arXiv:2110.06652v3

On the existence of scaling multi-centered black holes

Pierre Descombes and Boris Pioline

Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589 CNRS-
Sorbonne Université, Campus Pierre et Marie Curie, 4 place Jussieu,
F-75005 Paris, France

<EMAIL_ADDRESS>

###### Abstract

For suitable charges of the constituents, the phase space of multi-centered
BPS black holes in $\mathcal{N}=2$ four-dimensional supergravity famously
exhibits scaling regions where the distances between the centers can be made
arbitrarily small, so that the bound state becomes indistinguishable from a
single-centered black hole. In this note we establish necessary conditions on
the Dirac product of charges for the existence of such regions for any number
of centers, generalizing the standard triangular inequalities in the three-
center case. Furthermore, we show the same conditions are necessary for the
existence of multi-centered solutions at the attractor point. We prove that
similar conditions are also necessary for the existence of self-stable Abelian
representations of the corresponding quiver, as suggested by the duality
between the Coulomb and Higgs branches of supersymmetric quantum mechanics.

###### Contents

  1. 1 Introduction
  2. 2 Existence of scaling and attractor solutions
    1. 2.1 Denef’s equations as current conservation
    2. 2.2 Biconnectedness and strong connectedness
    3. 2.3 Cuts, weak cuts and R-charge
    4. 2.4 Constraints on the existence of scaling and attractor solutions
    5. 2.5 Non-Abelian scaling and attractor solutions
  3. 3 Existence of self-stable representations
    1. 3.1 Conserved current on the Higgs branch
    2. 3.2 Stronger constraints in the Abelian case
    3. 3.3 Stronger constraints in the non-Abelian case
  4. A Proofs
    1. A.1 Conserved currents and graph homology
    2. A.2 Biconnected components of a quiver
    3. A.3 Cuts and R-charge
    4. A.4 Current decomposition for Abelianized quivers
    5. A.5 Higgs branch

##  1 Introduction

In string theory vacua with $\mathcal{N}=2$ supersymmetry in 3+1 dimensions,
determining the exact number of single-particle BPS states with total
electromagnetic charge $\gamma$ and arbitrary values of the moduli $z$ is
still a daunting problem. Much progress has been made on computing the BPS
index $\Omega(\gamma,z)$, which counts the same states weighted by a sign
$(-1)^{2J_{3}}$, where $J_{3}$ is the angular momentum. As a result,
$\Omega(\gamma,z)$ becomes independent of the hypermultiplet moduli (in
particular, of the string coupling in type II string theories compactified on
a Calabi-Yau threefold), but it may still jump across certain codimension-one
walls in vector moduli space – a phenomenon known as wall-crossing [1], first
identified in the context of supersymmetric gauge theories [2] and studied
independently in the mathematical literature on Donaldson-Thomas invariants
[3, 4].

At strong coupling, the jumps of $\Omega(\gamma,z)$ reflect the appearance or
disappearance of multi-centered black hole solutions, where the total charge
$\gamma=\sum_{i=1}^{n}\gamma_{i}$ is distributed over $n$ centers with charge
$\gamma_{i}$ [5, 1]. The number of configurational bound states of such
constituents can be determined by quantizing the phase space of such solutions
[6], providing a transparent derivation of the wall-crossing formula [7]. More
generally, the knowledge of the number of configurational bound states allows
to express the total index $\Omega(\gamma,z)$ in terms of more elementary
indices $\Omega_{\rm S}(\gamma_{i})$ which count single-centered black holes
[8], or in the context of quiver quantum mechanics, pure-Higgs states [9, 10,
11]. However, this procedure is complicated by the fact that for certain
configurations of charges $\\{\gamma_{i}\\}$, the space of solutions is non-
compact, due to the existence of ‘scaling regions’ where some of the centers
become arbitrarily close to each other, and hard to distinguish from single-
centered black holes. The modest goal of this note is to establish some
necessary conditions on the charges $\gamma_{i}$ for such a scaling region to
arise.

The distance between the centers of a multi-centered black hole solution are
constrained by Denef’s equations [5]

$\forall i=1\dots n,\qquad\sum_{j\neq i}\frac{\kappa_{ij}}{r_{ij}}=\zeta_{i}$
(1.1)

where $r_{ij}=|\vec{r}_{i}-\vec{r}_{j}|$ and
$\kappa_{ij}=\langle\gamma_{i},\gamma_{j}\rangle$ is the Dirac-Schwinger-
Zwanziger product of electromagnetic charges, and $\zeta_{i}$ are real
parameters determined by the vector multiplet moduli. In order to analyze
solutions to (1.1), it is useful to introduce the quiver $Q$, obtained by
associating one node $i$ to each center $\vec{r}_{i}$, and $\kappa_{ij}$
arrows from node $i$ to node $j$ whenever $\kappa_{ij}>0$. Indeed, the same
equations (1.1) describe supersymmetric solutions on the Coulomb branch of the
supersymmetric quantum mechanics specified by the quiver $Q$ [12]. We assume
that the quiver $Q$ is connected, since otherwise the $n$-body problem reduces
to the same problem for each connected component.

When non-empty, the space of solutions to (1.1) modulo overall translations,
which we denote by $S_{Q}^{\zeta}$, is a real symplectic space of dimension
$2n-2$ [6]. For generic non-zero values of $\zeta_{i}$ away from walls of
marginal stability, it is easy to see that the maximal distance $r_{+}={\rm
sup}\\{r_{ij},i<j\\}$ is bounded from above, as appropriate for a classical
bound state. In contrast, the minimal distance $r_{-}={\rm
inf}\\{r_{ij},i<j\\}$ may vanish, due to some regions in $S_{Q}^{\zeta}$ where
a subset of the centers become arbitrarily close to each other [12, 13, 1].
The existence of such ‘scaling regions’ requires certain conditions on the
signs of the Dirac products $\kappa_{ij}$, as well as on their magnitude. For
example, if the $i$-th center satisfies $\kappa_{ij}\geq 0$ for all $j\neq i$,
or $\kappa_{ij}\leq 0$ for all $j\neq i$, then it is clear from (1.1) that the
distances $r_{ij}$ cannot be arbitrarily small. Thus, in order for all centers
to coalesce at one point, the quiver $Q$ must have no source nor sink. In
particular, scaling solutions never exist for $n=2$ centers. For $n=3$
centers, it is well-known [13, 1] that scaling solutions exist whenever
$\kappa_{12},\kappa_{23},\kappa_{31}$ have the same sign (say positive) and
moreover satisfy the triangular inequalities,

$\kappa_{12}\leq\kappa_{23}+\kappa_{31},\quad\kappa_{23}\leq\kappa_{31}+\kappa_{12},\quad\kappa_{31}\leq\kappa_{12}+\kappa_{23}$
(1.2)

In that case, the equations (1.1) are solved by setting
$r_{ij}=\lambda\kappa_{ij}+\dots$ with $\lambda\to 0$, with the dots
corresponding to $\zeta$-dependent corrections which become irrelevant as
$\lambda\to 0$.

In general, the existence of scaling regions where all centers coalesce is
independent of the parameters $\zeta_{i}$, and can be analyzed by setting
$\zeta_{i}=0$ in (1.1), obtaining the ‘conformal Denef equations’

$\forall i=1\dots n,\qquad\sum_{j\neq i}\frac{\kappa_{ij}}{r_{ij}}=0$ (1.3)

We refer to solutions of (1.3) as ‘scaling solutions’. The case where only a
subset of the $n$ centers coalesce can be analyzed by restricting (1.3) to
those centers. We shall also be interested in solutions of (1.1) at the
special point $\zeta_{i}=-\sum_{j\neq i}\kappa_{ij}$, which, as argued in [14,
15], corresponds to the attractor point $z_{\gamma}$ for a black hole with
total charge $\gamma=\sum_{i}\gamma_{i}$ [16]. At that point, one obtains the
‘attractor Denef equations’

$\forall i=1\dots n,\qquad\qquad\sum_{j\neq
i}\kappa_{ij}\left(1+\frac{1}{r_{ij}}\right)=0$ (1.4)

We denote by $S_{Q}^{0}$ and $S_{Q}^{\star}$ the set of solutions to (1.3) and
(1.4) modulo overall translations.

We shall now state a set of necessary conditions on $\kappa_{ij}$ for the
existence of solutions to (1.3). These conditions follow from geometric
constraints on the centers, similarly to the three-center case, and are
slightly more general than the conditions that were conjectured (and proven in
some special cases) in [17, §2.3]. Quite remarkably, we shall see that the
same conditions follow from the existence of solutions to (1.4) at the
attractor point. This is in agreement with the expectation that the only
multi-centered solutions allowed at the attractor point are scaling solutions
[15]. We anticipate that these conditions will prove useful in further studies
of black hole micro-states in Calabi-Yau compactifications (see [15, 18, 19,
20] for recent progress in this direction).

The first condition constrains the signs of the Dirac product $\kappa_{ij}$,
and is easily stated in terms of the quiver $Q$: the quiver should be strongly
connected, which means that for any pair of vertices $i,j$, there should be a
path going from $i$ to $j$ and and a path going from $j$ to $i$. In
particular, it implies that the $n$ centers cannot be separated into two
disjoint groups $S$ and $\bar{S}$ such that all arrows go from $S$ to
$\bar{S}$ [14] (in particular $Q$ should have no source nor sink).

The second condition constrains the magnitude of the Dirac products
$\kappa_{ij}$, but requires some additional terminology and notation. Given a
quiver $Q$, let $Q_{0}$ be the set of vertices, $Q_{1}$ the set of arrows and
$Q_{2}$ the set of simple oriented cycles. A cut (respectively, a weak cut) is
a subset $I\subset Q_{1}$ of the set of arrows such that each simple oriented
cycle $w\in Q_{2}$ contains exactly (respectively, at most) one arrow. The
second condition is that, for any weak cut $I$, the ’generalized triangular
inequalities’

$|I|\leq|Q_{1}-I|$ (1.5)

are satisfied, where $|I|$ denotes the number of arrows $i\to j$ in the weak
cut $I$ (counted with multiplicity $\kappa_{ij}$), and similarly for the
complement $Q_{1}-I$. In the case where one of these inequalities is
saturated, there are no attractor solutions, and if furthermore $Q$ is
biconnected (also known as non-separable, see §2.2 for the relevant
definitions), the only scaling solutions are collinear.

For example, in the special case of a cyclic quiver $v_{1}\to v_{2}\to\dots\to
v_{n}\to v_{n+1}=v_{1}$, each set of arrows $i\to i+1$ provides one possible
cut, and the condition (1.5) reduces to the well-known conditions

$\kappa_{n1}\leq\kappa_{12}+\kappa_{23}+\dots+\kappa_{n-1,n}$ (1.6)

and cyclic permutations thereof [11]. More generally, if the quiver admits a
cut as well as a cycle passing through all the nodes (which can be taken to be
$v_{1}\to v_{2}\to\dots\to v_{n}\to v_{1}$ at the expense of relabelling the
nodes), then the condition (1.5) reduces to the conditions
$\sum_{i<j}\kappa_{ij}\geq 0$ (and cyclic permutations thereof) conjectured in
[17, §2.3]. We also study a natural generalization of these inequalities in
the non-Abelian case, but we show that they cannot hold in full generality.

We further observe that the conditions stated above have a simple
interpretation in terms of the Higgs branch of the supersymmetric quantum
mechanics associated to the quiver $Q$ with generic potential $W$. In that
description, BPS states correspond to cohomology classes of the moduli space
of stable representations $Q$ satisfying the potential equations
$\partial_{a}W=0$ for each arrow $a\in Q_{1}$, where the stability condition
is determined by the parameters $\zeta_{i}$ [12]. In the attractor (or self-
stability) chamber $\zeta^{i}=-\sum_{j\neq i}\kappa_{ij}$, it is immediate to
show that stable representations cannot exist unless $Q$ is strongly
connected. Moreover, for a quiver $(Q,W)$ with a cut and a generic potential,
every cycle vanishes in any Abelian representation, hence each cycle contains
a vanishing arrow. Let $I$ be the set of arrows such that the corresponding
chiral fields vanish, and suppose $I$ is a cut 111In the appendix, we adapt
the proof of the final inequalities when this condition is not satisfied, i.e.
when there are more vanishing arrows than necessary: in this case, there are
less than $|I|$ independent potential constraints. The dimension of the moduli
space of stable Abelian representations is then given by

$\dim_{\mathbb{C}}\mathcal{M}^{\zeta,s}_{Q,W}=|Q_{1}-I|-|I|-(|Q_{0}|-1)$ (1.7)

where the first term corresponds to the non-vanishing chiral fields, the
second to the potential constraints induced by the vanishing chiral fields
(which can be proven to be independent), and the last term to the quotient by
the complexified gauge group $(\mathbb{C}^{\ast})^{Q_{0}}$ modulo the diagonal
$\mathbb{C}^{\ast}$ subgroup (which acts trivially). Requiring that the moduli
space is not empty for some cut $I$, hence has positive dimension, we arrive
at the inequality

$|I|\leq|Q_{1}-I|-|Q_{0}|+1$ (1.8)

In §3 we show that the only cut which is compatible with the stability
parameters $\zeta^{i}=-\sum_{j\neq i}\kappa_{ij}$ (corresponding to the
attractor or self-stability condition) is the one which minimizes the expected
dimension (1.7).222This minimization property was first observed in examples
of quivers associated to non-compact Calabi-Yau threefolds in [21]. In that
case, the condition (1.8) holds for any cut, and produces a stronger333The
fact that the inequalities for the existence of self-stable representations
are stronger than the one for the existence of scaling solutions is consistent
with the fact that the contribution of scaling solutions may cancel against
the contribution of regular collinear solutions when computing the equivariant
Dirac index of the phase space of multi-centered configurations using
localisation [22].  version of the condition (1.5) on the Coulomb branch.
Moreover, we prove Proposition A.10 which says that the same inequality
continues to holds when $I$ is a weak cut, as on the Coulomb branch side. We
also examine a natural non-Abelian generalization of these inequalities, but
conclude that it cannot hold in full generality.

