1.1 Background

A quiver with potential $(Q,W)$ is given by a finite oriented graph $Q$, and a finite formal linear combination $W$ of oriented cycles in $Q$. We assume that $Q$ does not contain oriented $2$-cycles, and we denote by $Q_0$ the set of vertices of $Q$. For every dimension vector $\gamma \in N := \mathbb {Z}^{Q_0}$ and stability parameter

(1.1)\begin{equation} \theta \in \gamma^{\perp} \subset M_\mathbb{R} := \operatorname{Hom}(N,\mathbb{R}), \end{equation}

where $\gamma ^{\perp } := \{\theta \in M_\mathbb {R}\,|\,\theta (\gamma )=0\}$, the theory of King's stability for quiver representations [Reference KingKin94] defines a quasiprojective variety $M_\gamma ^{\theta }$, parametrizing S-equivalence classes of $\theta$-semistable representations of $Q$ of dimension $\gamma$, and a regular function

(1.2)\begin{equation} \operatorname{Tr} (W)_\gamma^\theta \colon M_\gamma^\theta \longrightarrow \mathbb{C}. \end{equation}

Assuming that $\theta$ is $\gamma$-generic in the sense that $\theta (\gamma ')=0$ implies $\gamma '$ collinear with $\gamma$, the DT invariant $\Omega _\gamma ^{\theta }$ is an integer which is a virtual count of the critical points of $\operatorname {Tr} (W)_\gamma ^\theta$. Applying Hodge theory to the sheaf of vanishing cycles of $\operatorname {Tr} (W)_\gamma ^\theta$, the integer $\Omega _\gamma ^\theta$ can be refined into a Laurent polynomial $\Omega _\gamma ^\theta (y,t)$ in two variables $y$ and $t$ and with integer coefficients, referred to as refined DT invariants [Reference Joyce and SongJS12, Reference Kontsevich and SoibelmanKS08, Reference ReinekeRei11, Reference ReinekeRei10, Reference Meinhardt and ReinekeMR19, Reference Davison and MeinhardtDM15, Reference Davison and MeinhardtDM20]. It is often convenient to use the rational functions $\overline {\Omega }_\gamma ^\theta (y,t) \in \mathbb {Q}(y,t)$ defined as in [Reference Joyce and SongJS12, Reference Kontsevich and SoibelmanKS08, Reference Manschot, Pioline and SenMPS11] by

(1.3)\begin{equation} \overline{\Omega}_{\gamma}^\theta(y,t):= \sum_{\substack{\gamma' \in N\\ \gamma=k \gamma',\ k\in \mathbb{Z}_{\geq 1}}} \frac{1}{k} \frac{y-y^{-1}}{y^k - y^{-k}} \Omega_{\gamma'}^{\theta}(y^k,t^k), \end{equation}

and referred to as rational DT invariants.

The DT invariants $\Omega _\gamma ^\theta (y,t)$ are locally constant functions of the $\gamma$-generic stability parameter $\theta \in \gamma ^{\perp }$ and their jumps across the loci of non-$\gamma$-generic stability parameters are given by the wall-crossing formula of Joyce and Song [Reference Joyce and SongJS12] and Kontsevich and Soibelman [Reference Kontsevich and SoibelmanKS08]. Using the wall-crossing formula, the DT invariants can be computed in terms of the simpler attractor DT invariants, which are DT invariants at specific values of the stability parameter.

Let $\langle -,-\rangle \colon N \times N \rightarrow \mathbb {Z}$ be the skew-symmetric form given by

(1.4)\begin{equation} \langle \gamma, \gamma'\rangle = \sum_{i,j\in Q_0}(a_{ij}-a_{ji})\gamma_i\gamma_j', \end{equation}

where $a_{ij}$ is the number of arrows in $Q$ from the vertex $i$ to the vertex $j$. The specific point $\langle \gamma,- \rangle \in \gamma ^{\perp } \subset M_{\mathbb R }$ is called the attractor point for $\gamma$ (see [Reference Alexandrov and PiolineAP19, Reference Mozgovoy and PiolineMP20]). In general, the attractor point $\langle \gamma,-\rangle$ is not $\gamma$-generic and we define the attractor DT invariants $\Omega _\gamma ^{*}(y,t)$ by

