that the ‘electric potential’

(4.10) $\varphi_{s}(\gamma)\coloneqq 2(d-sr)$

can only increase as $t$ increases (for $r=0$, it remains constant since the
ray is vertical). The fact that $\Im Z(\gamma)=t\varphi_{s}(\gamma)$ increases
with $t$ is an obvious consequence of the monotonicity of $|Z(\gamma)|$ along
the attractor flow, but the point is that $\Im Z(\gamma)$ increases faster
than $t$. These properties will be instrumental in the next subsection for
obtaining bounds on allowed constituents for a given total charge $\gamma$.

  * •

In the simplest case of a collision of two incoming rays $\mathcal{R}^{\rm
eff}_{0}(\gamma_{1})$ and $\mathcal{R}^{\rm eff}_{0}(\gamma_{2})$ with
primitive vectors $\gamma_{i}$, each of them having
$\Omega(k\gamma_{i})=\delta_{k=1}$, there is an infinite fan of outgoing
effective rays of the form $\mathcal{R}^{\rm
eff}_{0}(n_{1}\gamma_{1}+n_{2}\gamma_{2})$, carrying an index
$\Omega(n_{1}\gamma_{1}+n_{2}\gamma_{2})=K_{\kappa}(n_{1},n_{2})$ given by the
DT invariant of a Kronecker quiver with
$\kappa=|\langle\gamma_{1},\gamma_{2}\rangle|$ arrows (a multiple of 3 due to
(2.3)) and dimension vector $(n_{1},n_{2})$ (see Table 2 for a table of some
relevant values). More generally, the outgoing rays at each collision are
determined from the incoming rays by requiring that the product (3.16) of
automorphisms $\mathcal{U}({\gamma_{i}})^{\epsilon_{i}}$ around the collision
point is equal to one, or equivalently by using the flow tree formula of [14,
22], with the indices associated to incoming rays playing the role of
attractor indices.

###  4.2. Initial rays and scattering sequences

The main result of [26] was to identify the initial rays of the scattering
diagram $\mathcal{D}^{\rm LV}_{0}$. Namely, it was shown that the initial rays
form two infinite families associated to $\mathcal{O}(m)$ and
$\mathcal{O}(m)[1]$, with charge $\gamma_{m}=[1,m,\frac{1}{2}m^{2}]$ and
$-\gamma_{m}$ respectively, emitted from the points $(m,0)$ along the
boundary, for any $m\in\mathds{Z}$. Moreover, the automorphism
$\mathcal{U}^{\rm eff}(\gamma)$ on each such ray is specified by

(4.11) $\Omega_{\star}(\pm k\gamma_{m})=\delta_{k=1}\
,\quad\bar{\Omega}_{\star}(\pm k\gamma_{m})=\frac{y-y^{-1}}{k(y^{k}-y^{-k})}$

for every $k\geq 1$. The fact that initial rays can only emanate from integer
points along the boundary follows by noting that in any triangle bounded by
$t<\max(s-m,m+1-s)$ in the $(s,t)$ plane, $\sigma$-semistable objects must be
in the heart of the quiver associated to the tilting sequence $\langle
E_{1}(m-1),E_{2}(m-1),E_{3}(m-1)\rangle$ (see (1.8)), but such objects
necessarily have $\Re Z(\gamma)\neq 0$. The invariance of the initial rays
under translations $(s,t)\mapsto(s+1,t)$ is a straightforward consequence of
the action of auto-equivalences $E\mapsto E(1)$ on $\mathcal{C}$, while the
invariance under $(s,t)\mapsto(-s,t)$ is consistent with the derived duality
(2.32) in the large volume limit.

Given this simple structure for the initial rays, the consistency requirement
(3.16) at any intersection determines in principle the automorphism
$\mathcal{U}^{\rm eff}_{\sigma}(\gamma)$ at any point $(s_{0},t_{0})$ along
the geometric ray $\mathcal{R}^{\rm geo}_{0}(\gamma)$ for any primitive charge
vector $\gamma$, and therefore the indices $\Omega_{(s,t)}(k\gamma)$ along the
same ray. In particular, the invariance of initial rays under translations
$(s,t)\mapsto(s+k,t)$ makes it obvious that the resulting indices will be
invariant under the action of auto-equivalences $E\mapsto E(k)$,

(4.12)
$\Omega_{(s,t)}\left([r,d,\chi)\right)=\Omega_{(s+k,t)}\left([r,d+kr,\chi+k(d+\tfrac{3}{2}r)+\tfrac{k^{2}}{2}r)\right)\
,\quad k\in\mathds{Z}$

while the invariance of initial rays under $(s,t)\mapsto(-s,t)$ implies the
symmetry under derived duality

(4.13) $\Omega_{(s,t)}([r,d,\chi))=\Omega_{(-s,t)}([-r,d,3d-\chi))$

For $r\neq 0$ and $2d$ divisible by $r$, this can be combined with (4.12) to
conclude that $\Omega_{(s,t)}([r,d,\chi))$ is invariant under
$s\mapsto\frac{2d}{r}-s$. Similarly, for $r=0$ and $2\chi$ divisible by $d$
one concludes that $\Omega_{(s,t)}([0,d,\chi))$ is invariant under
$s\mapsto\frac{2\chi}{d}-3-s$.

In order to compute the indices in practice, the difficulty is to determine
which rays, among the infinite set of initial rays associated to
$\mathcal{O}(m)$ and $\mathcal{O}(m)[1]$, can produce an outgoing ray with the
desired charge $\gamma$ passing through the desired point $(s_{0},t_{0})$. The
electromagnetic analogy mentioned in the previous subsection gives a way to
tackle this apparently formidable problem. Indeed, the problem amounts to
determining the set of all possible particles of charge
$\gamma_{i}=k_{i}\operatorname{ch}\mathcal{O}(m_{i})$ ($i=1,\dots n$) and
anti-particles of charge
$\gamma^{\prime}_{j}=-k^{\prime}_{j}\operatorname{ch}\mathcal{O}(m^{\prime}_{j})$
($j=1,\dots n^{\prime}$), emitted from the boundary $t=0$ at spatial positions
$s=m_{i}$ and $s=m^{\prime}_{j}$, respectively, such that their scattering
products contains a particle of charge $\gamma$ going through $(s_{0},t_{0})$.

A necessary condition is of course that all initial particles lie in the past
light-cone of $(s_{0},t_{0})$ and their global charges add up to the desired
charge,

(4.14) $s_{0}-t_{0}\leq m_{i},m^{\prime}_{j}\leq s_{0}+t_{0}$ (4.15)
$[r,d,\chi)=\sum_{i=0}^{n-1}k_{i}\left[1,m_{i},1+\frac{m_{i}(m_{i}+3)}{2}\right)-\sum_{j=0}^{n^{\prime}-1}k^{\prime}_{j}\left[1,m^{\prime}_{j},1+\frac{m^{\prime}_{j}(m^{\prime}_{j}+3)}{2}\right)$

but there can be cancellations between the charges of
$k_{i}\mathcal{O}(m_{i})$ and $k^{\prime}_{j}\mathcal{O}(m^{\prime}_{j})[1]$
so these requirements alone do not yet give a finite set of decompositions.
One can further reduce the set of possible decompositions to a finite list by
exploiting the monotonicity of the ‘electric potential’ defined in (4.10).
Indeed, since each of the initial particles and anti-particles is emitted
along the ‘light-cone’ $t=|s-m_{i}|$ or $t=|s-m^{\prime}_{j}|$ at the
boundary, the first scattering can only take place after a time
$t\geq\frac{1}{2}$, by which time the electric potential
$\varphi_{s}(\gamma_{i})$ has increased from 0 to $k_{i}$, and similarly for
$\gamma^{\prime}_{j}$. This immediately gives a bound

(4.16)
$\sum_{i=0}^{n-1}k_{i}+\sum_{j=0}^{n^{\prime}-1}k^{\prime}_{j}\leq\varphi_{s_{0}}(\gamma)$

In particular, since the multiplicities $k_{i}$ and $k^{\prime}_{j}$ are $\geq
1$, the number $n+n^{\prime}$ of possible constituents is bounded by
$\varphi_{s_{0}}(\gamma)$. Moreover, ordering the slopes
$m_{i},m^{\prime}_{j}$ such that

(4.17) $m^{\prime}_{0}\leq m^{\prime}_{1}\leq\dots\leq
m^{\prime}_{n^{\prime}-1}\ ,\quad m_{n-1}\leq\cdots\leq m_{1}\leq m_{0}$

with $k_{i}\leq k_{i-1}$ whenever $m_{i}=m_{i-1}$, and similarly
$k^{\prime}_{j}\leq k^{\prime}_{j-1}$ whenever
$m^{\prime}_{j}=m^{\prime}_{j-1}$, it is clear that the scattering between the
left-moving particles $k_{i}\mathcal{O}(m_{i})$ and right-moving anti-
particles $k^{\prime}_{j}\mathcal{O}(m^{\prime}_{j})[1]$ can only take place
if $m^{\prime}_{n^{\prime}-1}<m_{0}$ and $m_{0}^{\prime}<m_{n-1}$. Denoting by
$(s_{1},t_{1})$ the point where the last scattering takes place, such that
$m_{0}^{\prime}<s_{1}<m_{0}$, the left-most anti-particle must accumulate at
least $2s_{1}-2m^{\prime}_{0}-1$ electric potential in going from
$m^{\prime}_{0}+\frac{1}{2}$ to $s_{1}$, while the right-most particle must
accumulate at least $2m_{0}-2s_{1}-1$ electric potential in going from
$m_{0}-\frac{1}{2}$ to $s_{1}$. Thus, the bound (4.16) can be strengthened to

(4.18)
$\sum_{i=0}^{n-1}k_{i}+\sum_{j=0}^{n^{\prime}-1}k^{\prime}_{j}+2(m_{0}-m^{\prime}_{0}-1)\leq\varphi_{s_{0}}(\gamma)$

This restricts the initial positions $m_{i},m^{\prime}_{j}$ to a finite
interval around $s_{1}$ that is typically tighter than the causality bound
(4.14) for large $t$. At any rate, the set of allowed initial integer
positions $m_{i},m^{\prime}_{j}$ and multiplicities $k_{i},k^{\prime}_{j}$ is
finite. Thus, the list of possible decompositions having a particle of charge
$\gamma$ at position $(s_{0},t_{0})$ among all their scattering products is
finite, and the SAFC (in its strong form) holds along the large volume slice,
at least for trees rooted at a point where $\Re Z(\gamma)=0$ (we relax this
restriction in §4.5).

Unfortunately, many of decompositions
$\gamma=\sum_{i=1}^{n}\gamma_{i}+\sum_{=1}^{n^{\prime}}\gamma^{\prime}_{j}$
which satisfy (4.15) and (4.18) turn out to not include $\gamma$ among their
scattering products, as the worldlines of the constituents may fail to
intersect. To determine which of them do, one way is to construct all possible
binary trees with constituents of charge $\gamma_{i}$ and
$\gamma^{\prime}_{j}$, and retain those that satisfy the conditions (4.6) at
each vertex. Identifying vertices connected by edges of vanishing length (i.e.
mapped to the same point in $(s,t)$ plane), one obtains a set of attractor
flow trees associated to the decomposition. The contribution of each attractor
flow tree to the total index $\Omega_{(s_{0},t_{0})}(\gamma)$ can then be
computed by applying the (attractor or flow) tree formulae locally at each
vertex, as explained in §3.4, or more globally by perturbing the charges of
the initial constituents
$\gamma_{i}\to\gamma_{i}+\epsilon_{i}\delta,\gamma^{\prime}_{j}\to\gamma^{\prime}_{j}+\epsilon^{\prime}_{j}\delta$
where $\delta$ is the D0-brane charge vector and
$\epsilon_{i},\epsilon^{\prime}_{j}$ are small enough and generic.

While the resulting (unperturbed) attractor flow trees typically involve
vertices of higher valency and tend to proliferate, it is sufficient to keep
track of scattering sequences, where descendent edges whose charges are
multiples of the same primitive charge $\gamma_{e}$ are aggregated together as
a single edge of charge $k\gamma_{e}$. Conversely, the original family of
attractor flow trees can be recovered from the aggregated flow trees by
replacing each edge of charge $k\gamma_{e}$ by edges of charge
$\\{k_{1}\gamma_{e},k_{2}\gamma_{e},\dots\\}$ where $\\{k_{i}\\}$ runs over
all integer partitions of $k$. We shall make use of this bookkeeping device in
the next subsection, in order to keep the list of attractor flow trees (or
rather, scattering sequences) within reasonable length.

###  4.3. Examples: Hilbert scheme of $n$ points on $\mathds{P}^{2}$

We now apply the procedure outlined above to the case of rank 1 sheaves with
Chern vector $\gamma=[1,0,1-n)$. At large volume $t\to\infty$, the moduli
space $\mathcal{M}_{(s,t)}(\gamma)$ reduces to the Hilbert scheme of $n$
points on $\mathds{P}^{2}$, for which the Poincaré polynomials are well-known
(seee (2.13)). From the analysis in [32, 57], it follows that the chamber
structure is given by a set of nested walls, the largest of which being the
Gieseker wall corresponding to the destabilizing object $\mathcal{O}(-1)$.
From the discussion below (4.3), this is the half-circle
$\mathcal{C}(-n-\frac{1}{2},n-\frac{1}{2})$. Thus it suffices to consider all
scattering sequences which go through the points
$(s,t)=(-n-\frac{1}{2},n-\frac{1}{2})$, with constituents with slopes in the
range $-2n\leq m\leq-1$, and electric potential bounded by
$\varphi_{-n-\frac{1}{2}}(\gamma)=2n+1$.

We find that for any $n$, there is indeed a scattering sequence whose first
splitting lies on the Gieseker wall
$\mathcal{C}(-n-\frac{1}{2},n-\frac{1}{2})$, namely

(4.19) $T_{\rm
Gieseker}=\begin{cases}\\{-\mathcal{O}(-2),2\mathcal{O}(-1)\\}&n=1\\\
\\{\\{-\mathcal{O}(-n-1),\mathcal{O}(-n)\\},\mathcal{O}(-1)\\}&n\geq
2\end{cases}$

For $n=1$ the scattering sequence is a shorthand for two distinct attractor
flow trees with a single splitting into either two or three descendants,
namely $\\{\gamma_{1},2\gamma_{2}\\}$ and
$\\{\gamma_{1},\gamma_{2},\gamma_{2}\\}$ with
$\gamma_{1}=-\mathcal{O}(-2),\gamma_{2}=\mathcal{O}(-1)$. The contributions of
the two trees sum up to

(4.20)
$\frac{1}{2}\kappa(\langle\gamma_{1},\gamma_{2}\rangle)^{2}\bar{\Omega}(\gamma_{1})\bar{\Omega}(\gamma_{2})^{2}+\kappa(\langle\gamma_{1},2\gamma_{2}\rangle)\bar{\Omega}(\gamma_{1})\bar{\Omega}(2\gamma_{2})=\frac{1}{2}(y^{2}+1+y^{-2})^{2}-\frac{y^{6}-y^{-6}}{y-y^{-1}}\frac{y-y^{-1}}{2(y^{2}-y^{-2})}$

reproducing the index $K_{3}(1,2)=y^{2}+1+1/y^{2}$ of a Kronecker quiver with
3 arrows and dimension vector $(1,2)$ (see Table 2). For $n=2$, there is a
single attractor flow tree with two vertices
$\\{\\{\gamma_{1},\gamma_{2}\\},\gamma_{3}\\}$ contributing

(4.21)
$\kappa(\langle\gamma_{1},\gamma_{2}\rangle)\kappa(\langle\gamma_{1}+\gamma_{2},\gamma_{3}\rangle)=K_{3}(1,1)^{2}=(y^{2}+1+1/y^{2})^{2}$

For $n=1$ or $n=2$, there is a single wall of marginal stability associated to
either of the two sequences, and the total index agrees with the prediction
(2.13) for the Gieseker index outside the wall, and vanishes inside. This is
consistent with the fact that the moduli space $\mathcal{M}_{\infty}(\gamma)$
for $n\leq 2$ has a single stratum.

Figure 15. Scattering sequences for Hilbert scheme of 1 and 2 points on
$\mathds{P}^{2}$.

For $n\geq 3$, there are additional walls of the form
$\mathcal{C}(s_{\gamma,\gamma^{\prime}},R_{\gamma,\gamma^{\prime}})$,
associated to scattering sequences with final vertex of the form
$\gamma^{\prime}+(\gamma-\gamma^{\prime})\to\gamma$. Below we list the
corresponding scattering sequences for $n\leq 7$, along with their respective
contribution to the index in the region above the respective wall (for
brevity, we only indicate the unrefined limit $y\to 1$, but the full refined
index can be obtained from the Kronecker indices in Table 2). In all cases, we
find agreement with (2.13):

  * •

For $n=3$, there are two nested walls, each of which is associated to one
sequence,

$\begin{array}[]{l@{\hskip 5mm}l@{\hskip
5mm}l}\mathcal{C}(-\frac{7}{2},\frac{5}{2})\hfil\hskip
14.22636pt&\\{\\{-\mathcal{O}(-4),\mathcal{O}(-3)\\},\mathcal{O}(-1)\\}\hfil\hskip
14.22636pt&K_{3}(1,1)^{2}\\\ \mathcal{C}(-\frac{5}{2},\frac{1}{2})\hfil\hskip
14.22636pt&\\{-2\mathcal{O}(-3),3\mathcal{O}(-2)\\}\hfil\hskip
14.22636pt&K_{3}(2,3)\end{array}$

Again we stress that the scattering sequence
$\\{-2\mathcal{O}(-3),3\mathcal{O}(-2)\\}$ stands for 6 different attractor
flow trees, corresponding to the $2\times 3$ integer partitions of $2$ and
$3$, which all split at the point and add up to the index of a Kronecker
quiver with 3 arrows and dimension vector $(2,3)$. In the Gieseker chamber,
the contributions of the two sequences add up to $9+13=22$ as $y\to 1$.

  * •

For $n=4$ there are 3 nested walls, and correspondingly 3 sequences,

$\begin{array}[]{l@{\hskip 5mm}l@{\hskip
5mm}l}\mathcal{C}(-\frac{9}{2},\frac{7}{2})\hfil\hskip
14.22636pt&\\{\\{-\mathcal{O}(-5),\mathcal{O}(-4)\\},\mathcal{O}(-1)\\}\hfil\hskip
14.22636pt&K_{3}(1,1)^{2}\\\
\mathcal{C}(-\frac{7}{2},\frac{\sqrt{17}}{2})\hfil\hskip
14.22636pt&\\{\\{-\mathcal{O}(-4),\mathcal{O}(-3)\\},\\{-\mathcal{O}(-3),2\mathcal{O}(-2)\\}\\}\hfil\hskip
14.22636pt&K_{3}(1,1)^{2}K_{3}(1,2)\\\ \mathcal{C}(-3,1)\hfil\hskip
14.22636pt&\\{-\mathcal{O}(-4),2\mathcal{O}(-2)\\}\hfil\hskip
14.22636pt&K_{6}(1,2)\end{array}$

In the Gieseker chamber, their contributions add up to $9+27+15=51$ as $y\to
1$.

Figure 16. Scattering sequences for Hilbert scheme of 3 and 4 points on
$\mathds{P}^{2}$.

  * •

For $n=5$ there are 3 nested walls associated to 3 sequences,

$\begin{array}[]{l@{\hskip 5mm}l@{\hskip
5mm}l}\mathcal{C}(-\frac{11}{2},\frac{9}{2})\hfil\hskip
14.22636pt&\\{\\{-\mathcal{O}(-6),\mathcal{O}(-5)\\},\mathcal{O}(-1)\\}\hfil\hskip
14.22636pt&K_{3}(1,1)^{2}\\\
\mathcal{C}(-\frac{9}{2}.\frac{\sqrt{41}}{2})\hfil\hskip
14.22636pt&\\{\\{-\mathcal{O}(-5),\mathcal{O}(-4)\\},\\{-\mathcal{O}(-3),2\mathcal{O}(-2)\\}\\}\hfil\hskip
14.22636pt&K_{3}(1,1)^{2}K_{3}(1,2)\\\
\mathcal{C}(-\frac{7}{2},\frac{3}{2})\hfil\hskip
14.22636pt&\\{\\{-2\mathcal{O}(-4),2\mathcal{O}(-3)\\},\mathcal{O}(-2)\\}\hfil\hskip
14.22636pt&K_{-3,3,6}(2,2,1)\end{array}$

It is worth explaining in some detail the computation of the index associated
to the last sequence $\\{\\{2\gamma_{1},2\gamma_{2}\\},\gamma_{3}\\}$ with
$\gamma_{1}=-\mathcal{O}(-4),\gamma_{2}=\mathcal{O}(-3),\gamma_{3}=\mathcal{O}(-2)$
(which we have denoted by $K_{a,b,c}(2,2,1)$ with
$a=\langle\gamma_{1},\gamma_{2}\rangle$,
$b=\langle\gamma_{2},\gamma_{3}\rangle$,
$c=\langle\gamma_{3},\gamma_{1}\rangle$). This sequence actually stands for 5
different attractor flow trees, namely

(4.22) $\displaystyle\\{\\{2\gamma_{1},2\gamma_{2}\\},\gamma_{3}\\},\quad$
$\displaystyle\\{\\{\gamma_{1},\gamma_{1},2\gamma_{2}\\},\gamma_{3}\\},$
$\displaystyle\\{\\{2\gamma_{1},\gamma_{2},\gamma_{2}\\},\gamma_{3}\\},\quad$
$\displaystyle\\{\\{\gamma_{1},\gamma_{1},\gamma_{2},\gamma_{2}\\},\gamma_{3}\\},$
$\displaystyle\\{\\{\gamma_{1},\gamma_{2}\\},\\{\\{\gamma_{1},\gamma_{2}\\},\gamma_{3}\\}\\}$

The first four of those combine to produce the rational index
${\overline{K}}_{3}(2,2)\to-\frac{21}{4}$ of a Kronecker quiver with $3$
arrows and dimension vector $(2,2)$. The fifth involves two copies of the
Kronecker quiver with $3$ arrows and dimension vector $(1,1)$. The total index
is therefore

(4.23) $\displaystyle K_{-3,3,6}(2,2,1)$ $\displaystyle=\kappa(\langle
2\gamma_{1}+2\gamma_{2},\gamma_{3}\rangle){\overline{K}}_{3}(2,2)+\frac{1}{2}\kappa(\langle\gamma_{1}+\gamma_{2},\gamma_{3}\rangle)^{2}K_{3}(1,1)^{2}$
$\displaystyle\to 6\frac{21}{4}+\frac{1}{2}3^{2}3^{2}=72$

Equivalently, this factor arises by applying the flow tree formula to a local
scattering diagram with two incoming rays of charge
$\alpha=\gamma_{1}+\gamma_{2}$ and $\beta=\gamma_{3}$ with
$\Omega^{-}(\alpha)=K_{3}(1,2)=y^{2}+1+1/y^{2},\Omega^{-}(2\alpha)=K_{3}(2,2)=-y^{5}-y^{3}-y-1/y-1/y^{3}-1/y^{5}$
and $\Omega^{-}(\beta)=1$, selecting the outgoing ray of charge
$2\alpha+\beta$.