The remainder of this note is organized as follows. In Section 2, we derive
general constraints for the existence of scaling or attractor solutions on the
Coulomb branch of Abelian quivers, and conjecture similar constraints in the
non-Abelian case. In Section 3 we discuss similar conditions for the existence
of self-stable representations, which arise on the Higgs branch of the quiver
quantum mechanics at the attractor point. Mathematical proofs of some
technical results are relegated to the Appendix.

##  2 Existence of scaling and attractor solutions

In this section, we derive general conditions for the existence of scaling or
attractor solutions. The main idea is to reinterpret Denef’s equations as
current conservation, decompose each current into a sum of positive currents
running around the simple cycles, and enforce generalized triangular
inequalities on each cycle. In order to implement this idea, we need the
notions of biconnectedness, strong connectedness, cuts and R-charge, which we
introduce along the way.

###  2.1 Denef’s equations as current conservation

Consider a quiver $Q=(Q_{0},Q_{1})$. Here $Q_{0}$ denotes the set of nodes of
the quiver, and $Q_{1}$ denotes the set of arrows $a:i\to j$ of sources
$s(a)=i\in Q_{0}$ and target $t(a)=j\in Q_{0}$. Let $Q_{2}$ be the set of
simple oriented cycles of $Q$, i.e. oriented cycles passing at most once
through each node of $Q$. Consider the sequence:

$\displaystyle\mathbb{R}^{Q_{2}}\overset{\partial_{2}}{\to}\mathbb{R}^{Q_{1}}\overset{\partial_{1}}{\to}\mathbb{R}^{Q_{0}}\overset{\partial_{0}}{\to}\mathbb{R}$
(2.1)

with $\partial_{2}$, $\partial_{1}$ and $\partial_{0}$ defined by

$\partial_{2}(w)=\sum_{a\in w}a\ ,\qquad\partial_{1}(a)=t(a)-s(a)\
,\qquad\partial_{0}(i)=1$ (2.2)

It is immediate to check that $\partial_{1}\circ\partial_{2}=0$ and
$\partial_{0}\circ\partial_{1}=0$, so (2.1) is a complex444This complex is
related to the cellular homology complex
$0{\to}\mathbb{R}^{Q_{1}}\overset{\partial_{1}}{\to}\mathbb{R}^{Q_{0}}\to 0$
of the graph considered as a cellular space. The cellular homology group
$H^{0}$ gives the number of connected components of the graph, see [23, Sec
2.2].. We refer to elements of $\mathbb{R}^{Q_{1}}$ as ‘currents’, and
elements of $\ker(\partial_{1})$ as ‘conserved currents’. The complex (2.1) is
exact at $\mathbb{R}^{Q_{0}}$ if and only if $Q$ is connected: we shall assume
that $Q$ is connected unless otherwise specified.

The Denef equations (1.1) for general stability parameters $\zeta_{i}$ can be
recast as current conservation by rewriting them as follows:

$\displaystyle\forall\;i\in Q_{0}\quad\sum_{(a:i\to j)\in
Q_{1}}\frac{1}{r_{ij}}-\sum_{(a:j\to i)\in Q_{1}}\frac{1}{r_{ij}}=\zeta_{i}$
$\displaystyle\iff\partial_{1}\left((\frac{1}{r_{ij}})_{(a:i\to j)\in
Q_{1}}\right)=(\zeta_{i})_{i\in Q_{0}}$ (2.3)

Assuming (without loss of generality) that the quiver $Q$ is connected, and
summing over $i\in Q_{0}$, it is clear that solutions only exist if
$\partial_{0}((\zeta_{i})_{i\in Q_{0}})=\sum_{i\in Q_{0}}\zeta_{i}=0$. We can
therefore choose $(\chi_{a})_{a\in Q_{1}}$ such that
$\partial_{1}((\chi_{a})_{a\in Q_{1}})=(-\zeta_{i})_{i\in Q_{0}}$. We define
the current $\lambda^{\chi}_{a}$ running along the arrow $a:i\to j$ as

$\displaystyle\lambda^{\chi}_{a}:=\chi_{a}+\frac{1}{r_{ij}}$ (2.4)

The equations (2.1) then amount to conservation
$\partial_{1}(\lambda^{\chi})=0$ of the current $\lambda^{\chi}$. For given
$\zeta_{i}$, the $\chi_{a}$ are in general not unique, but for the conformal
(resp. attractor) Denef equations there is a canonical choice, namely

$\displaystyle\lambda^{0}_{a}:=\frac{1}{r_{ij}}\
,\qquad\lambda^{\star}_{a}:=1+\frac{1}{r_{ij}}$ (2.5)

such that the conserved current $\lambda^{0}$ and $\lambda^{\star}$ are
strictly positive. This positivity will be crucial for deriving constraints on
the existence of scaling and attractor solutions. This property is preserved
for a small perturbation of the attractor stability parameters, obtained by
replacing $\lambda^{\star}_{a}$ by $1+\delta_{a}+\frac{1}{r_{ij}}$ with
$|\delta_{a}|\ll 1$.

###  2.2 Biconnectedness and strong connectedness

In order to study the existence of conformal or attractor solutions, it will
be convenient to decompose the connected quiver $Q$ in two steps: first into
biconnected components, and then into strongly connected components.555The
notions of biconnected graph (also known as ’non-separable’), biconnected
components of a graph (also known as ‘blocks’), strongly connected directed
graph and strongly connected components are standard in graph theory, see for
example [24, Sec 3,6].

For the first step, a quiver is said to be biconnected if it cannot be
disconnected by removing one node. In general, a quiver can be decomposed into
maximal biconnected subquivers called biconnected components joined by a
shared node (see Figure 1). Consider the unoriented graph $K$ with one node
for each biconnected component and one node for each node of the quiver shared
between different biconnected components, and an edge between the node $i$ and
the biconnected component $b$ if $i\in b$. In lemma A.2 we show that $K$ is a
connected tree, i.e. has no cycle. Given a solution of Denef’s equations at
any stability for each biconnected component, one can join these solutions at
shared nodes by identifying the corresponding centers: the fact that $K$ has
no cycle ensures that this can be done consistently. According to lemma A.3, a
conserved current running on a connected quiver is also conserved on each of
its biconnected components, and the converse is also true: a solution of
Denef’s equations on a connected quiver is obtained by gluing together
solutions for each of the biconnected components. The gluing at a common point
in $\mathbb{R}^{3}$ freezes the relative translational degrees of freedom.
Denoting by $B$ the set of biconnected components, one has:

$\displaystyle S^{\zeta}_{Q}=\prod_{b\in B}S^{\zeta}_{Q^{b}}$ (2.6)

One can also decompose $Q$ into strongly connected components, defined as
follows. A quiver is strongly connected if for each pair of nodes $i,j\in
Q_{0}$ there is an oriented path from $i$ to $j$ and from $j$ to $i$. We
define an equivalence relation between nodes by writing $i\sim j$ if there is
a path from $i$ to $j$ and from $j$ to $i$: this relation is automatically
symmetric, reflexive by considering the trivial path, and transitive by
concatenation of paths. The equivalence classes under this relation are called
strongly connected components. Let us now draw an arrow from a strongly
connected component to another if there is an arrow in $Q$ from one node in
the first equivalence class to a node of the second equivalence class. The
resulting graph $\Gamma$ is connected and has no cycle because there cannot
exists paths going both ways between two strongly connected components. Thus
$\Gamma$ defines a poset. In Figure 2, we show an example of a quiver with its
graph $\Gamma$ of strongly connected components.

From lemma A.1, a strictly positive conserved current can exist on a quiver
$Q$ only if $Q$ is strongly connected, therefore:

######  Proposition 2.1.

A quiver which admits scaling or attractor solutions must be strongly
connected.

Figure 1: Constructing the unoriented graph of biconnected components.
$\Rightarrow$ Figure 2: Constructing the graph of strongly connected
components.

Notice that by $i)$ of lemma A.4, a quiver is strongly connected if and only
if its biconnected components are strongly connected. The biconnected
components of a strongly connected quiver $Q$ have an alternative description
as follows. Consider the equivalence relation $\sim$ on the arrows of $Q$
generated by $a\sim b$ if they are in a simple oriented cycle $w\in Q_{2}$.
From $ii)$ of lemma A.4, the equivalence classes of $\sim$ are identical to
the biconnected components of $Q$, therefore $Q$ is biconnected if and only if
$\sim$ has a single equivalence class.

###  2.3 Cuts, weak cuts and R-charge

Consider a map $R:\mathbb{R}^{Q}_{1}\to\mathbb{R}$ assigning a charge $R(a)$
to each arrow $a\in Q_{1}$. For any simple cycle $w\in Q_{2}$, one consider
the charge of the cycle $R\circ\partial_{2}(w)=\sum_{a\in w}R(a)$. We say that
$R$ is an R-charge if all simple cycles have charge $R\circ\partial_{2}(w)=2$.
For any subset $I\subset Q_{1}$ of arrows of $Q$, we define the charge $R_{I}$
by $R_{I}(a)=2$ if $a\in I$, and $R_{I}(a)=0$ if $a\in Q_{1}-I$.

We define a cut (resp. a weak cut, a strong cut) as a subset $I$ of arrows of
the quiver such that each simple oriented cycle $w\in Q_{2}$ contains exactly
(resp. at most, at least) one arrow of $I$, or equivalently
$R_{I}\circ\partial_{2}(w)=2$ (resp. $R_{I}\circ\partial_{2}(w)\leq 2$,
$R_{I}\circ\partial_{2}(w)\geq 2$). We shall consider maximal weak cuts, i.e.
weak cuts which are maximal for the inclusion. There are two possibilities, as
proven in lemma A.5:

  * •

either $Q$ admits a cut, and then all maximal weak cuts are cuts, and $Q$
admits an $R$-charge,

  * •

or $Q$ admits no $R$-charge and in particular no cut.

While quivers with few arrows typically have cuts (see Fig. 3), it is easy to
find examples of quivers which do not have any cuts. Our main examples are the
’pentagram’ and ’hexagram’ quivers, shown in Figure 4 and 5. In both cases,
the relations between the cycles forbids the existence of an R-charge, and
therefore of a cut (see Figures 4 and 5). Each of these quivers nonetheless
admits maximal weak cuts, as shown in dashed red in the respective figure.

######  Remark 2.2.

The notion of R-charge is standard in the physics literature on quiver gauge
theories, but the details may vary. Here we do not impose any condition on the
total R-charge of the arrows ending or starting at a given node. The notion of
cut is also standard in the mathematical and physic literature about quivers
with potential. The notions of weak and strong cut introduced here appear to
be new. R-charges and cuts are usually defined with respect to all cycles
appearing in the potential, not only the simple cycles. Here we always assume
that the potential is a generic sum of simple cycles, so this distinction is
unessential.

Figure 3: Two examples of quivers with cuts. Arrows in the cut are drawn in
dashed red. \+ 4 circular permutations=3+ Figure 4: The pentagram quiver does
not admit any cut, due to a relation between simple cycles. The maximal weak
cuts are shown in red, up to cyclic permutation of the nodes.  +++= Figure 5:
The hexagram quiver does not admit any cut, due to a relation between simple
cycles. The maximal weak cuts are shown in red, up to cyclic permutation of
the nodes.

The cuts of a quiver are particularly easy to describe when $Q$ contains a
cycle $w_{0}$ passing through all the nodes, and admits an R-charge. In
particular $Q$ is strongly connected in this case, and also biconnected: if
one removes a node $i$ of the quiver, two other nodes are still connected by
the path on the cycle $w_{0}$ avoiding $i$. One can then label the nodes using
$w_{0}$, such that this cycle passes by the nodes $1\to 2\to...\to n\to 1$.
According to lemma A.6, the cuts are then given by $I=\\{a:i\to j|i>j\\}$ and
circular permutations thereof (i.e. when we choose the order
$r<r+1<...<n<1<...<r-1$ on $\\{1,...,n\\}$). Said differently, the cuts
consist of the arrows ’going the wrong way’ according to one of the $n$
choices of circular ordering on the nodes. An example is shown in Figure 6.

1432143214321432 Figure 6: Structure of the cuts for a quiver with a cycle
$w_{0}$ passing through all the nodes. For each choice of origin along the
cycles, (marked by a black node), arrows in the cut (which are dashed) are
those which go backward along the cycle.

