aainstitutetext: Centre de Recherches Mathématiques, Université de Montréal,
C. P. 6128, Succ. Centre Ville, Montréal, QC H3C 3J7 Canadabbinstitutetext:
Department of Mathematics and Statistics, Concordia University, 1455 de
Maisonneuve Blvd. W. Montréal, QC H3G 1M8 Canadaccinstitutetext: Institute for
Theoretical Physics, ETH Zurich, 8093, Zurich, Switzerland

# Quiver symmetries and wall-crossing invariance

Fabrizio Del Monte c Pietro Longhi<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the BPS particle spectrum of five-dimensional superconformal field
theories (SCFTs) on $\mathbb{R}^{4}\times S^{1}$ with one-dimensional Coulomb
branch, by means of their associated BPS quivers. By viewing these theories as
arising from the geometric engineering within M-theory, the quivers are
naturally associated to the corresponding local Calabi-Yau threefold. We show
that the symmetries of the quiver, descending from the symmetries of the
Calabi-Yau geometry, together with the affine root lattice structure of the
flavor charges, provide equations for the Kontsevich-Soibelman wall-crossing
invariant. We solve these equations iteratively: the pattern arising from the
solution is naturally extended to an exact conjectural expression, that we
provide for the local Hirzebruch $\mathbb{F}_{0}$, and local del Pezzo
$dP_{3}$ and $dP_{5}$ geometries. Remarkably, the BPS spectrum consists of two
copies of suitable $4d$ $\mathcal{N}=2$ spectra, augmented by Kaluza-Klein
towers.

##  1 Introduction

The BPS spectral problem in supersymmetric Quantum Field Theories with eight
supercharges is a rich subject, with connections to various branches of
theoretical and mathematical physics, such as the WKB approximation and Stokes
phenomena for Schrödinger equations (and higher order analogues), Donaldson-
Thomas invariants in enumerative geometry, and integrable systems both of
continuous and discrete types. The past decade has brought remarkable progress
on the case of four-dimensional $\mathcal{N}=2$ theories Gaiotto:2008cd ;
Gaiotto:2009hg ; Gaiotto:2012rg ; Alim:2011ae ; Alim:2011kw ; Manschot:2010qz
; Manschot:2011xc ; Manschot:2012rx ; Manschot:2013sya ; Manschot:2014fua ,
driven by seminal advances on wall-crossing phenomena Kontsevich:2008fj ;
Joyce:2008pc . More recently, various groups have started to tackle the
problem of describing the spectrum of BPS states of five-dimensional SCFTs,
arising from the compactification of M-theory on local Calabi-Yau threefolds,
where the BPS states are to be understood as branes wrapping compact cycles of
the Calabi-Yau Eager:2016yxd ; Banerjee:2018syt ; Banerjee:2019apt ;
Banerjee:2020moh ; Closset:2019juk ; Bousseau:2019ift ; Alexandrov:2018iao ;
Arguz:2021zpx ; Beaujard:2020sgs ; Mozgovoy:2020has ; Mozgovoy:2021iwz ;
Descombes:2021snc . These are the theories we will discuss in the present
paper.

Both in four and five dimensions, a key feature of BPS spectra is the wall-
crossing phenomenon. As we move along the moduli space of vacua of a theory,
the BPS spectrum is only piecewise constant, with discontinuities on
codimension-one walls. These walls are loci where stable particles can decay,
and they subdivide the moduli space into chambers with different BPS spectra
in each of them. Even though the BPS spectrum itself is not preserved on the
whole moduli space, there are still wall-crossing invariant quantities, that
are constant and chamber-independent. An important example is the Konstevich-
Soibelman (KS) invariant $\mathbb{U}$, also known as quantum monodromy or
motivic spectrum generator Kontsevich:2008fj . On the one hand, it is always
possible to derive the BPS spectrum in any chamber from $\mathbb{U}$. On the
other hand, $\mathbb{U}$ can be computed from knowledge of the BPS spectrum at
some point in moduli space. Thanks to its invariance property, it is
sufficient to compute $\mathbb{U}$ in a specific chamber where the spectrum
admits a simple description, in order to obtain the spectrum in any other
chamber. Our goal will be to compute the spectrum generator for certain five-
dimensional SCFTs, by using symmetries of the underlying Calabi-Yau
geometry.111For BPS spectra of 4d $\mathcal{N}=2$ theories of class
$\mathcal{S}$, this task has been accomplished in Gaiotto:2009hg  for theories
of type $A_{1}$ and a general approach for all class $\mathcal{S}$ theories
was developed in Longhi:2016wtv . The latter approach should generalize to 5d
$\mathcal{N}=1$ theories via exponential networks, as observed in
Banerjee:2019apt ; Banerjee:2020moh . However, this requires finding a special
point in moduli space (the Roman locus of Gabella:2017hpz ). Existence of such
a locus has been discussed recently in the literature Closset:2021lhd , it
would be interesting to pursue this approach for more general models than
those considered here.

Our main tool will be the so-called BPS quiver of the theory Douglas:1996sw ;
Douglas:2000ah ; Douglas:2000qw ; Denef:2002ru ; Alim:2011ae ; Alim:2011kw ;
Chuang:2013wt . This can be thought of as the quiver characterizing the
Supersymmetric Quantum Mechanics describing the low-energy dynamics of the BPS
states of the theory. The spectrum is then encoded by the associated
representation theory for suitable stability conditions. As it turns out, this
approach is generically difficult for quivers of the five-dimensional theories
considered in this paper, see e.g. Closset:2019juk  for a recent discussion.

An alternative way to obtain the BPS spectrum from a quiver, which has been
used to great effect in the four-dimensional case, is the so-called “mutation
method” Alim:2011ae ; Alim:2011kw . The idea behind it is that every node in
the BPS quiver represents a hypermultiplet BPS state, characterized by a ray
in the complex plane of central charges. Since the central charge of an
antiparticle is the opposite of the central charge of the corresponding
particle, the whole spectrum is encoded by the rays within a half-plane. A
choice of half-plane corresponds to a choice of quiver description of the BPS
spectrum, for a given point in the moduli space of stability conditions. If
one starts to tilt the choice of half-plane, at some point the ray of a BPS
state will exit the half-plane, and this will induce a change in the quiver
description corresponding to a mutation of the BPS quiver at the corresponding
node Berenstein:2002fi . We have however done nothing physically meaningful,
so the new quiver just corresponds to a dual description of the same physics,
so that the charges of the new nodes of the quiver must have been also in the
original spectrum. By iterating this procedure, one produces stable
hypermultiplet states in a given chamber: multiplets with higher spin content
appear as limiting vectors of infinite sequences of mutations, and different
chambers in moduli space will correspond to different relative orderings of
the phases of central charges. In chambers where all stable BPS states
correspond to hypermultiplets, the mutation method exhausts the whole BPS
spectrum. This is sometimes the case for BPS quivers of four-dimensional
$\mathcal{N}=2$ theories, however it is never the case for five-dimensional
SCFTs.

While the mutation method is not as powerful in five dimensions as it is in
four dimensions, it is nevertheless quite useful in favourable circumstances.
On the one hand, it was observed in Closset:2019juk  that certain SCFTs admit
choices of stability conditions for which the mutation method _almost_
captures the full BPS spectrum, leaving out only BPS states with central
charges aligned along a single ray in the complex plane. Moreover, it was
proposed in Bonelli:2020dcp  that the mutation method should be generalized by
exploiting symmetries of the quivers, that can be used to define discrete
integrable systems giving rise to q-Painlevé equations Goncharov:2011hp ;
Fock:2014ifa ; Bershtein:2017swf ; mizuno2020q , whose connection with five-
dimensional theories was first recognized in a different but related context
in Bershtein:2016aef ; Bonelli:2017gdk . In (Longhi:2021qvz, , Appendix) the
symmetries of the quiver were used to complete the description of BPS states
derived via mutations in Closset:2019juk , for the case of local
$\mathbb{F}_{0}$. In this paper we elaborate on these ideas in a more precise
and systematic way, and apply them to compute the motivic spectrum generator
for certain local Calabi-Yau threefolds whose BPS charges have an underlying
structure of an affine root lattice, as it happens for all local del
Pezzos222Local del Pezzos are open Calabi-Yau threefolds arising as total
space of the canonical bundle over a del Pezzo surface. The del Pezzo surface
$dP_{n}$ is a two-dimensional complex surface obtained by blowing up
$\mathbb{P}^{2}$ at $n$ generic points.. We will focus on the specific
examples local $\mathbb{F}_{0}$, and the local del Pezzos $dP_{3},dP_{5}$, the
latter example being non-toric.

Our main result will be to show that symmetries of a local CY geometry (not
necessarily toric) together with wall-crossing invariance, are sometimes
enough to constrain the BPS spectrum entirely, leading to exact expressions
for $\mathbb{U}$ and the spectrum, as well as highly nontrivial wall-crossing
identities. Two important caveats apply. First, wall-crossing invariance only
constrains BPS states whose charges are not pure-flavor, i.e. whose Dirac-
Schwinger-Zwanziger (DSZ) pairing with other charges is not identically zero.
Our method does not determine BPS states with pure-flavor charges (such as
boundstates of D0 branes), these must be obtained by other methods. Our
results should be regarded as partially conjectural, admitting the possibility
of additional BPS states with pure-flavor charges that we have not
detected.333For the case of local $\mathbb{F}_{0}$ we believe the result to be
exhaustive, based on comparisons with exponential networks in Longhi:2021qvz .
More generally, the ambiguity on pure-flavor states may be related to the fact
that our approach is independent of a choice of superpotential associated to a
quiver. It is expected that different choices of superpotentials would result
in different BPS spectra. The difference should then lie in the spectrum of
pure-flavor states, while the part of the spectrum that we compute should be
independent of such a choice. Second, we use symmetries of the quiver and
wall-crossing invariance to derive exact equations for $\mathbb{U}$, which we
solve iteratively, leading to partial descriptions of the BPS spectrum. We
then complete these descriptions by observing certain patterns in the
spectrum, namely certain periodicities, leading to a _conjectural_ expression
for $\mathbb{U}$.

With these caveats in mind, we arrive at explicit and exact (conjectural)
descriptions of the BPS spectrum. The starting point for our approach is the
observation that the so-called Cremona group $\mathrm{Cr}(dP_{n})$ of del
Pezzo surfaces $dP_{n}$ 2001CMaPh.220..165S , are realized by the extended
affine Weyl group $\widetilde{W}(E_{n}^{(1)})$ (there can be an extra finite
group factor, see Section 3.4 for details) acting on the BPS charge lattice of
the theory through sequences of mutations and permutations that leave the
quiver invariant. These transformations form the automorphism group of the
quiver $\mathrm{Aut}_{Q}$, so that

$\mathrm{Aut}_{Q}\approx\mathrm{Cr}(dP_{n})\supseteq\widetilde{W}(E_{n}^{(1)})=\left(W(E_{n})\ltimes\mathcal{T}(E_{n}^{(1)})\right)\times\mathrm{Out}(E_{n}^{(1)}),$
(1)

where we decomposed the extended affine Weyl group into affine translations
$\mathcal{T}$, simple reflections $W$, and Dynkin diagram automorphisms
$\mathrm{Out}(E_{n}^{(1)})$. Among these transformations, a special role is
taken by affine translations $\mathcal{T}(E_{n}^{(1)})$, that can be thought
of as discrete time evolutions on the BPS charges, and by the subgroup
$\Pi_{Q}$ of quiver automorphisms consisting only of permutations on the
nodes.444In the context of 4d $\mathcal{N}=2$ theories, the time evolution of
discrete integrable systems associated to BPS quivers was related to the study
of BPS spectra in Cecotti:2014zga ; Cirafici:2017iju ; Cirafici:2020qlf

We will show that it is possible to identify certain affine translations with
tiltings of the upper half-plane corresponding to an appropriate stability
condition, so that we associate to an affine translation a chamber in the
moduli space of the theory. Furthermore, we will show that the chambers
obtained these way have the “collimation”property: this means that their
spectrum is organized into towers of hypermultiplets, with all the higher-spin
BPS states aligned on the real axis of the complex plane of BPS central
charges. The motivic spectrum generator in such a chamber is factorized as

$\mathbb{U}=\mathbb{U}(\measuredangle^{+})\cdot\mathbb{U}(\mathbb{R}^{+})\cdot\mathbb{U}(\measuredangle^{-}),$
(2)

where $\mathbb{U}(\measuredangle^{\pm})$ are the contributions from BPS states
with central charges lying inside the first quadrant and fourth quadrant
respectively, while $\mathbb{U}(\mathbb{R}^{+})$ contains the contribution of
the rays lying on the positive real axis. We will show that the affine
translations fully determine $\mathbb{U}(\measuredangle^{\pm})$, while
$\mathbb{U}(\mathbb{R}^{+})$ can be fixed using the remaining transformations
of $Aut_{Q}\approx\mathrm{Cr}(dP_{n})$. This is done by noting that the
permutation symmetry group $\Pi_{Q}\subset\mathrm{Aut}_{Q}$ of the quiver
contains transformations that are automorphisms of the Dynkin diagram.
Denoting such a transformation by $\pi$, this has the effect of relating
different affine translations $T,T^{\prime}$ as

$T^{\prime}=\pi^{-1}T\pi.$ (3)

As a result, the collimation chambers $\mathcal{C}$, $\mathcal{C}^{\prime}$
constructed from the two translations $T$ and $T^{\prime}$ will be simply
related by the permutation $\pi$ of the charge vectors. This statement,
combined with wall-crossing invariance and the factorization (2), allows us to
compute $\mathbb{U}(\mathbb{R}^{+})$ recursively from

$\mathbb{U}[\mathcal{C}]=\mathbb{U}[\mathcal{C}^{\prime}]=\mathbb{U}[\pi(\mathcal{C})]\,.$
(4)

For the cases we consider, the affine translations naturally split the BPS in
quiver into two four-dimensional quivers: this was first noticed in
Bonelli:2020dcp , and is shown in Figure 1. As we will discuss shortly, this
is reflected in the full BPS spectrum.

### Main results

Solving equation (4) for suitable chambers of local
$\mathbb{F}_{0},dP_{3},dP_{5}$ we obtain the following conjectural BPS
spectra.555 As mentioned earlier, BPS states with pure-flavor charges go
undetected by the wall-crossing identity (4). These states, which include
boundstates of D0 branes, can’t be determined by wall-crossing invariance,
since such states do not participate in wall-crossing.

For local $\mathbb{F}_{0}$ we choose generators $\gamma_{i}$ of the charge
lattice corresponding to nodes of the BPS quivers shown in Figure 1(a). In a
chamber where

$Z_{\gamma_{1}}=Z_{\gamma_{3}}\,,\quad
Z_{\gamma_{2}}=Z_{\gamma_{4}}\,,\quad\arg Z_{\gamma_{1}}>\arg
Z_{\gamma_{2}}\,,\quad Z_{\gamma_{1}}+Z_{\gamma_{2}}\in\mathbb{R}^{+}\,,$ (5)

we find the following BPS spectrum

$\begin{array}[]{|c|c|}\hline\cr\gamma&\Omega(\gamma;y)\\\
\hline\cr\hline\cr\gamma_{1}+k(\gamma_{1}+\gamma_{2})&1\\\
-\gamma_{1}+(k+1)(\gamma_{1}+\gamma_{2})&1\\\
\gamma_{3}+k(\gamma_{3}+\gamma_{4})&1\\\
-\gamma_{3}+(k+1)(\gamma_{3}+\gamma_{4})&1\\\
\hline\cr\gamma_{1}+\gamma_{2}+k\gamma_{D0}&y+y^{-1}\\\
-\gamma_{1}-\gamma_{2}+(k+1)\gamma_{D0}&y+y^{-1}\\\
(k+1)\gamma_{D0}&y^{3}+2y+y^{-1}\\\ \hline\cr\end{array}$ (6)

where $\gamma_{D0}=\sum_{i=1}^{4}\gamma_{i}$. The spectrum also includes the
respective antiparticles obtained by sending $\gamma\to-\gamma$. Note that
this spectrum consists of two copies of the weakly coupled spectrum of pure
$SU(2)$ SYM in 4d Alim:2011kw , with the extra KK towers in lower part of the
table. We also observe that the spectrum is organized into towers over
$\gamma_{1},\gamma_{3}$ and $\gamma_{1}+\gamma_{2},\gamma_{D0}$ with steps
$\gamma_{1}+\gamma_{2},\gamma_{3}+\gamma_{4}$ and $\gamma_{D0}$. This
structure is strongly reminiscent of the “peacock patterns” observed in
Gu:2021ize , where the possibility of a relation to BPS/DT invariants of local
CY threefolds is also contemplated. (Also see Figures 4, 7).

For local $dP_{3}$ we choose generators $\gamma_{i}$ of the charge lattice
corresponding to nodes of the BPS quivers shown in Figure 1(b). In a chamber
where

$\begin{split}&Z_{\gamma_{1}}=Z_{\gamma_{4}}\,,\quad
Z_{\gamma_{2}}=Z_{\gamma_{5}}\,,\quad Z_{\gamma_{3}}=Z_{\gamma_{6}}\,,\\\
&\arg Z_{\gamma_{1}}>\arg Z_{\gamma_{2}}>\arg Z_{\gamma_{3}}\,,\quad
Z_{\gamma_{1}}+Z_{\gamma_{3}}=Z_{\gamma_{2}}\in\mathbb{R}^{+}\,.\end{split}$
(7)

we find the following BPS spectrum

$\begin{array}[]{|c|c|}\hline\cr\gamma&\Omega(\gamma;y)\\\
\hline\cr\hline\cr\gamma_{r}+k(\gamma_{1}+\gamma_{2}+\gamma_{3})&1\\\
-\gamma_{r}+(k+1)(\gamma_{1}+\gamma_{2}+\gamma_{3})&1\\\
\gamma_{s}+k(\gamma_{4}+\gamma_{5}+\gamma_{6})&1\\\
-\gamma_{s}+(k+1)(\gamma_{4}+\gamma_{5}+\gamma_{6})&1\\\
\hline\cr\gamma_{t}+k\gamma_{D0}&1\\\ -\gamma_{t}+(k+1)\gamma_{D0}&1\\\
\gamma_{a}+\gamma_{b}+k\gamma_{D0}&1\\\
-\gamma_{a}-\gamma_{b}+(k+1)\gamma_{D0}&1\\\
\gamma_{1}+\gamma_{2}+\gamma_{3}+k\gamma_{D0}&y+y^{-1}\\\
-\gamma_{1}-\gamma_{2}-\gamma_{3}+(k+1)\gamma_{D0}&y+y^{-1}\\\
(k+1)\gamma_{D0}&y^{3}+4y+y^{-1}\\\ \hline\cr\end{array}$ (8)

with $k\geq 0$ and

$r\in\\{1,3\\}\qquad s\in\\{4,6\\}\qquad
t\in\\{2,5\\}\qquad(a,b)\in\\{(1,3),(4,6)\\}\,.$ (9)

where $\gamma_{D0}=\sum_{i=1}^{6}\gamma_{i}$, and again the spectrum also
includes the respective antiparticles obtained by sending $\gamma\to-\gamma$.
This spectrum consists of two copies of the weakly coupled spectrum of $SU(2)$
SQCD with one fundamental hypermultiplet in 4d Alim:2011kw , with the extra KK
towers in lower part of the table. This is quite striking, since the low-
energy gauge theory phase of this SCFT is five-dimensional $N_{f}=2$ $SU(2)$
gauge theory. As for local $\mathbb{F}_{0}$, we observe that the spectrum is
organized into towers, over $\gamma_{r},\gamma_{s}$ and
$\gamma_{t},\gamma_{a}+\gamma_{b},\gamma_{1}+\gamma_{2}+\gamma_{3},\gamma_{D0}$
with steps $\gamma_{1}+\gamma_{2}+\gamma_{3},\gamma_{4}+\gamma_{5}+\gamma_{6}$
and $\gamma_{D0}$. This structure is reminiscent of the “peacock patterns”
observed in Gu:2021ize .

