BPS Dendroscopy on Local $\mathbb {P}^2$

Abstract

The spectrum of BPS states in type IIA string theory compactified on a Calabi–Yau threefold famously jumps across codimension-one walls in complexified Kähler moduli space, leading to an intricate chamber structure. The Split Attractor Flow Conjecture posits that the BPS index $\Omega _z(\gamma )$ for given charge $\gamma$ and moduli z can be reconstructed from the attractor indices $\Omega _\star (\gamma _i)$ counting BPS states of charge $\gamma _i$ in their respective attractor chamber, by summing over a finite set of decorated rooted flow trees known as attractor flow trees. If correct, this provides a classification (or dendroscopy) of the BPS spectrum into different topologies of nested BPS bound states, each having a simple chamber structure. Here we investigate this conjecture for the simplest, albeit non-compact, Calabi–Yau threefold, namely the canonical bundle over $\mathbb {P}^2$. Since the Kähler moduli space has complex dimension one and the attractor flow preserves the argument of the central charge, attractor flow trees coincide with scattering sequences of rays in a two-dimensional slice of the scattering diagram ${\mathcal {D}}_\psi$ in the space of stability conditions on the derived category of compactly supported coherent sheaves on $K_{\mathbb {P}^2}$. We combine previous results on the scattering diagram of $K_{\mathbb {P}^2}$ in the large volume slice with an analysis of the scattering diagram for the three-node quiver valid in the vicinity of the orbifold point $\mathbb {C}^3/\mathbb {Z}_3$, and prove that the Split Attractor Flow Conjecture holds true on the physical slice of $\Pi$-stability conditions. In particular, while there is an infinite set of initial rays related by the group $\Gamma _1(3)$ of auto-equivalences, only a finite number of possible decompositions $\gamma =\sum _i \gamma _i$ contribute to the index $\Omega _z(\gamma )$ for any $\gamma$ and z, with constituents $\gamma _i$ related by spectral flow to the fractional branes at the orbifold point. We further explain the absence of jumps in the index between the orbifold and large volume points for normalized torsion free sheaves, and uncover new ‘fake walls’ across which the dendroscopic structure changes but the total index remains constant.

Appendices

Appendix A. Periods as Eichler Integrals

In this section, we review the modular description of the Kähler moduli space of $K_{\mathbb {P}^2}$, 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 [31, 47, 66,67,68,69,70] for earlier studies in the literature.³¹

1.1 A.1. Kähler moduli space as the modular curve $X_1(3)$

Recall that the mirror of $K_{\mathbb {P}^2}$ is a family of genus-one curves $\Sigma (z)=x_1^3+x_2^3+x_3^3-z^{-1/3}x_1x_2x_3=0$ in $\mathbb {P}^2$ parametrized by the complex structure modulus $z\in {\mathcal {M}}_K=\mathbb {C}\backslash \{0,-\frac{1}{27}\}$, which is identified as the complexified Kähler moduli space of $K_{\mathbb {P}^2}$. 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 $\Sigma '(z'):x+y+1-z' x^3/y=0$ in $\mathbb {C}^\times _x \times \mathbb {C}^\times _y$, related to $\Sigma (z)$ with $z'=-z-\frac{1}{27}$ by a 3-isogeny. The points $z'=0,-\frac{1}{27}, \infty$ then correspond to the conifold, large radius and orbifold points, respectively.

The space ${\mathcal {M}}_K$ is isomorphic to the modular curve $X_1(3)=\mathbb {H}/\Gamma _1(3)$, where $\Gamma _1(3)$ is the group of integer matrices $({\begin{matrix} a &{} b \\ c &{} d \end{matrix}})$ such that $a,d=1 \mod 3$ and $c=0\mod 3$. This is an index 4 subgroup of $PSL(2,\mathbb {Z})$, 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 $\tau _o=\frac{1}{\sqrt{3}} e^{5\pi \textrm{i}/6}=(-\frac{1}{2},\frac{1}{2\sqrt{3}})$, shifted horizontally by $-1/2$ compared to (A.2),

$$\begin{aligned} {\mathcal {F}}_o :=\Bigl \{\tau : \, -1\le \tau _1<0, (\tau _1+\frac{2}{3})^2+\tau _2^2\ge \frac{1}{9}, (\tau _1+\frac{1}{3})^2+\tau _2^2> \frac{1}{9}\Bigr \} \end{aligned}$$

(A.1)

This domain (or rather its translate ${\mathcal {F}}_{o'}:={\mathcal {F}}_o(1)$ under $\tau \mapsto \tau +1$) is depicted in Fig. 32, along with several of its images under $\Gamma _1(3)$. Alternatively, one may choose a fundamental domain centered around $\tau =0$,

$$\begin{aligned} {\mathcal {F}}_C :=\Bigl \{\tau : \, -\frac{1}{2}\le \tau _1<\frac{1}{2}, (\tau _1+\frac{1}{3})^2+\tau _2^2\ge \frac{1}{9}, (\tau _1-\frac{1}{3})^2+\tau _2^2> \frac{1}{9}\Bigr \} \end{aligned}$$

(A.2)

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

Fig. 32

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

The isomorphism ${\mathcal {M}}_K \simeq X_1(3)$ is given explicitly by

$$\begin{aligned} J_3(\tau ) = -15 - \frac{1}{z} = \frac{12-405 z'}{1+27 z'} \end{aligned}$$

(A.3)

where $J_3(\tau )$ is the normalized Hauptmodul of $\Gamma _1(3)$,

$$\begin{aligned} J_3 :=\left( \frac{\eta (\tau )}{\eta (3\tau )}\right) ^{12} + 12 = \frac{1}{q}+54 q-76 q^2 -243 q^3 + 1188 q^4 - 1384 q^5 - 2916 q^6+ \dots \nonumber \\ \end{aligned}$$

(A.4)

where $q=e^{2\pi \textrm{i}\tau }$. The latter maps the points $\tau =\textrm{i}\infty , 0$ 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 $z\mapsto z'$. Plugging the q-expansion (A.4) into (A.3), we get

$$\begin{aligned} \begin{aligned} q&=-z + 15 z^2 - 279 z^3 + 5729 z^4 - 124554 z^5+2810718 z^6 + \dots \\ z&= -q + 15 q^2 - 171 q^3 + 1679 q^4 - 15054 q^5 + 126981 q^6 +\dots \end{aligned} \end{aligned}$$

(A.5)

The Klein invariant is expressed in terms of $J_3$ via

$$\begin{aligned} J:=\frac{E_4^3(\tau )}{\eta ^{24}(\tau )} = J_3 + 744+ \frac{196830}{J_3-12} + \frac{19131876}{(J_3-12)^2} + \frac{387420489}{(J_3-12)^3} \end{aligned}$$

(A.6)

leading to

$$\begin{aligned} J = \frac{(216 z-1)^3}{z(1+27 z)^3} = -\frac{(1+24 z')^3}{z'^3(1+27 z')} \end{aligned}$$

(A.7)

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 ${\mathcal {F}}_C$.

1.2 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

$$\begin{aligned} \bigl [ \Theta ^2+3z (3\Theta +2)(3\Theta +1) \bigr ] \Theta \cdot \begin{pmatrix} \varpi \\ \varpi _D \end{pmatrix}=0 \end{aligned}$$

(A.8)

where $\Theta =z\partial _z$. On the other hands, the periods $(\varpi ',\varpi '_D)=(\int _{\ell } \omega ,\int _{\ell _D} \omega )$ of the unique (up to scale) holomorphic differential $\omega$ on the mirror curve satisfy the degree 2 Picard-Fuchs equation

$$\begin{aligned} \left[ \Theta ^2+3z (3\Theta +2)(3\Theta +1) \right] \cdot \begin{pmatrix} \varpi ' \\ \varpi '_D \end{pmatrix}=0 \end{aligned}$$

(A.9)

It follows that

$$\begin{aligned} \Theta \cdot \begin{pmatrix} \varpi \\ \varpi _D \end{pmatrix}= \begin{pmatrix} \varpi ' \\ \varpi '_D \end{pmatrix} \end{aligned}$$

(A.10)

Identifying the modular parameter as the ratio $\tau =\frac{\varpi '_D}{\varpi '}$, we find that $\varpi '_D$ 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 [71]),

$$\begin{aligned} \varpi '(\tau ) = 1+6\sum _{m\ge 1} \biggl ( \sum _{n|m} \chi (n) \biggr ) q^m = 1 + 6 q+ 6 q^3+ 6 q^4 +12 q^7+ \dots \end{aligned}$$

(A.11)

where $\chi (n)=\left( {\begin{array}{c}n\\ -3\end{array}}\right)$ 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,

$$\begin{aligned} C(\tau ) =1 -9 \sum _{m\ge 1} \biggl ( \sum _{n|m} n^2 \chi (n) \biggr ) q^m = 1-9 \sum _{n\ge 1} \frac{n^2 \chi (n) q^n }{1-q^n} \end{aligned}$$

(A.12)

This modular form can also be written as an eta product

$$\begin{aligned} C (\tau ) :=\frac{\eta (\tau )^9}{\eta (3\tau )^3}= \sum _{n=1}^{\infty } c_n q^n = 1 - 9 q + 27 q^2 -9 q^3 - 117 q^4+ \dots \end{aligned}$$

(A.13)

which makes it clear that it does not vanish anywhere in $\mathbb {H}$. The ratio $\frac{C-\varpi '^3}{27 \varpi '^3}$ is a meromorphic function on $X_1(3)$, which can be shown to coincide with z. Using the differential identities [Eq. (51), [71]] in $\frac{C-\varpi '^3}{27 \varpi '^3}$, we obtain immediately that $z\partial _z=\frac{\varpi '}{C} \partial _\tau$. Substituting into (A.10), we get

$$\begin{aligned} \frac{d}{d\tau } \begin{pmatrix} T \\ T_D \end{pmatrix} = C \begin{pmatrix} 1 \\ \tau \end{pmatrix} \end{aligned}$$

(A.14)

Using the values $(T,T_D)=(-\frac{1}{2},\frac{1}{3})$ at the orbifold point $\tau _o$ [67] we can write the periods as a holomorphic Eichler integral

$$\begin{aligned} \begin{pmatrix} T \\ T_D \end{pmatrix} = \begin{pmatrix} -\frac{1}{2} \\ \frac{1}{3} \end{pmatrix} + \int _{\tau _o}^{\tau } \begin{pmatrix} 1 \\ u \end{pmatrix} C(u)\, \text {d}u \end{aligned}$$

(A.15)

Equivalently, using the identity

$$\begin{aligned} \int _{\tau _o}^{\tau _{o'}} \begin{pmatrix} 1 \\ u \end{pmatrix} \, C(u) \, \text {d}u = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{aligned}$$

(A.16)

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

$$\begin{aligned} \begin{pmatrix} T \\ T_D \end{pmatrix} = \begin{pmatrix} \frac{1}{2} \\ \frac{1}{3} \end{pmatrix} + \int _{\tau _{o'}}^{\tau } \begin{pmatrix} 1 \\ u \end{pmatrix} \, C(u) \, \text {d}u \end{aligned}$$

(A.17)

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

$$\begin{aligned} \overline{T(\tau )} = - T(-{\bar{\tau }})\, ,\quad \overline{T_D(\tau )} = T_D(-{\bar{\tau }})\, \end{aligned}$$

(A.18)

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

1.3 A.3. Expansion around large radius

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

$$\begin{aligned} \int _{\tau _o}^{\tau } \begin{pmatrix} 1 \\ u \end{pmatrix} \, C(u) \textrm{d}u = \begin{pmatrix} \tau -\tau _o + \frac{1}{2\pi \textrm{i}} \left( {{\bar{f}}}_1(\tau ) - {{\bar{f}}}_1(\tau _o) \right) \\ \frac{1}{2} \tau ^2-\frac{1}{2} \tau _o^2 + \frac{1}{2\pi \textrm{i}} \left( \tau \, {{\bar{f}}}_1(\tau ) - \tau _o {{\bar{f}}}_1(\tau _o)\right) + \frac{1}{(2\pi \textrm{i})^2} \left( {{\bar{f}}}_2(\tau ) - {{\bar{f}}}_2(\tau _o) \right) \end{pmatrix}\nonumber \\ \end{aligned}$$

(A.19)

where

$$\begin{aligned} {{\bar{f}}}_1 (\tau ) :=\sum _{n=1}^{\infty } \frac{c_n}{n} q^n\, ,\quad {{\bar{f}}}_2(\tau ) :=- \sum _{n=1}^{\infty } \frac{c_n}{n^2} q^n \end{aligned}$$

(A.20)

We observe numerically that for $\tau =\tau _o$,

$$\begin{aligned} \tau _o + \frac{1}{2\pi \text {i}} {{\bar{f}}}_1(\tau _o) = - \frac{1}{2}\, ,\quad - \frac{1}{2} \tau _o^2 - \frac{\tau _o}{2\pi \text {i}} {{\bar{f}}}_1(\tau _o) + \frac{1}{(2\pi \text {i})^2} {{\bar{f}}}_2(\tau _o) = -\frac{1}{8} \end{aligned}$$

(A.21)