###  2.4 Constraints on the existence of scaling and attractor solutions

Having introduced the necessary notions, we are now ready to derive necessary
conditions for the existence of scaling and attractor solutions. The key point
is the positivity of the currents $\lambda^{\bullet}$ in (2.5), whose
conservation corresponds with the conformal ($\bullet=0$) or attractor
($\bullet=\star$) Denef equation. We can then apply part $ii)$ of lemma A.1,
to conclude that $\lambda^{\bullet}$ is a sum of positive currents circulating
on simple cycles of the quiver, namely there exists $(\mu_{w})_{w\in Q_{2}}$
such that

$\lambda^{\bullet}_{a}=\sum_{w\ni a}\mu_{w}\ ,\qquad\mu_{w}>0\quad\forall\
w\in Q_{2}$ (2.7)

Now, for each simple cycle $w$, and every arrow $(a:i\to j)\in w$, we have the
(generalized) triangular inequality

$\displaystyle r_{ij}\leq\sum_{(b:k\to l)\in w|b\neq a}r_{kl}$ (2.8)

Since the function $f:x\mapsto(1+\frac{1}{x})^{-1}$ is increasing on
$\mathbb{R}^{\star}_{+}$, we deduce

$\displaystyle\left(1+\frac{1}{r_{ij}}\right)^{-1}$
$\displaystyle\leq\left(1+\frac{1}{\sum_{(b:k\to l)\in w|b\neq
a}r_{kl}}\right)^{-1}=\frac{\sum_{(b:k\to l)\in w|b\neq
a}r_{kl}}{1+\sum_{(b:k\to l)\in w|b\neq a}r_{kl}}$
$\displaystyle<\sum_{(b:k\to l)\in w|b\neq
a}\frac{r_{kl}}{1+r_{kl}}=\sum_{(b:k\to l)\in w|b\neq
a}\left(1+\frac{1}{r_{kl}}\right)^{-1}$ (2.9)

Thus, we have

$\displaystyle\frac{1}{\lambda^{\bullet}_{a}}$ $\displaystyle\leq\sum_{b\in
w|b\neq a}\frac{1}{\lambda^{\bullet}_{b}}$ (2.10)

both for $\bullet=0$ and $\bullet=\star$. The data of a (weak) cut $I$ allows
one to choose consistently an arrow in each simple cycle (resp. in a set of
cycles). One denotes for convenience:

$\displaystyle\epsilon_{a}^{I}=\left\\{\begin{array}[]{ll}-1&\mbox{if}\;a\in
I\\\ 1&\mbox{if}\;a\in Q_{1}-I\end{array}\right.$ (2.13)

It then follows from (2.10) that, for any weak cut,

$\displaystyle\sum_{a\in w}\frac{\epsilon^{I}_{a}}{\lambda^{\bullet}_{a}}\geq
0$ (2.14)

the inequality being strict when $w$ contains no arrow of $I$. One can then
take a linear combination of the inequalities (2.14) with the same strictly
positive coefficients $\mu_{w}$ appearing in (2.7), obtaining

$\displaystyle 0\leq$ $\displaystyle\sum_{w\in Q_{2}}\mu_{w}\sum_{a\in
w}\frac{\epsilon_{a}^{I}}{\lambda^{\bullet}_{a}}=\sum_{a\in
Q_{1}}\frac{\epsilon_{a}^{I}}{\sum_{w\ni a}\mu_{w}}\left(\sum_{w\ni
a}\mu_{w}\right)=\sum_{a\in Q_{1}}\epsilon_{a}^{I}=|Q_{1}-I|-|I|$ (2.15)

Hence, the existence of scaling or attractor solutions implies, for every weak
cut, the inequality

$\displaystyle|I|\leq|Q_{1}-I|$ (2.16)

A few remarks are in order:

  * •

The inequality (2.16) can only be saturated for scaling solutions
($\bullet=0$), and if each simple cycle contains an arrow of $I$, i.e. $I$ is
a cut. In that case, (2.8) is an equality for each $w\in Q_{2},a\in I\cap w$.
In particular, the centers along each oriented cycle of the scaling solution
must be collinear. One has in particular that (2.8) is a strict inequality for
$a\in w-(I\cap w)$, and therefore $|J|<|Q_{1}-J|$ for all cuts $J$ distinct
from $I$. Moreover, two arrows contained in the same cycle $w$ are necessarily
collinear. If $Q$ is biconnected and strongly connected, from lemma A.4 one
can connect any two arrows by a sequence of arrows such that two consecutive
arrows are contained in a common cycle, and in that case the whole scaling
solution is collinear.

  * •

For a small perturbation of the attractor stability condition, such that the
current is perturbed to $\lambda_{a}=1+\delta_{a}+1/r_{ij}$ with
$|\delta_{a}|\ll 1$, the inequality (2.4) remains true up to small corrections
of order $\delta_{a}(1+1/r_{kl})^{-2}$. After multiplying by
$\mu^{\star}_{w}$, which is of order $1+1/r_{kl}$, the resulting correction to
(2.16) remains small even for $r_{kl}\to 0$. Since the leading terms are
integer, this correction does not affect the inequality (2.16), although it
affects the analysis of the cases where the inequality is saturated.

We thus obtain the main result of this note:

######  Proposition 2.3.

If a quiver $Q$ admits attractor or scaling solutions, then for each weak cut
$I$ one has:

$\displaystyle|I|\leq|Q_{1}-I|$ (2.17)

If scaling solutions exist, the inequality can be saturated for at most one
cut. When this is the case and if $Q$ is biconnected, then the scaling
solutions are collinear.

We shall now compare our result to the conjecture put forward (and proven for
some simple cases) in [17]. In that reference, it was assumed that $Q$ admits
an R-charge and contains a cycle $w_{0}\in Q_{2}$ passing through all the
nodes, giving a cyclic ordering $Q_{0}\simeq\mathbb{Z}/n\mathbb{Z}$. As show
in lemma A.6, all the cuts of $Q$ are then of the form $I=\\{a:i\to j|j>i\\}$
and cyclic permutations. The conjecture in [17] is therefore a consequence of
Proposition 2.3,

######  Proposition 2.4.

For a quiver with an R-charge and a cycle passing through all the nodes,
scaling or attractor solutions can only exist if

$\displaystyle\sum_{i<j}\kappa_{ij}\geq 0\qquad\mbox{and cyclic permutations}$
(2.18)

###  2.5 Non-Abelian scaling and attractor solutions

We now consider a non-Abelian quiver $Q$ with dimension vector $(d_{i})_{i\in
Q_{0}}$, supposing $d_{i}\geq 1$ for $i\in Q_{0}$. On the Coulomb branch, the
gauge group $\prod_{i\in Q_{0}}U(d_{i})$ is broken to a Cartan subgroup
$\prod_{i\in Q_{0}}U(1)^{d_{i}}$, and the scalar fields in the vector
multiplets associated to this subgroup satisfy similar equations (1.1) as in
the Abelian case with $n=\sum_{i\in Q_{0}}d_{i}$ centers, with $d_{i}$ of them
carrying the same charge $\gamma_{i}$ and stability parameter $\zeta_{i}$ for
each $i\in Q_{0}$. Labelling by $(i,k)$ the $k$-th center with charge
$\gamma_{i}$, for $1\leq k\leq d_{i}$, the distances between centers satisfy a
non-Abelian version of Denef’s equations:

$\displaystyle\forall i=1\dots n,\quad\forall k=1\dots d_{i},\quad\sum_{j\neq
i}\sum_{k^{\prime}=1}^{d_{j}}\frac{\kappa_{ij}}{|\vec{r}_{(i,k)}-\vec{r}_{(j,k^{\prime})}|}=\zeta_{i}$
(2.19)

As before, solutions may only exist if $\sum_{i\in Q_{0}}\zeta_{i}d_{i}=0$.
The same equations arise from the ‘totally Abelianized quiver’ $Q^{d}$ with
nodes $Q^{d}_{0}:=\\{(i,k)|i\in Q_{0},1\leq k\leq d_{i}\\}$, and arrows
$Q^{d}_{1}:=\\{(a,k,k^{\prime}):(i,k)\to(j,k^{\prime}))|(a:i\to j)\in
Q_{1},1\leq k\leq d_{i},1\leq k^{\prime}\leq d_{j}\\}$. On special loci where
$m$ of the centers attached to the same node $i\in Q_{0}$ coincide, the gauge
factor $U(1)^{d_{i}}$ is enhanced to $U(m)\times U(1)^{d_{i}-m}$, leading to
partially Abelianized quivers with one node carrying charge $m\gamma_{i}$ and
$d_{i}-m$ nodes carrying charge $\gamma_{i}$ [7]. We can focus on the Denef
equations associated to the totally Abelianized quiver $Q^{d}$, since
solutions relevant for partially Abelianized quivers arise as special cases.

It is clear from this construction that the Abelianized quiver $Q^{d}$ is
connected (resp. strongly connected) if and only if $Q$ is. In particular, if
$Q$ admits a non-Abelian $d$-dimensional scaling or attractor solution, then
$Q^{d}$ is strongly connected, and therefore also $Q$:

######  Proposition 2.5.

A non-Abelian quiver which admits scaling or attractor solutions must be
strongly connected.

If $Q$ is biconnected, then $Q^{d}$ is biconnected too, but the converse is
not true: if $i$ is a node shared between two biconnected components of $Q$,
those two components merge into a single biconnected component in $Q^{d}$ when
$d_{i}\geq 2$ (see the example of the butterfly quiver with dimension vector
$(2,1,1,1,1)$, Figure 8). In particular, the decomposition into biconnected
components of the solutions to Denef’s equations does not hold in the non-
Abelian case, because biconnected components of $Q$ are not in correspondence
with those of $Q^{d}$.

We shall now attempt to derive constraints on the magnitude of the Dirac
products $\kappa_{ij}$ for the existence of non-Abelian scaling or attractor
solutions. Let $\lambda^{\bullet}$ be the corresponding strictly positive
conserved current on $Q^{d}$. For $a$ an arrow on $Q^{d}$, we denote by $p(a)$
its projection to $Q$; similarly, for a path $v$ on $Q^{d}$, we denote by
$p(v)$ its projection to $Q$. For $I$ a weak cut of $Q$, each cycle of
$p^{-1}(Q_{2})$ (i.e. projecting on a simple cycle of $Q$) contains at most
one arrow of $p^{-1}(I)$. By the same argument as in the proof of proposition
2.3, the triangular inequality in the corresponding polygon in the non-Abelian
scaling or attractor solutions gives:

$\displaystyle\sum_{a\in
w}\frac{\epsilon^{p^{-1}(I)}_{a}}{\lambda^{\bullet}_{a}}\geq 0$ (2.20)

Now, suppose that the current could be expressed as a sum
$\lambda^{\bullet}_{a}=\sum_{w\in p^{-1}(Q_{2})|w\in a}\mu_{w}$ of positive
currents $\mu_{w}$ circulating on simple cycles of $Q^{d}$ which project to
simple cycles of $Q$. By taking the linear combination of the inequalities
(2.20) with coefficients $\mu_{w}$, one would conclude that
$|p^{-1}(I)|\leq|Q^{d}_{1}-p^{-1}(I)|$, or equivalently

$\displaystyle\sum_{(a:i\to j)\in I}d_{i}d_{j}\leq\sum_{(a:i\to j)\in
Q_{1}-I}d_{i}d_{j}$ (2.21)

In the special case where $p^{-1}(Q_{2})=Q^{d}_{2}$, i.e. when each cycle of
$Q^{d}$ projects to a simple cycle of $Q$, the existence of the positive
currents $\mu_{w}$ follows directly from $ii)$ of lemma A.1, and the therefore
the condition (2.21) must hold. This the case for example for the triangular
quiver where one of the entries in the dimension vector is equal to one, as in
the cases considered in [11, §6][25, §3] [26, §4] [27]. More generally, when
the quiver $Q$ is biconnected, lemma A.7 allows to express $\lambda^{\bullet}$
as a sum of (possibly negative) currents circulating on the cycles of
$p^{-1}(Q_{2})$. While we are not able to show that these currents can be
taken to be positive, it seems plausible to propose that for a biconnected
quiver, the existence of a $d$-dimensional scaling or attractor solution
implies that for each weak cut $I$, the condition (2.21) is satisfied.

pp12345 Figure 7: An example of non biconnected quiver: the butterfly quiver.
y$\displaystyle x$$\displaystyle p$$\displaystyle p$1$\displaystyle
1^{\prime}$2345 Figure 8: Scaling solution for the butterfly quiver with
$d=(2,1,1,1,1)$.

