in
Q_{0}$. These correspond to nonzero maps $P_{i}[n]\to x$ and $x\to P_{j}[n]$
in $\mathcal{C}$. We see that

$n\leq\phi_{\tau^{\prime}}(P_{i}[n])\leq\phi_{\tau^{\prime}}(x),\quad\phi_{\tau^{\prime}}(x)\leq\phi_{\tau^{\prime}}(P_{j}[n])<n+1.$

These inequalities pin down $\phi_{\tau^{\prime}}(x)$ to lie in the interval
$[n,n+1)$, and $\phi_{\tau}(x)$ lies in the same interval. Combined with the
fact that $\phi_{\tau^{\prime}}(x)$ is an integer shift of $\phi_{\tau}(x)$,
we have $\phi_{\tau^{\prime}}(x)=\phi_{\tau}(x)$.

We have shown that up to an overall rotation or scaling, $\tau^{\prime}$ and
$\tau$ have the same central charges and the same stable objects with the same
phases. We conclude that $\tau=\tau^{\prime}$ in
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$, and this completes the proof of
injectivity. ∎

###  4.2. Homeomorphism when there is an automaton

Assuming the existence of a suitable HN automaton, we can prove that the mass
map
$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathbf{S}}$
is a homeomorphism onto its image. Let $\tau$ be a stability condition such
that there exists an $\tau$-mass automaton $\Theta$ with objects $S$ and
representation $M$.

We need to make a further assumption on the automaton, namely that it is a
degenerating automaton. We now describe this assumption. Recall that for an
object $x$ supported at a state $v$, the symbol $[x]$ denotes the image of $x$
in $M(v)$ under the map $S\to M$.

Let $x\to y\to z\xrightarrow{+1}$ be a distinguished triangle. By [13], we
have the triangle inequality

$m_{\tau}(y)\leq m_{\tau}(x)+m_{\tau}(z).$

If the inequality is strict, we say that the triangle is _non-degenerate_. Let
$e\colon v\to w$ be an arrow of $\Theta$. We say that the $e$ _degenerates_
the triangle if:

  1. (1)

$e\cdot x$, $e\cdot y$, and $e\cdot z$ are direct sums of objects supported at
$w$,

  2. (2)

we have

$[e\cdot y]=[e\cdot x]+[e\cdot z].$

The second condition implies that

$m_{\tau}(e\cdot y)=m_{\tau}(e\cdot x)+m_{\tau}(e\cdot z).$

Linearity ensures that this degeneration of the triangle inequality persists:
for any arrow $f\colon w\to u$ of $\Theta$, we have

$m_{\tau}(fe\cdot y)=m_{\tau}(fe\cdot x)+m_{\tau}(fe\cdot z).$

######  Definition 4.2.

Suppose $\Theta$ is a $\tau$-mass automaton. We say that $\Theta$ is
_degenerating_ if there exists a finite set $T$ of non-degenerate triangles
such that for every $g\in G$, either $\tau$ and $g^{-1}\tau$ have the same
stable objects or $g=\sigma(e)$ for some arrow $e$ in $\Theta$ that
degenerates a triangle in $T$.

Let $\mathcal{C}$ be the 2-CY category associated to a connected quiver
$\Gamma$. Let $\mathbf{S}\subset\mathcal{C}$ be the set of spherical objects
of $\mathcal{C}$, and $\tau$ a standard stability condition.

######  Proposition 4.3.

In the setup above, suppose we have a degenerating $\tau$-mass automaton
$\Theta$. Then the map
$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathbf{S}}$
is a local homeomorphism onto its image at $\tau$. That is, there exist open
sets $U\subset\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ containing $\tau$
and $V\subset\mathbb{P}^{\mathbf{S}}$ containing $m(\tau)$ such that

$m|_{U}\colon U\to V\cap\operatorname{im}(m)$

is a homeomorphism.

###### Proof of 4.3.

Let $U\subset\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ be the set of
stability conditions with the same stable objects as $\tau$. As in the proof
of injectivity (4.1), order the vertices of $\Gamma$ so that for every $i>1$,
there exists $j<i$ such that the sub-quiver $\\{i,j\\}$ forms a doubled quiver
of type $A_{2}$. Let $S\subset\mathbf{S}$ be the finite set

$S=\\{P_{i}\mid i\in Q_{0}\\}\cup\\{P_{j}\to P_{i}\mid\\{i,j\\}\cong
A_{2}\\}.$

Consider the restricted mass map $m\colon U\to\mathbb{P}^{S}$. In the proof of
injectivity, we re-constructed $\tau\in U$ from $m(\tau)\in\mathbb{P}^{S}$.
Observe that the inverse map $m(\tau)\mapsto\tau$ is continuous. In other
words, $U\to\mathbb{P}^{S}$ is a homeomorphism onto its image, and hence, so
is $U\to\mathbb{P}^{\mathbf{S}}$.

We have seen that the map

$m\colon U\to m(U)$

is a homeomorphism. To conclude the proposition, it remains to show that there
exists an open subset $V\subset\mathbb{P}^{\mathbf{S}}$ containing $m(\tau)$
such that

$V\cap\operatorname{im}(m)\subset m(U).$

To construct $V$, we use that $\Theta$ is mass degenerating. Let $T$ be a
finite set of non-degenerate triangles as required for this property. Let
$V\subset\mathbb{P}^{\mathbf{S}}$ be defined by the inequalities

$m(y)<m(x)+m(z)$

for a triangle $x\to y\to z\xrightarrow{+1}$ in $T$. Then
$V\cap\operatorname{im}(m)$ is contained in $m(U)$, as desired. ∎

###  4.3. Pre-compactness

Recall that $\mathcal{C}$ be the 2-CY category associated to a connected
quiver $Q$, and $\mathbf{S}\subset\mathcal{C}$ is the set of spherical objects
of $\mathcal{C}$, and we have the projectivised mass map

$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathbf{S}}.$

######  Proposition 4.4 (Pre-compactness).

In the setup above, the closure of the image of $m$ is compact.

###### Proof.

Let
$\widetilde{m}\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\operatorname{\mathbb{R}}^{\mathbf{S}}$
be the lift of $m$ characterised by

$\sum\widetilde{m}(P_{i})=1.$

Let $\widetilde{B}$ be the closure in $\operatorname{\mathbb{R}}^{\mathbf{S}}$
of the image of $\widetilde{m}$.

Recall that for an exact triangle

$x\to y\to z\xrightarrow{+1},$

we have the following triangle inequality [13, Proposition 3.3]:

$\widetilde{m}_{\tau}(y)\leq\widetilde{m}_{\tau}(x)+\widetilde{m}_{\tau}(z).$

For every $s\in\mathbf{S}$, there exists an $n=n(s)$ and a filtration

$0=s_{0}\to s_{1}\to\cdots\to s_{n}=s,$

where the sub-quotients are twists of $P_{i}$. By the triangle inequality, and
the normalisation $\sum\widetilde{m}(P_{i})=1$, we have

$\widetilde{m}_{\tau}(s)\leq n=n(s).$

In other words, $\widetilde{m}$ maps
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ to the product $\prod_{s\in
S}[0,n(s)]$. By the Tychonoff theorem, the product is compact, and hence
$\widetilde{B}$ is compact. Thanks to the normalisation
$\sum\widetilde{m}(P_{i})=1$, we see that $\widetilde{B}$ is contained in
$\operatorname{\mathbb{R}}^{\mathbf{S}}\setminus 0$. Finally, note that the
closure of the image of $m$ in $\mathbb{P}^{\mathbf{S}}$ is contained in the
image of $\widetilde{B}$ under the projection map, which is a compact set.
Hence, the closure of the image of $m$ is also compact. ∎

###  4.4. Hom functionals are in the boundary

Let $a$ be a spherical object. The goal of this section is to prove that the
projectivized functional $\operatorname{\overline{hom}}(a)$ lies in the
closure of the image of $\operatorname{Stab}(\mathcal{C})$. The key ingredient
is the following.

######  Proposition 4.5.

Let $\tau$ be a stability condition in which $a$ is stable. For every
$x\in\mathbf{S}$, we have

$\lim_{n\to\infty}\frac{m_{\tau}(\sigma_{a}^{n}x)}{n}=m(a)\cdot\operatorname{\overline{hom}}(a,x).$

######  Corollary 4.6.

In $\mathbb{P}^{\mathbf{S}}$ we have

$\lim_{n\to\infty}m(\sigma_{a}^{-n}\tau)=\operatorname{\overline{hom}}(a).$

In particular, the projectivized functional $\operatorname{\overline{hom}}(a)$
is in the closure of the image of $\operatorname{Stab}(\mathcal{C})$.

For the proof of 4.5, we need some preparatory lemmas.

######  Lemma 4.7.

Let $x\in\mathcal{C}$ be any object that does not contain any shift of $a$ as
a direct summand. Consider the filtration

$x\to\sigma_{a}x\to\sigma^{2}_{a}x$

with subquotients $\hom(a,x)\otimes a[1]$ and $\hom(a,\sigma_{a}x)\otimes
a[1]$. Then the connecting map

$\hom(a,\sigma_{a}x)\otimes a[1]=\hom(a,x)\otimes a\to\hom(a,x)\otimes a[2]$

is equal to $\operatorname{id}\otimes\operatorname{loop}_{a}$.

###### Proof.

Let $x$ be any object. The connecting map

(3) $\hom(a,x)\otimes a\to\hom(a,x)\otimes a[2]$

is adjoint to a map

(4) $\hom(a,x)\otimes\hom(a,x)^{*}\otimes a\to a[2].$

By the 2-CY property, we have $\hom(a,x)^{*}=\hom(x,a)[2]$, and thus the map
in (4) is equivalent to a map

(5) $\hom(a,x)\otimes\hom(x,a)\otimes a\to a.$

We leave it to the reader to check that the map (5) is the composite

$\hom(a,x)\otimes\hom(x,a)\otimes a\to\hom(a,a)\otimes a\to a,$

where the first map is the composition of the first two factors and the second
map is evaluation.

Now suppose that $x$ does not have shifts of $a$ as summands. Then, it is easy
to check that the image of the composition map

$\hom(a,x)\otimes\hom(x,a)\to\hom(a,a)$

is the one-dimensional subspace spanned by $\operatorname{loop}_{a}$, and that

$\hom(a,x)\otimes\hom(x,a)\to\langle\operatorname{loop}_{a}\rangle$

is a perfect pairing. By undoing the adjunctions, we conclude that (3) is
$\operatorname{id}\otimes\operatorname{loop}_{a}$. ∎

######  Lemma 4.8.

Let $x\in\mathcal{C}$ be any object that does not contain any shift of $a$ as
a direct summand. There exists $N$ (depending on $x$ and $a$) such that for
every $n\geq N$, the filtration

$0\to\sigma_{a}^{n}x\to\sigma_{a}^{n+1}x$

is geodesic.

###### Proof.

We need to show that the connecting map

$\hom(a,\sigma_{a}^{n}x)\otimes a[1]\to\sigma_{a}^{n}x[1]$

is rectifiable. After rewriting $\hom(a,\sigma_{a}^{n}x)$ as $\hom(a,x)[-n]$
and eliminating the shift by $[1]$, the map becomes

$\chi\colon\hom(a,x)\otimes a[-n]\to\sigma_{a}^{n}x.$

Define $\ell_{n}$ by the exact triangle

$x\to\sigma_{a}^{n}x\to\ell_{n}\xrightarrow{+1}.$

Thanks to repeated applications of 4.7, it is easy to see that we have

$\ell_{n}=\hom(a,x)\otimes a_{n},$

where $a_{n}$ is an iterated extension of twists of $a$ by loop maps. More
precisely, the objects $a_{n}$ are defined inductively as follows. Set
$a_{1}=a[1]$ and observe that we have the map $a[-1]\to a[1]$ given by
$\operatorname{loop}_{a}$. Assuming we have defined $a_{n-1}$ along with a map
$a[-n+1]\to a_{n-1}$, define $a_{n}$ by the triangle

$a[-n+1]\to a_{n-1}\to a_{n}\xrightarrow{+1}.$

To continue the inductive definition, we need to provide a map $a[-n]\to
a_{n}$. Consider the loop map $a[-n]\to a[-n+2]$. It is easy to check that the
composite $a[-n]\to a[-n+2]\to a_{n-1}[1]$ vanishes, and thus yields a map
$a[-n]\to a_{n}$.

By definition, we have the filtration

$a_{1}\to a_{2}\to\dots\to a_{n}\to a_{n+1}$

with subquotients $a,a[-1],\dots,a[-n+1]$. This filtration is non-overlapping,
and hence geodesic. As a result, the filtration

(6) $\ell_{1}\to\ell_{2}\to\dots\to\ell_{n}\to\ell_{n+1},$

obtained by applying $\hom(a,x)\otimes-$ to the previous filtration, is also
geodesic.

Let $\alpha$ be a real number. We need to show that the map

(7) $\left(\hom(a,x)\otimes
a[-n]\right)_{\geq\alpha}\to(\sigma_{a}^{n}x)_{<\alpha+1}$

induced by $\chi$ is zero. By shifting $x$ if necessary, assume that all the
HN factors of $x$ have phase greater than $0$. Let $N$ be such that for any
$n\geq N$, all the HN factors of $\hom(a,x)\otimes a[-n]$ have phase less than
$-1$. As a result, if $\alpha\geq-1$, then $\left(\hom(a,x)\otimes
a[-n]\right)_{\geq\alpha}=0$ and hence the map in (7) is zero. If $\alpha<-1$,
then $x_{<\alpha+1}=0$, and the triangle

$x\to\sigma^{n}_{a}x\to\ell_{n}$

induces an isomorphism

(8)
$\left(\sigma^{n}_{a}x\right)_{<\alpha+1}\xrightarrow{\sim}\left(\ell_{n}\right)_{<\alpha+1}.$

The map $\left(\hom(a,x)\otimes
a[-n]\right)_{\geq\alpha}\to\left(\ell_{n}\right)_{<\alpha+1}$ is zero since
the filtration (6) is geodesic. In view of the isomorphism (8), we get that
the map in (7) is also zero. ∎

###### Proof of 4.5.

If $x$ is a shift of $a$, then both sides are zero. Assume that $x$ is not a
shift of $a$. Since $x$ is indecomposable, it does not contain any shift of
$a$ as a summand. Let $N$ be as guaranteed by 4.8. Then, for every $n\geq N$,
we have

$m_{\tau}(\sigma_{a}^{n}x)=(n-N)m(a)\hom(a,x)+m_{\tau}(\sigma_{a}^{N}x).$

Dividing by $n$ and taking the limit as $n\to\infty$ completes the proof. ∎

######  Remark 4.9 (Degenerate stability conditions).

4.5 describes one way to approach the boundary of the stability manifold in
$\mathbb{P}^{\mathbf{S}}$, namely by applying powers of a spherical twist.
There are more direct ways to approach the boundary. Start with a stability
condition $\tau$ with heart $\heartsuit$ and central charge $Z\colon
K(\mathcal{C})\to\mathbb{C}$. Recall that $Z$ must map non-zero objects of
$\heartsuit$ to non-zero complex numbers. Take a deformation $Z_{t}$ of $Z$
such that $Z_{0}$ sends some objects of $\heartsuit$ to $0$. Then the limit of
the stability condition described by $(\heartsuit,Z_{t})$ is not a stability
condition; it must be on the boundary. We expect such limits to be degenerate
stability conditions in the sense of [2]. We expect these to form a measure
zero, but dense subset of the boundary.

