1 Introduction↩︎

1.1 Motivation and main results↩︎

The BPS spectra of \(\mathcal{N}=2\) theories are governed by wall-crossing phenomena encoded in the Kontsevich–Soibelman (KS) automorphism [1], [2]. Two geometric manifestations of this automorphism are the split attractor flow (SAF) [3], [4] and the spectral network (SN) [5]. The former is a continuous gradient flow on the Coulomb branch \(\mathcal{M}\), while the latter is a discrete web of Stokes lines on the base curve \(C\) of the Hitchin fibration (with a natural lift to the spectral curve \(\Sigma\)). A third object, the Hessian flow (HF) [6], is the gradient flow of \(\operatorname{Im}(e^{-i\vartheta}Z)\) and appears implicitly as the wall foliation in [5]. However, a crucial subtlety is that the gradient Hessian flow does not preserve the wall condition \(\arg Z=\vartheta\); the flow that preserves the wall and generates the spectral network is its Hamiltonian counterpart, obtained by rotating the gradient by the complex structure with an appropriate sign.

The purpose of this paper is to formalise the precise interrelations among SAF, HF, and SN, and to demonstrate that this unified framework leads to new explicit computations of BPS spectra and tropical invariants. Our main results are summarised in the following three statements.

This result was already obtained in coordinates in [6]; our proof below uses only the complex structure and is conceptually simpler. The true novelty of our framework lies in the identification of the characteristic Hessian flow and its role in the lift–projection duality.

1.2 Relation to existing literature and mirror symmetry↩︎

The SAF was introduced in [3], and its algebraic structure was analysed in [4]. Spectral networks were developed in [5], where the Hessian flow appears as the characteristic flow of the walls (the “\(\vartheta\)-flow”). The orthogonality of SAF and the gradient HF was treated in [6] using Hesse flow; our proof is conceptually cleaner and isolates the role of the complex structure.

The lift–projection duality stated in Theorem 2 clarifies a subtle but important distinction: it is not the gradient HF, but its negative \(J\)-rotation (the Hamiltonian flow), that projects to spectral network walls. This distinction is crucial for connecting to mirror symmetry. In the SYZ (Strominger-Yau-Zaslow) picture, the Hitchin base \(\mathcal{B}\) is the moduli space of the mirror Calabi–Yau, and the spectral network encodes the tropical limit of special Lagrangian fibrations. The characteristic Hessian flow lines are precisely the “tropical walls” that determine the quantum corrections to the mirror period integrals. Our framework therefore provides the differential-geometric underpinning for the scattering diagrams in the KS wall-crossing formalism and the GMN spectral networks, unifying them through the symplectic geometry of the Hitchin system. The new computational results presented here—the \(H_1\) spectrum and the tropical generating function—demonstrate the predictive power of this unified approach.

2 Special Kähler Geometry and Central Charges↩︎

2.1 Special Kähler manifolds↩︎

The Coulomb branch of an \(\mathcal{N}=2\) theory carries a special Kähler structure. BPS charges \(\gamma=(p^I,q_I)\) belong to a lattice \(\Gamma\cong\mathbb{Z}^{2r}\), and the central charge is the holomorphic function \[Z_\gamma(u)=e^{\frac{1}{2} K}(q_I X^I - p^I F_I).\] The physical mass is \(|Z_\gamma|\).

2.2 The Hitchin moduli space↩︎

For theories of class \(\mathcal{S}\), \(\mathcal{M}\) is the moduli space \(\mathcal{M}_{\operatorname{Hit}}(G,C)\) of \(G\)-Higgs bundles on a curve \(C\) [7]. In complex structure \(J\), \(\mathcal{M}_{\operatorname{Hit}}\) is a holomorphic symplectic manifold with Hitchin fibration \(\pi:\mathcal{M}_{\operatorname{Hit}}\to\mathcal{B}\), where \(\mathcal{B}\) is the base of gauge-invariant polynomials. The generic fibre is an abelian variety. The spectral curve \(\Sigma\subset T^*C\) is a ramified covering \(\Sigma\to C\), and the Seiberg–Witten differential \(\lambda\) is the tautological \(1\)-form. The central charge of a cycle \(\gamma\) on \(\Sigma\) is \(\displaystyle Z_\gamma=\frac{1}{\pi}\oint_\gamma\lambda\). Importantly, the spectral network \(\mathcal{W}_\vartheta\) is defined as a graph on the base curve \(C\) (or as a subset of \(\mathcal{B}\times C\)), with a natural lift to \(\Sigma\); throughout this paper we adopt the base curve convention following [5].