(1.5)\begin{equation} \Omega_\gamma^{*}(y,t):= \Omega_\gamma^{\theta_\gamma}(y,t), \end{equation}

where $\theta _\gamma$ is a small $\gamma$-generic perturbation of $\langle \gamma,- \rangle$ in $\gamma ^{\perp }$ (see [Reference Alexandrov and PiolineAP19, Reference Mozgovoy and PiolineMP20]). One can check that $\Omega _\gamma ^{*}(y,t)$ is independent of the choice of the small perturbation [Reference Alexandrov and PiolineAP19, Reference Mozgovoy and PiolineMP20].

For an acyclic quiver $Q$ (and so $W=0$), or more generally for a quiver $Q$ with a non-degenerate potential $W$ admitting a green-to-red sequence [Reference MouMou19], the attractor DT invariants are as simple as possible:

(1.6)\begin{equation} \Omega_\gamma^{*}(y,t)= \begin{cases} 1 & \mathrm{if}\ \gamma=(\delta_{ij})_{i\in Q_0}\ \mathrm{for\ some}\ j \in Q_0 \\ 0 & \mathrm{otherwise}, \end{cases} \end{equation}

where $\delta _{ij}$ is the Kronecker delta. Similarly, for a quiver with potential $(Q,W)$ describing the derived category of coherent sheaves on a local del Pezzo surface, it has recently been conjectured [Reference Beaujard, Manschot and PiolineBMP21, Reference Mozgovoy and PiolineMP20] that $\Omega _\gamma ^{*}(y,t)=0$ unless $\gamma =(\delta _{ij})_{i\in Q_0}$ for some $j \in Q_0$ or unless $\gamma$ is the class of the skyscraper sheaf of a point. However, for quivers with potential $(Q,W)$ describing interesting parts of the derived category of coherent sheaves on a compact Calabi–Yau 3-fold, the attractor DT invariants are expected to be non-vanishing and to typically exhibit an exponential growth. We refer to [Reference Denef and MooreDM11, Reference Manschot, Pioline and SenMPS12, Reference Lee, Wang and YiLWY12b, Reference Lee, Wang and YiLWY12a, Reference Bena, Berkooz, de Boer, El-Showk and Van den BleekenBBdB⁺12] for some explicit examples involving $n$-gon quivers.

The rational DT invariants $\overline {\Omega }_\gamma ^{\theta }(y,t)$ for general $\gamma$-generic stability parameters $\theta \in \gamma ^{\perp }$ are expressed in terms of the rational attractor DT invariants $\overline {\Omega }_\gamma ^{*}(y,t)$ by a formula of the form

(1.7)\begin{equation} \overline{\Omega}_\gamma^\theta(y,t) = \sum_{r\geq 1} \sum_{\substack{\{\gamma_i\}_{1\leq i\leq r}\\ \sum_{i=1}^r \gamma_i = \gamma}} \frac{1}{|\operatorname{Aut}(\{\gamma_i\}_i)|} F_r^{\theta}(\gamma_1,\ldots,\gamma_r) \prod_{i=1}^r \overline{\Omega}_{\gamma_i}^{*}(y,t), \end{equation}

where the second sum is over the multisets $\{\gamma _i\}_{1\leq i\leq r}$ with $\gamma _i \in N$ and $\sum _{i=1}^r \gamma _i=\gamma$. Here, the denominator $|\operatorname {Aut}(\{\gamma _i\}_i)|$ is the order of the symmetry group of $\{\gamma _i\}$: if $m_{\gamma '}$ is the number of times that $\gamma ' \in N$ appears in $\{\gamma _i\}_i$, then $|\operatorname {Aut}(\{\gamma _i\}_i)|=\prod _{\gamma '\in N}m_{\gamma '}!$. The coefficients $F_r^{\theta }(\gamma _1,\ldots,\gamma _r)$ are element of $\mathbb {Q}(y)$ and are universal in the sense that they depend on $(Q,W)$ only through the skew-symmetric form $\langle -,-\rangle$ on $N$. Our main result is the proof of an explicit formula, called the flow tree formula and conjectured by Alexandrov and Pioline [Reference Alexandrov and PiolineAP19], which computes the coefficients $F_r^{\theta }(\gamma _1,\ldots,\gamma _r)$ in (1.7) combinatorially in terms of a sum over binary rooted trees, and where the contribution of each tree is computed following the flow on the tree starting at the root and ending at the leaves.