In the analogy with the Teichmüller space, applying a spherical twist
corresponds to applying a Dehn twist. On the other hand, degenerating $Z$
corresponds to contracting a (collection of) curve(s). In geometry, both
operations yield the same limit, in contrast with the categorical analogues.

##  5\. The $A_{2}$ case

The aim of this section is to construct the Thurston compactification of
stability space for 2-Calabi–Yau category $\mathcal{C}_{\Gamma}$ where
$\Gamma$ is the $A_{2}$ quiver.

###  5.1. Overview

Let $\mathcal{C}$ be the 2-Calabi–Yau category associated to the $A_{2}$
quiver, as defined in § 2. It is a graded, $\mathbf{k}$-linear triangulated
category classically generated by two spherical objects, which we denote
$P_{1}$ and $P_{2}$.

The extension closure of $P_{1}$ and $P_{2}$ in $\mathcal{C}$ is an abelian
category; it is the heart of a (bounded) $t$-structure. We refer to it as the
_standard heart_ $\heartsuit_{\operatorname{std}}$. It has two simple objects,
$P_{1}$ and $P_{2}$, and two additional indecomposable objects, denoted by
$P_{1}\to P_{2}$ and $P_{2}\to P_{1}$. The object $P_{i}\to P_{j}$ is the
unique extension of $P_{i}$ by $P_{j}$. In terms of complexes, it is the
complex

$P_{i}\to P_{j}=Ae_{i}\langle-1\rangle\xrightarrow{f_{ij}}Ae_{j},\quad\text{in
homological degrees $-1$ and $0$}$

where $f_{ij}$ is right multiplication by the path $i\to j$.

###  5.2. The spherical objects and the boundary circle

As discussed in § 2.2, the spherical twists in the objects $P_{i}$ satisfy the
defining relations of the Artin–Tits braid group of the $A_{2}$ quiver. In
this case, this group is the three-strand braid group $B_{3}$:

$B_{3}=\langle\sigma_{1},\sigma_{2}\mid\sigma_{1}\sigma_{2}\sigma_{1}=\sigma_{2}\sigma_{1}\sigma_{2}\rangle.$

Denoting the twist in $P_{i}$ as $\sigma_{P_{i}}$, we have the relation
$\sigma_{P_{1}}\sigma_{P_{2}}\sigma_{P_{1}}\cong\sigma_{P_{2}}\sigma_{P_{1}}\sigma_{P_{2}}.$
The weak action of $B_{3}$ on $\mathcal{C}$ via the homomorphism

$\sigma_{i}\mapsto\sigma_{P_{i}}$

is faithful [17].

Let $\mathbf{S}$ be the set of spherical objects of $\mathcal{C}$, up to
shifts. It turns out that $B_{3}$ acts transitively on the set of all the
spherical objects of $\mathcal{C}$, and hence on $\mathbf{S}$ [1].

The centre of $B_{3}$ is generated by $(\sigma_{2}\sigma_{1})^{3}$. The
central element $(\sigma_{2}\sigma_{1})^{3}$ acts by a triangulated shift,
precisely by $x\mapsto x[-2]$. The action of $B_{3}$ on $\mathbf{S}$ therefore
descends to an action of $B_{3}/Z(B_{3})$ on $\mathbf{S}$.

We have an isomorphism
$B_{3}/Z(B_{3})\to\operatorname{PSL}_{2}(\operatorname{\mathbb{Z}})$ given by

$\displaystyle\sigma_{1}\mapsto\begin{pmatrix}1&1\\\
0&1\end{pmatrix},\quad\sigma_{2}\mapsto\begin{pmatrix}1&0\\\
-1&1\end{pmatrix}.$

The stabiliser of an element $x\in\mathbf{S}$ is the subgroup of
$\operatorname{PSL}_{2}(\operatorname{\mathbb{Z}})$ generated by the image of
$\sigma_{x}$. In particular, the stabiliser of $P_{1}$ is generated by
$\sigma_{1}$. As a result, we have a
$\operatorname{PSL}_{2}(\operatorname{\mathbb{Z}})$-equivariant bijection

(9) $i\colon\mathbf{S}\to\mathbb{P}^{1}(\operatorname{\mathbb{Z}})$

defined uniquely by the choice

$P_{1}\mapsto[1:0].$

Under this bijection, we can calculate the pre-image in $\mathbf{S}$ of a
point $[a:c]\in\mathbb{P}^{1}(\operatorname{\mathbb{Z}})$ as follows. Assume
$c\neq 0$. Write the rational number $a/c$ as a continued fraction with an odd
number of terms:

$\frac{a}{c}=n_{0}+\cfrac{1}{n_{2}+\cfrac{1}{\ddots+\cfrac{1}{n_{2k}}}}.$

Here, each $n_{i}$ is an integer, with $n_{i}>0$ for $i=1,\ldots,2k$. Then
$[a:c]$ corresponds to the object of $\mathbf{S}$ given by

(10) $\sigma_{1}^{n_{0}}\sigma_{2}^{-n_{1}}\cdots\sigma_{1}^{n_{2k}}(P_{2}).$

For example, we get

$P_{1}\mapsto[1:0]\mapsto P_{1},\quad P_{2}\mapsto[0:1],\quad(P_{2}\to
P_{1})\mapsto[1:-1],\text{ and }(P_{1}\to P_{2})\mapsto[1:1].$

###  5.3. Standard stability conditions

We say that a stability condition $\tau$ is _standard_ if its $[0,1)$ heart is
the standard heart $\heartsuit_{\rm std}$. More generally, we say that an
element of $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ is _standard_ if one
of its representatives is standard. For a standard $\tau$, both $P_{1}$ and
$P_{2}$ must be stable. Let $\phi$ denote the phase function for $\tau$. Then
we either have $\phi(P_{1})\leq\phi(P_{2})$ or $\phi(P_{2})\leq\phi(P_{1})$.
In the first case, we say $\tau$ is standard of _type A_ , and in the latter
case, of _type B_. In type A (resp. type B), the object $X=(P_{2}\to P_{1})$
(resp. $X^{\prime}=(P_{1}\to P_{2})$) is (semi)-stable (stable if the
inequality is strict). If the inequality is strict, we say that $\tau$ is
_non-degenerate_. More generally, a stability condition is _non-degenerate_ if
it does not have strictly semi-stable objects. See Figure 2 for a sketch of a
standard, type A, non-degenerate $\tau$.

By [6, Proposition 4.2], every stability condition can be written as
$\beta\tau$, where $\beta\in B_{3}$ and $\tau$ is a standard stability
condition of type A.

The set of standard, type A stability conditions in
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ is a locally closed subset. Let
$\Lambda$ be its closure. This includes stability conditions where $P_{1}$ and
$P_{2}$ are stable, and $\phi(P_{2})=\phi(P_{1})+1$. We continue to call the
stability conditions in $\Lambda$ as of _type A_ , reserving the adjective
standard for those where $\phi(P_{2})<\phi(P_{1})+1$. Note that the non-
standard type A stability conditions are degenerate.

The set $\Lambda$ tessellates $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$
under the action of $B_{3}$. More precisely, $\Lambda$ satisfies the following
properties (see, e.g., [6, Proposition 4.2]).

  1. (1)

Each point of $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ lies in the
$B_{3}$-orbit of a point of $\Lambda$.

  2. (2)

The stabiliser of $\Lambda$ is the subgroup generated by
$\gamma=\sigma_{2}\sigma_{1}$ in $B_{3}$.

  3. (3)

For any $g\in B_{3}$ not in the stabiliser, the interiors of $\Lambda$ and
$g\Lambda$ have empty intersection.

See Figure 6 for a picture of the tessellation of
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ by the orbit of $\Lambda$. The
interior of $\Lambda$ is the set of non-degenerate standard stability
conditions of type A.

$P_{1}$$P_{2}$$P_{2}\to P_{1}$$\bullet$$P_{1}$$P_{2}$$P_{2}\to
P_{1}$$\bullet$$P_{1}$$P_{2}$$P_{2}\to P_{1}$$\bullet$standardtype
A$P_{1}$$P_{2}$$P_{2}\to P_{1}$$P_{1}\to
P_{2}$$\bullet$$P_{2}$$P_{1}$$P_{1}\to P_{2}$$\bullet$type
B$P_{2}$$P_{1}$$P_{1}\to P_{2}$$\bullet$$P_{2}$$P_{1}$$P_{1}\to
P_{2}$$\bullet$ Figure 2. Central charges in a stability condition of type A
and B

###  5.4. The automaton

Fix a non-degenerate standard stability condition $\tau$ of type A. Define the
braid $\gamma\in B_{3}$ by

$\gamma=\sigma_{2}\sigma_{1}=\sigma_{x}\sigma_{2}=\sigma_{1}\sigma_{X}.$

$[P_{1},P_{2}]$$[X,P_{1}]$$[P_{2},X]$$\sigma_{1}=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}$$\sigma_{X},\begin{pmatrix}1&0\\\
1&1\end{pmatrix}$$\gamma^{-1}$$\gamma$$\sigma_{X},\begin{pmatrix}1&1\\\
0&1\end{pmatrix}$$\sigma_{2},\begin{pmatrix}1&1\\\
1&0\end{pmatrix}$$\gamma^{-1}$$\gamma$$\sigma_{2},\begin{pmatrix}1&1\\\
0&1\end{pmatrix}$$\sigma_{1},\begin{pmatrix}1&0\\\
1&1\end{pmatrix}$$\gamma$$\gamma^{-1}$ Figure 3. An automaton describing the
dynamics of Harder–Narasimhan filtrations in a stability condition with stable
objects $P_{1}$, $P_{2}$, and $X=P_{2}\to P_{1}$.

Figure 3 shows a $\tau$-HN automaton $\Theta$ that computes HN multiplicities
up to shift. Formally, the automaton $\Theta$ is defined by the $G$-labelled
graph with three vertices and three edges from each vertex, as shown in Figure
3. The three states support the objects $\\{X,P_{1}\\}$, $\\{P_{1},P_{2}\\}$,
and $\\{P_{2},X\\}$, and the $\Theta$-set
$S\subset\operatorname{ob}\mathcal{C}$ is the one generated by this data. The
representation $M$ associates the free abelian group $\mathbb{Z}^{\\{A,B\\}}$
to the state labelled $[A,B]$ . The linear maps $M(e)$ are given by $2\times
2$ matrices in the standard basis, as shown on $e$ in Figure 3. (The arrows
labelled $\gamma^{\pm}$ correspond to the identity matrix, which is not
shown.) The map $S\to M$ is the unique $\Theta$-equivariant map that sends the
objects $\\{A,B\\}$ supported at the state $[A,B]$ to the standard basis
vectors. Note that $\Sigma=\\{P_{1},P_{2},X\\}$ is the set of semi-stable (=
stable) objects of $\tau$ (up to shift). For $v=[A,B]$, the map $HN_{v}\colon
Z^{\\{A,B\\}}\to\mathbb{Z}^{\Sigma}$ is simply the inclusion induced by the
inclusion $\\{A,B\\}\to\Sigma$. It is not _a priori_ clear why $S\to M$ is
well-defined and why $\Theta$ computes $\tau$-HN multiplicities (up to shift).
Both are consequences of 5.1.

######  Proposition 5.1.

Let $e\colon v\to w$ be an arrow of $\Theta$, where $v=[A,B]$ and $w=[C,D]$.
Let $x\in\mathcal{C}$ be any object whose HN filtration consists of shifts of
$A$ and $B$. The following hold.

  1. (1)

The HN filtration of $e\cdot x$ consists of shifts of $C$ and $D$.

  2. (2)

Consider $HN_{\tau}(x)\in\mathbb{Z}^{\\{A,B\\}}$ and $HN_{\tau}(e\cdot
x)\in\mathbb{Z}^{\\{C,D\\}}$. Then

$HN_{\tau}(e\cdot x)=M(e)\cdot HN_{\tau}(x).$

###### Proof.

It suffices to treat the case where $e$ is an arrow shown in Figure 3; the
rest follows by composition. Let

$0\to x_{0}\to x_{1}\to\dots\to x_{n}=x$

be an HN-filtration of $x$ with stable factors
$z_{i}=\operatorname{Cone}(x_{i-1}\to x_{i})$. By applying $\sigma=\sigma(e)$,
we get a filtration

(11) $0\to\sigma(x_{0})\to\sigma(x_{1})\to\dots\to\sigma(x_{n})=\sigma(x)$

with factors $\sigma(z_{i})$.

We check that the filtration (11) is geodesic in the sense of A.6. Suppose
$\sigma=\gamma$. Recall that $\gamma$ simply permutes the three stable objects
and preserves the ordering of the phases. So, (11) is already an HN filtration
(and hence obviously geodesic, see A.7). For the others, we use A.14.