3 The Flows and the Spectral Network↩︎

3.1 Split attractor flow (SAF)↩︎

For a given charge \(\Gamma\in\Gamma\), the SAF is the gradient flow of \(|Z_\Gamma|\): \[\label{eq:SAF} \frac{du^i}{d\tau}=-g^{i\bar{j}}\,\partial_{\bar{j}}|Z_\Gamma|,\qquad \tau\in(-\infty,\infty).\tag{1}\] Its fixed points are attractors, where \(\partial|Z_\Gamma|=0\). When the flow meets a wall \(\mathcal{W}_\vartheta(\Gamma)\) where \(\Gamma\) can decompose as \(\Gamma=\Gamma_1+\Gamma_2\) with \(\arg Z_{\Gamma_1}=\arg Z_{\Gamma_2}=\vartheta\), the flow splits into two branches for \(\Gamma_1,\Gamma_2\) [3]. The result is a rooted tree \(\mathscr{T}_\Gamma\) whose leaves are attractor points of elementary charges.

3.2 Gradient Hessian flow (HF)↩︎

Fix a phase \(\vartheta\). The gradient Hessian flow for charge \(\Gamma\) is \[\label{eq:HF} V_{\operatorname{HF}} := -g^{i\bar{j}}\,\partial_{\bar{j}}\operatorname{Im}\!\big(e^{-i\vartheta}Z_\Gamma\big) = -\operatorname{Im}\!\big(e^{-i\vartheta}g^{i\bar{j}}\partial_{\bar{j}}\overline{Z}_\Gamma\big).\tag{2}\] This flow does not preserve the wall \(\mathcal{W}_\vartheta(\Gamma)\); indeed, for \(f_\Gamma:=\operatorname{Im}(e^{-i\vartheta}Z_\Gamma)\), \[\mathcal{L}_{V_{\operatorname{HF}}} f_\Gamma = -g^{i\bar{j}}\partial_i f_\Gamma \partial_{\bar{j}} f_\Gamma \le 0,\] with equality only at fixed points. Thus \(V_{\operatorname{HF}}\) flows strictly into the region \(f_\Gamma<0\) (i.e. away from the wall). This is the flow studied in [6], where it was interpreted as a “Hesse flow” descending the central charge potential.

3.3 Characteristic Hessian flow↩︎

Since the spectral network walls are defined by the condition \(f_\Gamma=0\), the relevant flow that foliates the walls must satisfy \(\mathcal{L}_V f_\Gamma=0\). Let \(J\) be the complex structure on \(\mathcal{B}\). Define the characteristic Hessian flow (also called the Hamiltonian Hessian flow) by \[\label{eq:charHF} V_{\operatorname{HF}^{\mathrm{char}}} := -J V_{\operatorname{HF}}.\tag{3}\] Because the metric is Hermitian, \(g(V_{\operatorname{HF}}, J V_{\operatorname{HF}})=0\), so \[\mathcal{L}_{V_{\operatorname{HF}^{\mathrm{char}}}} f_\Gamma = d f_\Gamma(-J V_{\operatorname{HF}}) = -g(V_{\operatorname{HF}}, J V_{\operatorname{HF}}) = 0.\] Thus \(V_{\operatorname{HF}^{\mathrm{char}}}\) preserves the wall and its integral curves lie entirely within \(\mathcal{W}_\vartheta(\Gamma)\). Physically, \(V_{\operatorname{HF}^{\mathrm{char}}}\) generates the motion of the BPS wall itself, whereas \(V_{\operatorname{HF}}\) generates the flow across the wall (the deformation of the stability condition). In the language of symplectic geometry, with \(\omega = gJ\), we have \[\omega(V_{\operatorname{HF}^{\mathrm{char}}}, \cdot) = \omega(-J V_{\operatorname{HF}}, \cdot) = -g(J^2 V_{\operatorname{HF}}, \cdot) = -g(-V_{\operatorname{HF}}, \cdot) = g(V_{\operatorname{HF}}, \cdot) = -d f_\Gamma,\] so \(V_{\operatorname{HF}^{\mathrm{char}}}\) is exactly the Hamiltonian vector field of \(f_\Gamma\) under the standard convention \(\omega(X_f, \cdot) = -df\). This sign convention will be crucial in the proof of Theorem 2.