In the Gieseker chamber, the contributions of the three sequences add up to
$9+27+72=108$ as $y\to 1$.

  * •

For $n=6$ there are 5 nested walls associated to 5 sequences,

$\begin{array}[]{c@{\hskip 5mm}l@{\hskip
5mm}l}\mathcal{C}(-\frac{12}{2},\frac{11}{2})\hfil\hskip
14.22636pt&\\{\\{-\mathcal{O}(-7),\mathcal{O}(-6)\\},\mathcal{O}(-1)\\}\hfil\hskip
14.22636pt&K_{3}(1,1)^{2}\\\
\mathcal{C}(-\frac{11}{2},\frac{\sqrt{73}}{2})\hfil\hskip
14.22636pt&\\{\\{-\mathcal{O}(-6),\mathcal{O}(-5)\\},\\{-\mathcal{O}(-3),2\mathcal{O}(-2)\\}\\}\hfil\hskip
14.22636pt&K_{3}(1,1)^{2}K_{3}(1,2)\\\
\mathcal{C}(-\frac{9}{2},\frac{\sqrt{33}}{2})\hfil\hskip
14.22636pt&\\{\\{-\mathcal{O}(-5),\mathcal{O}(-4)\\},\\{\\{-\mathcal{O}(-4),\mathcal{O}(-3)\\},\mathcal{O}(-2)\\}\\}\hfil\hskip
14.22636pt&K_{3}(1,1)^{4}\\\ \mathcal{C}(-4,2)\hfil\hskip
14.22636pt&\\{\\{-\mathcal{O}(-5),\mathcal{O}(-3)\\},\mathcal{O}(-2)\\}\hfil\hskip
14.22636pt&K_{6}(1,1)^{2}\\\ \mathcal{C}(-\frac{7}{2},\frac{1}{2})\hfil\hskip
14.22636pt&\\{-3\mathcal{O}(-4),4\mathcal{O}(-3)\\}\hfil\hskip
14.22636pt&K_{3}(3,4)\end{array}$

In the Gieseker chamber, their contributions add up to $9+27+81+36+68=221$ as
$y\to 1$.

Figure 17. Scattering sequences for Hilbert scheme of 5 and 6 points on
$\mathds{P}^{2}$.

  * •

For $n=7$ there are 6 nested walls, but one of them is associated to 2
sequences:

$\begin{array}[]{cll}\mathcal{C}(-\frac{15}{2},\frac{13}{2})&\\{\\{-\mathcal{O}(-8),\mathcal{O}(-7)\\},\mathcal{O}(-1)\\}&K_{3}(1,1)^{2}\\\
\mathcal{C}(-\frac{13}{2},\frac{\sqrt{113}}{2})&\\{\\{-\mathcal{O}(-7),\mathcal{O}(-6)\\},\\{-\mathcal{O}(-3),2\mathcal{O}(-2)\\}\\}&K_{3}(1,1)^{2}K_{3}(1,2)\\\
\mathcal{C}(-\frac{11}{2},\frac{\sqrt{65}}{2})&\\{\\{-\mathcal{O}(-6),\mathcal{O}(-5)\\},\\{\\{-\mathcal{O}(-4),\mathcal{O}(-3)\\},\mathcal{O}(-2)\\}\\}&K_{3}(1,1)^{4}\\\
\mathcal{C}(-\frac{9}{2},\frac{5}{2})&\\{\\{-2\mathcal{O}(-5),2\mathcal{O}(-4)\\},\mathcal{O}(-2)\\}&K_{-3,6,9}(2,2,1)\\\
"&\\{\\{-\mathcal{O}(-5),\mathcal{O}(-4)\\},\\{-2\mathcal{O}(-4),3\mathcal{O}(-3)\\}\\}&K_{3}(1,1)^{2}K_{3}(2,3)\\\
\mathcal{C}(-4,\sqrt{2})&\\{\\{-\mathcal{O}(-5),\mathcal{O}(-3)\\},\\{-\mathcal{O}(-4),2\mathcal{O}(-3)\\}\\}&K_{3}(1,2)K_{6}(1,1)^{2}\\\
\mathcal{C}(-\frac{39}{10},\frac{11}{10})&\\{-\mathcal{O}(-5),\\{-\mathcal{O}(-4),3\mathcal{O}(-3)\\}\\}&K_{3}(1,3)K_{15}(1,1)\end{array}$

The index for the first tree associated to the wall
$\mathcal{C}(-\frac{9}{2},\frac{5}{2})$ is computed in the same way as
explained around (4.23). Note also that the scattering sequence associated to
$\mathcal{C}(-4,\sqrt{2})$ is non-planar, as the ray $\mathcal{O}(-3)$ has to
cross through $-\mathcal{O}(-4)$ before colliding with $-\mathcal{O}(-5)$. The
contributions of the 7 sequences to the Gieseker index add up to
$9+27+81+72+117+108+15=429$ as $y\to 1$, in agreement with (2.13).

Figure 18. Scattering sequences for Hilbert scheme of 7 points on
$\mathds{P}^{2}$.

###  4.4. Examples: D2-D0 indices

We now turn to the case of rank 0 sheaves with Chern vector
$\gamma=[0,d,\chi)$ with $d>0$. At large volume $t\to\infty$, the index
$\Omega_{(s,t)}(\gamma)$ is determined in terms of refined Gopakumar-Vafa
invariants via (2.15), in particular it is independent of $\chi$. This is in
general no longer true for finite $t$, due to the existence of a sequence of
nested walls of marginal stability. The outermost wall (or Gieseker wall) is
determined as follows [58, Prop. 7.5]. Let $\eta$ be an integer in the range
$-\frac{d}{2}\leq\eta\leq\frac{d}{2}$ such that
$\chi+\frac{1}{2}d(d-3)=kd+\eta$ with $k\in\mathds{Z}$. Then the Gieseker wall
is the half-circle $\mathcal{C}(s_{0},R_{0})$ of radius
$R_{0}=\frac{d}{2}-\frac{|\eta|}{d}$ centered at
$s_{0}=\frac{\chi}{d}-\frac{3}{2}=k+\frac{\eta}{d}-\frac{d}{2}$. When
$\eta\geq 0$, the destabilizing sub-object along the wall is $\mathcal{O}(k)$,
whereas for $\eta\leq 0$ it is the ideal sheaf $\mathcal{I}_{Z(-\eta)}(k)$
where $Z(w)$ is a scheme of dimension 0 and length $w$. For $\eta=0$, these
two notions agree, while for $|\eta|=\frac{d}{2}$ they do not, but
destabilizing subobjects of both types may occur. It follows that all
constituents of the trees contributing to $\Omega_{\infty}(\gamma)$ must be
emitted in an interval of width $\frac{d}{2}-\frac{|\eta|}{d}$ centered around
$s_{0}$, and electric potential bounded by $\varphi_{s_{0}}(\gamma)=2d$.

We shall now discuss examples with low degree $d$, making use of the symmetry
properties (4.12) and (4.13) to restrict the values of $\chi$. In all cases,
the wall structure matches with [33] and the index outside the Gieseker wall
agrees with the GV invariants in (2.16). The trees arising at each wall also
agree with the stratification in [59, 60, 61], with one exception for
$(d,\chi)=(5,3)$ mentioned below.

  * •

For $(d,\chi)=(1,1)$, such that $(k,\eta)=(0,0)$, there is only one wall,

$\begin{array}[]{c@{\hskip 10mm}l@{\hskip 5mm}l@{\hskip
5mm}l}\chi=1:\hfil\hskip
28.45274pt&\mathcal{C}(-\tfrac{1}{2},\tfrac{1}{2})\qquad\\{-\mathcal{O}(-1),\mathcal{O}\\}\qquad
K_{3}(1,1)\hfil\hskip 14.22636pt\end{array}$

The tree corresponds to the fact that the structure sheaf of a curve
$\mathcal{O}_{C}$ is given by the short exact sequence
$0\to\mathcal{O}(-1)\to\mathcal{O}_{C}\to\mathcal{O}\to 0$, and its index
$K_{3}(1,1)=y^{2}+1+1/y^{2}$ reproduces the Poincaré polynomial of the linear
system $[C]=\mathds{P}^{2}$.

  * •

For $(d,\chi)=(2,0),(2,1)$, such that $(k,\eta)=(0,-1),(0,0)$, respectively,
there is a single wall

$\begin{array}[]{c@{\hskip 10mm}l@{\hskip 5mm}l@{\hskip
5mm}l}\chi=0:\hfil\hskip
28.45274pt&\mathcal{C}(-\frac{3}{2},\frac{1}{2})\hfil\hskip
14.22636pt&\\{-2\mathcal{O}(-2),2\mathcal{O}(-1)\\}\hfil\hskip
14.22636pt&K_{3}(2,2)\\\\[5.69054pt] \chi=1:\hfil\hskip
28.45274pt&\mathcal{C}(-1,1)\hfil\hskip
14.22636pt&\\{-\mathcal{O}(-2),\mathcal{O}\\}\hfil\hskip
14.22636pt&K_{3}(1,2)\end{array}$

giving the same index $-y^{5}-y^{3}-y-\dots$ in the Gieseker chamber in either
case.

  * •

For $(d,\chi)=(3,0),(3,1)$, such that $(k,\eta)=(0,0),(0,1)$, respectively,
there are two walls in the first case, and a single wall in the second case,

$\begin{array}[]{c@{\hskip 10mm}l@{\hskip 5mm}l@{\hskip
5mm}l}\chi=0:\hfil\hskip
28.45274pt&\mathcal{C}(-\frac{3}{2},\frac{3}{2})\hfil\hskip
14.22636pt&\\{-\mathcal{O}(-3),\mathcal{O}(0)\\}\hfil\hskip
14.22636pt&K_{9}(1,1)\\\\[1.0pt] "\hfil\hskip
28.45274pt&\mathcal{C}(-\frac{3}{2},\frac{1}{2})\hfil\hskip
14.22636pt&\\{-3\mathcal{O}(-2),3\mathcal{O}(-1)\\}\hfil\hskip
14.22636pt&K_{3}(3,3)\\\\[5.69054pt] \chi=1:\hfil\hskip
28.45274pt&\mathcal{C}(-\frac{7}{6},\frac{7}{6})\hfil\hskip
14.22636pt&\\{\\{-2\mathcal{O}(-2),\mathcal{O}(-1)\\},\mathcal{O}\\}\hfil\hskip
14.22636pt&K_{3}(1,2)K_{9}(1,1)\end{array}$

contributing $27=18+9$ as $y\to 1$ in the Gieseker chamber in either case.

Figure 19. Scattering sequences for $(d,\chi)=(3,0),(3,1)$.

  * •

For $(d,\chi)=(4,0),(4,1),(4,2)$, such that $(k,\eta)=(0,2),(1,-1),(1,0)$,
respectively, there are two walls in the first case, and three in the third
case,

$\begin{array}[]{c@{\hskip 10mm}l@{\hskip 5mm}l@{\hskip
5mm}l}\chi=0:\hfil\hskip
28.45274pt&\mathcal{C}(-\frac{3}{2},\frac{3}{2})\hfil\hskip
14.22636pt&\\{-\mathcal{O}(-3),\\{\\{-\mathcal{O}(-2),\mathcal{O}(-1)\\},\mathcal{O}\\}\\}\hfil\hskip
14.22636pt&K_{3}(1,1)^{2}K_{12}(1,1)\\\\[1.0pt] "\hfil\hskip
28.45274pt&\mathcal{C}(-\frac{3}{2},\frac{1}{2})\hfil\hskip
14.22636pt&\\{-4\mathcal{O}(-2),4\mathcal{O}(-1)\\}\hfil\hskip
14.22636pt&K_{3}(4,4)\\\\[5.69054pt] \chi=1:\hfil\hskip
28.45274pt&\mathcal{C}(-\frac{5}{4},\frac{7}{4})\hfil\hskip
14.22636pt&\\{-\mathcal{O}(-3),\\{-\mathcal{O}(-1),2\mathcal{O}\\}\\}\hfil\hskip
14.22636pt&K_{3}(1,2)K_{12}(1,1)\\\\[1.0pt] "\hfil\hskip
28.45274pt&\mathcal{C}(-\frac{5}{4},\frac{5}{4})\hfil\hskip
14.22636pt&\\{\\{-3\mathcal{O}(-2),2\mathcal{O}(-1)\\},\mathcal{O}\\}\hfil\hskip
14.22636pt&K_{3}(2,3)K_{12}(1,1)\\\\[5.69054pt] \chi=2:\hfil\hskip
28.45274pt&\mathcal{C}(-1,2)\hfil\hskip
14.22636pt&\\{-\mathcal{O}(-3),\mathcal{O}(1)\\}\hfil\hskip
14.22636pt&K_{12}(1,1)\\\\[1.0pt] "\hfil\hskip
28.45274pt&\mathcal{C}(-1,\sqrt{2})\hfil\hskip
14.22636pt&\\{\\{-2\mathcal{O}(-2),\mathcal{O}(-1)\\},\\{-\mathcal{O}(-1),2\mathcal{O}\\}\\}\hfil\hskip
14.22636pt&K_{3}(1,2)^{2}K_{12}(1,1)\\\\[1.0pt] "\hfil\hskip
28.45274pt&\mathcal{C}(-1,1)\hfil\hskip
14.22636pt&\\{-2\mathcal{O}(-2),2\mathcal{O}\\}\hfil\hskip
14.22636pt&K_{6}(2,2)\end{array}$

contributing $-108-84=-36-156=-108-72-12=-192$ in the Gieseker chamber in each
case. Note that for $\chi=0$, the first wall involves three charges
$-\mathcal{O}(-3),-\mathcal{O}(-2)+\mathcal{O}(-1)$ and $\mathcal{O}$
colliding at the same point, which can be resolved into two successive
collisions by perturbing the incoming rays. For each value of $\chi$, the
sequences match with the stratification in [59].

Figure 20. Scattering sequences for $(d,\chi)=(4,0),(4,1),(4,2)$.

  * •

For $(d,\chi)=(5,0),(5,1),(5,3)$, such that $(k,\eta)=(1,0),(1,1),(2,-2)$, we
find

$\begin{array}[]{c@{\hskip 3mm}l@{\hskip 3mm}l@{\hskip 3mm}l}\chi=0\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{3}{2},\frac{5}{2})\hfil\hskip
8.53581pt&\\{-\mathcal{O}(-4),\mathcal{O}(1)\\}\hfil\hskip
8.53581pt&K_{15}(1,1)\\\\[1.0pt] "\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{3}{2},\frac{\sqrt{17}}{2})\hfil\hskip
8.53581pt&\\{\\{-2\mathcal{O}(-3),\mathcal{O}(-2)\\},\\{-\mathcal{O}(-1),2\mathcal{O}\\}\\}\hfil\hskip
8.53581pt&K_{3}(1,2)^{2}K_{15}(1,1)\\\\[1.0pt] "\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{3}{2},\frac{3}{2})\hfil\hskip
8.53581pt&\\{-\mathcal{O}(-3),\\{\\{-2\mathcal{O}(-2),2\mathcal{O}(-1)\\},\mathcal{O}\\}\\}\hfil\hskip
8.53581pt&K_{15}(1,1)K_{-3,3,6}(2,2,1)\\\\[1.0pt] \hfil\hskip
8.53581pt&\mathcal{C}(-\frac{3}{2},\frac{1}{2})\hfil\hskip
8.53581pt&\\{-5\mathcal{O}(-2),5\mathcal{O}(-1)\\}\hfil\hskip
8.53581pt&K_{3}(5,5)\\\\[5.69054pt] \chi=1\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{13}{10},\frac{23}{10})\hfil\hskip
8.53581pt&\\{\\{-2\mathcal{O}(-3),\mathcal{O}(-2)\\},\mathcal{O}(1)\\}\hfil\hskip
8.53581pt&K_{3}(1,2)K_{15}(1,1)\\\\[1.0pt] "\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{13}{10},\frac{\sqrt{329}}{10})\hfil\hskip
8.53581pt&\begin{array}[t]{@{}r@{}}\\{\\{-\mathcal{O}(-3),\\{-\mathcal{O}(-2),\mathcal{O}(-1)\\}\\},\\\
\\{-\mathcal{O}(-1),2\mathcal{O}\\}\\}\end{array}\hfil\hskip
8.53581pt&K_{3}(1,1)^{2}K_{3}(1,2)K_{15}(1,1)\\\\[1.0pt] "\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{13}{10},\frac{17}{10})\hfil\hskip
8.53581pt&\\{-\mathcal{O}(-3),\\{-\mathcal{O}(-2),2\mathcal{O}\\}\\}\hfil\hskip
8.53581pt&K_{6}(1,2)K_{15}(1,1)\\\\[1.0pt] \hfil\hskip
8.53581pt&\mathcal{C}(-\frac{13}{10},\frac{13}{10})\hfil\hskip
8.53581pt&\\{\\{-4\mathcal{O}(-2),3\mathcal{O}(-1)\\},\mathcal{O}\\}\hfil\hskip
8.53581pt&K_{3}(3,4)K_{15}(1,1)\\\\[5.69054pt] \chi=3\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{9}{10},\frac{21}{10})\hfil\hskip
8.53581pt&\\{-\mathcal{O}(-3),\\{\\{-\mathcal{O}(-1),\mathcal{O}\\},\mathcal{O}(1)\\}\\}\hfil\hskip
8.53581pt&K_{3}(1,1)^{2}K_{15}(1,1)\\\\[1.0pt] "\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{9}{10},\frac{19}{10})\hfil\hskip
8.53581pt&\\{\\{-3\mathcal{O}(-2),2\mathcal{O}(-1)\\},\mathcal{O}(1)\\}\hfil\hskip
8.53581pt&K_{3}(2,3)K_{15}(1,1)\\\\[1.0pt] "\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{9}{10},\frac{\sqrt{241}}{10})\hfil\hskip
8.53581pt&\\{\\{-2\mathcal{O}(-2),\mathcal{O}(-1)\\},\\{-2\mathcal{O}(-1),3\mathcal{O}\\}\\}\hfil\hskip
8.53581pt&K_{3}(1,2)K_{3}(2,3)K_{15}(1,1)\\\\[1.0pt] "\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{9}{10},\frac{\sqrt{161}}{10})\hfil\hskip
8.53581pt&\\{\\{-2\mathcal{O}(-2),\mathcal{O}\\},\\{-\mathcal{O}(-1),2\mathcal{O}\\}\\}\hfil\hskip
8.53581pt&K_{3}(1,2)K_{6}(1,2)K_{15}(1,1)\\\\[1.0pt] "\hfil\hskip
8.53581pt&\mathcal{C}(-\frac{9}{10},\frac{11}{10})\hfil\hskip
8.53581pt&\\{-2\mathcal{O}(-2),\\{-\mathcal{O}(-1),3\mathcal{O}\\}\\}\hfil\hskip
8.53581pt&K_{3}(1,3)K_{15}(1,2)\end{array}$

contributing $15+135+1080+465=45+405+225+1020=135+195+585+675+105=1695$ in the
Gieseker chamber. For $\chi=0$, the third wall involves three charges
$-\mathcal{O}(-3),-2\mathcal{O}(-2)+2\mathcal{O}(-1)$ and $\mathcal{O}$
colliding at the same point; it can be treated by perturbing in the same way
as for $(d,\chi)=(4,0)$ and using the same reasoning as explained below
(4.23). Note that the fourth sequence for $\chi=3$ is non-planar, and
corresponds to a codimension one stratum, which appears to be missing in [60].

Figure 21. Scattering sequences for $(d,\chi)=(5,0),(5,1),(5,3)$

  * •

For $(d,\chi)=(6,1)$, corresponding to $(k,\eta)=(2,-2)$,

$\begin{array}[]{lll}\mathcal{C}(-\frac{4}{3},\frac{8}{3})&\\{-\mathcal{O}(-4),\\{\\{-\mathcal{O}(-1),\mathcal{O}\\},\mathcal{O}(1)\\}\\}&K_{3}(1,1)^{2}K_{18}(1,1)\\\\[1.0pt]
\mathcal{C}(-\frac{4}{3},\frac{7}{3})&\\{\\{\\{-2\mathcal{O}(-3),\mathcal{O}(-2)\\},\\{-\mathcal{O}(-2),\mathcal{O}(-1)\\}\\},\mathcal{O}(1)\\}&K_{3}(1,1)^{2}K_{3}(1,2)K_{18}(1,1)\\\\[1.0pt]
&\\{\\{-2\mathcal{O}(-3),\mathcal{O}(-1)\\},\mathcal{O}(1)\\}&K_{6}(1,2)K_{18}(1,1)\\\\[1.0pt]
\mathcal{C}(-\frac{4}{3}\frac{\sqrt{46}}{3})&\\{\\{-2\mathcal{O}(-3),\mathcal{O}(-2)\\},\\{-2\mathcal{O}(-1),3\mathcal{O}\\}\\}&K_{3}(1,2)K_{3}(2,3)K_{18}(1,1)\\\\[1.0pt]
\mathcal{C}(-\frac{4}{3},\frac{\sqrt{31}}{3})&\\{\\{-\mathcal{O}(-3),\\{-2\mathcal{O}(-2),2\mathcal{O}(-1)\\}\\},\\{-\mathcal{O}(-1),2\mathcal{O}\\}\\}&K_{-3,6,3}(2,2,1)K_{3}(1,2)K_{18}(1,1)\\\\[1.0pt]
\mathcal{C}(-\frac{4}{3},\frac{2\sqrt{7}}{3})&\\{\\{-\mathcal{O}(-3),\\{-\mathcal{O}(-2),\mathcal{O}(-1)\\}\\},\\{-\mathcal{O}(-2),2\mathcal{O}\\}\\}&K_{3}(1,1)^{2}K_{6}(1,2)K_{18}(1,1)\\\\[1.0pt]
\mathcal{C}(-\frac{4}{3},\frac{5}{3})&\\{-\mathcal{O}(-3),\\{\\{-2\mathcal{O}(-2),\mathcal{O}(-1)\\},2\mathcal{O}\\}\\}&K_{3}(1,2)K_{9}(1,2)K_{18}(1,1)\\\\[1.0pt]
\mathcal{C}(-\frac{4}{3},\frac{4}{3})&\\{\\{-5\mathcal{O}(-2),4\mathcal{O}(-1)\\},\mathcal{O}\\}&K_{3}(4,5)K_{18}(1,1)\end{array}$

contributing $-162-270-486-702-2430-2430-1944-7182=-17064$ in the Gieseker
chamber (here the factor $K_{-3,6,3}(2,2,1)$ in the fourth sequence can be
computed as explained below (4.23), and still evaluates to 72). Note that the
sequence associated to $\mathcal{C}(-\frac{4}{3},\frac{2\sqrt{7}}{3})$ is non-
planar. We refrain from considering other values of $\chi$ in this case.