We outline the case $v=[P_{1},P_{2}]$, leaving the others to the reader. To
apply A.14 to the filtration (11), we must check the following: for $i<j$,
either $\operatorname{Hom}(\sigma z_{j},\sigma z_{i}[1])=0$ or $\lfloor\sigma
z_{i}\rfloor\geq\lceil\sigma z_{j}\rceil$. By assumption, $z_{i}$ and $z_{j}$
are shifts of $P_{1}$ or $P_{2}$, and since $i<j$, we have
$\phi(z_{i})>\phi(z_{j})$. The only pairs $(z_{i},z_{j})$ with
$\operatorname{Hom}(z_{j},z_{i}[1])\neq 0$ are $(P_{1}[1],P_{1})$,
$(P_{2}[1],P_{2})$, and $(P_{2},P_{1})$. The two possible $\sigma$ at the
vertex $v$ are $\sigma=\sigma_{1}$ and $\sigma=\sigma_{X}$. We must now check
that $\lfloor\sigma z_{i}\rfloor\geq\lceil\sigma z_{j}\rceil$ holds for six
possibilities: 2 for $\sigma$ and 3 for the pairs $(z_{i},z_{j})$. This is
straightforward, so we leave it out.

Since the filtration (11) is geodesic, an HN filtration of $\sigma(x)$ is
simply a rearrangement of the HN filtrations of $\sigma(z_{i})$. In
particular, the HN multiplicity vector of $\sigma(x)$ is the sum of those of
$\sigma(z_{i})$. As a result, it suffices to prove both parts of the
proposition for $x=P_{1}$ and $x=P_{2}$. This is another straightforward
calculation. ∎

The next proposition shows that the automaton recognises every braid and every
object of $\mathbf{S}$.

######  Proposition 5.2.

Let $\beta\in B_{3}$ and let $s\in\mathbf{S}$ be arbitrary.

  1. (1)

The automaton in Figure 3 recognises $\beta$. More precisely, $\beta$ has a
recognised expression of the form

$\beta=\gamma^{n}\sigma_{a_{1}}^{m_{1}}\sigma_{a_{2}}^{m_{2}}\cdots\sigma_{a_{k}}^{m_{k}},$

where $n$ is an integer, $k$ is a non-negative integer, the $m_{i}$ are
positive integers, and the sequence $(a_{1},a_{2}\dots,a_{k})$ is a contiguous
subsequence of the sequence $(\dots,X,1,2,X,1,2,\dots)$.

  2. (2)

The automaton in Figure 3 recognises $s$.

  3. (3)

The $\tau$-HN filtration of $s$ consists of at most two objects out of
$\\{P_{1},P_{2},X\\}$.

We call the writing of $\beta$ described in 5.1 an _admissible cyclic
writing_.

###### Proof.

For (1), we repeatedly use the commutation relations

$\gamma\sigma_{2}\gamma^{-1}=\sigma_{1},\quad\gamma\sigma_{X}\gamma^{-1}=\sigma_{2},\quad\gamma\sigma_{1}\gamma^{-1}=\sigma_{X}.$

Begin by writing $\beta$ as any product of the generators $\sigma_{1}$ and
$\sigma_{2}$, along with their inverses. Eliminate the inverses of the
generators by rewriting as follows:

$\sigma_{1}^{-1}=\sigma_{X}\gamma^{-1},\quad\sigma_{2}^{-1}=\sigma_{1}\gamma^{-1}.$

Next, use the commutation relations to rewrite

$\gamma^{i}\sigma_{X}=\sigma_{2}\gamma^{i},\quad\gamma^{i}\sigma_{1}=\sigma_{X}\gamma^{i},\quad\gamma^{i}\sigma_{2}=\sigma_{1}\gamma^{i},$

and thus move instances of $\gamma^{-1}$ to the left as a single power of
$\gamma^{-1}$. The rest of $\beta$ is now a product of elements from
$\\{\sigma_{1},\sigma_{2},\sigma_{X}\\}$.

Finally, replace any occurrences of $\sigma_{2}\sigma_{1}$,
$\sigma_{1}\sigma_{X}$, or $\sigma_{X}\sigma_{2}$ by $\gamma$ and move
$\gamma$ to the left, again using the commutation relations. Each such
operation decreases the length of the braid in the elements
$\\{\sigma_{1},\sigma_{2},\sigma_{X},\gamma^{\pm}\\}$. Therefore, the
procedure terminates and $\beta$ reaches the desired form.

For (2), write $s$ as $s=\beta P_{1}$, and then write

$\beta=\gamma^{n}\sigma_{a_{1}}^{m_{1}}\sigma_{a_{2}}^{m_{2}}\cdots\sigma_{a_{k}}^{m_{k}}$

as in (1). Note that $P_{1}$ is supported at two states, namely
$[P_{1},P_{2}]$ and $[X,P_{1}]$. Each of the letters
$\\{\sigma_{1},\sigma_{2},\sigma_{X}\\}$ is applicable at least at one of the
two states above, so we can apply $\sigma_{a_{k}}$. The condition on our
cyclic writing guarantees that we can apply each subsequent letter. Therefore,
$\beta P_{1}$ is recognised by the automaton.

Finally, (3) follows from (2) and 5.1. ∎

######  Remark 5.3.

Let $\tau$ be a degenerate stability condition of type A. Then the automaton
in Figure 3 does not compute $\tau$-HN-multiplicities. Indeed, a degenerate
$\tau$ has additional semi-stable objects than those in
$\Sigma=\\{P_{1},P_{2},X\\}$, which are not accounted for by the automaton.
Nevertheless, it _does_ compute $\tau$-masses, with the mass functions
$m_{v}\colon\mathbb{Z}^{\Sigma}\to\mathbb{R}$ defined as usual by

$m_{v}\colon s\mapsto m_{\tau}(s)\text{ for }s\in\Sigma.$

######  Remark 5.4.

Let $\tau$ be a degenerate standard stability condition of type A. Recall that
$\gamma=\sigma_{2}\sigma_{1}$. Let

$\gamma^{\prime}=\sigma_{1}\sigma_{2},\quad
w_{0}=\sigma_{1}\sigma_{2}\sigma_{1}=\sigma_{2}\sigma_{1}\sigma_{2}.$

Figure 4 describes an automaton that computes $\tau$-HN multiplicities,
continuing to recognises all braids and spherical objects. Since the automaton
in Figure 3 suffices for mass computations, we omit the details.

######  Remark 5.5.

For every non-degenerate stability condition, we have an HN automaton obtained
by applying an appropriate braid to the automaton in (3). Likewise, for a
degenerate stability condition, we have an HN automaton obtained by applying a
braid to the automaton in (4). Thus, we have an automaton at every point of
the stability space. Observe that these automata exhibit a wall-and-chamber
behaviour: they remain constant in a chamber, get more complicated along a
wall, and flip to a different automaton as we cross the wall.

$[X^{\prime},P_{2}]$$[P_{1},X^{\prime}]$$[X,P_{1}]$$[P_{2},X]$$\sigma_{1}$$\gamma^{\prime}$$\sigma_{1}$$\sigma_{2}$$\gamma$$\sigma_{2}$$\gamma$$\sigma_{1}$$\gamma^{\prime}$$\sigma_{2}$$\gamma$,
$w_{0}^{\pm}$$\gamma^{\prime}$, $w_{0}^{\pm}$$\gamma^{\prime}$,
$w_{0}^{\pm}$$\gamma$, $w_{0}^{\pm}$ Figure 4. An automaton describing the
dynamics of HN filtrations in a degenerate stability condition with semistable
objects $P_{1}$, $P_{2}$, $X=P_{2}\to P_{1}$, and $X^{\prime}=P_{1}\to P_{2}$.

We observe that the automaton in Figure 3 is mass degenerating.

######  Proposition 5.6.

Let $\tau$ be a standard stability condition of type A, possibly degenerate.
The automaton $\Theta$ is mass degenerating for $\tau$.

###### Proof.

By 5.1, every braid has a writing recognised by $\Theta$ that ends with either
$\sigma_{1}$, $\sigma_{2}$, or $\sigma_{X}$. Therefore, it suffices to find
non-degenerate triangles that are made degenerate by each of these letters.

Suppose $\tau$ is non-degenerate; that is $\phi(P_{1})<\phi(P_{2})$. The non-
degenerate triangle

$P_{1}\to X\to P_{2}\xrightarrow{+1}$

is made degenerate by $\sigma_{2}$. The non-degenerate triangle

$X\to P_{2}\to P_{1}[1]\xrightarrow{+1}$

is made degenerate by $\sigma_{1}$. And the non-degenerate triangle

$P_{2}[-1]\to P_{1}\to X\xrightarrow{+1}$

is made degenerate by $\sigma_{X}$.

If $\tau$ is degenerate, that is $\phi(P_{1})=\phi(P_{2})$, then
$m_{\tau}(X)=m_{\tau}(P_{1})+m_{\tau}(P_{2})$. In this case the first triangle
is already degenerate, and must be replaced. Set
$X^{\prime}=\sigma_{1}(P_{2})=(P_{1}\to P_{2})$. Then the non-degenerate
triangle

$X^{\prime}\to P_{1}\to P_{2}[1]$

is made degenerate by $\sigma_{2}$. ∎

###  5.5. Consequences of the automaton

In this section, we use the automaton from § 5.4 to prove results about our
projective embedding of $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$.

######  Proposition 5.7.

The map
$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathbf{S}}$
is a homeomorphism onto its image.

###### Proof.

We have already shown in 4.3 that the map $m$ is a local homeomorphism onto
its image at $\tau$, for any standard stability condition $\tau$ of type A.
Since every stability condition is in the $B_{3}$ orbit of a standard
stability condition of type A, and the map $m$ is $B_{3}$-equivariant, we see
that $m$ is a local homeomorphism onto its image at every point
$\tau\in\operatorname{Stab}(\mathcal{C})/\mathbb{C}$. By 4.1, the map $m$ is
also injective. This proves that $m$ is a homeomorphism onto its image. ∎

Recall that we have a surjective map
$B_{3}\to\operatorname{PSL}_{2}(\mathbb{Z})$. Via this map, the
$B_{3}$-automaton $\Theta$ becomes a
$\operatorname{PSL}_{2}(\mathbb{Z})$-automaton. Consider
$\mathbb{P}^{1}(\mathbb{Z})$ as a $\operatorname{PSL}_{2}(\mathbb{Z})$-set in
the standard way, and hence also as a $\Theta$-set. Let $\mathbb{P}M$ be the
projectivisation of the representation $M$. Then $\mathbb{P}M$ is also a
$\Theta$-set.

######  Proposition 5.8.

We have an isomorphism of $\Theta$-sets
$\phi\colon\mathbb{P}M\to\mathbb{P}^{1}(\mathbb{Z})$.

###### Proof.

The isomorphism $\phi$ is induced by linear maps $M(v)\to\mathbb{Z}^{2}$
defined at each state as follows

$\displaystyle\phi_{[P_{1},P_{2}]}$ $\displaystyle:P_{1}\mapsto(1,0)\text{ and
}P_{2}\mapsto(0,1),$ $\displaystyle\phi_{[P_{2},X]}$
$\displaystyle:P_{2}\mapsto(0,-1)\text{ and }X\mapsto(1,-1),$
$\displaystyle\phi_{[X,P_{1}]}$ $\displaystyle:X\mapsto(-1,1)\text{ and
}P_{1}\mapsto(-1,0).$

To check that $\phi$ is $\Theta$-equivariant, we must check that for every
arrow $e\colon v\to w$ in Figure 3, we have

$\sigma(e)\phi_{v}=\phi_{w}M(e)$

in $\operatorname{PSL}_{2}(\mathbb{Z})$. We omit this straightforward
verification. ∎

Recall that we have a $B_{3}$-equivariant map
$i\colon\mathbf{S}\to\mathbb{P}^{1}(\operatorname{\mathbb{Z}})$ defined in
(9). Thanks to 5.8, we get another map: send an object $s\in\mathbf{S}$
supported at the vertex $v$ to $\phi([s])\in\mathbb{P}^{1}(\mathbb{Z})$ (note
that the definition does not create conflicts for objects supported at two
states). It is not hard to see that the new map agrees with the old map.
Indeed, to see this, note that every object $s\in\mathbf{S}$ has an expression
$s=g_{1}\cdots g_{n}\cdot P_{1}$ recognised by $\Theta$. Since the map
$s\mapsto\phi([s])$ is $\Theta$-equivariant, we have

$\phi([s])=g_{1}\cdots g_{n}\phi([P_{1}]).$

Since the map $i$ is $B_{3}$-equivariant, the same equation holds with $i$
instead of $\phi$. Since $i(P_{1})=\phi([P_{1}])$, we see that
$i(s)=\phi([s])$.

The division of $\mathbf{S}$ according to the support corresponds nicely to a
geometric division of the circle $\mathbb{P}^{1}(\operatorname{\mathbb{R}})$,
which we now describe. The three objects $P_{1}$, $P_{2}$, and $X$ divide
$\mathbb{P}^{1}(\operatorname{\mathbb{R}})$ into three closed arcs (see Figure
5). We denote these arcs by $[P_{1},P_{2}]$, $[P_{2},X]$, and $[X,P_{1}]$.

$X=[-1:1]$$P_{1}=[1:0]$$P_{2}=[0:1]$ Figure 5. The points $P_{1}$, $P_{2}$,
and $X=P_{2}\to P_{1}$ divide $\mathbb{P}^{1}(\mathbb{R})$ into three arcs.
The Harder–Narasimhan pieces of an object only include the two endpoints of
the arc on which the object lies.

######  Proposition 5.9.

The objects of $\mathbf{S}$ supported at the state $[P_{1},P_{2}]$ lie on the
arc $[P_{1},P_{2}]$, and likewise for the other two arcs.

###### Proof.

We use that the map $\mathbf{S}\to\mathbb{P}^{1}(\mathbb{R})$ is given by
$s\mapsto\phi([s])$. For $s$ supported at $[P_{1},P_{2}]$, the vector $[s]$
lies in the non-negative cone in $\mathbb{Z}^{\\{P_{1},P_{2}\\}}$, which is
mapped by $\phi_{[P_{1},P_{2}]}$ to the arc $[P_{1},P_{2}]$. The argument for
the other two states is similar. ∎

The following proposition is a theorem of Rouquier and Zimmermann [16,
Proposition 4.8]. We give a new and simpler proof using the automaton.

######  Proposition 5.10.

Let $s$ be a spherical object corresponding to
$[a:c]\in\mathbb{P}^{1}(\mathbb{Z})$, where $a,c$ are relatively prime
integers. Every minimal complex of projective modules representing $s$ has
exactly $|a|$ occurrences of $P_{1}$ and $|c|$ occurrences of $P_{2}$.

###### Proof.

Recall that $\tau$ is a standard stability condition of type A. The objectcs
$P_{1}$ and $P_{2}$ are the simple objects of the standard heart of $\tau$.
The $\tau$-HN filtration is a refinement of the cohomology filtration, and on
each cohomology factor it is a coarsening of a Jordan–Hölder filtration.
Therefore, the number of occurrences of $P_{i}$ is the sum of the number of
occurrences of $P_{i}$ in the $\tau$-HN factors.