For local $dP_{5}$ we choose generators $\gamma_{i}$ of the charge lattice
corresponding to nodes of the BPS quivers shown in Figure 1(c). In a chamber
where

$\begin{split}&Z_{\gamma_{2}}=Z_{\gamma_{6}}=c\cdot Z_{\gamma_{1}}=c\cdot
Z_{\gamma_{5}}\,,\quad Z_{\gamma_{4}}=Z_{\gamma_{8}}=c\cdot
Z_{\gamma_{3}}=c\cdot Z_{\gamma_{7}}\,,\\\
&Z_{\gamma_{1}}=\overline{Z}_{\gamma_{3}}\,,\quad\arg Z_{\gamma_{1}}>\arg
Z_{\gamma_{3}}\,,\end{split}$ (10)

with an arbitrary real constant $c>1$, needed to ensure that $Z_{\alpha_{i}}$
belong to the positive half-plane, we find the following BPS spectrum

$\begin{array}[]{|c|c|}\hline\cr\gamma&\Omega(\gamma;y)\\\
\hline\cr\hline\cr\gamma_{r}+k(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})&1\\\
-\gamma_{r}+(k+1)(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})&1\\\
\gamma_{s}+k(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})&1\\\
-\gamma_{s}+(k+1)(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})&1\\\
\hline\cr\gamma_{a}+\gamma_{b}+k\gamma_{D0}&1\\\
-\gamma_{a}-\gamma_{b}+(k+1)\gamma_{D0}&1\\\
\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}+k\gamma_{D0}&y+y^{-1}\\\
-\gamma_{1}-\gamma_{2}-\gamma_{3}-\gamma_{4}+(k+1)\gamma_{D0}&y+y^{-1}\\\
(k+1)\gamma_{D0}&y^{3}+6y+y^{-1}\\\ \hline\cr\end{array}$ (11)

with $k\geq 0$ and

$\begin{split}&r\in\\{1,2,3,4\\},\quad s\in\\{5,6,7,8\\}\\\
&(a,b)\in\\{(1,3),(1,4),(2,3),(2,4),(5,7),(5,8),(6,7),(6,8)\\}\\\ \end{split}$
(12)

where $\gamma_{D0}=\sum_{i=1}^{8}\gamma_{i}$, and again the spectrum also
includes the respective antiparticles obtained by sending $\gamma\to-\gamma$.
This spectrum consists of two copies of the weakly coupled spectrum of $SU(2)$
SQCD with two fundamental hypermultiplets in 4d Alim:2011kw , with the extra
KK towers in lower part of the table. Again, this differs from what could be
expected from the low-energy gauge theory phase of this SCFT, which is five-
dimensional $N_{f}=4$ $SU(2)$ gauge theory. As for local $\mathbb{F}_{0}$ and
$dP_{3}$, we observe once again that the spectrum is organized into towers,
over $\gamma_{r},\gamma_{s}$ and
$\gamma_{t},\gamma_{a}+\gamma_{b},\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4},\gamma_{D0}$
with steps
$\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4},\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8}$
and $\gamma_{D0}$, in a way that is reminiscent of “peacock patterns”
Gu:2021ize .

A byproduct of these results are new and highly nontrivial wall-crossing
identities. For example, for local $\mathbb{F}_{0}$ we find

$\begin{split}&\prod_{k\geq
0}^{\nearrow}\Phi(X_{\gamma_{1}+k(\gamma_{1}+\gamma_{2})})\Phi(X_{\gamma_{3}+k(\gamma_{3}+\gamma_{4})})\\\
&\qquad\times\prod_{s=\pm 1}\prod_{k\geq
0}\Phi((-y)^{s}X_{\gamma_{1}+\gamma_{2}+k\gamma_{D0}})^{-1}\cdot\Phi((-y)^{s}X_{\gamma_{3}+\gamma_{4}+k\gamma_{D0}})^{-1}\\\
&\qquad\qquad\times\prod_{k\geq
0}^{\searrow}\Phi(X_{\gamma_{2}+k(\gamma_{1}+\gamma_{2})})\Phi(X_{\gamma_{4}+k(\gamma_{3}+\gamma_{4})})\\\
=&\prod_{k\geq
0}^{\nearrow}\Phi(X_{\gamma_{2}+k(\gamma_{2}+\gamma_{3})})\Phi(X_{\gamma_{4}+k(\gamma_{4}+\gamma_{1})})\\\
&\qquad\times\prod_{s=\pm 1}\prod_{k\geq
0}\Phi((-y)^{s}X_{\gamma_{4}+\gamma_{1}+k\gamma_{D0}})^{-1}\cdot\Phi((-y)^{s}X_{\gamma_{2}+\gamma_{3}+k\gamma_{D0}})^{-1}\\\
&\qquad\qquad\times\prod_{k\geq
0}^{\searrow}\Phi(X_{\gamma_{3}+k(\gamma_{2}+\gamma_{3})})\Phi(X_{\gamma_{1}+k(\gamma_{4}+\gamma_{1})}).\\\
\end{split}$ (13)

Here

$\displaystyle\Phi(x)=(-yx;y^{2})_{\infty}^{-1},$
$\displaystyle(x;y)_{\infty}\equiv\prod_{n=1}^{\infty}\left(1-y^{n}x\right),$
(14)

and $\gamma_{1}\dots,\gamma_{4}$ are generators of the charge lattice
$\Gamma\simeq\mathbb{Z}^{4}$, $X_{\gamma}$ are quantum-torus algebra variables
with product twisted by the DSZ pairing
$X_{\gamma}X_{\gamma^{\prime}}=y^{\langle\gamma,\gamma^{\prime}\rangle}X_{\gamma+\gamma^{\prime}}$,
and $\nearrow$ ($\searrow$) denotes increasing (decreasing) values of $k$ from
left to right. The analogues of this identity for the case of $dP_{3}$,
$dP_{5}$ are contained in equations (149) and (204) respectively.

(a) Four-dimensional subquivers for the flow $T_{1}$ of $\mathbb{F}_{0}$. The
towers (73) are two copies of the dyons for 4d $SU(2)$ pure SYM, with
additional $D0$ towers over the 4d vector multiplets.

(b) Four-dimensional subquivers for the flow $T_{1}$ of $dP_{3}$. The towers
(LABEL:eq:T1T2-dP3-charges) are two copies of the dyons for 4d $SU(2)$ SYM
with $N_{f}=1$, with the addition of $D0$ towers over the 4d quarks and vector
multiplets.

(c) Four-dimensional subquivers for the flow $T_{5}$ of $dP_{5}$. The towers
(164) are two copies of the dyons for 4d $SU(2)$ SYM with $N_{f}=2$, with the
addition of $D0$ towers over the 4d quarks and vector multiplets.

Figure 1:

### Contents of the paper

This paper is structured as follows. We introduce the necessary background in
Section 2; after recalling the notion of BPS quiver and mutation, we define
what is meant by a collimation chamber, giving the example of pure $SU(2)$
super Yang-Mills and $\mathcal{N}=2^{*}$ theory in 4d. After this, we
introduce the KS invariant $\mathbb{U}$, and discuss how it can be constrained
by using permutation symmetries of the quiver. In section 3 we review how the
lattice of BPS charges can be identified with the even-dimensional cohomology
of the local Calabi-Yau, and for rank-1 SCFTs the sublattice of flavor charges
is organized as an affine root lattice of type $E_{n}^{(1)}$. We then discuss
how to extend to the whole BPS charge lattice of the local Calabi-Yau the
action of the Cremona isometries of the del Pezzo surface, which are famously
related to an action of the affine Weyl group on the $E_{n}^{(1)}$ root
lattice. We identify this isometry group with the group of self-dualities of
the five-dimensional theory, and formulate in section 3.4 a set of criteria to
associate a collimation chamber to an affine Weyl translation. In section 4 we
test our ideas in three specific examples, that of local $\mathbb{F}_{0}$,
local $dP_{3}$ and local $dP_{5}$, for all of which we find appropriate
collimation chambers and conjectural expressions for the wall-crossing
invariants, leading to novel wall-crossing identities.

## Acknowledgements

We thank Sibasish Banerjee, Giulio Bonelli, Michele Cirafici, Cyril Closset,
Michele Del Zotto, Alba Grassi, Boris Pioline, Mauricio Romo and Alessandro
Tanzini for discussions and correspondence. The work of FDM is supported by
the NSERC through a CRM-ISM postdoctoral fellowship. The work of PL is
supported by NCCR SwissMAP, funded by the Swiss National Science Foundation.

##  2 Symmetry constraints on wall-crossing invariants

###  2.1 Quivers, stability conditions and BPS states

The notion of moduli space of stability conditions is a fundamental piece of
data in the description of the spectrum of BPS states, in the context of SUSY
gauge theories, string theory, and mathematics. In this paper we focus mainly
on BPS states described by quivers, for which a notion of stability condition
is provided by King’s $\theta$-stability King1994MODULIOR .666All quivers
considered in this paper arise in the study of D-branes on local Calabi-Yau
threefolds. See e.g. 2002math…..12237B ; 2016arXiv160300416B ; Douglas:2002fj
; 2000math……9209D ; Douglas:2000gi  for further work on the relation to
$\Pi$-stability and Bridgeland stability.

Recall that a quiver $Q$ is an oriented graph, consisting of a set of $m$
vertices $Q_{0}$ and a set of oriented edges $Q_{1}$. In what follows we will
restrict to quivers without loops (arrows from a node to itself) or 2-cycles
(i.e. where a pair of nodes is connected by at least one arrow in each
direction). In this case the structure of $(Q_{0},Q_{1})$ may be encoded by an
antisymmetric adjacency matrix $B_{ij}=-B_{ji}$ whose non-negative entries are
the number of arrows from node $j$ to node $i$.777This convention follows
Alim:2011ae  and other papers in the physics literature. It is opposite to the
convention often used in mathematics, where $B_{ij}>0$ would encode the number
of arrows $i\to j$. A quiver representation of dimension
$\vec{d}=(d_{1},\dots,d_{m})$ is a collection of vector spaces with dimension
$\dim V_{i}=d_{i}$, and linear maps $\phi_{a}\in{\rm Hom}(V_{i},V_{j})$ for
each arrow $a:i\to j$. The stability data for the representation
$R_{\vec{d}}=(\\{V_{i}\\},\\{\phi_{a}\\})$ is a collection of reals
$\theta_{i}$ for each $i\in Q_{0}$. The role of $\theta_{i}$ in physics is
that of Fayet-Iliopoulous couplings for an $\mathcal{N}=4$ quantum mechanics
associated to $R_{\vec{d}}$ in the Higgs branch regime, see Douglas:1996sw ;
Denef:2002ru ; Alim:2011ae  for details. This quantum mechanics descibes the
worldvolume dynamics of a collection of fractional branes with charges
$\gamma_{i}$ for $i=1,\dots,m$. Let $\Gamma=\oplus_{i}\gamma_{i}\mathbb{Z}$ be
the charge lattice generated by nodes of the quiver, and recall the central
charge is a linear function on $\Gamma$

$Z\in{\rm Hom}(\Gamma,\mathbb{C})\,.$ (15)

In a regime when central charges are nearly aligned, the FI couplings are
related to the central charges as

$\theta_{i}=|Z_{\gamma_{i}}|\left(\arg Z_{\gamma_{i}}-\arg Z_{\gamma}\right)$
(16)

where $\gamma=\sum_{i}d_{i}\gamma_{i}$ is the total charge of $R_{\vec{d}}$,
and $Z_{\gamma}\in\mathbb{C}$ is the central charge of a BPS state with charge
$\gamma$. The Dirac-Schwinger-Zwanziger pairing of basic charges ie encoded by
the quiver adjacency matrix

$\langle\gamma_{i},\gamma_{j}\rangle=B_{ij}\,.$ (17)

Thanks to linearity of central charge as a function of $\gamma$, the relation
(16) allows to trade $\theta_{i}$ for $Z_{\gamma_{i}}$ in discussing stability
conditions.

The values of FI couplings $\theta_{i}$, or equivalently central charges
$Z_{\gamma_{i}}$, determine whether a representation $R_{\vec{d}}$ is
(semi-)stable or not. In the case when (semi-)stable representations with
given dimension $\vec{d}$ exist, we denote by $\Omega(\gamma,y)$ the shifted
Poincaré polynomial of the moduli space of such representations modulo gauge
equivalence, where $\gamma=\sum_{i}d_{i}\gamma_{i}$.888The Poincaré polynomial
is defined with respect to compactly supported de Rham cohomology of the
moduli space $\mathcal{M}_{\vec{d}}(\vec{\theta})$ of (semi-)stable quiver
representations. This is the appropriate definition for counting BPS states in
string theory, but differs from the one that would be appropriate to discuss
BPS states in the geometrically engineered gauge theory. For the latter, the
appropriate notion is $L^{2}$ cohomology, which has the property of respecting
Poincaré symmetry. See Yi:1997eg ; Duan:2020qjy ; Mozgovoy:2020has  for
details. In physics $\Omega(\gamma,y)$ coincides with (an uplift of) the
Protected Spin Character introduced in Gaiotto:2010be , which counts BPS
states with different spin. A convenient pictorial description of the BPS
spectrum for a given choice of stability data is provided by _ray diagrams_ ,
where each BPS state is represented by a vector in the $Z$-plane, see Figure
4.

#### Mutations.

Sometimes it is useful to consider how the quiver description of BPS states
changes with different choices of positive half-plane (corresponding to ${\rm
Re}Z>0$ in Figure 4). If we keep the stability condition fixed, the BPS rays
do not move. However the ray diagram changes whenever the choice of half-plane
is tilted so that one of the rays exits from one side or the other (the CPT-
conjugate ray with $Z_{\gamma}\to-Z_{\gamma}$ enters on the other side). The
quiver description of the BPS spectrum also changes, by a left or right
mutation of $Q$ Berenstein:2002fi ; Alim:2011ae ; Alim:2011kw .

Note that the BPS rays which are closest to either half-boundary of the
positive $Z$ half-plane always correspond to central charges of some of the
quiver nodes. When the half-plane is tilted clockwise, so that a single ray
exists, the quiver changes by a mutation on the node corresponding to the
exiting ray. A mutation $\mu_{k}$, acting on the node $k$ of $Q$, produces a
new BPS quiver $Q^{\prime}$ with adjacency matrix $B^{\prime}_{ij}$ defined as
follows

$B^{\prime}_{ij}=\begin{cases}-B_{ij},&i=k\text{ or }j=k,\\\
B_{ij}+\frac{B_{ik}|B_{kj}|+B_{kj}|B_{ik}|}{2},\end{cases}$ (18)

At the same time, the charge vectors $\\{\gamma_{i}\\}$ labeling nodes of $Q$
also change to new labels $\\{\gamma^{\prime}_{i}\\}$ for the nodes of
$Q^{\prime}$ as follows

$\mu_{k}(\gamma_{j})=\begin{cases}-\gamma_{j},&j=k,\\\
\gamma_{j}+[B_{jk}]_{+}\,\gamma_{k},&\text{otherwise}.\end{cases}$ (19)

If the half-plane is instead rotated counterclockwise, the mutation takes the
form

$\widetilde{\mu_{k}}(\gamma_{j})=\begin{cases}-\gamma_{j},&j=k,\\\
\gamma_{j}+[-B_{jk}]_{+}\,\gamma_{k},&\text{otherwise}.\end{cases}$ (20)

The simultaneous changes in $B_{ij}$ and labels of nodes are compatible, they
preserve the property (17).

###  2.2 Collimation chambers

In this work we focus on a specific class of stability conditions, which we
name _collimation chambers_. The essential property of collimation chambers is
that central charges of BPS states only accumulate along
$\mathbb{R}^{\pm}e^{i\vartheta_{0}}$, if at all. For quivers of toric CY
threefolds the real ray $\vartheta_{0}=0$ is canonically an accumulation ray,
since the BPS spectrum always features towers of D0 branes with central charge
$Z_{nD0}=n\,2\pi/R$ for $n\in\mathbb{Z}\setminus\\{0\\}$, where $R$ is the
radius of the M-theory circle. An equivalent definition of collimation
chambers, at least in all examples we considered, is that these are stability
conditions for which the only BPS states with $Z_{\gamma}\notin\mathbb{R}$ are
hypermultiplets, i.e. their BPS index is $\Omega(\gamma)=1$.

The most basic example of a collimation chamber with at least one accumulation
ray appears for the Kronecker quiver $K_{2}$ with
$\langle\gamma_{1},\gamma_{2}\rangle=2$ shown in Figure 2. For stability
conditions within the chamber

$\mathcal{C}_{1}[K_{2}]:\quad\arg Z_{\gamma_{2}}>\arg Z_{\gamma_{1}}$ (21)

the ray diagram is the one of Figure 4(a). This is the celebrated weak-
coupling spectrum of 4d $\mathcal{N}=2$ super Yang-Mills theory obtained by
Seiberg and Witten Seiberg:1994rs .

Figure 2: $K_{2}$: BPS quiver for 4d pure $SU(2)$ super Yang-Mills.

Another example of collimation chamber arises for the Markov quiver $M_{2}$
with $\langle\gamma_{i},\gamma_{i-1}\rangle=2$ shown in Figure 3. The ray
diagram for this chamber is shown in Figure 4(b), and corresponds to the
chamber

$\mathcal{C}_{1}[M_{2}]:\quad\arg Z_{\gamma_{1}}>\arg
Z_{\gamma_{3}}=\arg(Z_{\gamma_{1}+\gamma_{2}})>\arg Z_{\gamma_{2}}\,.$ (22)

This chamber was studied in unpublished joint work by Greg Moore and the
second-named author, and was used to obtain a closed-form description of the
BPS spectrum of $\mathcal{N}=2^{*}$ super Yang-Mills theory Longhi:2015ivt ;
Longhi:2016wtv . The $M_{2}$ quiver has two more collimation chambers
$\mathcal{C}_{2}[M_{2}],\mathcal{C}_{3}[M_{2}]$ related by cyclic permutations
of the central charges in (22).

Figure 3: $M_{2}$: BPS quiver for the $SU(2)$ $\mathcal{N}=2^{*}$ theory

The Markov example is especially interesting, because it turns out that the
spectrum of $\mathcal{N}=2^{*}$ theory is generically very intricate,
featuring infinitely many accumulation rays. The only known exception is
precisely the collimation chamber of Figure 4(b) (or its $\mathbb{Z}_{3}$
images $\mathcal{C}_{2},\mathcal{C}_{3}$), where the BPS spectrum admits a
simple exact description. In fact this correlation between collimation
chambers and nice BPS spectra turns out to generalize to (at least certain)
quivers of toric Calabi-Yau threefolds, suggesting that closed-form exact
descriptions of the spectrum can be obtained by studying such chambers. In the
rest of this paper we explore and realize this idea for a few selected
examples.

(a) Collimation chamber $\mathcal{C}_{1}[K_{2}]$.

(b) Collimation chamber $\mathcal{C}_{1}[M_{2}]$.

Figure 4: Ray diagrams of collimation chambers.

###  2.3 The wall-crossing invariant $\mathbb{U}$

The moduli space of stability conditions has a wall-and-chamber structure,
with different BPS spectra in different chambers. Jumps of the spectrum are
described by the wall-crossing formulae Kontsevich:2008fj ; Joyce:2008pc . We
will adopt here the framework of Kontsevich and Soibelman, whose formula can
be stated as a conservation law for a certain invariant $\mathbb{U}$ built
from the BPS spectrum.