1.2 Main result: the flow tree formula

We introduce some notation which is necessary to state precisely the flow tree formula in Theorem 1.1. We fix $\gamma \in N$, a $\gamma$-generic stability parameter $\theta \in \gamma ^{\perp }$, and $\gamma _1,\ldots,\gamma _r \in N$ such that $\sum _{i=1}^r \gamma _i=\gamma$.

An essential ingredient in the formulation of the flow tree formula for $F_r^\theta (\gamma _1,\ldots,\gamma _r)$ is the choice of a generic skew-symmetric perturbation $(\omega _{ij})_{1\leq i,j\leq r}$ of the skew-symmetric matrix $(\langle \gamma _i,\gamma _j\rangle )_{1\leq i,j \leq r}$. The matrix $(\omega _{ij})_{1\leq i,j\leq r}$ cannot be viewed in general as a skew-symmetric bilinear form on the sublattice of $N$ generated by $\gamma _1,\ldots,\gamma _r$ because $\gamma _1,\ldots,\gamma _r$ are not necessarily linearly independent in $N$. Nevertheless, the matrix $(\omega _{ij})_{1\leq i,j \leq r}$ can always be interpreted as a skew-symmetric bilinear form $\omega$ on a rank $r$ free abelian group $\mathcal {N} := \bigoplus _{i=1}^r \mathbb {Z} e_i$ with a basis $\{e_i\}_{1\leq i\leq r}$ and such that $\omega _{ij}=\omega (e_i,e_j)$. From this point of view, there is a natural additive map

(1.8)\begin{align} \begin{aligned} p \colon \mathcal{N} \longrightarrow N \nonumber\\ e_i \longmapsto \gamma_i, \end{aligned} \end{align}

which enables us to define a skew-symmetric bilinear form $\eta$ on $\mathcal {N}$ as being the pullback of $\langle -,-\rangle$ on $N$, that is, $\eta (e_i,e_j) := \langle \gamma _i,\gamma _j\rangle$, and we consider a real-valued skew-symmetric form $\omega$ on $\mathcal {N}$ obtained as a small enough generic perturbation of $\eta$. Let $\mathcal {M}_\mathbb {R} := \operatorname {Hom}(\mathcal {N},\mathbb {R})$ and $q \colon M_\mathbb {R} \rightarrow \mathcal {M}_\mathbb {R}$ be the map induced from $p \colon \mathcal {N} \rightarrow N$ by duality. We denote by

(1.9)\begin{equation} \alpha := q(\theta) \end{equation}

the image in $\mathcal {M}_\mathbb {R}$ of the stability parameter $\theta \in M_\mathbb {R}$ by the map $q$.

The flow tree formula in Theorem 1.1 takes the form of a sum over trees. More precisely, we consider rooted trees which apart from the root vertex have $r$ univalent vertices, or leaves, decorated by the basis elements $e_1,\ldots,e_r$ of $\mathcal {N}$. For such a tree $T$, we denote by $V_T^{\circ }$ the set of interior, that is, non-univalent, vertices. We endow each such tree with the flow from the root to the leaves. Given a vertex $v$ in a tree, the vertex adjacent to $v$ coming before $v$ along the flow is referred to as the parent of $v$ and denoted by $p(v)$, and the vertices adjacent to $v$ and coming after $v$ along the flow are referred to as the children of $v$, as illustrated in Figure 1.1. Any vertex that comes after $v$ along the flow is a descendent of $v$. Let $\mathcal {T}_r$ be the set of such trees which are binary, that is such that each interior vertex $v$ of a tree $T \in \mathcal {T}_r$ has exactly two children. For every tree $T \in \mathcal {T}_r$ and $v$ a vertex of $T$, we define $e_v \in \mathcal {N}$ as the sum of all elements that appear as decorations on the leaves which are descendent of a vertex $v$. We denote by $\mathcal {T}_r^{\eta }$ the set of trees $T \in \mathcal {T}_r$ such that $\eta (e_{v'},e_{v''}) \neq 0$ where $v$ is the child of the root and $v',v''$ are the children of $v$.