Figure 22. Scattering sequences for $(d,\chi)=(6,1)$

###  4.5. Generalization to $\psi\neq 0$

We now discuss how this picture generalizes to an arbitrary phase
$\psi\in(-\frac{\pi}{2},\frac{\pi}{2})$. For $r\neq 0$, the geometric ray
$\mathcal{R}^{\rm geo}_{\psi}(\gamma)$ is now a branch of hyperbola
(degenerating to a straight line for vanishing discriminant $\Delta=0$)

(4.24)
$s=\tfrac{d}{r}-t\tan\psi-\frac{\operatorname{sign}r}{\cos\psi}\sqrt{t^{2}+2\Delta\cos^{2}\psi}$

with $t>0$, while for $r=0$, it is the line
$s=\frac{\textnormal{ch}_{2}}{d}-t\tan\psi$ when $d>0$ (or the empty set when
$d\leq 0$). In particular, the electromagnetic analogy of §4.1 breaks down for
$\psi\neq 0$, since one of the two rays emanating from $s=m$ is no longer
inside the light-cone (indeed, the angle between the two rays remains equal to
$\frac{\pi}{2}$ independently of $\psi$).

Moreover, under the same conditions (4.6) two rays $\mathcal{R}^{\rm
geo}_{\psi}(\gamma)$ and $\mathcal{R}^{\rm geo}_{\psi}(\gamma^{\prime})$ still
intersect on the same wall $\mathcal{W}(\gamma,\gamma^{\prime})$, but at the
point
$(s_{\gamma,\gamma^{\prime}}-R_{\gamma,\gamma^{\prime}}\sin\psi,R_{\gamma,\gamma^{\prime}}\cos\psi)$
at an angle $\psi$ from the top of the half-circle
$\mathcal{C}(s_{\gamma,\gamma^{\prime}},R_{\gamma,\gamma^{\prime}})$. In fact,
observing that

(4.25) $\Re\left(e^{-\mathrm{i}\psi}Z^{\rm
LV}_{(s,t)}(\gamma)\right)=\cos\psi\,\Re\left(Z^{\rm
LV}_{(s_{\psi},t_{\psi})}(\gamma)\right)$

where $(s_{\psi},t_{\psi})$ are related to $(s,t)$ through

(4.26) $s_{\psi}=s+t\tan\psi,\quad t_{\psi}=\frac{t}{\cos\psi}$

we see that the loci $\Re\left(e^{-\mathrm{i}\psi}Z(\gamma)\right)=0$ in the
$(s,t)$ upper half-plane are mapped to the loci $\Re(Z(\gamma))=0$ in the
$(s_{\psi},t_{\psi})$ upper half-plane. The relation for the imaginary parts
is however more complicated,

(4.27) $\Im\bigl{(}e^{-\mathrm{i}\psi}Z^{\rm
LV}_{(s,t)}(\gamma)\bigr{)}=\Im\bigl{(}Z^{\rm
LV}_{(s_{\psi},t_{\psi})}(\gamma)\bigr{)}-\sin\psi\,\Re\bigl{(}Z^{\rm
LV}_{(s_{\psi},t_{\psi})}(\gamma)\bigr{)}+r(\sin\psi)t_{\psi}^{2}$

In particular, the transformation $e^{-\mathrm{i}\psi}Z^{\rm
LV}_{(s,t)}(\gamma)\rightarrow Z^{\rm LV}_{(s_{\psi},t_{\psi})}(\gamma)$ is
not a $\widetilde{GL^{+}}(2,\mathds{R})$ transformation, due to the last term
in (4.27). Rather, one should view $(s_{\psi},t_{\psi})$ as variants of the
affine coordinates $(x,y)$ defined in (1.10), indeed
$(x,y)=(s_{\psi},\frac{1}{2}(t_{\psi}^{2}-s_{\psi}^{2}))$.

Thus, we conclude that whenever $\psi\in(-\frac{\pi}{2},\frac{\pi}{2})$, the
scattering diagram $\mathcal{D}^{\rm LV}_{\psi}$ in the $(s,t)$ upper half-
plane coincides with the scattering diagram $\mathcal{D}^{\rm LV}_{0}$ in the
$(s_{\psi},t_{\psi})$ upper half-plane, and both have the same image in the
$(x,y)$ plane. Thus, the SAFC holds on the large volume slice, as long as
$\cos\psi\neq 0$. For $\psi=\pm\frac{\pi}{2}$, the scattering diagram
$\mathcal{D}^{\rm LV}_{\psi}$ becomes degenerate, as all loci $\Im
Z(\gamma)=0$ become either vertical lines $s=d/r$ for $r\neq 0$ or the
horizontal line $t=0$ on the boundary. As we shall see in §6, the scattering
diagram $\mathcal{D}^{\Pi}_{\psi}$ along the slice of $\Pi$-stability
conditions is better behaved in this limit.

##  5\. The orbifold scattering diagram

In this section, we construct the scattering diagram $\mathcal{D}_{Q}$ for the
orbifold quiver shown in Figure 4, following the general construction of §3.1.
In §5.2, we identify the initial rays, giving a rigorous proof of the
Attractor Conjecture for this case. In §5.3, we restrict $\mathcal{D}_{Q}$ to
a two-dimensional slice $\mathcal{D}_{o}$, which we shall identify later in
§6.3 with a subset of the exact scattering diagram $\mathcal{D}^{\Pi}_{\psi}$.
In §5.4 we illustrate the use of the diagram $\mathcal{D}_{o}$ for computing
the index for small dimension vectors.

###  5.1. Quiver descriptions

As explained for example in [20, §4.2], any tilting sequence
$\mathcal{S}=(E_{1},E_{2},E_{3})$ in the derived category
$\mathcal{C}=D^{b}\operatorname{Coh}(K_{\mathds{P}^{2}})$ (in the sense that
$\mathcal{S}$ generates $\mathcal{C}$ and
$\operatorname{Ext}^{k}(E_{i},E_{j})=0$ for $k\neq 0$ and all $i,j$) provides
an isomorphism $\mathcal{C}\sim D^{b}(\operatorname{Rep}J(Q,W))$ with the
derived category of representations of the Jacobian algebra $J(Q,W)$ of a
certain quiver with potential $(Q,W)$ associated to $\mathcal{S}$. A tilting
sequence can be obtained from any strong exceptional collection
$(F_{1},F_{2},F_{3})$ on $\mathds{P}^{2}$ by setting
$\mathcal{S}=(i_{*}(F_{1}[1]),i_{*}(F_{2}),i_{*}(F_{3})[-1])$. Starting from
the strong exceptional collection $(\mathcal{O}(-1),\Omega(1),\mathcal{O})$,
one arrives at the tilting sequence in (1.8), associated to the quiver with
potential depicted in Figure 4. All other strong exceptional collections are
obtained by successive mutations of $(\mathcal{O}(-1),\Omega(1),\Omega)$, and
similarly lead to 3-node quivers $(Q_{a,b,c},W_{a,b,c})$ with $a$ arrows from
node 1 to node 2, $b$ arrows from 2 to 3 and $c$ arrows from 3 to 1, with
$(a,b,c)$ any set of positive integers satisfying the Markov condition
$a^{2}+b^{2}+c^{2}=abc$.

While the isomorphism $\mathcal{C}\simeq D^{b}(\operatorname{Rep}J(Q,W))$
holds for any choice of tilting sequence $\mathcal{S}=(E_{1},E_{2},E_{3})$,
the heart $\mathcal{A}_{\sigma}$ only coincides (up to the action of
$\widetilde{GL^{+}}(2,\mathds{R})$) with the Abelian category
$\operatorname{Rep}J(Q,W)$ of representations of $J(Q,W)$ in a region
$\mathds{H}_{\mathcal{S}}$ of $\operatorname{Stab}\mathcal{C}$ where the
objects $E_{i}$ are stable and the phases of their central charges $Z(E_{i})$
lie in a common half-space (see for example Lemma 3.16 in [62]). In that case,
the moduli space of $\sigma$-semistable objects $E$ of Chern vector $\gamma$
in $\mathcal{C}$ coincides with the moduli space of semi-stable
representations of $J(Q,W)$, with dimension vector $(n_{1},n_{2},n_{3})$ and
King parameters $(\theta_{1},\theta_{2},\theta_{3})$ determined by

(5.1) $\gamma=\sum_{i=1}^{3}n_{i}\operatorname{ch}E_{i}\
,\quad\theta_{i}=\lambda\,\Re[e^{-\mathrm{i}\psi}Z(E_{i})]$

with $\psi=\arg Z(E)-\frac{\pi}{2}$ such that
$n_{1}\theta_{1}+n_{2}\theta_{2}+n_{3}\theta_{3}=0$, and $\lambda$ is an
arbitrary non-negative factor. At the boundary of the region
$\mathds{H}_{\mathcal{S}}$, one of the objects $E_{i}$ exits from the common
half-space and one apply a left or right mutation so as to obtain a new quiver
description.

Let us consider the exceptional collection in (1.8), corresponding to the
fractional branes on the orbifold $\mathds{C}^{3}/\mathds{Z}_{3}$:

(5.2) $E_{1}=i_{*}(\mathcal{O})[-1],\quad E_{2}=i_{*}(\Omega(1)),\quad
E_{3}=i_{*}(\mathcal{O}(-1))[1]$

with charges $\gamma_{i}=\operatorname{ch}E_{i}$ given by

(5.3)
$\gamma_{1}=[-1,0,0],\quad\gamma_{2}=[2,-1,-\frac{1}{2}],\quad\gamma_{3}=[-1,1,-\frac{1}{2}]$

The corresponding quiver, shown in Figure 4, has $3$ arrows $E_{i}\to E_{i-1}$
for each $i$ modulo 3, consistently with
$\langle\gamma_{i},\gamma_{i-1}\rangle=-3$. Using the Eichler integral
representation (1.5), it is easy to check that the central charges of the
simple objects coincide at the orbifold point $\tau_{o}$, namely
$Z_{\tau_{o}}(\gamma_{i})=1/3$. Thus, this quiver description will be
appropriate212121 Note that the objects $E_{i}$ are not stable at
$\tau=\tau_{o}$ strictly, since their central charge is not in the half-plane
$\mathds{H}_{B}$; the objects $E_{i}[1]$ are stable, and lead to the same
quiver but opposite dimension vector. in a region $\mathds{H}^{o}$ around
$\tau_{o}$ in the physical slice $\Pi$, which we identify in §6.3. The
dimension vector associated to an object of charge
$\gamma=[r,d,\textnormal{ch}_{2}]$ is given by

(5.4)
$(n_{1},n_{2},n_{3})=(-\frac{3}{2}d-\textnormal{ch}_{2}-r,-\frac{1}{2}d-\textnormal{ch}_{2},\frac{1}{2}d-\textnormal{ch}_{2})=(-\chi,r+d-\chi,r+2d-\chi)$

or conversely

(5.5) $r=2n_{2}-n_{1}-n_{3},\quad
d=n_{3}-n_{2},\quad\textnormal{ch}_{2}=-\frac{1}{2}(n_{2}+n_{3})$

Abusing notation once again, we shall write $\gamma=(n_{1},n_{2},n_{3})$,
using round brackets on both sides to distinguish it from the other notations
$\gamma=[r,d,\textnormal{ch}_{2}]=[r,d,\chi)$.

A similar quiver description holds true around any image of $\tau_{o}$ under
an element $g\in\Gamma_{1}(3)$, with the tilting sequence being replaced by
its image $(g(E_{1}),g(E_{2}),g(E_{3}))$. In particular, around
$\tau=\tau_{o}+k$, the tilting sequence (1.8) is shifted to

(5.6) $E_{1}(k)=i_{*}(\mathcal{O}(k))[-1],\quad
E_{2}(k)=i_{*}(\Omega(k+1)),\quad E_{3}(k)=i_{*}(\mathcal{O}(k-1))[1]$

with Chern vectors $\gamma_{1}(k),\gamma_{2}(k),\gamma_{3}(k)$. The relevant
dimension vector is then

(5.7)
$\begin{cases}n_{1}(k)=n_{1}-\frac{1}{2}k(3n_{1}-4n_{2}+n_{3})+\frac{1}{2}k^{2}(n_{1}-2n_{2}+n_{3})\\\
n_{2}(k)=n_{2}-\frac{1}{2}k(n_{1}-n_{3})+\frac{1}{2}k^{2}(n_{1}-2n_{2}+n_{3})\\\
n_{3}(k)=n_{3}-\frac{1}{2}k(4n_{2}-n_{1}-3n_{3})+\frac{1}{2}k^{2}(n_{1}-2n_{2}+n_{3})\end{cases}$

where $(n_{1},n_{2},n_{3})$ denotes the value for $k=0$, given in (5.4). We
note that for fixed $\gamma=[r,d,\textnormal{ch}_{2}]$ and large $k$, the
entries $n_{i}(k)$ grow like $-\frac{1}{2}k^{2}r$ for $r\neq 0$, or like $kd$
for $r=0$, in particular they all have the same sign for large $k$.

###  5.2. Initial rays for the orbifold scattering diagram

In this subsection, we prove that for the quiver with potential $(Q,W)$ shown
in Figure 4, the attractor invariant $\Omega_{\star}(\gamma)$ vanishes all
dimension vectors $\gamma=(n_{1},n_{2},n_{3})$ except

(5.8) $\gamma\in\\{(1,0,0),(0,1,0),(0,0,1),(n,n,n)|n\geq 1\\}$

where it takes the values stated in (1.9). This result was conjectured in
[23], but the proof outlined in that paper was not mathematically rigorous.
Here we complete the proof, combining ideas from [23, Section 3.3] and [25,
§3.2]. We denote by $a_{i}\colon 2\to 1$, $b_{i}\colon 3\to 2$ and
$c_{i}\colon 1\to 3$ the arrows of the quiver.

Proof: Consider a representation $\phi$ of $(Q,W)$ with dimension vector
$(n_{1},n_{2},n_{3})$, assumed to be stable for a King stability condition
$\theta=(\theta_{1},\theta_{2},\theta_{3})$. Consider a cycle $w$ of the
quiver: because $(Q,W)$ gives a noncommutative crepant resolution of
$\mathds{C}^{3}/\mathds{Z}_{3}$ [63, Prop. 3.13], the cycle $w$ is a central
element of the Jacobian algebra of $(Q,W)$, hence it defines an endomorphism
of $\phi$. Because $\phi$ is stable, its only automorphisms are rescalings,
hence $w$ acts as a scalar on $\phi$. Suppose now that $\phi$ is not of
dimension vector $(n,n,n)$ for $n\in\mathds{N}^{\ast}$: $w$ cannot be an
automorphism (otherwise all its arrows would be isomorphisms, hence one would
have $n_{1}=n_{2}=n_{3}$), hence $w$ vanishes on $\phi$.

We suppose now that the King stability parameters satisfy $\theta_{1}<0$, and
$\theta_{3}>0$ (since $\theta_{1}+\theta_{2}+\theta_{3}=0$, other cases follow
by circular permutation of the nodes). We now show by contradiction that
$\phi_{c_{j}}$ vanishes for every arrow $c_{j}:1\to 3$. Suppose that there
exists $x\in V_{1}$ and $j\in\\{1,2,3\\}$ such that $\phi_{c_{j}}(x)\in V_{3}$
does not vanish. Because $\theta_{1}<0$ and $\phi$ is stable, one must have
$x\in\bigoplus_{k=1}^{3}{\rm Im}(\phi_{a_{k}})$. Because $\theta_{3}>0$ and
$\phi$ is stable, $\langle\phi_{c_{j}}(x)\rangle$ cannot be a sub-
representation of $\phi$, hence there must be an arrow $b_{i}\colon 3\to 2$
such that $\phi_{b_{i}}\phi_{c_{j}}(x)\in V_{2}$ does not vanish. Because all
the cycles of $Q$ vanish on $\phi$, one must have
$\phi_{b_{i}}\phi_{c_{j}}(\bigoplus_{k=1}^{3}{\rm Im}(\phi_{a_{k}}))=0$, a
contradiction. Hence the $\phi_{c_{j}}$ vanish on $\phi$ for all $j$. The set
$I=\\{c_{1},c_{2},c_{3}\\}$ of vanishing arrows provides a cut of the
potential $W$, in the sense that each cycle of $W$ contains exactly one arrow
of $I$.

The stable representation $\phi$ is then a representation of the quiver with
relation $(Q_{I},\partial_{I}W)$, where the arrows $c_{j}\in I$ have been
traded for relations $\partial_{c_{j}}W=0$ for $c_{j}\in I$. From general
arguments of geometric invariant theory, the moduli
$\mathcal{M}^{\theta,s}_{Q_{I}}$ of stable representations of $Q_{I}$ is
smooth of dimension $3n_{3}n_{2}+3n_{2}n_{1}-n_{1}^{2}-n_{2}^{2}-n_{3}^{2}+1$.
The moduli space $\mathcal{M}^{\theta,s}_{Q_{I},\partial_{I}W}$ of stable
representations of $(Q_{I},\partial_{I}W)$ is cut out by $3n_{3}n_{1}$
bilinear relations $\partial_{c_{j}}W=0$ inside
$\mathcal{M}^{\theta,s}_{Q_{I}}$. We denote the infinitesimal versions of
these relations on the tangent space of $\phi$ by
$\delta(\partial_{c_{j}}W)\colon V_{3}\to V_{1}$. A linear dependence between
these infinitesimal relations would be given by maps
$\tilde{\phi}_{c_{j}}\colon V_{1}\to V_{3}$ such that:

(5.9)
$\displaystyle\mbox{Tr}(\tilde{\phi}_{c_{j}}\delta(\partial_{c_{j}}W))=0$

As shown in [25, §3.2], this equation is equivalent to the fact that the
representation $\bar{\phi}\coloneqq(\phi_{a},\phi_{b},\tilde{\phi}_{c})$ of
$Q$ (which, like $\phi$, satisfies the relations $\delta_{c_{j}}W=0$)
satisfies the relations $\delta_{a_{i}}W=0$ and $\delta_{b_{i}}W=0$. Hence
$\tilde{\phi}$ is a representation of $(Q,W)$, which is $\theta$-stable
(because it has fewer subobjects than the $\theta$-stable representation
$\phi$) of dimension $\gamma\neq(n,n,n)$. By the above arguments
$\tilde{\phi}_{c_{j}}=0$, therefore the relations $\partial_{c_{j}}W=0$ are
transverse, and $\mathcal{M}^{\theta,s}_{Q_{I},\partial_{I}W}$ is smooth of
dimension
$3n_{3}n_{2}+3n_{2}n_{1}-3n_{3}n_{1}-n_{1}^{2}-n_{2}^{2}-n_{3}^{2}+1$. In
particular, because this dimension is positive, one has

(5.10) $\displaystyle
n_{1}^{2}+n_{2}^{2}+n_{3}^{2}-3n_{3}n_{2}-3n_{2}n_{1}-3n_{1}n_{3}+6n_{3}n_{1}\leq
1$

Now the proof proceeds exactly as in [23, §3.3]. Assume that $\theta$ is an
attractor stability condition, hence a small deformation of the self-stability
condition $\langle-,\gamma\rangle$ for some dimension vector
$\gamma\neq(n,n,n)$. Then $\theta_{1}<0$ implies that $n_{1}\geq n_{2}$, and
$\theta_{3}>0$ implies that $n_{3}\geq n_{2}$: hence $6n_{3}n_{1}\geq
2n_{3}n_{2}+2n_{2}n_{1}+2n_{1}n_{3}$, and then, using (5.10):

(5.11) $\displaystyle q(\gamma)\coloneqq{}$
$\displaystyle\frac{1}{2}((n_{1}-n_{2})^{2}+(n_{2}-n_{3})^{2}+(n_{3}-n_{1})^{2})=n_{1}^{2}+n_{2}^{2}+n_{3}^{2}-n_{1}n_{2}-n_{2}n_{3}-n_{3}n_{1}$
$\displaystyle\leq{}$ $\displaystyle
n_{1}^{2}+n_{2}^{2}+n_{3}^{2}-3n_{1}n_{2}-3n_{2}n_{3}-3n_{2}n_{3}+6n_{3}n_{1}\leq
1$

The kernel of the quadratic form $q(\gamma)$ is given by dimensions vectors
$(n,n,n)$, but we have assumed that $\gamma$ was not of this form. One has
$q(\gamma)=1$ only for $\gamma=(n+1,n,n)$ and circular permutations thereof,
but the inequality is strict in the second line of (5.11) unless $n=0$. Hence,
$\gamma$ must be $(1,0,0)$, $(0,1,0)$, or $(0,0,1)$, with attractor invariants
equal to $1$. The DT invariants for $\gamma=(n,n,n)$ are independent of the
stability conditions, and given by the shifted Poincaré polynomial
$(-y)^{-3}P(K_{\mathds{P}^{2}},y)=-y^{3}-y-1/y$ by [20, Remark 5.2]. $\Box$

Figure 23. Quiver scattering diagram $\mathcal{D}_{Q}$, showing only the
initial rays $\mathcal{R}_{o}(\gamma_{i})$ and $\mathcal{R}_{o}(\delta)$ and
secondary rays $\mathcal{R}_{o}(\gamma_{j}+\gamma_{k})$.