Assume ${\rm Re}Z_{\gamma_{i}}>0$, $\forall i\in Q_{0}$, we shall refer to BPS
particles as those BPS states whose central charge lies in the right-half of
the $Z$-plane, and anti-particles as those in the other half999We make this
choice of half-plane instead of the more traditional upper half-plane so as to
have the D0-brane central charge, which is real and positive, at the center of
the domain.. Let $\Gamma=\oplus_{i}\gamma_{i}\,\mathbb{Z}$ denote the lattice
of charges generated by quiver nodes, and consider the quantum torus algebra
of formal variables $X_{\gamma}$ obeying

$X_{\gamma}X_{\gamma^{\prime}}=y^{\langle\gamma,\gamma^{\prime}\rangle}X_{\gamma+\gamma^{\prime}}\,.$
(23)

The Kontsevich-Soilbeman invariant of wall-crossing is a product of quantum
dilogarithms

$\mathbb{U}=\prod_{\gamma\in\Gamma_{+}}^{\curvearrowleft}\prod_{m\in\mathbb{Z}}\Phi((-y)^{m}X_{\gamma})^{a_{m}(\gamma)}$
(24)

where

  * •

$\Gamma_{+}=\\{\gamma\in\Gamma,{{\rm Re}Z_{\gamma}}>0\\}$

  * •

the ordering $\curvearrowleft$ is by increasing $\arg(Z_{\gamma})$ to the left

  * •

$\Omega(\gamma,y)=\sum_{m}(-y)^{m}a_{m}(\gamma)$ is the Protected Spin
Character Gaiotto:2010be , which recovers the BPS index for $y\to-1$.

  * •

$\Phi(x)=(-yx;y^{2})_{\infty}^{-1}=\prod_{s\geq 0}(1+y^{2s+1}x)^{-1}$

  * •

$m\in\mathbb{Z}$ is the eigenvalue of $2J_{3}$, a Cartan generator of the
rotation group in three dimensions101010A standard physical realization of BPS
quivers we study below in Section (4) is M-theory on a CY threefold $X$ times
$S^{1}\times\mathbb{R}^{4}$. In this setup the quiver describes the spectrum
of M2 and M5 branes wrapping along $\mathbb{R}\subset\mathbb{R}^{4}$, breaking
$\mathfrak{so}(4)$ to $\mathfrak{so}(3)$.

When the stability condition belongs to a collimation chamber, the wall-
crossing invariant takes the form

$\mathbb{U}=\mathbb{U}(\measuredangle^{+})\cdot\mathbb{U}(\mathbb{R}^{+})\cdot\mathbb{U}(\measuredangle^{-})$
(25)

where $\measuredangle^{+}$ and $\measuredangle^{-}$ correspond respectively to
angular sectors $0<\arg Z<\pi/2$ and $-\pi/2\leq\arg Z<0$. The terms
$\mathbb{U}(\measuredangle^{\pm})$ contain only (possibly countably infinitely
many) hypermultiplet factors $\Phi(X_{\gamma})$. 111111This is based on the
alternative characterization of collimation chambers mentioned in Section 2.2.
A consequence of this, is that in collimations chambers it is straightforward
to compute $\mathbb{U}(\measuredangle^{\pm})$ by standard techniques based on
quiver mutations Alim:2011ae ; Alim:2011kw ; Closset:2019juk . The only
nontrivial task is the computation of $\mathbb{U}(\mathbb{R}^{+})$.

######  Remark 1.

For the BPS spectrum to be well-defined in a collimation chamber, one must
ensure that the marginal stability condition is not verified. This implies
that for any pair of $\gamma,\gamma^{\prime}\in\Gamma$ such that
$Z_{\gamma}//Z_{\gamma^{\prime}}$ (either on the real ray, or away from it)
their pairing vanishes $\langle\gamma,\gamma^{\prime}\rangle=0$. However, this
requirement could also be slightly relaxed. For example, if one is willing to
forgo a factorization of $\mathbb{U}(\mathbb{R}^{+})$ into quantum
dilogarithms, then one may still make sense of this factor as a formal series
in $X_{\gamma}$ even if the states with real central charge are at marginal
stability with respect to each other.

###  2.4 Quiver permutation symmetry

Before proceeding with the task of determining $\mathbb{U}(\mathbb{R}^{+})$,
we take a brief technical detour to introduce the notion of permutation
symmetries of a quiver.

Let $\Pi_{Q}$ be the group of permutations acting on the labels of $Q_{0}$
which preserves the structure of the quiver. This is the subgroup of
$S_{|Q_{0}|}$ of elements

$\pi:\gamma_{i}\mapsto\gamma_{i}^{\prime}=\gamma_{\pi(i)}$ (26)

such that the quiver $(Q,\\{\gamma^{\prime}_{i}\\})$ is identical to the
original one $(Q,\\{\gamma_{i}\\})$. For example the $M_{2}$ quiver of Figure
3 has a permutation symmetry $\Pi_{M_{2}}\simeq\mathbb{Z}_{3}$, acting as
$\gamma_{i}\mapsto\gamma_{i+1\,{\rm mod}\,3}$. An example of a quiver with
trivial symmetry group is the Kronecker quiver (2).

Later we will encounter a more general definition of quiver symmetries,
involving mutations. The group of $\Pi_{Q}$ is a subgroup of the full
automorphism group ${\rm Aut}_{Q}$ of the quiver, whose definition will be
introduced in section 3.3). For later convenience, we highlight a few
important features of mutations and permutations. Here we simply state these
as (partly obvious) facts, their scope will become more clear with examples
below.

  * •

A mutation cannot change the stability condition. A permutation can. The
action of $\pi\in\Pi_{Q}$ restricts to permutations of collimation chambers
$\mathcal{C}_{\alpha}\mapsto\mathcal{C}_{\pi(\alpha)}$. This action is
generically not faithful. For an example we refer to Observation 1 for local
$\mathbb{F}_{0}$.

  * •

Stability conditions related by a permutation are always connected by a
continuous path along which no BPS ray exits the positive half-plane of
central charges (in our case, this is the half-plane ${\rm Re}Z_{\gamma}\geq
0$). For an example see Observation 2 for local $\mathbb{F}_{0}$.

  * •

A mutation changes the choice of positive half-plane of central charges by
tilting it (until the BPS ray of the mutant note exists the half-plane
Alim:2011ae ). A permutation does not.

  * •

The group $\Pi_{Q}$ isn’t preserved by mutations, because mutations
generically change the structure of a quiver, hence its automorphism group. We
will work with a fixed representative $Q$ in the mutation class $[Q]$, and
consider its symmetry group $\Pi_{Q}$.

###  2.5 Constraining $\mathbb{U}$

The outstanding task of computing $\mathbb{U}(\mathbb{R}^{+})$ in a
collimation chamber can be achieved by taking advantage of the permutation
symmetry of a quiver $Q$, whenever $\Pi_{Q}$ is nontrivial. A prototypical
example is the $\mathbb{Z}_{4}$ symmetry of the BPS quiver of local
$\mathbb{F}_{0}$ in Figure 6. In this case, it was shown in Longhi:2021qvz
that constraints from this symmetry are strong enough to determine
$\mathbb{U}(\mathbb{R}^{+})$.121212More precisely, up to factors involving
pure-flavor states, such as $D0$ branes in this case. The reason is that wall-
crossing invariance doesn’t constrain these spectrum of purer-flavor states.
Here we review the basic idea, and explain how it its generalizes to other
quivers with nontrivial symmetries.

When a labeled quiver $(Q,\\{\gamma_{i}\\})$ has a nontrivial permutation
symmetry $\Pi_{Q}$, this imposes constraints on the wall-crossing invariant
$\mathbb{U}$. To illustrate this, let us write the latter as a formal series
in $X_{\gamma}$

$\mathbb{U}=\sum_{\gamma\in\Gamma_{+}}u_{\gamma}X_{\gamma}$ (27)

where coefficients $u_{\gamma}$ are Laurent polynomials in $y$ with integer
coefficients. Acting with $\pi$ induces a permutation of the $\gamma_{i}$,
which induces a relabeling of formal variables $X_{\gamma}$

$\mathbb{U}\quad\mathop{\longrightarrow}^{\pi}\quad\mathbb{U}^{\prime}=\sum_{\gamma}u_{\gamma}X_{\pi(\gamma)}=\sum_{\gamma}u_{\pi^{-1}(\gamma)}X_{\gamma}=\sum_{\gamma}u^{\prime}_{\gamma}X_{\gamma}$
(28)

where we used the fact that
$\gamma\in\Gamma_{+}\leftrightarrow\pi(\gamma)\in\Gamma_{+}$.131313Note that
$\Gamma_{+}=\\{\gamma\in\Gamma|\ \gamma=\sum_{i=1}^{Q_{0}}n_{i}\gamma_{i},\
n_{i}\geq 0\\}$. By definition $\Gamma_{+}$ is the subset of charges with
${\rm Re}Z_{\gamma}>0$, which is the quiver half-plane. It follows that any
such $\gamma$ is a non-negative integer linear combination of the charges
$\gamma_{i}$ associated to the vertices of $Q$.

Next, recall from Section 2.4 that the symmetry $\pi$ may change the stability
condition by permuting the central charges associated to the nodes, but the
new configuration of central charges is connected to the old one by a smooth
motion in the moduli space of $\\{Z_{\gamma_{i}}\\}\ \simeq\mathbb{C}^{Q_{0}}$
throughout which no rays exit or enter the half-plane ${\rm ReZ}>0$. This
implies that $\mathbb{U}^{\prime}=\mathbb{U}$ by wall-crossing invariance, and
leads to the non-trivial constraint

$u_{\pi(\gamma)}=u_{\gamma}\,.$ (29)

Taken alone, these constraints are generically too weak to fully determine
$\mathbb{U}$ itself. But in a collimation chamber we can factorize
$\mathbb{U}$ as in (25) into $\mathbb{U}(\measuredangle^{\pm})$ and
$\mathbb{U}(\mathbb{R}^{+})$. As mentioned previously, it is often easy to
compute $\mathbb{U}(\measuredangle^{\pm})$ by standard techniques such as the
mutation algorithm. With this input, (29) becomes powerful enough to determine
$\mathbb{U}(\mathbb{R}^{+})$ entirely, or almost entirely (details below). If
we further require $\mathbb{U}(\mathbb{R}^{+})$ to factorize into quantum
dilogarithms, we typically find that symmetries determine the exponents
$\Omega(\gamma;y)$ uniquely.

Concretely, we shall often adopt a formal series expression for
$\mathbb{U}(\mathbb{R}^{+})$

$\mathbb{U}(\mathbb{R}^{+})=\sum_{\gamma\in\Gamma_{\mathbb{R}^{+}}}c_{\gamma}\,X_{\gamma}$
(30)

where $\Gamma_{\mathbb{R}^{+}}=\\{\gamma\in\Gamma|\
Z_{\gamma}\in\mathbb{R}^{+}\\}$ and $c_{\gamma}\in\mathbb{Z}[y,y^{-1}]$. We
then compute the coefficients $u_{\gamma}(y)$ of $\mathbb{U}$ as a function of
$c_{\gamma}(y)$ through (25), and finally impose the symmetry equations (29)
on the $u_{\gamma}(y)$ to fix the coefficients $c_{\gamma}(y)$ of
$\mathbb{U}(\mathbb{R}^{+})$. We will illustrate this in Section 4 with
several examples.

##  3 Affine root lattices and stability conditions

In the previous section we discussed collimation chambers
$\mathcal{C}_{\alpha}$, and explained how they can be used in conjunction with
quiver permutation symmetries $\Pi_{Q}$ to obtain equations for $\mathbb{U}$
which determine the invariant. Here we discuss in more detail the larger
automorphism group $\mathrm{Aut}_{Q}$ of a quiver $Q$, of which permutations
are generically a subgroup

$\Pi_{Q}\subset\mathrm{Aut}_{Q}\,.$ (31)

In the cases of interests to this paper, $\mathrm{Aut}_{Q}$ contains a
subgroup isomorphic to the extended affine Weyl group141414This is defined as
the semidirect product of the affine Weyl group of the Lie algebra and its
outer automoprhisms. for the affine Lie algebra associated to $Q$
(Bershtein:2017swf, , Theorem 3.1) (also see mizuno2020q  for a comprehensive
discussion). The affine root lattice can be naturally embedded within the
charge lattice $\Gamma$ Bonelli:2020dcp . One of the main points of this
section is to illustrate how the action of affine Weyl translations on
$\Gamma$ can be used, at least in some cases, to define stability conditions
corresponding to collimation chambers.

###  3.1 Charge lattices

Affine root lattices naturally describe the flavor symmetry group of five-
dimensional SCFTs, from the geometry of the Calabi-Yau threefold from which
the field theory is obtained in M-theory compactification Seiberg:1996bd ;
Intriligator:1997pq . In the case of rank-1 theories, the relevant geometry
consists of local del Pezzos $X=K_{S}$ with $S=dP_{n}$ $n=1,\dots,8$, together
with local $S=\mathbb{F}_{0}$151515More precisely, the local del Pezzo
corresponds to a gauge theory phase, while its blowdown to the UV
superconformal fixed point.. These are local Calabi-Yau threefolds defined as
total space of the canonical line bundle of a suface $S$, obtained by blowing
up $\mathbb{P}^{2}$ at $n$ points or $\mathbb{P}^{1}\times\mathbb{P}^{1}$ at
$n-1$ points, $n=1,\dots,8$. The cases with $S=dP_{0,1,2,3}$ are toric, while
the others are not (although they still have many properties of toric
manifolds, in particular shrinking one-cycles).

We now recall a few well-known facts about BPS states of type IIA string
theory on $X$, see e.g. Beaujard:2020sgs ; Mozgovoy:2020has  for recent
accounts with additional details. The physical moduli space of the theory is
called the “extended Coulomb branch” Closset:2018bjz , including Coulomb
moduli, mass parameters, and the D0 brane mass. This moduli space is
geometrically realized by the Kähler parameters of the Calabi-Yau,
parametrizing the resolution of its singularities and the volume of the only
exceptional four-cycle, and by the radius of the M-theory circle.161616We
slightly abuse terminology in calling this the “extended Coulomb branch”,
since in Closset:2018bjz  the M-theory circle radius is not included as a
parameter of the theory. Rather the redius is just seen as setting a scale,
which is then used to multiply masses to produce dimensionless Kähler
parameters. In connection to 4d limits, form the viewpoint of field theory,
one can consider various scalings where the radius is taken to zero, but they
are equivalently described as limits in the Kahler moduli space. In a 5d
theory on $S^{1}\times\mathbb{R}^{4}$ however, where we are interested in the
moduli space parametrizing BPS central charges, it makes sense to include the
M-theory radius. Its role is to set the overall scale of central charges, such
a rescaling isn’t necessarily included in the parametrization by Kahler
moduli, at least a priori. We thank Cyril Closset for helpful comments on this
point.  BPS charges are Chern characters of compactly supported coherent
sheaves, with values in the charge lattice171717Actual charges would be
defined by K-theory classes. The distinction between the two involves issues
related to integrality of the charge lattice. For the purpose of this work we
can safely ignore these.

$\Gamma:=H^{{\rm even}}_{{\rm cpt}}=H^{0}(X,\mathbb{Z})\oplus
H^{2}_{cpt}(X,\mathbb{Z})\oplus H^{4}_{cpt}(X,\mathbb{Z})\,.$ (32)

The lattice $\Gamma$ is endowed with a skew-symmetric bilinear form
$\langle\,\cdot\,,\,\cdot\,\rangle$, arising as the antisymmetrized Euler
form. Namely, for sheaves $E_{\gamma}$ with charge $\gamma$, the pairing is

$\langle
E_{\gamma},E_{\gamma^{\prime}}\rangle:=\chi(E_{\gamma},E_{\gamma^{\prime}})-\chi(E_{\gamma^{\prime}},E_{\gamma})$
(33)

where

$\chi(E_{\gamma},E_{\gamma^{\prime}})=\int_{S}\mathrm{ch}(E_{\gamma}^{\vee})\,\mathrm{ch}(E_{\gamma^{\prime}})\,\mathrm{Td}(S)$
(34)

where $E^{\vee}$ is the dual sheaf to $E$, and $\mathrm{Td}$ denotes the Todd
characteristic class. In gauge theory descriptions the pairing
$\langle\,\cdot\,,\,\cdot\,\rangle$ agrees with the Dirac-Schwinger-Zwanziger
pairing (17), encoded in the quiver structure. Quiver nodes correspond to a
choice of exceptional collection for $S$. For $X=K_{S}$ with $S$ a del Pezzo
surface, the DSZ pairing matrix has rank two. This induces a (noncanonical)
splitting of the charge lattice into flavor and gauge charges

$\Gamma\simeq\Gamma_{f}\oplus\Gamma_{g},$ (35)

locally on moduli space. The flavor sublattice is characterized by the
vanishing of $\langle\,\cdot\,,\,\cdot\,\rangle$ when restricted to
$\Gamma_{f}\simeq\mathbb{Z}^{f}$. We also note that in all examples of this
class $H^{4}_{{\rm cpt}}\simeq\mathbb{Z}$ is a sublattice of
$\Gamma_{g}\simeq\mathbb{Z}^{2}$. When $S=dP_{n}$, we have $\dim
H^{2}(S,\mathbb{Z})=n+1$. Since $\dim H^{0}(X,\mathbb{Z})=\dim H^{4}_{{\rm
cpt}}(X,\mathbb{Z})=1$, these add up to $\mathrm{rk}\,\Gamma=n+3$. Moreover
since $\mathrm{rk}\,\Gamma_{g}=2$, it follows that
$\mathrm{rk}\,\Gamma_{f}=n+1$. Clearly all the interesting information about
$\Gamma$ is contained in $H^{2}_{{\rm cpt}}(S,\mathbb{Z})$. We will discuss
its structure in greater detail next.

###  3.2 Affine symmetry from geometry

While $X$ is noncompact, by restricting to compactly supported sheaves we have
$H^{2}_{cpt}(X,\mathbb{Z})\simeq H^{2}(S,\mathbb{Z})$, and furthermore we can
invoke Poincaré duality to identify

$H_{2}(S,\mathbb{Z})\simeq H^{2}(S,\mathbb{Z})=\text{Pic}(S)\,.$ (36)

Under this identification, generators of $H_{2}^{cpt}(S,\mathbb{Z})$ are
mapped to divisors of $S$ (i.e. line bundles) in $\text{Pic}(S)$. As observed
in Iqbal:2001ye ; Hanany:2001py , a natural basis of
$H_{2}^{cpt}(S,\mathbb{Z})$ is provided by homology classes of the exceptional
curves $\mathcal{L}_{1},\dots,\mathcal{L}_{n}$ obtained by blowing up points
in $\mathbb{P}^{2}$, as well as the pull-back of the hyperplane
$\mathcal{L}_{0}\in\text{Pic}(\mathbb{P}^{2},\mathbb{Z})$. If we view instead
$dP_{n}$ as obtained by blowups on $\mathbb{F}_{0}$, for $n\geq 1$ we can
consider the basis of $\text{Pic}(dP_{n})$ given by
$\mathcal{H}_{1},\mathcal{H}_{2}$ (which are total transforms of lines in
$\mathbb{F}_{0}$), and $\mathcal{E}_{1},\dots,\mathcal{E}_{n-1}$, which are
the exceptional classes of the blown-up points. Then

$\text{Pic}(S)=\bigoplus_{i=0}^{n}\mathcal{L}_{i}=\left(\mathcal{H}_{1}\oplus\mathcal{H}_{2}\right)\oplus\bigoplus_{i=1}^{n-1}\mathcal{E}_{i}.$
(37)

The intersection form between the cycles is

$\displaystyle\chi\left(\mathcal{L}_{0},\mathcal{L}_{0}\right)=1,$
$\displaystyle\chi\left(\mathcal{L}_{0},\mathcal{L}_{i}\right)=0,$
$\displaystyle\chi\left(\mathcal{L}_{i},\mathcal{L}_{j}\right)=-\delta_{ij},$
$\displaystyle i,j=1,\dots,n,$ (38)
$\displaystyle\chi\left(\mathcal{H}_{m},\mathcal{H}_{k}\right)=1-\delta_{km},$
$\displaystyle\chi\left(\mathcal{H}_{m},\mathcal{E}_{k}\right)=0,$
$\displaystyle\chi\left(\mathcal{E}_{m},\mathcal{E}_{k}\right)=-\delta_{mk},$
(39)

and the anti canonical class is

$\mathcal{K}_{S}=3\mathcal{L}_{0}-\mathcal{L}_{1}\dots-\mathcal{L}_{n}=2\mathcal{H}_{1}+2\mathcal{H}_{2}-\mathcal{E}_{1}\dots-\mathcal{E}_{n-1}.$
(40)

A remarkable fact about these geometries Iqbal:2001ye ; Hanany:2001py  is that
there exist a subset of $n$ generators $\\{\alpha_{i}\\}$ of
$\text{Pic}(dP_{n})$ for which the intersection form is (minus) the Cartan
matrix of the $E_{n}$ algebra:

$\left(\alpha_{i},\alpha_{j}\right)=-C_{ij}^{(E_{n})},$ (41)

so that $\\{\alpha_{i}\\}$ can be regarded as the simple roots of $E_{n}$.