Figure 1.1.

A binary tree $T$ with five leaves for $\mathcal {N}=\mathbb {Z} e_1 \oplus \mathbb {Z} e_2 \oplus \mathbb {Z} e_3 \oplus \mathbb {Z} e_4 \oplus \mathbb {Z} e_5$.

For every tree $T \in \mathcal {T}_r^\eta$ and $v$ a vertex of $T$ distinct from the leaves, we define $\theta _{T,v}^{\alpha,\omega } \in \mathcal {M}_\mathbb {R}$ recursively as follows: if $v$ is the root vertex, then set $\theta _{T,v}^{\alpha,\omega }:= \alpha$. If $v$ is not the root, let $p(v)$ be the parent of $v$, and for any of the children, say $v'$ of $v$, and $\iota _{e_v} \omega := \omega (e_v,-) \in \mathcal {M}_{\mathbb {R}}$, define

(1.10)\begin{equation} \theta_{T,v}^{\alpha,\omega} :=\theta_{T,p(v)}^{\alpha,\omega} -\frac{\theta_{T,p(v)}^{\alpha,\omega}(e_{v'})}{ \omega(e_v,e_{v'})} \iota_{e_v} \omega. \end{equation}

We show in Lemma 2.12 that this definition is independent of the choice of the child $v'$ of $v$. Following [Reference Alexandrov and PiolineAP19], we call $v \mapsto \theta _{T,v}^{\alpha,\omega }$ the discrete attractor flow.

For every tree $T \in \mathcal {T}_r^\eta$ and interior vertex $v \in V_T^\circ$, we fix a labeling $v'$ and $v''$ of the two children of $v$, and we define

(1.11)\begin{equation} \epsilon_{T,v}^{\alpha, \omega} := - \frac{\operatorname{sgn} (\theta_{T,p(v)}^{\alpha,\omega}(e_{v'})) + \operatorname{sgn} (\omega(e_{v'}, e_{v''}))}{2} \in \{0,1,-1\}, \end{equation}

where $\operatorname {sgn}(x) \in \{\pm 1\}$ is the sign of $x \in \mathbb {R}-\{0\}$. We show in § 2 that for generic $\omega \in \bigwedge ^2\mathcal {M}_{\mathbb R }$, we have $\theta _{T,p(v)}^{\alpha,\omega }(e_{v'}) \neq 0$ and $\omega (e_{v'}, e_{v''}) \neq 0$ and so the definition of $\epsilon _{T,v}^{\alpha,\omega }$ indeed makes sense. Our main result is the following flow tree formula, conjectured in [Reference Alexandrov and PiolineAP19], which enables us to determine the coefficients $F_r^{\theta }(\gamma _1,\ldots,\gamma _r)$ in (1.7) expressing the DT invariants $\overline {\Omega }_\gamma ^{\theta }(y,t)$ in terms of the attractor DT invariants $\overline {\Omega }_{\gamma _i}^{*}(y,t)$.

Theorem 1.1 For every choice a small enough generic perturbation $\omega \in \bigwedge ^2 \mathcal {M}_\mathbb {R}$ of the skew-symmetric bilinear form $\eta$, the universal coefficient $F_r^{\theta }(\gamma _1,\ldots,\gamma _r)$ in ( 1.7) is given by the flow tree formula:

(1.12)\begin{equation} F_r^\theta(\gamma_1,\ldots,\gamma_r)= \sum_{T \in \mathcal{T}_r^{\eta}} \prod_{v \in V_T^\circ} \epsilon_{T,v}^{\alpha,\omega} \kappa(\eta(e_{v'},e_{v''})), \end{equation}

where $\epsilon _{T,v}^{\alpha,\omega }$ is as in ( 1.11) and

(1.13)\begin{equation} \kappa(x):= {(-1)^x} \cdot \frac{y^x-y^{-x}}{y-y^{-1}} \end{equation}

for every $x \in \mathbb {Z}$.