Having determined the initial rays for the orbifold quiver, it is now
straightforward at least in principle to construct the stability scattering
diagram $\mathcal{D}_{Q}$. The result is represented in Figure 23, including
only initial and some secondary rays. The full diagram is dense except in a
cone which includes the positive and negative octant, as will become clear
shortly.

Figure 24. Two-dimensional section $\mathcal{D}_{o}$ of the orbifold
scattering diagram $\mathcal{D}_{Q}$ along the hyperplane
$\theta_{1}+\theta_{2}+\theta_{3}=1$. We show only the restriction to the
triangle $\Delta_{\psi}$ defined in §6 bounded by the points
$p_{i}(\mathcal{V}_{\psi})$ (with $\psi=-1.4$ for illustration), but the
scattering diagram $\mathcal{D}_{o}$ in the space of King stability conditions
is unbounded. Regions with a dense set of rays are shown in gray. The rays in
red correspond to the discrete set of real roots of the Kronecker quiver with
3 arrows, with dimension vector $(F_{2k},F_{2k+2},0)$ or permutations thereof,
see §3.2.

###  5.3. Restricted scattering diagram

Due to the rescaling symmetry in the space of King stability parameters, there
is no significant loss of information222222The price to pay is that the ray
$\mathcal{R}^{\rm eff}_{o}(\delta)$ is no longer visible, but this causes no
problem since the automorphism $\mathcal{U}^{\rm eff}(\delta)$ is central;
moreover, the self-stability condition $\langle-,\gamma\rangle$ is now pushed
to infinity in the direction opposite to the vector $\nu(\gamma)$ introduced
below.  in restricting the scattering diagram $\mathcal{D}_{Q}$ to its
intersection $\mathcal{D}_{o}$ with the hyperplane
$\mathcal{H}=\\{\theta_{1}+\theta_{2}+\theta_{3}=1\\}$. It is furthermore
convenient to parametrize this hyperplane by $(u,v)\in\mathds{R}^{2}$ such
that

(5.12)
$\theta_{1}=-u+v\sqrt{3}+\tfrac{1}{3},\quad\theta_{2}=2u+\tfrac{1}{3},\quad\theta_{3}=-u-v\sqrt{3}+\tfrac{1}{3}$

In these coordinates, the $\mathds{Z}_{3}$ symmetry permuting the nodes of the
quiver cyclically acts by a rotation of angle $2\pi/3$ around the origin (see
Figure 5). The geometric rays $\mathcal{R}^{\rm
geo}_{o}(\gamma)=\\{(u,v)\in\mathds{R}^{2}:\sum_{i}n_{i}\theta_{i}(u,v)=0\\}$
are given by straight lines

(5.13)
$(n_{1}+n_{3}-2n_{2})u+(n_{3}-n_{1})v\sqrt{3}-\frac{1}{3}(n_{1}+n_{2}+n_{3})=0$

oriented along the vector
$\nu(\gamma)=(\sqrt{3}(n_{3}-n_{1}),2n_{2}-n_{1}-n_{3})$ pointing from the
attractor stability condition $\theta_{i}=\lambda(n_{i-1}-n_{i+1})$ towards
the anti-attractor stability condition $\theta_{i}=\lambda(n_{i+1}-n_{i-1})$,
with indices $i=1,2,3$ taken modulo 3. In particular, the initial rays
intersect at $\mathcal{R}^{\rm geo}_{o}(\gamma_{i})\cap\mathcal{R}^{\rm
geo}_{o}(\gamma_{i+1})=\\{p_{i}\\}$ with

(5.14) $p_{1}=\left(-\frac{1}{6},-\frac{1}{2\sqrt{3}}\right),\quad
p_{2}=\left(-\frac{1}{6},\frac{1}{2\sqrt{3}}\right),\quad
p_{3}=\left(\frac{1}{3},0\right)$

Interpreting the ray as the worldline of a fiducial particle traveling at
point $r=(u,v)$ with velocity $\nu(\gamma)$, the condition (5.13) implies that
the particle has angular momentum
$r\wedge\nu(\gamma)=-\frac{1}{3}(n_{1}+n_{2}+n_{3})$ with respect to the
origin $(0,0)$ in the $(u,v)$ plane, therefore rotates clockwise around the
origin (assuming that $n_{i}\geq 0$). In fact, (5.13) implies that the
particle has angular momentum

(5.15) $(r-p_{i})\wedge\nu(\gamma)=-n_{i-1}$

with respect to any of the points $p_{i}$, so the particle rotates clockwise
around each $p_{i}$, and can only pass through it if $n_{i-1}=0$ (this
generalizes the previous statement since $p_{1}+p_{2}+p_{3}=0$).

Another consequence of (5.13) is that the linear function on $\Gamma$

(5.16)
$\varphi_{o}(\gamma)=(n_{3}-n_{1})u\sqrt{3}+(2n_{2}-n_{1}-n_{3})v=(r,\nu(\gamma))$

increases monotonically along the ray and is additive at each vertex, similar
to the electric potential (4.10) in the large volume scattering diagram.
Unlike the latter however, the function (5.16) is not positive, in particular
the first scattering between $\mathcal{R}_{o}(\gamma_{i})$ and
$\mathcal{R}_{o}(\gamma_{i+1})$ may take place at arbitrary negative values of
$\varphi_{o}(\gamma_{i}),\varphi_{o}(\gamma_{i+1})$. Fortunately, we do not
need to rely on such a cost function to define the quiver scattering diagram
and enumerate all possible scattering sequences, instead we can use the fact
that only positive dimension vectors support non-zero DT invariants. In the
next subsection, we use the orbifold scattering diagram to compute DT
invariants for some simple dimension vectors.

###  5.4. Examples

We first consider the case $\gamma=[1,0,1-n)$, corresponding to the Hilbert
scheme of $n$ points on $\mathds{P}^{2}$. The corresponding dimension vector
is $\gamma=(n-1,n,n)$. As in §4.3 we find that the index in the anti-attractor
chamber, which we denote by
$\Omega_{c}(\gamma)\coloneqq\Omega_{\langle\gamma,-\rangle}(\gamma)$, agrees
with the Gieseker index (2.13) for all cases considered, although the set of
scattering sequences for the large volume and orbifold scattering diagrams may
differ. As explained in §6.4.4, the absence of walls between the anti-
canonical chamber and the large volume chamber is a general property of
objects with slope in $\mu\in[-1,0]$.

Figure 25. Scattering sequences for dimension vector $(n-1,n,n)$ corresponding
the Hilbert scheme of $n$ points on $\mathds{P}^{2}$, with $n=1,2,3,4$

  * •

For $n=1$ we find a single scattering sequence $\\{\gamma_{2},\gamma_{3}\\}$
contributing $K_{3}(1,1)=y^{2}+1+1/y^{2}\to 3$ in the anti-attractor chamber.

  * •

For $n=2$, we find a single scattering sequence
$\\{\\{\gamma_{1},2\gamma_{2}\\},2\gamma_{3}\\}\\}$ contributing
$K_{3}(1,2)^{2}=(y^{2}+1+1/y^{2})^{2}\to 9$.

  * •

For $n=3$ we find two scattering sequences,

(5.17) $\begin{array}[]{l@{\hskip
5mm}l}\left\\{\left\\{2\gamma_{1},3\gamma_{2}\right\\},3\gamma_{3}\right\\}\hfil\hskip
14.22636pt&K_{3}(1,3)K_{3}(2,3)\to 13\\\
\left\\{\left\\{\left\\{2\gamma_{1},\gamma_{3}\right\\},3\gamma_{2}\right\\},2\gamma_{3}\right\\}\hfil\hskip
14.22636pt&K_{3}(1,2){}^{2}K_{3}(1,3)\to 9\end{array}$

for a total index of $y^{6}+2y^{4}+5y^{2}+6+\dots\to 22$.

  * •

For $n=4$ we find 3 scattering sequences,

(5.18) $\begin{array}[]{l@{\hskip
5mm}l}\left\\{\left\\{\left\\{3\gamma_{1},\gamma_{3}\right\\},4\gamma_{2}\right\\},3\gamma_{3}\right\\}\hfil\hskip
14.22636pt&K_{3}(1,3){}^{2}K_{6}(1,4)\to 15\\\
\left\\{\left\\{\left\\{3\gamma_{1},\left\\{\gamma_{2},2\gamma_{3}\right\\}\right\\},3\gamma_{2}\right\\},2\gamma_{3}\right\\}\hfil\hskip
14.22636pt&K_{3}(1,2){}^{2}K_{3}(1,3){}^{2}\to 9\\\
\\{\\{\\{\\{2\gamma_{1},\gamma_{3}\\},3\gamma_{2}\\},\\{\gamma_{1},\gamma_{2}\\}\\},3\gamma_{3}\\}\hfil\hskip
14.22636pt&K_{3}(1,1){}^{2}K_{3}(1,2)K_{3}(1,3){}^{2}\to 27\end{array}$

for a total index of $y^{8}+2y^{6}+6y^{4}+10y^{2}+13+\dots\to 51$.

Next we consider $\gamma=[0,1,1-n)$, corresponding to a D2-brane bound to $n$
anti-D0-branes. The corresponding dimension vector is $\gamma=(n,n+1,n+2)$.
Unlike the Gieseker index at large volume, the index $\Omega_{c}(\gamma)$ in
the anti-attractor chamber depends on $n$, though it is still invariant under
$n\mapsto-n-2$. After a circular permutation, the dimension vector becomes
$\gamma^{\prime}=(n+1,n+2,n)=[3,-2,-1-n)$, with slope $\mu=-2/3\in[0,1]$, and
$\Omega_{c}(\gamma^{\prime})$ agrees with the Gieseker index for rank 3
sheaves with $d=-2$ and $\chi=-1-n$.

  * •

For $n=-1$, the dimension vector $(-1,0,1)$ has mixed signs so the index
vanishes.

  * •

For $n=0$, we find a single scattering sequence $\\{\gamma_{2},2\gamma_{3}\\}$
contributing $K_{3}(1,2)=y^{2}+1+1/y^{2}\to 3$.

  * •

For $n=1$, we find $2$ scattering sequences,

(5.19)
$\begin{array}[]{ll}\\{\gamma_{1},\\{2\gamma_{2},3\gamma_{3}\\}\\}&K_{3}(2,3)K_{3}(1,1)\to
39\\\ \\{\\{\gamma_{1},2\gamma_{2}\\},3\gamma_{3}\\}&K_{3}(1,2)K_{3}(1,3)\to
3\end{array}$

giving a total index of $y^{8}+2y^{6}+5y^{4}+8y^{2}+10+\dots\to 42$.

  * •

For $n=2$, we find $3$ scattering sequences,

(5.20)
$\begin{array}[]{ll}\\{2\gamma_{1},\\{3\gamma_{2},4\gamma_{3}\\}\\}&K_{3}(1,2)K_{3}(3,4)\to
204\\\
\\{\\{\\{2\gamma_{1},\gamma_{3}\\},3\gamma_{2}\\},3\gamma_{3}\\}&K_{3}(1,2)K_{3}(1,3)^{2}\to
3\\\
\\{\\{\\{\gamma_{1},3\gamma_{2}\\},4\gamma_{3}\\},\gamma_{1}\\}&K_{3}(1,1)K_{3}(1,3)K_{6}(1,4)\to
45\\\
\\{\\{\\{\gamma_{1},2\gamma_{2}\\},3\gamma_{3}\\},\\{\gamma_{2},\gamma_{3}\\}\\},\gamma_{1}\\}&K_{3}(1,1)^{3}K_{3}(1,2(K_{3}(1,3)\to
81\end{array}$

giving a total index of $333$.

  * •

For $n=3$, we find $7$ scattering sequences,

$\begin{array}[]{ll}\left\\{3\gamma_{1},\left\\{4\gamma_{2},5\gamma_{3}\right\\}\right\\}&K_{3}(1,3)K_{3}(4,5)\to
399\\\
\left\\{2\gamma_{1},\left\\{\left\\{\left\\{\gamma_{1},3\gamma_{2}\right\\},5\gamma_{3}\right\\},\gamma_{2}\right\\}\right\\}&K_{3}(1,2)K_{3}(1,3)K_{6}(1,5)K_{12}(1,1)\to
216\\\
\left\\{2\gamma_{1},\\{\\{\left\\{\gamma_{1},3\gamma_{2}\right\\},4\gamma_{3}\\},\\{\gamma_{2},\gamma_{3}\\}\\}\right\\}&K_{3}(1,1){}^{2}K_{3}(1,2)K_{3}(1,3)K_{6}(1,4)\to
405\\\
\left\\{2\gamma_{1},\\{\\{\left\\{\gamma_{1},2\gamma_{2}\right\\},3\gamma_{3}\\},\\{2\gamma_{2},2\gamma_{3}\\}\\}\right\\}&K_{3}(2,2,1)K_{3}(1,2){}^{2}K_{3}(1,3)\to
648\\\
\left\\{\left\\{\left\\{3\gamma_{1},\left\\{\gamma_{2},2\gamma_{3}\right\\}\right\\},3\gamma_{2}\right\\},3\gamma_{3}\right\\}&K_{3}(1,2)K_{3}(1,3){}^{3}\to
3\\\
\left\\{\left\\{\left\\{2\gamma_{1},4\gamma_{2}\right\\},5\gamma_{3}\right\\},\gamma_{1}\right\\}&K_{6,6,3}(2,2,5)K_{3}(1,1)\to
216\\\
\left\\{\\{\\{\left\\{\left\\{2\gamma_{1},\gamma_{3}\right\\},3\gamma_{2}\right\\},3\gamma_{3}\\},\\{\gamma_{2},\gamma_{3}\\}\\},\gamma_{1}\right\\}&K_{3}(1,1){}^{3}K_{3}(1,2)K_{3}(1,3){}^{2}\to
81\end{array}$

giving a total index of $1968$. Similar to the discussion below (4.23), the
factor $K_{6,6,3}(2,2,5)$ for the penultimate sequence is computed by applying
the flow tree formula to a local scattering diagram with two incoming rays of
charge $\alpha=\gamma_{1}+2\gamma_{2}$ and $\beta=\gamma_{3}$ with
$\bar{\Omega}^{-}(\alpha)=K_{3}(1,2)=y^{2}+1+1/y^{2},\bar{\Omega}^{-}(2\alpha)=K_{3}(2,4)=-y^{5}-y^{3}-y-1/y-1/y^{3}-1/y^{5}$
and $\bar{\Omega}^{-}(k\beta)=\delta_{k=1}$, selecting the outgoing ray of
charge $2\alpha+5\beta$.

##  6\. The exact scattering diagram

In this section, we determine the scattering diagram
$\mathcal{D}_{\psi}^{\Pi}$ along the slice of $\Pi$-stability conditions, by
combining results on the large volume and orbifold scattering diagrams
constructed in §4 and §5 with invariance under $\Gamma_{1}(3)$. We start by
analyzing the attractor flow for the exact central charge (1.4), first in the
Poincaré upper half-plane (§6.1) and then in affine coordinates (§6.2). In
§6.3 we then identify the orbifold scattering diagram $\mathcal{D}_{o}$ as a
particular subset of $\mathcal{D}_{\psi}^{\Pi}$ in a region around the
orbifold point. In §6.4 we outline a proof of the SAFC, leaving details to a
separate article [41].

###  6.1. Exact attractor flow

In this section, we study the attractor flow (3.19) for the central charge
(1.4) along the slice of $\Pi$-stability conditions, equipped with an general
hermitean metric
$\mathrm{d}s^{2}=g_{\tau\bar{\tau}}\mathrm{d}\tau\mathrm{d}\bar{\tau}$. Using
the Eichler integral representation (1.5) for the coefficients $T,T_{D}$ in
the central charge, the attractor flow ${\rm AF}(\gamma)$ (3.19) reduces to

(6.1)
$\frac{\mathrm{d}\tau}{\mathrm{d}\mu}=-g^{\tau\bar{\tau}}\partial_{\bar{\tau}}|Z_{\tau}(\gamma)|^{2}=-g^{\tau\bar{\tau}}(d-r\bar{\tau})\,\overline{C(\tau)}Z_{\tau}(\gamma)$

The flow is only meaningful in the region where the charge $\gamma$ is
populated, $\bar{\Omega}_{\tau}(\gamma)\neq 0$. As already noted in §3.5, the
modulus $|Z_{\tau}(\gamma)|=\Im[e^{-\mathrm{i}\psi}Z_{\tau}(\gamma)]$
decreases along the flow while the argument of $Z_{\tau}(\gamma)$ is
preserved, so the trajectories of (6.1) are included in the active ray
$\mathcal{R}_{\psi}(\gamma)$ defined in (3.12) as the locus
$\\{\Re[e^{-\mathrm{i}\psi}Z_{\tau}(\gamma)]=0,\Im(e^{-\mathrm{i}\psi}Z_{\tau}(\gamma))>0,\bar{\Omega}_{\tau}(\gamma)\neq
0\\}$ inside $\mathds{H}$, where the phase $\psi$ is determined by the
argument of $Z_{\tau_{0}}(\gamma)$ at the starting point $\tau_{0}$. The
attractor flow stops when either a) the flow crosses a wall of marginal
stability, after which $\bar{\Omega}_{\tau}(\gamma)$ jumps, b)
$|Z_{\tau}(\gamma)|$ reaches a local minimum in the interior of the upper
half-plane, or c) $\tau$ reaches the boundary $\Im\tau=0$.

Case a) arises at a point $\mu_{1}$ where the quantity
$\tilde{W}_{\tau}(\gamma,\gamma^{\prime})\coloneqq\Im[Z_{\tau}(\gamma^{\prime})\overline{Z_{\tau}(\gamma)}]$
vanishes for some charge $\gamma^{\prime}$. Since
$\tilde{W}_{\tau}(\gamma,\gamma^{\prime})$ varies along the flow according to

(6.2)
$\frac{\mathrm{d}\tilde{W}_{\tau}(\gamma,\gamma^{\prime})}{\mathrm{d}\mu}=-g^{\tau\bar{\tau}}|C|^{2}\left[2\tau_{2}(rd^{\prime}-r^{\prime}d)|Z_{\tau}(\gamma)|^{2}+\tilde{W}_{\tau}(\gamma,\gamma^{\prime})|d-r\tau|^{2}\right]$

the sign of
$(rd^{\prime}-r^{\prime}d)\tilde{W}_{\tau}(\gamma,\gamma^{\prime})$ is
positive for $\mu<\mu_{1}$ and negative for $\mu>\mu_{1}$. Thus, the flow
crosses from the side of the wall where a two-particle bound state of charges
$\gamma$ and $\gamma-\gamma^{\prime}$ is stable, into the side where the bound
state decays. In particular, it follows from (6.2) that the flow cannot cross
the same wall more than once, and that BPS states do not decay as one follows
the attractor flow in reverse [9].

As for case b), since $C(\tau)\neq 0$ for $\Im\tau>0$, $|Z_{\tau}(\gamma)|$
can only reach a local minimum at a point $\tau_{i}$ in the interior of
$\mathds{H}$ if $Z_{\tau_{i}}(\gamma)=0$, but this is ruled out by the
assumption that $\bar{\Omega}_{\tau}(\gamma)\neq 0$ along the flow. This
leaves case c) with $\Im\tau_{i}=0$. Consider the image of the attractor flow
on the quotient $\mathds{H}/\Gamma_{1}(3)$. Since the quotient can be
compactified by adding the large volume and conifold points, $\tau_{i}$ is
either in the orbit of a large volume point or a conifold point, i.e.
$\tau_{i}=p/q$ with $(p,q)$ coprime, $q=0\mod 3$ in the first case and $q\neq
0\mod 3$ in the second case. Using suitable bounds on the central charge of
semi-stable objects in the large volume region, we show in [41] that
$\tau_{i}$ can only be a conifold point, where some spherical object $E$ of
charge $\gamma_{C}$ becomes massless, $Z_{\tau_{i}}(\gamma_{C})=0$.

To determine $E$, it suffices to find an element $g\in\Gamma_{1}(3)$ which
maps $\tau=0$ to $\tau_{i}=p/q$, and act with the corresponding auto-
equivalence on the object $\mathcal{O}$ which is massless at $\tau=0$. Since
$(T,T_{D})=(\mathrm{i}\mathcal{V},0)$ at $\tau=0$, where $\mathcal{V}$ is the
quantum volume defined in (1.12), the periods at $\tau_{i}$ are then

(6.3) $T(\tau_{i})=m+\mathrm{i}q\mathcal{V},\quad
T_{D}(\tau_{i})=m_{D}+\mathrm{i}p\mathcal{V}$

where $(m,m_{D})$ are the off-diagonal elements of the monodromy matrix $M(g)$
in (A.50). The charge of the massless object $E$ at $\tau=\tau_{i}$ is then
$\gamma_{C}=[q,p,pm-qm_{D}]$ up to overall sign, see Table 1 for examples with
$0\leq p\leq q\leq 5$.

Returning to the active ray $\mathcal{R}_{\psi}(\gamma)$, it follows from
(6.3) that it can only reach the conifold point $\tau_{i}=p/q$ if

(6.4) $0=\Re[e^{-\mathrm{i}\psi}Z_{p/q}(\gamma)]=(dm-
rm_{D}-\textnormal{ch}_{2})\cos\psi+(dq-pr)\mathcal{V}\sin\psi$

For generic $\psi$, this is only possible if $\gamma$ is a multiple of
$\gamma_{C}$, in which case $\mathcal{R}_{\psi}(\gamma)$ coincides with the
ray $\mathcal{R}_{\psi}(\gamma_{C})$ originating from $\tau_{i}=p/q$. For
special values of $\psi$ such that
$\mathcal{V}_{\psi}\coloneqq\mathcal{V}\tan\psi$ is rational, namely

(6.5) $\mathcal{V}_{\psi}=\frac{\textnormal{ch}_{2}+rm_{D}-dm}{dq-pr}$

it is possible that a ray $\mathcal{R}_{\psi}(\gamma)$ with
$\langle\gamma_{C},\gamma\rangle=dq-pr\neq 0$ originates from the point
$\tau_{i}=\frac{p}{q}$, even though $Z_{\tau_{i}}(\gamma)\neq 0$. Conversely,
for the same critical values of $\psi$ there can also be rays which terminate
at a conifold point, rather than escaping at a large volume point (this can be
seen by taking the $\psi\to\psi^{\rm cr}_{-1/2}$ limit of Figure 8). In the
next subsection, we introduce adapted coordinates where these critical rays
become much easier to detect.