Together with the only generator $\delta$ of $H^{0}(X,\mathbb{Z})$, with which
they all have vanishing pairing (34) suitably extended to $X$, they generate
the flavor lattice $\Gamma_{f}$. Physically $\delta$ is identified with the D0
brane charge. This is therefore isomorphic to the root lattice
$\mathcal{Q}(\widehat{\mathfrak{g}})$ of the affine Lie algebra $E_{n}^{(1)}$,
where $\delta$ is the null root

$\Gamma_{f}\,\simeq\,\mathcal{Q}(\widehat{\mathfrak{g}})\,.$ (42)

The gauge charge lattice $\Gamma_{g}\simeq\mathbb{Z}^{2}$ is instead generated
by the remaining cycle in $\text{Pic}(S)$ and by the unique generator of
$H^{4}_{{\rm cpt}}(X,\mathbb{Z})$.

######  Remark 2.

The DSZ pairing descends from the Euler form (33) by antisymmetrization, and
it is trivial on $\Gamma_{f}$. The DSZ pairing plays a prominent role in the
dynamics of BPS states, and their wall-crossing phenomena, and encodes the
structure of the BPS quiver $Q$. On the other hand, the information encoded by
the Euler form, which is hidden by the antisymmetrization, contains key
information about the symmetries of the underlying theory embedded in the
structure of $\Gamma_{f}$ Iqbal:2001ye ; Hanany:2001py . Remarkably,
information about these symmetries is not quite lost. As we will review below,
they are encoded in $\mathrm{Aut}_{Q}$.

$E_{8}^{(1)}$ | $E_{7}^{(1)}$ | $E_{6}^{(1)}$ | $E_{5}^{(1)}$ | $E_{4}^{(1)}$ | $E_{3}^{(1)}$ | $E_{2}^{(1)}$ | $E_{1}^{(1)}$
---|---|---|---|---|---|---|---
$E_{8}^{(1)}$ | $E_{7}^{(1)}$ | $E_{6}^{(1)}$ | $D_{5}^{(1)}$ | $A_{4}^{(1)}$ | $(A_{2}+A_{1})^{(1)}$ | $(A_{1}+A_{1})^{(1)}$ | $A_{1}^{(1)}$
Table 1: Notation for affine exceptional algebras

###  3.3 Cremona isometries, affine translations, and $\mathrm{Aut}_{Q}$

The Cremona isometries $\mathrm{Cr}(S)$ automorphisms of $\text{Pic}(S)$ that
preserve the intersection form, the canonical class and the semi-group of
effective divisors of $\text{Pic}(S)$ 2001CMaPh.220..165S ; joshi2019discrete
. Since they preserve the canonical class, they are also the automorphisms of
the local del Pezzo CY manifolds (that are the total space of the canonical
bundle), preserving the intersection form and the semi-group of effective
divisors of the base.181818As we have reviewed above, from the point of view
of the local Calabi-Yau, in order to get the affine root lattice it is
necessary to consider not just $Pic(S)$, but also $H^{0}(X)$, which contains
the null root. However, the del Pezzo contains all the information about the
local Calabi-Yau through its canonical divisor. Consequently, in the
literature on q-Painlevé equations, where affine root lattices are
traditionally seen as arising from del Pezzo surfaces, the null root is
identified with the canonical divisor of $dP_{9}$ (note that equation (40)
implies that $\chi(\mathcal{K}_{dP_{9}},\mathcal{K}_{dP_{9}})=0$), which is
then mapped to lower del Pezzos through a blowdown procedure
2001CMaPh.220..165S .  It is well-known that such transformation are realized
as the extended affine Weyl group $\widetilde{W}(\widehat{g})$ acting not only
on the root lattice $\Gamma_{f}$, but on the whole lattice $\Gamma$ GHK2015 ;
mizuno2020q .

It was shown in Bershtein:2017swf ; mizuno2020q  that $\mathrm{Aut}_{Q}$
contains a subgroup isomorphic to the extended affine Weyl group
$\widetilde{W}(\widehat{\mathfrak{g}})$ for all quivers of del Pezzo surfaces
up to $dP_{8}$ and of $\mathbb{F}_{0}$

$\mathrm{Aut}_{Q}\supseteq\widetilde{W}(\widehat{\mathfrak{g}})\,.$ (43)

Thus elements $\mathrm{Aut}_{Q}$, which are realized by specific sequences of
mutations, permutations and inversion, provide a realization of the Cremona
group of $dP_{n}$ on the associated BPS charge lattice $\Gamma$. Since
$\mathrm{Aut}_{Q}$ preserves the form of the quiver, it may be regarded as the
_self-duality_ group of the quiver quantum mechanics.

When extending the representation of $\widetilde{W}(\widehat{g})$ from
$\Gamma_{f}$, where it is naturally defined, to $\Gamma$, there are subtleties
to deal with: in order to have a proper representation of
$\widetilde{W}(\widehat{g})$ on $\Gamma$, it seems necessary to use
transformations that include both clockwise and anticklockwise mutations, as
was done in mizuno2020q . However this is unnatural from a physical point of
view: we require that the action must include only left (or only right)
mutations, in keeping with our goal of connecting to tilting induced by
stability conditions as in the mutation method of Berenstein:2002fi ;
Alim:2011ae ; Alim:2011kw ; Cecotti:2011rv . Instead, we will adopt a
realization of $\widetilde{W}(\widehat{g})$ involving only left mutations as
in Bershtein:2017swf .191919Clearly, it would be important to determine to
what extent the condition of using only left (or right) mutations determines
_uniquely_ the realization of
$\widetilde{W}(\widehat{g})\subset\mathrm{Aut}_{Q}$. This will result in an
action on $\Gamma$ which is generally _not_ a proper representation of
$\widetilde{W}(\widehat{g})$, but restricts to a proper representation on the
root lattice $\Gamma_{f}$ (on $\Gamma$ we have a representation “modulo seed
isomorphisms” GHK2015 ; mizuno2020q ). Recall that affine Weyl groups have the
following decomposition into reflections and translations

$W(\widehat{\mathfrak{g}})=W(\mathfrak{g})\ltimes\mathcal{T}(\widehat{\mathfrak{g}})\,.$
(44)

In the context of discrete Painlevé equations the group
$\mathcal{T}(\widehat{g})$ corresponds to the flow described by the equations
themselves, while other elements of $\widetilde{W}(\widehat{g})$ act as
symmetries of the equation. Following the general philosophy of
Bonelli:2020dcp , we want to study the action of affine translations on the
lattice $\Gamma$ of BPS charges, to study the BPS spectral problem. To this
end, we recall the quiver description of BPS states reviewed in Section 2.1.
For general toric geometries, BPS quivers can be determined by resorting to
brane tiling techniques (see Franco:2017jeo  for a recent review). Other
options include: local mirror symmetry Hanany:2001py , studying calibrated
one-cycles on the mirror curve Klemm:1996bj ; Eager:2016yxd ; Banerjee:2019apt
; Banerjee:2020moh , directly deriving the adjacency matrix directly from
$\text{Pic}(S)$ as in mizuno2020q , and the approach based on the geometry of
loop groups of Fock:2014ifa . We collect the relevant quivers in Figure 5.

$E_{8}^{(1)}$$E_{7}^{(1)}$$E_{6}^{(1)}$$E_{5}^{(1)}$$E_{4}^{(1)}$$E_{3}^{(1)}$$E_{2}^{(1)}$$A_{1}^{(1)}$$A_{0}^{(1)}$$\begin{matrix}A_{1}^{(1)}\\\
|\alpha|^{2}=8\end{matrix}$ Figure 5: Symmetry algebra of 5d SCFTs (or
symmetry type of q-Painlevé equations) with BPS quivers for local
$dP_{5},dP_{3},\mathbb{F}_{0}$.

Having identified the relevant BPS quivers (labeled by $\gamma_{i}$), the next
step is to explain how these are related the affine Weyl group action on
$\Gamma$. Recall the definition of mutations from section 2.1. In addition to
this we’ll need another important transformation, called inversion and denoted
by $\iota$. This operation reverses the orientation of all arrows in $Q$, and
flips the signs of all label charges

$\displaystyle\iota(B)=-B,$ $\displaystyle\iota(\gamma_{i})=-\gamma_{i}.$ (45)

The automorphism group $\mathrm{Aut}_{Q}$ of the quiver $Q$ consists of
sequences of mutations, permutations of the nodes, and inversions, that
preserve the quiver $Q$.

As already mentioned, affine Weyl translations (44) have a distinguished role
through their appearance as time flows in the theory of discrete Painlevé
equations. In general, a translation $T\in\mathcal{T}(\widehat{\mathfrak{g}})$
acts on simple roots $\alpha_{i}$ as follows 2015arXiv150908186K

$\displaystyle T_{\vec{n}}(\alpha_{i})=\alpha_{i}+n_{i}\delta,$
$\displaystyle\delta=\sum_{i}m_{i}\alpha_{i},$ $\displaystyle\sum
n_{i}m_{i}=0\,,$ (46)

for some choice of $n_{i}$. For $dP_{n}$ there are $n$ solutions to the last
equation above, corresponding to the existence of $n$ independent flows

$\mathrm{rk}\,\mathcal{T}(E_{n}^{(1)})=n\,.$ (47)

The action of a generic translation can be obtained as a sequence of
elementary reflections on the root lattice, possibly followed by a permutation
of the roots. See (2015arXiv150908186K, , Section 3.2) for a convenient way to
compute the translation in terms of reflections.

Once a translation $T\in\mathcal{T}(E_{n}^{(1)})$ is realized as an element of
$\mathrm{Aut}_{Q}$, its action gets extended from $\Gamma_{f}$ to the whole
$\Gamma$. The key data encoding this extended action is a collection of
_limiting rays_ in $\Gamma$. A limiting ray is defined as
$\delta_{j}\mathbb{Z}\subset\Gamma$, where

$\delta_{j}=\lim_{k\to\infty}\frac{1}{k}T^{k}(\gamma_{i})$ (48)

for some charge $\gamma_{i}$ labeling the $i$-th node of the quiver. The
collection of all limiting rays is obtained by applying the above definition
to all $\gamma_{i}$ (some $\gamma_{i}$’s may have the same limiting ray, other
maybe left invariant by translations).

###  3.4 Stability condition for local del Pezzos from affine translations

We now come to the main point of this section, namely to explain the relevance
of (extended) affine Weyl group symmetries arising within the Cremona group of
$X$, for the study of BPS states encoded by a quiver $Q$. As emphasized in
Section 3.3, there is an important distinction between affine Weyl
translations $\mathcal{T}$ and the rest of $\mathrm{Aut}_{Q}$, which arises
naturally in the study of discrete Painlevé equations. Furthermore these
equations and the time evolution described by
$\mathcal{T}(\widehat{\mathfrak{g}})$ also play a role in the study of BPS
spectrum of the quiver $Q$. In particular, Bonelli:2020dcp  showed explicitly
how affine translations 202020More precisely, the double cover
$2\cdot\mathcal{T}$, see below. $\mathcal{T}(A_{1}^{(1)})$ reproduce the
sequence of mutations for the BPS quiver of $\mathbb{F}_{0}$ discussed in
Closset:2019juk , and conjectured that this observation should generalize to
the other del Pezzo geometries by studying the action of discrete flows on the
BPS charges of local $\mathbb{F}_{1}$ and local $dP_{3}$.

In this paper we wish to make the above proposal more precise, identifying a
relation between affine translations and certain stability conditions.
Furthermore, in all examples we considered it turns out that the class of
stability conditions arising from affine Weyl translations often belongs to a
collimation chamber, as introduced in Section 2. This fact will allow us to
compute the wall-crossing invariant order by order as outlined in Section 2.5.
Our goal is to associate uniquely a collimation chamber to an appropriate Weyl
translation

$T\mapsto\mathcal{C}_{T}\,.$ (49)

A priori it is unclear whether a Weyl translation $T$ should correspond to a
stability condition. In fact, through (43) a translation $T$ may generically
be realized as a sequence of mutations, permutations, and possibly certain
involutions on $Q$. There are three basic conditions that $T$ must satisfy, in
order for (49) to make sense:

  1. (c1)

There is a positive integer $k\in\mathbb{N}$ such that
$T^{k}=\mu_{i_{1}}\dots\mu_{i_{\ell}}$ admits an expression involving only
left mutations.

  2. (c2)

There exists a configuration of $Z_{\gamma_{i}}\in\mathbb{C}$ such that
clockwise tilting the positive half-plane induces a sequence of mutations
identical to iterations of $(T^{k})^{n}$ for $n\geq 1$.

  3. (c3)

The above configuration of central charges belongs to a collimation chamber.

If criteria (c1)-(c2) are satisfied, we can associate a stability condition to
$T$. The third requirement is only needed to restrict to stability conditions
of collimation type.

In general it is not always clear how to verify whether an affine translation
$T$ (more precisely, its realization in $\mathrm{Aut}_{Q}$) satisfies the
above criteria. (c1) is the easiest to verify: for example if $T$ only
involves left mutations and pemutations one takes $k$ to be the order of the
permutation $\pi^{k}=1$. Then one uses $\mu_{i}\pi=\pi\mu_{\pi(i)}$ to take
all instances of $\pi$ together so as to cancel them, leaving mutations only.
Criteria (c2)-(c3) are more difficult to verify, and we will now outline a
procedure tailored to this purpose.

We shall begin with some technical assumptions:

  1. (a1)

In the mapping (42), each simple root
$\alpha_{i}\in\mathcal{Q}(\hat{\mathfrak{g}})$ is realized as a linear
combination of basic charges

$\alpha=\sum_{i}c_{\alpha,i}\gamma_{i}$ (50)

with nonzero coefficients only for a subset of mutually local $\gamma_{i}$.

  2. (a2)

We require that $\langle\delta_{i},\delta_{j}\rangle=0$ for all charges of
limiting rays of $T$.

  3. (a3)

There is a nontrivial sublattice whose roots are given by only mutually local
charges.

If the first condition is not satisfied, we can still define a virtual
stability condition, but it will not necessarily correspond to a collimation
chamber. This second assumption is only technical, and we hope to be able to
lift it in future work. It certainly does not hold in the four-dimensional
case, see e.g. the example of the Kronecker quiver in Section 2.2. For the
case of local del Pezzos, it fails for $dP_{1},dP_{2},dP_{4}$. We now explain
the procedure that associates a collimation chamber to each affine traslation.

#### Procedure.

Given the above assumptions (a1)-(a3), we can now state a procedure to verify
whether conditions (c2)-(c3) are verified. We shall proceed constructively,
_i.e._ by stating a set of constraints on the possible configurations of
$Z_{\gamma_{i}}$ that would realize (c2)-(c3). Let
$\varphi_{i}=\arg(Z_{\gamma_{i}})$, then we require

  1. 1.

The D0-brane central charge is real

$\arg(Z_{\delta})=0.$ (51)

  2. 2.

The limiting rays of $T$ should lie on the real axis

$\arg Z_{\delta_{i}}=0\qquad\forall i.$ (52)

Thanks to (a2), this does not correspond to a wall of marginal stability.

  3. 3.

Since roots $\alpha_{i}$ are made of mutually local charges (see (a1)), we
demand that $Z_{\gamma_{i}}$ of the central charges of $\gamma_{i}$
contributing to (50) with nonzero coefficients are mutually aligned

$\varphi_{\alpha,1}=\dots=\varphi_{\alpha,k}\,.$ (53)

In the examples that we consider this procedure constrains the
$Z_{\gamma_{i}}$ to the extent that it completely fixes the map (49).

##  4 Examples

###  4.1 $\mathbb{F}_{0}$

Figure 6: BPS quiver for local $\mathbb{F}_{0}$.

The Cremona group of $\mathbb{F}_{0}$ is

$\mathrm{Cr}(\mathbb{F}_{0})=W(A_{1}^{(1)})\rtimes{\rm Dih}_{4}\,.$ (54)

The extended affine Weyl group

$\widetilde{W}(A_{1}^{(1)})=W(A_{1}^{(1)})\rtimes\mathrm{Out}(A_{1}^{(1)})$
(55)

is a subgroup of $\mathrm{Cr}(\mathbb{F}_{0})$, since
$\mathrm{Out}(A_{1}^{(1)})\simeq\mathbb{Z}^{2}\subset{\rm Dih}_{4}$ as
depicted in Figure 8.

On the other hand, the full Cremona group is visible on the BPS quiver, which
is depicted in Figure 6. The adjacency matrix is

$B=\left(\begin{array}[]{cccc}0&-2&0&2\\\ 2&0&-2&0\\\ 0&2&0&-2\\\
-2&0&2&0\end{array}\right).$ (56)

The full automorphism group of $Q$ coincides with the Cremona group
Bershtein:2017swf

$\mathrm{Aut}_{Q}=\mathrm{Cr}(\mathbb{F}_{0})\,,$ (57)

with generators

$\displaystyle s_{0}=(1,3)\mu_{1}\mu_{3},$ $\displaystyle
s_{1}=(2,4)\mu_{2}\mu_{4},$ $\displaystyle\pi=(1,2,3,4),$
$\displaystyle\sigma=\iota(1,3).$ (58)

Automorphisms $\mathrm{Aut}_{Q}$ have a permutation subgroup
$\Pi_{Q}\simeq\mathbb{Z}_{4}\subset{\rm Dih_{4}}$.

The rank of $B$ is $2$, therefore

$\mathrm{rk}\,\Gamma_{g}=\mathrm{rk}\,\Gamma_{f}=2\,.$ (59)

A positive integral basis for $\Gamma_{f}^{+}=\\{\gamma\in\Gamma_{f},{\rm
Re}Z_{\gamma}>0\\}$ is given by

$\gamma_{f,0}=\gamma_{1}+\gamma_{3}\,,\qquad\gamma_{f,1}=\gamma_{2}+\gamma_{4}\,.$
(60)

#### Collimation chambers.

The moduli space of stability conditions for the quiver of Figure 6 features
collimation chambers. The first one of these, found in Closset:2019juk , is a
chamber containing the following configuration of central charges212121Recall
that by convention all $Z_{\gamma_{i}}$ in this paper are understood to have
positive real part. The full extent of the chamber depends on its BPS
spectrum, in this case $\mathcal{C}_{1}$ also contains more general
configurations, such as $Z_{\gamma_{1}}\neq Z_{\gamma_{3}}$ as long as their
phases are the same.

$\mathcal{C}_{1}[\mathbb{F}_{0}]\supset\quad
Z_{\gamma_{1}}=Z_{\gamma_{3}}\,,\quad
Z_{\gamma_{2}}=Z_{\gamma_{4}}\,,\quad\arg Z_{\gamma_{1}}>\arg
Z_{\gamma_{2}}\,,\quad Z_{\gamma_{1}}+Z_{\gamma_{2}}\in\mathbb{R}^{+}\,.$ (61)

The authors of Closset:2019juk  observed that tilting the positive half-plane
of central charges induces infinite sequences of left- and right-mutations,
corresponding to the following towers of hypermultiplets

$\begin{split}&\Omega(\gamma_{1}+n(\gamma_{1}+\gamma_{2}),\mathcal{C}_{1})=1\qquad\Omega(\gamma_{2}+n(\gamma_{1}+\gamma_{2}),\mathcal{C}_{1})=1\\\
&\Omega(\gamma_{3}+n(\gamma_{3}+\gamma_{4}),\mathcal{C}_{1})=1\qquad\Omega(\gamma_{4}+n(\gamma_{3}+\gamma_{4}),\mathcal{C}_{1})=1\end{split}$
(62)

These BPS states are those that lie off the real axis in the ray diagram of
Figure 7. As $n\to\infty$ rays accumulate towards
$Z_{\gamma_{1}+\gamma_{2}}=Z_{\gamma_{3}+\gamma_{4}}\in\mathbb{R}^{+}$ as
required by the definition of collimation chamber.