Suppose $s$ is supported at the state $v$ of $\Theta$. By 5.1, $[s]\in M(v)$
is the $\tau$-HN multiplicity vector of $s$. By the argument above, the number
of occurrences of $P_{i}$ in the minimal complex of $s$ is a linear function
of $[s]$.

By the discussion following 5.8, we know that $[a:c]=\phi([s])$. The proof of
5.8 gives an isomorphism $\widetilde{\phi}\colon M\to\mathbb{Z}^{2}$, which is
$\Theta$-equivariant up to sign, and sends $[s]$ to $(a,c)$. By looking at the
basis vectors, we see that the number of occurrences of $P_{i}$ is given by
the absolute value of the $i$-th coordinate of $\widetilde{\phi}$. The
statement follows. ∎

The next proposition relates the minimal complex and the hom functionals, as a
step towards describing the closure of the image of the mass map $m$.

######  Proposition 5.11.

Let $x$ be a spherical object corresponding to
$[a:c]\in\mathbb{P}^{1}(\mathbb{Z})$, where $a,c$ are relatively prime
integers. Then $\operatorname{\overline{hom}}(x,P_{2})=|a|$ and
$\operatorname{\overline{hom}}(x,P_{1})=|c|$.

###### Proof.

We prove the assertion for $\operatorname{\overline{hom}}(x,P_{1})$; the other
case follows by applying $\gamma$.

If $x=P_{1}$, then the statement is evident. Suppose $x$ is supported at
$[P_{2},X]$. Consider the Harder–Narasimhan filtration of $x$:

$0=x_{0}\to\cdots\to x_{n}=x,$

with sub-quotients $z_{i}=\operatorname{Cone}(x_{i}\to x_{i+1})$. Note that
$z_{i}$ is isomorphic to $P_{2}$ or $X$, up to shift.

By applying $\operatorname{Hom}(P_{1},-)$, we obtain a filtration

(12)
$0=\operatorname{Hom}(P_{1},x_{0})\to\cdots\to\operatorname{Hom}(P_{1},x_{n})=\operatorname{Hom}(P_{1},x),$

in the bounded derived category of (graded) vector spaces. We claim that each
of the triangles

$\operatorname{Hom}(P_{1},x_{i})\to\operatorname{Hom}(P_{1},x_{i+1})\to\operatorname{Hom}(P_{1},z_{i})\xrightarrow{+1}$

is split; that is, all possible boundary maps between the sub-quotients of
(12) are zero. Recall that the filtration of $x$ has three possible non-zero
boundary maps:

$P_{2}\to P_{2}[2],\quad X\to X[2],\text{ and }P_{2}\to X[2].$

It is easy to verify that all three are killed by the functor
$\operatorname{Hom}(P_{1},-)$.

In particular, we have

$\operatorname{Hom}(P_{1},x)=\bigoplus\operatorname{Hom}(P_{1},z_{i}).$

For $z_{i}=P_{2}$ or $z_{i}=X$, we have

$\operatorname{\overline{hom}}(z_{i},P_{1})=\operatorname{\overline{hom}}(P_{1},z_{i})=1=\text{number
of $P_{2}$'s in the minimal complex of $z_{i}$}.$

Hence, by linearity, we conclude

$\operatorname{\overline{hom}}(x,P_{1})=\operatorname{\overline{hom}}(P_{1},x)=\text{number
of $P_{2}$'s in the minimal complex of $x$.}$

We now treat the case of $x$ supported at the other two vertices. By 5.10, $x$
and $\sigma_{1}^{r}x$ have the same number of occurrences of $P_{2}$, and both
also have the same value for $\operatorname{\overline{hom}}(P_{1})$. Hence,
the proposition for $x$ implies the proposition for $\sigma_{1}^{r}x$. If $x$
is supported at $[X,P_{1}]$, then $\sigma_{1}^{-1}x=\sigma_{2}\gamma^{-1}x$ is
supported at $[P_{2},X]$; hence the proposition holds for $x$. If $x$ is
supported at $[P_{1},P_{2}]$, then it is easy to check that $x$ has a
recognised writing of the form

$x=\sigma_{1}^{a}\beta y,$

where $y\in\\{P_{1},P_{2},X\\}$ and $\beta y$ is not supported at
$[P_{1},P_{2}]$. Then $y=\sigma_{1}^{-a}x$ is supported at $[P_{2},X]$ or
$[X,P_{1}]$, for which the proposition holds; hence the proposition holds for
$x$. ∎

######  Corollary 5.12.

The values of the basic hom functions on $x=[a:c]$ are given by

  1. (1)

$\operatorname{\overline{hom}}(P_{1},x)=|c|$,

  2. (2)

$\operatorname{\overline{hom}}(P_{2},x)=|a|$,

  3. (3)

$\operatorname{\overline{hom}}(X,x)=|a+c|$,

  4. (4)

$\operatorname{\overline{hom}}(P_{1}\to P_{2},x)=|a-c|$.

###### Proof.

The first two follow directly from 5.11. For the last two, use

$\operatorname{\overline{hom}}(X,x)=\operatorname{\overline{hom}}(P_{1},\sigma_{2}^{-1}x),$

and

$\operatorname{\overline{hom}}(P_{1}\to
P_{2},x)=\operatorname{\overline{hom}}(P_{2},\sigma_{1}^{-1}x).$

The result follows by using the action of $\sigma_{i}^{\pm 1}$ on $(a,c)$ and
applying 5.10. ∎

Recall the map
$\operatorname{\overline{hom}}\colon\mathbf{S}\to\mathbb{R}^{\mathbf{S}}$
defined by

$\operatorname{\overline{hom}}(x)\colon
y\mapsto\operatorname{\overline{hom}}(x,y).$

The map $h\colon\mathbf{S}\to\mathbb{P}^{\mathbf{S}}$ is the composition of
$\operatorname{\overline{hom}}$ and the projection
$\operatorname{\mathbb{R}}^{\mathbf{S}}\setminus 0\to\mathbb{P}^{\mathbf{S}}$.
Consider $\mathbf{S}$ as a subset of
$\mathbb{P}^{1}(\operatorname{\mathbb{R}})$ via the identification
$\mathbf{S}=\mathbb{P}^{1}(\operatorname{\mathbb{Z}})$ as defined in (9).

######  Proposition 5.13.

The map
$h\colon\mathbf{S}\to\mathbb{P}(\operatorname{\mathbb{R}}^{\mathbf{S}})$
extends to a continuous map
$h\colon\mathbb{P}^{1}(\operatorname{\mathbb{R}})\to\mathbb{P}^{\mathbf{S}}$.
The extension maps $\mathbb{P}^{1}(\operatorname{\mathbb{R}})$
homeomorphically onto its image.

###### Proof.

Let $s\in\mathbf{S}$ be a spherical object (up to shift). Write $s=\beta
P_{2}$ for some braid $\beta$. Let $\overline{\beta}$ be the image of $\beta$
in $\operatorname{PSL}_{2}(\operatorname{\mathbb{Z}})$. Then
$\overline{\beta}$ is uniquely determined up to right multiplication by powers
of $\overline{\sigma}_{2}$. Consider a point
$[a:c]\in\mathbb{P}^{1}(\operatorname{\mathbb{Z}})$, where $a$ and $c$ are
relatively prime integers, and let $t\in\mathbf{S}$ be the corresponding
spherical object. We have

$\displaystyle\operatorname{\overline{hom}}(s,t)$
$\displaystyle=\operatorname{\overline{hom}}(\beta P_{2},t)$
$\displaystyle=\operatorname{\overline{hom}}(P_{2},\beta^{-1}t)$
$\displaystyle=|(1,0)\cdot\overline{\beta}^{-1}\cdot(a,c)^{t}|\text{ by
\autoref{prop:homs}}.$

We define
$h\colon\mathbb{P}^{1}(\operatorname{\mathbb{R}})\to\mathbb{P}(\operatorname{\mathbb{R}}^{\mathbf{S}})$
by using the final expression for an arbitrary $[a:c]$. That is, we set
$h([a:c])$ to be the (projectivised) function whose value at $s=\beta P_{2}$
is

$|(1,0)\cdot\overline{\beta}^{-1}\cdot(a,c)^{t}|.$

Plainly,
$h\colon\mathbb{P}^{1}(\operatorname{\mathbb{R}})\to\mathbb{P}^{\mathbf{S}}$
is a continuous extension of the original map
$h\colon\mathbf{S}\to\mathbb{P}^{\mathbf{S}}$.

We now check that the extended map is a homeomorphism onto its image. Since
the domain $\mathbb{P}^{1}(\operatorname{\mathbb{R}})$ is compact and the
target $\mathbb{P}(\operatorname{\mathbb{R}}^{\mathbf{S}})$ is Hausdorff, it
suffices to check that it is injective. Let

$T=\\{P_{1},P_{2},X\\}.$

Consider the restricted map
$h_{T}\colon\mathbb{P}^{1}(\operatorname{\mathbb{R}})\to\mathbb{P}(\mathbb{R}^{T})$
obtained as the composition of $h$ and the projection to $\mathbb{P}^{T}$.
From 5.12, $h_{T}$ is given in coordinates by

$h_{T}\colon[a:c]\mapsto[|a|:|c|:|a+c|].$

This map is injective, and hence so is $h$. ∎

###  5.6. Gromov coordinates

Let $\tau$ be a stability condition of type A. Since the three positive real
numbers $m_{\tau}(P_{1})$, $m_{\tau}(P_{2})$, and $m_{\tau}(X)$ satisfy the
triangle inequalities, there exist non-negative real numbers $x$, $y$, $z$
such that

$m_{\tau}(P_{1})=y+z,\quad m_{\tau}(P_{2})=z+x,\quad m_{\tau}(X)=x+y.$

We call the $x$, $y$, $z$, the _Gromov coordinates_ of $\tau$. Note that if
$\tau$ is non-degenerate, then the Gromov coordinates are all positive.
Otherwise, one of the coordinates is zero. Two of the coordinates cannot be
zero.

Recall that $m_{\tau}\colon\mathbf{S}\to\mathbb{R}$ is the mass function
associated to $\tau$ and $\operatorname{\overline{hom}}(s)$ the reduced $\hom$
functional associated to an object $s$.

######  Proposition 5.14 (Linearity).

We have

$m_{\tau}=x\operatorname{\overline{hom}}(P_{1})+y\operatorname{\overline{hom}}(P_{2})+z\operatorname{\overline{hom}}(X).$

###### Proof.

Let $s\in\mathbf{S}$ be a spherical object. Suppose $\tau$ is non-degenerate.
Observe that the HN filtration of $s$ is the same for all non-degenerate type
A stability conditions, and its stable factors are $P_{1}$, $P_{2}$, and $X$,
up to shift. Denoting by $a(s)$, $b(s)$, and $c(s)$ the multiplicities of
these three, we have

$m_{\tau}(s)=(y+z)a(s)+(x+z)b(s)+(x+y)c(s).$

In particular, $m_{\tau}(s)$ is linear in the Gromov coordinates. Using 5.10,
we have

$\displaystyle b(s)+c(s)$ $\displaystyle=\text{number of occurrences of
$P_{2}$ in the minimal complex of $s$}$
$\displaystyle=\operatorname{\overline{hom}}(P_{1},s),$ $\displaystyle
a(s)+c(s)$ $\displaystyle=\text{number of occurrences of $P_{1}$ in the
minimal complex of $s$}$
$\displaystyle=\operatorname{\overline{hom}}(P_{2},s),\text{ and }$
$\displaystyle a(s)+b(s)$ $\displaystyle=\operatorname{\overline{hom}}(X,s).$

The first two equalities are evident. The third is obtained by applying
$\gamma$ to the second. We conclude that

$m_{\tau}(s)=x\operatorname{\overline{hom}}(P_{1},s)+y\operatorname{\overline{hom}}(P_{2},s)+z\operatorname{\overline{hom}}(X,s).$

The case of a degenerate $\tau$ follows by continuity. ∎

Let $\tau$ be an arbitrary stability condition. Then there are three $\tau$
semi-stable objects, say $A$, $B$, $C$, whose classes in the Grothendieck
group are (up to sign) the classes of $P_{1}$, $P_{2}$, and $X$. These are
obtained simply by applying an appropriate braid to $P_{1}$, $P_{2}$, and $X$.
Define the Gromov coordinates $x$, $y$, $z$ for $\tau$ by the condition

$m_{\tau}(A)=y+z,\quad m_{\tau}(B)=x+z,\quad m_{\tau}(C)=x+y.$

Then 5.14 implies that we have

$m_{\tau}=x\operatorname{\overline{hom}}(A)+y\operatorname{\overline{hom}}(B)+z\operatorname{\overline{hom}}(C).$

The Gromov coordinates give a nice geometric picture of
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}\subset\mathbb{P}^{\mathbf{S}}$.
Let $\Delta$ denote the following clipped triangle:

$\displaystyle\Delta$ $\displaystyle=\\{(x,y,z)\in\mathbb{R}_{\geq
0}^{3}\mid\text{at least two coordinates non-
zero}\\}/\operatorname{\mathbb{R}}_{>0}$ $\displaystyle\cong\text{a closed
planar triangle minus the three vertices}.$

Recall that $\Lambda\subset\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ is the
set of type A stability conditions. We have a homeomorphism
$\Delta\to\Lambda\subset\mathbb{P}^{\mathbf{S}}$ given by the Gromov
coordinates:

$(x,y,z)\mapsto
x\cdot\operatorname{\overline{hom}}(P_{1})+y\cdot\operatorname{\overline{hom}}(P_{2})+z\cdot\operatorname{\overline{hom}}(X).$

Note that the homeomorphism extends to the closed (unclipped) triangle
$\overline{\Delta}$, under which the three vertices are mapped to the points
$P_{1}$, $P_{2}$, and $X$. By applying the $B_{3}$ action, we obtain the
picture as shown in Figure 6.