###  6.2. Affine coordinates

While the attractor flow ${\rm AF}(\gamma)$ is complicated on the
$\tau$-plane, it becomes a straight line in affine coordinates $(x,y)$ defined
globally by232323The affine coordinate $y$ should not be confused with the
refinement variable $y$ in the definition of motivic DT invariants.

(6.6) $x=\frac{\Re\left(e^{-\mathrm{i}\psi}T\right)}{\cos\psi}\ ,\quad
y=-\frac{\Re\left(e^{-\mathrm{i}\psi}T_{D}\right)}{\cos\psi}$

It is useful to introduce the dual coordinates

(6.7)
$\tilde{x}=\frac{\Im(e^{-\mathrm{i}\psi}T)}{\cos\psi},\quad\tilde{y}=-\frac{\Im(e^{-\mathrm{i}\psi}T_{D})}{\cos\psi}$

such that the geometric ray $\mathcal{R}^{\rm geo}_{\psi}(\gamma)$, oriented
along the direction of decreasing
$\Im[e^{-\mathrm{i}\psi}Z(\gamma)]=r\tilde{y}+d\tilde{x}$, are given by
oriented straight lines

(6.8)
$ry+dx-\textnormal{ch}_{2}=0,\quad\frac{\mathrm{d}}{\mathrm{d}\mu}(r\tilde{y}+d\tilde{x})<0$

Notice that for the large volume central charge $Z^{\rm LV}_{(s,t)}$, the
coordinates $(x,y)$ coincide with
$(s_{\psi},\frac{1}{2}(t_{\psi}^{2}-s_{\psi}^{2}))$ defined in (4.26) and
generalize the coordinates $(x,y)$ of [26] to any
$\psi\in(-\frac{\pi}{2},\frac{\pi}{2})$. Notably, the image of the upper half-
plane $t>0$ lies above the parabola $y=-\frac{1}{2}x^{2}$.

In contrast, for the exact central charge (1.4), the image of the fundamental
domain $\mathcal{F}_{C}$ and its translates in the $(x,y)$-plane extend below
the parabola $y=-\frac{1}{2}x^{2}$, while remaining above the parabola
$y=-\frac{1}{2}x^{2}-\frac{5}{24}$ passing through the images of the orbifold
points at $\tau=\tau_{o}+n$ (see Figure 26)

(6.9)
$(x_{o(n)},y_{o(n)})=\left(n-\tfrac{1}{2},-\tfrac{1}{3}-\tfrac{1}{2}n(n-1)\right)$

For $\psi=0$, the preimage of the region $y>-\frac{1}{2}x^{2}$ includes the
domain $\mathds{H}^{\rm LV}$ defined by the condition $w>\frac{1}{2}s^{2}$ (or
equivalently $t>0$). In fact, the curves $t=0$ and $y=-\frac{1}{2}x^{2}$ are
tangent to the same point $\tau\simeq\frac{1}{2}+0.559926\mathrm{i}$
corresponding to $(s,w)=(\frac{1}{2},\frac{1}{8})$ or
$(x,y)=(\frac{1}{2},-\frac{1}{8})$, and are nearly indistinguishable for any
$\tau_{1}$, with the former lying below the latter. For general $\psi$, using
(A.25) along the line $\tau_{1}=\frac{1}{2}$ we get

(6.10) $y+\frac{1}{2}x=w-\frac{1}{2}s^{2}+\frac{1}{2}\Im T\tan\psi(1+\Im
T\tan\psi)$

so the relative position of the two curves depends on the sign of $\Im
T\tan\psi$.

Figure 26. Image of the fundamental domain $\mathcal{F}_{C}$ and its
translates $\mathcal{F}_{C}(m)$ in the $(x,y)$-plane for $\psi=0$ (left) and
$\psi=-0.9<-\psi^{\rm cr}_{1/2}$ (right). The parabolas $y=-\frac{1}{2}x^{2}$
and $y=-\frac{1}{2}x^{2}-\frac{5}{24}$ is shown for reference. For any
$\psi\in(-\frac{\pi}{2},\frac{\pi}{2})$, the map $\tau\mapsto(x,y)$ is
injective in $\bigcup_{m\in\mathds{Z}}\mathcal{F}_{C}(m)$.

Geometrically, the coordinates $(x,y)$ and $(\tilde{x},\tilde{y})$ provide two
systems of local Darboux coordinates for the Kähler form

(6.11)
$\mathrm{d}x\wedge\mathrm{d}y=\mathrm{d}\tilde{x}\wedge\mathrm{d}\tilde{y}=-\frac{1}{4}(\mathrm{d}T\wedge\mathrm{d}\bar{T}_{D}+\mathrm{d}\bar{T}\wedge\mathrm{d}T_{D})$

Hence, locally there exists a function $H_{\psi}(x,y)$ such that
$\tilde{x}=\partial_{y}H_{\psi},\tilde{y}=-\partial_{x}H_{\psi}$, satisfying
the Monge-Ampère type equation

(6.12)
$(\partial^{2}_{x}H_{\psi})(\partial^{2}_{y}H_{\psi})-(\partial_{x}\partial_{y}H_{\psi})^{2}=1$

One easily checks that the Legendre transform to the imaginary part of the
tree-level prepotential (A.30), sometimes known as the Hesse potential [64],
provides such a function,

(6.13)
$H_{\psi}(x,y)=\langle\Im\left[-\frac{2\pi\mathrm{i}}{3}\frac{e^{-2\mathrm{i}\psi}}{\cos^{2}\psi}F_{0}\left(\cos\psi
e^{\mathrm{i}\psi}(x+\mathrm{i}\tilde{x})\right)\right]+\tilde{x}y\rangle_{\tilde{x}}$

where $F_{0}$ is normalized such that
$T_{D}=-\frac{3}{2\pi\mathrm{i}}\partial_{T}F_{0}$ (see (A.30)). In the large
volume limit, (6.13) evaluates to

(6.14) $\displaystyle H^{\rm LV}_{\psi}(x,y)$
$\displaystyle=\frac{1}{3\cos\psi}(x^{2}+2y)^{\frac{3}{2}}-\frac{1}{3}x(x^{2}+3y)\tan\psi$
$\displaystyle=\frac{t^{3}}{3\cos\psi}+\frac{s}{6}(s^{2}-3t^{2})\tan\psi-\frac{t}{6}(t^{2}-3s^{2})\tan^{2}\psi$

An explicit computation shows that the Hessian matrix of $H_{\psi}$ with
respect to $(x,y)$ coincides with that of $H^{\rm LV}_{\psi}$ (i.e. the
Hessian is insensitive to the factor $C(u)$ appearing in the Eichler integral
(1.5)), namely

(6.15) $\begin{pmatrix}\partial_{xx}H_{\psi}&\partial_{xy}H_{\psi}\\\
\partial_{xy}H_{\psi}&\partial_{yy}H_{\psi}\end{pmatrix}=\frac{1}{\tau_{2}}\begin{pmatrix}|\tau|^{2}&\tau_{1}\\\
\tau_{1}&1\end{pmatrix}$

In particular it is positive definite. This observation allows to write the
orientation condition in (6.8) purely in terms of the coordinates $(x,y)$:
indeed, it becomes

(6.16)
$-\frac{1}{r}\left[r^{2}\partial_{x}^{2}H_{\psi}-2dr\partial_{x}\partial_{y}H_{\psi}+d^{2}\partial_{y}^{2}H_{\psi}\right]\frac{\mathrm{d}x}{\mathrm{d}\mu}<0$

Thus we conclude that for $r\neq 0$ the ray is oriented such that
$r\frac{\mathrm{d}x}{\mathrm{d}\mu}>0$. In particular, the electric potential
$\varphi_{x}(\gamma)=2(d-xr)$ (generalizing (4.10) for the exact central
charge) decreases along the ray. When $r=0$, we get instead
$\frac{\mathrm{d}(d\tilde{x})}{\mathrm{d}\mu}=\frac{d}{\tau_{2}}\frac{\mathrm{d}y}{\mathrm{d}\mu}<0$,
so the vertical ray is oriented towards decreasing $y$ for $d>0$ (or
increasing $y$ for $d<0$).

We now determine the images of the conifold points in the $(x,y)$ plane. Using
(6.3), we find that the point $\tau=p/q$ where the object of charge
$\gamma_{C}=[q,p,pm-qm_{D}]$ becomes massless maps to

(6.17) $(x_{C},y_{C})=(m+q\mathcal{V}_{\psi},-m_{D}-p\mathcal{V}_{\psi})$

where $\mathcal{V}_{\psi}=\mathcal{V}\tan\psi$. In particular, the conifold
point $\tau=m$ where the object $\mathcal{O}(m)$ becomes massless is mapped to
the point

(6.18)
$(x_{\mathcal{O}(m)},y_{\mathcal{O}(m)})=\left(m+\mathcal{V}_{\psi},-\frac{1}{2}{m^{2}}-m\mathcal{V}_{\psi}\right)$

on the parabola $y=-\frac{1}{2}x^{2}+\frac{1}{2}\mathcal{V}_{\psi}^{2}$.
Moreover, the associated rays $\mathcal{R}^{\rm
geo}_{\psi}(\pm\mathcal{O}(m))$ are contained in the tangent to the parabola
$y=-\frac{1}{2}x^{2}$ at $x=m$, just as for the large volume scattering
diagram. Similarly, the point $\tau=n-\frac{1}{2}$ where $\Omega(n+1)$ becomes
massless is mapped to

(6.19)
$(x_{\Omega(n+1)},y_{\Omega(n+1)})=\left(n-\frac{1}{2}-2\mathcal{V}_{\psi},-\frac{1}{2}(n^{2}-n+1)+(2n-1)\mathcal{V}_{\psi}\right)\quad$

on the parabola $y=-\frac{1}{2}x^{2}+2\mathcal{V}_{\psi}^{2}-\frac{3}{8}$, and
the corresponding rays $\mathcal{R}^{\rm geo}_{\psi}(\pm\Omega(n+1))$ are
contained in the tangent to the parabola $y=-\frac{1}{2}x^{2}-\frac{3}{8}$ at
$x=n-\frac{1}{2}$. The corresponding points are marked in Figure 27.

Figure 27. Images of the fundamental domains $\mathcal{F}_{o}$,
$\mathcal{F}_{o^{\prime}}$ and their $\mathds{Z}_{3}$ images in the $(x,y)$
plane for $\psi=0$. The vertical black lines correspond to the semi-infinite
lines $(-1,0,1)+\frac{\mathrm{i}}{2\sqrt{3}}+\mathrm{i}\mathds{R}^{+}$, while
the blue and purple lines correspond to their images under $\mathds{Z}_{3}$.

###  6.3. Orbifold region

We now discuss the domain of validity of the orbifold quiver description
inside the space of $\Pi$-stability conditions, and how the two-dimensional
scattering diagram constructed in §5.3 fits in the scattering diagram for the
exact central charge.

As explained in §5.1, the quiver description holds in the region
$\mathds{H}^{o}\subset\Pi$ where the central charges $Z_{\tau}(\gamma_{i})$ of
the simple objects belong to a common half-plane. In that region, the heart
$\mathcal{A}(\tau)$ is equivalent under $\widetilde{GL^{+}}(2,\mathds{R})$ to
the Abelian category of representations of $J(Q,W)$, and the moduli space of
semi-stable objects of phase $\arg Z_{\tau}(\gamma)=\psi+\frac{\pi}{2}$ in
$\mathcal{A}(\tau)$ coincides with the moduli space of semi-stable
representations for King stability parameters
$\theta_{i}\propto\Re[e^{-\mathrm{i}\psi}Z_{\tau}(\gamma_{i})]$. Choosing the
scale such that
$\theta_{i}(\gamma)=\Im[Z(\gamma_{i})\bar{Z}(\gamma)]=\tilde{W}(\gamma,\gamma_{i})$,
we can apply the relation (6.2) to obtain the variation of $\theta_{i}$ along
the attractor flow ${\rm AF}(\gamma)$,

(6.20)
$\frac{\mathrm{d}\theta_{i}}{\mathrm{d}\mu}=-g^{\tau\bar{\tau}}|C|^{2}\left[2\tau_{2}\langle\gamma,\gamma_{i}\rangle|Z(\gamma)|^{2}+|d-r\tau|^{2}\theta_{i}\right]$

The second term simply amounts to a rescaling of $\theta_{i}$, while the first
term shows that $\theta_{i}$ flows towards the self-stability condition
$\langle\gamma_{i},\gamma\rangle$. On the other hand, since
$Z_{\tau}(\gamma_{1})+Z_{\tau}(\gamma_{2})+Z_{\tau}(\gamma_{3})=Z_{\tau}(\delta)=1$
for any $\tau\in\mathds{H}$, it makes more sense to fix the scale by requiring

(6.21)
$\theta_{i}=\frac{\Re[e^{-\mathrm{i}\psi}Z_{\tau}(\gamma_{i})]}{\cos\psi}$

such that $\theta_{1}+\theta_{2}+\theta_{3}=1$. The King stability parameters
(6.21) are then linear combinations of the affine coordinates defined in
(6.6), such that the lines of gradient flow (6.8) corresponds to straight
lines $n_{1}\theta_{1}+n_{2}\theta_{2}+n_{3}\theta_{1}=0$.

Since the central charges $Z_{\tau}(\gamma_{i})$ are all equal at the orbifold
point $\tau_{o}=\frac{1}{\sqrt{3}}e^{5\pi\mathrm{i}/6}$, the region
$\mathds{H}^{o}\subset\Pi$ includes an open set around $\tau=\tau_{o}$. Using
the Eichler integral representation (1.5), it is immediate to compute the
central charge to first order in $\tau-\tau_{o}$,

(6.22) $Z_{\tau}(\gamma)\simeq
Z_{\tau}^{o}(\gamma)\coloneqq-\frac{r}{3}-\frac{d}{2}-\textnormal{ch}_{2}+C(\tau_{0})(d-r\tau_{o})\,(\tau-\tau_{o})$

leading to the central charges for the simple objects

(6.23)
$\begin{cases}Z_{\tau}(\gamma_{1})\simeq\tfrac{1}{3}+\tau_{o}\,C(\tau_{o})\,(\tau-\tau_{o})\\\
Z_{\tau}(\gamma_{2})\simeq\tfrac{1}{3}-(2\tau_{o}+1)\,C(\tau_{o})\,(\tau-\tau_{o})\\\
Z_{\tau}(\gamma_{3})\simeq\tfrac{1}{3}+(\tau_{o}+1)\,C(\tau_{o})\,(\tau-\tau_{o})\\\
\end{cases}$

Computing the stability parameters via (6.21), one finds that the coordinates
$(u,v)$ on the two-dimensional section (5.12) are related to the distance away
from the orbifold point via

(6.24)
$\tau-\tau_{o}\simeq\frac{2\mathrm{i}\sqrt{3}}{C(\tau_{o})}e^{\mathrm{i}\psi}(u+\mathrm{i}v)\cos\psi$

In particular, as $\psi\to\pm\frac{\pi}{2}$, the scattering diagram
$\mathcal{D}_{o}$ is mapped to an infinitesimal neighbourhood of $\tau_{o}$
(see Figure 10).

Going away from the orbifold point, the region $\mathds{H}^{o}$ is bounded in
the fundamental domain $\mathcal{F}_{o}$ by the condition

(6.25) $\Im(\overline{Z_{\tau}(E_{1})}Z_{\tau}(E_{3}))>0$

and similarly by conditions
$\Im(\overline{Z_{\tau}(E_{2})}Z_{\tau}(E_{1}))>0$,
$\Im(\overline{Z_{\tau}(E_{3})}Z_{\tau}(E_{2}))>0$ in the images
$M_{o}\mathcal{F}_{o}$ and $M_{o}^{2}\mathcal{F}_{o}$ under the
$\mathds{Z}_{3}$ symmetry. In terms of the variables $(s,w)$ defined in (1.7),
where walls of (anti)marginal stability are straight lines (see (2.29)), the
inequality (6.25) in $\mathcal{F}_{o}$ amounts to

(6.26) $w+\frac{1}{2}s<0$

In that region, the coordinates $(u,v)$ on the two-dimensional section of the
quiver scattering diagram can be expressed in terms of the affine coordinates
(6.6),

(6.27) $u=\frac{1}{2}x-y-\frac{1}{12},\quad
v=\frac{1}{2\sqrt{3}}\left(x+\frac{1}{2}\right)$

such that the rays in the exact scattering diagram map to segments contained
in the geometric rays (5.13).

However, while the initial rays $\mathcal{R}_{o}(\gamma_{i})$ in the quiver
scattering diagram extend to infinity inside the space of King stability
conditions, they actually originate from conifold points at
$\tau=0,-\frac{1}{2},-1$ in the slice of $\Pi$-stability conditions. Using the
values of the periods in Table 3, one finds that these conifold points are
mapped to points at finite distance in the coordinates defined by (6.27). To
make this precise, let us parametrize the initial rays
$\mathcal{R}_{o}(\gamma_{i})$ in the quiver scattering diagram as

(6.28)
$p_{1}(\lambda)=\Bigl{(}\frac{1}{12}-\frac{\lambda}{2},\frac{-2\lambda-1}{4\sqrt{3}}\Bigr{)},\qquad
p_{2}(\lambda)=\Bigl{(}-\frac{1}{6},\frac{\lambda}{\sqrt{3}}\Bigr{)},\qquad
p_{3}(\lambda)=\Bigl{(}\frac{1}{12}+\frac{\lambda}{2},\frac{-2\lambda+1}{4\sqrt{3}}\Bigr{)}.$

with $\lambda\in\mathds{R}$. In this parametrization, the ray
$\mathcal{R}_{o}(\gamma_{i})$ starts from $\lambda=-\infty$ and intersects
$\mathcal{R}_{o}(\gamma_{i-1})$ and $\mathcal{R}_{o}(\gamma_{i+1})$ at
$p_{i}(-1/2)$ and $p_{i}(1/2)$, respectively, reproducing the intersection
points in (5.14).

In contrast to the rays $\mathcal{R}_{o}(\gamma_{i})$ in the orbifold
scattering diagram, the rays $\mathcal{R}_{\psi}(\gamma_{i})$ in the slice of
$\Pi$-stability conditions start from the images of the conifold points at
$\tau=0,-\frac{1}{2},-1$ in the $(u,v)$-plane, given by the points
$p_{i}(\mathcal{V}_{\psi})$ with $\lambda=\mathcal{V}_{\psi}$. For
$\psi=-\frac{\pi}{2}+\epsilon$ with $\epsilon\to 0^{+}$, these initial points
recess to infinity, and one recovers the infinite two-dimensional scattering
diagram constructed above. Similarly, for $\psi=\frac{\pi}{2}-\epsilon$ with
$\epsilon\to 0^{+}$, the initial points of the rays
$\mathcal{R}_{\psi}(-\gamma_{i})$ associated to the exceptional collection
$E_{i}[1]$ recess to infinity. On the contrary, for $\psi$ small enough such
that $|\mathcal{V}_{\psi}|<\frac{1}{2}$, the initial points are such that the
initial rays $\mathcal{R}_{\psi}(\gamma_{i})$ can no longer interact among
themselves (see Figure 28).

In between these two regimes, there are critical phases $\psi$ such that some
ray in the scattering diagram $\mathcal{D}_{o}$ passes through one of the
initial points $p_{i}(\mathcal{V}_{\psi})$ (see Figure 24). The first of these
critical values arise from the discrete series of rays
$\mathcal{R}_{\psi}(F_{2k}\gamma_{i}+F_{2k+2}\gamma_{i+1})$ emitted in the
scattering of $\gamma_{i}$ and $\gamma_{i+1}$, where $F_{k}$ are the Fibonacci
numbers. As mentioned above (1.13) in the introduction, this leads to jumps in
the topology of the scattering sequence contributing to the index
$\Omega_{\infty}(\gamma)$ along the ray $\mathcal{R}_{\psi}(\gamma)$ as a
function of the phase $\psi$.

Figure 28. Initial rays around the orbifold point in $(u,v)$ plane for
$\psi=0$ (left), $\psi=-\psi^{\rm cr}_{1/2}$ (middle) and $\psi<-\psi^{\rm
cr}_{1/2}$ (right). The dashed lines represent the $u=0$ and $v=0$ axis. The
green curves correspond to the boundary of the region $\mathds{H}^{o}$ defined
around (6.25). As $\psi\to-\frac{\pi}{2}$, the initial points recess to
infinity, and the scattering diagram reduces to Figure 5.

###  6.4. Exact scattering diagram

Having identified the image of the large volume and orbifold region in affine
coordinates, we are ready to describe the exact scattering diagram
$\mathcal{D}_{\psi}^{\Pi}$ in the same coordinates. Statements in this section
are proven in our upcoming work [41]. We outline the proof of the Split
Attractor Flow Conjecture on the slice of $\Pi$-stability conditions, and give
an algorithm to list the finite set of trees that contribute to a given
(rational) DT invariant.242424To be precise, the algorithm presented here
gives the list of active attractor flow trees, which is sufficient for the
weak form of SAFC. In [41] we prove that the strong form of SAFC also holds.
Depending on the phase $\psi$, we find that $\mathcal{D}_{\psi}^{\Pi}$ has the
following structure, modulo the action of $\Gamma_{1}(3)$ :

  * •

For small phases $|\psi|<\psi^{\rm cr}_{1/2}$ (where $\psi^{\rm cr}_{\alpha}$
was defined in (1.13)), $\mathcal{D}_{\psi}^{\Pi}$ coincides with the large
volume diagram $\mathcal{D}^{\rm LV}_{0}$ in the $(x,y)$ plane, except for a
shift in the origin of the initial rays.

  * •

For large phases $\psi^{\rm cr}_{1/2}<|\psi|<\pi/2$ that are not critical (as
defined at the end of §6.3), $\mathcal{D}^{\rm\Pi}_{\psi}$ includes a
triangular portion of the orbifold diagram, whose outbound rays seed
scattering diagrams in the large volume region.

  * •

For $\psi=\pi/2$, $\mathcal{D}^{\rm\Pi}_{\psi}$ coincides with the
$\theta_{1}+\theta_{2}+\theta_{3}=0$ slice of the quiver scattering diagram
$\mathcal{D}_{Q}$ concentrated at the orbifold point $\tau_{o}$ in the
$\tau$-plane, supplemented with a vertical ray $\mathcal{R}(\mathcal{O}[1])$
(and $\Gamma_{1}(3)$ images thereof)

We refrain from discussing critical phases, since additional rays emanating
from the conifold points complicate the analysis.