Figure 7: BPS ray diagram for the collimation chamber
$\mathcal{C}_{1}[\mathbb{F}_{0}]$.

There is also a second collimation chamber, characterized by

$\mathcal{C}_{2}[\mathbb{F}_{0}]\supset\quad
Z_{\gamma_{1}}=Z_{\gamma_{3}}\,,\quad
Z_{\gamma_{2}}=Z_{\gamma_{4}}\,,\quad\arg Z_{\gamma_{1}}<\arg
Z_{\gamma_{2}}\,,\quad Z_{\gamma_{1}}+Z_{\gamma_{2}}\in\mathbb{R}^{+}\,.$ (63)

Again there are infinite towers of hypermultiplets accumulating towards
$\mathbb{R}^{+}$, however their charges are now

$\begin{split}&\Omega(\gamma_{1}+n(\gamma_{1}+\gamma_{4}),\mathcal{C}_{2})=1\qquad\Omega(\gamma_{2}+n(\gamma_{2}+\gamma_{3}),\mathcal{C}_{2})=1\\\
&\Omega(\gamma_{3}+n(\gamma_{2}+\gamma_{3}),\mathcal{C}_{2})=1\qquad\Omega(\gamma_{4}+n(\gamma_{1}+\gamma_{4}),\mathcal{C}_{2})=1\,.\end{split}$
(64)

In either of these chambers the spectrum is not exhausted by hypermultiplets,
as there are BPS states with $Z\in\mathbb{R}_{+}$ that need to be determined.
We’ll return to this below.

Clearly, the two chambers $\mathcal{C}_{1},\mathcal{C}_{2}$ in (61) and (63)
are related to each other by exchanging $\\{\gamma_{1},\gamma_{3}\\}$ with
$\\{\gamma_{2},\gamma_{4}\\}$ (as unordered sets). This is a $\mathbb{Z}_{2}$
subgroup of $\Pi_{Q}$, whose action is therefore not faithful on
$\mathcal{C}_{i}$.

#### Translations of the $A_{1}^{(1)}$ lattice.

Before proceeding with the analysis of BPS states on the real ray, let us
pause to explain how the collimation chambers described above could have been
found by studying translations of the affine root lattice associated to the
quiver. The relation between the chamber (61) and lattice translations was
first observed by Bonelli:2020dcp . We begin by reviewing how translations of
the root lattice are realized by mutations and permutations on quiver
vertices.

The low-energy gauge theory phase is $5d$ $\mathcal{N}=1$ pure $SU(2)$ Yang-
Mills. The isomorphism (42) identifies the flavor sublattice $\Gamma_{f}$ with
the root lattice $\mathcal{Q}(A_{1}^{(1)})$. Simple roots are identified with
the generators (60)

$\displaystyle\alpha_{0}=\gamma_{f,0},$
$\displaystyle\alpha_{1}=\gamma_{f,1}.$ (65)

Therefore the null root $\delta:=\sum_{i=1}^{4}\gamma_{i}$ decomposes into
$\delta=\sum_{i}m_{i}\alpha_{i}$ with $(m_{0},m_{1})=(1,1)$. Applying the
general definition (46) of affine Weyl translations we therefore find a single
translation generator $T_{1}$, defined by $(n_{0},n_{1})=(1,-1)$ acting as
follows on the simple roots

$T^{n}_{1}(\vec{\alpha})=\left(\begin{array}[]{c}\alpha_{0}+n\delta\\\
\alpha_{1}-n\delta\end{array}\right)\,.$ (66)

Therefore in the case of local $\mathbb{F}_{0}$

$\mathcal{T}(A_{1}^{(1)})\simeq\mathbb{Z}\,.$ (67)

Next recall the relation (43) between the extended Weyl group of the affine
root lattice, and quiver automorphisms. The translation subgroup
$\mathcal{T}(A_{1}^{(1)})\subset\widetilde{W}({A_{1}^{(1)}})$ is realized in
$\mathrm{Aut}_{Q}$ by Bershtein:2017swf

$T_{1}=(1,2)(3,4)\mu_{1}\mu_{3}$ (68)

Note that while the action of $\mathcal{T}(A_{1}^{(1)})$ is only defined on
$\mathcal{Q}(A_{1}^{(1)})\simeq\Gamma_{f}$, the realization (68) extends to
all of $\Gamma$, as follows

$\displaystyle
T^{n}_{1}(\vec{\gamma})=\left(\begin{array}[]{c}\gamma_{1}+n(\gamma_{1}+\gamma_{2})\\\
\gamma_{2}-n(\gamma_{1}+\gamma_{2})\\\ \gamma_{3}+n(\gamma_{3}+\gamma_{4})\\\
\gamma_{4}-n(\gamma_{3}+\gamma_{4})\end{array}\right)\,.$ (73)

As a check, when restricted to $\gamma_{f,i}$ with $i=0,1$, the action
reproduces that on $\alpha_{i}$ for $i=0,1$.

The affine Weyl group contains only one set of translations (67). However its
realization on the BPS charge lattice $\Gamma$ isn’t unique. This is because
the map (65) can be composed with the nontrivial outer automorphism
$\pi\in\mathrm{Out}(A_{1}^{(1)})\simeq\mathbb{Z}_{2}$, which exchanges the two
roots

$\pi:\qquad\alpha_{0}\mapsto\alpha_{1}\,,\ \ \alpha_{1}\mapsto\alpha_{0}\,.$
(74)

Note that $\mathrm{Out}(A_{1}^{(1)})\subset\widetilde{W}(A_{1}^{(1)})$ belongs
to the extended affine Weyl group, and therefore it must admit a realization
as an element of $\mathrm{Aut}_{Q}$, by virtue of (43), so that $\pi$ is
really the one defined in (58). This means that $\mathrm{Out}(A_{1}^{(1)})$ is
realized as a subgroup of quiver permutations

$\mathbb{Z}_{2}\approx\mathrm{Out}(A_{1}^{(1)})\subset\Pi_{Q}\approx\mathbb{Z}_{4}\,,$
(75)

since $\pi^{4}=id$, but $\pi^{2}|_{\Gamma_{f}}=id$.

$\alpha_{0}$$\alpha_{1}$$\pi$$\pi$ Figure 8: $A_{1}^{(1)}$ Dynkin diagram and
automorphisms

Conjugation by $\pi\in\mathrm{Out}(A_{1}^{(1)})$ leads to the _same_ group of
affine translations, but with a _different_ realization in $\mathrm{Aut}_{Q}$.
In fact the generator

$T_{2}\equiv(1,4)(2,3)\mu_{2}\mu_{4}=\pi^{-1}T_{1}\pi.$ (76)

acts simply as the inverse of $T_{1}$ on $\mathcal{Q}(A_{1}^{(1)})$

$T^{n}_{2}(\vec{\alpha})=\left(\begin{array}[]{c}\alpha_{0}-n\delta\\\
\alpha_{1}+n\delta\end{array}\right)\,,$ (77)

which is the image under $\pi$ of (66). However for the action on $\Gamma$ we
obtain

$\displaystyle
T^{n}_{2}(\vec{\gamma})=\left(\begin{array}[]{c}\gamma_{1}-n(\gamma_{1}+\gamma_{4})\\\
\gamma_{2}+n(\gamma_{3}+\gamma_{2})\\\ \gamma_{3}-n(\gamma_{3}+\gamma_{2})\\\
\gamma_{4}+n(\gamma_{1}+\gamma_{4})\end{array}\right)\,,$ (82)

which is the image of (73) under $\pi$.

Note that, while $T_{2}=T_{1}^{-1}$ on
$\mathcal{Q}(A_{1}^{(1)})\simeq\Gamma_{f}$, this is not so on $\Gamma$. This
is due to the fact that we chose a realization of
$\mathrm{Cr}(\mathbb{F}_{0})$ in $\mathrm{Aut}_{Q}$ which only involves _left_
mutations, as explained in Section 3.3.

#### Collimation chambers from affine Weyl translations.

Having reviewed the realization of affine Weyl translations, we are now in a
position to explain how these give rise to a sequence of mutations which
corresponds to a stability condition. Namely, the sequence of mutations
induced by lattice translations coincides with the sequence of mutations
induced by tilting the positive half-plane of central charges for a certain
choice of stability condition. Furthermore, this stability condition belongs
to one of the collimation chambers $\mathcal{C}_{i}$.

The conditions (51) and (53) coming from the root lattice structure of
$\Gamma$ read in this case

$\begin{cases}\varphi_{1}+\varphi_{2}+\varphi_{3}+\varphi_{4}=0,\\\
\varphi_{1}=\varphi_{3},\\\ \varphi_{2}=\varphi_{4}.\end{cases}$ (83)

To discuss (52) we must determine the charges $\delta_{i}$ of limiting rays
(48). For the flow $T_{1}$, it is clear from (73) that

$\delta_{1}{(T_{1})}=\gamma_{1}+\gamma_{2}\,,\qquad\delta_{2}{(T_{1})}=\gamma_{3}+\gamma_{4}\,,$
(84)

whereas for the flow $T_{2}$ in (82) we have instead

$\delta_{1}{(T_{2})}=\gamma_{4}+\gamma_{1}\,,\qquad\delta_{2}{(T_{2})}=\gamma_{2}+\gamma_{3}\,.$
(85)

In either case we find that $Z_{\delta_{i}}\in\mathbb{R}^{+}$ thanks to (83).
Therefore (52) is automatically satisfied for both flows. The general solution
to (83) is

$\varphi_{1}=\varphi_{3}=-\varphi_{2}=-\varphi_{4}.$ (86)

However equation (86) does not yet determine completely a stability condition,
as it does not fully specify the phase ordering of the central charges. This
last piece of information is fixed in different ways for the two flows
$T_{1},T_{2}$, by viewing each of them as tiltings of the positive half
$Z$-plane. Indeed, it was observed in Bonelli:2020dcp  that

$\displaystyle T_{1}^{2}=\mu_{2}\mu_{4}\mu_{3}\mu_{1}\equiv\textbf{m}_{1}$
$\displaystyle T_{2}^{2}=\mu_{1}\mu_{3}\mu_{2}\mu_{4}\equiv\textbf{m}_{2}\,.$
(87)

For $T_{1}$ the first mutations occur on $\gamma_{1},\gamma_{3}$, implying

$\displaystyle\varphi_{1}=\varphi_{3}>\varphi_{2}=\varphi_{4}=-\varphi_{1}\,.$
(88)

The configuration of BPS rays for simple roots is shown in Figure 9(a). This
class of stability conditions corresponds exactly to the collimation chamber
defined in (61)

$\text{Stability conditions for }T_{1}\equiv\text{collimation chamber
}\mathcal{C}_{1}[\mathbb{F}_{0}]\,.$ (89)

Conversely, the first charges mutated by the flow $T_{2}$ are
$\gamma_{2},\gamma_{4}$, so that

$\displaystyle\varphi_{2}=\varphi_{4}>\varphi_{1}=\varphi_{3}=-\varphi_{2}\,.$
(90)

The configuration of BPS rays for simple roots is shown in Figure 9(b). This
class of stability conditions corresponds exactly to the collimation chamber
defined in (63)

$\text{Stability conditions for }T_{2}\equiv\text{collimation chamber
}\mathcal{C}_{2}[\mathbb{F}_{0}]\,.$ (91)

(a) Stability condition associated to $T_{1}$

(b) Stability condition associated to $T_{2}$

Figure 9: Stability conditions for $\mathbb{F}_{0}$, corresponding to
$\mathcal{C}_{1}[\mathbb{F}_{0}]$ and $\mathcal{C}_{2}[\mathbb{F}_{0}]$
respectively.

#### Wall-crossing invariant.

Let us now return to the analysis of the two collimation chambers (61) and
(63), and to the computation of the wall-crossing invariant $\mathbb{U}$.
First we make two observations

  1. 1.

The chamber $\mathcal{C}_{2}$ in (63) and the states (64) are related to
$\mathcal{C}_{1}$ in (61) and its BPS states (62) by a
$\Pi_{Q}\simeq\mathbb{Z}_{4}$ shift in charge labels
$\pi:\gamma_{i}\mapsto\gamma_{i+1\,{\rm mod}4}$. Note that $\pi^{2}$ acts
trivially on stability conditions, so the action of
$\Pi_{Q}\simeq\mathbb{Z}_{4}$ is not faithful on $\\{\mathcal{C}_{i}\\}$.
Indeed, the subgroup $\mathrm{Out}(A_{1}^{(1)})\simeq\mathbb{Z}_{2}$ is the
one that acts faithfully on $\\{\mathcal{C}_{1},\mathcal{C}_{2}\\}$.

  2. 2.

Stability conditions (62) and (63) are connected by a path in moduli space
along which no BPS rays exit the half-plane ${\rm Re}Z>0$. Concretely, let
$Z^{(0)}_{\gamma_{i}}$ be arbitrary representatives for the chamber (61). Then
a suitable path would be

$\begin{split}Z_{\gamma_{1}}(t)=Z_{\gamma_{3}}(t)=t\cdot
Z^{(0)}_{\gamma_{1}}+(1-t)\cdot Z^{(0)}_{\gamma_{2}}\\\
Z_{\gamma_{2}}(t)=Z_{\gamma_{4}}(t)=t\cdot Z^{(0)}_{\gamma_{2}}+(1-t)\cdot
Z^{(0)}_{\gamma_{1}}\\\ \end{split}\qquad t\in[0,1]$ (92)

Existence of a path with these properties implies that $\mathbb{U}$ remains
unchanged in moving from one stability condition to the other. Therefore

$\mathbb{U}(\mathcal{C}_{1})=\mathbb{U}(\mathcal{C}_{2})$ (93)

where each side admits a factorization of the form (25)

$\mathbb{U}(\mathcal{C}_{i})=\mathbb{U}(\measuredangle^{+},\mathcal{C}_{i})\cdot\mathbb{U}(\mathbb{R}^{+},\mathcal{C}_{i})\cdot\mathbb{U}(\measuredangle^{-},\mathcal{C}_{i})\,.$
(94)

The BPS rays off the real axis contribute, respectively in each chamber

$\begin{split}\mathbb{U}(\measuredangle^{+},\mathcal{C}_{1})&=\prod_{k\geq
0}^{\nearrow}\Phi(X_{\gamma_{1}+k(\gamma_{1}+\gamma_{2})})\Phi(X_{\gamma_{3}+k(\gamma_{3}+\gamma_{4})})\\\
\mathbb{U}(\measuredangle^{-},\mathcal{C}_{1})&=\prod_{k\geq
0}^{\searrow}\Phi(X_{\gamma_{2}+k(\gamma_{1}+\gamma_{2})})\Phi(X_{\gamma_{4}+k(\gamma_{3}+\gamma_{4})})\end{split}$
(95)

and

$\begin{split}\mathbb{U}(\measuredangle^{+},\mathcal{C}_{2})&=\prod_{k\geq
0}^{\nearrow}\Phi(X_{\gamma_{2}+k(\gamma_{2}+\gamma_{3})})\Phi(X_{\gamma_{4}+k(\gamma_{4}+\gamma_{1})})\\\
\mathbb{U}(\measuredangle^{-},\mathcal{C}_{2})&=\prod_{k\geq
0}^{\searrow}\Phi(X_{\gamma_{3}+k(\gamma_{2}+\gamma_{3})})\Phi(X_{\gamma_{1}+k(\gamma_{4}+\gamma_{1})})\end{split}$
(96)

where $\nearrow$ ($\searrow$) denotes increasing (decreasing) values of $k$ to
the right.

To compute $\mathbb{U}(\mathbb{R}^{+},\mathcal{C}_{1})$ and
$\mathbb{U}(\mathbb{R}^{+},\mathcal{C}_{2})$, consider a formal series
expansion like (30) for each of them. Plugging these into (94), and imposing
the equality (93) leads to an equation involving $c_{\gamma}(\mathcal{C}_{1})$
on the _l.h.s._ and $c_{\gamma}(\mathcal{C}_{2})$ on the _r.h.s._ Next we use
the fact that factorizations $\mathbb{U}(\mathcal{C}_{1})$ and
$\mathbb{U}(\mathcal{C}_{2})$ correspond to stability conditions related by a
$\mathbb{Z}_{4}$ relabeling of $\gamma_{i}$ on quiver nodes. In particular
this implies
$c_{\gamma}(\mathcal{C}_{2})=c_{\pi^{-1}(\gamma)}(\mathcal{C}_{1})$ (cf. (28))
and we can express both $\mathbb{U}(\mathcal{C}_{i})$ in terms of
$c_{\gamma}(\mathcal{C}_{1})\equiv c_{\gamma}$ only. Wall-crossing invariance
(93) then imposes equations on these coefficients: namely

$\mathbb{U}(\measuredangle^{+},\mathcal{C}_{1})\left(\sum_{\gamma\in\Gamma_{\mathbb{R}^{+}}}c_{\gamma}\,X_{\gamma}\right)\mathbb{U}(\measuredangle^{-},\mathcal{C}_{1})=\mathbb{U}(\measuredangle^{+},\mathcal{C}_{2})\left(\sum_{\gamma\in\Gamma_{\mathbb{R}^{+}}}c_{\pi^{-1}(\gamma)}\,X_{\gamma}\right)\mathbb{U}(\measuredangle^{-},\mathcal{C}_{2})\,.$
(97)

This is the equation we needed to constrain $c_{\gamma}$ and compute
$\mathbb{U}(\mathbb{R}^{+},\mathcal{C}_{i})$. Expanding each side, this is
precisely the symmetry constraint derived in (29). Note that this condition
holds for coefficients $u_{\gamma}$, which are nontrivial functions of the
$c_{\gamma}$ through (25) and the known factors (95).

To solve equation (97) we expand each side as a series in $X_{\gamma}$ and
compare coefficients of the monomials. This leads to infinitely many equations
which can be solved recursively by introducing a filtration in
$|\gamma|=\sum_{i}d_{i}$. Such a procedure can be easily implemented on a
computer, and yields

$\begin{split}\mathbb{U}(\mathbb{R}^{+},\mathcal{C}_{1})&=1-\frac{1+y^{2}}{1-y^{2}}\left(X_{\gamma_{1}+\gamma_{2}}+X_{\gamma_{3}+\gamma_{4}}\right)+\frac{2y^{2}+y^{4}+y^{6}}{(1-y^{2})(1-y^{4})}\left(X_{2\gamma_{1}+2\gamma_{2}}+X_{2\gamma_{3}+2\gamma_{4}}\right)\\\
&-\frac{\left(y^{4}+2y^{6}+y^{10}\right)}{(1-y^{2})^{2}(1-y^{6})}\left(X_{3\gamma_{1}+3\gamma_{2}}+X_{3\gamma_{3}+3\gamma_{4}}\right)\\\
&-\frac{\left(1-y^{2}\right)\left(1-y^{4}\right)c_{\gamma_{D0}}-2y^{2}-3y^{4}+y^{6}}{\left(1-y^{2}\right)^{3}}\left(X_{\gamma_{1}+\gamma_{2}+\gamma_{D0}}+X_{\gamma_{3}+\gamma_{4}+\gamma_{D0}}\right)\\\
&+c_{\gamma_{D0}}X_{\gamma_{D0}}+\dots\end{split}$ (98)

where we introduced the shorthand notation $\gamma_{D0}=\sum_{i}\gamma_{i}$.