Let us describe the transformations indicated in Figure 6 more explicitly.
Recall that the element $\gamma=\sigma_{2}\sigma_{1}$ generates the stabiliser
of $\Lambda$ in $\operatorname{PSL}_{2}(\mathbb{Z})$. So, we can label the
various (distinct) translates of $\Lambda$ by $\langle\gamma\rangle$-cosets.
Two translates of $\Lambda$ are either disjoint or intersect along an edge,
which is a copy of the open interval. The three translates that intersect
$\Lambda$ along its three edges are $\sigma_{1}\Lambda$, $\sigma_{X}\Lambda$,
and $\sigma_{2}\Lambda$. Consequently, the three translates that intersect
$\beta\Lambda$ are $\beta\sigma_{1}\Lambda$, $\beta\sigma_{X}\Lambda$, and
$\beta\sigma_{2}\Lambda$. We can encode the translates and their intersections
in a graph (called the _exchange graph_ in [6]). Its vertices are left
$\gamma$-cosets in $\operatorname{PSL}_{2}(\operatorname{\mathbb{Z}})$, and a
coset $\beta\langle\gamma\rangle$ is connected by an edge with
$\beta\sigma_{1}\langle\gamma\rangle$, $\beta\sigma_{X}\langle\gamma\rangle$,
and $\beta\sigma_{2}\langle\gamma\rangle$.

$P_{1}$$P_{2}$$X$$\Lambda$$\sigma_{1}\Lambda$$\sigma_{2}\Lambda$$\sigma_{X}\Lambda$$\sigma_{1}^{2}\Lambda$$\sigma_{1}\sigma_{2}\Lambda$$\sigma_{2}^{2}\Lambda$$\sigma_{2}\sigma_{X}\Lambda$$\sigma_{X}^{2}\Lambda$$\sigma_{X}\sigma_{1}\Lambda$$\sigma_{1}$$\sigma_{X}$$\sigma_{2}$$\sigma_{X}$$\sigma_{2}$$\sigma_{1}$$\sigma_{1}$$\sigma_{X}$$\sigma_{2}$$\sigma_{1}$$\sigma_{2}$$\sigma_{1}$$\sigma_{X}$$\sigma_{2}$$\sigma_{X}$$\sigma_{2}$$\sigma_{1}$$\sigma_{X}$
Figure 6.  The tessellation of $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$
by the braid group orbit of $\Lambda$. The vertices of the triangles
correspond to spherical objects.

######  Remark 5.15.

The complement of $\Lambda$ in $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$
has three connected components. We can describe these three components in two
ways. For the first, write $\tau^{\prime}$ in the complement of $\Lambda$ as
$\tau^{\prime}=\beta\tau$, where $\beta\in B_{3}$ and $\tau\in\Lambda$. Write
$\beta$ in a form recognised by the automaton in Figure 3. Then the three
components correspond to the three possible end states of the automaton when
it reads the writing of $\beta$. For the second, recall from the proof of 5.6
that the three numbers

$m_{\tau^{\prime}}(P_{1}),m_{\tau^{\prime}}(P_{2}),\text{ and
}m_{\tau^{\prime}}(X)$

satisfy a degenerate triangle inequality—one is the sum of the other two. The
three connected components correspond to the ways in which the inequality
degenerates.

###  5.7. The closure

We know by 4.4 that the closure of
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ in $\mathbb{P}^{\mathbf{S}}$ is
compact. The goal of this section is to identify this closure. Set

$\displaystyle P$
$\displaystyle=\overline{h(\mathbf{S})}=h(\mathbb{P}^{1}(\operatorname{\mathbb{R}}))\subset\mathbb{P}^{\mathbf{S}}$
$\displaystyle M$
$\displaystyle=m\left(\operatorname{Stab}(\mathcal{C})/\mathbb{C}\right)\subset\mathbb{P}^{\mathbf{S}}.$

The main goal of this section is to prove that $\overline{M}=M\cup P$.

Let

$\overline{\Delta}=\left(\operatorname{\mathbb{R}}_{\geq 0}^{3}\setminus
0\right)/\operatorname{\mathbb{R}}_{>0}.$

Then $\overline{\Delta}$ is homeomorphic to a closed planar triangle. An
immediate consequence of the linearity (from 5.14) is the following.

######  Proposition 5.16 (Closure of $\Lambda$).

The closure of $\Lambda$ in $\mathbb{P}^{\mathbf{S}}$ is

$\overline{\Lambda}=\Lambda\cup\left\\{\operatorname{\overline{hom}}(P_{1}),\operatorname{\overline{hom}}(P_{2}),\operatorname{\overline{hom}}(X)\right\\}.$

This closure is homeomorphic to $\overline{\Delta}$.

###### Proof.

Identify $\Lambda$ with $\Delta$ using the Gromov coordinates. From 5.14, we
see that the map $m\colon\Delta\to\mathbb{P}^{\mathbf{S}}$ extends to a
continuous map $\overline{\Delta}\to\mathbb{P}^{\mathbf{S}}$. Its image is
closed, and equals the union of $\Lambda$ and the three additional points
described above. The map from $\overline{\Delta}$ is injective, and hence an
isomorphism onto its image. ∎

It is convenient to have a measure of closeness for two elements of a
projective space. Given two non-zero vectors in
$\operatorname{\mathbb{R}}^{n}$ for some positive integer $n$, denote by
$\sphericalangle(v,w)$ the (acute) angle between them. By a slight abuse of
notation, we use $\sphericalangle(v,w)$ also for two points
$v,w\in\mathbb{P}^{n-1}$; this is just the angle between any two
representatives in $\operatorname{\mathbb{R}}^{n}$.

Given a finite subset $T\subset\mathbf{S}$, let
$h_{T}\colon\mathbf{S}\to\mathbb{P}^{T}$ be the composition of
$h\colon\mathbf{S}\to\mathbb{P}^{\mathbf{S}}$ and the projection onto the
$T$-coordinates.

######  Proposition 5.17 (Group action contracts).

Fix two elements $A,B\in\mathbf{S}$ and a finite subset $T\subset\mathbf{S}$.
Given an $\epsilon>0$, for all but finitely many elements $L$ of
$\operatorname{PSL}_{2}(\mathbb{Z})$, we have

$\sphericalangle(h_{T}(LA),h_{T}(LB))<\epsilon.$

###### Proof.

In the identification
$S=\mathbb{P}^{1}(\mathbb{Z})\subset\mathbb{P}^{1}(\operatorname{\mathbb{R}})$,
let $A$ and $B$ be represented by two integer vectors $a$ and $b$ in
$\mathbb{Z}^{2}$. The map
$h_{T}\colon\mathbb{P}^{1}(\operatorname{\mathbb{R}})\to\mathbb{P}^{T}$ is
continuous, and hence uniformly continuous. Therefore, it suffices to check
that for all but finitely many $L$, the angle between $La$ and $Lb$ in
$\operatorname{\mathbb{R}}^{2}$ is small. To see this, observe that we have

$\sin\left(\sphericalangle(La,Lb)\right)=\frac{|\det L|\cdot|\det(a\mid
b)|}{|La|\cdot|Lb|}=\frac{|\det(a\mid b)|}{|La|\cdot|Lb|}.$

Since $L$ is required to have integer entries, the quantity $|L|\cdot|Lb|$ is
large—greater than $\epsilon|\det(a\mid b)|$—for all but finitely many $L$.
The claim follows. ∎

######  Proposition 5.18 (Closure of $M$).

The sets $M$ and $P$ are disjoint and their union is the closure of $M$.

###### Proof.

The mass of every object in a stability condition is positive. On the other
hand, $\operatorname{\overline{hom}}(x,x)=0$, by definition. Therefore, $M$
and $P$ are disjoint.

By 5.16, we know that $P$ is contained in the closure of $M$. We now prove
that $M\cup P$ is closed. Let $\tau_{n}$ be a sequence in
$\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ whose images $m(\tau_{n})$ in
$M$ approach a limit $z\in\mathbb{P}^{\mathbf{S}}$. We must show that $z$ lies
in $M\cup P$.

Write $\tau_{n}=\beta_{n}\tau^{\prime}_{n}$, where $\beta_{n}$ is a braid and
$\tau^{\prime}_{n}\in\Lambda$ is a stability condition of type A. Let
$\overline{\beta_{n}}$ be the image of $\beta_{n}$ in
$\operatorname{PSL}_{2}(\operatorname{\mathbb{Z}})$. We have two
possibilities: the set $\\{\overline{\beta}_{n}\\}$ is finite or infinite.

If $\\{\overline{\beta}_{n}\\}$ is finite, then the sequence $\\{\tau_{n}\\}$
is contained in the union of finitely many translates of the set of standard
stability conditions $\Lambda$. By 5.16, the closure of $\Lambda$ is contained
in $M\cup P$. Hence, the closure of the union of finitely many translates of
$\Lambda$ is also contained in $M\cup P$. So the limit $z$ lies in $M\cup P$.

Suppose $\\{\overline{\beta}_{n}\\}$ is infinite. Set
$A_{n}=\beta_{n}(P_{1})$, $B_{n}=\beta_{n}(P_{2})$, and $C_{n}=\beta_{n}(X)$.
Since $P$ is compact, we may assume, after passing to a subsequence if
necessary, that $h(A_{n})$, $h(B_{n})$, and $h(C_{n})$ have limits in $P$, say
$A$, $B$, and $C$.

We first observe that $A=B=C$. To see this, let $T\subset\mathbf{S}$ be an
arbitrary finite set, and use the subscript $T$ to denote projections to
$\mathbb{P}^{T}$. It suffices to show that $A_{T}=B_{T}=C_{T}$. Since
$\\{\overline{\beta}_{n}\\}$ is infinite, by 5.17, we note that the angles
between $h_{T}(A_{n})$, $h_{T}(B_{n})$, and $h_{T}(C_{n})$ approach $0$ as $n$
approaches $\infty$. Therefore, the three sequences have the same limits. But
these limits are $A_{T}$, $B_{T}$, and $C_{T}$.

Next, we claim that $z=h(A)=h(B)=h(C)$. Indeed, by linearity (5.14), we know
that there exist non-negative real numbers $x_{n}$, $y_{n}$, $z_{n}$ such that

$m_{T}(\tau_{n})=x_{n}h_{T}(A_{n})+y_{n}h_{T}(B_{n})+z_{n}h_{T}(C_{n}).$

Taking the limit as $n\to\infty$ in projective space yields

$z=h(A)=h(B)=h(C).$

In particular, $z$ lies in $M\cup P$. The proof is thus complete. ∎

###  5.8. Homeomorphism to the closed disk

We take up the final part of the main theorem, namely that $(\overline{M},P)$
is a manifold with boundary homeomorphic to the unit disk. We explicitly
construct a homeomorphism from $\overline{M}$ to the disk, using the
unprojectivised, but suitably normalised, mass and hom functions.

Fix

$T=\\{P_{1},P_{2},X=P_{2}\to P_{1}\\}.$

Define a map

$\mu\colon\operatorname{Stab}(\mathcal{C})/i\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}^{T}=\operatorname{\mathbb{R}}^{3}$

by

$\mu\colon\tau\mapsto(m_{\tau}(P_{1}),m_{\tau}(P_{2}),m_{\tau}(X)).$

Similarly, define

$\eta\colon\mathbf{S}\to\operatorname{\mathbb{R}}^{T}=\operatorname{\mathbb{R}}^{3}$

by

$\eta(s)=(\operatorname{\overline{hom}}(s,P_{1}),\operatorname{\overline{hom}}(s,P_{2}),\operatorname{\overline{hom}}(s,X)).$

Thinking of $\mathbf{S}$ as
$\mathbb{P}^{1}(\operatorname{\mathbb{Z}})\subset\mathbb{P}^{1}(\operatorname{\mathbb{R}})$,
it is easy to see that $\eta$ extends to a continuous map

$\eta\colon\mathbb{P}^{1}(\operatorname{\mathbb{R}})\to\operatorname{\mathbb{R}}^{3}.$

Indeed, using 5.12, we get

$\eta\colon[a:c]\mapsto(|c|,|a|,|a+c|).$

Let $\tau$ be a stability condition with (semi)-stable objects $A$, $B$, and
$C$ of class $[P_{1}]$, $[P_{2}]$, and $[X]$ in the Grothendieck group. Let
$x$, $y$, $z$ be the Gromov coordinates of $\tau$, namely the non-negative
real numbers such that

$m_{\tau}(A)=y+z,\quad m_{\tau}(B)=z+x,\quad m_{\tau}(C)=x+y.$

Denote by $|-|$ the standard Euclidean norm in
$\operatorname{\mathbb{R}}^{3}$. We say that $\tau$ is _normalised_ if

(13) $x|\eta(A)|+y|\eta(B)|+z|\eta(C)|=1.$

######  Remark 5.19.

We point out one subtle aspect of the definition. If $\tau$ is degenerate—that
is, it has strictly semi-stable objects—then one of the $A$, $B$, or $C$ is
not uniquely determined, even up to shift. In this case, however, the
corresponding Gromov coordinate is $0$, and hence (13) is well-posed.

The normalisation yields a continuous section of

$\operatorname{Stab}(\mathcal{C})/i\operatorname{\mathbb{R}}\to\operatorname{Stab}(\mathcal{C})/\mathbb{C}.$

That is, for every stability condition $\tau$ up to rotation and scaling,
there is a unique normalised stability condition $\tau^{\nu}$ up to rotation,
and furthermore the map $\tau\to\tau^{\nu}$ is continuous.

We now identify $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ with its image
$M$ in $\mathbb{P}^{\mathbf{S}}$ and
$\mathbb{P}^{1}(\operatorname{\mathbb{R}})$ with its image $P$ in
$\mathbb{P}^{\mathbf{S}}$. Define

(14) $\pi\colon\overline{M}=M\cup P\to\operatorname{\mathbb{R}}^{3}$

as follows. For $\tau\in M$, set

(15) $\pi(\tau)=\mu(\tau^{\nu}),$

and for $s\in P$, set

$\pi(s)=\eta(s)/|\eta(s)|.$

By construction, $\pi(a:c)$ is the unit vector in the direction of

$\eta(a:c)=(|a|,|c|,|a+c|).$

That is, $\pi$ maps $P$ to the unit sphere in $\operatorname{\mathbb{R}}^{3}$.
It is easy to see that $\pi|_{P}$ is injective, and hence a homeomorphism onto
its image. The image consists of three circular arcs, one for each pair of
end-points from the three points

$\frac{1}{\sqrt{2}}(0,1,1),\quad\frac{1}{\sqrt{2}}(1,0,1),\text{ and
}\frac{1}{\sqrt{2}}(1,1,0).$

The arc joining each pair is a geodesic arc on the unit sphere; that is, the
plane it spans passes through the origin.