Thus we find that

$$\begin{aligned} T=\frac{f_1}{2\pi \text {i}}-\frac{1}{2}\, ,\quad T_D = \frac{f_2}{(2\pi \text {i})^2} -\frac{f_1}{4\pi \text {i}} +\frac{1}{4} \end{aligned}$$

(A.22)

with

$$\begin{aligned} f_1 :=\log (- q) + {{\bar{f}}}_1 \,\quad f_2 :=\frac{1}{2}[\log (-q) ]^2 + {{\bar{f}}}_1 \log (-q) + {{\bar{f}}}_2\,\quad \end{aligned}$$

(A.23)

Consequently, we have

$$\begin{aligned} T = \tau + \frac{{{\bar{f}}}_1}{2\pi \textrm{i}}\,\quad T_D = \frac{1}{2}\tau ^2+\frac{1}{8} + \frac{\tau \, {{\bar{f}}}_1}{2\pi \textrm{i}} + \frac{{{\bar{f}}}_2}{(2\pi \textrm{i})^2} \end{aligned}$$

(A.24)

In particular, on any vertical line with $2\tau _1\in \mathbb {Z}$, we have

$$\begin{aligned} \Re T = \frac{\Im T_D}{\Im T}= \tau _1 \end{aligned}$$

(A.25)

since ${{\bar{f}}}_1, {{\bar{f}}}_2$ are real. In the limit $\tau \rightarrow \textrm{i}\infty$, ${{\bar{f}}}_1(\tau ), {{\bar{f}}}_2(\tau )$ 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 [67]

$$\begin{aligned} {f_{1(z)}}&= - \frac{1}{\Gamma ({\frac{1}{3}})\Gamma ({\frac{2}{3}})} {G_{{3,3}}^{2,2}} \left( \begin{array}{lll} \frac{1}{3} &{} \frac{2}{3} &{} 1 \\ 0&{} 0 &{} 0 \end{array} \bigg | 27 z \right) \end{aligned}$$

(A.26)

$$\begin{aligned} {f_{2(z)}}&= \frac{1}{2} \left[ G_{3,3}^{3,1} \left( \begin{array}{lll} \frac{1}{3} &{} \frac{2}{3} &{} 1 \\ 0&{} 0 &{} 0 \end{array} \bigg | 27 z \right) + G_{3,3}^{3,1} \left( \begin{array}{lll} \frac{2}{3} &{} \frac{1}{3} &{} 1 \\ 0 &{} 0 &{} 0 \end{array} \bigg | 27 z \right) \right] - \frac{\pi ^2}{3} \end{aligned}$$

(A.27)

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,

$$\begin{aligned} T_D{} & {} =\frac{T^2}{2} + \frac{1}{8} + \frac{1}{(2\pi \textrm{i})^2} \left( -9 Q + \frac{135}{4} Q^2 +244 Q^3 + \frac{36999}{16} Q^4 + \frac{635634}{25} Q^5 \right. \nonumber \\{} & {} \left. \quad + 307095 Q^6 + \dots \right) \end{aligned}$$

(A.28)

where

$$\begin{aligned} Q :=e^{2\pi \textrm{i}T} = q + 9 q^2 + 54 q^3 + 246 q^4 + 909 q^5 + 2808 q^6 +\dots \end{aligned}$$

(A.29)

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

$$\begin{aligned} F_0= & {} -\frac{T^3}{18} - \frac{T}{24} -\frac{1}{36} + \frac{1}{(2\pi \textrm{i})^3} \left( 3 Q - \frac{45}{8} Q^2 + \frac{244}{9} Q^3 -\frac{12333 Q^4}{64}+\frac{211878 Q^5}{125}\right. \nonumber \\{} & {} \left. \quad -\frac{102365 Q^6}{6}+ \dots \right) \end{aligned}$$

(A.30)

such that $T_D=-3 \partial _T F_0$.

1.4 A.4. Expansion around conifold point

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

$$\begin{aligned} C'(\tau ') := \frac{\eta (3\tau ')^9}{\eta (\tau ')^3} = \sum _{n=1}^{\infty } c'_n q'^n = q' + 3 q'^2 + 9 q'^3 + 13 q'^4 + 24 q'^5 + 27 q'^6 + \dots \qquad \end{aligned}$$

(A.31)

This is again recognized as an Eisenstein series,

$$\begin{aligned} C'(\tau ') = \sum _{m\ge 1} \big ( \sum _{n|m} n^2 \chi (m/n) \big ) q'^m = \sum _{n\ge 1} \chi (n) \frac{ q'^n(1+q'^n) }{(1-q'^n)^2} \end{aligned}$$

(A.32)

Changing variables $u\rightarrow u'=-1/(3u)$ in (A.15) and using $\eta (-1/\tau )=\sqrt{-\textrm{i}\tau }\eta (\tau )$, we get

$$\begin{aligned} \begin{pmatrix} T \\ T_D \end{pmatrix} = \begin{pmatrix} -\frac{1}{2} \\ \frac{1}{3} \end{pmatrix} + \text {i}\, 3^{\frac{5}{2}} \int _{\tau _{o'}}^{\tau '} \begin{pmatrix} 3 u' \\ -1 \end{pmatrix} C'(u')\,\text {d}u ' \end{aligned}$$

(A.33)

Integrating terms by terms we get

$$\begin{aligned} \int _{\tau _{o'}}^{\tau } \begin{pmatrix} 3 u' \\ -1 \end{pmatrix} C'(u)\,\text {d}u = \begin{pmatrix} \frac{3}{2\pi \text {i}} \left( \tau ' f^c_1(\tau ') - \tau _{o'} f^c_1(\tau _{o'})\right) + \frac{3}{(2\pi \text {i})^2} \left( \bar{f}^c_2(\tau ')- {{\bar{f}}}^c_2(\tau _{o'}) \right) \\ - \frac{1}{2\pi \text {i}} \left( f^c_1(\tau ') - f^c_1(\tau _{o'}) \right) \end{pmatrix}\nonumber \\ \end{aligned}$$

(A.34)

where we defined

$$\begin{aligned} \begin{aligned} f^c_1(\tau ')&:=\sum _{n=1}^{\infty } \frac{c'_n}{n} q'^n = q' + \frac{3}{2} q'^2 + 3 q'^3 + \frac{13}{4} q'^4 + \dots \\ f^c_2(\tau ')&:=f_1^c \log q' + {{\bar{f}}}_2^c(\tau ')\\ {{\bar{f}}}_2^c(\tau ')&:=- \sum _{n=1}^{\infty } \frac{c'_n}{n^2} q'^n = -q'- \frac{3}{4} q'^2 - q'^3 - \frac{13}{16} q'^4 - \dots \end{aligned} \end{aligned}$$

(A.35)

We observe numerically that for $\tau =\tau _o$,

$$\begin{aligned} \kappa \, f_1^c(\tau _o) = -1, \quad \frac{\kappa }{2\pi \textrm{i}} f_2^c(\tau _o) = \tfrac{1}{2} -\textrm{i}\,{\mathcal {V}}\end{aligned}$$

(A.36)

where

$$\begin{aligned} \kappa :=\frac{27\sqrt{3}}{2\pi }, \quad {\mathcal {V}}:=\frac{27}{4\pi ^2} \Im {{\,\textrm{Li}\,}}_2\left( e^{2\pi \textrm{i}/3}\right) \simeq 0.462758 \end{aligned}$$

(A.37)

Thus we find that $T, T_D$ can be expressed as

$$\begin{aligned} T= \frac{\kappa }{2\pi \textrm{i}} f_2^c +\textrm{i}\,{\mathcal {V}}\,\quad T_D = -\frac{\kappa }{3} f_1^c \end{aligned}$$

(A.38)

In particular, $(T,T_D)=(\textrm{i}{\mathcal {V}},0)$ at the conifold point. Substituting $q'$ in terms of $z'$ using (A.5) (with $q\rightarrow q', z\rightarrow z')$, we recover the usual expression in terms of Meijer G-functions, and arrive at an alternative representation for the ‘quantum volume’ [66],

$$\begin{aligned} {\mathcal {V}}= \frac{G_{3,3}^{2,2} \left( {\begin{matrix} {\frac{1}{3}} &{} {\frac{2}{3}} &{} 1 \\ 0&{} 0 &{} 0 \end{matrix}} \bigg | -1 \right) }{2\pi \, \Gamma (\frac{1}{3})\Gamma (\frac{2}{3})} +\frac{{\textrm{i}}}{2} \end{aligned}$$

(A.39)

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

$$\begin{aligned} L_{{{\bar{f}}}_1}(s) :=\sum _{m\ge 1} \frac{c_m}{m^{1+s}} = -9 \zeta (s+1)\, L(s-1)\, ,\quad L_{{{\bar{f}}}_2}(s) :=-\sum _{m\ge 1} \frac{c_m}{m^{2+s}} = 9 \zeta (s+2)\, L(s)\ \nonumber \\ \end{aligned}$$

(A.40)

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

$$\begin{aligned} L^\star (s) = \left( \tfrac{3}{\pi } \right) ^{\frac{s+1}{2}} \Gamma \left( \tfrac{s+1}{2}\right) L_s(s) \end{aligned}$$

(A.41)

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 $s\rightarrow 0$,

$$\begin{aligned} {{\bar{f}}}_1(0)&= -9 \lim _{s\rightarrow 0} \zeta (s+1)\, L(s-1) = -\frac{27 \sqrt{3} L(2)}{8\pi ^2} \end{aligned}$$

(A.42)

$$\begin{aligned} {{\bar{f}}}_2(0)&= 9 \lim _{s\rightarrow 0} \zeta (s+2)\, L(s) = \frac{\pi ^2}{2} \end{aligned}$$

(A.43)

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

1.5 A.5. Orbifold point

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

$$\begin{aligned} T&= -\frac{1}{2} +C(\tau _o) (\tau -\tau _o) + \frac{1}{2} C'(\tau _o)(\tau -\tau _o)^2 + \dots \end{aligned}$$

(A.44)

$$\begin{aligned} T_D&= \frac{1}{3} + \frac{1}{2} C(\tau _o) (\tau ^2-\tau _o^2)+\frac{1}{6} C'(\tau _o) (\tau -\tau _o)^2(2\tau +\tau _o) + \dots \end{aligned}$$

(A.45)

We observe numerically that

$$\begin{aligned} C(\tau _o)&= \frac{\Gamma (\frac{1}{3})^3}{\Gamma (\frac{2}{3})^6}\simeq 3.1185\ ,\quad C'(\tau _o) = -\textrm{i}\frac{729}{4\pi ^{\frac{9}{2}}} \Gamma (\tfrac{1}{3})^3 \Gamma (\tfrac{7}{6})^3\simeq 16.2043\, \textrm{i}\ ,\quad \end{aligned}$$

(A.46)

$$\begin{aligned} J_3'''(\tau _o)&= \frac{18\sqrt{3} \Gamma (\frac{1}{3})^9}{\textrm{i}\Gamma (\frac{2}{3})^9}\simeq -14474\, \textrm{i}\end{aligned}$$

(A.47)

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

$$\begin{aligned} w= - J_3 - 15 = \frac{3\sqrt{3} \Gamma (\frac{1}{3})^9}{\textrm{i}\, \Gamma (\frac{2}{3})^9} (\tau -\tau _o)^3 +{\mathcal {O}}( (\tau -\tau _o)^6) \end{aligned}$$

(A.48)

1.6 A.6. Monodromies

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

$$\begin{aligned} \begin{pmatrix} T+\frac{1}{2} \\ T_D-\frac{1}{3} \end{pmatrix}\left( \frac{a\tau +b}{c\tau +d} \right) = \begin{pmatrix} d &{}{} c \\ b &{}{} a \end{pmatrix} \begin{pmatrix} T+\frac{1}{2} \\ T_D-\frac{1}{3} \end{pmatrix}\left( \tau \right) + \int _{\frac{d\tau _o-b}{a-c\tau _o}}^{\tau _o} \begin{pmatrix} c u+d \\ a u+b \end{pmatrix} \, C(u)\; \text {d}u\nonumber \\ \end{aligned}$$

(A.49)

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

$$\begin{aligned} \Pi ^t \mapsto M \Pi ^t \, , \quad M=\begin{pmatrix} 1 &{}{} 0 &{}{} 0 \\ m &{}{} d &{}{} c \\ m_D &{}{} b &{}{} a \end{pmatrix}\,\quad \end{aligned}$$

(A.50)

where

$$\begin{aligned} \begin{pmatrix} m \\ m_D \end{pmatrix} = \begin{pmatrix} \frac{1}{2}(d-1)-\frac{c}{3} \\ \frac{1}{3}(1-a)+\frac{b}{2} \end{pmatrix} + \int _{\frac{d\tau _o-b}{a-c\tau _o}}^{\tau _o} \begin{pmatrix} c u+d \\ a u+b \end{pmatrix} \, C(u)\, \text {d}u \end{aligned}$$

(A.51)

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

$$\begin{aligned} s\mapsto \frac{as+b}{cs+d}, \quad w\mapsto \frac{w}{cs+d} + \frac{as+b}{cs+d} m - m_D \end{aligned}$$

(A.52)

Under the $\Gamma _1(3)$ transformations

$$\begin{aligned} \tau \mapsto \tau +1,\,\quad \tau \mapsto -\frac{\tau }{3\tau -1},\,\quad \tau \mapsto -\frac{\tau +1}{3\tau +2} \end{aligned}$$

(A.53)

corresponding to monodromies around $\textrm{i}\infty$, 0 and $\tau _o$, we find, in agreement with [40]³²

$$\begin{aligned} M_{\textrm{LV}}=\begin{pmatrix} 1 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 \\ \frac{1}{2} &{} 1 &{} 1 \end{pmatrix}, \, \quad M_{C}=\begin{pmatrix} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} -3 \\ 0 &{} 0 &{} 1 \end{pmatrix}, \, \quad M_{o} =\begin{pmatrix} 1 &{} 0 &{} 0 \\ -\frac{1}{2} &{} -2 &{} -3 \\ \frac{1}{2} &{} 1 &{} 1 \end{pmatrix} \end{aligned}$$

(A.54)

satisfying $M_o=M_C M_{\textrm{LV}}, M_o^3=\mathbb {1}$.