The solution can be obtained to arbitrary orders in $|\gamma|$, and we find
that it is compatible with the factorization

$\mathbb{U}_{0}(\mathbb{R}^{+},\mathcal{C}_{1})=\prod_{s=\pm 1}\prod_{k\geq
0}\Phi((-y)^{s}X_{\gamma_{1}+\gamma_{2}+k\gamma_{D0}})^{-1}\cdot\Phi((-y)^{s}X_{\gamma_{3}+\gamma_{4}+k\gamma_{D0}})^{-1}$
(99)

where $\gamma_{D0}=\sum_{i=1}^{4}\gamma_{i}$. We can write this in a more
suggestive way by recalling the D-brane charges associated with $\gamma_{i}$
as follows

$\mathbb{U}_{0}(\mathbb{R}^{+},\mathcal{C}_{1})=\prod_{s=\pm 1}\prod_{k\geq
0}\Phi((-y)^{s}X_{\gamma_{1}+\gamma_{2}+k\gamma_{D0}})^{-1}\cdot\Phi((-y)^{s}X_{-\gamma_{1}-\gamma_{2}+(k+1)\gamma_{D0}})^{-1}\,.$
(100)

Indeed recall that $\gamma_{1}+\gamma_{2}$ can be identified with a D2 brane
wrapping the fiber $\mathbb{P}^{1}$ in $\mathbb{F}_{0}$, then the above
expression makes it evident that these are towers of $D2\text{-}D0$
boundstates as well as the CPT conjugate $\overline{D2}\text{-}D0$ states
which belong to the positive half $Z$-plane Longhi:2021qvz . We will see this
pattern arising in greater generality in other examples below.

This corresponds to a BPS spectrum along the real ray consisting of
vectormultiplets

$\Omega(\gamma_{1}+\gamma_{2}+n\gamma_{D0},\mathcal{C}_{1})=\Omega(-\gamma_{1}-\gamma_{2}+(n+1)\gamma_{D0},\mathcal{C}_{1})=y+y^{-1}$
(101)

As observed in Longhi:2021qvz , this formula misses contributions from D0
branes, an ambiguity reflected in the fact that the coefficients
$c_{n\gamma_{D0}}$ remain undetermined. Since $\gamma_{D0}$ has trivial
pairing with any other charge $\langle\gamma_{D0},\gamma\rangle=0$, it does
not participate in wall-crossing. It follows that pure D0 states cannot be
fixed by our methods, which are fundamentally based on symmetries of $Q$ in
conjunction with wall-crossing invariance. The D0 contributions can be
obtained with other techniques, such as exponential networks Eager:2016yxd ;
Banerjee:2018syt ; Banerjee:2019apt ; Banerjee:2020moh ; Longhi:2021qvz  or a
combination of results from Attractor Flow Tree formulae and known results
about DT invariants Alexandrov:2018iao ; Beaujard:2020sgs ; Mozgovoy:2020has ;
Mozgovoy:2021iwz . The expected motivic DT invariant for any number of D0
branes is Mozgovoy:2020has

$\Omega(n\gamma_{D0};y)=y^{3}+2y+y^{-1}\,.$ (102)

The corresponding contribution to $\mathbb{U}(\mathbb{R}^{+})$ is a product of
quantum dilogarithms as outlined in (24) _et seq._

$\mathbb{U}(\mathbb{R}^{+},\mathcal{C}_{1})=\mathbb{U}_{0}(\mathbb{R}^{+},\mathcal{C}_{1})\cdot\prod_{n\geq
1}\Phi((-y)^{-1}X_{n\gamma_{D0}})^{-1}\Phi((-y)X_{n\gamma_{D0}})^{-2}\Phi((-y)^{3}X_{n\gamma_{D0}})^{-1}\,.$
(103)

It is worth recalling a subtle distinction in the counting of D0 branes: the
Donaldson-Thomas counting is based on compactly-supported de Rham cohomolgy of
quiver representation varieties, while the physical counting of BPS states
(pertaining e.g. to gauge theory in the context of geometric engineering) is
based on the $L^{2}$ cohomology Lee:1997vp ; Yi:1997eg . The latter typically
contains fewer states, and leads to Poincaré-symmetric expressions for
$\Omega(\gamma,y)$ while the former does not, see Duan:2020qjy ;
Mozgovoy:2020has  for a discussion.

In conclusion, the spectrum we obtain for local $\mathbb{F}_{0}$ in chamber
$\mathcal{C}_{1}$ is

$\begin{array}[]{|c|c|}\hline\cr\gamma&\Omega(\gamma;y)\\\
\hline\cr\hline\cr\gamma_{1}+k(\gamma_{1}+\gamma_{2})&1\\\
-\gamma_{1}+(k+1)(\gamma_{1}+\gamma_{2})&1\\\
\gamma_{3}+k(\gamma_{3}+\gamma_{4})&1\\\
-\gamma_{3}+(k+1)(\gamma_{3}+\gamma_{4})&1\\\
\hline\cr\gamma_{1}+\gamma_{2}+k\gamma_{D0}&y+y^{-1}\\\
-\gamma_{1}-\gamma_{2}+(k+1)\gamma_{D0}&y+y^{-1}\\\
(k+1)\gamma_{D0}&y^{3}+2y+y^{-1}\\\ \hline\cr\end{array}$ (104)

with $k\geq 0$, plus the respective antiparticles obtained by sending
$\gamma\to-\gamma$. This manifestly agrees with the explicit expression in
Longhi:2021qvz , it can also be shown to agree with the generating function of
stacky invariants obtained in Mozgovoy:2020has .222222We thank Sergey Mozgovoy
and Boris Pioline for correspondence on this check.

It should be noted that the spectrum is organized as _two_ copies of the
spectrum of 4d pure $SU(2)$ SYM from the subquivers shown in Figure 18, in
which the local $\mathbb{F}_{0}$ quiver is divided by the time flows, with
additional towers of D0 branes over the vector multiplet states.

(a) Four-dimensional subquivers for the flow $T_{1}$ of $\mathbb{F}_{0}$.

(b) Four-dimensional subquivers for the flow $T_{2}$ of $\mathbb{F}_{0}$.

Figure 10:

######  Remark 3.

Equation (93) translates into the following wall-crossing identity

$\begin{split}&\prod_{k\geq
0}^{\nearrow}\Phi(X_{\gamma_{1}+k(\gamma_{1}+\gamma_{2})})\Phi(X_{\gamma_{3}+k(\gamma_{3}+\gamma_{4})})\\\
&\qquad\times\prod_{s=\pm 1}\prod_{k\geq
0}\Phi((-y)^{s}X_{\gamma_{1}+\gamma_{2}+k\gamma_{D0}})^{-1}\cdot\Phi((-y)^{s}X_{\gamma_{3}+\gamma_{4}+k\gamma_{D0}})^{-1}\\\
&\qquad\qquad\times\prod_{k\geq
0}^{\searrow}\Phi(X_{\gamma_{2}+k(\gamma_{1}+\gamma_{2})})\Phi(X_{\gamma_{4}+k(\gamma_{3}+\gamma_{4})})\\\
=&\prod_{k\geq
0}^{\nearrow}\Phi(X_{\gamma_{2}+k(\gamma_{2}+\gamma_{3})})\Phi(X_{\gamma_{4}+k(\gamma_{4}+\gamma_{1})})\\\
&\qquad\times\prod_{s=\pm 1}\prod_{k\geq
0}\Phi((-y)^{s}X_{\gamma_{4}+\gamma_{1}+k\gamma_{D0}})^{-1}\cdot\Phi((-y)^{s}X_{\gamma_{2}+\gamma_{3}+k\gamma_{D0}})^{-1}\\\
&\qquad\qquad\times\prod_{k\geq
0}^{\searrow}\Phi(X_{\gamma_{3}+k(\gamma_{2}+\gamma_{3})})\Phi(X_{\gamma_{1}+k(\gamma_{4}+\gamma_{1})})\\\
\end{split}$ (105)

where we factored out the contribution of pure-flavor states (pure D0 branes),
which do not participate in wall-crossing. We have tested this as an identity
in the numerical ($y\to-1$) setting, by acting on $X_{\gamma_{i}}$ and
expanding as formal series in $X_{\gamma}$ with $|\gamma|\leq 12$.

###  4.2 $dP_{3}$

Figure 11: BPS quiver for local $\text{dP}_{3}$

The BPS quiver for the theory corresponding to local $dP_{3}$ is depicted in
Figure 11, and has adjacency matrix

$B=\left(\begin{array}[]{cccccc}0&-1&-1&0&1&1\\\ 1&0&-1&-1&0&1\\\
1&1&0&-1&-1&0\\\ 0&1&1&0&-1&-1\\\ -1&0&1&1&0&-1\\\ -1&-1&0&1&1&0\\\
\end{array}\right)$ (106)

The permutation symmetry of this quiver is $\Pi_{Q}\simeq\mathbb{Z}_{6}$. The
adjacency matrix has rank 2, hence $\mathrm{rk}\,\Gamma_{f}=4$.

#### Translations of the $(A_{1}+A_{2})^{(1)}$ lattice.

The low-energy gauge theory phase of this theory is $SU(2)$ Super Yang-Mills
with two fundamental hypermultiplets, and the affine root lattice is
$\Gamma_{f}\simeq\mathcal{Q}((A_{1}+A_{2})^{(1)})$. We can take as its
generators the following vectors in $\Gamma$:

$\displaystyle\alpha_{0}=\gamma_{3}+\gamma_{6},$
$\displaystyle\alpha_{1}=\gamma_{1}+\gamma_{4},$
$\displaystyle\alpha_{2}=\gamma_{2}+\gamma_{5},$ (107)
$\displaystyle\beta_{0}=\gamma_{2}+\gamma_{4}+\gamma_{6},$
$\displaystyle\beta_{1}=\gamma_{1}+\gamma_{3}+\gamma_{5}.$ (108)

The null root is again $\delta=\sum_{i}\gamma_{i}$.

Note how the affine root lattice $\mathcal{Q}((A_{1}+A_{2})^{(1)})$ has rank
four, with generators the simple roots of $A_{2}$, the simple root of $A_{1}$
and the null root $\delta$ which is ‘shared’. This is in line with the
statement that such an affine root lattice is isomorphic to $\Gamma_{f}$ which
also has rank four. As a matter of fact, this is reflected in the explicit
form of the isomorphism (42) given above, where it is clear that the five
generators obey the relation

$\alpha_{0}+\alpha_{1}+\alpha_{2}=\beta_{0}+\beta_{1}\equiv\delta\,.$ (109)

In this case the Cremona group coincides with the extended Weyl group

$\mathrm{Cr}(dP_{3})\simeq\widetilde{W}((A_{1}+A_{2})^{(1)})=W((A_{1}+A_{2})^{(1)})\rtimes{\rm
Dih}_{6}\,.$ (110)

This is generated by the reflections on the $A_{2},A_{1}$ sublattices

$\displaystyle s_{0}=(3,6)\mu_{6}\mu_{3}$ $\displaystyle
s_{1}=(1,4)\mu_{4}\mu_{1},$ $\displaystyle s_{2}=(2,5)\mu_{5}\mu_{2},$ (111)
$\displaystyle r_{0}=(4,6)\mu_{2}\mu_{4}\mu_{6}\mu_{2},$ $\displaystyle
r_{1}=(3,5)\mu_{1}\mu_{3}\mu_{5}\mu_{1},$ (112)

and by the outer automorphisms generating ${\rm Dih}_{6}$

$\displaystyle\pi=(1,2,3,4,5,6),$
$\displaystyle\sigma=(1,4)(2,3)(5,6)\iota\,.$ (113)

Of these, $\pi$ is an order-six Dynkin diagram automorphism that permutes
simple roots, see Figure 12. This is the generator of $\Pi_{Q}\subset{\rm
Dih}_{6}$ that we’ll use to constrain $\mathbb{U}$.

$\beta_{0}$$\beta_{1}$$\pi$$\pi$

$\alpha_{2}$$\alpha_{1}$$\alpha_{0}$$\pi$$\pi$$\pi$

Figure 12: $(A_{2}+A_{1})^{(1)}$ Dynkin diagrams and automorphisms.
$\pi^{2}=id$ on the $A_{1}^{(1)}$ sublattice, while $\pi^{3}=id$ on the
$A_{2}^{(1)}$ sublattice. The two combine to give the $\mathbb{Z}_{6}$
permutation symmetry of the quiver.

We will focus here only on the $A_{2}^{(1)}$ sublattice spanned by
$\alpha_{1},\alpha_{2},\alpha_{3}$, since it is the only one whose roots
contain only locally commuting charges, so that it gives rise to a collimation
chamber as discussed in Section 3.4. (The sublattice $A_{1}^{(1)}$ does not
satisfy our technical assumption (a1), we leave its discussion to future
work.) The subgroup of affine Weyl translations

$\mathbb{Z}^{3}\simeq\mathcal{T}(A_{2}^{(1)})\subset\mathcal{T}((A_{1}+A_{2})^{(1)})$
(114)

is realized in $\mathrm{Aut}_{Q}$ by the following generators

$\displaystyle T_{1}=\pi^{2}s_{2}s_{1},$ $\displaystyle
T_{2}=\pi^{2}s_{0}s_{2},$ $\displaystyle T_{3}=\pi^{2}s_{1}s_{0}\,.$ (115)

Its action on the sublattice of the affine root lattice $\Gamma_{f}$
corresponding to $\mathcal{Q}(A_{2}^{(1)})$ is

$\displaystyle
T_{1}^{n}(\vec{\alpha})=\left(\begin{array}[]{c}\alpha_{0}-n\delta,\\\
\alpha_{1}+n\delta,\\\ \alpha_{2}\end{array}\right),$ $\displaystyle
T_{2}^{n}(\vec{\alpha})=\left(\begin{array}[]{c}\alpha_{0},\\\
\alpha_{1}-n\delta,\\\ \alpha_{2}+n\delta\end{array}\right)$ $\displaystyle
T_{3}^{n}(\vec{\alpha})=\left(\begin{array}[]{c}\alpha_{0}+n\delta,\\\
\alpha_{1},\\\ \alpha_{2}-n\delta\end{array}\right).$ (125)

This action extends to $\Gamma$ as follows

$\begin{split}&T_{1}^{n}(\vec{\gamma})=\left(\begin{array}[]{c}\gamma_{1}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}),\\\
\gamma_{2},\\\ \gamma_{3}-n(\gamma_{1}+\gamma_{2}+\gamma_{3}),\\\
\gamma_{4}+n(\gamma_{4}+\gamma_{5}+\gamma_{6}),\\\ \gamma_{5},\\\
\gamma_{6}-n(\gamma_{4}+\gamma_{5}+\gamma_{6})\end{array}\right)\,,\qquad
T_{2}^{n}(\vec{\gamma})=\left(\begin{array}[]{c}\gamma_{1}-n(\gamma_{1}+\gamma_{5}+\gamma_{6})\\\
\gamma_{2}+n(\gamma_{2}+\gamma_{3}+\gamma_{4})\\\ \gamma_{3}\\\
\gamma_{4}-n(\gamma_{2}+\gamma_{3}+\gamma_{4})\\\
\gamma_{5}+n(\gamma_{1}+\gamma_{5}+\gamma_{6})\\\
\gamma_{6}.\end{array}\right)\,,\\\ &\qquad\qquad\qquad\qquad
T_{3}^{n}(\vec{\gamma})=\left(\begin{array}[]{c}\gamma_{1}\\\
\gamma_{2}-n(\gamma_{1}+\gamma_{2}+\gamma_{6})\\\
\gamma_{3}+n(\gamma_{3}+\gamma_{4}+\gamma_{5})\\\ \gamma_{4}\\\
\gamma_{5}-n(\gamma_{3}+\gamma_{4}+\gamma_{5})\\\
\gamma_{6}+n(\gamma_{1}+\gamma_{2}+\gamma_{6}).\end{array}\right)\,.\end{split}$
(126)

As in the case of local $\mathbb{F}_{0}$, we observe that the three flows are
related by the action of $\mathbb{Z}_{3}\subset\mathrm{Out}(A_{2}^{(1)})$
permuting the three simple roots $\alpha_{i}$.232323Note that $\pi$ is a
$\mathbb{Z}_{6}\simeq\Pi_{Q}$ generator on the full charge lattice $\Gamma$,
and also on the affine root lattice $\mathcal{Q}((A_{1}+A_{2})^{(1)})$. This
$\mathbb{Z}_{6}$ is contained in the outer automorphism group
$\mathrm{Out}((A_{1}+A_{2})^{(1)})\simeq{\rm Dih}_{6}$. Then $\pi^{2}$
generates a $\mathbb{Z}_{3}$ outer automorphisms from
$\mathrm{Out}(A_{2}^{(1)})\simeq{\rm Dih}_{3}$, which acts trivially on
$\mathcal{Q}(A_{1}^{(1)})$.

$\begin{split}T_{1}&=\pi^{-2}T_{2}\pi^{2}=\pi^{-4}T_{3}\pi^{4}\,.\end{split}$
(127)

(a) Stability condition for $T_{1}$

(b) Stability condition for $T_{2}$

(c) Stability condition for $T_{3}$

Figure 13: Stability conditions for $dP_{3}$.

#### Collimation chambers.

Having identified the relevant flows (115), we now study the associated
stability conditions following the procedure outlined in Section 3.4. Recall
that $\delta=\sum_{i=1}^{6}\gamma_{i}$, while $\alpha_{i}$ are expressed in
terms of $\gamma_{i}$ as in (107). Then the conditions corresponding to
equations (51) and (53) translate here into

$\begin{cases}\varphi_{1}+\varphi_{2}+\varphi_{3}+\varphi_{4}+\varphi_{5}+\varphi_{6}=0\\\
\varphi_{3}=\varphi_{6},\\\ \varphi_{1}=\varphi_{4},\\\
\varphi_{2}=\varphi_{5}.\\\ \end{cases}$ (128)

This system by itself does not yet determine the stability condition. Recall
indeed that we impose an additional requirement (52), corresponding to
demanding that limiting rays lie on the real axis. The accumulation rays for
the flow $T_{1}$ in (48) can be read off from (LABEL:eq:T1T2-dP3-charges)

$\delta_{1}(T_{1})=\gamma_{1}+\gamma_{2}+\gamma_{3}\,,\qquad\delta_{2}(T_{1})=\gamma_{4}+\gamma_{5}+\gamma_{6}\,,$
(129)

therefore we demand

$\varphi_{1}+\varphi_{2}+\varphi_{3}=\varphi_{4}+\varphi_{5}+\varphi_{6}=0.$
(130)

As $\gamma_{2},\gamma_{5}$ are invariant under $T_{1}$, we can think of them
in a certain sense as limiting vectors as well. It is then quite natural to
also impose

$\varphi_{2}=\varphi_{5}=0.$ (131)

Finally, observing that the first charge mutated by $T_{1}$ in (115) is
$\gamma_{1}$, the stability condition corresponding to the first flow is

$\displaystyle\text{Stability condition for }T_{1}:$
$\displaystyle\varphi_{1}=\varphi_{4}>\varphi_{2}=\varphi_{5}=0>\varphi_{3}=\varphi_{6}=-\varphi_{1},$
(132)

as in Figure 13(a).

######  Remark 4.

The condition $\theta_{2}=\theta_{5}=0$ is not strictly necessary, but we need
it to fix everything since $A_{2}^{(1)}$ is only a sublattice of the full root
system. It is suggested by the fact that these states are left invariant by
the flow $T_{1}$ (and so they are strictly speaking limiting vectors for the
flow, although they are hypermultiplets). It is consistent since

$\begin{split}\langle\gamma_{2},\gamma_{5}\rangle&=\langle\gamma_{2},\gamma_{1}+\gamma_{2}+\gamma_{3}\rangle=\langle\gamma_{5},\gamma_{1}+\gamma_{2}+\gamma_{3}\rangle\\\
&=\langle\gamma_{2},\gamma_{4}+\gamma_{5}+\gamma_{6}\rangle=\langle\gamma_{5},\gamma_{4}+\gamma_{5}+\gamma_{6}\rangle=0.\end{split}$
(133)

The conditions for the flows $T_{2}$ and $T_{3}$ respectively are obtained by
applying $\pi^{2}$ and $\pi^{4}$ to the charges, so we have the other two
stability conditions

$\displaystyle\text{Stability condition for }T_{2}:$
$\displaystyle\varphi_{3}=\varphi_{6}>\varphi_{4}=\varphi_{1}=0>\varphi_{2}=\varphi_{5}=-\varphi_{3},$
(134) $\displaystyle\text{Stability condition for }T_{3}:$
$\displaystyle\varphi_{2}=\varphi_{5}>\varphi_{3}=\varphi_{6}=0>\varphi_{1}=\varphi_{4}=-\varphi_{2},$
(135)

illustrated in Figure 13(b) and 13(c) respectively.