Equation 13 and the triangle inequality imply that for every stability
condition $\tau$, we have $|\pi(\tau)|\leq 1$. In fact, for every stability
condition, at least two of the Gromov coordinates are non-zero. Therefore, we
actually have a strict inequality $|\pi(\tau)|<1$.

To get a better understanding of $\pi$, let us study it on the translates of
the fundamental domain $\Lambda$. Let $\beta$ be a braid. Consider the
translate $\beta\overline{\Lambda}$. Set

$A=\beta(P_{1}),\quad B=\beta(P_{2}),\quad C=\beta(X).$

These are the (semi)-stable objects for the stability conditions in
$\beta\Lambda$. Recall that $\beta\overline{\Lambda}$ is homeomorphic to the
projectivised octant $\overline{\Delta}=\left(\operatorname{\mathbb{R}}_{\geq
0}^{3}\setminus 0\right)/\operatorname{\mathbb{R}}_{>0}$; the homeomorphism is
given by the Gromov coordinates. The normalisation condition (13) is a linear
condition on the Gromov coordinates. It cuts out an affine hyperplane slice of
the octant, and gives a section of the projectivisation map. Since $\pi$ is
linear in the Gromov coordinates for normalised stability conditions, it maps
$\beta\overline{\Lambda}$ linearly, and homeomorphically, onto the triangle in
$\operatorname{\mathbb{R}}^{3}$ with vertices $\pi(A)$, $\pi(B)$, and
$\pi(C)$.

Let $\Phi\subset\operatorname{\mathbb{R}}^{3}$ be the union of the triangle

$\\{(x,y,z)\mid x+y+z=\sqrt{2},\,x+y-z\geq 0,\,y+z-x\geq 0,\,z+x-y\geq 0\\},$

and the three circular segments, each bounded by an edge of the triangle above
and the arc in the image of $\pi(P)$ with the same end-points as the edge (see
Figure 7).

$x$$y$$z$ Figure 7. The homeomorphic image
$\Phi\subset\operatorname{\mathbb{R}}^{3}$ of the compactified stability
manifold.

Observe that $\pi$ maps $\overline{\Lambda}$, the triangle formed by the
standard stablity conditions, to the central triangle of $\Phi$. On the other
hand, the translates of $\overline{\Lambda}$ on the three sides of the
identity on the exchange graph (Figure 6) are mapped to the three circular
segments. Indeed, the segment to which $\pi$ sends a non-standard stability
condition $\tau$ depends on how the triangle inequality degenerates among the
$\tau$-masses of $P_{1}$, $P_{2}$, and $X$, and this in turn, is determined by
where $\tau$ lies on the exchange graph.

######  Proposition 5.20.

The map $\pi\colon\overline{M}\to\Phi$ is a homeomorphism, where
$\overline{M}$ is given the subspace topology from $\mathbb{P}^{\mathbf{S}}$.

###### Proof.

Let us first show that $\pi$ is continuous. Since $\pi$ is linear on the
translates of $\overline{\Lambda}$, and these translates cover $M$, it follows
that $\pi$ is continuous on $M$. The restriction of $\pi$ to $P$ is given by

$\pi\colon[a:c]\mapsto\frac{1}{\sqrt{a^{2}+c^{2}+(a+c)^{2}}}\left(|a|,|c|,|a+c|\right),$

which is continuous. We must show that $\pi$ is continuous on $\overline{M}$
at a point $p\in P$.

Let $\tau_{n}$ be a sequence of normalised stability conditions converging to
$p\in P$. We already know that $m_{T}(\tau_{n})$ converges to $h_{T}(p)$ in
the projective space $\mathbb{P}^{T}$. Therefore, it suffices to show that
$\pi(\tau_{n})$ approaches a vector of norm 1 in
$\operatorname{\mathbb{R}}^{3}$.

Let $\epsilon>0$ be given. Write $\tau_{n}=\beta_{n}\tau^{\prime}_{n}$ for
some braid $\beta_{n}$ and standard stability condition $\tau_{n}$. Denote by
$\overline{\beta}_{n}$ the image of $\beta_{n}$ in
$\operatorname{PSL}_{2}(\operatorname{\mathbb{Z}})$. If the set
$\\{\overline{\beta}_{n}\\}$ is finite, then the sequence $\tau_{n}$ lies in
finitely many translates of $\Lambda$. Without loss of generality, we may
assume that it lies in one translate, say $\beta\Lambda$. Recall that
$\pi\colon\overline{\beta}\Lambda\to\operatorname{\mathbb{R}}^{3}$ is linear
in the Gromov coordinates and hence continuous. Therefore,
$\pi(\tau_{n})\to\pi(p)$ as $n\to\infty$.

The harder case is when the set $\\{\overline{\beta}_{n}\\}$ is infinite. Set
$A_{n}=\beta_{n}(P_{1})$, $B_{n}=\beta_{n}(P_{2})$, $C_{n}=\beta_{n}(X)$, and
let $x_{n}$, $y_{n}$, $z_{n}$ be the Gromov coordinates. But in this case, we
know by 5.17 that the angle between $\eta(A_{n})$, $\eta(B_{n})$, and
$\eta(C_{n})$ approaches $0$ as $n$ approaches $\infty$. Therefore, the
difference between

$|x_{n}\eta(A_{n})+y_{n}\eta(B_{n})+z_{n}\eta(C_{n})|\text{ and
}x_{n}|\eta(A_{n})|+y_{n}|\eta(B_{n})|+z_{n}|\eta(C_{n})|$

approaches $0$ as $n$ approaches $\infty$. Since $\tau_{n}$ is normalised, the
right-hand quantity is $1$, and the left-hand quantity is $|\pi(\tau_{n})|$.

We have now proved that $\pi\colon\overline{M}\to\Phi$ is continuous. Since
$\overline{M}$ is compact, $\pi$ is a homeomorphism once we know that it is a
bijection. We know that the map

$\pi\colon P\to\partial\Phi=\Phi\cap\\{v\mid|v|=1\\}$

is a bijection and $\pi$ maps $M$ to

$\Phi^{\circ}=\Phi\cap\\{v\mid|v|<1\\}.$

Recall that $M$ is the union of the translates of the fundamental domains
$\beta\Lambda$, and each fundamental domain is homeomorphic to a clipped
triangle (a planar triangle minus the vertices). The map $\pi$ maps each
translate bijectively to a (clipped) triangle in $\Phi^{\circ}$. To check that
$\pi$ is an injection, we must check that the clipped triangles
$\pi(\beta\Lambda)$ have disjoint interiors, and two of them, say
$\pi(\beta\Lambda)$ and $\pi(\beta^{\prime}\Lambda)$, intersect along an edge
if and only if $\beta$ and $\beta^{\prime}$ are adjacent in the exchange
graph. To check that $\pi$ is a surjection, we must check that the union
$\bigcup_{\beta}\pi(\beta\Lambda)$ is $\Phi^{\circ}$.

Take $\beta=\operatorname{id}$. Then $\pi(\Lambda)$ is the central triangle of
$\Phi^{\circ}$. The only other triangles that intersect this triangle are
$\pi(\sigma_{1}\Lambda)$, $\pi(\sigma_{2}\Lambda)$, and
$\pi(\sigma_{X}\Lambda)$, and the intersections are along the three edges, as
required.

The exchange graph divides the non-trivial translates of $\Lambda$ into three
connected components, and the translates in each of the three components map
to the three distinct circular segments in $\Phi^{\circ}$. So it suffices to
restrict our attention to one component and the corresponding circular
segment. Since $\gamma$ permutes the components, it suffices to look at only
one of them.

Let us consider the component containing $\sigma_{1}\Lambda$. The translates
in this component are $A\Lambda$ where
$A\in\operatorname{PSL}_{2}(\operatorname{\mathbb{Z}})$ is a matrix that has
an admissible cyclic writing that starts with $\sigma_{1}$. Suppose
$A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$. Then $\pi(A\Lambda)$ is the
clipped triangle with vertices

(16) $\pi([a:c]),\quad\pi([b:d]),\quad\pi([a-b:c-d]).$

Since we are considering the translates in the $\sigma_{1}\Lambda$ component,
the points $[a:c]$, $[b:d]$, and $[a-b:c-d]$ lie in the arc $[P_{1},P_{2}]$ of
$\mathbb{P}^{1}(\mathbb{Z})\subset\mathbb{P}^{1}(\operatorname{\mathbb{R}})$.
(So, under the bijection
$\mathbb{P}^{1}(\mathbb{Z})=\operatorname{\mathbb{Q}}\cup\\{\infty\\}$ given
by $[a:c]\to a/c$, they correspond to $\operatorname{\mathbb{Q}}_{\geq
0}\cup\\{\infty\\}$). By construction, the points $\pi([a:c])$ and
$\pi([b:d])$ form an edge of a clipped triangle if and only if $|ad-bc|=1$.
Thus, the triangles (16) form the Farey tessellation [3, Chapter 8] of the
circular segment, which has the intersection properties as dictated by the
exchange graph. ∎

##  6\. The $\widehat{A}_{1}$ case

The aim of this section is to construct the Thurston compactification of
stability space for 2-Calabi–Yau category $\mathcal{C}_{\Gamma}$ where
$\Gamma$ is the $\widehat{A}_{1}$ quiver.

###  6.1. Overview

Let $\mathcal{C}$ be the 2-Calabi–Yau category associated to the
$\widehat{A}_{1}$ quiver, as defined in § 2. It is a graded,
$\mathbf{k}$-linear triangulated category classically generated by two
spherical objects, which we denote $P_{0}$ and $P_{1}$.

The extension closure of $P_{0}$ and $P_{1}$ in $\mathcal{C}$ is an abelian
category; it is the heart of a (bounded) $t$-structure. We refer to it as the
_standard heart_ $\heartsuit_{\operatorname{std}}$.

###  6.2. The spherical objects and the boundary circle

As discussed in § 2.2, the spherical twists in the objects $P_{i}$ satisfy the
defining relations of the Artin–Tits braid group of the $\widehat{A}_{1}$
quiver. In this case, this group is the free group on two letters $F_{2}$:

$F_{2}=\langle\sigma_{0},\sigma_{1}\rangle.$

Denoting the twist in $P_{i}$ as $\sigma_{P_{i}}$, we have a weak action of
$F_{2}$ on $\mathcal{C}$ given by

$\sigma_{i}\to\sigma_{P_{i}}.$

Let $G$ be the image of $F_{2}$ in $\operatorname{Aut}(\mathcal{C})$; it is
precisely the subgroup generated by $\sigma_{P_{0}}$ and $\sigma_{P_{1}}$.

Define $\gamma$ as the following auto-equivalence:

$\gamma=\sigma_{P_{1}}\sigma_{P_{0}}.$

One can check that for any integer $n$, we can write

$\displaystyle\sigma_{1}\gamma^{n}(P_{0})$
$\displaystyle=|2n+2|P_{1}\to|2n+1|P_{0},\text{ and }$
$\displaystyle\gamma^{n}P_{1}$ $\displaystyle=|2n+1|P_{1}\to|2n|P_{0}.$

The objects obtained as above are all spherical; let $T$ be the set containing
all of them. Then $T$ contains are exactly the objects of the form
$nP_{1}\to(n\pm 1)P_{0}$ for any integer $n$. There is a bijection
$\mathbb{Z}\to T$ given by

$n\mapsto|n|P_{1}\to|n-1|P_{0}.$

Let $P_{n}$ denote the object $|n|P_{1}\to|n-1|P_{0}$.

It is easy to check that $\gamma$ preserves $T$, and in fact acts by
translation by $2$ up to shift:

$\gamma(P_{n})=\begin{cases}P_{n+2},&n\neq 0,-1,\\\
P_{n+2}[-1],&n=0,-1.\end{cases}$

Regard $\mathbb{Z}$ as the subset of $\mathbb{P}^{1}({\mathbb{Z}})$ consisting
of points of the form $[k:1]$. We can write a homomorphism
$G\to\operatorname{PSL}_{2}(\mathbb{Z})$ compatible with the action of
$\gamma$ on $\mathbb{Z}$, as follows:

$\sigma_{P_{0}}\mapsto\begin{pmatrix}1&0\\\
2&1\end{pmatrix},\quad\sigma_{P_{1}}\mapsto\begin{pmatrix}-3&2\\\
-2&1\end{pmatrix}.$

Indeed, under the homomorphism above, $\gamma$ is the matrix of translation by
$2$:

$\gamma\mapsto\begin{pmatrix}1&2\\\ 0&1\end{pmatrix}.$

Let $\mathbf{S}$ be the $G$-orbit of $\\{P_{0},P_{1}\\}$. We have a
$G$-equivariant map $i\colon\mathbf{S}\to\mathbb{P}^{1}(\mathbb{Z})$
characterised by $P_{n}\mapsto[n:1]$. It is easy to check that $i$ is
injective. For convenience, let $P_{\infty}$ be any one of the indecomposable
extensions of $P_{0}$ by $P_{1}$, written as $P_{1}\to P_{0}$. We extend the
embedding above to $\mathbf{S}\cup\\{P_{\infty}\\}$ by sending $P_{\infty}$ to
$[1:0]$, which is the point at infinity.

The element $\gamma=\sigma_{1}\sigma_{0}$ acts on the objects of $T$ by an
infinite-order cyclic rotation by two steps, lowering the phases. More
precisely,

$\displaystyle\gamma$ $\displaystyle\colon P_{n}\mapsto P_{n+2}\text{ for
}n\neq 0,-1,\infty,$ $\displaystyle\gamma$ $\displaystyle\colon P_{n}\mapsto
P_{n+2}[-1]\text{ for }n=0,-1,$ $\displaystyle\gamma$ $\displaystyle\colon
P_{\infty}\mapsto P_{\infty}.$

We see that the object $P_{k}\in\mathbf{S}$ maps to the point $k=[k:1]$ of
$\mathbb{P}^{1}(\mathbb{Z})$. For each $k\in\mathbb{Z}$, let $\sigma_{k}$
denote the spherical twist in the object $P_{k}$. We have

$\sigma_{2k}=\gamma^{k}\sigma_{0}\gamma^{-k}\text{ and
}\sigma_{2k+1}=\gamma^{k}\sigma_{1}\gamma^{-k}.$

The image of $\sigma_{k}$ in $\operatorname{PSL}_{2}(\mathbb{Z})$ is given by

(17) $\sigma_{k}\mapsto\begin{pmatrix}2k+1&-2k^{2}\\\ 2&-2k+1\end{pmatrix}.$

###  6.3. Standard stability conditions

We say that a stability condition $\tau$ is _standard_ if its heart
$\heartsuit([0,1))$ is the standard heart $\heartsuit_{\rm std}$. Let $\phi$
denote the phase function for $\tau$. Then we either have
$\phi(P_{0})\leq\phi(P_{1})$ or $\phi(P_{1})\leq\phi(P_{0})$. In the first
case, we say $\tau$ is standard of _type A_ , and in the latter case, of _type
B_. Note that, in type A, all objects of $T$ are $\tau$-semistable.