Theorem 5.5 presents a version of Theorem 1.1 in which we phrase more explicitly the condition that $\omega$ should be a small enough generic perturbation of $\eta$.

We also prove a variant of the flow tree formula recently conjectured by Mozgovoy [Reference MozgovoyMoz22], which relies on a perturbation of points in $\mathcal {M}_{\mathbb R }$ rather than the skew-symmetric form. We first remark that $\theta \in \gamma ^{\perp }$ implies that $\alpha \in \mathcal {M}_\mathbb {R}$ defined in (1.9) satisfies $\alpha \in (\sum _{i=1}^r e_i)^{\perp }$. For $\beta$ a small perturbation of $\alpha$ in the hyperplane $(\sum _{i=1}^r e_i)^{\perp }$, we define $\theta _{T,v}^{\beta,\eta } \in \mathcal {M}_{\mathbb R }$ and $\epsilon _{T,v}^{\beta,\eta } \in \{0,1,-1\}$ by replacing $\alpha$ by $\beta$ and $\omega$ by $\eta$ in (1.10) and (1.11).

Theorem 1.2 For every choice $\beta \in (\sum _{i=1}^r e_i)^{\perp }$ of small enough generic perturbation of $\alpha := q(\theta )$ in the hyperplane $(\sum _{i=1}^r e_i)^{\perp }$, the universal coefficient $F_r^{\theta }(\gamma _1,\ldots,\gamma _r)$ is given by

(1.14)\begin{equation} F_r^\theta(\gamma_1,\ldots,\gamma_r)= \sum_{T \in \mathcal{T}_r^{\eta}} \prod_{v \in V_T^\circ} \epsilon_{T,v}^{\beta,\eta} \kappa(\eta(e_{v'},e_{v''})), \end{equation}

where $\epsilon _{T,v}^{\alpha,\omega }$ is as in (1.11) and $\kappa$ is as in (1.13).

In Theorem 5.6, we present a version of Theorem 1.2 in which we state more precisely the condition that $\beta$ should be a small enough generic perturbation of $\alpha$.

1.3 Structure of the proof

The proof of Theorems 1.1 and 1.2 relies on the notion of a scattering diagram, introduced in [Reference Gross and SiebertGS11], based on the insights of [Reference Kontsevich and SoibelmanKS06], to provide an algebro-geometric understanding of the mirror symmetry phenomenon in physics. To give the rough idea of a scattering diagram, which we elaborate further in § 3.1, fix a nilpotent $N^+$-graded Lie algebra $\mathfrak g=\bigoplus _{n \in N^+} \mathfrak g_n$. There is an associated unipotent algebraic group $G$ with a bijective exponential map $\exp :\mathfrak g \to G$ defined using the Baker–Campbell–Hausdorff formula. Given these data, a $(N^+,\mathfrak g)$-scattering diagram is defined as the collection of real codimension-one cones in $M_{\mathbb R }$, called walls, which are decorated by wall-crossing automorphisms, that are elements of $G$. We focus attention on scattering diagrams relevant to DT and cluster theory, which have wall-crossing automorphism preserving a holomorphic symplectic form as in [Reference Gross, Pandharipande and SiebertGPS10, Reference Gross and PandharipandeGP10, Reference Kontsevich and SoibelmanKS14, Reference BridgelandBri17, Reference Gross, Hacking, Keel and KontsevichGHKK18, Reference MouMou19, Reference MandelMan21, Reference Cheung and MandelCM20, Reference Davison and MandelDM21], and not on the more general scattering diagrams that have wall-crossing automorphisms preserving a holomorphic volume form, and which appear frequently in the context of mirror symmetry [Reference Gross and SiebertGS11, Reference Gross, Hacking and SiebertGHS22, Reference Argüz and GrossAG20, Reference Keel and YuKY19].