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 $\tau _{o'}=\tau _o+1$, while preserving $T_D$. Thus, (A.15) implies the second equation in (A.16), namely

$$\begin{aligned} \int _{\tau _{o}}^{\tau _{o'}} u \, C(u)\, \text {d}u=0 \end{aligned}$$

(A.55)

As a result, $T_D(\tau _{o'})=T_D(\tau _{o})=1/3$. Similarly, $M_C$ maps T to $T-3T_D$, therefore $T(\tau _{o})=T(\tau _{o'})-3T_D(\tau _{o'})$, which implies the first equation in (A.16),

$$\begin{aligned} \int _{\tau _{o}}^{\tau _{o'}} C(u)\, \text {d}u= T(\tau _{o'}) - T(\tau _{o}) = 3T_D(\tau _{o'}) = 1\nonumber \\ \end{aligned}$$

(A.56)

Secondly, since $M_C$ sends $\tau =\tau _1+\textrm{i}\tau _2$ to

$$\begin{aligned} - \frac{\tau }{3 \tau -1} =-\frac{3\tau _1^2+3\tau _2^2-\tau _1}{3(3\tau _1^2+3\tau _2^2-2\tau _1)+1} +\textrm{i}\frac{\tau _2}{3(3\tau _1^2+3\tau _2^2-2\tau _1)+1} \end{aligned}$$

(A.57)

we see that on the half circle ${\mathcal {C}}(-1/3,1/3)$ defined by $3\tau _1^2+3\tau _2^2-2\tau _1=0$, the action of $M_C$ restricts to $\tau \mapsto -{\bar{\tau }}$. Since $T_D$ is invariant under $M_C$, it follows that for any $\tau \in {\mathcal {C}}(-1/3,1/3)$

$$\begin{aligned} T_D(\tau )=T_D(-\bar{\tau }) \end{aligned}$$

(A.58)

Since $\overline{T_D(\tau )} = T_D(-{\bar{\tau }})$ by (2.32), it follows that $\Im T_D=0$ on the half circle ${\mathcal {C}}(-1/3,1/3)$. $\square$

Appendix B. Massless Objects at Conifold Points

The structure sheaf ${\mathcal {O}}$ of $\mathbb {P}^2$ is a spherical object in the derived category $D^b({{\,\textrm{Coh}\,}}_c K_{\mathbb {P}^2})$, whose central charge vanishes at the conifold point $\tau =0$. The action of the group $\Gamma _1(3)$ on the $\tau$ upper half-plane lifts to an action by auto-equivalences on the derived category of $K_{\mathbb {P}^2}$. Thus, for every $g \in \Gamma _1(3)$, $E=g({\mathcal {O}})$ is a spherical object whose central charge vanishes at the conifold point $\tau =g(0)=p/q$ with $q\ne 0\mod 3$. In this section we compute the object E for low values of p, q. The results are summarized in Table 1 on page 11.

For example, the element $V=({\begin{matrix} 1 &{} 0 \\ -3 &{} 1 \end{matrix}})$ (equal to the monodromy around the conifold point $\tau =0$) acts on $\tau$ via $V :\tau \mapsto -\frac{\tau }{3\tau -1}$, and on the derived category via the spherical twist ${{\,\textrm{ST}\,}}_{\mathcal {O}}$ around the spherical object ${\mathcal {O}}$. The latter is given by the exact triangle (see (2.20) and [31, §9.1])

$$\begin{aligned} {{\textrm{Hom}^\bullet }}_{K_{\mathbb {P}^2}}({\mathcal {O}},E)\otimes {\mathcal {O}} \xrightarrow {\textrm{ev}} E \rightarrow {{\,\textrm{ST}\,}}_{\mathcal {O}}( E ) \xrightarrow {+1} \end{aligned}$$

(B.1)

Similarly for $U=V^{-1}$ the action on the derived category is the inverse of the spherical twist ${{\,\textrm{ST}\,}}_{\mathcal {O}}$ around the spherical object ${\mathcal {O}}$, which is given, following Proposition 2.10 of [50], by the exact triangle

$$\begin{aligned} {{{\,\textrm{ST}\,}}^{-1}}_{\mathcal {O}}(E) \rightarrow E\xrightarrow {\textrm{coev}} {{\,\textrm{Hom}\,}}({{{\,\textrm{Hom}\,}}^{\bullet }}_{K_{\mathbb {P}^{2}}}(E,{\mathcal {O}}),{\mathcal {O}}) \xrightarrow {+1} \end{aligned}$$

(B.2)

At the point $\tau =-1/2$, obtained by acting with VT on $\tau =0$, the spherical object becoming massless is ${{\,\textrm{ST}\,}}_{{\mathcal {O}}}({\mathcal {O}}(1))$. We have ${{\,\textrm{Hom}\,}}^0_{\mathbb {P}^2}({\mathcal {O}},{\mathcal {O}}(1))=\mathbb {C}^3$, ${{\,\textrm{Hom}\,}}^k_{\mathbb {P}^2}({\mathcal {O}},{\mathcal {O}}(1))=0$ for $k>0$, and ${{\,\textrm{Hom}\,}}^k_{\mathbb {P}^2}({\mathcal {O}}(1), {\mathcal {O}})=0$ for $k \ge 0$. Hence, using (2.18),

$$\begin{aligned} {{\,\textrm{Hom}\,}}^0_{K_{\mathbb {P}^2}}({\mathcal {O}},{\mathcal {O}}(1))={{\,\textrm{Hom}\,}}^0_{\mathbb {P}^2}({\mathcal {O}}, {\mathcal {O}}(1))=\mathbb {C}^3\,\quad {{\,\textrm{Hom}\,}}^{k\ne 0}_{K_{\mathbb {P}^2}}({\mathcal {O}},{\mathcal {O}}(1))=0 \end{aligned}$$

(B.3)

Finally, from the Euler exact sequence

$$\begin{aligned} 0 \rightarrow {\mathcal {O}} \rightarrow {\mathcal {O}}(1)^{\oplus 3} \rightarrow T \rightarrow 0 \end{aligned}$$

(B.4)

where T is the tangent bundle of $\mathbb {P}^2$, we obtain the exact sequence

$$\begin{aligned} 0 \rightarrow \Omega (1) \rightarrow {\mathcal {O}}^{\oplus 3} \rightarrow {\mathcal {O}}(1) \rightarrow 0 \end{aligned}$$

(B.5)

and so an exact triangle ${\mathcal {O}}^{\oplus 3} \rightarrow {\mathcal {O}}(1) \rightarrow \Omega (1)[1] \xrightarrow {+1}$. Hence the massless object at $\tau =-\frac{1}{2}$ is

$$\begin{aligned} {{\,\textrm{ST}\,}}_{{\mathcal {O}}}({\mathcal {O}}(1))=\Omega (1)[1] \end{aligned}$$

(B.6)

For the point $\tau =4/5$, obtained by acting by TV on $\tau =-1/2$, the spherical object becoming massless is ${{\,\textrm{ST}\,}}_{{\mathcal {O}}}(\Omega (1)[1])(1)$. Using (B.5) and the Bott vanishing theorem, we obtain ${{\,\textrm{Hom}\,}}^0_{\mathbb {P}^2}(\Omega (1), {\mathcal {O}})=\mathbb {C}^3$. On the other hand, using the Bott vanishing theorem and Riemann–Roch formula, we find that ${{\,\textrm{Hom}\,}}^k_{\mathbb {P}^2}(\Omega (1),{\mathcal {O}})=0$ for all $k \ne 0$, and ${{\,\textrm{Hom}\,}}^k_{\mathbb {P}^2}({\mathcal {O}}, \Omega (1))=0$ for all k. Hence, by (2.18),

$$\begin{aligned}{} & {} {{{\,\textrm{Hom}\,}}^{\bullet }}_{K_{\mathbb {P}^2}}({\mathcal {O}},\Omega (1)) ={\mathbb {C}^{3{[}-3{]}}} \end{aligned}$$

(B.7)

$$\begin{aligned} {}&{} {{{\,\text {Hom}\,}}^{3}}_{K_{\mathbb {P}^2}}({\mathcal {O}},\Omega (1))={\mathbb {C}^3}\, ,\quad {{{\,\text {Hom}\,}}^{k\ne 3}}_{K_{\mathbb {P}^2}}({\mathcal {O}},\Omega (1)) \end{aligned}$$

(B.8)

It follows that the exact triangle defining $E={{\,\textrm{ST}\,}}_{\mathcal {O}}(\Omega (1)[1])(1)$ is of the form

$$\begin{aligned} {\mathcal {O}}^{\oplus 3}(1)[-2] \rightarrow \Omega (2)[1] \rightarrow E \xrightarrow {+1} \end{aligned}$$

(B.9)

Note in particular that E has rank $-5$ and degree $-4$, as expected.

It is important to remark that the map ${\mathcal {O}}^{\oplus 3}(1)[-2] \rightarrow \Omega (2)[1]$ in the derived category of $K_{\mathbb {P}^2}$ does not come from maps in the derived category of $\mathbb {P}^2$: indeed all extension groups ${{\,\textrm{Hom}\,}}^3_{\mathbb {P}^2}$ between sheaves are zero since $\mathbb {P}^2$ is of dimension 2. As mentioned at the beginning of Sect. 2.2, in general, an object E in the derived category of coherent sheaves on $K_{\mathbb {P}^2}$ supported set-theoretically on the zero section can be viewed as a pair $(F,\phi )$ with F an object in the derived category of $\mathbb {P}^2$ and $\phi :F \rightarrow F \otimes K_{\mathbb {P}^2}$ a nilpotent Higgs field. For the object E defined by (B.9), since every map ${\mathcal {O}}^{\oplus 3}(1)[-2] \rightarrow \Omega (2)[1]$ in the derived category of $\mathbb {P}^2$ is zero, the underlying object F is simply the direct sum

$$\begin{aligned} F=\Omega (2)[1] \oplus {\mathcal {O}}^{\oplus 3}(1)[-1] \end{aligned}$$

(B.10)

but there is a non-trivial nilpotent Higgs field on F coming from a map ${\mathcal {O}}^{\oplus 3}(1)[-1] \rightarrow \Omega (-1)[1]$ in $D^b({{\,\textrm{Coh}\,}}\mathbb {P}^2)$. Note that by Serre duality, we have

$$\begin{aligned} {{\,\textrm{Hom}\,}}^2_{\mathbb {P}^2}({\mathcal {O}}(1), \Omega (-1)) ={{\,\textrm{Hom}\,}}_{\mathbb {P}^2}(\Omega (2),{\mathcal {O}}(1))^\vee \end{aligned}$$

(B.11)

which is indeed non-zero.

The same type of analysis provides the exact triangles giving the following massless objects at other conifold points.

• $\tau =1/5$, $g=U^2T^{-1}$. Because of the relations $(VT)^3=1$ and $U=V^{-1}$,

  $$\begin{aligned} UT^{-1}({\mathcal {O}})=TVTV({\mathcal {O}})=TVT({\mathcal {O}}[-2])=\Omega (2)[-1] \end{aligned}$$

  (B.12)

  In the quiver associated to the exceptional collection $({\mathcal {O}},\Omega (2)[-1],{\mathcal {O}}(1)[-2])$ there are 3 relations and no arrows from the second to the first node, so we deduce successively

  $$\begin{aligned} {{\,\textrm{Hom}\,}}^\bullet _{K_{\mathbb {P}^2}}(\Omega (2)[-1],{\mathcal {O}})&= \mathbb {C}^3[-2] \end{aligned}$$

  (B.13)
  $$\begin{aligned} {{\,\textrm{Hom}\,}}( {{\,\textrm{Hom}\,}}^\bullet _{K_{\mathbb {P}^2}}(\Omega (2)[-1],{\mathcal {O}}),{\mathcal {O}})&= {\mathcal {O}}^{\oplus 3}[2] \end{aligned}$$

  (B.14)

  The object $E=U(\Omega (2)[-1])$ is then given by the exact triangle

  $$\begin{aligned} E\rightarrow \Omega (2)[-1]\rightarrow {\mathcal {O}}^{\oplus 3}[2] \xrightarrow {+1} \end{aligned}$$

  (B.15)

• $\tau =1/4$, $g=UT$. One has ${{\,\textrm{Hom}\,}}({{\,\textrm{Hom}\,}}^\bullet _{K_{\mathbb {P}^2}}({\mathcal {O}}(1),{\mathcal {O}}),{\mathcal {O}}) = {{\,\textrm{Hom}\,}}(\mathbb {C}^3[-3],{\mathcal {O}}) = {\mathcal {O}}^{\oplus 3}[3]$, thus the object $E=U({\mathcal {O}}(1))$ is given by the exact triangle

  $$\begin{aligned} E\rightarrow {\mathcal {O}}(1)\rightarrow {\mathcal {O}}^{\oplus 3}[3] \xrightarrow {+1} \end{aligned}$$

  (B.16)

• $\tau =2/5$, $g=UT^{-2}$. One has ${{\,\textrm{Hom}\,}}({{\,\textrm{Hom}\,}}^\bullet _{K_{\mathbb {P}^2}}({\mathcal {O}}(-2),{\mathcal {O}}),{\mathcal {O}}) = {{\,\textrm{Hom}\,}}(\mathbb {C}^6,{\mathcal {O}}) = {\mathcal {O}}^{\oplus 6}$, thus $E=U({\mathcal {O}}(-2))$ is given by the exact triangle

  $$\begin{aligned} E\rightarrow {\mathcal {O}}(-2)\rightarrow {\mathcal {O}}^{\oplus 6} \xrightarrow {+1} \end{aligned}$$

  (B.17)

• $\tau =1/2$, $g=TVT$. This is a translate of (B.6), namely $E=\Omega (2)[1]$.