We can construct a counterexample to the inequalities (2.21) for a non
biconnected quiver, namely the ’butterfly’ quiver shown in Figure 7. We choose
multiplicities $\kappa_{23}=\kappa_{45}=p\geq 1$, and multiplicity $1$ for
other arrows, and dimension vector $(2,1,1,1,1)$. The corresponding
Abelianized quiver $Q^{d}$ is obtained by splitting the central node $1$ into
two nodes which we denote by $1$ and $1^{\prime}$. In particular for the
simple cycle $w:1\to 2\to 3\to 1^{\prime}\to 4\to 5\to 1$, its projection on
$Q$ $p(w):=1\to 2\to 3\to 1\to 4\to 5\to 1$ is not simple, and $w$ cannot be
expressed as a linear combination of cycles projection to a simple cycle of
$Q$. We consider solutions inscribed in a rectangle with height $y$ and length
$2x$ as shown in Figure 8. The scaling (resp. attractor) Denef equations are
trivially verified at the node $1$ and $1^{\prime}$, and require

$\frac{p}{y}=\frac{1}{x}+\frac{1}{\sqrt{x^{2}+y^{2}}}\
,\qquad\mbox{resp.}\qquad
p\left(1+\frac{1}{y}\right)=2+\frac{1}{x}+\frac{1}{\sqrt{x^{2}+y^{2}}}$ (2.22)

at the other nodes. Solutions manifestly exist for any $p\geq 1$. In contrast,
the inequalities (2.21) for the cut $I=\\{2\to 3,4\to 5\\}$ would require

$\displaystyle\sum_{(a:i\to j)\in I}d_{i}d_{j}=2p\leq\sum_{(a:i\to j)\in
Q_{1}-I}d_{i}d_{j}=8$ (2.23)

which is false. This shows that the assumption of biconnectedness is important
for the inequalities (2.21) to hold. Whether the inequalities are verified in
the biconnected case is left as an interesting problem for future work.

##  3 Existence of self-stable representations

In this section, we discuss constraints for the existence of vacuum solutions
on the Higgs branch at the attractor point, which mathematically correspond to
self-stable representations. Physically, we expect that such solutions only
exist when the Coulomb branch admits scaling solutions (except for simple
representations associated to the nodes of the quiver, which exist for
arbitrary stability condition). Indeed, we shall demonstrate that for quivers
with generic potential, the existence of self-stable Abelian representations
requires similar but slightly stronger conditions as for existence of scaling
solutions on the Coulomb branch.

###  3.1 Conserved current on the Higgs branch

We consider the quiver quantum mechanics associated to a connected quiver $Q$,
dimension vector $(d_{i})\in(\mathbb{N}^{\star})^{Q_{0}}$, Fayet-Iliopoulos
parameters $(\zeta_{i})\in\mathbb{R}^{Q_{0}}$ and a generic potential $W$. On
the Higgs branch, the expectation value of the chiral multiplets associated to
the arrows $a\in Q_{1}$ breaks the gauge group $\prod_{i\in Q_{0}}U(d_{i})$ to
the center $U(1)^{Q_{0}}$. After integrating out the vector multiplets one is
left with an effective quantum mechanics for the chiral multiplet scalars
$(\phi_{a})_{a\in Q_{1}}$. Supersymmetric vacua are given by solutions of the
D-term and F-term equations,

$\displaystyle\sum_{(a:i\to j)\in
Q_{1}}\phi_{a}^{\dagger}\phi_{a}-\sum_{(a:j\to i)\in
Q_{1}}\phi_{a}\phi_{a}^{\dagger}=\zeta_{i}\,{\rm Id}_{d_{i}}\quad\forall\;i\in
Q_{0}$ (3.1) $\displaystyle\partial_{\phi_{a}}W=0$ (3.2)

modulo the action of the gauge group. From [28, Prop 6.5], such solutions are
in one-to-one correspondence with polystable representations of the quiver,
i.e. direct sums of stable representations, where the stability condition is
determined by the slope $\sum_{i\in Q_{0}}d_{i}\zeta_{i}/\sum_{i\in
Q_{0}}d_{i}$. For primitive dimension vector and generic stability parameters,
polystable representations are automatically stable.

Summing up the relations (3.1) over $i\in Q_{0}$, it is clear that solutions
only exist when $\partial_{0}((d_{i}\zeta_{i})_{i\in Q_{0}}=\sum_{i\in
Q_{0}}d_{i}\zeta_{i}=0$. Thus, one can choose $(\chi_{a})_{a\in
Q_{1}}\in\mathbb{R}^{Q_{1}}$ such that $\partial_{1}((\chi_{a})_{a\in
Q_{1}}=(-d_{i}\zeta_{i})_{i\in Q_{0}}$. The trace of the D-term equations
(3.1) then implies the conservation at each node of $Q$ of the current:

$\displaystyle\lambda_{a}:=\chi_{a}+{\rm Tr}(\phi_{a}\phi_{a}^{\dagger})$
(3.3)

The choice of $\chi_{a}$ is non unique, unless $Q$ has no oriented cycle. For
the attractor stability condition (also known as self-stability), one may
choose $\chi_{a}=d_{i}d_{j}$ such that the current $\lambda$ is strictly
positive (arrows $a:i\to j$ such that $d_{i}d_{j}=0$ do not support any chiral
multiplet and may be removed from the quiver $Q$). By applying $i)$ of lemma
A.1, it then immediately follows that, similarly to the Coulomb branch case,

######  Proposition 3.1.

A quiver which admits self-stable representation of dimension
$(d_{i})\in(\mathbb{N}^{\star})^{Q_{0}}$ must be strongly connected.

We note that for Abelian representations, the D-terms equations are equivalent
to the conservation of the current $\lambda$. As on the Coulomb branch, one
has then, according to lemma (A.3), that an Abelian representation of $Q$
satisfies the D-term equations at the attractor stability condition if and
only if the induced representations on the biconnected components of $Q$
satisfy the induced D-terms equations. Moreover, by definition, the F-term
equations $\partial_{a}W=0$ constrain only the arrows which share a cycle with
$a$, and therefore are in the same biconnected component as $a$. Thus, a
representation satisfies the F-terms equations if and only if the induced
representations satisfy the induced F-terms equations on each biconnected
component. To summarize, stable Abelian representations of the quiver with
potential $(Q,W)$ at any stability condition are obtained by gluing such
representations for the biconnected components of $Q$. As we shall see below,
the example of the ’butterfly quiver’ shows that this is no longer true in the
non-Abelian case.

###  3.2 Stronger constraints in the Abelian case

In order to obtain conditions on the number of chiral fields, we shall rely on
lemma A.8, which applies to any quiver with potential $(Q,W)$ such that
$W=\sum_{w\in C}\nu_{w}w$ is a linear combination of a subset of simple cycles
$C\subset Q_{2}$ with generic coefficients, and such that there exists a
subset of arrows $I\subset Q_{2}$ such that each cycle of $C$ contains exactly
one arrow of $I$. Under these assumptions, any Abelian representation of
$(Q,W)$ is such that every each cycle in $C$ vanishes when evaluated on this
representation. The proof relies on Bertini’s theorem for linear systems on
algebraic varieties, and generalizes the informal argument outlined in [1,
§5.2.3] in the case of a triangular cyclic quiver.

Now, consider a quiver $Q$ with a cut $I$ with potential $W=\sum_{w\in
Q_{2}}\nu_{w}w$ which is generic in the sense of definition A.9, and $\phi$ an
Abelian representation of $(Q,W)$. From lemma A.8, it follows that there is a
strong cut $J$ such that the arrows of $J$ vanish in $\phi$ while no other
arrows vanish: for simplicity, we consider here the case where $J$ is a cut,
the general case being treated in the appendix. The representation $\phi$ is
therefore a representation of the quiver with relations
$(Q_{J},\partial_{J}W)$, where $Q_{J}$ is the subquiver of $Q$ where one has
removed the arrows of $J$, and the relations are $\partial_{a}W=0$ for $a\in
J$.

Suppose that $\phi$ is $\zeta$-stable. The moduli space $M^{\zeta,s}_{Q_{J}}$
of $\zeta$-stable representations of $Q_{J}$ is smooth of dimension
$|Q_{1}-J|-|Q_{0}|+1$. Since the moduli space
$M^{\zeta,s}_{Q_{J},\partial_{J}W_{J}}$ of $\zeta$-stable Abelian
representations of $(Q_{J},\partial_{J}W)$ is given by the zero locus of the
$|J|$ relations $\partial_{a}W$ in $M^{\zeta,s}_{Q_{J}}$, one expects then
that its complex dimension is given by

$\displaystyle d_{J}=|Q_{1}-J|-|Q_{0}|+1-|J|\ ,$ (3.4)

unless there are linear relations between the relations. We shall now prove
that there a no relation between the relations infinitesimally near $\phi$.

For a cycle $w$ of $W$, we denote its arrows by $w=b_{1}^{w}...b_{r_{w}}^{w}$.
The linearization of $\partial_{a}W$ on the tangent space at $\phi$ gives:

$\displaystyle\delta(\partial_{a}W):(\delta\phi_{b})_{b\in
Q_{1}}\mapsto\sum_{w,i|b^{w}_{i}=a}\nu_{w}\sum_{i+1\leq j\leq
i-1}\phi_{b^{w}_{i+1}}\cdots\phi_{b^{w}_{j-1}}\,\delta\phi_{b^{w}_{j}}\,\phi_{b^{w}_{j+1}}\cdots\phi_{b^{w}_{i-1}}$
(3.5)

Here we use the cyclic ordering of the cycle. Consider a linear relation
$(\tilde{\phi}_{a})_{a\in Q_{1}}$ between these relations such that:

$\displaystyle\sum_{a\in Q_{1}}\tilde{\phi}_{a}\delta(\partial_{a}W)=0$ (3.6)

One computes:

$\displaystyle 0=\sum_{a\in Q_{1}}\tilde{\phi}_{a}\delta(\partial_{a}W)$
$\displaystyle=\sum_{w\in Q_{2}}\nu_{w}\sum_{1\leq i\neq j\leq
r_{w}}\tilde{\phi}_{b^{w}_{i}}\phi_{b^{w}_{i+1}}\cdots\phi_{b^{w}_{j-1}}\,\delta\phi_{b^{w}_{j}}\,\phi_{b^{w}_{j+1}}\cdots\phi_{b^{w}_{i-1}}$
$\displaystyle=\sum_{w\in Q_{2}}\nu_{w}\sum_{1\leq i\neq j\leq
r_{w}}\phi_{b^{w}_{j+1}}\cdots\phi_{b^{w}_{i-1}}\tilde{\phi}_{b^{w}_{i}}\phi_{b^{w}_{i+1}}\cdots\phi_{b^{w}_{j-1}}\,\delta\phi_{b^{w}_{j}}$
(3.7)

Consider now the representation of $Q$ $\bar{\phi}=\phi+\epsilon\tilde{\phi}$
with $\epsilon^{2}=0$ (hence it is a representation over the ring
$\mathbb{C}[\epsilon]/\epsilon^{2}$). Because $\phi$ satisfies the equation of
the potential, one has, multiplying by $\epsilon$:

$\displaystyle 0$ $\displaystyle=\sum_{w\in Q_{2}}\nu_{w}\sum_{1\leq j\leq
r_{w}}(\phi+\epsilon\tilde{\phi})_{b^{w}_{j+1}}\cdots\cdots(\phi+\epsilon\tilde{\phi})_{b^{w}_{j-1}}\,\delta\phi_{b^{w}_{j}}$
$\displaystyle=\sum_{b\in Q_{1}}\partial_{b}W|_{\bar{\phi}}\delta\phi_{b}$
(3.8)

Now we consider a linear relation between the relations $\sum_{a\in
J}\tilde{\phi}_{a}\delta(\partial_{a}W)=0$ where one restricts to deformations
with vanishing arrows of $J$, hence $\delta\phi_{b}=0$ for $b\in J$. One
obtains in this case
$\bar{\phi}=(\epsilon\tilde{\phi}_{a},\phi_{b})_{a_{\in}J,b\in Q_{1}-J}$, and
$\partial_{b}W|_{\bar{\phi}}$ for $b\in Q_{1}-J$. Moreover, for $a\in J$,
$\partial_{a}W_{J}|_{\bar{\phi}}=\partial_{a}W|_{\phi}=0$, hence $\bar{\phi}$
is a representation of $(Q,W)$. Because $W$ is generic, according to Lemma
A.8, the cycles of $Q_{2}$ vanish in $\bar{\phi}$. Each cycle in $Q_{2}$
contains exactly one arrow in $J$ and other arrows in $Q_{1}-J$: because the
arrows of $Q_{1}-J$ do not vanish in $\phi$, it follows that the arrows of $J$
vanish in $\tilde{\phi}$. Hence there are no nontrivial linear relations
between the $|J|$ differential forms, and the dimension of the tangent space
at $\phi$ in $M^{\zeta,s}_{Q_{J},\partial_{I}W_{J}}$ is:

$\displaystyle 0\leq d=|Q_{1}-J|-|Q_{0}|+1-|J|$ (3.9)

Now $R_{J}\circ\partial_{2}=R_{I}\circ\partial_{2}$. Suppose that $\zeta$ is
the attractor stability condition: the positive conserved current
$\lambda=(1+|\phi_{a}|^{2})_{a\in Q_{1}}$ can then be written as
$\lambda=\sum_{w}\mu_{w}\partial_{2}(w)$ with $\mu_{w}>0$ from $ii)$ of lemma
A.1, and then because the arrows of $J$ vanish on $\phi$:

$2|I|\leq R_{I}(\lambda)=R_{J}(\lambda)=2|J|$ (3.10)

hence

$2|I|\leq|Q_{1}|-|Q_{0}|+1$ (3.11)

We show in the appendix how to remove the assumption that $J$ is a cut, and
how to generalize the above argument when $I$ is only a weak cut, and obtain
finally an equivalent of Proposition 2.3 on the Higgs branch:

######  Proposition 3.2.

(Proposition A.10) If a quiver with generic potential $(Q,W)$ admits a self-
stable Abelian representation, then for each weak cut $I$:

$\displaystyle|I|\leq|Q_{1}-I|-|Q_{0}|+1$ (3.12)

The case were $I$ is a cut but $J$ is not necessarily a cut is easy to deal
with. Namely, because there is too much arrows vanishing, some relations of
the potential $\delta_{a}W=0$ for $a\in J$ can become trivial, but this is
counterpoised exactly by the fact that there is more relations of the form
$\phi_{a}=0$ for $a\in J$. The case where $I$ is only a weak cut and not a cut
case is more subtle, because for a quiver without cut and generic potential,
the cycles do not necessarily vanish in an Abelian representation, hence the
first part of the proof does not work. One must set some arrows to zero to
apply a similar argument, but one must ensure that the quiver stays connected
during this procedure (such that the action of the gauge group up to global
scaling is still free), and that it is done in a consistent way (such that one
does not lose too many relations of the quiver). We refer to the poof in the
Appendix for all the details.