Let $\Lambda\subset\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ be the closure
of the set of standard stability conditions of type A. Then $\Lambda$ includes
non-standard degenerate stability conditions where
$\phi(P_{1})=\phi(P_{0})+1$. We continue to call these stability conditions as
of _type A_ , reserving the adjective standard for those where
$\phi(P_{1})<\phi(P_{0})+1$. Figure 8 shows the central charges of the objects
of $T$ in a standard non-degenerate stability condition of type A. The dotted
line is the direction of the sum of the central charges of $P_{0}$ and
$P_{1}$, which is also the central charge of $P_{\infty}$.

$P_{0}$$P_{1}$$P_{2}$$P_{3}$$\cdots$$P_{\infty}$$\ddots$$P_{-1}$$P_{-2}$
Figure 8. The central charges of the objects of $T$ in a standard non-
degenerate stability condition of type A.

###  6.4. The automaton

Fix a non-degenerate stability condition $\tau$ of type A. Figure 9 describes
a $\tau$-HN-automaton $\Theta$ by depicting some of the incoming and outgoing
edges at one of the vertices.

$[P_{k},P_{k+1}]$$[P_{k-1},P_{k}]$$[P_{k-2},P_{k-1}]$$[P_{k-3},P_{k-2}]$$[P_{k+1},P_{k+2}]$$[P_{k+2},P_{k+3}]$$[P_{k+3},P_{k+4}]$$\sigma_{k}$$\sigma_{k-1}$$\sigma_{k-2}$$\sigma_{k-3}$$\sigma_{k+2}$$\sigma_{k+3}$$\gamma$$\gamma^{-1}$
Figure 9. An HN automaton for a nondegenerate type A stability condition for
$\widehat{A}_{1}$, showing the outgoing edges from $[P_{k},P_{k+1}]$.

Here is a more formal description of the automaton.

  1. (1)

The automaton $\Theta$ is defined by the $G$-labelled graph with vertices
indexed by $\mathbb{Z}$ and edges as indicated in Figure 9: an edge from state
$k$ to state $j$ labelled $\sigma_{P_{j}}$ (unless $j=k+1$); an edge from
state $k$ to state $(k+2)$ labelled $\gamma$; and an edge from state $k$ to
state $(k-2)$ labelled $\gamma^{-1}$.

  2. (2)

The state $k$ supports the objects $P_{k}$ and $P_{k+1}$. The $\Theta$-set
$S\subset\operatorname{ob}\mathcal{C}$ is the one generated by this data.

  3. (3)

The representation $M$ associates to the state $k$ the free abelian group
$\mathbb{Z}^{\\{P_{k},P_{k+1}\\}}$. The map associated to the edge
$\sigma_{P_{j}}\colon k\to j$ is represented by the matrix

(18) $\begin{pmatrix}|j-k-1|&|j-k-2|\\\ |j-k|&|j-k-1|\end{pmatrix}$

in the standard basis. The maps associated to the edges labelled
$\gamma^{\pm}$ are represented by the identity matrix.

  4. (4)

The map $S\to M$ is the unique one that sends $\\{P_{k},P_{k+1}\\}$ to the
corresponding standard basis vectors. The map
$HN_{k}\colon\mathbb{Z}^{\\{P_{k},P_{k+1}\\}}\to\mathbb{Z}^{\Sigma}$ is
induced by the inclusion $\\{P_{k},P_{k+1}\\}\to\Sigma$.

6.1 shows that $S\to M$ is well-defined and that $HN_{k}$ indeed computes the
HN-multiplicities.

######  Proposition 6.1.

Let $e\colon[P_{k},P_{k+1}]\to[P_{j},P_{j+1}]$ be an edge of the automaton.
Let $x\in\mathcal{C}$ be an object whose HN filtration consists of shifts of
$P_{k}$ and $P_{k+1}$.

  1. (1)

The HN filtration of $e\cdot x$ consists of shifts of $P_{j}$ and $P_{j+1}$.

  2. (2)

Let $v\in\mathbb{Z}^{\\{P_{k},P_{k+1}\\}}$ be the HN multiplicity vector of
$x$. Then $M(e)v$ is the HN multiplicity vector of $e\cdot x$.

###### Proof.

Let $\sigma$ be the auto-equivalence associated to $e$. By an argument similar
to 5.1, the image under $\sigma$ of the HN filtration of $x$ is geodesic. So
it suffices to prove the proposition for $x=P_{k}$ and $x=P_{k+1}$.

Suppose $\sigma=\gamma^{\pm}$ so that $j=k\pm 2$. Since $\gamma^{\pm}$ sends
$P_{i}$ to $P_{i\pm 2}$ (up to shift), preserving the ordering of the phases,
it is clear that $\sigma(x)$ is supported at $[P_{j},P_{j+1}]$, and that the
HN multiplicities transform by the identity matrix.

Suppose that $k=0$. We can check explicitly that if $j\neq 1$, we have

(19) $\begin{split}\sigma_{j}P_{0}&=P_{j}^{\oplus|j-1|}[-1]\to
P_{j+1}^{\oplus|j|},\text{ and }\\\
\sigma_{j}P_{1}&=P_{j}^{\oplus|j-2|}[-1]\to P_{j+1}^{\oplus|j-1|}.\end{split}$

As a result, both $\sigma_{j}(P_{0})$ and $\sigma_{j}(P_{1})$ are both
supported at state $j$ and their HN multiplicities transform according to the
matrix $M(e)$.

Let $k$ be arbitrary. Observe that we have

$\sigma_{j}(P_{k})=\begin{cases}\gamma^{m}\sigma_{j-2m}P_{0},&\text{if
}k=2m\text{ is even},\\\ \gamma^{m}\sigma_{j-2m}P_{1},&\text{if }k=2m+1\text{
is odd}.\end{cases}$

Using the above, we see that for any $k$ except $k=j-1$, we have

(20) $\sigma_{j}P_{k}=P_{j}^{\oplus|j-k-1|}[-1]\to P_{j+1}^{\oplus|j-k|}.$

As a result, $\sigma P_{k}$ and $\sigma P_{k+1}$ are both supported at state
$j$ and their HN multiplicities transform according to the matrix $M(e)$. ∎

The next proposition shows that the automaton recognises every braid and every
object of $\mathbf{S}$.

######  Proposition 6.2.

Let $\beta\in F_{2}$ and let $s\in\mathbf{S}$ be arbitrary.

  1. (1)

The automaton in Figure 9 recognises $\beta$. More precisely, $\beta$ has the
recognised expression

$\beta=\gamma^{n}\sigma_{a_{1}}\sigma_{a_{2}}\dots\sigma_{a_{k}},$

such that $a_{i}-a_{i+1}\neq 1$ for each $i$.

  2. (2)

The automaton in Figure 9 recognises $s$.

  3. (3)

The $\tau$-HN filtration of $s$ consists of at most two objects out of $T$, up
to shift.

###### Proof.

For (1), first write $\beta$ as any expression in the generators
$\\{\sigma_{0}^{\pm 1},\sigma_{1}^{\pm 1}\\}$. For any $i\in\mathbb{Z}$, we
have

$\sigma_{i}^{-1}=\sigma_{i-1}\gamma^{-1},\quad\gamma^{-1}\sigma_{i}=\sigma_{i-2}\gamma^{-1},\quad\gamma=\sigma_{i}\sigma_{i-1}.$

Using these relations, we first rewrite $\beta$ solely in terms of the
elements $\sigma_{i}$ for $i\in\mathbb{Z}$ and $\gamma^{\pm}$. Then we
successively replace any instances of $\sigma_{i}\sigma_{i-1}$ by $\gamma$,
and commute all powers of $\gamma$ to the left. This process terminates,
resulting in an expression of the desired form.

For (2), write $s=\beta P_{i}$ where $i$ is either $0$ or $1$. Assume that
$i=0$; the other case is similar. Rewrite

$\beta=\gamma^{n}\sigma_{a_{1}}\sigma_{a_{2}}\dots\sigma_{a_{k}}$

as in (1). Note that the object $P_{0}$ is supported at two states, namely
$[P_{0},P_{1}]$ and $[P_{-1},P_{0}]$. We can apply $\sigma_{a_{k}}$ at least
at one of these two states. From the condition on our cyclic writing, it is
clear that all subsequent letters are applicable. To finish, any power of
$\gamma$ is applicable at any vertex. Therefore $s$ is recognised by the
automaton.

Finally, (3) follows from (2) and from 6.1 ∎

######  Remark 6.3.

Let $\tau$ be a degenerate stability condition of type A. Then $\Theta$ does
not compute the $\tau$-HN multiplicities, but it does compute $\tau$-masses,
as explained in 5.3

######  Remark 6.4.

Let $\tau$ be a degenerate standard stability condition of type A. Set
$P_{i}^{\prime}$ to be the object $\sigma_{0}P_{i-1}$. The objects
$P_{i}^{\prime}$ are also semistable in $\tau$, in addition to the objects
$P_{i}$. Note that up to shift, we have $P_{1}^{\prime}=P_{0}$ and
$P_{0}^{\prime}=P_{1}$. Recall that $\gamma=\sigma_{1}\sigma_{0}$. Let
$\gamma^{\prime}=\sigma_{0}\sigma_{1}$.

Figure 10 shows an automaton that computes $\tau$-HN multiplicities. Only the
outgoing arrows labelled $\sigma_{0}$ and $\sigma_{1}$ are depicted. The
remaining possible labels for outgoing arrows are $\gamma^{\pm}$ and
$\gamma^{\prime\pm}$, which act as follows:

$\displaystyle\gamma$
$\displaystyle\colon[P_{i},P_{i+1}]\to[P_{i+2},P_{i+3}]\text{ unless
}i=-2,\quad\gamma\colon[P^{\prime}_{i},P^{\prime}_{i+1}]\to[P_{1},P_{2}];$
$\displaystyle\gamma^{-1}$
$\displaystyle\colon[P_{i},P_{i+1}]\to[P_{i-2},P_{i-3}],\text{ unless
}i=2,\quad\gamma^{-1}\colon[P^{\prime}_{i},P^{\prime}_{i+1}]\to[P_{-2},P_{-1}];$
$\displaystyle\gamma^{\prime}$
$\displaystyle\colon[P_{i}^{\prime},P_{i+1}^{\prime}]\to[P_{i+2}^{\prime},P_{i+3}^{\prime}]\text{
unless
}i=-2,\quad\gamma^{\prime}\colon[P_{i},P_{i+1}]\to[P^{\prime}_{1},P^{\prime}_{2}];$
$\displaystyle\gamma^{\prime-1}$
$\displaystyle\colon[P_{i}^{\prime},P_{i+1}^{\prime}]\to[P_{i-2}^{\prime},P_{i-3}^{\prime}]\text{
unless
}i=2,\quad\gamma^{\prime-1}\colon[P_{i},P_{i+1}]\to[P^{\prime}_{-2},P^{\prime}_{-1}].$

Since $\Theta$ suffices for mass computations, we omit further details.

$[P_{-1},P_{0}]$$[P_{-2},P_{-1}]$$\cdots$$\cdots$$[P_{2},P_{3}]$$[P_{1},P_{2}]$$[P^{\prime}_{-1},P^{\prime}_{0}]$$[P^{\prime}_{-2},P^{\prime}_{-1}]$$\cdots$$\cdots$$[P^{\prime}_{2},P^{\prime}_{3}]$$[P^{\prime}_{1},P^{\prime}_{2}]$$\sigma_{1}$$\sigma_{0}$$\sigma_{0}$$\sigma_{0}$$\sigma_{0}$$\sigma_{0}$$\sigma_{0}$$\sigma_{0}$$\sigma_{1}$$\sigma_{1}$$\sigma_{1}$$\sigma_{1}$$\sigma_{1}$$\sigma_{1}$
Figure 10. An HN automaton for a degenerate standard stability condition of
type A, depicting outgoing arrows labelled $\sigma_{0}$ and $\sigma_{1}$.

We observe that $\Theta$ is degenerating.

######  Proposition 6.5.

Let $\tau$ be a stability condition of type A. Then $\Theta$ is degenerating
for $\tau$.

###### Proof.

By 6.1, every braid has a recognised expression that ends in $\sigma_{i}$ for
some $i\in\mathbb{Z}$. Therefore it suffices to find a finite set of non-
degenerate triangles, such that each of these letters makes at least one of
the triangles degenerate. Let us exhibit two triangles that suffice.

The non-degenerate distinguished triangle

$P_{1}[-1]\to P_{0}\oplus P_{0}\to P_{-1}\xrightarrow{+1}$

is made degenerate by $\sigma_{j}$ for all $j\in\mathbb{Z}\setminus\\{0,1\\}$.
This can be checked directly using the formulas (20). For $j=0$ or $j=1$, we
can use any $\gamma$-translate of the above triangle. In particular, the non-
degenerate triangle

$P_{3}[-1]\to P_{2}\oplus P_{2}\to P_{1}\xrightarrow{+1}$

is made degenerate by both $\sigma_{0}$ and $\sigma_{1}$. ∎

###  6.5. Consequences of the automaton

In this section, we use the automaton from § 6.4 to prove results about our
projective embedding of $\operatorname{Stab}(\mathcal{C})/\mathbb{C}$.

######  Proposition 6.6.

The map
$m\colon\operatorname{Stab}(\mathcal{C})/\mathbb{C}\to\mathbb{P}^{\mathbf{S}}$
is a homeomorphism onto its image.

###### Proof.

The proof is entirely analogous to the proof of 5.7. ∎

As in the $A_{2}$-case, the representation $M$ of $\Theta$ is projectively
equivalent to the standard representation of
$\operatorname{PSL}_{2}(\mathbb{Z})$.

######  Proposition 6.7.

We have an isomorphism of $\Theta$-sets
$\phi\colon\mathbb{P}M\to\mathbb{P}^{1}(\mathbb{Z})$.