A codimension-two locus in $M_{\mathbb R }$ along which distinct walls intersect is called a joint. A scattering diagram is said to be consistent if for any joint, the path-ordered product of all wall-crossing automorphisms of walls that are adjacent to the joint is identity. It is shown in [Reference Kontsevich and SoibelmanKS06, Reference Gross and SiebertGS11] that there is an algorithmic prescription for constructing a consistent scattering diagram from the data of an initial set of walls. This prescription is based on inserting new walls, along with wall-crossing automorphisms, which order-by-order decrease the divergence of the path-ordered products of wall-crossing automorphisms around joints from being identity.

Given a quiver with potential $(Q,W)$, Bridgeland [Reference BridgelandBri17] constructed from the DT invariants of $(Q,W)$ a consistent scattering diagram in $M_\mathbb {R}$, called the stability scattering diagram, whose initial walls are determined by the attractor DT invariants. The stability scattering diagram is a very useful tool to study DT invariants of quivers. For example, the transformation properties of DT invariants under mutations of a quiver with potential, conjectured in [Reference Manschot, Pioline and SenMPS14] and [Reference Mozgovoy and PiolineMP20, Conjecture 3.14], are proved in [Reference MouMou19, Theorem 4.22] by a study of the corresponding transformation of the stability scattering diagram.

The main technical goal of the paper is to prove Theorems 4.22 and 4.24: they are flow tree formulas for consistent scattering diagrams which express as a sum over binary trees the wall-crossing automorphism attached to a general wall in terms of the wall-crossing automorphisms attached to the initial walls. In § 5, we then derive Theorems 1.1 and 1.2 from the flow tree formulas for scattering diagram applied to the stability scattering diagram.

The proof of Theorems 4.22 and 4.24 is given in § 4 and consists of two parts. In the first part of the proof, described in § 4.2, we relate the $(N^+,\mathfrak g)$-scattering diagrams, which live in $M_\mathbb {R}$, to auxiliary $(\mathcal {N}^+,\mathfrak h)$-scattering diagrams which live in $\mathcal {M}_\mathbb {R}$, where $\mathfrak h$ is a $\mathcal {N}^+$-graded Lie algebra constructed from $\mathfrak g$. In the second part of the proof in § 4.3, we show that the discrete attractor flow naturally defines a embedding of the binary rooted trees inside the walls of the auxiliary scattering diagrams in $\mathcal {M}_\mathbb {R}$. The images of the trees in $\mathcal {M}_\mathbb {R}$ are embedded graphs in $\mathcal {M}_\mathbb {R}$ with a balancing condition satisfied at each vertex distinct from the root, that is, essentially tropical disks in $\mathcal {M}_\mathbb {R}$ (see [Reference Nishinou and SiebertNS06, Reference GrossGro10, Reference Carl, Pumperla and SiebertCPS22]). The generic perturbation of either the skew-symmetric form or the position in $\mathcal {M}_\mathbb {R}$ of the root of the embedded trees guarantees that the vertices of the embedded trees are always contained in double intersections of walls, but never in triple intersections. The iteration of the local consistency condition around double intersection of walls determines the contribution of each tree. In the language of DT invariants, this reduces the general wall-crossing formula to an iteration of the much simpler primitive wall-crossing formula.

We note in Remark 4.25 that the perturbation of the position in $\mathcal {M}_\mathbb {R}$ of the root of the trees used in the formulation of Theorems 1.2 and 4.24 is related to a way of perturbing scattering diagrams going back to the work of Gross, Pandharipande, and Siebert [Reference Gross, Pandharipande and SiebertGPS10]. However the perturbation of the skew-symmetric form used in the formulation of Theorems 1.1 and 4.22 seems to be a completely new way to study scattering diagrams. Thus, most of the paper is focused on the study of this perturbation of the skew-symmetric form and on the proof of Theorems 1.1 and 4.22.

1.4 Related work

1.5 Plan of the paper

In § 2, we introduce our notation for trees and the discrete attractor flow, and we prove the existence of suitably generic perturbations of the skew-symmetric form. In § 3, we first review the reconstruction of consistent scattering diagrams from initial data, and then we state the flow tree formula for scattering diagrams. The technical heart of the paper is § 4 in which we prove the flow tree formula for scattering diagrams. Finally, we prove in § 5 the flow tree formula for DT invariants of quivers with potentials by applying the flow tree formula for scattering diagrams to the stability scattering diagram.