###  3.3 Stronger constraints in the non-Abelian case

One can try to obtain stronger constraints for the existence of non-Abelian
self-stable representations, by assuming that non-Abelian analogue of Lemma
A.8:

######  Assumption 3.3.

In a stable representation $\phi$ of a quiver with potential $(Q,W)$ with $W$
generic, each cycle contains an arrow which vanishes on $\phi$.

If this assumptions was true, one could generalize the inequalities for cuts
from proposition A.10 to the non-Abelian setting as follows. Let $\phi$ be a
$d$-dimensional self-stable representation of a quiver $Q$ with generic
potential $W$, and $I$ a cut of $Q$: under assumption 3.3, the arrows of a
strong cut $J$ of $Q$ vanish in $\phi$. We consider the simpler situation
where $J$ is a cut, the general case being treated as in A.10. $\phi$ is a
$d$-dimensional stable representation of the quiver with relation
$(Q_{J},\partial_{J}W)$. The moduli space $M^{\zeta,s}_{Q_{J},d}$ of
$d$-dimensional representations of $Q_{J}$ is smooth of dimension
$\sum_{(a:i\to j)\in Q_{1}-J}d_{i}d_{j}-\sum_{i\in Q_{0}}d_{i}^{2}+1$, and the
space $M^{\zeta,s}_{Q_{J},\partial_{J}W,d}$ of stable representations of
$(Q,\partial_{J}W)$ is the zero set of $\sum_{(a:i\to j)\in J}d_{i}d_{j}$
equations inside $M^{\zeta,s}_{Q_{J},d}$. Using the cyclicity of the trace,
one can derive non-Abelian analogues of equations 3.2, giving that a linear
relation $(\tilde{\phi}_{a})_{a\in J}$ between these equations at the tangent
space of $\phi$ gives a representation
$\bar{\phi}=(\phi_{a},\epsilon\tilde{\phi}_{b})_{a\in Q_{1}-J,b\in J}$ of
$(Q_{J},W)$ over $\mathbb{C}[\epsilon]/\epsilon^{2}$, which is still self-
stable because some vanishing arrows in the self-stable representation $\phi$
have been set to a non-vanishing value. Then under the assumption 3.3, the
cycles of $Q_{2}$ contain an arrow vanishing in $\bar{\phi}$, hence
$\tilde{\phi}_{a}=0$ for $a\in J$, hence there are no relations between the
relations. The dimension of the tangent space at $\phi$ inside
$M^{\zeta,s}_{Q_{J},\partial_{J}W,d}$ is therefore:

$\displaystyle 0\leq d=\sum_{(a:i\to j)\in Q_{1}}d_{i}d_{j}-\sum_{i\in
Q_{0}}d_{i}^{2}+1-\sum_{(a:i\to j)\in J}d_{i}d_{j}-\sum_{(a:i\to j)\in
J}d_{i}d_{j}$ (3.13)

Applying the same arguments as in the proof of Proposition A.10 using the
conserved current $(d_{i}d_{j}+{\rm Tr}(\phi_{a}^{\dagger}\phi_{a}))_{(a:i\to
j)\in Q_{1}}$, one obtains

$2\sum_{(a:i\to j)\in I}d_{i}d_{j}\leq 2\sum_{(a:i\to j)\in J}d_{i}d_{j}$
(3.14)

hence for any cut $I$:

$\displaystyle\sum_{(a:i\to j)\in I}d_{i}d_{j}\leq\sum_{(a:i\to j)\in
Q_{1}-I}d_{i}d_{j}-\sum_{i\in Q_{0}}d_{i}^{2}+1$ (3.15)

However, Assumption 3.3 fails for non-Abelian quivers which are not
biconnected. Consider as in subsection 2.5 the ’butterfly’ quiver with $p$
arrows $2\to 3$ and $p\geq 2$ arrows $4\to 5$, and dimension vector $2$ on the
central node $1$, and $1$ on the other nodes. For a generic potential $W$, the
$p$ equations corresponding with the arrows $2\to 3$ (resp. $4\to 5$) impose
that $\phi_{12}\phi_{31}=0$ (resp. $\phi_{14}\phi_{51}=0$). Denote $w^{i}$
(resp. $\bar{w}^{i}$) the cycle $a_{31}a^{i}_{23}a_{12}$ (resp.
$a_{51}a^{i}_{45}a_{14}$), and
$W=\sum_{i=1}^{p}\nu_{i}w_{i}+\sum_{i=1}^{p}\bar{\nu}_{i}\bar{w}_{i}$. Take
$(\phi^{i}_{23})_{1\leq i\leq p}$ (resp. $(\phi^{i}_{45})_{1\leq i\leq p}$) as
a generic vector satisfying $\sum_{i}\nu_{i}\phi^{i}_{23}=0$ (resp.
$\sum_{i}\nu_{i}\phi^{i}_{45}=0$), in particular all these arrows are non-
vanishing because $W$ is generic. We fix then:

$\displaystyle\phi_{51}=\begin{pmatrix}1\\\
0\end{pmatrix}\;,\;\phi_{12}=(1,0)\;,\;\phi_{31}=\begin{pmatrix}0\\\
1\end{pmatrix}\;,\;\phi_{14}=(0,1)$ (3.16)

The resulting representation $\phi$ satisfies the F-term relations for the
potential $W$. As can be seen on the figure 9, where the arrows correspond to
nonzero matrix elements and the points $1,1^{\prime}$ correspond to the two
elements of the basis of the two dimensional vector space of the node $1$,
each vector generates the whole representation, i.e. the only subobjects of
$\phi$ are the trivial ones, and then $\phi$ is stable for any stability
condition. We have therefore constructed a self-stable representation of a
quiver with generic potential $(Q,W)$ admitting a cut such that all the arrows
are non-vanishing. Moreover, considering the cut $I$ containing the arrows
$2\to 3$ and $4\to 5$: for $p\geq 1$, one has:

$\displaystyle\sum_{(a:i\to j)\in I}d_{i}d_{j}=2p>1=\sum_{(a:i\to j)\in
Q_{1}-I}d_{i}d_{j}-\left(\sum_{i\in Q_{0}}d_{i}^{2}-1\right)$ (3.17)

i.e. the stronger triangular inequalities are not necessarily true for non
biconnected quivers. Notice the similarity of this construction with the
counterexample discussed in §2.5.

123451’$\displaystyle p$$\displaystyle p$ Figure 9:  A stable representation
of the butterfly quiver with dimension vector $(2,1,1,1,1)$.

We leave it as an interesting problem to study if the assumption 3.3, and then
also the stronger inequalities (3.15) in the non-Abelian case, hold when one
restricts to biconnected quivers.

Acknowledgments: We are grateful to Guillaume Beaujard, Jan Manschot,
Swapnamay Mondal and Olivier Schiffmann for useful discussions, and to the
anonymous referees for their careful reading and valuable suggestions. The
research of BP is supported by Agence Nationale de la Recherche under contract
number ANR-21-CE31-0021.

##  Appendix A Proofs

In this Appendix, we provide mathematical proofs for some of the technical
results used in the body of the paper.

###  A.1 Conserved currents and graph homology

A strictly positive conserved current is an element of
$\ker(\partial_{1})\cap(\mathbb{R}^{\star}_{+})^{Q_{1}}$. A quiver is strongly
connected if and only if for any two nodes $i,j\in Q_{0}$ , there is an
oriented path from $i$ to $j$ and from $j$ to $i$. The key result which allows
us to derive constraints on the existence of scaling or attractor solutions is
the following:

######  Lemma A.1.

$i)$ If $Q$ admits a strictly positive conserved current, then $Q$ is strongly
connected.

$ii)$ A strictly positive conserved current $\lambda$ can be expressed as a
sum of strictly positive conserved currents circulating on all simple oriented
cycles of the quiver, i.e. $\lambda=\sum_{w\in Q_{2}}\mu_{w}\partial_{2}(w)$,
with $\mu_{w}>0$.

Proof:

i)

Suppose that $Q$ is not strongly connected. The strongly connected components
of $Q$ form a connected tree with at least two nodes: one has then a
nontrivial partition $Q_{0}=Q_{0}^{\prime}\sqcup Q_{0}^{\prime\prime}$ such
that there is at least one arrow $Q_{0}^{\prime}\to Q_{0}^{\prime\prime}$, and
no arrow $Q_{0}^{\prime\prime}\to Q_{0}^{\prime}$. Consider
$\lambda\in\ker(\partial_{1})\cap\mathbb{R}^{Q_{1}}$: by summing the
conservation of $\lambda$ at each node of $Q_{0}^{\prime}$, one has
$\sum_{(a:Q_{0}^{\prime}\to Q_{0}^{\prime\prime})\in Q_{1}}\lambda_{a}=0$, and
then $\lambda\not\in(\mathbb{R}^{\star}_{+})^{Q_{1}}$.

ii)

Suppose now that $Q$ is strongly connected. We must show that
$\ker(\partial_{1})\cap(\mathbb{R}^{\star}_{+})^{Q_{1}}\subset\partial_{2}((\mathbb{R}^{\star}_{+})^{Q_{2}})$.
We begin by showing
$\ker(\partial_{1})\cap(\mathbb{R}_{+})^{Q_{1}}\subset\partial_{2}((\mathbb{R}_{+})^{Q_{2}})$:
for this we reason inductively on the number of arrows which carry a non-
vanishing current. For
$\lambda\in\ker(\partial_{1})\cap(\mathbb{R}_{+})^{Q_{1}}$, we consider
$Q^{\lambda}=(Q_{0},\\{a\in Q_{1}|\lambda_{a}>0\\})$, the (possibly non
connected) quiver where we keep only the arrows with a non-vanishing current.
If $Q^{\lambda}_{1}\neq\emptyset$, take $(a:i\to j)\in Q^{\lambda}_{1}$ such
that $\lambda_{a}>0$ is minimal. One has
$\lambda\in\ker(\partial_{1})\cap(\mathbb{R}^{\star}_{+})^{Q^{\lambda}_{1}}$,
and then each connected component of $Q^{\lambda}$ is strongly connected from
$i)$: there is then a path $v:j\to i$ in $Q^{\lambda}$, and then a simple
oriented cycle $w:=av\in Q_{2}$ containing $a$, such that the arrows of $w$
are in $Q^{\lambda}$. One has
$\lambda^{\prime}:=\lambda-\partial_{2}(\lambda_{a}w)\in\ker(\partial_{1})$,
and, because $\lambda_{a}$ is minimal,
$\lambda^{\prime}\in(\mathbb{R}_{+})^{Q_{1}}$. Moreover,
$\lambda^{\prime}_{a}=0$, and then $Q^{\lambda^{\prime}}_{1}\subsetneq
Q^{\lambda}_{1}$: by induction on the number of arrows of $Q^{\lambda}$, one
has $\lambda^{\prime}\in\partial_{2}((\mathbb{R}_{+})^{Q_{2}})$, and then
$\lambda\in\partial_{2}((\mathbb{R}_{+})^{Q_{2}})$. We have then
$\ker(\partial_{1})\cap(\mathbb{R}_{+})^{Q_{1}}\subset\partial_{2}((\mathbb{R}_{+})^{Q_{2}})$.
Consider $\lambda\in\ker(\partial_{1})\cap(\mathbb{R}^{\star}_{+})^{Q_{1}}$:
one has $0<\epsilon\ll 1$ such that $\lambda-\partial_{2}(\epsilon\sum_{w\in
Q_{2}}w)\in\ker(\partial_{1})\cap(\mathbb{R}_{+})^{Q_{1}}$, and then
$\lambda=\partial_{2}\sum_{w\in Q_{2}}(\mu_{w}+\epsilon)w$ with
$\mu_{w}\in\mathbb{R}_{+}$, i.e.
$\lambda\in\partial_{2}((\mathbb{R}^{\star}_{+})^{Q_{2}})$.