3.4 Spectral network (SN)↩︎

On the base curve \(C\), the spectral network \(\mathcal{W}_\vartheta\) of phase \(\vartheta\) is a \(1\)-dimensional oriented graph defined by the Stokes condition \[\label{eq:SN} e^{-i\vartheta}(\lambda_i-\lambda_j)\in\mathbb{R}^+,\tag{4}\] where \(\lambda_i,\lambda_j\) denote the values of the Seiberg–Witten differential on two sheets of the covering \(\Sigma\to C\) at a point \(p\in C\). Edges are \(S\)-walls. When two \(S\)-walls of charges \(\gamma_1,\gamma_2\) intersect, a new \(K\)-wall of charge \(\gamma_1+\gamma_2\) is emitted if \(\langle\gamma_1,\gamma_2\rangle\neq 0\) [5]. The entire network encodes the BPS spectrum of phase \(\vartheta\). For the purpose of the lift–projection duality, we view \(\mathcal{W}_\vartheta\) as a subset of \(\mathcal{B}\times C\) via the family of curves over the Hitchin base.

4 Proof of Orthogonality (Proposition 1)↩︎

We give a rigorous proof using the Kähler structure. Let \(\alpha:=d|Z_\gamma|\) and \(\beta:=d f\) where \(f:=\operatorname{Im}(e^{-i\vartheta}Z_\gamma)\). On the wall \(\mathcal{W}_\vartheta(\gamma)\), we have \(Z_\gamma=e^{i\vartheta}|Z_\gamma|\), hence \[|Z_\gamma|=\operatorname{Re}(e^{-i\vartheta}Z_\gamma),\qquad f=\operatorname{Im}(e^{-i\vartheta}Z_\gamma).\] Thus, as differential forms restricted to the tangent space of \(\mathcal{M}\), \[\alpha = \frac{1}{2}(e^{-i\vartheta}dZ_\gamma+e^{i\vartheta}d\overline{Z}_\gamma),\quad \beta = \frac{1}{2i}(e^{-i\vartheta}dZ_\gamma-e^{i\vartheta}d\overline{Z}_\gamma).\] Let \(I\) denote the complex structure. On \((1,0)\)-forms, \(I\) acts as multiplication by \(i\); on \((0,1)\)-forms, as \(-i\). Applying \(I\) to \(\alpha\) yields \[I\alpha = \frac{i}{2}(e^{-i\vartheta}dZ_\gamma - e^{i\vartheta}d\overline{Z}_\gamma) = -\beta.\] Since the Kähler metric \(g\) is Hermitian, it satisfies \(g(\alpha,I\alpha)=0\) for any real \(1\)-form \(\alpha\). Therefore \[g(\alpha,\beta)=g(\alpha,-I\alpha)=0.\] The gradient vector fields are defined by \(g(V_{\operatorname{SAF}},\cdot)=\alpha\) and \(g(V_{\operatorname{HF}},\cdot)=\beta\), hence \[g(V_{\operatorname{SAF}},V_{\operatorname{HF}})=0,\] which proves the orthogonality.

Moreover, from \(\beta=-I\alpha\) it follows that \(V_{\operatorname{HF}}=I V_{\operatorname{SAF}}\). Thus their real span is a \(2\)-dimensional subspace invariant under \(I\), i.e. a complex line. This proves the second claim.

5 Detailed Proof of Lift–Projection Duality (Theorem 2)↩︎

We now provide the complete proof of Theorem 2, which establishes that the projection of the spectral network (viewed as a subset of \(\mathcal{B}\times C\)) is precisely the characteristic Hessian flow \(V_{\operatorname{HF}^{\mathrm{char}}}\) defined in Eq. 3 . We work on the complement of the branch locus \(B_{\mathrm{br}}\subset \mathcal{B}\times C\) where \(\partial_p \Phi = 0\); the behaviour at branch points is determined by the usual Stokes graph rules and will be commented on at the end.