###### Proof.

The isomorphism $\phi$ is given by the linear map
$\mathbb{Z}^{\\{P_{k},P_{k+1}\\}}\to\mathbb{Z}^{2}$ defined by

$\phi_{k}\colon P_{k}\mapsto(k,1),\quad P_{k+1}\mapsto(k+1,1).$

Let us check that $\phi$ is $\Theta$-equivariant. It is evident that
$\phi_{k}$ intertwines the edges labelled $\gamma^{\pm}$. To check that it
also intertwines $e\colon k\to j$ labelled $\sigma_{P_{j}}$, we must check
that

$\sigma(P_{j})\cdot\phi_{k}=\phi_{j}\cdot M(e).$

Recalling the matrix of $\sigma_{P_{j}}$ from (17) and $M(e)$ from (18), we
must check

$\begin{pmatrix}2j+1&-2j^{2}\\\ 2&-2j+1\end{pmatrix}\begin{pmatrix}k&k+1\\\
1&1\end{pmatrix}=\pm\begin{pmatrix}j-k-1&j-k-2\\\
j-k&j-k-1\end{pmatrix}\begin{pmatrix}j&j+1\\\ 1&1\end{pmatrix}.$

This is straightforward. ∎

Recall that we have a $G$-equivariant map
$i\colon\mathbf{S}\to\mathbb{P}^{1}(\mathbb{Z})$. 6.7 gives another map,
defined by $s\mapsto\phi([s])$. By the same argument as in the $A_{2}$ case,
these two maps agree.

The division of $\mathbf{S}$ according to the HN support corresponds nicely to
a geometric division of the circle $\mathbb{P}^{1}(\mathbb{R})$, which we now
describe. The objects $P_{i}$ divide $\mathbb{P}^{1}(\mathbb{R})$ into a
collection of closed arcs, namely $\\{[P_{i},P_{i+1}]\mid i\in\mathbb{Z}\\}$.
Together with $P_{\infty}$, these arcs cover all of
$\mathbb{P}^{1}(\mathbb{R})$. We show that this decomposition of the circle is
compatible with the automaton.

######  Proposition 6.8.

For each $i\in\mathbb{Z}$, the objects supported at state $i$ are mapped to
the arc $[P_{i},P_{i+1}]$.

###### Proof.

Entirely analogous to that of 5.9. ∎

We obtain a Rouquier–Zimmermann theorem for $\widehat{A}_{1}$.

######  Proposition 6.9.

Let $x\in\mathbf{S}$ such that $i(x)=[a:c]\in\mathbb{P}^{1}(\mathbb{Z})$, with
$a$ and $c$ relatively prime integers. Then the minimal complex of projective
modules representing $x$ has exactly $|a-c|$ occurrences of $P_{0}$ and $|a|$
occurrences of $P_{1}$.

###### Proof.

Entirely analogous to that of 5.10. ∎

######  Proposition 6.10.

Let $x\in\mathbf{S}$ such that any minimal complex of $x$ has exactly $n_{0}$
occurrences of $P_{0}$ and $n_{1}$ occurrences of $P_{1}$. Then
$\operatorname{\overline{hom}}(x,P_{1})=2n_{0}$ and
$\operatorname{\overline{hom}}(x,P_{0})=2n_{1}$.

###### Proof.

We prove the assertion for $\operatorname{\overline{hom}}(x,P_{1})$. The proof
for $\operatorname{\overline{hom}}(x,P_{0})$ is similar. Fix a non-degenerate
stability condition of type A.

If $x=P_{1}$, the statement is evident. Suppose that $x$ is supported at
$[P_{k},P_{k+1}]$ for $k\neq 0,1$. Consider the Harder–Narasimhan filtration
of $x$:

$0=x_{0}\to x_{1}\to\cdots\to x_{n}=x,$

with sub-quotients $z_{i}=\operatorname{Cone}(x_{i}\to x_{i+1})$. Each $z_{i}$
is either $P_{k}$ or $P_{k+1}$, up to shift.

By applying $\operatorname{Hom}(P_{1},-)$, we obtain the following filtration
in the bounded derived category of graded vector spaces:

(21)
$0=\operatorname{Hom}(P_{1},x_{0})\to\cdots\to\operatorname{Hom}(P_{1},x_{n})=\operatorname{Hom}(P_{1},x).$

We claim that each of the triangles

$\operatorname{Hom}(P_{1},x_{i})\to\operatorname{Hom}(P_{1},x_{i+1})\to\operatorname{Hom}(P_{1},z_{i})\xrightarrow{+1}$

is split; that is, all possible boundary maps between the sub-quotients of
(21) are zero. The filtration of $x$ has three possible non-zero boundary
maps:

$P_{k}\to P_{k}[2],\quad P_{k+1}\to P_{k+1}[2],\quad P_{k}\to P_{k+1}[1].$

It is easy to verify that all three are killed by the functor
$\operatorname{Hom}(P_{1},-)$ when $k\neq 0,1$. In particular, we have

$\operatorname{Hom}(P_{1},x)=\bigoplus\operatorname{Hom}(P_{1},z_{i}).$

We can compute directly using minimal complexes that
$\operatorname{\overline{hom}}(P_{1},P_{k})=2|k-1|$ if $k\neq 1$. So when
$k\neq 0,1$, we have for $z_{i}=P_{k}$ that

$\operatorname{\overline{hom}}(z_{i},P_{1})=\operatorname{\overline{hom}}(P_{1},z_{i})=2|k-1|,$

and for $z_{i}=P_{k+1}$, that

$\operatorname{\overline{hom}}(z_{i},P_{1})=\operatorname{\overline{hom}}(P_{1},z_{i})=2|k|.$

In each case, $z_{i}$ contributes twice the number of occurrences of $P_{0}$
in its minimal complex. By linearity, we conclude that
$\operatorname{\overline{hom}}(x,P_{1})$ is twice the number of occurrences of
$P_{0}$ in its minimal complex.

We now treat the cases where $x$ is supported either at $[P_{0},P_{1}]$ or at
$[P_{1},P_{2}]$. By 6.9, the objects $x$ and $\sigma_{1}^{r}x$ have the same
number of occurrences of $P_{2}$, and both also have the same value for
$\operatorname{\overline{hom}}(-,P_{1})$. Hence the proposition for $x$
implies the proposition for $\sigma_{1}^{r}x$.

If $x$ is supported at $[P_{0},P_{1}]$, then it is easy to check that $x$ has
a recognised writing of the form

$x=\sigma_{0}^{a}\beta y,$

where $y\in\\{P_{0},P_{1}\\}$, and $\beta y$ is supported neither at
$[P_{-1},P_{0}]$ nor at $[P_{0},P_{1}]$. Then it is easy to check that
$\sigma_{1}x$ is supported neither at $[P_{0},P_{1}]$ nor at $[P_{1},P_{2}]$,
hence the proposition follows for $\sigma_{1}x$, and hence for $x$, from the
previous case.

If $x$ is supported at $[P_{1},P_{2}]$, then it is easy to check that $x$ has
a recognised writing of the form

$x=\sigma_{1}^{a}\beta y,$

where $y\in\\{P_{0},P_{1}\\}$, and $\beta y=\sigma_{1}^{-a}x$ is not supported
at $[P_{1},P_{2}]$. Then the proposition follows for $\beta y$, and hence for
$x$, from the previous cases.

∎

###  6.6. Gromov coordinates

######  Definition 6.11 (Gromov coordinates).

Let $\tau$ be a stability condition of type A. For each integer
$i\in\mathbb{Z}$, let $x_{i}(\tau)$ be the rational number defined as follows:

$x_{i}(\tau)=\frac{m_{\tau}(P_{i-1})+m_{\tau}(P_{i+1})-2m_{\tau}(P_{i})}{2}.$

We call the numbers $(x_{i}(\tau))$ the _Gromov coordinates_ of $\tau$.

Using the triangle inequality [13, Proposition 3.3] on the triangles

(22) $\begin{split}P_{2i+1}[-1]\to P_{2i}\oplus P_{2i}\to
P_{2i-1}\xrightarrow{+1},\\\ P_{2i+2}\to P_{2i+1}\oplus P_{2i+1}\to
P_{2i}[1]\xrightarrow{+},\end{split}$

we deduce that all Gromov coordinates of $\tau$ are non-negative. If $\tau$ is
non-degenerate, they are all positive.

######  Proposition 6.12 (Linearity).

Let $\tau$ be a stability condition of type A with Gromov coordinates
$(x_{i})$. For any object $X\in\mathbf{S}$, we have

(23)
$m_{\tau}(X)=\frac{1}{2}\sum_{i\in\mathbb{Z}}x_{i}\operatorname{\overline{hom}}(P_{i},X).$

###### Proof.

Recall that for any $i\in\mathbb{Z}$, we have

$P_{2i}=\gamma^{i}P_{0},\quad P_{2i+1}=\gamma^{i}P_{1}.$

Therefore, we get

$\operatorname{\overline{hom}}(P_{2i},X)=\operatorname{\overline{hom}}(P_{0},\gamma^{-i}X),\quad\operatorname{\overline{hom}}(P_{2i+1},X)=\operatorname{\overline{hom}}(P_{1},\gamma^{-i}X).$

6.10 allows us to compute $\operatorname{\overline{hom}}(P_{0},\gamma^{-i}X)$
and $\operatorname{\overline{hom}}(P_{1},\gamma^{-i}X)$ from the HN
multiplicities of $\gamma^{-i}X$. In turn, the automaton Figure 9 allows us to
compute the HN multiplicities of $\gamma^{-i}X$ from the HN multiplicities of
$X$. Suppose $X$ is supported at the $k$th state of the automaton. Suppose the
HN filtration of $X$ consists of $\alpha$ copies of $P_{k}$ and $\beta$ copies
of $P_{k+1}$, up to shift. Then $\gamma^{-i}X$ is supported on the $(k-2i)$th
state, and its HN filtration contains $\alpha$ copies of $P_{k-2i}$ and
$\beta$ copies of $P_{k+1-2i}$. In particular, for any $i\in\mathbb{Z}$, the
number of instances of $P_{1}$ and $P_{0}$ in the minimal complex of
$\gamma^{-i}X$ are exactly

$\left(\alpha|k-2i|+\beta|k-2i+1|\right)\text{ and
}\left(\alpha|k-2i-1|+\beta|k-2i|\right)$

respectively. Using 6.10, we deduce that for any $j\in\mathbb{Z}$, we have

(24)
$\operatorname{\overline{hom}}(P_{j},X)=2\left(\alpha|k-j|+\beta|k-j+1|\right).$

Substitute

$x_{i}=\frac{m_{\tau}(P_{i-1})+m_{\tau}(P_{i+1})-2m_{\tau}(P_{i})}{2}$

in

$\frac{1}{2}\sum_{i\in\mathbb{Z}}x_{i}\operatorname{\overline{hom}}(P_{i},X).$

Then the coefficient of $m_{\tau}(P_{j})$ is exactly

$\frac{\operatorname{\overline{hom}}(P_{j-1},X)+\operatorname{\overline{hom}}(P_{j+1},X)-2\operatorname{\overline{hom}}(P_{j},X)}{4}.$

Using (24), we see that the coefficient is $\alpha$ if $j=k$, is $\beta$ if
$j=k+1$, and is $0$ otherwise. Therefore we see that

$\frac{1}{2}\sum_{i\in\mathbb{Z}}x_{i}\operatorname{\overline{hom}}(P_{i},X)=\alpha\cdot
m_{\tau}(P_{k})+\beta\cdot m_{\tau}(P_{k+1}),$

which is precisely the $\tau$-mass of $X$. ∎

Next we study the topology of the space $\Lambda$, and compute the closure of
$m(\Lambda)$ in $\mathbb{P}^{\mathbf{S}}$.

######  Proposition 6.13.

A stability condition in $\Lambda$ is specified uniquely by a central charge
$Z$ for which $Z(P_{1})/Z(P_{0})$ lies in the closed upper half plane and
avoids the set

$L=\\{-1\\}\cup\left\\{\frac{1-n}{n}\mid
n\in\mathbb{Z}\setminus\\{0\\}\right\\}.$

###### Proof.

Recall that $\Lambda$ is the closure of non-degenerate standard stability
conditions of type A, namely those for which we have the phase inequalities

$\phi(P_{0})<\phi(P_{1})<\phi(P_{0})+1.$

It follows that for any stability condition in the closure,
$\omega=Z(P_{1})/Z(P_{0})$ must lie in the closed upper half plane. Also
recall that a stability condition is uniquely specified by its heart and its
central charge.

To see why $\omega$ avoids $L$, note that every stability condition in
$\operatorname{Stab}(\mathcal{C})$ contains semi-stable objects of class
$[P_{n}]$ for $n\in\mathbb{Z}$ and $n=\infty$. Therefore, we must have
$Z([P_{n}])\neq 0$ for $n\in\mathbb{Z}\cup\\{\infty\\}$. Since

$[P_{n}]=|n|[P_{1}]+|n-1|[P_{0}]\text{ and }[P_{\infty}]=[P_{0}]+[P_{1}]$

, we see that $\omega$ must avoid $L$.

We now prove that if $\omega=Z(P_{1})/Z(P_{0})$ avoids $L$ then there is a
unique $\tau\in\operatorname{Stab}(\mathcal{C})/\mathbb{C}$ with $Z$ as the
central charge. If $\omega$ lies in the (open) upper half plane, then $\tau$
must be standard and non-degenerate. Since the standard heart is the extension
closure of $P_{0}$ and $P_{1}$, specifying $Z(P_{0})$ and $Z(P_{1})$ in the
upper half plane uniquely determines the stability condition $\tau$, and
specifying their ratio $\omega$ determines $\tau$ uniquely up to the
$\mathbb{C}$ action. Since $\omega$ is in the upper half plane, $\tau$ is of
type A.

In the rest of the proof, uniquely means uniquely up to the $\mathbb{C}$
action.

We now treat the degenerate cases, namely those where $\omega$ is a real
number. The set $\mathbb{R}\setminus L$ is a disjoint union of open intervals
of two kinds:

$\mathbb{R}\setminus L=\bigcup_{n\geq
0}\left(-\frac{n}{n+1},-\frac{n-1}{n}\right)\cup\bigcup_{n\geq
0}\left(-\frac{n+1}{n},-\frac{n+2}{n+1}\right).$

The heart of the stability condition will depend on the interval that contains
$\omega$.