$\Box$

###  A.2 Biconnected components of a quiver

A quiver is biconnected if there is no node $i$ of the quiver such that
removing $i$ (and then also the arrows with source or target $i$) disconnects
the quiver. On a quiver $Q$, the biconnected components are defined as the
maximal subquivers of $Q$ being biconnected. We prove the following fact about
biconnected components:

######  Lemma A.2.

$i)$ The biconnected components give a partition of the arrows of the quiver
$Q$, and two different biconnected components of a quiver can share at most
one node.

$ii)$ Define $K$ as the unoriented graph with one node for each biconnected
component and one node for each node of the quiver shared between different
biconnected components, and an edge between the node $i$ and the biconnected
component $b$ if $i\in b$. Then $K$ is a connected tree, i.e. has no loops.

Proof: This result can be deduced from [24, Prop 3.5], but we prove it here
from clarity and completeness. Suppose that there is a sequence of distinct
biconnected components $b_{1},...,b_{p}$ and a sequence of distinct nodes
$i_{1},...,i_{p}$ such that $i_{k}\in b_{k}\cup b_{k+1}$, $i_{p}\in b_{p}\cup
b_{1}$, with $p\geq 2$. Consider the subquiver $b_{1}\cup...\cup b_{p}$ of
$Q$, and remove a node $i$: one can consider up to a circular permutation that
$i\neq i_{1}$,…, $i\neq i_{p-1}$. Consider two nodes $j,j^{\prime}$ such that
$j\in b_{k}$, $j^{\prime}\in b_{k^{\prime}}$, and suppose up to exchanging $j$
and $j^{\prime}$ that $k\leq k^{\prime}$. Because the $b_{k^{\prime\prime}}$
are biconnected, there is an unoriented path between $j$ and $i_{k}$ in
$b_{k}$ avoiding $i$, unoriented paths between $i_{k^{\prime\prime}}$ and
$i_{k^{\prime\prime}+1}$ in $b_{k^{\prime\prime}}$ avoiding $i$ for $k\leq
k^{\prime\prime}\leq k^{\prime}$ and an unoriented path between
$i_{k^{\prime}}$ and $j^{\prime}$ in $b_{k^{\prime}}$ avoiding $i$. By
concatenation, these give an unoriented path between $j$ and $j^{\prime}$ in
$b_{1}\cup...\cup b_{p}$ avoiding $i$, i.e. $b_{1}\cup...\cup b_{p}$ is still
connected when one has removed $i$. One obtains then that $b_{1}\cup...\cup
b_{p}$ is biconnected, but it is strictly bigger that the $b_{k}$, which were
assumed to be maximal biconnected subquivers of $Q$, giving a contradiction.

In particular, for $p=2$, one obtains that two different biconnected
components can share at most one node: because an arrow is adjacent to two
nodes, two different biconnected components cannot share an arrow, proving
$i)$. The argument above for general $p$ gives exactly that $K$ has no cycle,
i.e. is a tree. Consider two biconnected components $b,b^{\prime}$, and two
nodes $i\in b,i^{\prime}\in b^{\prime}$: because $Q$ is connected, there is an
unoriented path in $Q$ between $i$ and $i^{\prime}$: by denoting
$b,b_{1},...,b_{n},b^{\prime}$ the biconnected components crossed by these
paths, $b$ and $b^{\prime}$ are then connected by a sequence of biconnected
components sharing a node, i.e. the graph $K$ is connected. This concludes the
proof of $ii)$. $\Box$

Denote by $B$ the set of biconnected components of a quiver, and $Q^{b}$ the
quiver associated to a biconnected component. Two biconnected components have
distinct arrows, and then one has a decomposition
$\mathbb{R}^{Q_{1}}=\bigoplus_{b\in B}\mathbb{R}^{Q^{b}_{1}}$. Consider a
simple unoriented cycle of $Q$: it cannot pass through different biconnected
components, since otherwise it would project to a cycle in $K$, which is
forbiddent by the above lemma, i.e. it is included in a single biconnected
component. In particular this applies to simple oriented cycles, and then one
has $\mathbb{R}^{Q_{2}}=\bigoplus_{b\in B}\mathbb{R}^{Q^{b}_{2}}$, giving a
decomposition of the complex:

$\displaystyle\bigoplus_{b\in B}\mathbb{R}^{Q^{b}_{2}}\overset{\sum_{b\in
B}\partial^{b}_{2}}{\to}\bigoplus_{b\in
B}\mathbb{R}^{Q^{b}_{1}}\overset{\sum_{b\in
B}\partial^{b}_{1}}{\to}\mathbb{R}^{Q_{0}}\overset{\partial_{0}}{\to}\mathbb{R}\to
0$ (A.1)

The cellular homology group $H^{1}$ of the graph gives the loops of the quiver
(see [23, Sec 2.2]), hence $\ker(\partial_{1})$ is generated by the simple
unoriented cycles of $Q$, each one lying in a single biconnected component. It
gives then the decomposition $\ker(\partial_{1})=\bigoplus_{b\in
B}\ker(\partial^{b}_{1})$:

######  Lemma A.3.

A current on a quiver is conserved if and only if its restriction to each
biconnected component is a conserved current.

######  Lemma A.4.

$i)$ A quiver is strongly connected if and only if all its biconnected
components are strongly connected.

$ii)$Suppose that $Q$ is strongly connected. Consider the equivalence class
$\sim$ on the arrows of $Q$ generated by $a\sim b$ if $a,b\in w$ for $w\in
Q_{2}$. The equivalence classes of $\sim$ correspond to the biconnected
components of $Q$.

Proof:

  1. $i)$

Suppose that $Q$ is strongly connected. Consider $i,i^{\prime}$ in a
biconnected component $b$. There is an oriented path $v:i\to i^{\prime}$ in
$Q$, giving a cycle $\bar{v}:b\to b$ on the graph $K$. If this cycle is
trivial, then $v$ stays in the biconnected component $b$. If it is not
trivial, since $K$ is a tree, the cycle $\bar{v}$ must be of the form $b\to
j\to...\to j\to b$, i.e. the path $v$ can be decomposed as
$i\overset{v_{1}}{\to}j\overset{v_{2}}{\to}j\overset{v_{3}}{\to}i^{\prime}$,
with $v_{3}$ and $v_{1}$ oriented paths in $b$. One obtains then an oriented
path $v_{3}v_{1}:i\to i^{\prime}$ in $b$, i.e. $b$ is strongly connected.

Suppose that each biconnected component of $Q$ is strongly connected. Consider
$i,i^{\prime}$ two nodes of $Q$, with $i\in b,i^{\prime}\in b^{\prime}$. The
graph $K$ is connected, consider a path $b\ni i_{1}\in b_{1}...i_{n-1}\in
b_{n-1}\ni i_{n}\in b^{\prime}$. Because $b,b_{k},b^{\prime}$ are strongly
connected, consider oriented paths $v:i\to i_{1},v_{k}:i_{k}\to
i_{k+1},v^{\prime}:i_{n}\to i^{\prime}$ respectively in $b,b_{k},b^{\prime}$.
This gives an oriented path $v^{\prime}v_{n-1}...v_{1}v:i\to i^{\prime}$ in
$Q$, i.e. $Q$ is strongly connected.

  2. $ii)$

Because a simple oriented cycle $w\in Q_{2}$ is contained in a single
biconnected component of $Q$, an equivalence class of $\sim$ is contained in a
single biconnected component of $Q$. One has then to show that a biconnected
quiver contains only one single class of $\sim$. Two equivalence classes do
not share any arrows by definition: one can construct an unoriented graph
$\tilde{K}$ with a node corresponding to each equivalence class $s$ of $\sim$,
and a node for each node $i\in Q_{0}$ shared between different equivalence
classes, and an edge between $i$ and $s$ when $i\in s$.

Considering two nodes $i,j$, there is a path between $i$ and $j$ in $Q$: the
sequence of equivalence classes of $\sim$ crossed by this path gives a path in
$\tilde{K}$ between $i$ and $j$, i.e. $\tilde{K}$ is connected. Consider a
cycle $i_{1}\in s_{1}\ni i_{2}...i_{n}\in s_{n}\ni i_{1}$ with no node or edge
repeated in $\tilde{K}$. Because each equivalence class is strongly connected,
one can choose an oriented path $v_{k}:i_{k}\to i_{k+1}$ with no node repeated
in each equivalence class $s_{k}$, the concatenation giving an oriented cycle
$v_{n}...v_{1}$ in $Q$. Because each $v_{k}$ contains no repeated node, there
is a subcycle $w$ of $v_{n}...v_{1}$ which is simple and contains arrows of
different equivalence classes of $\sim$, giving a contradiction. The graph
$\tilde{K}$ is then a connected tree. If there were more than one equivalence
classes of $\sim$, then there would be a node $i\in\tilde{K}$ such that
removing $i$ disconnects $\tilde{K}$, and then disconnects the quiver $Q$, a
contradiction because $Q$ was assumed to be biconnected. There is therefore a
single equivalence class of $\sim$ in a biconnected quiver $Q$.

$\Box$

###  A.3 Cuts and R-charge

######  Lemma A.5.

The following assertions are equivalent:
$i)$ $Q$ admits a cut.
$ii)$ $Q$ admits an $R$-charge.
$iii)$ Each maximal weak cut of $Q$ is a cut.

Proof: $iii)\Rightarrow i)$: $\emptyset$ is a weak cut, and the poset of weak
cuts is finite, then $Q$ admits a maximal weak cut, which is a cut by
assumption.

$i)\Rightarrow ii)$ Consider a cut $I$ of $Q$. Recall that we have defined the
homomorphism $R_{I}:\mathbb{R}^{Q_{1}}\mapsto\mathbb{R}$ by $R_{I}(a)=2$ if
$a\in I$ and $R_{I}(a)=0$ otherwise. Each cycle $w\in Q_{2}$ contains exactly
one arrow $a\in I$, and $R_{I}\circ\partial_{2}(w)=R_{I}(a)=2$, i.e. $R_{I}$
is a R-charge.

$ii)\Rightarrow iii)$. Consider a maximal weak cut $I$, and
$R:\mathbb{R}^{Q_{1}}\to\mathbb{R}$ such that $R\circ\partial_{2}\geq
R_{I}\circ\partial_{2}$, and $R_{I}\circ\partial_{2}(w)=2\Rightarrow
R\circ\partial_{2}(w)=2$. We must show that
$R\circ\partial_{2}=R_{I}\circ\partial_{2}$. Consider $(w=a_{n}...a_{1})\in
Q_{2}$ such that $R_{I}\circ\partial_{2}(w)=0$ i.e. $a_{k}\not\in I$ for each
$k$ (see Figure 10): in particular one has $R\circ\partial_{2}(w)\geq 0$.
Because $I$ is maximal, each arrow $a_{k}\in w$ is contained in a simple cycle
$w_{k}=v_{k}a_{k}\in Q_{2}$ such that $v_{k}$ contains exactly one arrow of
$I$ (since otherwise $I\cup\\{a_{k}\\}$ would be a larger weak cut). The
oriented cycle $\bar{w}=v_{1}...v_{n}$ is a product of simple oriented cycles,
and satisfies:

$\displaystyle R\circ\partial_{2}(\bar{w})\geq
R_{I}\circ\partial_{2}(\bar{w})=2n$
$\displaystyle\partial_{2}(w)=\sum_{k}\partial_{2}(w_{k})-\partial_{2}(\bar{w})$
$\displaystyle\Rightarrow$ $\displaystyle
R\circ\partial_{2}(w)=2n-R\circ\partial_{2}(\bar{w})\leq 0$
$\displaystyle\Rightarrow$ $\displaystyle
R\circ\partial_{2}(w)=0=R_{I}\circ\partial_{2}(w)$ (A.2)

where the third line holds because $R_{I}\circ\partial_{2}(w_{k})=2$. By
disjunction of cases, one has then
$R\circ\partial_{2}=R_{I}\circ\partial_{2}$. When $R$ is an R-charge, one
obtains then that $R_{I}$ is a R-charge, i.e. $I$ is a cut. $\Box$

$\displaystyle a_{3}$$\displaystyle a_{5}$$\displaystyle a_{4}$$\displaystyle
a_{2}$$\displaystyle a_{1}$$\displaystyle w$$\displaystyle
w_{5}$$\displaystyle w_{4}$$\displaystyle w_{3}$$\displaystyle
w_{2}$$\displaystyle w_{1}$$\displaystyle v_{1}$$\displaystyle
v_{5}$$\displaystyle v_{4}$$\displaystyle v_{3}$$\displaystyle v_{2}$ Figure
10: If $Q$ admits an R-charge, then each maximal weak cut is a cut.

######  Lemma A.6.

For a quiver with an R-charge and a cycle $w_{0}$ passing through all the
nodes, the cuts are given by $I=\\{a:i\to j|i>j\\}$ for each labelling of the
nodes such that $w_{0}:1\to 2\to...\to n\to 1$.

Consider a cut $I$ of $Q$: it contains exactly one arrow $a\in w_{0}$: we
choose the labelling of $Q_{0}$ such that $a:n\to 1$ (see Figure 11). Consider
an arrow $(b:i\to j)\in Q_{1}$, and denote by $v_{ji}$ the minimal path going
from $j$ to $i$ on the cycle $w_{0}$: $v_{ji}$ contains an arrow of $I$ if and
only if $i<j$, and the simple oriented cycle $v_{ji}b\in Q_{2}$ contains
exactly one arrow of the cut $I$, i.e. $b\in I$ if and only if $j<i$. One
concludes that each cut of $Q$ is of the form $I=\\{b:i\in j|j<i$. for a
specific cyclic ordering of $w_{0}$.