####  6.4.1. Initial rays and a tessellation

A basic constraint on the scattering diagram $\mathcal{D}_{\psi}^{\Pi}$ is
that it is invariant under the group $\Gamma_{1}(3)$ of auto-equivalences of
$D^{b}\operatorname{Coh}(K_{\mathds{P}^{2}})$, generated by tensor product by
$\mathcal{O}_{X}(1)$ (acting by $T\colon\tau\mapsto\tau+1$) and by the
spherical twist $ST_{\mathcal{O}}$ (acting by
$V\colon\tau\mapsto\frac{\tau}{1-3\tau}$). The quotient $\Pi/\Gamma_{1}(3)$
has two boundary points from which initial rays could emanate: the large
volume point, which is ruled out because central charges grow without bound in
this limit, and the conifold point. Without loss of generality, we can thus
restrict to initial rays emanating from the point $\tau=0$ in $\Pi$. The
analysis in (6.4) (with $\tau=p/q=0/1$ hence $m=m_{D}=0$) shows a ray with
charge $\gamma=[r,d,\textnormal{ch}_{2}]$ can only emanate from $\tau=0$ if
$d\mathcal{V}_{\psi}=\textnormal{ch}_{2}$. If $\mathcal{V}_{\psi}$ is
irrational, this is only possible when $d=0=\textnormal{ch}_{2}$. In order to
determine the associated attractor invariant, we use the fact that the
orbifold and large volume descriptions valid in regions $\mathds{H}^{o}$ and
$\mathds{H}^{\rm LV}$ that cover a neighborhood of $\tau=0$. The outcome of
this analysis is that the initial rays of the exact scattering diagram
are252525To lighten the notation, we omit the $\psi$-dependence of the rays.

(6.29) $\displaystyle\mathcal{R}^{\rm eff}(\mathcal{O})\colon$
$\displaystyle\Omega([k,0,0])$ $\displaystyle=\delta_{k=1},\quad$
$\displaystyle\text{and its }\Gamma_{1}(3)\text{ images,}$
$\displaystyle\mathcal{R}^{\rm eff}(\mathcal{O}[1])\colon$
$\displaystyle\Omega([-k,0,0])$ $\displaystyle=\delta_{k=1},\quad$
$\displaystyle\text{and its }\Gamma_{1}(3)\text{ images,}$

provided $\psi$ is _not critical_ in a sense we have defined at the end of
§6.3. In particular this holds for all small phases $|\psi|<\psi^{\rm
cr}_{1/2}$. Notably, since $V\in\Gamma_{1}(3)$ maps $\mathcal{O}(0)[n]$ to
$\mathcal{O}(0)[n+2]$, the initial rays in $\mathcal{D}^{\Pi}_{\psi}$ include
infinitely many effective rays starting from $\tau=0$, which lie in images of
$\mathcal{F}_{C}$ that border this point. By a mild abuse of notation we will
denote by $\mathcal{R}^{\rm eff}(\mathcal{O}(0)[n])$ these distinct rays, even
though their charge vector only depends on the parity of $n$. More generally,
there are an infinite set of rays emanating from every rational number
$\tau=\frac{p}{q}$ with $q\neq 0\mod 3$, corresponding to the images of
$\mathcal{O}(0)[n]$ under $\Gamma_{1}(3)$. We denote by $\mathcal{R}^{\rm
eff}(\mathcal{O}(p/q)[n])$ the corresponding effective ray. For non-critical
$\psi$, these images exhaust all initial rays. In particular, there are no
rays emanating from $\tau=\frac{p}{q}$ with $q=0\mod 3$, nor from points with
$\tau\notin\mathds{Q}$.

Figure 29. The upper half plane $\mathds{H}$ is partitioned into regions
$g\cdot\diamondsuit_{\psi}$ around images of the large volume point
$\mathrm{i}\infty$ and $g\cdot\Delta_{\psi}$ around images of the orbifold
point $\tau_{o}$. The region $\Delta_{\psi}$ and its translates
$\Delta_{\psi}(k)$ map to triangles in the $(x,y)$ plane, while the region
$\diamondsuit_{\psi}$ above all blue curves maps to the region above the
“jagged parabola” $\mathfrak{p}$ defined in (6.30). The figures are drawn for
$\psi=-0.2$. While the figure in the $\tau$ plane is rather insensitive to the
value of $\psi\in[0,2\pi]$, the images of these regions in the $(x,y)$ plane
depend significantly on it: for instance the images of $\diamondsuit_{\psi}$
and $\Delta_{\psi}$ overlap for $\psi<-\psi^{\rm cr}_{1/2}$.

For any $\psi\in(-\pi/2,\pi/2)$ we introduce a $\Gamma_{1}(3)$-invariant
tessellation of $\Pi$. Edges of the tiles consist of all $\Gamma_{1}(3)$
images of the straight line joining the conifold points
$(x_{\mathcal{O}(-1)},y_{\mathcal{O}(-1)})$ to
$(x_{\mathcal{O}(0)},y_{\mathcal{O}(0)})$ in the $(x,y)$ plane. There are two
types of tiles: an orbifold region around each $\Gamma_{1}(3)$ image of
$\tau_{o}$ and a large volume region around each image of
$\tau=\mathrm{i}\infty$. The orbifold region around $\tau_{o}$ is the
aforementioned triangle $\Delta_{\psi}$, whose corners are images of
$\tau=0,-1/2,-1$. The large volume region $\diamondsuit_{\psi}$ around
$\tau=\mathrm{i}\infty$ maps in $(x,y)$ coordinates to the region above the
piecewise linear function (or ‘jagged parabola’, depicted in Figure 29)

(6.30)
$y\geq\mathsf{p}(x)\coloneqq\Bigl{(}m+\mathcal{V}_{\psi}+\frac{1}{2}\Bigr{)}(\mathcal{V}_{\psi}-x)+\frac{m(m+1)}{2}\quad\text{with
}m=\lfloor x-\mathcal{V}_{\psi}\rfloor$

zigzaging between262626To see this, rewrite
$\mathsf{p}(x)=\frac{(\mathcal{V}_{\psi})^{2}-x^{2}}{2}+\frac{(2\\{x-\mathcal{V}_{\psi}\\}-1)^{2}-1}{8}$
where $\\{x\\}=x-\lfloor x\rfloor$ denotes the fractional part. the parabolas
$y+\frac{1}{2}x^{2}-\frac{1}{2}(\mathcal{V}_{\psi})^{2}\in\\{0,\frac{1}{8}\\}$.
The boundary of $\diamondsuit_{\psi}$ joins consecutive conifold points
$(x_{\mathcal{O}(m)},y_{\mathcal{O}(m)})$, $m\in\mathds{Z}$, so that
$\diamondsuit_{\psi}$ borders all translates of $\Delta_{\psi}$, while
$\Delta_{\psi}$ borders three large volume regions. The initial rays (6.29)
emanate from vertices of the tessellation by construction, and they enter
different regions depending on $\psi$.

####  6.4.2. Small phases $|\psi|<\psi^{\rm cr}_{1/2}$

Within $\Delta_{\psi}$, the exact diagram $\mathcal{D}_{\psi}^{\Pi}$ coincides
with a triangular region of the orbifold diagram $\mathcal{D}_{o}$. For our
first case of interest, $|\mathcal{V}_{\psi}|<1/2$, this region lies inside
the empty central triangle of $\mathcal{D}_{o}$, thus $\Delta_{\psi}$ does not
contain any ray, as depicted in Figure 28. The large volume regions are thus
disconnected from each other in $\mathcal{D}^{\Pi}$, so that we can focus on
$\diamondsuit_{\psi}$.

The initial rays in that region are $\mathcal{R}^{\rm eff}(\mathcal{O}(m))$
and $\mathcal{R}^{\rm eff}(\mathcal{O}(m)[1])$, emanating from integer
conifold points $\tau=m\in\mathds{Z}$. In $(x,y)$ coordinates, the geometric
ray $\mathcal{R}^{\rm geo}(\pm[1,m,m^{2}/2])$ is contained in the line
$y+mx=m^{2}/2$ tangent to the parabola $y+x^{2}/2=0$ at $x=m$. The active rays
$\mathcal{R}^{\rm eff}(\mathcal{O}(m))$ and $\mathcal{R}^{\rm
eff}(\mathcal{O}(m)[1])$ emanate leftwards/rightwards from the point
$(x_{\mathcal{O}(m)},y_{\mathcal{O}(m)})=(m+\mathcal{V}_{\psi},-m^{2}/2-m\mathcal{V}_{\psi})$
on this line: for $\psi=0$ this coincides with the point of tangency with the
parabola, while for $\psi\lessgtr 0$ it is moved left/right along the tangent
line.

For $\psi=0$, the initial rays $\mathcal{R}^{\rm eff}(\mathcal{O}(m))$ and
$\mathcal{R}^{\rm eff}(\mathcal{O}(m)[1])$ for all $m\in\mathds{Z}$ coincide
with the initial rays for $\mathcal{D}^{\rm LV}_{0}$. Under conditions that
are fulfilled here, consistent scattering diagrams are uniquely determined by
the initial rays, so that $\mathcal{D}^{\Pi}_{0}\cap\diamondsuit_{\psi}$
coincides with $\mathcal{D}^{\rm LV}_{0}$ up to the map $\tau\mapsto(x,y)$.
The rest of the exact diagram is completed by $\Gamma_{1}(3)$ invariance,
namely $\mathcal{D}^{\Pi}_{\psi=0}$ consists of all images of
$\mathcal{D}^{\rm LV}$ under $\Gamma_{1}(3)$, in which translations simply map
$\mathcal{D}^{\rm LV}$ to itself.

In the large volume diagram $\mathcal{D}^{\rm LV}$, the first intersection
along each initial ray is the intersection with the neighboring initial ray,
with intersection points

(6.31) $\mathcal{R}^{\rm geo}(\mathcal{O}(m))\cap\mathcal{R}^{\rm
geo}(\mathcal{O}(m-1))=\Bigl{\\{}(x,y)=\Bigl{(}m-\tfrac{1}{2},-\tfrac{1}{2}m(m-1)\Bigr{)}\Bigr{\\}}.$

In other words, along each geometric ray there is a neighborhood of
$(x,y)=(m,-m^{2}/2)$ in which there is no intersection. When $\psi$ is varied
away from zero, the initial point
$(m+\mathcal{V}_{\psi},-m^{2}/2-m\mathcal{V}_{\psi})$ remains in this
intersection-free portion as long as $|\mathcal{V}_{\psi}|<1/2$. We conclude
that for small phases $|\psi|<\psi^{\rm cr}_{1/2}$, the exact diagram
$\mathcal{D}^{\Pi}_{\psi}$ consists of disjoint copies (images under
$\Gamma_{1}(3)$) of $\mathcal{D}^{\rm LV}$ with initial points shifted
horizontally by $\mathcal{V}_{\psi}$ along the geometric rays.

The SAFC for $\mathcal{D}^{\Pi}_{\psi}$ with $|\psi|<\psi^{\rm cr}_{1/2}$ is
then a consequence of the SAFC for $\mathcal{D}^{\rm LV}$ since trees
contributing to a given $\bar{\Omega}_{\tau}(\gamma)$ are simply trees in one
of the copies of $\mathcal{D}^{\rm LV}$. Assuming without loss of generality
(up to $\Gamma_{1}(3)$ transformation of $\tau$ and $\gamma$) that
$\tau\in\diamondsuit_{\psi}$, the electric potential
$\varphi_{\tau}(\gamma)=2(d-rx)$ bounds the number of leaves of the tree as in
§4.2, while a causality argument (4.14) bounds the slope $m\in\mathds{Z}$ of
the initial rays which which can contribute.

####  6.4.3. Large phases $\psi^{\rm cr}_{1/2}<|\psi|<\pi/2$

More generally, the exact diagram combines aspects of the orbifold and large
volume diagrams. For definiteness, we consider $-\pi/2<\psi<-\psi^{\rm
cr}_{1/2}$, namely $\mathcal{V}_{\psi}<-1/2$. Then, the $\Gamma_{1}(3)$ images
of $\mathcal{R}^{\rm eff}(\mathcal{O})$ enter large volume regions while those
of $\mathcal{R}^{\rm eff}(\mathcal{O}[1])$ enter orbifold regions.
Specifically, $\mathcal{R}^{\rm eff}(\mathcal{O})$ enters
$\diamondsuit_{\psi}$ while its homological shift $\mathcal{R}^{\rm
eff}(\mathcal{O}[-1])$ enters $\Delta_{\psi}$.272727For
$\mathcal{V}_{\psi}>1/2$, the situation is analogous: $\mathcal{R}^{\rm
eff}(\mathcal{O}[1])$ enters $\diamondsuit_{\psi}$ while $\mathcal{R}^{\rm
eff}(\mathcal{O})$ enters $\Delta_{\psi}$. In the region $\Delta_{\psi}$ the
exact scattering diagram coincides with the corresponding portion of
$\mathcal{D}_{o}$, as argued in §6.3 on general grounds.282828An alternative
proof of the matching is that the three $\mathds{Z}_{3}$ images of
$\mathcal{R}^{\rm eff}(\mathcal{O}[-1])$ carry the same initial data as in the
orbifold diagram. The initial ray $\mathcal{R}^{\rm eff}(\mathcal{O})$ and
rays exiting from $\Delta_{\psi}$ to $\diamondsuit_{\psi}$, and all of their
translates, then serve as incoming rays for the scattering diagram in the
large volume region $\diamondsuit_{\psi}$.

In $(x,y)$ coordinates, active rays in $\diamondsuit_{\psi}$ are “outbound”,
namely straight line segments pointing away from the boundary
$\mathfrak{p}=\partial\diamondsuit_{\psi}$: this is manifestly true of the
rays that enter $\diamondsuit_{\psi}$, and the scattering of multiple outbound
rays produces only rays within a certain angular sector that ensures that they
are still outbound. In particular, none of the active rays in
$\diamondsuit_{\psi}$ can ever “go back” and enter $\Delta_{\psi}$. The
resulting picture for $\mathcal{D}^{\Pi}$ is that three initial rays scatter
within $\Delta_{\psi}$, producing some rays that leave $\Delta_{\psi}$ towards
$\diamondsuit_{\psi}$; these rays scatter further with $\mathcal{R}^{\rm
eff}(\mathcal{O})$ and translates of all of these rays so as to produce an
ever denser set of outbound rays, isimilarly to the $|\psi|<\psi^{\rm
cr}_{1/2}$ case but with additional incoming rays from the orbifold regions.
Throughout this process, only a limited set of initial rays participate in the
construction of the scattering diagram in $\diamondsuit_{\psi}$:

(6.32) $\mathcal{R}^{\rm eff}(\mathcal{O}(m)[-1]),\ \mathcal{R}^{\rm
eff}(\Omega(m+1)),\ \mathcal{R}^{\rm eff}(\mathcal{O}(m-1)[1])\ \ \text{and}\
\ \mathcal{R}^{\rm eff}(\mathcal{O}(m))$

for each $m\in\mathds{Z}$, which amount to one initial ray from each conifold
point $\tau=m-1/2$ and three from each integer conifold point.

We now outline the proof of the Split Attractor Flow Conjecture, by providing
a finite algorithm to list all trees contributing to a given
$\bar{\Omega}_{\tau}(\gamma)$ and made of active rays (henceforth “active flow
trees”). For $\tau\in\Delta_{\psi}$ the problem reduces to the orbifold
diagram, where the dimension vector $\gamma=(n_{1},n_{2},n_{3})$ has a finite
number of decompositions into elementary charges of the form $k\gamma_{i}$,
and each decomposition leads to a finite number of trees with those leaves.

For $\tau\in\diamondsuit_{\psi}$, infinitely many leaf rays (6.32) could in
principle contribute. The informal discussion above (6.32) implies that active
flow trees can be sawed into a trunk within $\diamondsuit_{\psi}$ rooted at
$\tau$, with leaves that are initial rays $\mathcal{R}(\mathcal{O}(m))$, and
various shrubs in orbifold regions $\Delta_{\psi}(m)$. This is exemplified in
Figure 9. The trunk must lie in $H\cap\diamondsuit_{\psi}$, where $H$ is the
convex hull of $\mathfrak{p}=\partial\diamondsuit_{\psi}$ and of $\tau$: this
is easily seen by induction starting from the root and noting that the ray
reaching it is active hence outbound. Since $H$ has a finite extent in $x$, we
learn that the set of orbifold regions that can be reached by the attractor
flow starting from $\tau$ is finite, therefore active flow trees have finitely
many possible constituents.

This does not suffice for establishing finiteness however: as in the large
volume diagram, some charges of possible constituents are opposite to each
other, hence there are still infinitely many decompositions of $\gamma$. The
proof relies on essentially the same electric potential

(6.33) $\varphi_{x}(\gamma)=2(d-rx)$

as in the large volume diagram, but only using it in $\diamondsuit_{\psi}$. It
is manifestly additive in the charge $\gamma$, and we saw in (6.16) that it is
monotonically decreasing along the attractor flow. We now explain how it
measures the complexity of all rays entering $\diamondsuit_{\psi}$.

Since $\varphi_{x}(\gamma)$ is invariant under translations
$T\in\Gamma_{1}(3)$, we can focus on the effective ray $\mathcal{R}^{\rm
eff}(\mathcal{O})$ and rays crossing from $\Delta_{\psi}$ to
$\diamondsuit_{\psi}$. At the starting point $(x,y)=(\mathcal{V}_{\psi},0)$ of
$\mathcal{R}^{\rm eff}(\mathcal{O})$, its active rays $\mathcal{R}([k,0,0])$
for $k\geq 1$ obey

(6.34) $\varphi_{x}([k,0,0])=-2k\mathcal{V}_{\psi}\geq 2|\mathcal{V}_{\psi}|$

thus $\varphi_{x}$ simply measures (up to a scaling) the multiplicity $k$ of
the primitive charge vector $[1,0,0]$. Consider next an active ray
$\mathcal{R}(\gamma)$ crossing from $\Delta_{\psi}$ to the large volume region
$\diamondsuit_{\psi}$ at some point $(x,y)$ of their boundary. An analysis of
active rays of $\mathcal{D}_{o}$ in [41] proves that the charge vector
$\gamma=[r,d,\textnormal{ch}_{2}]=(n_{1},n_{2},n_{3})$ obeys

(6.35) $\varphi_{x}(\gamma)=2(d-rx)\geq c_{\psi}(n_{1}+n_{2}+\sqrt{n_{3}})\geq
c_{\psi}>0$

where $(x,y)$ is the point where the ray enters $\diamondsuit_{\psi}$ and
$c_{\psi}=(1+2|\mathcal{V}_{\psi}|)^{-2}\in(0,1/4)$ is a constant. Thus, the
electric potential controls the total dimension $n_{1}+n_{2}+n_{3}$.

The first consequence is that all rays entering $\diamondsuit_{\psi}$ obey

(6.36)
$\varphi_{x}(\gamma)\geq\varphi_{\min}=\min(2|\mathcal{V}_{\psi}|,c_{\psi}).$

Since $\varphi_{x}(\gamma)$ is additive and positive, its value at the root
bounds the sum of contributions of all rays entering $\diamondsuit_{\psi}$,
therefore the total multiplicity $\sum k_{i}$ for all leaves $k_{i}\gamma_{i}$
is bounded above by $(\varphi_{x}(\gamma)/\varphi_{\min})^{2}$, where the
square accounts for the square root in (6.35).

The SAFC thus follows: the trees contributing to $\bar{\Omega}_{\tau}(\gamma)$
are made of a bounded set of possible constituents with a bounded number of
each species, hence the number of attractor flow trees is finite.

####  6.4.4. Behavior as $\psi\to\pm\pi/2$

In the limit $\psi\to\pi/2$, the scattering diagram $\mathcal{D}^{\Pi}_{\psi}$
drastically simplifies. In that case, geometric rays $\mathcal{R}^{\rm
geo}(\gamma)_{\psi}$ satisfy

(6.37) $\Im Z_{\tau}(\gamma)=-r\Im T_{D}+d\Im T,$

thus are included in contours of constant slope $s=\frac{\Im T_{D}}{\Im
T}=\frac{d}{r}$, independent of the value of $\textnormal{ch}_{2}$ (see Figure
10). The contours only intersect at points where $s$ is ill-defined, i.e. when
$\Im T=\Im T_{D}=0$. These two curves intersect at the orbifold point
$\tau_{o}$ and $\Gamma_{1}(3)$ images thereof. At $\tau_{o}$, there are three
incoming rays

(6.38) $\mathcal{R}^{\rm eff}(\mathcal{O}),\ \mathcal{R}^{\rm
eff}(\Omega(1)[1]),\ \mathcal{R}^{\rm eff}(\mathcal{O}(-1)[2])$

The first and last emanate from $\tau=0$ and $\tau=-1$ and lie along the
boundary of the fundamental domain $\mathcal{F}_{o}$, where $s=0$ and $s=-1$,
respectively while $\mathcal{R}^{\rm eff}(\Omega(1)[1])$ is a vertical line
emanating from $\tau=-1/2$, with slope $s=-1/2$. In addition, there are
vertical rays $\mathcal{R}^{\rm eff}(\mathcal{O}(m)[1])$ for any
$m\in\mathds{Z}$, which do not interact with the rays (6.39). Consistency at
$\tau_{o}$ fixes uniquely the outgoing rays, which then continue on towards a
large volume limit, either at $\tau\to-\frac{1}{2}+\mathrm{i}\infty$,
$\tau\to-\frac{2}{3}+\mathrm{i}0$ or $\tau\to-\frac{1}{3}+\mathrm{i}0$. The
rays (6.38) are associated to the homological shifts $E_{i}[1]$ of the objects
$E_{i}$ in the exceptional collection (1.8). Their scattering diagram around
$\tau_{o}$, which we denote by $\mathcal{D}^{\vee}_{o}$, is simply obtained
from $\mathcal{D}_{o}$ by sending $(u,v)\mapsto(-u,-v)$ and reversing the
direction of the arrows. Equivalently, we can send $(u,v)\mapsto(-u,v)$ and
exchange $\gamma_{1}$ and $\gamma_{3}$, which amounts to applying the derived
duality (2.32) and a suitable translation. Thus, the scattering diagram
$\mathcal{D}^{\Pi}_{\pi/2}$ consists of the scattering diagram
$\mathcal{D}^{\vee}_{o}$ concentrated at the point $\tau_{o}$, the vertical
ray $\mathcal{R}^{\rm eff}(\mathcal{O}[1])$ and all $\Gamma_{1}(3)$ images
thereof.