• $\tau =3/5$, $g=TVT^2$. One has ${{\,\textrm{Hom}\,}}^\bullet _{K_{\mathbb {P}^2}}({\mathcal {O}},{\mathcal {O}}(2))=\mathbb {C}^6$, hence $E=TV({\mathcal {O}}(2))$ is given by the exact triangle

  $$\begin{aligned} {\mathcal {O}}(1)^{\oplus 6}\rightarrow {\mathcal {O}}(3)\rightarrow E \xrightarrow {+1} \end{aligned}$$

  (B.18)

• $\tau =3/4$, $g=TVT^{-1}$. One has ${{\,\textrm{Hom}\,}}^\bullet _{K_{\mathbb {P}^2}}({\mathcal {O}},{\mathcal {O}}(-1))=\mathbb {C}^3[-3]$, hence $E=TV({\mathcal {O}}(-1))$ is given by the exact triangle

  $$\begin{aligned} {\mathcal {O}}(1)^{\oplus 3}[-3]\rightarrow {\mathcal {O}}\rightarrow E \xrightarrow {+1} \end{aligned}$$

  (B.19)

• $\tau =1$, $g=T$. Trivially, $E={\mathcal {O}}(1)$.

These results are summarized in Table 1 on page 11.

Appendix C. Endpoints of Attractor Flows for Local $\mathbb {P}^2$

In this section we derive several bounds on the behaviour of the attractor flow on the slice of $\Pi$-stability conditions in the case ${\mathfrak {Y}}=K_{\mathbb {P}^2}$, which we used in Sect. 6.1. As explained there, the central charge $Z_\tau (\gamma )$ is a holomorphic function of $\tau \in \mathbb {H}$ with no critical point, so that the attractor flow can either end at a marginal stability wall, end at a conifold point, end at a large volume point, or continue indefinitely. In §C.1 and §C.2 we rule out the last two cases in turn. In §C.3, §C.4 and §C.5 we determine conditions for an attractor flow to end or start at a conifold point, which leads to the equivalent definitions of the critical phase $\psi$ in Definition 2.

1.1 C.1. Attractor flows avoid large volume points

Let us assume that an attractor flow associated to some $\gamma \in \Gamma$ ends at a large volume point, which we take without loss of generality to be $\tau =\textrm{i}\infty$. Such a flow $\mu \mapsto \tau (\mu )\in \mathbb {H}$ would at late times lie in the fundamental domain ${\mathcal {F}}_C$ centered around the conifold point $\tau =0$, or its translates centered around $\tau =n$. By applying a transformation $g(\mu )\in \Gamma _1(3)$ (which depends discretely on the flow parameter $\mu$), we can map $\tau (\mu )$ into ${{\tilde{\tau }}} = g\cdot \tau \in {\mathcal {F}}_C$, at the cost of also mapping $\gamma$ to ${{\tilde{\gamma }}}=g\cdot \gamma$. The central charge is unchanged, and in particular the mass $|Z_{{{\tilde{\tau }}}(\mu )}({{\tilde{\gamma }}}(\mu ))| = |Z_{\tau (\mu )}(\gamma )|$ is monotonically decreasing, hence lower than its initial value. We now exclude such attractor flows by proving that this upper bound on $|Z_{{{\tilde{\tau }}}}({{\tilde{\gamma }}})|$ would translate into an upper bound on $\Im \tau$, regardless of the charge vector ${{\tilde{\gamma }}}$ (provided the corresponding DT invariant is nonzero). Passing to the contrapositive statement and dropping tildes,³³ we shall prove the following statement (recall that $\delta$ denotes the D0-brane charge vector).

Proposition 4

For every $M>0$, there exists $D>0$ such that for every $\tau \in {\mathcal {F}}_C$ with $\Im (\tau )>D$, and every $\gamma \in \Gamma {\setminus } \mathbb {Z}\delta$ with $\Omega _\tau (\gamma ) \ne 0$, one has $|Z_\tau (\gamma )|>M$.

The proof relies on approximating the exact central charge Z by the large volume central charge (1.6) $Z_{(s,t)}^\textrm{LV}(\gamma ) = - \frac{r}{2} (s+\textrm{i}t)^2+ d (s+\textrm{i}t) -\text {ch}_2$, and we begin by proving bounds on it. We denote

$$\begin{aligned} Y = \frac{1}{2}(t^2-s^2) \end{aligned}$$

(C.1)

Lemma 5

For every (s, t) with $Y \ge 0$, and every $\gamma \in \Gamma$ such that $r\ge 0$ and $d^2-2r \text {ch}_2\ge 0$, one has

$$\begin{aligned} |Z_{(s,t)}^{\textrm{LV}}(\gamma )|^2 \ge r^2 Y^2+\frac{d^2}{2}Y \end{aligned}$$

(C.2)

Proof

We have

$$\begin{aligned} \begin{aligned} |Z_{(s,t)}^{\textrm{LV}}(\gamma )|^2&= (r Y + d s -\text {ch}_2)^2 + (2Y+s^2) (d - rs)^2 \\&\ge r^2 Y^2+2rY(ds-\text {ch}_2) +2Y(d-rs)^2 \end{aligned} \end{aligned}$$

(C.3)

If $r=0$, this gives $|Z_{(s,t)}^{\textrm{LV}}(\gamma )|^2 \ge 2d^2Y$ and so in particular (C.2). If $r>0$, we rephrase the inequality in terms of the slope $\mu =d/r$ and use the assumption $-\text {ch}_2/r \ge -\mu ^2/2$ to obtain

$$\begin{aligned} |Z_{(s,t)}^{\textrm{LV}}(\gamma )|^2 \ge r^2 \bigl ( Y^2 + Y (2\mu s - \mu ^2) + 2Y(\mu -s)^2\bigr ) \end{aligned}$$

(C.4)

The coefficient of Y is $\mu ^2-2s\mu +2s^2 \ge \mu ^2/2$, which yields (C.2). $\square$

Proof of Proposition 4

We write $\tau =\tau _1+i\tau _2$ with $\tau _1=\Re \tau$ and $\tau _2=\Im \tau$. We denote by ${\mathcal {O}}(1)$ any function on ${\mathcal {F}}_C$ which is bounded for $\tau _2$ large enough, uniformly in r and d. For instance, $1/\tau _2={\mathcal {O}}(1)$ and $\tau _1={\mathcal {O}}(1)$, as $-\frac{1}{2} \le \tau _1\le \frac{1}{2}$ on ${\mathcal {F}}_C$. According to the large volume expansion of the periods in §A.3, $T=\tau +{\mathcal {O}}(1)$ and $T_D=\frac{\tau ^2}{2}+{\mathcal {O}}(1)$, hence

$$\begin{aligned} \begin{aligned} \Re T&= {\mathcal {O}}(1),&\quad \Re T_D&= - \tau _2^2/2 + {\mathcal {O}}(1) \\ \Im T&= \tau _2 + {\mathcal {O}}(1),&\quad \Im T_D&= \tau _1 \tau _2 + {\mathcal {O}}(1) = {\mathcal {O}}(1) \tau _2 \end{aligned} \end{aligned}$$

(C.5)

As a result

$$\begin{aligned} \begin{aligned} s&= \frac{\Im T_D}{\Im T} = \tau _1 + {\mathcal {O}}(1)/\tau _2 = {\mathcal {O}}(1) \\ w&= - \Re T_D + s \Re T = \tau _2^2 / 2 + {\mathcal {O}}(1) \\ t&= \sqrt{2w - s^2} = \sqrt{\tau _2^2 + {\mathcal {O}}(1)} = \tau _2 + {\mathcal {O}}(1)/\tau _2 \\ Y&= w - s^2 = \tau _2^2 / 2 + {\mathcal {O}}(1) \end{aligned} \end{aligned}$$

(C.6)

In particular, large enough $\tau _2$, t or Y are synonymous within ${\mathcal {F}}_C$. Another consequence is

$$\begin{aligned} \tau = s + \textrm{i}t + {\mathcal {O}}(1)/\tau _2, \qquad \tau ^2 = (s+\textrm{i}t)^2 + {\mathcal {O}}(1) \end{aligned}$$

(C.7)

which implies that the large volume central charge is a good approximation of the central charge in the sense that

$$\begin{aligned} \begin{aligned} Z_\tau (\gamma )&= -rT_D + dT - \text {ch}_2 \\&= -\frac{r}{2} (s+\textrm{i}t)^2 + d (s+\textrm{i}t) - \text {ch}_2 + |r| {\mathcal {O}}(1) + |d| {\mathcal {O}}(1) \\&= Z_{(s,t)}^{\textrm{LV}}(\gamma ) + |r| {\mathcal {O}}(1) + |d| {\mathcal {O}}(1) \end{aligned} \end{aligned}$$

(C.8)

Next, we seek to apply Lemma 5. For large enough $\tau _2$, the point $\tau \in {\mathcal {F}}_C$ does not belong to the lower boundary of ${\mathcal {F}}_C$, thus $\tau$ defines a geometric stability condition. Hence, by [51] (Corollary 1.33 in published version, or Corollary 1.30 in arXiv version), if $\Omega _\tau (\gamma ) \ne 0$, then $\gamma =n\gamma '$ is a multiple $n\in \mathbb {Z}\setminus \{0\}$ of the class $\gamma '$ of a Gieseker semistable sheaf. The latter obeys $r'\ge 0$ and $d'^2-2r' {\text {ch}}'_2 \ge 0$ (see for example [41, Lemma 3.4]). Up to replacing $\gamma$ by $-\gamma$, which does not change the mass $|Z_\tau (\gamma )|$, one can assume that $n>0$, so that $r\ge 0$ and $d^2-2r \text {ch}_2 \ge 0$. Lemma 5 thus applies (for large enough $\tau _2$ to ensure $Y\ge 0$): for every $\gamma \in \Gamma {\setminus } \mathbb {Z}\delta$ such that $\Omega _\tau (\gamma )\ne 0$, we have

$$\begin{aligned} |Z_{(s,t)}^{\textrm{LV}}(\gamma )| \ge \sqrt{r^2 Y^2 + \frac{1}{2} d^2 Y} \ge \frac{1}{2} |r| Y +\frac{1}{2} |d| \sqrt{Y} \end{aligned}$$

(C.9)

where we did not try to optimize the constants. As a result,

$$\begin{aligned} |Z_\tau (\gamma )| \ge |Z_{(s,t)}^{\textrm{LV}}(\gamma )| - |r {\mathcal {O}}(1)| - |d {\mathcal {O}}(1)| \ge \frac{1}{3} |r| Y/ +\frac{1}{3} |d| \sqrt{Y} \end{aligned}$$

(C.10)

for large enough Y. Since we restrict to $\gamma \notin \mathbb {Z}\delta$, one has $(r,d)\ne (0,0)$, so that we have proven $|Z_\tau (\gamma )|\ge \sqrt{Y}/3$ for large enough Y, or equivalently for large enough $\tau _2$. This ends the proof of Proposition 4, which confines any attractor flow away from all large volume points. $\square$

1.2 C.2. Attractor flows end at walls or conifold points

We are now ready to prove that an attractor flow cannot continue indefinitely. We denote by ${\overline{\mathbb {H}}}=\mathbb {H}\cup \mathbb {R}$ the closed upper half plane.

Proposition 6

For a charge vector $\gamma \in \Gamma \setminus \mathbb {Z}\delta$ and a starting point $\tau (\mu _0)\in \mathbb {H}$, consider the attractor flow $[\mu _0,\mu _\infty )\ni \mu \mapsto \tau (\mu )$ that is maximally extended subject to the condition $\Omega _{\tau (\mu )}(\gamma )\ne 0$. Then the limit $\tau (\mu _\infty ) = \lim _{\mu \rightarrow \mu _\infty }\tau (\mu )\in {\overline{\mathbb {H}}}$ exists and lies either at a conifold point or on a wall of marginal stability of $\gamma$.