So far we applied the criteria of Section 3.4 to deduce the stability
conditions associated to flows $T_{i}$. As a check, we observe that iterations
of the translation generators indeed admit a realization that only involves
left-mutations Note that

$\displaystyle T_{i}^{2}=\textbf{m}_{i}^{3},$ (136)

where

$\displaystyle\textbf{m}_{1}=\mu_{6}\mu_{3}\mu_{5}\mu_{2}\mu_{4}\mu_{1},$
$\displaystyle\textbf{m}_{2}=\pi^{2}\textbf{m}_{1}\pi^{-2},$
$\displaystyle\textbf{m}_{3}=\pi^{4}\textbf{m}_{1}\pi^{-4},$ (137)

Moreover we claim that iterations of these mutations correspond to tiltings of
the positive half-plane for choices of stability data as given in (132), (134)
and (134). Once again, the time flows are permuted version of charge tiltings,
and contain the full information about the stability condition.

#### BPS spectrum and Wall-crossing invariant.

A configuration of central charges satisfying conditions (132) corresponding
to $T_{1}$ can be taken as follows

$\begin{array}[]{cc}\mathcal{C}_{1}[dP_{3}]\supset&\begin{split}&Z_{\gamma_{1}}=Z_{\gamma_{4}}\,,\quad
Z_{\gamma_{2}}=Z_{\gamma_{5}}\,,\quad Z_{\gamma_{3}}=Z_{\gamma_{6}}\,,\\\
&\arg Z_{\gamma_{1}}>\arg Z_{\gamma_{2}}>\arg Z_{\gamma_{3}}\,,\quad
Z_{\gamma_{1}}+Z_{\gamma_{3}}=Z_{\gamma_{2}}\in\mathbb{R}^{+}\,.\end{split}\end{array}$
(138)

Similarly we denote by $\mathcal{C}_{i}[dP_{3}]$ the chambers corresponding to
flows $T_{i}$ with $i=2,3$. Since the limiting rays of $T_{1}$ are (129), and
since we demanded that their central charges are real (130), the flow is
expected to produce towers of states asymptotic to the real axis (also see
Bonelli:2020dcp  for a discussion). This matches the definition of collimation
chamber.

Having identified the choice of stability condition corresponding to a
(putative) collimation chamber, we may deduce parts of the spectrum contained
in the sectors $\measuredangle^{\pm}$ of the $Z$-plane (corresponding to ${\rm
Re}Z>0$ and ${\rm Im}Z>0$ or $<0$) by tilting the positive half-plane and
following the induced sequences of left- and right-mutations. The mutation
sequences are, respectively 242424These coincide with the mutation sequences
observed in (Bonelli:2020dcp, , eq. (2.36)). To see it, it suffices to
separate all permutations (taking them to the left) from all mutations (taken
to the right) in their formulae.

$\begin{split}\text{CW
tilt}&\quad\textbf{m}_{1}=\mu_{6}\mu_{3}\mu_{5}\mu_{2}\mu_{4}\mu_{1}\\\
\text{CCW
tilt}&\quad\widetilde{\textbf{m}}_{1}=\widetilde{\mu}_{4}\widetilde{\mu}_{1}\widetilde{\mu}_{5}\widetilde{\mu}_{2}\widetilde{\mu}_{6}\widetilde{\mu}_{3}\end{split}$
(139)

Acting on $Q$ with $\textbf{m}_{1}^{n}$ and $\widetilde{\textbf{m}}_{1}^{n}$
yields respectively the following towers of hypermultiplets

$\begin{split}\Omega(\gamma_{1}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}))&=1\\\
\Omega(\gamma_{3}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}))&=1\\\
\Omega(-\gamma_{1}+(n+1)(\gamma_{1}+\gamma_{2}+\gamma_{3}))&=1\\\
\Omega(-\gamma_{3}+(n+1)(\gamma_{1}+\gamma_{2}+\gamma_{3}))&=1\end{split}$
(140)

for $n\geq 0$, and similarly with $\\{1,2,3\\}\to\\{4,5,6\\}$. As promised,
their central charges accumulate towards $\mathbb{R}^{+}$ from both sides for
$n\to+\infty$. It is then straightforward to write down
$\mathbb{U}(\measuredangle^{\pm})$ explicitly as a product of phase-ordered
quantum dilogarithms.

$\begin{split}\mathbb{U}(\measuredangle^{+},\mathcal{C}_{1})&=\prod^{\nearrow}_{k\geq
0}\Bigg{(}\Phi\left(X_{\gamma_{1}+k(\gamma_{1}+\gamma_{2}+\gamma_{3})}\right)\Phi\left(X_{\gamma_{4}+k(\gamma_{4}+\gamma_{5}+\gamma_{6})}\right)\\\
&\qquad\qquad\times\Phi\left(X_{\gamma_{1}+\gamma_{2}+k(\gamma_{1}+\gamma_{2}+\gamma_{3})}\right)\Phi\left(X_{\gamma_{4}+\gamma_{5}+k(\gamma_{4}+\gamma_{5}+\gamma_{6})}\right)\bigg{)}\\\
\mathbb{U}(\measuredangle^{-},\mathcal{C}_{1})&=\prod^{\searrow}_{k\geq
0}\Bigg{(}\Phi\left(X_{\gamma_{5}+\gamma_{6}+k(\gamma_{4}+\gamma_{5}+\gamma_{6})}\right)\Phi\left(X_{\gamma_{2}+\gamma_{3}+k(\gamma_{1}+\gamma_{2}+\gamma_{3})}\right)\\\
&\qquad\qquad\times\Phi\left(X_{\gamma_{6}+k(\gamma_{4}+\gamma_{5}+\gamma_{6})}\right)\Phi\left(X_{\gamma_{3}+k(\gamma_{1}+\gamma_{2}+\gamma_{3})}\right)\Bigg{)}\end{split}$
(141)

To determine $\mathbb{U}(\mathbb{R}^{+})$ we adopt a formal series expansion
as in (30) and express $\mathbb{U}$ as a function of the coefficients
$c_{\gamma}$ of $\mathbb{U}(\mathbb{R}^{+})$ through (25). We then impose (29)
using the $\Pi_{Q}\simeq\mathbb{Z}_{6}$ symmetry of the quiver of Figure 11

$\pi:\,\gamma_{i}\mapsto\gamma_{i+1\,{\rm mod}\,6}.$ (142)

Solving for the coefficients $c_{\gamma}$ we obtain a series in $X_{\gamma}$
up to arbitrary order $|\gamma|$, which is compatible with the following
factorization

$\begin{split}\mathbb{U}_{0}(\mathbb{R}^{+},\mathcal{C}_{1})=\prod_{k\geq
0}\Bigg{(}&\Phi(X_{\gamma_{2}+k\gamma_{D0}})\cdot\Phi(X_{\gamma_{1}+\gamma_{3}+k\gamma_{D0}})\\\
\cdot&\Phi(X_{-\gamma_{2}+(k+1)\gamma_{D0}})\cdot\Phi(X_{-\gamma_{1}-\gamma_{3}+(k+1)\gamma_{D0}})\\\
\cdot&\prod_{s=\pm
1}\Phi((-y)^{s}X_{\gamma_{1}+\gamma_{2}+\gamma_{3}+k\gamma_{D0}})^{-1}\Bigg{)}\\\
&\times\left(\\{1,2,3\\}\to\\{4,5,6\\}\right)\end{split}$ (143)

where $\gamma_{D0}=\sum_{i}\gamma_{i}$. This corresponds to a spectrum along
the real ray consisting of

$\begin{split}&\Omega(\gamma_{2}+n\gamma_{D0})=\Omega(-\gamma_{2}+(n+1)\gamma_{D0})=1\\\
&\Omega(\gamma_{5}+n\gamma_{D0})=\Omega(-\gamma_{5}+(n+1)\gamma_{D0})=1\\\
&\Omega(\gamma_{1}+\gamma_{3}+n\gamma_{D0})=\Omega(-\gamma_{1}-\gamma_{3}+(n+1)\gamma_{D0})=1\\\
&\Omega(\gamma_{4}+\gamma_{6}+n\gamma_{D0})=\Omega(-\gamma_{4}-\gamma_{6}+(n+1)\gamma_{D0})=1\\\
&\Omega(\gamma_{1}+\gamma_{2}+\gamma_{3}+n\gamma_{D0})=\Omega(-\gamma_{1}-\gamma_{2}-\gamma_{3}+(n+1)\gamma_{D0})=y+y^{-1}\end{split}$
(144)

As in the case of local $\mathbb{F}_{0}$, the spectrum of BPS states on the
real ray features towers of BPS states corresponding to Kaluza-Klein towers of
charges $\gamma_{2},\gamma_{5},\gamma_{1}+\gamma_{3},\gamma_{4}+\gamma_{6}$,
and their CPT conjugate KK towers. From the viewpoint of the splitting into
two 4d $SU(2)$ $N_{f}=1$ subquivers, these KK towers correspond to bound
states of $N_{f}=1$ quarks with D0 branes.

(a) Four-dimensional subquivers for the flow $T_{1}$ of $dP_{3}$.

(b) Four-dimensional subquivers for the flow $T_{2}$ of $dP_{3}$.

(c) Four-dimensional subquivers for the flow $T_{3}$ of $dP_{3}$.

Figure 14:

Once again this misses the contributions from pure D0 branes. We refer to the
remarks below (101) for a discussion. In this case the expected motivic DT
invariant for any number of D0 branes is Mozgovoy:2020has

$\Omega(n\gamma_{D0};y)=y^{3}+4y+y^{-1}\,.$ (145)

The corresponding contribution to $\mathbb{U}(\mathbb{R}^{+})$ is a product of
quantum dilogarithms as outlined in (24) _et seq._

$\mathbb{U}(\mathbb{R}^{+},\mathcal{C}_{1})=\mathbb{U}_{0}(\mathbb{R}^{+},\mathcal{C}_{1})\cdot\prod_{n\geq
1}\Phi((-y)^{-1}X_{n\gamma_{D0}})^{-1}\Phi((-y)X_{n\gamma_{D0}})^{-4}\Phi((-y)^{3}X_{n\gamma_{D0}})^{-1}\,.$
(146)

In conclusion, the spectrum we obtain for local $dP_{3}$ in chamber
$\mathcal{C}_{1}$ is

$\begin{array}[]{|c|c|}\hline\cr\gamma&\Omega(\gamma;y)\\\
\hline\cr\hline\cr\gamma_{r}+k(\gamma_{1}+\gamma_{2}+\gamma_{3})&1\\\
-\gamma_{r}+(k+1)(\gamma_{1}+\gamma_{2}+\gamma_{3})&1\\\
\gamma_{s}+k(\gamma_{4}+\gamma_{5}+\gamma_{6})&1\\\
-\gamma_{s}+(k+1)(\gamma_{4}+\gamma_{5}+\gamma_{6})&1\\\
\hline\cr\gamma_{t}+k\gamma_{D0}&1\\\ -\gamma_{t}+(k+1)\gamma_{D0}&1\\\
\gamma_{a}+\gamma_{b}+k\gamma_{D0}&1\\\
-\gamma_{a}-\gamma_{b}+(k+1)\gamma_{D0}&1\\\
\gamma_{1}+\gamma_{2}+\gamma_{3}+k\gamma_{D0}&y+y^{-1}\\\
-\gamma_{1}-\gamma_{2}-\gamma_{3}+(k+1)\gamma_{D0}&y+y^{-1}\\\
(k+1)\gamma_{D0}&y^{3}+4y+y^{-1}\\\ \hline\cr\end{array}$ (147)

with $k\geq 0$ and

$r\in\\{1,3\\}\qquad s\in\\{4,6\\}\qquad
t\in\\{2,5\\}\qquad(a,b)\in\\{(1,3),(4,6)\\}\,.$ (148)

The spectrum also includes the respective antiparticles obtained by sending
$\gamma\to-\gamma$.

######  Remark 5.

The wall-crossing constraint that we solved is the following identity

$\begin{split}&\prod^{\nearrow}_{k\geq
0}\Bigg{(}\Phi\left(X_{\gamma_{1}+k(\gamma_{1}+\gamma_{2}+\gamma_{3})}\right)\Phi\left(X_{\gamma_{4}+k(\gamma_{4}+\gamma_{5}+\gamma_{6})}\right)\\\
&\qquad\qquad\times\Phi\left(X_{-\gamma_{3}+(k+1)(\gamma_{1}+\gamma_{2}+\gamma_{3})}\right)\Phi\left(X_{-\gamma_{6}+(k+1)(\gamma_{4}+\gamma_{5}+\gamma_{6})}\right)\bigg{)}\\\
&\times\prod_{k\geq
0}\Bigg{(}\Phi(X_{\gamma_{2}+k\gamma_{D0}})\Phi(X_{-\gamma_{2}+(k+1)\gamma_{D0}})\Phi(X_{\gamma_{1}+\gamma_{3}+k\gamma_{D0}})\Phi(X_{-\gamma_{1}-\gamma_{3}+(k+1)\gamma_{D0}})\\\
&\qquad\times\Phi(X_{\gamma_{5}+k\gamma_{D0}})\Phi(X_{-\gamma_{5}+(k+1)\gamma_{D0}})\Phi(X_{\gamma_{4}+\gamma_{6}+k\gamma_{D0}})\Phi(X_{-\gamma_{4}-\gamma_{6}+(k+1)\gamma_{D0}})\\\
&\qquad\times\left(\prod_{s=\pm
1}\Phi((-y)^{s}X_{\gamma_{1}+\gamma_{2}+\gamma_{3}+k\gamma_{D0}})^{-1}\Phi((-y)^{s}X_{-\gamma_{1}-\gamma_{2}-\gamma_{3}+(k+1)\gamma_{D0}})^{-1}\right)\Bigg{)}\\\
&\times\prod^{\searrow}_{k\geq
0}\Bigg{(}\Phi\left(X_{-\gamma_{4}+(k+1)(\gamma_{4}+\gamma_{5}+\gamma_{6})}\right)\Phi\left(X_{-\gamma_{1}+(k+1)(\gamma_{1}+\gamma_{2}+\gamma_{3})}\right)\\\
&\qquad\qquad\times\Phi\left(X_{\gamma_{6}+k(\gamma_{4}+\gamma_{5}+\gamma_{6})}\right)\Phi\left(X_{\gamma_{3}+k(\gamma_{1}+\gamma_{2}+\gamma_{3})}\right)\Bigg{)}\\\
=&\left(\\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5},\gamma_{6}\\}\to\\{\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5},\gamma_{6},\gamma_{1}\\}\right)\end{split}$
(149)

where we factored out the contribution of pure-flavor states (pure D0 branes),
which do not participate in wall-crossing. We have tested this as an identity
in the numerical ($y\to-1$) setting, by acting on $X_{\gamma_{i}}$ and
expanding as formal series in $X_{\gamma}$ with $|\gamma|\leq 12$.

###  4.3 $dP_{5}$

Figure 15: BPS quiver for $\text{dP}_{5}$

The BPS quiver for the theory corresponding to local $dP_{5}$ is depicted in
Figure 15, and has adjacency matrix252525This quiver is related by mutations
to the one obtained in (Hanany:2001py, , Figure 19). It also matches with the
quiver in (Wijnholt:2002qz, , Figure 8A). As pointed out in Wijnholt:2002qz
this is also equivalent to the quiver in Figure 7 of the same paper, which
further coincides with the recent derivation in (Closset:2021lhd, , eq.
(8.44)). Although local $dP_{5}$ is not toric, this quiver can also be
obtained by techniques for toric CY threefolds, see model 4a in Hanany:2012hi
, which corresponds to a _pseudo_ del Pezzo surface $PdP_{5}$. The distinction
between quivers of $dP_{5}$ and $PdP_{5}$ lies in the possible presence of
bidirectional arrows for the latter Beaujard:2020sgs , however this depends on
the superpotential, which plays no role in our construction.

$B=\left(\begin{array}[]{cccccccc}0&0&-1&-1&0&0&1&1\\\ 0&0&-1&-1&0&0&1&1\\\
1&1&0&0&-1&-1&0&0\\\ 1&1&0&0&-1&-1&0&0\\\ 0&0&1&1&0&0&-1&-1\\\
0&0&1&1&0&0&-1&-1\\\ -1&-1&0&0&1&1&0&0\\\ -1&-1&0&0&1&1&0&0\\\
\end{array}\right)$ (150)

The permutation symmetry of this quiver is
$\Pi_{Q}\simeq(\mathbb{Z}_{2})^{\times 4}\rtimes\mathbb{Z}_{4}$. The adjacency
matrix of this quiver has rank two, so $\mathrm{rk}\,\Gamma_{f}=6$.

#### Translations of the $D_{5}^{(1)}$ lattice.

The low-energy gauge theory phase of this theory is $SU(2)$ Super Yang-Mills
with four fundamental hypermultiplets, and the affine root lattice is
$\mathcal{Q}(D_{5}^{(1)})\simeq\Gamma_{f}$. We can take as its generators the
following vectors in $\Gamma$:

$\displaystyle\alpha_{0}=\gamma_{2}-\gamma_{1},$
$\displaystyle\alpha_{1}=\gamma_{6}-\gamma_{5},$
$\displaystyle\alpha_{2}=\gamma_{1}+\gamma_{5},$ (151)
$\displaystyle\alpha_{3}=\gamma_{3}+\gamma_{7},$
$\displaystyle\alpha_{4}=\gamma_{4}-\gamma_{3},$
$\displaystyle\alpha_{5}=\gamma_{8}-\gamma_{7}.$ (152)

The null root is again $\delta=\sum_{i}\gamma_{i}$.

The Cremona group coincides with the extended affine Weyl group

$\mathrm{Cr}(dP_{5})=\widetilde{W}(D_{5}^{(1)})={W}(D_{5}^{(1)})\rtimes\mathrm{Out}(D_{5}^{(1)})\,.$
(153)

Note that $\mathrm{Out}(D_{5}^{(1)})$ contains the permutation symmetries
$\Pi_{Q}$ as a subgroup. Indeed, $\mathrm{Cr}(dP_{5})$ can be realized in
$\mathrm{Aut}_{Q}$ as follows

$\displaystyle s_{0}=(1,2),$ $\displaystyle s_{1}=(5,6),$ $\displaystyle
s_{2}=(1,5)\mu_{1}\mu_{5},$ (154) $\displaystyle s_{3}=(3,7)\mu_{3}\mu_{7},$
$\displaystyle s_{4}=(3,4),$ $\displaystyle s_{5}=(7,8),$ (155)
$\displaystyle\pi=(1,3,5,7)(2,4,6,8),$
$\displaystyle\sigma=(1,7)(2,8)(3,5)(4,6)\iota,$ (156)

where $s_{i}$ is an elementary reflection along the root $\alpha_{i}$, and

$\pi(\vec{\alpha})=\left(\begin{array}[]{c}\alpha_{4}\\\ \alpha_{5}\\\
\alpha_{3}\\\ \alpha_{2}\\\ \alpha_{1}\\\ \alpha_{0}\end{array}\right)$ (157)

is the Dynkin diagram automorphism of Figure 16. All affine simple roots
$\alpha_{i}$ correspond to sums of mutually local charges, in fulfillment of
assumption (a3) in Section 3.4.