Similarly, as $\psi\to-\pi/2+\epsilon$ with $\epsilon\to 0^{+}$, the rays
associated to the objects $E_{i}$ in the exceptional collection (1.8).

(6.39) $\mathcal{R}^{\rm eff}(\mathcal{O}[-1]),\ \mathcal{R}^{\rm
eff}(\Omega(1)),\ \mathcal{R}^{\rm eff}(\mathcal{O}(-1)[1])$

intersect in a region of size $\epsilon$ around $\tau_{o}$. As $\epsilon\to
0$, the scattering diagram $\mathcal{D}^{\Pi}_{-\pi/2+\epsilon}$ reduces to
the orbifold scattering diagram ${\mathcal{D}}_{o}$ concentrated in a region
of size $\epsilon$ at the point $\tau_{o}$, the vertical ray $\mathcal{R}^{\rm
eff}(\mathcal{O})$ and all their $\Gamma_{1}(3)$ images.

In either case, the SAFC on $\mathcal{D}^{\Pi}_{\pi/2}$ reduces to the flow
tree formula for quivers, which is proven in [22]. Moreover, the fact that
rays do not intersect away from orbifold points makes it clear that the index
$\Omega_{c}(\gamma)$ computed using the quiver associated to the exceptional
collection $E_{i}(k)$ will coincide with the Gieseker index
$\Omega_{\infty}(\gamma)$ along the ray $\mathcal{R}_{\pm\pi/2}(\gamma)$
provided $k-1\leq\frac{d}{r}\leq k$ for $r\neq 0$, as observed in [40, 23].

###  6.5. Case studies

In this final section, we study the scattering sequences which contribute to
the Gieseker index $\Omega_{\infty}(\gamma)$ for two simple choices of
charges, as function of the phase $\psi$. The initial rays in these sequences
turn out to jump at a subset of the critical values (1.14), while the index
remains unaffected. This is reminiscent of the ‘fake walls’ encountered in
earlier studies [65, 66, 14], where the structure of the bound state changes
while the index remains unchanged.

We first consider $\gamma=[0,1,1)$, corresponding to the Chern vector of the
structure sheaf $\mathcal{O}_{C}$ of a rational curve in the hyperplane class.
As noted below (2.14), the moduli space $\mathcal{M}_{\infty}(\gamma)$ is
$\mathds{P}^{2}$ itself, so $\Omega_{\tau}(\gamma)$ should equal
$y^{2}+1+1/y^{2}$ for $\tau_{2}\gg 1$ and arbitrary values of $\tau_{1}$.
Thus, for any $\psi\in(-\pi/2,\pi/2)$ the scattering diagram should contain a
ray $\mathcal{R}^{\rm eff}_{\psi}(\gamma)$ reaching
$\tau_{2}=\mathrm{i}\infty$ with the above index. The scattering sequences
which contribute to this index however depend sensitively on the value of
$\psi$ (which we can assume to be negative, using the reflection symmetry of
the scattering diagram):

  * •

$-\psi^{\rm cr}_{\frac{1}{2}}<\psi<\psi^{\rm cr}_{\frac{1}{2}}\simeq 0.82406$:
$T_{0}=\\{\mathcal{O}(-1)[1],\mathcal{O}(0)\\}$ contributing $K_{3}(1,1)$;

  * •

$-\psi^{\rm cr}_{\frac{3}{2}}\simeq-1.27155<\psi<-\psi^{\rm
cr}_{\frac{1}{2}}$: $T_{1}=\\{2\mathcal{O}(0)[1],\Omega(2)\\}$ contributing
$K_{3}(1,2)$;

  * •

$-\psi^{\rm cr}_{\frac{5}{2}}\simeq-1.38766<\psi<-\psi^{\rm
cr}_{\frac{3}{2}}$:
$T_{2}=\\{3\mathcal{O}(1)[1],\\{2\Omega(3),\mathcal{O}(2)[-1]\\}\\}$
contributing $K_{3}(1,2)K_{3}(1,3)$;

  * •

$-\psi^{\rm cr}_{\frac{7}{2}}\simeq-1.43934<\psi<-\psi^{\rm
cr}_{\frac{5}{2}}$:
$T_{3}=\\{3\mathcal{O}(2)[1],\\{3\Omega(4),\\{2\mathcal{O}(3)[-1],\mathcal{O}(2)[1]\\}\\}\\}$
contributing $K_{3}(1,2)K_{3}(1,3)$;

In all cases, the tree produces the desired index $y^{2}+1+1/y^{2}$. More
generally, when

(6.40) $-\psi^{\rm cr}_{n+1/2}<\psi<-\psi^{\rm cr}_{n-1/2}\ ,$

we find a single tree $T_{n}$ with constituents in the positive cone spanned
by the exceptional collection
$\langle\mathcal{O}(n)[-1],\Omega(n+1),\mathcal{O}(n-1)[1]\rangle$ with Chern
vectors $\gamma_{1}(n),\gamma_{2}(n),\gamma_{3}(n)$, obtained from (1.8) by
$n$ units of spectral flow (and homology shift $[-1]$). The tree $T_{n}$ has
$n+1$ constituents and is of the iterated form

(6.41)
$T_{n}=\\{k_{1}\gamma_{3}(n),\\{k_{2}\gamma_{2}(n),\\{k_{3}\gamma_{1}(n),\\{k_{4}\gamma_{3}(n),\\{k_{5}\gamma_{2}(n),\dots\\}\\}\\}\\},$

where $k_{i}=3$ for $i=1,\dots n-1$ and $k_{n}=2,k_{n+1}=1$. Equivalently,
$T_{n}$ is defined inductively by

(6.42) $T_{n}=\\{3\gamma_{3}(n),\sigma_{n}^{-1}\cdot T_{n-1}(1)\\}\ ,\quad
T_{1}=\\{2\gamma_{3}(1),\gamma_{2}(1)\\}$

where $T(1)$ is the tree $T$ shifted by one unit of spectral flow, and
$\sigma_{n}$ acts by cyclic permutation

(6.43) $\sigma_{n}\colon\gamma_{1}(n)\mapsto\gamma_{2}(n)\
,\quad\gamma_{2}(n)\mapsto\gamma_{3}(n)\
,\quad\gamma_{3}(n)\mapsto\gamma_{1}(n)$

The charges indeed add up to
$(n-1)\gamma_{1}(n)+n\gamma_{2}(n)+(n+1)\gamma_{3}(n)=[0,1,1)$, and the index
evaluates to $K_{3}(1,2)K_{3}(1,3)^{n-1}=y^{2}+1+1/y^{2}$. In Figure 30 we
plot the scattering sequences for the first few values of $n$.

Figure 30. Attractor flow trees contributing to
$\gamma=[0,1,-\frac{1}{2}]=[0,1,1)$ for $\psi$ varying from $-\frac{\pi}{2}$
(right) to $\frac{\pi}{2}$ (left). The dashed lines corresponding to the
incoming rays for $\psi=\psi^{\rm cr}_{\alpha}$ with
$\alpha\in\\{\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{5}{2},\dots\\}$.

We now consider $\gamma=[1,0,1)$, the Chern vector of the structure sheaf
$\mathcal{O}$. This object is spherical and stable throughout the large volume
region. In particular, for large $\tau_{2}$ and any $\tau_{1}<0$, the index
$\Omega_{\tau}(\gamma)$ should equal $1$.

  * •

In the range $-\frac{\pi}{2}<\psi<\psi^{\rm cr}_{1/2}$, there is a single ray
originating from the conifold point $\mathcal{O}(0)=\gamma_{1}(0)[1]$;

  * •

In the range $\psi^{\rm cr}_{\frac{1}{2}}<\psi<\psi^{\rm cr}_{1}$, we find a
single sequence $\\{3\mathcal{O}(-1),\Omega(0)[1]\\}$ contributing
$K_{3}(1,3)=1$;

  * •

In the range $\psi^{\rm cr}_{1}<\psi<\psi^{\rm cr}_{\frac{3}{2}}$, we find a
single sequence
$\\{6\mathcal{O}(-2),\\{3\Omega(-1)[1],\mathcal{O}(-3)[2]\\}\\}$ contributing
$K_{3}(1,3)K_{6}(1,6)=1$;

More generally, for $\psi^{\rm cr}_{\frac{n}{2}}<\psi<\psi^{\rm
cr}_{\frac{n+1}{2}}$ there is a sequence $T_{n}$ obtained inductively through

(6.44) $T_{n}=\\{(3n)\gamma_{1}(-n)[1],\sigma_{-n}\cdot T_{n-1}(-1)\\}\ ,\quad
T_{0}=\\{\gamma_{1}(0)[1]\\}$

with total charge $[1,0,1)$ contributing 1 to the index. In Figure 31 we plot
the scattering sequences for the first few values of $n$.

Figure 31. Attractor flow trees contributing to $\gamma=[1,0,0]=[1,0,1)$ for
$\psi$ varying from $-\frac{\pi}{2}$ (right) to $\frac{\pi}{2}$ (left). The
dashed lines corresponding to the incoming rays for $\psi=\psi^{\rm
cr}_{\alpha}$ with $\alpha\in\\{\frac{1}{2},1,\frac{3}{2},\dots\\}$.

##  Appendix A Periods as Eichler integrals

In this section, we review the modular description of the Kähler moduli space
of $K_{\mathds{P}^{2}}$, derive the Eichler integral representation (1.5) of
the periods $(T,T_{D})$, and use it to obtain asymptotic expansions around the
large volume, conifold and orbifold points and the behavior under monodromies.
We refer to [67, 68, 48, 69, 70, 31, 71] for earlier studies in the
literature.292929We are grateful to Thorsten Schimannek for his help about the
material in this section.

###  A.1. Kähler moduli space as the modular curve $X_{1}(3)$

Recall that the mirror of $K_{\mathds{P}^{2}}$ is a family of genus-one curves
$\Sigma(z)=x_{1}^{3}+x_{2}^{3}+x_{3}^{3}-z^{-1/3}x_{1}x_{2}x_{3}=0$ in
$\mathds{P}^{2}$ parametrized by the complex structure modulus
$z\in\mathcal{M}_{K}=\mathds{C}\backslash\\{0,-\frac{1}{27}\\}$, which is
identified as the complexified Kähler moduli space of $K_{\mathds{P}^{2}}$.
The points $z=0,-\frac{1}{27},\infty$ then correspond to the large radius,
conifold and orbifold points, respectively. Alternatively, we can consider the
family of genus-one curves $\Sigma^{\prime}(z^{\prime})\colon
x+y+1-z^{\prime}x^{3}/y=0$ in
$\mathds{C}^{\times}_{x}\times\mathds{C}^{\times}_{y}$, related to $\Sigma(z)$
with $z^{\prime}=-z-\frac{1}{27}$ by a 3-isogeny. The points
$z^{\prime}=0,-\frac{1}{27},\infty$ then correspond to the conifold, large
radius and orbifold points, respectively.

Figure 32. Fundamental domain $\mathcal{F}_{o^{\prime}}$ centered around the
orbifold point $\tau_{o^{\prime}}=e^{\mathrm{i}\pi/6}/\sqrt{3}$, and some of
its images. Here $g=ST^{-2}STS\colon\tau\mapsto\frac{\tau-1}{3\tau-2}$ and
$g^{-1}=TST^{3}S\colon\tau\mapsto\frac{2\tau-1}{3\tau-1}$. The fundamental
domain $\mathcal{F}_{o}$ is the same domain translated to the interval
$[-1,0]$.

The space $\mathcal{M}_{K}$ is isomorphic to the modular curve
$X_{1}(3)=\mathds{H}/\Gamma_{1}(3)$, where $\Gamma_{1}(3)$ is the group of
integer matrices $(\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix})$ such that
$a,d=1\mod 3$ and $c=0\mod 3$. This is an index 4 subgroup of
$PSL(2,\mathds{Z})$, with two cusps of width 1 and 3, corresponding to the
large volume and conifold points, respectively, and one elliptic point of
order 3 corresponding to the orbifold point. A convenient choice of
fundamental domain is the domain centered around the orbifold point at
$\tau_{o}=\frac{1}{\sqrt{3}}e^{5\pi\mathrm{i}/6}=(-\frac{1}{2},\frac{1}{2\sqrt{3}})$,
shifted horizontally by $-1/2$ compared to (A.2),

(A.1)
$\mathcal{F}_{o}\coloneqq\Bigl{\\{}\tau:\,-1\leq\tau_{1}<0,(\tau_{1}+\frac{2}{3})^{2}+\tau_{2}^{2}\geq\frac{1}{9},(\tau_{1}+\frac{1}{3})^{2}+\tau_{2}^{2}>\frac{1}{9}\Bigr{\\}}$

This domain (or rather its translate under $\tau\mapsto\tau+1$) is depicted in
Figure 32, along with several of its images under $\Gamma_{1}(3)$.
Alternatively, one may choose a fundamental domain centered around $\tau=0$,

(A.2)
$\mathcal{F}_{C}\coloneqq\Bigl{\\{}\tau:\,-\frac{1}{2}\leq\tau_{1}<\frac{1}{2},(\tau_{1}+\frac{1}{3})^{2}+\tau_{2}^{2}\geq\frac{1}{9},(\tau_{1}-\frac{1}{3})^{2}+\tau_{2}^{2}>\frac{1}{9}\Bigr{\\}}$

shown in Figure 6. It has the nice property of being invariant under the
Fricke involution $\tau\to-1/(3\tau)$.

The isomorphism $\mathcal{M}_{K}\simeq X_{1}(3)$ is given explicitly by

(A.3) $J_{3}(\tau)=-15-\frac{1}{z}=\frac{12-405z^{\prime}}{1+27z^{\prime}},$

where $J_{3}(\tau)$ is the normalized Hauptmodul of $\Gamma_{1}(3)$,

(A.4)
$J_{3}\coloneqq\left(\frac{\eta(\tau)}{\eta(3\tau)}\right)^{12}+12=\frac{1}{q}+54q-76q^{2}-243q^{3}+1188q^{4}-1384q^{5}-2916q^{6}+\dots$

where $q=e^{2\pi\mathrm{i}\tau}$. The latter maps the points
$\tau=\mathrm{i}\infty,0$ and $\tau_{o}$ to $J_{3}=\infty,12$ and $-15$,
corresponding to the large volume, conifold and orbifold points, respectively.
One easily checks that the Fricke involution maps
$J_{3}\mapsto\frac{729}{J_{3}-12}+12$ and $z\mapsto z^{\prime}$. Plugging the
$q$-expansion (A.4) into (A.3), we get

(A.5) $\displaystyle q$
$\displaystyle=-z+15z^{2}-279z^{3}+5729z^{4}-124554z^{5}+2810718z^{6}+\dots$
$\displaystyle z$
$\displaystyle=-q+15q^{2}-171q^{3}+1679q^{4}-15054q^{5}+126981q^{6}+\dots$

The Klein invariant is expressed in terms of $J_{3}$ via

(A.6)
$J\coloneqq\frac{E_{4}^{3}(\tau)}{\eta^{24}(\tau)}=J_{3}+744+\frac{196830}{J_{3}-12}+\frac{19131876}{(J_{3}-12)^{2}}+\frac{387420489}{(J_{3}-12)^{3}}$

leading to

(A.7)
$J=\frac{(216z-1)^{3}}{z(1+27z)^{3}}=-\frac{(1+24z^{\prime})^{3}}{z^{\prime
3}(1+27z^{\prime})}$

Using the isomorphism $\mathcal{M}_{K}\simeq X_{1}(3)$, the universal cover of
$\mathcal{M}_{K}$ is therefore identified with the Poincaré upper half plane,
tesselated by an infinite number of copies of the fundamental domain
$\mathcal{F}_{C}$.

###  A.2. Periods as Eichler integrals

Under mirror symmetry, the central charges $(T(\tau),T_{D}(\tau))$ of the
D2-brane and D4-brane are identified with periods
$(\varpi,\varpi_{D})=(\int_{\ell}\lambda,\int_{\ell_{D}}\lambda)$ of a
suitable meromorphic differential $\lambda$ over a basis of one cycles
$(\ell,\ell_{D})$ on the mirror curve. Each of these periods satisfies the
degree $3$ Picard-Fuchs equation

(A.8)
$\bigl{[}\Theta^{2}+3z(3\Theta+2)(3\Theta+1)\bigr{]}\Theta\cdot\begin{pmatrix}\varpi\\\
\varpi_{D}\end{pmatrix}=0$

where $\Theta=z\partial_{z}$. On the other hands, the periods
$(\varpi^{\prime},\varpi^{\prime}_{D})=(\int_{\ell}\omega,\int_{\ell_{D}}\omega)$
of the unique (up to scale) holomorphic differential $\omega$ on the mirror
curve satisfy the degree $2$ Picard-Fuchs equation

(A.9)
$\left[\Theta^{2}+3z(3\Theta+2)(3\Theta+1)\right]\cdot\begin{pmatrix}\varpi^{\prime}\\\
\varpi^{\prime}_{D}\end{pmatrix}=0$

It follows that

(A.10) $\Theta\cdot\begin{pmatrix}\varpi\\\
\varpi_{D}\end{pmatrix}=\begin{pmatrix}\varpi^{\prime}\\\
\varpi^{\prime}_{D}\end{pmatrix}$

Identifying the modular parameter as the ratio
$\tau=\frac{\varpi^{\prime}_{D}}{\varpi^{\prime}}$, we find that
$\varpi^{\prime}_{D}$ transforms as a modular form of weight 1 under
$\Gamma_{1}(3)$, hence it is given (up to normalization) by the weight 1
Eisenstein series (denoted by $A$ in [72]),

(A.11) $\varpi^{\prime}(\tau)=1+6\sum_{m\geq
1}\biggl{(}\sum_{n|m}\chi(n)\biggr{)}q^{m}=1+6q+6q^{3}+6q^{4}+12q^{7}+\dots$

where $\chi(n)=\binom{n}{-3}$ is the Dirichlet character equal to $+1$ for
$n=1\mod 3$, $-1$ for $n=2\mod 3$ and 0 otherwise. Let $C$ be the weight $3$
Eisenstein series with the same character,

(A.12) $C(\tau)=1-9\sum_{m\geq
1}\biggl{(}\sum_{n|m}n^{2}\chi(n)\biggr{)}q^{m}=1-9\sum_{n\geq
1}\frac{\chi(n)q^{n}}{1-q^{n}}$

This modular form can also be written as an eta product

(A.13)
$C(\tau)\coloneqq\frac{\eta(\tau)^{9}}{\eta(3\tau)^{3}}=\sum_{n=1}^{\infty}c_{n}q^{n}=1-9q+27q^{2}-9q^{3}-117q^{4}+\dots$

which makes it clear that it does not vanish anywhere in $\mathds{H}$. The
ratio $\frac{C-\varpi^{\prime 3}}{27\varpi^{\prime 3}}$ is a meromorphic
function on $X_{1}(3)$, which can be shown coincides with $z$. Using the
differential identities in $\frac{C-\varpi^{\prime 3}}{27\varpi^{\prime 3}}$,
we obtain immediately that
$z\partial_{z}=\frac{\varpi^{\prime}}{C}\partial_{\tau}$. Substituting into
(A.10), we get

(A.14) $\frac{d}{d\tau}\begin{pmatrix}T\\\
T_{D}\end{pmatrix}=C\begin{pmatrix}1\\\ \tau\end{pmatrix}$

Using the values $(T,T_{D})=(-\frac{1}{2},\frac{1}{3})$ at the orbifold point
$\tau_{o}$ [68] we can write the periods as a holomorphic Eichler integral

(A.15) $\begin{pmatrix}T\\\ T_{D}\end{pmatrix}=\begin{pmatrix}-\frac{1}{2}\\\
\frac{1}{3}\end{pmatrix}+\int_{\tau_{o}}^{\tau}\begin{pmatrix}1\\\
u\end{pmatrix}\,C(u)\mathrm{d}u$

Equivalently, using the identity

(A.16) $\int_{\tau_{o}}^{\tau_{o^{\prime}}}\begin{pmatrix}1\\\
u\end{pmatrix}\,\,C(u)\mathrm{d}u=\begin{pmatrix}1\\\ 0\end{pmatrix}$

proven at the end of §A.6, one can also take $\tau_{o^{\prime}}=\tau_{o}+1$ as
a base point and write

(A.17) $\begin{pmatrix}T\\\ T_{D}\end{pmatrix}=\begin{pmatrix}\frac{1}{2}\\\
\frac{1}{3}\end{pmatrix}+\int_{\tau_{o^{\prime}}}^{\tau}\begin{pmatrix}1\\\
u\end{pmatrix}\,\,C(u)\mathrm{d}u$

This representation (or equivalently (A.15)) provides a global formula for the
analytic continuation of $T$ and $T_{D}$ throughout the upper half-plane,
which gives immediate access to the asymptotic expansions near all singular
points and monodromies around them. It also proves to be very efficient for
numerical evaluations. In Table 3 we record the values of the periods at some
special points. Using $\overline{C(\tau)}=C(-\bar{\tau})$ and (A.16), it is
easy to establish the reality properties

(A.18) $\overline{T(\tau)}=-T(-\bar{\tau})\
,\quad\overline{T_{D}(\tau)}=T_{D}(-\bar{\tau})\ ,$

At the end of §A.6, we use (A.18) to conclude that $\Im T_{D}=0$ on the semi-
circle $\mathcal{C}(-\frac{1}{3},\frac{1}{3})$ passing through the orbifold
point $\tau_{o}$. As a result, the slope $s=\Im T_{D}/\Im T$ vanishes on that
semi-circle.

Table 3. Periods at some special points.