Proof

The modular curve $X_1(3)=\Gamma _1(3)\backslash \mathbb {H}$ has a natural compactification $\overline{X_1(3)} \simeq \mathbb {P}^1$ obtained by adding the large volume point $z_{\textrm{LV}}$ and the conifold point $z_C$. Let $\pi :\mathbb {H}\rightarrow X_1(3)$ be the quotient map. Then $\pi \circ \tau :[\mu _0,\mu _\infty )\rightarrow X_1(3)$ takes values in the compact space $\overline{X_1(3)}$ hence admits at least one limit point $z_\infty \in \overline{X_1(3)}$. In other words there exists a sequence $(\mu _n)_{n=1,2,\dots }$ that tends to $\mu _\infty$ and such that $\pi (\tau (\mu _n))\rightarrow z_\infty$. For later purposes, it is useful to recall that the mass $|Z_{\tau (\mu )}(\gamma )|$ is monotonically decreasing hence has a limit as $\mu \rightarrow \mu _\infty$, which necessarily coincides with the limit of its subsequence $|Z_{\tau (\mu _n)}(\gamma )|$.

Consider the unique element $g_n\in \Gamma _1(3)$ that maps $\tau (\mu _n)$ to a point $\tau '_n=g_n\cdot \tau (\mu _n)$ in the fundamental domain ${\mathcal {F}}_C$ centered on the conifold point, and consider the corresponding charge $\gamma '_n=g_n\cdot \gamma$. By $\Gamma _1(3)$-equivariance,

$$\begin{aligned} \Omega _{\tau '_n}(\gamma '_n) = \Omega _{\tau (\mu _n)}(\gamma )\ne 0, \qquad |Z_{\tau '_n}(\gamma '_n)| = |Z_{\tau (\mu )}(\gamma )| \le |Z_{\tau (\mu _0)}(\gamma )| \end{aligned}$$

(C.11)

Proposition 4 applied with $M=|Z_{\tau (\mu _0)}(\gamma )|$ implies that the imaginary parts $\Im \tau '_n\le D$ must be bounded above by some constant D. This bound excludes $\pi (\tau '_n)=\pi (\tau (\mu _n))$ from a neighborhood of the large volume point in $\overline{X_1(3)}$. Therefore, the large volume point cannot be a limit point $z_\infty$ of $\pi \circ \tau$.

Next, assume that the limit point $z_\infty$ lies in $X_1(3)$. Consider its lift $\tau _\infty \in {\mathcal {F}}_C$, and note that³⁴$\tau '_n\rightarrow \tau _\infty$. By the support property, $\tau _\infty$ admits an open neighbourhood U on which there are finitely many classes $\gamma ' \in \Gamma \setminus \mathbb {Z}\delta$ with non-zero DT invariants and with central charge less than the upper bound $|Z_{\tau (\mu _0)}(\gamma )|$. The point $\tau '_n$ lies in U for large enough n, hence $\gamma '_n$ takes finitely many values. Up to passing to a subsequence we can assume that all $\gamma '_n=\gamma '$ are equal to the same charge vector. Fix an arbitrary (Euclidean) norm $\Vert \ \Vert :\Gamma \rightarrow [0,+\infty )$. The support property ensures that $\Vert \gamma '\Vert \lesssim |Z_\tau (\gamma ')|$ whenever $\tau \in U$ and $\Omega _\tau (\gamma ')\ne 0$, with an implied constant that is uniform in $\tau \in U$. This gives a positive lower bound on $|Z_{\tau '_n}(\gamma ')|$, hence on its limit

$$\begin{aligned} m_\infty :=\lim _{\mu \rightarrow \mu _\infty } |Z_{\tau (\mu )}(\gamma )| = \lim _{n\rightarrow +\infty } |Z_{\tau (\mu _n)}(\gamma )| = \lim _{n\rightarrow +\infty } |Z_{\tau '_n}(\gamma ')| = |Z_{\tau _\infty }(\gamma ')| > 0\nonumber \\ \end{aligned}$$

(C.12)

We learn that the point $\tau _\infty \in \mathbb {H}$ is not a critical point of $Z_\tau (\gamma ')$ and the gradient flow is smooth near $\tau _\infty$. Therefore, near $\tau _\infty$ there exists local coordinates $m=|Z_\tau (\gamma ')|$ along attractor flow lines and $\ell$ parametrizing the different flow lines: the attractor flow keeps $\ell$ constant and decreases m. Consider a neighborhood that is rectangular in these coordinates,

$$\begin{aligned} V = (\ell _-,\ell _+)\times (m_-,m_+) \ni (\ell _\infty ,m_\infty ) = \tau _\infty \end{aligned}$$

(C.13)

For large enough n we have $\tau '_n=(\ell _n,m_n)\in V$. The gradient flow of $|Z_\tau (\gamma ')|$ starting from this point is $(\ell _n,m)$ with m decreasing from $m_n$ all the way to $m_-$ at the boundary of V. Since $m_-<m_\infty$, the attractor flow must stop before, and specifically (C.12) requires the attractor flow to stop precisely at $m=m_\infty$. Recall now that $\tau '_n=g_n\cdot \tau (\mu _n)$. The image $\{g_n^{-1}\cdot (\ell _n,m), m_\infty <m\le m_n\}$ of the gradient flow of $|Z_\tau (\gamma ')|$ is the gradient flow of $|Z_\tau (\gamma )|$ starting from $\tau (\mu _n)$, which is precisely the attractor flow. We have thus fully determined the end segment of the attractor flow: the end point $\tau (\mu _\infty )=g_n^{-1}\cdot (\ell _n,m_\infty )$ of the attractor flow exists. The gradient flow of $|Z_\tau (\gamma )|$ could continue unimpeded beyond $m=m_\infty$, hence what stops the attractor flow must be that $\Omega _\tau (\gamma )=0$ for $\tau =g_n^{-1}\cdot (\ell _n,m)$ with $m<m_\infty$ (at least, close to $m_\infty$). This means that $\tau (\mu _\infty )$ is on a wall of marginal stability.

It remains to treat the case where none of the limit points of $\pi \circ \tau$ are of the above type, in which case the only remaining possibility for the limit point is the conifold point $z_\infty =z_C\in \overline{X_1(3)}$. Since this is a unique limit point, we have $\lim _{\mu \rightarrow \mu _\infty }\pi (\tau (\mu ))=z_C$. For a constant $0<D<\Im \tau _o$, consider the (connected) set $W_D=\{\tau \in {\mathcal {F}}_C,\Im \tau <D\}$, its projection $V_D=\pi (W_D)\subset X_1(3)$, and the union $U_D=\pi ^{-1}(V_D)\subset \mathbb {H}$ of all of its $\Gamma _1(3)$ images. The sets $V_D$ and $U_D$ are manifestly open. In fact, $V_D\cup \{z_C\}$ is a neighborhood of $z_C$ in $\overline{X_1(3)}$, hence for large enough $\mu$, one has $\pi (\tau (\mu ))\in V_D$ thus $\tau (\mu )\in U_D$. We learn that $\tau (\mu )$ remains in a fixed connected component of $U_D$ for large enough $\mu$. For instance, the connected component of $U_D$ containing $W_D$ consists of the union of images $g\cdot W_D$ for all elements $g\in \Gamma _1(3)$ that leave $\tau =0$ invariant, so if $\tau (\mu )$ lies in this connected component, it can only tend to the conifold point $\tau =0$. Other connected components are $\Gamma _1(3)$ images of this one, which implies that $\tau (\mu )$ tends to a conifold point $\tau _C\in \mathbb {Q}\subset \partial {\overline{\mathbb {H}}}$. $\square$

1.3 C.3. Initial data of the exact diagram from the large volume diagram

A key ingredient when building scattering diagrams is the initial data: DT invariants along the initial rays, which are rays that do not arise from the scattering of other rays. Initial rays correspond to attractor flows that do not end on a wall of marginal stability, hence that end at a conifold point by Proposition 6. In this subsection we identify the DT invariants of such a flow (near a conifold point) to some DT invariants in the large volume diagram. This establishes the equivalence of the characterizations (2) and (3) of critical phases in Definition 2. Using $\Gamma _1(3)$ invariance we take the conifold point to be $\tau =0$. We assume $\psi \in (-\pi /2,\pi /2)$ in this section as the scattering diagram for $\psi =\pi /2$ is highly degenerate and best treated separately in Sect. 6.4.4. We recall ${\mathcal {V}}_\psi ={\mathcal {V}}\tan \psi$.

Proposition 7

Consider an attractor flow that ends at the conifold point $\tau =0$ and has $Z_\tau (\gamma )\in \textrm{i}e^{\textrm{i}\psi }[0,+\infty )$ with $\psi \in (-\pi /2,\pi /2)$. Then the point $(s,t) = ({\mathcal {V}}_\psi ,|{\mathcal {V}}_\psi |)$ lies on (the closure of) an active ray ${\mathcal {R}}^{\textrm{LV}}_0(\gamma )$ of charge $\gamma$ in the large volume scattering diagram ${\mathcal {D}}^{\textrm{LV}}_0$. Furthermore, each DT invariant $\Omega _\tau (k\gamma )$, $k\ge 1$, is eventually constant along the flow close to the conifold point, and coincides with the limit of $\Omega ^{\textrm{LV}}_{(s,t)}(k\gamma )$ as $(s,t)\rightarrow ({\mathcal {V}}_\psi ,|{\mathcal {V}}_\psi |)$ along ${\mathcal {R}}^{\textrm{LV}}_0(\gamma )$ in ${\mathcal {D}}^{\textrm{LV}}_0$. In particular, either $\gamma =[r,0,0]$ with DT invariants $\Omega _\tau ([k,0,0])=\delta _{k,{{\,\textrm{sign}\,}}r}$, or $\psi$ is a critical phase in the sense that $({\mathcal {V}}_\psi ,|{\mathcal {V}}_\psi |)$ is an intersection of active rays in ${\mathcal {D}}^{\textrm{LV}}_0$.

Proof

While the statement is expressed in a uniform way, we shall distinguish $d=0$ from $d\ne 0$ momentarily as they require very different approaches.

There is a $\mathbb {Z}$-worth of fundamental domains meeting at this conifold point, acted upon by the monodromy $V:\tau \mapsto \frac{\tau }{1-3\tau }$ around $\tau =0$. We use the coordinate $\tau '=-1/(3\tau )$, in which the conifold point lies at $\tau '\rightarrow +\textrm{i}\infty$. and the monodromy acts as $V:\tau '\mapsto \tau '+1$. In terms of $q'=e^{2\pi \textrm{i}\tau '}$ (which vanishes at the conifold point), the expansion (A.38) reads

$$\begin{aligned} T = \textrm{i}{\mathcal {V}}+ \kappa \tau ' q' (1+o(1)), \qquad T_D = -\frac{\kappa }{3} q' (1 + o(1)) \end{aligned}$$

(C.14)

where o(1) denotes any function of $\tau '$ (such as $1/\tau '$ or $q'$) that vanishes as $q'\rightarrow 0$. Within an attractor flow, the phase of $Z_\tau (\gamma )$ is fixed and its modulus is monotonically decreasing, hence $Z_\tau (\gamma )=-rT_D+dT-\text {ch}_2$ has a limit as $q'\rightarrow 0$.

If $d=0$ then $Z_\tau (\gamma )\rightarrow -\text {ch}_2$, which does not belong to the half-line $\textrm{i}e^{\textrm{i}\psi }[0,+\infty )$ unless $\text {ch}_2=0$, which corresponds to a charge $\gamma =[r,0,0]$, with $r\ne 0$ since $\gamma \ne 0$. Such a charge is a multiple of the class of sheaves ${\mathcal {O}}(0)[k]$ becoming massless at the conifold point. It is easy to see that $Z_\tau (\gamma ) = -rT_D = \frac{\kappa }{3} rq'(1+o(1))\in \textrm{i}e^{\textrm{i}\psi }(0,+\infty )$ requires

$$\begin{aligned} \tau '_1 = \Re \tau ' = -n + \frac{1}{4} {{\,\textrm{sign}\,}}r + \frac{\psi }{2\pi } + o(1) \end{aligned}$$

(C.15)

for some integer $n\in \mathbb {Z}$. Close enough to the conifold point, the o(1) term is less than 1/2, so along a given attractor flow n is eventually constant. Applying a $\Gamma _1(3)$ translation $V^n:\tau '\mapsto \tau '+n$ reduces the problem to the case $n=0$, for which $|\tau '_1|<1/2$ close enough to the conifold point. As the Fricke involution $\tau '=-1/(3\tau )$ maps the fundamental domain ${\mathcal {F}}_C$ to itself, we deduce that $\tau \in {\mathcal {F}}_C$ at late enough times along the flow. In addition,

$$\begin{aligned} {{\,\textrm{sign}\,}}(\Re \tau ) = {{\,\textrm{sign}\,}}\Bigl (\frac{-\tau '_1}{3|\tau '|^2}\Bigr ) = - {{\,\textrm{sign}\,}}(\tau '_1) = - {{\,\textrm{sign}\,}}r \end{aligned}$$

(C.16)

Let us determine the DT invariants along this attractor flow that eventually lies in ${\mathcal {F}}_C$ and ends at $\tau =0$. It follows from [51, Corollary 1.24] (Corollary 1.21 of the arXiv version) that $\Omega _\tau (\gamma )$ does not jump for $\gamma$ proportional to $\gamma ({\mathcal {O}})$ and $\tau$ geometric (apart from $\gamma \mapsto -\gamma$ across the vertical axis). So it is enough to work at large volume, namely with Gieseker stability. Let E be a Gieseker-stable sheaf of class $\gamma = k \gamma ({\mathcal {O}})$ with $k\ge 1$. It is of slope $\mu =0$ and of discriminant $\Delta =0$. Hence, E is exceptional in the sense of [41, Section 4.2]. By [41, Lemma 4.3], there is a unique exceptional sheaf of given slope. As ${\mathcal {O}}$ is exceptional of slope 0, we obtain $E={\mathcal {O}}$. The moduli space of stable objects is a point for $k=1$ and empty for $k>1$. So

$$\begin{aligned} \Omega _\tau (k \gamma ({\mathcal {O}})) = \delta _{k,1} \end{aligned}$$

(C.17)

for $k\in \mathbb {Z}$ and for every $\tau \in {\mathcal {F}}_C$ with $\Re \tau <0$, and likewise $\Omega _\tau (k \gamma ({\mathcal {O}})) = \delta _{k,-1}$ for $\Re \tau >0$. Thanks to (C.16) this translates to the sign condition in the statement of the Proposition.