$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\alpha_{4}$$\alpha_{0}$$\alpha_{5}$$\pi$$\pi$$\pi$$\pi$$\pi$
Figure 16: $D_{5}^{(1)}$ Dynkin diagram and automorphisms

Instead of discussing the whole subgroup of affine Weyl translations
$\mathcal{T}(D_{5}^{(1)})$, we just focus on the $\mathbb{Z}$-subgroup
generated by

$\displaystyle T_{1}^{n}(\vec{\alpha})=\left(\begin{array}[]{c}\alpha_{0}\\\
\alpha_{1}\\\ \alpha_{2}+n\delta\\\ \alpha_{3}-n\delta\\\ \alpha_{4}\\\
\alpha_{5}\end{array}\right).$ (164)

This admits the following realization in $\mathrm{Aut}_{Q}$

$T_{1}=s_{3}s_{4}s_{5}s_{3}s_{2}s_{1}s_{0}s_{2}(2,6)(1,5)(4,8)(7,3)\,,$ (165)

which extends the definition of $T_{1}$ to the whole charge lattice $\Gamma$

$\displaystyle
T_{1}^{n}(\vec{\gamma})=\left(\begin{array}[]{c}\gamma_{1}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})\\\
\gamma_{2}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})\\\
\gamma_{3}-n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})\\\
\gamma_{4}-n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})\\\
\gamma_{5}+n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})\\\
\gamma_{6}+n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})\\\
\gamma_{7}-n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})\\\
\gamma_{8}-n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})\end{array}\right).$
(174)

We can obtain another, independent flow, by applying the Dynkin diagram
automorphism

$T_{2}=\pi T_{1}\pi^{-1}$ (175)

acting on roots, and on the whole $\Gamma$, respectively as follows

$\displaystyle T_{2}^{n}(\vec{\alpha})=\left(\begin{array}[]{c}\alpha_{0}\\\
\alpha_{1}\\\ \alpha_{2}-n\delta\\\ \alpha_{3}+n\delta\\\ \alpha_{4}\\\
\alpha_{5}\end{array}\right),$ $\displaystyle
T_{2}^{n}(\vec{\gamma})=\left(\begin{array}[]{c}\gamma_{1}-n(\gamma_{1}+\gamma_{2}+\gamma_{7}+\gamma_{8})\\\
\gamma_{2}-n(\gamma_{1}+\gamma_{2}+\gamma_{7}+\gamma_{8})\\\
\gamma_{3}+n(\gamma_{3}+\gamma_{4}+\gamma_{5}+\gamma_{6})\\\
\gamma_{4}+n(\gamma_{3}+\gamma_{4}+\gamma_{5}+\gamma_{6})\\\
\gamma_{5}-n(\gamma_{3}+\gamma_{4}+\gamma_{5}+\gamma_{6})\\\
\gamma_{6}-n(\gamma_{3}+\gamma_{4}+\gamma_{5}+\gamma_{6})\\\
\gamma_{7}+n(\gamma_{1}+\gamma_{2}+\gamma_{7}+\gamma_{8})\\\
\gamma_{8}+n(\gamma_{1}+\gamma_{2}+\gamma_{7}+\gamma_{8})\end{array}\right).$
(190)

######  Remark 6.

Note that the Weyl translation naturally splits the BPS quiver into two four-
dimensional subquivers Bonelli:2020dcp . The towers of states that we find are
those appearing in the weakly-coupled spectrum of these subquivers, shown in
Figure 1

#### Collimation chambers.

We now turn to the discussion of collimation chambers associated to the flows
$T_{1}$ and $T_{2}$. Recalling that $\delta=\sum_{i=1}^{8}\gamma_{i}$, while
$\alpha_{i}$ are expressed in terms of $\gamma_{i}$ as in (151), conditions
corresponding to equations (51) and (53) translate here into262626This
explains why we considered just the flow $T_{1}$ instead of the whole
translation group $\mathcal{T}(D_{5}^{(1)})$. In principle the root lattice
$\mathcal{Q}(D_{5}^{(1)})$ allows for many different flows. However, the
technical condition (53), that requires all charges entering in a root in
(151) to lie on the same ray, greatly restricts our allowed choices of
tiltings. It is plausible that by relaxing this condition one could make sense
also of the other affine translations.

$\begin{cases}\sum_{i}\varphi_{i}=0,\\\ \varphi_{2}=\varphi_{1},\\\
\varphi_{6}=\varphi_{5},\\\ \varphi_{1}=\varphi_{5},\\\
\varphi_{3}=\varphi_{7},\\\ \varphi_{4}=\varphi_{3},\\\
\varphi_{8}=\varphi_{7}.\end{cases}$ (191)

which is solved by

$\varphi_{1}=\varphi_{2}=\varphi_{5}=\varphi_{6}=-\varphi_{3}=-\varphi_{4}=-\varphi_{7}=-\varphi_{8}.$
(192)

These constraints leave room for precisely two configurations of central
charges within the positive half-plane, namely

$\mathcal{C}_{1}[dP_{5}]:\quad\varphi_{1}>\varphi_{3}\,,\qquad\qquad\mathcal{C}_{2}[dP_{5}]:\quad\varphi_{1}<\varphi_{3}\,.$
(193)

The corresponding ray diagrams are shown in Figure 17.

So far we applied the criteria of Section 3.4 to deduce the stability
conditions associated to flows $T_{1}$ and $T_{2}$. As a check, we observe
that iterations of these translation generators indeed admit a realization
that only involves left-mutations

$\begin{split}T_{1}&=\textbf{m}_{1}:=\mu_{3}\mu_{4}\mu_{7}\mu_{8}\mu_{1}\mu_{2}\mu_{5}\mu_{6}\,,\\\
T_{2}&=\textbf{m}_{2}:=\mu_{1}\mu_{2}\mu_{5}\mu_{6}\mu_{3}\mu_{4}\mu_{7}\mu_{8}\,.\end{split}$
(194)

Moreover we claim that iterations of these mutations correspond to tiltings of
the positive half-plane for choices of stability data as given in (193).

(a) Stability condition associated to $T_{1}$

(b) Stability condition associated to $T_{2}$

Figure 17: Stability conditions for $dP_{5}$, corresponding to
$\mathcal{C}_{1}[dP_{5}]$ and $\mathcal{C}_{2}[dP_{5}]$.

#### BPS spectrum and Wall-crossing invariant.

A configuration of central charges satisfying conditions (192)-(193)
corresponding to $T_{1}$ can be taken as follows

$\begin{array}[]{cc}\mathcal{C}_{1}[dP_{5}]\supset&\begin{split}&Z_{\gamma_{2}}=Z_{\gamma_{6}}=c\cdot
Z_{\gamma_{1}}=c\cdot Z_{\gamma_{5}}\,,\quad
Z_{\gamma_{4}}=Z_{\gamma_{8}}=c\cdot Z_{\gamma_{3}}=c\cdot
Z_{\gamma_{7}}\,,\\\ &Z_{\gamma_{1}}=\overline{Z}_{\gamma_{3}}\,,\quad\arg
Z_{\gamma_{1}}>\arg Z_{\gamma_{3}}\,,\end{split}\end{array}$ (195)

with an arbitrary real constant $c>1$, needed to ensure that $Z_{\alpha_{i}}$
belong to the positive half-plane. Likewise, a configuration of central
charges in chamber $\mathcal{C}_{2}[dP_{5}]$ is relatetd to
$\mathcal{C}_{1}[dP_{5}]$ by switching the last condition with $\arg
Z_{\gamma_{1}}<\arg Z_{\gamma_{3}}$.

Focusing on $\mathcal{C}_{1}[dP_{5}]$, and tilting the positive half-plane of
central charges clockwise, induces infinite iterations of the sequence of
mutations $\textbf{m}_{1}$. Likewise, tilting CCW we find an infinite
repetition of the following sequence

$\widetilde{\bf{m}}_{1}:=\widetilde{\mu}_{6}\widetilde{\mu}_{5}\widetilde{\mu}_{2}\widetilde{\mu}_{1}\widetilde{\mu}_{8}\widetilde{\mu}_{7}\widetilde{\mu}_{4}\widetilde{\mu}_{3}\,.$
(196)

Together the mutation sequences ${\bf m}_{1},\widetilde{\bf m}_{1}$ generate
towers of hypermultiplets

$\begin{split}\Omega(\gamma_{1}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}))&=1\\\
\Omega(\gamma_{2}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}))&=1\\\
\Omega(\gamma_{3}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}))&=1\\\
\Omega(\gamma_{4}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}))&=1\\\
\Omega(\gamma_{1}+\gamma_{2}+\gamma_{3}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}))&=1\\\
\Omega(\gamma_{1}+\gamma_{2}+\gamma_{4}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}))&=1\\\
\Omega(\gamma_{3}+\gamma_{4}+\gamma_{1}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}))&=1\\\
\Omega(\gamma_{3}+\gamma_{4}+\gamma_{2}+n(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}))&=1\\\
\end{split}\qquad\begin{split}\Omega(\gamma_{5}+n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8}))&=1\\\
\Omega(\gamma_{6}+n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8}))&=1\\\
\Omega(\gamma_{7}+n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8}))&=1\\\
\Omega(\gamma_{8}+n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8}))&=1\\\
\Omega(\gamma_{5}+\gamma_{6}+\gamma_{7}+n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8}))&=1\\\
\Omega(\gamma_{5}+\gamma_{6}+\gamma_{8}+n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8}))&=1\\\
\Omega(\gamma_{7}+\gamma_{8}+\gamma_{5}+n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8}))&=1\\\
\Omega(\gamma_{7}+\gamma_{8}+\gamma_{6}+n(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8}))&=1\\\
\end{split}$ (197)

Since
$Z_{\gamma_{1}+\dots+\gamma_{4}}=Z_{\gamma_{5}+\dots+\gamma_{8}}\in\mathbb{R}^{+}$
the BPS rays of these hypermultiplets asymptote to $\mathbb{R}^{+}$ form both
sides, confirming that (195) defines a collimation chamber. It should be noted
that, although many central charges have the same phase, this does not
correspond to a wall of marginal stability because all relevant charges are
mutually local.

$\begin{split}\mathbb{U}(\measuredangle^{+},\mathcal{C}_{1})&=\prod^{\nearrow}_{k\geq
0}\left[\left(\prod_{\ell=1,2}\Phi\left(X_{\gamma_{\ell}+k(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})}\right)\right)\left(\prod_{\ell=5,6}\Phi\left(X_{\gamma_{\ell}+k(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})}\right)\right)\right.\\\
&\times\left.\left(\prod_{\ell=3,4}\Phi\left(X_{\gamma_{1}+\gamma_{2}+\gamma_{\ell}+k(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})}\right)\right)\left(\prod_{\ell=7,8}\Phi\left(X_{\gamma_{5}+\gamma_{6}+\gamma_{\ell}+k(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})}\right)\right)\right]\\\
\mathbb{U}(\measuredangle^{-},\mathcal{C}_{1})&=\prod^{\searrow}_{k\geq
0}\left[\left(\prod_{\ell=1,2}\Phi\left(X_{\gamma_{3}+\gamma_{4}+\gamma_{\ell}+k(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})}\right)\right)\left(\prod_{\ell=5,6}\Phi\left(X_{\gamma_{7}+\gamma_{8}+\gamma_{\ell}+k(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})}\right)\right)\right.\\\
&\times\left.\left(\prod_{\ell=3,4}\Phi\left(X_{\gamma_{\ell}+k(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})}\right)\right)\left(\prod_{\ell=7,8}\Phi\left(X_{\gamma_{\ell}+k(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})}\right)\right)\right]\end{split}$
(198)

Following the familiar script already used for $\mathbb{F}_{0}$ and $dP_{3}$,
to determine $\mathbb{U}(\mathbb{R}^{+})$ we adopt a formal series expansion
as in (30) and express $\mathbb{U}$ as a function of the coefficients
$c_{\gamma}$ of $\mathbb{U}(\mathbb{R}^{+})$ through (25). We then impose (29)
using the generator $\pi$ of the $\mathbb{Z}_{4}\subset\Pi_{Q}$
symmetry272727Note that $(\mathbb{Z}_{2})^{\times 4}$ acts trivially on
$\mathcal{C}_{1}$. Moreover, only a $\mathbb{Z}_{2}$ subgroup of
$\mathbb{Z}_{4}$ acts nontrivially on $\mathcal{C}_{1}$, exchanging stability
conditions in Figure 17. of the quiver of Figure 15 Solving for the
coefficients $c_{\gamma}$ we obtain a series in $X_{\gamma}$ up to arbitrary
order $|\gamma|$, which is compatible with the factorization

$\mathbb{U}(\mathbb{R}^{+})=\prod_{\gamma}\prod_{m\in\mathbb{Z}}\Phi((-y)^{m}X_{\gamma})^{a_{m}(\gamma)}$
(199)

where $\Omega(\gamma,y)=\sum_{m}(-y)^{m}a_{m}(\gamma)$ are as follows

$\begin{split}\Omega(\gamma_{1}+\gamma_{3}+n\gamma_{D0})=\Omega(-\gamma_{1}-\gamma_{3}+(n+1)\gamma_{D0})&=1\\\
\Omega(\gamma_{1}+\gamma_{4}+n\gamma_{D0})=\Omega(-\gamma_{1}-\gamma_{4}+(n+1)\gamma_{D0})&=1\\\
\Omega(\gamma_{2}+\gamma_{3}+n\gamma_{D0})=\Omega(-\gamma_{2}-\gamma_{3}+(n+1)\gamma_{D0})&=1\\\
\Omega(\gamma_{2}+\gamma_{4}+n\gamma_{D0})=\Omega(-\gamma_{2}-\gamma_{4}+(n+1)\gamma_{D0})&=1\\\
\Omega(\gamma_{5}+\gamma_{7}+n\gamma_{D0})=\Omega(-\gamma_{5}-\gamma_{7}+(n+1)\gamma_{D0})&=1\\\
\Omega(\gamma_{5}+\gamma_{8}+n\gamma_{D0})=\Omega(-\gamma_{5}-\gamma_{8}+(n+1)\gamma_{D0})&=1\\\
\Omega(\gamma_{6}+\gamma_{7}+n\gamma_{D0})=\Omega(-\gamma_{6}-\gamma_{7}+(n+1)\gamma_{D0})&=1\\\
\Omega(\gamma_{6}+\gamma_{8}+n\gamma_{D0})=\Omega(-\gamma_{6}-\gamma_{8}+(n+1)\gamma_{D0})&=1\\\
\Omega(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}+n\gamma_{D0})=\Omega(-\gamma_{1}-\gamma_{2}-\gamma_{3}-\gamma_{4}+(n+1)\gamma_{D0})&=y+y^{-1}\end{split}$
(200)

Once again this misses the contributions from pure D0 branes. We refer to the
remarks below (101) for a discussion. Based on the observation that the moduli
space of a D0 brane (or a boundstate of $n$ D0 branes) is the Calabi-Yau
itself, we expect that once again the Protected Spin Character of $n$ D0
equals the Poincaré polynomial of the local del Pezzo Mozgovoy:2020has .
Recalling from Section 3.1 that $\dim H^{2}(S,\mathbb{Z})=n+1$ we conjecture

$\Omega(n\gamma_{D0};y)=y^{3}+6y+y^{-1}\,.$ (201)

In conclusion, the spectrum we obtain for local $dP_{5}$ in chamber
$\mathcal{C}_{1}$ is

$\begin{array}[]{|c|c|}\hline\cr\gamma&\Omega(\gamma;y)\\\
\hline\cr\hline\cr\gamma_{r}+k(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})&1\\\
-\gamma_{r}+(k+1)(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})&1\\\
\gamma_{s}+k(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})&1\\\
-\gamma_{s}+(k+1)(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})&1\\\
\hline\cr\gamma_{a}+\gamma_{b}+k\gamma_{D0}&1\\\
-\gamma_{a}-\gamma_{b}+(k+1)\gamma_{D0}&1\\\
\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}+k\gamma_{D0}&y+y^{-1}\\\
-\gamma_{1}-\gamma_{2}-\gamma_{3}-\gamma_{4}+(k+1)\gamma_{D0}&y+y^{-1}\\\
(k+1)\gamma_{D0}&y^{3}+6y+y^{-1}\\\ \hline\cr\end{array}$ (202)

with $k\geq 0$ and

$\begin{split}&r\in\\{1,2,3,4\\},\quad s\in\\{5,6,7,8\\}\\\
&(a,b)\in\\{(1,3),(1,4),(2,3),(2,4),(5,7),(5,8),(6,7),(6,8)\\}\\\ \end{split}$
(203)

The spectrum also includes the respective antiparticles obtained by sending
$\gamma\to-\gamma$.

######  Remark 7.

The wall-crossing constraint that we solved is the following identity

$\begin{split}&\prod^{\nearrow}_{k\geq
0}\left[\left(\prod_{\ell=1,2}\Phi\left(X_{\gamma_{\ell}+k(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})}\right)\right)\left(\prod_{\ell=5,6}\Phi\left(X_{\gamma_{\ell}+k(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})}\right)\right)\right.\\\
&\times\left.\left(\prod_{\ell=3,4}\Phi\left(X_{\gamma_{1}+\gamma_{2}+\gamma_{\ell}+k(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})}\right)\right)\left(\prod_{\ell=7,8}\Phi\left(X_{\gamma_{5}+\gamma_{6}+\gamma_{\ell}+k(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})}\right)\right)\right]\\\
&\prod_{k\geq 0}\Bigg{[}\left(\prod_{s=\pm
1}\Phi\left((-y)^{s}X_{\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4}+k\gamma_{D0}}\right)^{-1}\Phi\left((-y)^{s}X_{-\gamma_{1}-\gamma_{2}-\gamma_{3}-\gamma_{4}+(k+1)\gamma_{D0}}\right)^{-1}\right)\\\
&\qquad\times\left(\prod_{(a,b)}\Phi\left(X_{\gamma_{a}+\gamma_{b}+k\gamma_{D0}}\right)\Phi\left(X_{-\gamma_{a}-\gamma_{b}+(k+1)\gamma_{D0}}\right)\right)\Bigg{]}\\\
&\times\prod^{\searrow}_{k\geq
0}\left[\left(\prod_{\ell=1,2}\Phi\left(X_{\gamma_{3}+\gamma_{4}+\gamma_{\ell}+k(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})}\right)\right)\left(\prod_{\ell=5,6}\Phi\left(X_{\gamma_{7}+\gamma_{8}+\gamma_{\ell}+k(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})}\right)\right)\right.\\\
&\times\left.\left(\prod_{\ell=3,4}\Phi\left(X_{\gamma_{\ell}+k(\gamma_{1}+\gamma_{2}+\gamma_{3}+\gamma_{4})}\right)\right)\left(\prod_{\ell=7,8}\Phi\left(X_{\gamma_{\ell}+k(\gamma_{5}+\gamma_{6}+\gamma_{7}+\gamma_{8})}\right)\right)\right]\\\
&=\left(\\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5},\gamma_{6},\gamma_{7},\gamma_{8}\\}\to\\{\gamma_{3},\gamma_{4},\gamma_{5},\gamma_{6},\gamma_{7},\gamma_{8},\gamma_{1},\gamma_{2}\\}\right)\end{split}$
(204)

where labels $(a,b)$ run over the same combinations as in (203). As we did
previously, we factored out the contribution of pure-flavor states (pure D0
branes), which do not participate in wall-crossing. We have tested this as an
identity in the numerical ($y\to-1$) setting, by acting on $X_{\gamma_{i}}$
and expanding as formal series in $X_{\gamma}$ with $|\gamma|\leq 10$.

(a) Four-dimensional subquivers for the flow $T_{1}$ of $dP_{5}$.

(b) Four-dimensional subquivers for the flow $T_{2}$ of $dP_{5}$.

Figure 18:

## References

  * (1) D. Gaiotto, G. W. Moore, and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun.Math.Phys. 299 (2010) 163–224, [arXiv:0807.4723].
  * (2) D. Gaiotto, G. W. Moore, and A. Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013) 239–403, [arXiv:0907.3987].
  * (3) D. Gaiotto, G. W. Moore, and A. Neitzke, Spectral networks, Annales Henri Poincare 14 (2013) 1643–1731, [arXiv:1204.4824].
  * (4) M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi, and C. Vafa, BPS Quivers and Spectra of Complete N=2 Quantum Field Theories, Commun. Math. Phys. 323 (2013) 1185–1227, [arXiv:1109.4941].
  * (5) M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi, and C. Vafa, $\mathcal{N}=2$ quantum field theories and their BPS quivers, Adv. Theor. Math. Phys. 18 (2014), no. 1 27–127, [arXiv:1112.3984].
  * (6) J. Manschot, B. Pioline, and A. Sen, Wall Crossing from Boltzmann Black Hole Halos, JHEP 07 (2011) 059, [arXiv:1011.1258].
  * (7) J. Manschot, B. Pioline, and A. Sen, A Fixed point formula for the index of multi-centered N=2 black holes, JHEP 05 (2011) 057, [arXiv:1103.1887].