$\begin{array}[]{|c|c|c|c|c|c|}\hline\cr\tau&z&j&J_{3}&T&T_{D}\\\ \hline\cr
n\in\mathds{Z}&-\frac{1}{27}&\infty&12&n+\mathrm{i}\,\mathcal{V}&\frac{n^{2}}{2}+n\mathrm{i}\,\mathcal{V}\\\
\hline\cr\frac{1}{2}&-\frac{1}{27}&\infty&12&\frac{1}{2}-2\mathrm{i}\,\mathcal{V}&\frac{1}{2}-\mathrm{i}\,\mathcal{V}\\\
\hline\cr-\frac{1}{2}&-\frac{1}{27}&\infty&12&-\frac{1}{2}-2\mathrm{i}\,\mathcal{V}&\frac{1}{2}+\mathrm{i}\,\mathcal{V}\\\
\hline\cr\frac{1}{4}&-\frac{1}{27}&\infty&12&\frac{5}{2}+4\mathrm{i}\,\mathcal{V}&\frac{1}{2}+\mathrm{i}\,\mathcal{V}\\\
\hline\cr\frac{3}{4}&-\frac{1}{27}&\infty&12&-\frac{3}{2}+4\mathrm{i}\,\mathcal{V}&-\frac{3}{2}+3\mathrm{i}\,\mathcal{V}\\\
\hline\cr\frac{1}{5}&-\frac{1}{27}&\infty&12&2-5\mathrm{i}\,\mathcal{V}&\frac{1}{2}-\mathrm{i}\,\mathcal{V}\\\
\hline\cr\frac{2}{5}&-\frac{1}{27}&\infty&12&4-5\mathrm{i}\,\mathcal{V}&2-2\mathrm{i}\,\mathcal{V}\\\
\hline\cr\frac{3}{5}&-\frac{1}{27}&\infty&12&-3-5\mathrm{i}\,\mathcal{V}&-\frac{3}{2}-3\mathrm{i}\,\mathcal{V}\\\
\hline\cr\frac{3}{7}&-\frac{1}{27}&\infty&12&3+7\mathrm{i}\,\mathcal{V}&\frac{3}{2}+3\mathrm{i}\,\mathcal{V}\\\
\hline\cr\tau_{o}+n=\frac{e^{\mathrm{i}\pi/6}}{\sqrt{3}}&\infty&0&-15&n-\frac{1}{2}&\frac{1}{3}+\frac{n(n-1)}{2}\\\
\hline\cr\frac{9+\mathrm{i}\sqrt{3}}{42}&\infty&0&-15&\frac{3}{2}&\frac{1}{3}\\\
\hline\cr\frac{33+\mathrm{i}\sqrt{3}}{42}&\infty&0&-15&-\frac{1}{2}&-\frac{2}{3}\\\
\hline\cr\frac{33\mathrm{i}\sqrt{3}}{78}&\infty&0&-15&\frac{5}{2}&\frac{4}{3}\\\
\hline\cr\frac{45+\mathrm{i}\sqrt{3}}{78}&\infty&0&-15&-\frac{3}{2}&-\frac{2}{3}\\\
\hline\cr
e^{2\pi\mathrm{i}/3}&\frac{1}{216}&0&-231&-\frac{1}{2}+0.859778\,\mathrm{i}&-0.118598-0.429889\,\mathrm{i}\\\
\hline\cr
e^{\pi\mathrm{i}/3}&\frac{1}{216}&0&-231&\frac{1}{2}+0.859778\,\mathrm{i}&-0.118598+0.429889\,\mathrm{i}\\\
\hline\cr\mathrm{i}&\frac{5-3\sqrt{3}}{108}&1728&255+162\sqrt{3}&1.00267\mathrm{i}&-0.378093\\\
\hline\cr\end{array}$

###  A.3. Expansion around large radius

In the large radius limit, integrating term-by-term the $q$-expansion of $C$,
we get

(A.19) $\int_{\tau_{o}}^{\tau}\begin{pmatrix}1\\\
u\end{pmatrix}\,C(u)\mathrm{d}u=\begin{pmatrix}\tau-\tau_{o}+\frac{1}{2\pi\mathrm{i}}\left(\bar{f}_{1}(\tau)-\bar{f}_{1}(\tau_{o})\right)\\\
\frac{1}{2}\tau^{2}-\frac{1}{2}\tau_{o}^{2}+\frac{1}{2\pi\mathrm{i}}\left(\tau\,\bar{f}_{1}(\tau)-\tau_{o}\bar{f}_{1}(\tau_{o})\right)+\frac{1}{(2\pi\mathrm{i})^{2}}\left(\bar{f}_{2}(\tau)-\bar{f}_{2}(\tau_{o})\right)\end{pmatrix}$

where

(A.20) $\bar{f}_{1}(\tau)\coloneqq\sum_{n=1}^{\infty}\frac{c_{n}}{n}q^{n}\
,\quad\bar{f}_{2}(\tau)\coloneqq-\sum_{n=1}^{\infty}\frac{c_{n}}{n^{2}}q^{n}$

We observe numerically that for $\tau=\tau_{o}$,

(A.21) $\tau_{o}+\frac{1}{2\pi\mathrm{i}}\bar{f}_{1}(\tau_{o})=-\frac{1}{2}\
,\quad-\frac{1}{2}\tau_{o}^{2}-\frac{\tau_{o}}{2\pi\mathrm{i}}\bar{f}_{1}(\tau_{o})+\frac{1}{(2\pi\mathrm{i})^{2}}\bar{f}_{2}(\tau_{o})=-\frac{1}{8}$

we find that

(A.22) $T=\frac{f_{1}}{2\pi\mathrm{i}}-\frac{1}{2}\ ,\quad
T_{D}=\frac{f_{2}}{(2\pi\mathrm{i})^{2}}-\frac{f_{1}}{4\pi\mathrm{i}}+\frac{1}{4}$

with

(A.23) $f_{1}\coloneqq\log(-q)+\bar{f}_{1}\ ,\quad
f_{2}\coloneqq\frac{1}{2}[\log(-q)]^{2}+\bar{f}_{1}\log(-q)+\bar{f}_{2}\
,\quad$

Consequently, we have

(A.24) $T=\tau+\frac{\bar{f}_{1}}{2\pi\mathrm{i}}\ ,\quad
T_{D}=\frac{1}{2}\tau^{2}+\frac{1}{8}+\frac{\tau\,\bar{f}_{1}}{2\pi\mathrm{i}}+\frac{\bar{f}_{2}}{(2\pi\mathrm{i})^{2}}$

In particular, on any vertical line with $2\tau_{1}\in\mathds{Z}$, we have

(A.25) $\Re T=\frac{\Im T_{D}}{\Im T}=\tau_{1}$

since $\bar{f}_{1},\bar{f}_{2}$ are real. In the limit
$\tau\to\mathrm{i}\infty$, $\bar{f}_{1}(\tau),\bar{f}_{2}(\tau)$ are
exponentially suppressed hence $(T,T_{D})\sim(\tau,\frac{1}{2}\tau^{2})$.
Substituting $q$ in terms of $z$ via (A.5), we find agreement with the usual
representations in terms of Meijer G-functions [68]

(A.26) $\displaystyle f_{1}(z)$
$\displaystyle=-\frac{1}{\Gamma(\frac{1}{3})\Gamma(\frac{2}{3})}G_{3,3}^{2,2}\left(\begin{smallmatrix}\frac{1}{3}&\frac{2}{3}&1\\\
0&0&0\end{smallmatrix}\middle|27z\right)$ (A.27) $\displaystyle f_{2}(z)$
$\displaystyle=\frac{1}{2}\left[G_{3,3}^{3,1}\left(\begin{smallmatrix}\frac{1}{3}&\frac{2}{3}&1\\\
0&0&0\end{smallmatrix}\middle|27z\right)+G_{3,3}^{3,1}\left(\begin{smallmatrix}\frac{2}{3}&\frac{1}{3}&1\\\
0&0&0\end{smallmatrix}\middle|27z\right)\right]-\frac{\pi^{2}}{3}$

Expressing $\tau$ and $T_{D}$ in terms of the flat coordinate $T$ by inverting
the $q$-expansions, we recover the usual expansion in terms of Gromov-Witten
invariants,

(A.28)
$T_{D}=(2\pi\mathrm{i})^{2}\left(\frac{T^{2}}{2}+\frac{1}{8}\right)-9Q+\frac{135}{4}Q^{2}+244Q^{3}+\frac{36999}{16}Q^{4}+\frac{635634}{25}Q^{5}+307095Q^{6}+\dots$

where

(A.29) $Q\coloneqq
e^{2\pi\mathrm{i}T}=q+9q^{2}+54q^{3}+246q^{4}+909q^{5}+2808q^{6}+\dots$

The expansion (A.28) can be integrated term-by-term to obtain the tree-level
prepotential

(A.30)
$F_{0}=(2\pi\mathrm{i})^{3}\left(-\frac{T^{3}}{18}-\frac{T}{24}-\frac{1}{36}\right)+3Q-\frac{45}{8}Q^{2}+\frac{244}{9}Q^{3}-\frac{12333Q^{4}}{64}+\frac{211878Q^{5}}{125}-\frac{102365Q^{6}}{6}+\dots$

such that $T_{D}=-\frac{3}{2\pi\mathrm{i}}\partial_{T}F_{0}$.

###  A.4. Expansion around conifold point

The expansion near the conifold can be obtained by applying the Fricke
involution $\tau\mapsto\tau^{\prime}=-1/(3\tau)$ which maps $\tau_{o}$ to
$\tau_{o^{\prime}}=\tau_{o}+1$ and $C$ to

(A.31)
$C^{\prime}(\tau^{\prime})\coloneqq\frac{\eta(3\tau^{\prime})^{9}}{\eta(\tau^{\prime})^{3}}=\sum_{n=1}^{\infty}c^{\prime}_{n}q^{\prime
n}=q^{\prime}+3q^{\prime 2}+9q^{\prime 3}+13q^{\prime 4}+24q^{\prime
5}+27q^{\prime 6}+\dots$

This is again recognized as an Eisenstein series,

(A.32) $C^{\prime}(\tau^{\prime})=\sum_{m\geq
1}\big{(}\sum_{n|m}n^{2}\chi(m/n)\big{)}q^{\prime m}=\sum_{n\geq
1}\chi(n)\frac{q^{\prime n}(1+q^{\prime n})}{(1-q^{\prime n})^{2}}$

Changing variables $u\to u^{\prime}=-1/(3u)$ in (A.15) and using
$\eta(-1/\tau)=\sqrt{-\mathrm{i}\tau}\eta(\tau)$, we get

(A.33) $\begin{pmatrix}T\\\ T_{D}\end{pmatrix}=\begin{pmatrix}-\frac{1}{2}\\\
\frac{1}{3}\end{pmatrix}+\mathrm{i}\,3^{\frac{5}{2}}\int_{\tau_{o^{\prime}}}^{\tau^{\prime}}C^{\prime}(u^{\prime})\begin{pmatrix}3u^{\prime}\\\
-1\end{pmatrix}\mathrm{d}u^{\prime}$

Integrating terms by terms we get

(A.34)
$\int_{\tau_{o^{\prime}}}^{\tau}C^{\prime}(u)\begin{pmatrix}3u^{\prime}\\\
-1\end{pmatrix}\mathrm{d}u=\begin{pmatrix}\frac{3}{2\pi\mathrm{i}}\left(\tau^{\prime}f^{c}_{1}(\tau^{\prime})-\tau_{o^{\prime}}f^{c}_{1}(\tau_{o^{\prime}})\right)+\frac{3}{(2\pi\mathrm{i})^{2}}\left(\bar{f}^{c}_{2}(\tau^{\prime})-\bar{f}^{c}_{2}(\tau_{o^{\prime}})\right)\\\
-\frac{1}{2\pi\mathrm{i}}\left(f^{c}_{1}(\tau^{\prime})-f^{c}_{1}(\tau_{o^{\prime}})\right)\end{pmatrix}$

where we defined

(A.35) $\displaystyle f^{c}_{1}(\tau^{\prime})$
$\displaystyle\coloneqq\sum_{n=1}^{\infty}\frac{c^{\prime}_{n}}{n}q^{\prime
n}=q^{\prime}+\frac{3}{2}q^{\prime 2}+3q^{\prime 3}+\frac{13}{4}q^{\prime
4}+\dots$ $\displaystyle f^{c}_{2}(\tau^{\prime})$ $\displaystyle\coloneqq
f_{1}^{c}\log q^{\prime}+\bar{f}_{2}^{c}(\tau^{\prime})$
$\displaystyle\bar{f}_{2}^{c}(\tau^{\prime})$
$\displaystyle\coloneqq-\sum_{n=1}^{\infty}\frac{c^{\prime}_{n}}{n^{2}}q^{\prime
n}=-q^{\prime}-\frac{3}{4}q^{\prime 2}-q^{\prime 3}-\frac{13}{16}q^{\prime
4}-\dots$

We observe numerically that for $\tau=\tau_{o}$,

(A.36)
$\kappa\,f_{1}^{c}(\tau_{o})=-1,\quad\frac{\kappa}{2\pi\mathrm{i}}f_{2}^{c}(\tau_{o})=\tfrac{1}{2}-\mathrm{i}\,\mathcal{V}$

where

(A.37)
$\kappa\coloneqq\frac{27\sqrt{3}}{2\pi}\,,\quad\mathcal{V}\coloneqq\frac{27}{4\pi^{2}}\Im\operatorname{Li}_{2}\left(e^{2\pi\mathrm{i}/3}\right)\simeq
0.462758$

we find that $T,T_{D}$ can be expressed as

(A.38) $T=\frac{\kappa}{2\pi\mathrm{i}}f_{2}^{c}+\mathrm{i}\,\mathcal{V}\
,\quad T_{D}=-\frac{\kappa}{3}f_{1}^{c}$

In particular, $(T,T_{D})=(\mathrm{i}\mathcal{V},0)$ at the conifold point.
Substituting $q^{\prime}$ in terms of $z^{\prime}$ using (A.5) (with $q\to
q^{\prime},z\to z^{\prime})$, we recover the usual expression in terms of
Meijer G-functions, and arrive at an alternative representation for the
‘quantum volume’ [67],

(A.39)
$\mathcal{V}=\frac{G_{3,3}^{2,2}\left(\begin{smallmatrix}\frac{1}{3}&\frac{2}{3}&1\\\
0&0&0\end{smallmatrix}\middle|-1\right)}{2\pi\,\Gamma(\frac{1}{3})\Gamma(\frac{2}{3})}+\frac{\mathrm{i}}{2}$

The value for $\mathcal{V}$ observed numerically in (A.37) can be determined
exactly by evaluating (A.24) at $\tau=0$ using zeta function regularization.
Indeed, the $L$-series of $\bar{f}_{1},\bar{f}_{2}$ easily evaluate to

(A.40) $L_{\bar{f}_{1}}(s)\coloneqq\sum_{m\geq
1}\frac{c_{m}}{m^{1+s}}=-9\zeta(s+1)\,L(s-1)\ ,\quad
L_{\bar{f}_{2}}(s)\coloneqq-\sum_{m\geq
1}\frac{c_{m}}{m^{2+s}}=9\zeta(s+2)\,L(s)\ $

where $L(s)\coloneqq\sum_{m\geq
1}\chi(m)m^{-s}=3^{-s}\left(\zeta(s,\frac{1}{3})-\zeta(s,\frac{2}{3})\right)$
is the Dirichlet $L$-series. Its completion

(A.41)
$L^{\star}(s)=\left(\tfrac{3}{\pi}\right)^{\frac{s+1}{2}}\Gamma\left(\tfrac{s+1}{2}\right)L_{s}(s)$

is analytic for $\Re(s)>1$ and invariant under $s\mapsto 1-s$, and so is the
completed Riemann zeta function
$\zeta^{\star}(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$. Using this, one can evaluate
the limit as $s\to 0$,

(A.42) $\displaystyle\bar{f}_{1}(0)$ $\displaystyle=-9\lim_{s\to
0}\zeta(s+1)\,L(s-1)=-\frac{27\sqrt{3}L(2)}{8\pi^{2}}$ (A.43)
$\displaystyle\bar{f}_{2}(0)$ $\displaystyle=9\lim_{s\to
0}\zeta(s+2)\,L(s)=\frac{\pi^{2}}{2}$

where we used $L(0)=1/3$. This reproduces the expected value
$(T,T_{D})=(\mathrm{i}\,\mathcal{V},0)$ at $\tau=0$.

###  A.5. Orbifold point

Near the orbifold point, integrating the Taylor expansion of $C$ term by term
we get

(A.44) $\displaystyle T$
$\displaystyle=-\frac{1}{2}+C(\tau_{o})(\tau-\tau_{o})+\frac{1}{2}C^{\prime}(\tau_{o})(\tau-\tau_{o})^{2}+\dots$
(A.45) $\displaystyle T_{D}$
$\displaystyle=\frac{1}{3}+\frac{1}{2}C(\tau_{o})(\tau^{2}-\tau_{o}^{2})+\frac{1}{6}C^{\prime}(\tau_{o})(\tau-\tau_{o})^{2}(2\tau+\tau_{o})+\dots$

We observe numerically that

(A.46) $\displaystyle C(\tau_{o})$
$\displaystyle=\frac{\Gamma(\frac{1}{3})^{3}}{\Gamma(\frac{2}{3})^{6}}\simeq
3.1185\ ,\quad
C^{\prime}(\tau_{o})=-\mathrm{i}\frac{729}{4\pi^{\frac{9}{2}}}\Gamma(\tfrac{1}{3})^{3}\Gamma(\tfrac{7}{6})^{3}\simeq
16.2043\,\mathrm{i}\ ,\quad$ (A.47) $\displaystyle
J_{3}^{\prime\prime\prime}(\tau_{o})$
$\displaystyle=\frac{18\sqrt{3}\Gamma(\frac{1}{3})^{9}}{\mathrm{i}\Gamma(\frac{2}{3})^{9}}\simeq-14474\,\mathrm{i}$

with the same values for $\tau=\tau_{o^{\prime}}$. The flat coordinate $w=1/z$
is obtained by expanding (A.3),

(A.48)
$w=-J_{3}-15=\frac{3\sqrt{3}\Gamma(\frac{1}{3})^{9}}{\mathrm{i}\,\Gamma(\frac{2}{3})^{9}}(\tau-\tau_{o})^{3}+\mathcal{O}((\tau-\tau_{o})^{6})$

###  A.6. Monodromies

The monodromies around the three singular points can be computed by using the
transformation property of the Eichler integral,

(A.49) $\begin{pmatrix}T+\frac{1}{2}\\\
T_{D}-\frac{1}{3}\end{pmatrix}\left(\frac{a\tau+b}{c\tau+d}\right)=\begin{pmatrix}c&d\\\
a&b\end{pmatrix}\begin{pmatrix}T+\frac{1}{2}\\\
T_{D}-\frac{1}{3}\end{pmatrix}\left(\tau\right)+\int_{\frac{d\tau_{o}-b}{a-c\tau_{o}}}^{\tau_{o}}\begin{pmatrix}cu+d\\\
au+b\end{pmatrix}\,\,C(u)\mathrm{d}u$

whenever $ad-bc=1$, $c=0\mod 3$. The last term is independent of $\tau$, and
is a degree 1 period polynomial for the weight 3 modular form $C$. It follows
from (A.15) and (A.49) that the period vector $\Pi=(1,T,T_{D})$ transforms as

(A.50) $\Pi^{t}\mapsto M\Pi^{t}\,\qquad M=\begin{pmatrix}1&0&0\\\ m&d&c\\\
m_{D}&b&a\end{pmatrix}\ ,\quad$

where

(A.51) $\begin{pmatrix}m\\\
m_{D}\end{pmatrix}=\begin{pmatrix}\frac{1}{2}(d-1)-\frac{c}{3}\\\
\frac{1}{3}(1-a)+\frac{b}{2}\end{pmatrix}+\int_{\frac{d\tau_{o}-b}{a-c\tau_{o}}}^{\tau_{o}}\begin{pmatrix}cu+d\\\
au+b\end{pmatrix}\,\,C(u)\mathrm{d}u$

Consequently, the coordinates $(s,w)$ defined in (1.7) transform as

(A.52) $s\mapsto\frac{as+b}{cs+d},\quad
w\mapsto\frac{w}{cs+d}+\frac{as+b}{cs+d}m-m_{D}$

Under the $\Gamma_{1}(3)$ transformations

(A.53) $\tau\mapsto\tau+1\ ,\quad\tau\mapsto-\frac{\tau}{3\tau-1}\
,\quad\tau\mapsto-\frac{\tau+1}{3\tau+2}$

corresponding to monodromies around $\mathrm{i}\infty$, $0$ and $\tau_{o}$, we
find, in agreement with [40]303030This also agrees with [68] upon conjugating
the matrices in by $\Bigl{(}\begin{smallmatrix}1&0&0\\\ \\!\\!1/2&1&0\\\
0&0&1\end{smallmatrix}\Bigr{)}$, due to a shift $T\to T+\frac{1}{2}$.

(A.54) $M_{\rm LV}=\begin{pmatrix}1&0&0\\\ 1&1&0\\\
\frac{1}{2}&1&1\end{pmatrix}\,\quad M_{C}=\begin{pmatrix}1&0&0\\\ 0&1&-3\\\
0&0&1\end{pmatrix}\,\quad M_{o}=\begin{pmatrix}1&0&0\\\ -\frac{1}{2}&-2&-3\\\
\frac{1}{2}&1&1\end{pmatrix}$

satisfying $M_{o}=M_{C}M_{\rm LV},M_{o}^{3}=\mathds{1}$.

In the remainder of this section, we prove the identity (A.16) and the
statement below (2.32) by studying the action of the monodromy $M_{C}$. First,
we observe that $M_{C}$ maps $\tau_{o}$ to $\tau_{o^{\prime}}=\tau_{o}+1$,
while preserving $T_{D}$. Thus, (A.15) implies the second equation in (A.16),
namely

(A.55) $\int_{\tau_{o}}^{\tau_{o^{\prime}}}u\,C(u)\mathrm{d}u=0$

As a result, $T_{D}(\tau_{o^{\prime}})=T_{D}(\tau_{o})=1/3$. Similarly,
$M_{C}$ maps $T$ to $T-3T_{D}$, therefore
$T(\tau_{o})=T(\tau_{o^{\prime}})-3T_{D}(\tau_{o^{\prime}})$, which implies
the first equation in (A.16),

(A.56)
$\int_{\tau_{o}}^{\tau_{o^{\prime}}}C(u)\mathrm{d}u=T(\tau_{o^{\prime}})-T(\tau_{o})=3T_{D}(\tau_{o^{\prime}})=1$

Secondly, since $M_{C}$ sends $\tau=\tau_{1}+\mathrm{i}\tau_{2}$ to

(A.57)