We henceforth assume that $d\ne 0$.

After proving that the phase must obey ${\mathcal {V}}_\psi =\text {ch}_2/d$, our strategy is to show that the attractor flow (near $\tau =0$) lies in the large volume region $\tau \in \mathbb {H}^{\textrm{LV}}$, map this point to large volume coordinates (s, t), and finally use that the large volume scattering diagram for non-zero phase $\psi ^{\textrm{LV}}=\arg (-\textrm{i}Z^{\textrm{LV}}_{(s,t)}(\gamma ))$ coincides with the diagram ${\mathcal {D}}^\textrm{LV}_0$ with zero phase up to a further change of coordinates (4.26),

$$\begin{aligned} \Omega _\tau (\gamma ) = \Omega ^{\textrm{LV}}_{(s,t)}(\gamma ) = \Omega ^\textrm{LV}_{({\tilde{s}},{\tilde{t}})}(\gamma ), \qquad {\tilde{s}} = s + t \tan \psi ^{\textrm{LV}}, \qquad {\tilde{t}}^2 = t^2 + \bigl (t \tan \psi ^\textrm{LV}\bigr )^2\nonumber \\ \end{aligned}$$

(C.18)

Our calculations show that $({\tilde{s}},{\tilde{t}})\rightarrow ({\mathcal {V}}_\psi ,|{\mathcal {V}}_\psi |)$ as $\tau \rightarrow 0$ along the flow, which suffices to conclude.

We start by determining how the conifold point is approached. The central charge has a limit, hence T has a limit, and equivalently $\tau ' q'\rightarrow c$ for some $c\in \mathbb {C}$. Decomposing $\tau '=\tau '_1+\textrm{i}\tau '_2$, we see that $\tau '_2 q' = {\mathcal {O}}(\tau '_2 e^{-2\pi \tau '_2})$ vanishes at the conifold point so $\tau '_1q'\rightarrow c$. If $c\ne 0$ then $|\tau '_1|\sim |c|/|q'| = |c|e^{2\pi \tau '_2}$, which means that the phase of $\tau '_1q'$ diverges, contradicting $\tau '_1q'\rightarrow c$. Thus $c=0$ and altogether $T\rightarrow \textrm{i}{\mathcal {V}}$. We learn that

$$\begin{aligned} Z_\tau (\gamma ) = \textrm{i}{\mathcal {V}}d - \text {ch}_2 + \kappa \tau ' q' (d + o(1)) \end{aligned}$$

(C.19)

with $\tau 'q'\rightarrow 0$. Along the flow, the central charge is fixed to lie in $Z_\tau (\gamma )\in \textrm{i}e^{\textrm{i}\psi }[0,+\infty )$ and to move towards 0 along this half-line, hence $Z_\tau (\gamma )-(\textrm{i}{\mathcal {V}}d - \text {ch}_2)$ lies in the same half-line. This implies (using $-\pi /2<\psi <\pi /2$ hence $\cos \psi >0$)

$$\begin{aligned} d > 0, \qquad \frac{\text {ch}_2}{d} = {\mathcal {V}}_\psi , \qquad 2\pi \tau '_1 = - 2\pi n + \psi + o(1) \end{aligned}$$

(C.20)

where ${\mathcal {V}}_\psi = {\mathcal {V}}\tan \psi$ and $n\in \mathbb {Z}$ is eventually constant (as in (C.15), $\tau \in {\mathcal {F}}_C$ if and only if $n=0$). We then evaluate the large-volume coordinates s, t,

$$\begin{aligned} \begin{aligned} s&= \frac{\Im T_D}{\Im T} = -\frac{\kappa }{3{\mathcal {V}}} e^{-2\pi \tau '_2} \bigl ( \sin \psi + o(1)\bigr ) \\ w&= - \Re T_D + s \Re T = \frac{\kappa }{3} e^{-2\pi \tau '_2} \bigl ( \cos \psi + o(1) \bigr )\\ t^2&= 2w - s^2 = \frac{2\kappa }{3} e^{-2\pi \tau '_2} \bigl ( \cos \psi + o(1)\bigr ) > 0 \end{aligned} \end{aligned}$$

(C.21)

Therefore, close enough to the conifold point, the ray lies in the large volume region $2w>s^2$, and one has $\Omega _\tau (\gamma ) = \Omega ^{\textrm{LV}}_{(s,t)}(\gamma )$.

Next we consider the phase $\psi ^{\textrm{LV}}=\arg (-\textrm{i}Z^\textrm{LV}_{(s,t)}(\gamma ))$ of the large volume central charge at (s, t), and evaluate its tangent since this is what appears in the change of coordinates (C.18):

$$\begin{aligned} t \tan \psi ^{\textrm{LV}} = - t \frac{\Re Z^{\textrm{LV}}_{(s,t)}(\gamma )}{\Im Z^{\textrm{LV}}_{(s,t)}(\gamma )} = \frac{\text {ch}_2 - d s + \frac{r}{2} (s^2-t^2)}{d-rs} = {\mathcal {V}}_\psi + {\mathcal {O}}(e^{-\pi \tau '_2}) \end{aligned}$$

(C.22)

Thus

$$\begin{aligned} \begin{aligned} {\tilde{s}}&= s + t \tan \psi ^{\textrm{LV}} = {\mathcal {V}}_\psi + {\mathcal {O}}(e^{-\pi \tau '_2}), \\ {\tilde{t}}&= \sqrt{t^2 + \bigl (t \tan \psi ^{\textrm{LV}}\bigr )^2} = |{\mathcal {V}}_\psi | + {\mathcal {O}}(e^{-\pi \tau '_2}) \end{aligned} \end{aligned}$$

(C.23)

As announced, we learn that $({\tilde{s}},{\tilde{t}})$ tends to the point $({\mathcal {V}}_\psi ,|{\mathcal {V}}_\psi |)$. By construction, $({\tilde{s}},{\tilde{t}})$ moves along the geometric ray ${\mathcal {R}}^\textrm{geo,LV}_0(\gamma )$ of the zero-phase large volume scattering diagram ${\mathcal {D}}^{\textrm{LV}}_0$. DT invariants $\Omega _\tau (\gamma )$ near the end of the attractor flow are thus given by DT invariants along the ray ${\mathcal {R}}^{\textrm{geo,LV}}_0(\gamma )$ near its intersection $({\mathcal {V}}_\psi ,|{\mathcal {V}}_\psi |)$ with the initial ray ${\mathcal {R}}_0^\textrm{LV}([{{\,\textrm{sign}\,}}{\mathcal {V}}_\psi ,0,0])$. The ray ${\mathcal {R}}^{\textrm{geo}}_\psi (\gamma )$ is thus active close to $\tau =0$ precisely when the ray ${\mathcal {R}}^\textrm{geo,LV}_0(\gamma )$ is active close to $({\mathcal {V}}_\psi ,|{\mathcal {V}}_\psi |)$. We conclude that there are non-trivial initial rays at $\tau =0$ if and only if $({\mathcal {V}}_\psi ,|{\mathcal {V}}_\psi |)$ is an intersection of active rays. $\square$

1.4 C.4. Initial data of the exact diagram from the orbifold diagram

In the previous section we have mapped DT invariants along an attractor flow ending at $\tau =0$ to DT invariants in the large volume scattering diagram. We now map the DT invariants to the orbifold diagram, thus proving the equivalence of the criteria (2) and (4) in Definition 2, in terms of attractor flows and of the orbifold diagram. The latter criterion involves the point $\theta =(0,\frac{1}{2}+|{\mathcal {V}}_\psi |,\frac{1}{2}-|{\mathcal {V}}_\psi |)$ whose (u, v) coordinates are determined from (5.12) to be

$$\begin{aligned} u = \frac{1}{12} + \frac{1}{2} |{\mathcal {V}}_\psi |, \qquad v = \frac{1}{4\sqrt{3}} (- 1 + 2|{\mathcal {V}}_\psi |) \end{aligned}$$

(C.24)

Recall the functions $p_j:\mathbb {R}\rightarrow \mathbb {R}^2$ given in (6.28) for $j=1,2,3$ parametrizing the three initial rays of the orbifold diagram. The point (C.24) involved in Definition 2 is $p_1(-|{\mathcal {V}}_\psi |)$. By $\mathbb {Z}_3$-invariance (cyclic permutations of the $\theta _j$) one can replace this point by $p_3(-|{\mathcal {V}}_\psi |)$, which will appear more naturally in this section. Rather than repeating what can already be learned about $\gamma =[k,0,0)$ from Proposition 7, we restrict our attention immediately to attractor flows with $\gamma \notin [1,0,0)$.

Proposition 8

Consider an attractor flow that ends at the conifold point $\tau =0$ and has $Z_\tau (\gamma )\in \textrm{i}e^{\textrm{i}\psi }[0,+\infty )$ with $-\pi /2<\psi <\pi /2$ and with $\gamma \notin [1,0,0)\mathbb {Z}$. Then the point $p_3(-|{\mathcal {V}}_\psi |)$ is a ray intersection in ${\mathcal {D}}_o$.

Proof

For $|{\mathcal {V}}_\psi |<1/2$ there are no such ray intersections in ${\mathcal {D}}_o$, and we have shown as part of Proposition 7 that there is no such attractor flow. We thus concentrate on ${\mathcal {V}}_\psi \le -1/2$, fixing the sign to be negative by using the $\psi \mapsto -\psi$ symmetry. In other words, $-\pi /2<\psi \le -\psi ^{\textrm{cr}}_{1/2}$.

A translation $V^n:\tau '\mapsto \tau '+n$ maps the attractor flow (C.20) to that with $n=0$,

$$\begin{aligned} \tau '_1 = \psi /(2\pi ) + o(1) \in (-1/2,0) \end{aligned}$$

(C.25)

As discussed below (C.15), this condition on $\tau '$ implies that $\tau \in {\mathcal {F}}_C$. Furthermore, the sign ${{\,\textrm{sign}\,}}(\Re \tau )=-{{\,\textrm{sign}\,}}(\tau '_1)=1$ implies that $\tau \in {\mathcal {F}}_{o'}={\mathcal {F}}_o(1)$. Within the orbifold fundamental domain ${\mathcal {F}}_o$, the region of validity $\mathbb {H}^o$ of the quiver description is described by the inequality (6.26) $2w+s<0$, namely $t^2<-s(1+s)$ for the point $\tau -1\in {\mathcal {F}}_o$. Since $s(\tau -1)=s(\tau )-1$ and $t^2$ is invariant under translations, this condition reduces to $t^2<(1-s)s$, namely $2w<s$, for the point $\tau \in {\mathcal {F}}_{o'}$. Then, thanks to the asymptotics (C.21), we evaluate

$$\begin{aligned} s - 2w = \frac{\kappa \cos \psi }{3{\mathcal {V}}^2} e^{-2\pi \tau '_2} \bigl ( - {\mathcal {V}}_\psi - 2 {\mathcal {V}}^2 + o(1)\bigr ) \end{aligned}$$

(C.26)

Since $-{\mathcal {V}}_\psi \ge 1/2 > 2{\mathcal {V}}^2 \simeq 0.4283$, this is positive, which ensures that the attractor flow lies in $\mathbb {H}^{o'}=\mathbb {H}^{o}(1)$ and its DT invariants are correctly given by the quiver scattering diagram.