$\displaystyle n$$\displaystyle 1$$\displaystyle i$$\displaystyle
j$$\displaystyle a$$\displaystyle b$$\displaystyle v_{j}{}_{i}$$\displaystyle
n$$\displaystyle 1$$\displaystyle i$$\displaystyle j$$\displaystyle
a$$\displaystyle b$$\displaystyle v_{j}{}_{i}$$\displaystyle
v_{j}{}_{i}$$\displaystyle i<j$$\displaystyle i>j$ Figure 11: Under
assumptions of lemma A.6, all cuts are of the form $I=\\{a:i\to j|i>j\\}$

Conversely, consider the set $I=\\{b:i\to j|j<i$}, and a cycle
$w:i_{1}\overset{a_{1}}{\to}i_{2}...i_{r}\overset{a_{r}}{\to}i_{1}\in Q_{2}$,
such that $R\circ\partial_{2}(w)=2$ (see Figure 12). We will show that $w$
contains exactly one arrow of $I$. The cycle $v_{i_{1}i_{r}}...v_{i_{2}i_{1}}$
is equal to the $m$-th iteration $w_{0}^{m}$, for some $m\in\mathbb{N}$. Since
$2=R\circ\partial_{2}(w)=\sum_{k=1}^{r}R\circ\partial_{2}(v_{i_{k+1}i_{k}}a_{k})-mR\circ\partial_{2}(w_{0})=2r-2m$,
the number of iterations is $m=r-1$. But $w_{0}\in Q_{2}$ contains exactly one
arrow of $I$, and each cycle $v_{i_{k+1}i_{k}}a_{k}\in Q_{2}$ contains exactly
one arrow of $I$. Indeed, there are two options: a) either $i_{k}>i_{k+1}$,
and then $a_{k}\in I$ but $v_{i_{k+1}i_{k}}$ does not contain any arrow of
$I$, or b) $i_{k}<i_{k+1}$, and then $a_{k}\not\in I$ but $v_{i_{k+1}i_{k}}$
contains the arrow $n\to 1\in I$. Evaluating the R-charge, we get
$R_{I}\circ\partial_{2}(w)=\sum_{k=1}^{r}R_{I}\circ\partial_{2}(v_{i_{k+1}i_{k}}a_{k})-(r-1)R_{I}\circ\partial_{2}(w_{0})=2$.
Therefore $w\in Q_{2}$ contains exactly one arrow of $I$, i.e. $I$ is a cut.
$\Box$

$\displaystyle a_{1}$$\displaystyle i_{1}$$\displaystyle
v_{i_{1}}{}_{i_{2}}$$\displaystyle a_{5}$$\displaystyle a_{4}$$\displaystyle
a_{3}$$\displaystyle a_{2}$$\displaystyle i_{5}$$\displaystyle
i_{4}$$\displaystyle i_{3}$$\displaystyle i_{2}$$\displaystyle
v_{i_{5}}{}_{i_{1}}$$\displaystyle v_{i_{4}}{}_{i_{5}}$$\displaystyle
v_{i_{3}}{}_{i_{4}}$$\displaystyle v_{i_{2}}{}_{i_{3}}$ Figure 12: Under
assumptions of lemma A.6, the set $I=\\{a:i\to j|i>j\\}$ is a cut.

When $Q$ admits no R-charge (see Figures 5, 4 for examples), the set
$I=\\{b:i\in j|j<i\\}$ is a minimal strong cut of $Q$ which is not a cut: each
simple oriented cycle contains at least one arrow of $I$, but there must be a
cycle $w\in Q_{2}$ containing more that one arrow of $Q$. In particular, if
$Q$ has an other cycle $w_{1}$ passing though all the nodes in a different
order than $w_{0}$, then it has no R-charge. Indeed, for a labelling of the
nodes such that $w_{0}:1\to 2\to...\to n\to 1$, the cycle $w_{1}$ is of the
form $1=\sigma(1)\to\sigma(2)\to...\to\sigma(n)\to\sigma(1)$, with $\sigma$ a
permutation of $\\{1,...,n\\}$ different from the identity. There must then be
at least two arrows of $w_{1}$ such that $\sigma(i)>\sigma(i+1)$, and then
$w_{1}$ contains at least two arrows of $I=\\{a:i\to j|i>j\\}$, i.e. $I$ is
not a cut, and then $Q$ has no R-charge. More generally, the criterion of the
above lemma gives a simple algorithm to check if a quiver with a cycle passing
through all the nodes has an R-charge.

###  A.4 Current decomposition for Abelianized quivers

######  Lemma A.7.

For $Q$ a biconnected quiver and $d$ a dimension vector with $d_{i}\geq 1$ for
$i\in Q_{0}$, a strictly positive conserved current on $Q^{d}$ can be
expressed as a sum of (not necessarily positive) currents circulating on the
cycles in $p^{-1}(Q_{2})$.

Proof: Consider a strictly positive conserved current $\lambda$ on $Q^{d}$: in
particular $Q^{d}$, and therefore $Q$, is strongly connected from $i)$ of
lemma A.1, and $\lambda$ can be expressed as a sum of positive currents
circulating on the cycles in $Q^{d}_{2}$. One must then show that for $Q$ a
biconnected strongly connected quiver each cycle in $Q^{d}_{2}$ can be
expressed as a linear combination of cycles in $p^{-1}(Q_{2})$, i.e.
$\partial_{2}(\mathbb{R}^{Q^{d}_{2}})\subset\partial_{2}(\mathbb{R}^{p^{-1}(Q_{2})})$.
One can construct the Abelianized quiver by using a finite sequence of
elementary steps, where one node is split into two nodes at each step. Each of
these steps preserves the fact that the quiver is biconnected and strongly
connected, thus it suffices to prove the statement for an elementary step.

Consider a biconnected strongly connected quiver $Q$, a node $i\in Q_{0}$ and
the dimension vector $d$ such that $d_{i}=2$ and $d_{j}=1$ for $j\neq i$.
Consider a cycle $w\in Q^{d}_{2}-p^{-1}(Q_{2})$: it necessarily passes through
the two nodes $(i,1),(i,2)$, i.e. it is of the form666Since $i$ is the only
non-Abelian node, we abuse notation and denote by $(a,k)$ and $(b,k)$ the lift
of the arrows $a:i\to j$ and $b:j\to i$ on $Q$ to the $k$-th copy of the node
$i$ on $Q^{d}$. $(d,1)v(c,2)(b,2)u(a,1)$, where $a,c$ are arrows of $Q$ with
source $i$, $b,d$ arrows of $Q$ with target $i$, and $u$ and $v$ paths of $Q$
avoiding $i$, such that $dvc,bua\in Q_{2}$ (see Figure 13). We claim that
there is a sequence of arrows $b_{0}=b,b_{1},...,b_{n}$ with target $i$, a
sequence of arrows $c_{0},...,c_{n}=c$ with source $i$, and sequences of paths
$u_{1},...,u_{n}$ and $v_{1},...,v_{n-1}$ in $Q$, such that $b_{k}u_{k}c_{k}$
and $b_{k+1}v_{k}c_{k}$ are simple oriented cycles in $Q_{2}$ for $1\leq k\leq
n-1$. Then:

$\displaystyle\partial_{2}\left((d,1)v(c,2)(b,2)u(a,1)\right)=$
$\displaystyle\partial_{2}\left((d,1)v(c,1)+(b,1)u(a,1)\right.$
$\displaystyle+\sum_{k=0}^{n-1}\left((b_{k},2)u_{k}(c_{k},2)-(b_{k},1)u_{k}(c_{k},1)+(b_{k+1},1)v_{k}(c_{k},1)-(b_{k+1},2)v_{k}(c_{k},2)\right)$
(A.3)

where each of the cycles appearing on the right belongs to $p^{-1}(Q_{2})$,
i.e. project to a simple cycle of $Q$. Thus
$\partial_{2}((d,1)v(c,2)(b,2)u(a,1))\in\partial_{2}(\mathbb{R}^{p^{-1}(Q_{2})})$,
and then
$\partial_{2}(\mathbb{R}^{Q^{d}_{2}})\subset\partial_{2}(\mathbb{R}^{p^{-1}(Q_{2})})$
which concludes the proof.

$\displaystyle(d,1)$$\displaystyle(a,1)$$\displaystyle
u$$\displaystyle(b_{0},2)$$\displaystyle
v$$\displaystyle(c_{1},2)$$\displaystyle(c_{0},2)$$\displaystyle(c_{0},1)$$\displaystyle(c_{1},1)$$\displaystyle(b_{0},1)$$\displaystyle(b_{1},1)$$\displaystyle(b_{1},2)$$\displaystyle
u_{0}$$\displaystyle u_{1}$$\displaystyle
v_{0}$+++++$-$$-$$-$$\displaystyle(i,1)$$\displaystyle(i,2)$ Figure 13: Proof
of the fact that a simple cycle of $Q^{d}$ can be expressed as a signed sum of
cycles in $p^{-1}(Q_{2})$.

Proof of the claim: Consider a strongly connected biconnected quiver $Q$, and
a node $i\in Q_{0}$. Consider the equivalence $\sim$ on the arrows of $Q$ with
source or target $i$, generated by $a\sim b$, with $a$ (resp. $b$) with source
(resp. target) $i$ if there is a path $v$ in $Q$ such that $bva$ is a simple
oriented cycle in $Q_{2}$. We need to show that there is a single equivalence
class under $\sim$. Let $S$ be the set of equivalence classes. To each $s\in
S$ of $\sim$ one can associate the subquiver $Q^{s}\subset Q-\\{i\\}$ whose
nodes are the nodes $j$ such that there is a path $v:j\to i$ passing only once
through $i$ and ending by an arrow in the given equivalence class.

Since $Q$ is strongly connected, for each $j\in Q_{0}-\\{i\\}$ there is a path
from $j$ to $i$, which up to truncating the path, can be considered as passing
only once through $i$, i.e. $j\in Q_{0}^{s}$ for at least one $s\in S$.
Consider $j\in Q_{0}^{s}$, $j^{\prime}\in Q_{0}^{s^{\prime}}$ and $u:j\to
j^{\prime}$ an oriented path of $Q$ avoiding $i$. Consider paths $v:j\to i$,
$v^{\prime}:j^{\prime}\to i$ passing only once through $i$ and ending
respectively by $b\in s$, $b^{\prime}\in s^{\prime}$. Because $Q$ is strongly
connected, there is a path $u^{\prime}:i\to j$ beginning by an arrow $a$ with
source $i$: the two cycles $vu^{\prime}$ (resp. $v^{\prime}uu^{\prime}$)
contain a simple cycle containing $ba$ (resp. $b^{\prime}a$), and then $b\sim
a\sim b^{\prime}$, i.e. $s=s^{\prime}$. For $j=j^{\prime}$ and $u$ the trivial
path at $j$, one obtains that the $Q_{0}^{s}$ form a partition of
$Q_{0}^{s}-\\{i\\}$, and for $u=a:j\to j^{\prime}$ an arrow of $Q$ one obtains
that $Q^{s}$ and $Q^{s^{\prime}}$ are disconnected in $Q-\\{i\\}$. If there
were different equivalence classes under $\sim$, $Q-\\{i\\}$ would be
disconnected, contradicting the assumption that $Q$ is biconnected. Thus there
is a single equivalence class under $\sim$, which proves the claim. $\Box$

###  A.5 Higgs branch

######  Lemma A.8.

Let $Q$ be a quiver, $C\subset Q_{2}$ a set of cycles of $Q$ such that there
exists a subset $I\subset Q_{1}$ of arrows such that each cycle of $C$
contains exactly one arrow of $I$. For any dimension vector
$d\in\mathbb{N}^{Q}_{0}$, there exists a dense open subset
$U_{C,d}\subset\mathbb{C}^{C}$ such that for any $(\nu_{w})_{w\in C}\in
U_{c,d}$ and any $d$-dimensional representation $\phi$ of the quiver $Q$ with
potential $W=\sum_{w\in C}\nu_{w}w$, the trace ${\rm Tr}(w)$ vanishes on the
representation $\phi$ for all cycles $w\in C$ appearing in the potential.

Proof: We follow an argument of Kontsevich quoted in the proof of [29, Prop.
3.1]. For $d\in\mathbb{N}^{Q_{0}}$, denote by $M_{d}$ the smooth quasi-
projective connected space of $d$-dimensional representations of $Q$. Now,
consider a potential cut $I$ of $W$, such that ${\rm Tr}(W)=\sum_{a\in I}{\rm
Tr}(a\partial_{a}W)$: by homogeneity, the critical points of ${\rm Tr}(W)$ on
$M_{d}$ necessarily have ${\rm Tr}(W)=0$. Let
$f:M_{d}\rightarrow\mathbb{C}^{C}$ be the map which associates to any
$d$-dimensional representation of $Q$ the vector $({\rm Tr}(w)_{w\in C}$. Away
from the locus $f^{-1}(0)$, the image of this map descends to the projective
space $\mathbb{P}^{|C|-1}$. The trace of the potential $W=\sum_{w\in
C}\nu_{w}w$ is obtained by composing $f$ with a linear form, corresponding to
a hyperplane section of $\mathbb{P}^{|C|-1}$. By applying Bertini’s theorem to
the complete linear system ${\rm Tr}(W)^{-1}(0)$, one finds that, for
$(\nu_{w})_{w\in C}$ in a dense open subset $U_{C,d}\subset\mathbb{C}^{C}$,
${\rm Tr}(W)^{-1}(0)$ is a smooth connected strict sub-variety of $M_{d}$ away
from the zero locus $f^{-1}(0)$. In particular, its tangent space at any point
$x\in{\rm Tr}(W)^{-1}(0)-f^{-1}(0))$ is strictly included in the tangent space
of $M_{d}$,

$\displaystyle T_{x}({\rm Tr}(W)^{-1}(0))\subsetneq T_{x}(M_{d})$ (A.4)

hence $\delta({\rm Tr}(W))|_{x}\neq 0$. It follows that the critical points of
$({\rm Tr}(W)$ lie in $f^{-1}(0)$, hence ${\rm Tr}(w)=0$ for $w\in C$ on a
$d$-dimensional representation of $(Q,W)$. Specializing to the dimension
vector $d=(1,...,1)$, one obtains that for $(\nu_{w})_{w\in C}\in U_{C}$ a
dense open subset of $\mathbb{C}^{C}$, and for any Abelian representation of
$(Q,W)$, the cycles $w$ vanish for all $w\in C$. $\Box$.

Now, for any weak cut $I\subset Q_{1}$, we denote by $Q_{2}^{I}$ the set of
cycles containing exactly one arrow of $I$, and for $W=\sum_{w\in
Q_{2}}\nu_{w}w$, we define $W_{I}:=\sum_{w\in Q_{2}^{I}}\nu_{w}w$. Let
$p_{I}:\mathbb{C}^{Q_{2}}\to\mathbb{C}^{Q_{2}^{I}}$ be the natural projection.

######  Definition A.9.

A potential $W=\sum_{w\in Q_{2}}\nu_{w}w$ is said to be generic if
$(\nu_{w})_{w\in Q_{2}}$ is in the dense open subset
$\bigcap_{I}p_{I}^{-1}(U_{Q_{2}^{I}})$.

######  Proposition A.10.

Consider a quiver with generic potential $(Q,W)$. If there exists a self-
stable Abelian representation of $(Q,W)$, then for each weak cut $I$:

$\displaystyle|I|\leq|Q_{1}-I|-|Q_{0}|+1$ (A.5)

Proof: Let $\phi$ be a $\zeta$-stable representation of $(Q,W)$, with
$\zeta,W$ generic, and $I$ a weak cut of $Q$. Denote by $J$ the set of arrows
of $Q$ vanishing in $\phi$. The representation $\phi$ gives a $\zeta$-stable
representation of $Q_{J}$, the quiver where the arrows of $J$ have been
removed: in particular, the stability of $\phi$ implies that $Q_{J}$ is
connected (otherwise, $\phi$ would be a direct sum of representations
supported on the connected components). Consider the set $K\subset I-J$ of
arrows $(a:i\to j)\in I-J$ such that $i$ and $j$ are connected in $Q_{J\cup
I}$. The quiver $Q_{J\cup K}$ obtained by removing all arrows in $K$ is then
still connected. Sending the arrows of $K$ to $0$ in $\phi$, one obtains a
representations $\psi$ of $Q_{J\cup K}$ without vanishing arrows. Because
$Q_{J\cup K}$ is connected, the gauge group $(\mathbb{C}^{\ast})^{Q_{0}}$
scaling the nodes acts on the space $(\mathbb{C}^{\ast})^{Q_{1}-J-K}$ of
representations of $Q_{J\cup K}$ without vanishing arrows with stabilizer
$\mathbb{C}^{\star}$, giving a smooth moduli space of dimension
$|Q_{1}-J-K|-|Q_{0}|+1$.

Consider the potential $W_{I}$ obtained from the generic potential $W$ by
keeping only the cycles $w\in Q_{2}$ which contain an arrow of $I$. Consider
$L\subset J\cup K$ the set of arrows contained in a cycle $w\in Q_{2}$ such
that $R_{J\cup K}\circ\partial_{2}(w)=R_{I}\circ\partial_{2}(w)=2$. Consider
$a\in L$:

  * •

If $a\in I$, because $I$ is a weak cut, any cycle $w\in Q_{2}$ which contains
$a$ is a cycle of $W_{I}$, and contains no other arrow of $I$, and in
particular no other arrows of $K$, i.e. :

$\displaystyle\partial_{a}W_{I}|_{\psi}=\partial_{a}W|_{\psi}=\partial_{a}W|_{\phi}=0$
(A.6)

  * •

Suppose $a\in J-I$. There is a cycle $w\in Q_{2}$ containing $a$ such that
$R_{J\cup K}\circ\partial_{2}(w)=R_{I}\circ\partial_{2}(w)=2$. Hence $w$
contains an arrow $b\in I$ different from $a$, which cannot be in $K$, hence
$b\in I-K$. Denote by $i\sim j$ the equivalence relation identifying vertices
$i$ and $j$ which are connected in $Q_{J\cup I}$: one has $t(b)\sim s(a)$ and
$t(a)\sim s(b)$ because $w$ contains no other arrows of $J\cup I$. Consider an
other cycle $w^{\prime}$ containing $a$.

    * –

If $w^{\prime}$ contains an other arrow of $J$, then
$0=\partial_{a}w^{\prime}|_{\psi}=\partial_{a}w^{\prime}|_{\phi}$.

    * –

If $w^{\prime}$ contains no other arrow of $J$ and an arrow $c\in K\subset
I-J$, by definition, $s(c)\sim t(c)$: using $t(a)\sim s(c)$ and $t(c)\sim
s(a)$ because $w^{\prime}$ contains no other arrows of $J\cup I$, one obtains
$t(b)\sim s(b)$, a contradiction because $b\in I-K$.

    * –

If $w^{\prime}$ contains no other arrow of $J$ and no arrow of $I$, then
$t(a)\sim s(a)$, and then $t(b)\sim s(b)$, a contradiction because $b\in I-K$.

    * –

In the remaining case $w^{\prime}$ contains one arrow of $J$ and one arrow of
$I-K$, then $w^{\prime}$ is a cycle of $W_{I}$ and
$\partial_{a}w^{\prime}|_{\psi}=\partial_{a}w^{\prime}|_{\phi}$

By disjunction of cases one has
$\partial_{a}W|_{\psi}=\partial_{a}W|_{\phi}=0$, and because the only cycles
contributing to $\partial_{a}W|_{\psi}$ are cycles of $W_{I}$, one has
$\partial_{a}W_{I}|_{\psi}=\partial_{a}W|_{\psi}=0$.

By disjunction on cases, $\partial_{a}W_{I}|_{\psi}=0$ for $a\in L$, hence
$\psi$ is a representation of the quiver with relations $(Q_{J\cup
K},\partial_{L}W_{I})$ without vanishing arrows. The tangent space of the
moduli space of representations of $(Q_{J\cup K},\partial_{L}W_{I})$ at $\psi$
is given by the intersection of the kernel of the $|L|$ differential forms
$\delta(\partial_{a}W_{I})$ for $a\in L$.

As in (3.8), a linear relation $\sum_{a\in
L}\tilde{\psi}_{a}\delta(\partial_{a}W_{I})=0$ between these differential
forms yields a representation
$\bar{\psi}=(\psi_{a},\epsilon\tilde{\psi}_{b})_{a\in Q_{1}-L,b\in L}$ over
$\mathbb{C}\epsilon/\epsilon^{2}$ such that $\delta_{b}W|_{\bar{\psi}}=0$ for
$b\in Q_{1}-J\cup K$. For an arrow $b\in J\cup K$, because
$\partial_{b}|_{\psi}=0$, $\partial_{b}|_{\bar{\psi}}$ is at most of order
$\epsilon$, and $\bar{\psi}_{b}$ is at most of order $\epsilon$, hence
$\bar{\psi}_{b}\partial_{b}|_{\bar{\psi}}=0$. Then
$\bar{\psi}_{b}\partial_{b}|_{\bar{\psi}}=0$ for $b\in Q_{1}$. Since the each
cycle of the potential $W_{I}$ contains exactly one arrow of $I$, Lemma A.8
implies that the cycles of $W_{I}$ vanish in $\bar{\psi}$. By definition, each
arrow $a\in L\subset J\cup K$ is contained in a cycle $w$ of $W_{I}$ in which
it is the only arrow of $J\cup K$. In particular all the other arrows are in
$Q_{1}-J-K$ and are non-vanishing: then $\epsilon\tilde{\psi}_{a}\prod_{a\neq
b\in w}\psi_{b}=0$, hence $\tilde{\psi}_{a}=0$. The $|L|$ differential forms
$\delta(\partial_{a}W_{I})$ for $a\in L$ are then independent, hence:

$\displaystyle|Q_{1}-J-K|-|Q_{0}|+1-|L|\geq 0$ (A.7)

Defining

$\displaystyle R(a)=\left\\{\begin{array}[]{ll}0&\mbox{if }a\in Q_{1}-J\\\
1&\mbox{if }a\in J-L\\\ 2&\mbox{if }a\in L\cap J\end{array}\right.\quad\quad
R^{\prime}(a)=\left\\{\begin{array}[]{ll}0&\mbox{if }a\in Q_{1}-K\\\
1&\mbox{if }a\in K-L\\\ 2&\mbox{if }a\in K\cap L\end{array}\right.$ (A.14)

one finds

$\displaystyle\sum_{a\in Q_{1}}(R+R^{\prime})(a)\leq|Q_{1}|-|Q_{0}|+1$ (A.15)

We will now show that the following inequality holds for all $w\in Q_{2}$:

$\displaystyle
R_{I}\circ\partial_{2}(w)\leq(R+R^{\prime})\circ\partial_{2}(w)$ (A.16)

By disjunction of cases,

  * •

If $w$ contains no arrow of $I$, one has directly
$R_{I}\circ\partial_{2}(w)=0\leq(R+R^{\prime})\circ\partial_{2}(w)$.

  * •

Suppose that $w$ contains an arrow $a\in I$. If $w$ contains no arrow of $J$,
then $t(a)$ and $s(a)$ are connected in $Q_{J\cup I}$, hence $a\in K$. Then in
any cases, $w$ contains an arrow of $J\cup K$. If it contains a single arrow
of $J\cup K$, then by definition it is an arrow of $L$, contributing to
$R+R^{\prime}$ by $2$. If it contains at least two arrows of $J\cup K$, then
each of them contributes to $R+R^{\prime}$ by $1$. By disjunction of cases
$R_{I}\circ\partial_{2}(w)=2\leq(R+R^{\prime})\circ\partial_{2}(w)$.

The inequality (A.16) follows, and can be rewritten:

$\displaystyle(R_{I}-R^{\prime})\circ\partial_{2}(w)\leq
R\circ\partial_{2}(w)\;\quad\forall\;w\in Q_{2}$ (A.17)

We now assume that $\zeta$ is a self-stability condition. Because the positive
conserved current $\lambda=(1+\delta_{a}+|\psi_{a}|^{2})_{a\in Q_{1}}$ (with
$\delta_{a}\ll 1$) corresponding with the self-stable representation $\psi$ is
a sum of positive currents supported of cycles in $Q_{2}$ from Lemma A.1, it
follows that:

$\displaystyle(R_{I}-R^{\prime})(\lambda)\leq R(\lambda)$ (A.18)

We deduce then:

$\displaystyle 2|I|+\sum_{a\in Q_{1}}(R_{I}-R^{\prime})(a)\delta_{a}$
$\displaystyle\leq\sum_{a\in
Q_{1}}(R_{I}-R^{\prime})(a)(1+\delta_{a}+|\phi_{a}|^{2})+\sum_{a\in
Q_{1}}R^{\prime}(a)$ $\displaystyle=(R_{I}-R^{\prime})(\lambda)+\sum_{a\in
Q_{1}}R^{\prime}(a)$ $\displaystyle\leq R(\lambda)+\sum_{a\in
Q_{1}}R^{\prime}(a)=\sum_{a\in
Q_{1}}R(a)(1+\delta_{a}+|\phi_{a}|^{2})+\sum_{a\in Q_{1}}R^{\prime}(a)$
$\displaystyle=\sum_{a\in Q_{1}}(R+R^{\prime})(a)+\sum_{a\in
Q_{1}}\delta_{a}R(a)$ $\displaystyle\leq|Q_{1}|-|Q_{0}|+1+\sum_{a\in
Q_{1}}\delta_{a}R(a)$ $\displaystyle\implies 2|I|$
$\displaystyle\leq|Q_{1}|-|Q_{0}|+1$ (A.19)

Here we have used the fact that $(R_{I}-R^{\prime})(a)\geq 0$ in the first
line, the fact that $\phi_{a}=0$ when $R(a)\neq 0$ in the fourth line, (A.15)
in the fifth line and the fact that $\delta\ll 1$. in the last line. $\Box$

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