The translated attractor flow $\mu \mapsto \tau (\mu )-1\in {\mathcal {F}}_o$ tends to the $\tau =-1$ conifold point, hence

$$\begin{aligned} (x,y) \longrightarrow (x_{{\mathcal {O}}(-1)}, y_{{\mathcal {O}}(-1)}) = ({\mathcal {V}}_\psi - 1, {\mathcal {V}}_\psi - 1/2) \end{aligned}$$

(C.27)

where coordinates of the conifold point were calculated in (6.18). The corresponding (u, v) coordinates are

$$\begin{aligned} (u,v){} & {} = \biggl ( \frac{1}{12} - \frac{1}{2} x + y, - \frac{2x + 1}{4\sqrt{3}}\biggr ) \longrightarrow \biggl ( \frac{1}{12} + \frac{1}{2} {\mathcal {V}}_\psi , \frac{1 - 2{\mathcal {V}}_\psi }{4\sqrt{3}}\biggr ) \nonumber \\{} & {} = p_3({\mathcal {V}}_\psi ) = p_3(-|{\mathcal {V}}_\psi |) \end{aligned}$$

(C.28)

where $p_3$ was defined in (6.28) and we used $\psi <0$. Along the attractor flow, the quiver description is valid in a neighborhood of this point, so DT invariants of the exact ray ${\mathcal {R}}_\psi (\gamma )$ starting at $\tau =0$ coincide with those of the ray ${\mathcal {R}}_o(\gamma )$ starting at $p_3(-|{\mathcal {V}}_\psi |)$ in the orbifold diagram. In particular, there exists an active ray (with $\gamma \notin [1,0,0]\mathbb {Z}$) ending at $\tau =0$ if and only if $p_3(-|{\mathcal {V}}_\psi |)$ is a ray intersection. $\square$

1.5 C.5. Attractor flows starting at conifold points

To complete our description of the neighborhood of the conifold point, we now also study the attractor flows that emanate from $\tau =0$, namely $\mu \mapsto \tau (\mu )$ with $\mu \in (\mu _0,\mu _\infty )$ such that $\lim _{\mu \rightarrow \mu _0}\tau (\mu )=0$. We establish the equivalence of the characterizations (1) and (4) in Definition 2 by mapping such flows to rays passing through $p_1(-|{\mathcal {V}}_\psi |)$ in the quiver.

As before, the central charge must have a limit as $\mu \rightarrow \mu _0$, but now this limit must be non-zero (as its modulus should decrease along the flow). This rules out the case $d=0$ because $Z_{\tau =0}(\gamma )=-\text {ch}_2$ cannot be in the open half-line $\textrm{i}e^{\textrm{i}\psi }(0,+\infty )$. Thus, $d\ne 0$. By symmetry under $\psi \mapsto -\psi$, we focus on $\psi \in (-\pi /2,0]$.

Returning to the expansion (C.19) of the central charge, we again find that $\tau ' q'$ has a limit and that this limit must vanish to avoid a divergent phase. Thus, $Z_\tau (\gamma )\rightarrow \textrm{i}{\mathcal {V}}d-\text {ch}_2$ as $\mu \rightarrow \mu _0$. Along the flow, the central charge is fixed to lie in $Z_\tau (\gamma )\in \textrm{i}e^{\textrm{i}\psi }[0,+\infty )$ and to move towards 0 along this half-line, hence $(\textrm{i}{\mathcal {V}}d - \text {ch}_2) - Z_\tau (\gamma )$ lies in the same half-line. This implies (C.20) with a constant shift of $\tau '_1$,

$$\begin{aligned} d > 0, \qquad \frac{\text {ch}_2}{d} = {\mathcal {V}}_\psi , \qquad 2\pi \tau '_1 = -2\pi n + \psi + \pi + o(1) \end{aligned}$$

(C.29)

The integer $n\in \mathbb {Z}$ can be eliminated by a $\Gamma _1(3)$ transformation $V^n:\tau '\rightarrow \tau '+n$. Then $\tau '_1\in (1/4,1/2]$, namely $\tau '$ is in the closure ${\overline{{\mathcal {F}}}}_C$, which is stable under the Fricke involution, so $\tau \in {\overline{{\mathcal {F}}}}_C$. In addition, the sign of $\tau '_1$ yields $\Re \tau <0$, so that $\tau$ lies in the closure ${\overline{{\mathcal {F}}}}_o$ of the orbifold fundamental domain.

The expansions of s and w are then the opposites of (C.21), so that for $\psi \in (-\pi /2,0]$ we have $w,s\le 0$ close to the conifold point, hence the inequality $2w\le -s$ defining the orbifold region $\mathbb {H}^o$ within ${\mathcal {F}}_o$ is satisfied. DT invariants along the flow are thus correctly given by those of the orbifold diagram ${\mathcal {D}}_o$ at a suitable point (u, v). The affine coordinates (6.6) are

$$\begin{aligned} x = \frac{\Re ( e^{-\textrm{i}\psi } T)}{\cos \psi } = {\mathcal {V}}_\psi + o(1) = -|{\mathcal {V}}_\psi | + o(1),\quad y = -\frac{\Re ( e^{-\textrm{i}\psi } T_D)}{\cos \psi } = o(1).\nonumber \\ \end{aligned}$$

(C.30)

Thus, (C.28) holds as well, and the attractor flow starts (in the conifold limit $\mu \rightarrow \mu _0$) at the point $p_1(-|{\mathcal {V}}_\psi |)$ lying on the initial ray ${\mathcal {R}}_o(\gamma _1)$ of the quiver scattering diagram.

As in the previous section, we find that flows with $\gamma \notin [1,0,0]\mathbb {Z}$ that start (rather than end) at $\tau =0$ are given by rays of the quiver scattering diagram that end³⁵ (rather than start) at $(u,v)=p_1(-|{\mathcal {V}}_\psi |)$. By consistency of the orbifold scattering diagram, the points $p_1(-|{\mathcal {V}}_\psi |)$ of ${\mathcal {R}}_o$ with incoming rays or with outgoing rays are the same, and correspond by Proposition 8 to critical phases. This situation, in which both incoming and outgoing rays at $\tau =0$ occur for a critical phase, is illustrated in Fig. 8 for ${\mathcal {V}}_\psi \simeq -1/2$.

Since both incoming and outgoing rays at $\tau =0$ are seen in the orbifold diagram, it should be interesting to translate the consistency of the orbifold scattering diagram at $p_1(-|{\mathcal {V}}_\psi |)$ into a notion of consistency of the exact diagram ${\mathcal {D}}^\Pi _\psi$ at the conifold point, which is a singular point in the moduli space.

Appendix D. On the Mathematical Definition of DT Invariants

Here we provide mathematical details on the definition of the DT invariants $\Omega _\sigma (\gamma )$. These invariants are a direct generalisation of the integer BPS invariants of [72]. From [73], the objects of a smooth CY3 dg category ${\mathcal {C}}$ (like the dg category of perfect complexes with compact support on a smooth CY3-fold) form a $-1$-shifted symplectic derived stack ${\mathcal {M}}$ in the sense of [74]. We suppose that ${\mathcal {M}}$ admits an orientation, i.e. a square root of the line bundle $\,\textrm{det}\, ({\mathbb {L}}_{\mathcal {M}})$ given by the determinant of the cotangent complex of ${\mathcal {M}}$ (a canonical orientation was constructed in the case of sheaves with compact support on noncompact CY3-fold in [75, Theorem 4.9]). For $\sigma$ a stability condition on ${\mathcal {C}}$, $\sigma$-semistability is a Zariski open condition, hence from [76, Proposition 2.1] there is an open $-1$-shifted symplectic substack ${\mathcal {M}}_\sigma \hookrightarrow {\mathcal {M}}$ of $\sigma$-semistable objects. From the definition of semistability, ${{\,\textrm{Ext}\,}}^i(E,E)=0$ for any $E\in {\mathcal {M}}_\sigma$ and $i<0$, hence by [73, Proposition 3.3] ${\mathbb {T}}_{{\mathcal {M}}_\sigma }|_E={{\,\textrm{Ext}\,}}(E,E)[1]$, ${\mathcal {M}}$ is a $-1$-shifted symplectic Artin–1 stack. We then define ${\mathcal {M}}_\sigma (\gamma )$ as the component of ${\mathcal {M}}_\sigma$ of objects of class $\gamma$ in the Grothendieck group of ${\mathcal {C}}$. Suppose now that $\sigma$ is generic, i.e. that two $\sigma$-semistable objects $E,E'$ of the same phase have collinear charges. For $\gamma$ primitive, we define the DT invariants $\Omega _\sigma (k\gamma ),k\ge 1$ by

$$\begin{aligned} {{\,\textrm{Exp}\,}}\biggl (\sum _{k=1}^\infty \frac{\Omega _\sigma (k\gamma )}{y^{-1}-y}x^k\biggr ):=\sum _{k=0}^\infty H_c({\mathcal {M}}_\sigma (k\gamma ),P_{{\mathcal {M}}_\sigma (k\gamma )})x^k \end{aligned}$$

(D.1)

where ${{\,\textrm{Exp}\,}}$ denotes the plethystic exponential, $P_{{\mathcal {M}}_\sigma (k\gamma )}$ the monodromic mixed Hodge module on ${\mathcal {M}}_\sigma (k\gamma )$ constructed in [77, Theorem 4.4] using the orientation data, and $H_c(M,P)$ the Hodge polynomial of the cohomology with compact support on M with values in P. In the case of quiver with potentials and King stability conditions, the invariants $\Omega _\sigma (k\gamma )$ are integer for any $\gamma$ and $k\ge 1$ by [72], and we conjecture that this remains true in this more general framework. In particular, we conjecture that, as in [72],

$$\begin{aligned} \Omega _\sigma (k\gamma )=H_c\bigl (M_\sigma (k\gamma ),{\mathcal {H}}^1({{\,\textrm{JH}\,}}_!P_{{\mathcal {M}}_\sigma (k\gamma )})\bigr ) \end{aligned}$$

(D.2)

where ${{\,\textrm{JH}\,}}:{\mathcal {M}}_\sigma (k\gamma )\rightarrow M_\sigma (k\gamma )$ denotes the Jordan-Hölder map to the coarse moduli space, ${{\,\textrm{JH}\,}}_!$ denotes the proper pushforward for the derived categories of monodromic mixed Hodge modules, and ${\mathcal {H}}^1$ denotes the first cohomology of a complex of monodromic mixed Hodge modules.

Appendix E. Gieseker Indices for Higher Rank Sheaves

In this section, we extend the list of examples in Sect. 4 and determine the trees contributing to the Gieseker index for some examples with higher rank. As explained in [56, 78], for (r, d) coprime and discriminant $\Delta (\gamma )\ge \Delta _1(r,d)$ large enough, the Gieseker wall is ${\mathcal {W}}(\gamma ,\gamma ')$ due to a subobject with Chern vector $\gamma '=[r',d',\chi ')$ uniquely determined by the following conditions:

• Whenever $0 < r' \le r,$    $\mu (\gamma ')<\mu (\gamma )$

• Every rational number in the interval $(\mu (\gamma '),\mu (\gamma ))$ has denominator greater than r

• The discriminant of any stable bundle of slope $\mu (\gamma ')$ and rank $\gamma ' \le r$ is $\ge \Delta (\gamma ')$

• The rank of any stable bundle of slope $\mu (\gamma ')$ and discriminant $\Delta (\gamma ')$ is $\ge r'$

The minimal value $\Delta _1(r,d)$ for which conditions are applicable and the rightmost point $x_+=s_{\gamma ,\gamma '}+R_{\gamma ,\gamma '}$ of the Gieseker wall for the lowest discriminant $\Delta _0\ge \Delta _1$ are tabulated in [78, Table 3] for $r\le 6$ and $0<\mu (\gamma )\le 1$.

1.1 E.1. Rank 2

We consider rank 2 sheaves with $\gamma =[2,-1,1-n)$, discriminant $\Delta =\frac{n}{2}-\frac{1}{8}$. The condition (2.10) gives $\Delta \ge \delta _{\textrm{LP}}(-\frac{1}{2})=\frac{5}{8}$ for non-exceptional sheaves. The generating function of Gieseker indices is given by [79] [23, (A.38)]

$$\begin{aligned} \begin{aligned} h_{2,-1}&= q + (y^2+1+1/y^2)^2 q^2 + (y^8+2y^6+6y^4+9y^2+12+\dots ) q^3 \\&\quad + \left( y^{12}+2 y^{10}+6 y^8+13 y^6+24 y^4+35 y^2+41+\dots \right) q^4+ \dots \\&\rightarrow q + 9 q^2 + 48 q^3 +203 q^4 + 729 q^5 + 2346 q^6 + \dots \end{aligned} \end{aligned}$$

(E.1)

• For $n=1$, corresponding to the exceptional sheaf $\Omega (1)$, there is a single wall ${\mathcal {C}}(-\frac{3}{2},\frac{1}{2})$ associated to the scattering sequence $\{-{\mathcal {O}}(-2), 3{\mathcal {O}}(-1)\}$ contributing $K_3(1,3)=1$.

• For $n=2$ there is a single wall ${\mathcal {C}}(-\frac{5}{2},\frac{3}{2})$ associated to $\{\{-{\mathcal {O}}(-3), {\mathcal {O}}(-2)\}, 2{\mathcal {O}}(-1)\}$ contributing $K_3(1,1)K_3(1,2)=(y^2+1+1/y^2)^2$. The rightmost point on the wall is at $s=-1$, consistent with the entries $\Delta _0=\frac{7}{8}, x_+=0$ in [78, Table 3].

• For $n=3$, there is a single wall associated to two scattering sequences

  $$\begin{aligned} \begin{array}{lll} {\mathcal {C}}(-\frac{7}{2},\frac{5}{2}) &{}\{\{-{\mathcal {O}}(-4),{\mathcal {O}}(-3)\},2{\mathcal {O}}(-1)\} &{} K_3(1,1) K_3(1,2) \\ &{} \{\{ -2{\mathcal {O}}(-3),3{\mathcal {O}}(-2)\},{\mathcal {O}}(-1)\} &{} K_3(1,1) K_3(2,3) \end{array} \end{aligned}$$

  contributing $9+39=48$ in the unrefined limit.

• For $n=4$, there are 2 walls associated to four scattering sequences

  $$\begin{aligned} \begin{array}{lll} {\mathcal {C}}(-\frac{9}{2},\frac{7}{2})&{} \{\{-{\mathcal {O}}(-5), {\mathcal {O}}(-4)\}, 2 {\mathcal {O}}(-1)\} &{} K_3(1,1) K_3(1,2) \\ &{} \{\{\{-{\mathcal {O}}(-4), {\mathcal {O}}(-3)\}, \{-{\mathcal {O}}(-3), 2 {\mathcal {O}}(-2)\}\}, {\mathcal {O}}(-1)\} &{} K_3(1,1)^2 K_3(1,2) \\ &{} \{\{-{\mathcal {O}}(-4), 2 {\mathcal {O}}(-2)\}, {\mathcal {O}}(-1)\} &{} K_3(1,1) K_6(1,2) \\ {\mathcal {C}}(-\frac{7}{2},\frac{1}{2}) &{} \{-3 {\mathcal {O}}(-3), 5 {\mathcal {O}}(-2)\} &{} K_3(3,5) \end{array} \end{aligned}$$

  contributing $9+81+45+ 68=203$ in the unrefined limit.

For $\gamma =[2,0,2-n)$, with discriminant $\Delta =n/2$, the expected generating function is

$$\begin{aligned} \begin{aligned} h_{2,0}&= -(y^5+y^3+y+\dots ) q^2 - (y^9+2y^7+4y^5+6y^3+6y+\dots ) q^3 \\&\quad - \left( y^{13} + 2 y^{11} + 6 y^9 + 11 y^7 + 19 y^5 + 24 y^3 + 27 y+\dots \right) q^4 - \dots \\&\rightarrow -6 q^2 -38 q^3 - 180 q^4 - 678 q^5- 2260q^6 - \dots \end{aligned} \end{aligned}$$

(E.2)

The condition (2.10) gives $\Delta \ge \delta _{\textrm{LP}}(0)=1$ for non-exceptional sheaves.

• For $n=2$ there is a single wall ${\mathcal {C}}(-\frac{3}{2},\frac{1}{2})$ associated to $\{-2{\mathcal {O}}(-2), 4{\mathcal {O}}(-1)\}$ contributing $K_3(2,4)\rightarrow -6$.

• For $n=3$ there are two walls

  $$\begin{aligned} \begin{array}{lll} {\mathcal {C}}(-\frac{5}{2},\frac{\sqrt{13}}{2}) &{}\{-{\mathcal {O}}(-3),3{\mathcal {O}}(-1)\} &{} K_6(1,3)\\ {\mathcal {C}}(-2,1) &{}\{\{ -{\mathcal {O}}(-3),{\mathcal {O}}(-2)\},\{-{\mathcal {O}}(-2),3{\mathcal {O}}(-1)\}\} &{} K_3(1,1) K_3(1,3) K_6(1,1) \end{array} \end{aligned}$$

  contributing $-20-18=-38$ in the unrefined limit. The Gieseker wall has rightmost point at $s=-\frac{5}{2}+\frac{\sqrt{13}}{2}$, consistent with the values $\Delta _0=\frac{3}{2}, x_+\simeq 0,30$ in [78, Table 3].

• For $n=4$ there are two walls

  $$\begin{aligned} \begin{array}{lll} {\mathcal {C}}(-\frac{7}{2},\frac{\sqrt{33}}{2}) &{} \{\{-{\mathcal {O}}(-4), {\mathcal {O}}(-3)\}, \{-{\mathcal {O}}(-2), 3 {\mathcal {O}}(-1)\}\} &{} K_3(1,1) K_3(1,3) K_6(1,1) \\ {\mathcal {C}}(-\frac{5}{2},\frac{3}{2}) &{} \{\{-2 {\mathcal {O}}(-3), 2 {\mathcal {O}}(-2)\}, 2 {\mathcal {O}}(-1)\} &{} K_{-3,3,6}(2,2,2) \end{array} \end{aligned}$$

  The index for the second scattering sequence is obtained in the same way as in (4.23), i.e. by applying the flow tree formula for 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 $\Omega ^-(\beta )=1$, and selecting the outgoing ray of charge $2\alpha +2\beta$. This leads to

  $$\begin{aligned} K_{-3,3,6}(2,2,2){} & {} =-y^{13} - 2 y^{11} - 6 y^9 - 10 y^7 - 17 y^5 - 21 y^3 - 24 y- \dots \nonumber \\{} & {} \quad \rightarrow -162 \end{aligned}$$

  (E.3)

  Adding up the contributions of the two scattering sequences, we get $-18-162=-180$ in the limit $y\rightarrow 1$, in agreement with (E.2).

1.2 E.2. Rank 3

We now turn to rank 3 sheaves, with $\gamma =[3,-1,2-n)$, discriminant $\Delta =\frac{n}{3}-\frac{1}{9}$. The condition (2.10) gives $\Delta \ge \delta _{\textrm{LP}}(-\frac{1}{3})=\frac{5}{9}$ for non-exceptional sheaves. The generating function is given by [80, Table 1] [23, (A.40)]

$$\begin{aligned} \begin{aligned} h_{3,-1}&= (y+1+1/y^2) q^2 + (y^8+2y^6+5y^4+8y^2+10+\dots ) q^3 + \dots \\&\rightarrow 3q^2 + 42 q^3 + 333 q^4 + 1968 q^5 + 9609 q^6+ \dots \end{aligned} \end{aligned}$$

(E.4)

• For $n=2$ there is a single wall ${\mathcal {C}}(-\frac{3}{2},\frac{1}{2})$ associated to $\{-2{\mathcal {O}}(-2), 5{\mathcal {O}}(-1)\}$ contributing $K_3(2,5)=y^2+1+1/y^2$.

• For $n=3$ there are three walls:

  $$\begin{aligned} \begin{array}{lll} {\mathcal {C}}(-\frac{7}{2},\frac{33}{2})&{}\{ \{\{-{\mathcal {O}}(-3), {\mathcal {O}}(-2)\}, {\mathcal {O}}(-1)\},\{-{\mathcal {O}}(-2), 3 {\mathcal {O}}(-1)\}\} &{} K_3(1,1)^3 K_3(1,3)\\ {\mathcal {C}}(-\frac{5}{2},\frac{\sqrt{35}}{2\sqrt{3}})&{}\{\{ -{\mathcal {O}}(-3),{\mathcal {O}}(-2)\},\{-{\mathcal {O}}(-2),4{\mathcal {O}}(-1)\}\} &{} K_3(1,1) K_3(1,4) K_9(1,1)\\ {\mathcal {C}}(-2,1)&{}\{{\mathcal {O}}(-3),4{\mathcal {O}}(-1)\} &{} K_6(1,4) \end{array} \end{aligned}$$

  contributing $27+0+15=42$ in the unrefined limit. The Gieseker wall ${\mathcal {C}}(-\frac{7}{2},\frac{33}{2})$ has rightmost point at $-\frac{5}{2}+\frac{1}{2} \sqrt{33}$, consistent with the values $\Delta _0=\frac{8}{9}, x_+\simeq 0,37$ quoted in [78, Table 3].

For $\gamma =[3,1,5-n)$ with $\Delta =\frac{n}{3}-\frac{1}{9}$, we find instead the following scattering sequences (not related to the previous ones by reflection)

• For $n=1$, there is a scattering sequence $\{-{\mathcal {O}}(-1),4{\mathcal {O}}\}$ but its index $K_3(1,4)$ vanishes.

• For $n=2$ there is a single wall

  $$\begin{aligned} {\mathcal {C}}(-\tfrac{3}{2},\tfrac{3}{2}) \quad \{\{-{\mathcal {O}}(-2), {\mathcal {O}}(-1)\},3{\mathcal {O}}(0)\} \quad K_3(1,3) K_3(1,1) \end{aligned}$$

  contributing $y^2+1+1/y^2$.

• For $n=3$ there is a single wall but two scattering sequences,

  $$\begin{aligned} \begin{array}{lll} {\mathcal {C}}(-\frac{5}{2},\frac{5}{2}) &{}\{ \{ -2{\mathcal {O}}(-2), 3{\mathcal {O}}(-1)\}, 2{\mathcal {O}}\} &{} K_3(1,2) K_3(2,3)\\ &{}\{\{ -{\mathcal {O}}(-3),{\mathcal {O}}(-2)\},3{\mathcal {O}}\} &{} K_3(1,1) K_3(1,3) \end{array} \end{aligned}$$

  The wall ${\mathcal {C}}(-\frac{5}{2},\frac{5}{2})$ has rightmost point at $x_+=0$, consistent with the values $\Delta _0=\frac{8}{9}, x_+=0$ quoted in [78, Table 3].

For $\gamma =[3,0,3-n)$ with discriminant $\Delta =n/3$, the generating function is [23, (A.40)]

$$\begin{aligned} \begin{aligned} h_{3,0}&= (y^{10}+y^8+2y^6+2y^4+2y^2+2+\dots ) q^3 \\&\quad + \left( y^{16}+2 y^{14}+5 y^{12}+9 y^{10}+15 y^8+19 y^6+22 y^4+23 y^2+24+\dots \right) q^4 + \dots \\&\rightarrow 18 q^3 +216 q^4+1512 q^5 + 8109 q^6 \dots \end{aligned}\nonumber \\ \end{aligned}$$

(E.5)

• For $n=3$, there is a single wall ${\mathcal {C}}(-\frac{3}{2},\frac{1}{2})$ associated to $\{-3{\mathcal {O}}(-2),6{\mathcal {O}}(-1)\}$ contributing $K_3(3,6) \rightarrow 18$.

• For $n=4$, there are two walls

  $$\begin{aligned} \begin{array}{lll} {\mathcal {C}}(-\frac{5}{2},\frac{\sqrt{43}}{2\sqrt{3}}) &{} \{\{-{\mathcal {O}}(-3), {\mathcal {O}}(-2)\}, \{-2 {\mathcal {O}}(-2), 5 {\mathcal {O}}(-1)\}\} &{} K_3(1,1) K_3(2,5) K_9(1,1) \\ {\mathcal {C}}(-\frac{13}{6},\frac{\sqrt{73}}{6}) &{} \{\{-{\mathcal {O}}(-3), 2 {\mathcal {O}}(-1)\}, \{-{\mathcal {O}}(-2), 3 {\mathcal {O}}(-1)\}\} &{} K_3(1,3) K_6(1,2) K_9(1,1) \end{array} \end{aligned}$$

  contributing $81+135=216$ in the unrefined limit. The Gieseker wall has rightmost point $-\frac{5}{2}+\frac{1}{2}\sqrt{\frac{43}{3}}= x_+-1$, consistent with the values $\Delta _0=\frac{4}{3}, x_+\simeq 0,39$ quoted in [78, Table 3].

Appendix F. Mathematica Package P2Scattering.m

The Mathematica package P2Scattering.m, available from

https://github.com/bpioline/P2Scattering

provides a suite of routines for analyzing the scattering diagrams considered in this work, both at large volume, around the orbifold and along the $\Pi$-stability slice. It was used extensively in order to generate the figures and arrive at the global picture presented in this article. A list of routines is provided in the documentation P2Scattering.pdf available in the GitHub repository, along with several demonstration worksheets. Here we simply give a taste of the package capabilities.

After copying file P2Scattering.m in the current directory, load the package via

For a given charge $\gamma =[r,d,\chi )$ and point (s, t) on the large volume slice, the scattering sequences contributing to the index $\Omega _{(s,t)}(\gamma )$ can be found by using the routine , for example for $\gamma =[3,0,0)$ through the point $(s,t)=(-\frac{3}{2},2)$,

reproducing the GV invariant $N_3^{(0)}=27$ (compare with Sect. 4.4). Note that the current implementation of the routine assumes that the index associated to each scattering sequence is a product of Kronecker indices associated to each vertex, and may give the wrong result if some of the edges carry non-primitive charges (see (4.23) for an example). In the case above, it does produce the correct results for both scattering sequences, $\Omega _\infty (\gamma )=K_9(1,1)+K_3(3,3)$. More generally, the routine computes the total rational index ${\bar{\Omega }}_{s,t}(\gamma )$ by decomposing each scattering sequence into attractor flow trees as explained at the end of Sect. 4.2, and perturbing the charges of the constituents $\gamma _i\rightarrow \gamma _i+\epsilon _i \delta$ such that only binary splittings remain:

consistent with ${\bar{\Omega }}_{s,t}(3\gamma )= \Omega _{s,t}(3\gamma )+\frac{1}{9} \Omega _{s,t}(\gamma )=27+\frac{1}{3}$.

Similarly, one can find the scattering sequences contributing near the orbifold point using : for the same charge, corresponding to dimension vector (0, 3, 6), a single scattering sequence contributes in the anti-attractor chamber, with index 18,

In this example, the index in the anti-attractor chamber differs from the one at large volume, due to wall-crossing along the circle ${\mathcal {C}}(-\frac{3}{2},\frac{1}{2})$.