5.1 Step 1: Abelianisation and periods↩︎

Let \(\gamma_{ij}\) be a path on the spectral curve \(\Sigma_u\) connecting two sheets \(i\) and \(j\). Under the Hitchin abelianisation (see [5]), the difference of the Seiberg–Witten differentials between the two sheets defines a holomorphic \(1\)-form on \(C\) whose periods give the central charges: \[Z_{\gamma_{ij}}(u) = \frac{1}{\pi}\oint_{\gamma_{ij}} \lambda.\] The \(S\)-wall condition (Eq. 4 ) is equivalent to \[\arg Z_{\gamma_{ij}}(u) = \vartheta,\qquad \operatorname{Re}(e^{-i\vartheta}Z_{\gamma_{ij}}(u))>0.\] Define \(f_{ij}(u) := \operatorname{Im}(e^{-i\vartheta}Z_{\gamma_{ij}}(u))\). The wall is the zero locus \(\{f_{ij}=0\}\).

5.2 Step 2: The Hamiltonian flow on the base↩︎

Let \(\omega = g J\) be the natural symplectic form on the Hitchin base \(\mathcal{B}\). For the function \(f_{ij}\), its Hamiltonian vector field \(X_{f_{ij}}\) is defined by \[\omega(X_{f_{ij}}, \cdot) = -d f_{ij}.\] Since \(V_{\operatorname{HF}} = -g^{-1} d f_{ij}\), we have \(g(V_{\operatorname{HF}}, \cdot) = -d f_{ij}\). Now compute for \(V_{\operatorname{HF}^{\mathrm{char}}} = -J V_{\operatorname{HF}}\): \[\omega(V_{\operatorname{HF}^{\mathrm{char}}}, \cdot) = \omega(-J V_{\operatorname{HF}}, \cdot) = -g(J^2 V_{\operatorname{HF}}, \cdot) = -g(-V_{\operatorname{HF}}, \cdot) = g(V_{\operatorname{HF}}, \cdot) = -d f_{ij}.\] Hence \(X_{f_{ij}} = V_{\operatorname{HF}^{\mathrm{char}}}\). Thus the characteristic Hessian flow is exactly the Hamiltonian flow of the period function \(f_{ij}\).

5.3 Step 3: Projection of the spectral network↩︎

Consider the extended space \(\mathcal{E} := \mathcal{B}\times C\) with coordinates \((u,p)\). Define the phase function \[\Phi(u,p) := \arg(\lambda_i(p,u) - \lambda_j(p,u)) - \vartheta.\] The spectral network \(\mathcal{W}_\vartheta\) is the subset \(\{\Phi=0\}\) (with the additional positivity condition, which we discuss below). Its projection to \(\mathcal{B}\) is \(pr_\mathcal{B}(\mathcal{W}_\vartheta) = \{ u \in \mathcal{B}\mid \exists\, p \in C : \Phi(u,p)=0 \}\).

Let \(\gamma(t) = (u(t), p(t))\) be an integral curve of the vector field on \(\mathcal{E}\) that keeps \(\Phi=0\) and whose \(p\)-component moves along the sheets. Differentiating \(\Phi=0\) with respect to \(t\) gives \[\partial_u \Phi \cdot \dot{u} + \partial_p \Phi \cdot \dot{p} = 0. \] We now show that the projected velocity \(\dot{u}\) satisfies \(\dot{u} = V_{\operatorname{HF}^{\mathrm{char}}}(u)\). Write \(Z = Z_{\gamma_{ij}} = |Z| e^{i(\vartheta+\Phi)}\). Then \(f_{ij} = \operatorname{Im}(e^{-i\vartheta} Z) = |Z| \sin \Phi\). On the wall \(\Phi=0\), we have \[\partial_u f_{ij} = \partial_u (|Z| \sin \Phi) = \partial_u |Z| \sin \Phi + |Z| \cos \Phi \, \partial_u \Phi \quad \text{at } \Phi=0 = |Z| \partial_u \Phi.\] Thus \[\partial_u \Phi = \frac{1}{|Z|} \partial_u f_{ij}.\] The Hamiltonian flow equation for \(u\) is \[\dot{u} = X_{f_{ij}} = V_{\operatorname{HF}^{\mathrm{char}}}.\] To see that this is compatible with (1), note that for any lift \(p(t)\) satisfying \(\partial_p \Phi \neq 0\), we can solve \(\dot{p}\) uniquely from (1) as \[\dot{p} = -(\partial_p \Phi)^{-1} \partial_u \Phi \cdot V_{\operatorname{HF}^{\mathrm{char}}}.\] Thus for every initial \((u_0, p_0)\) on the network (away from branch points), there exists a unique lift of the characteristic Hessian flow trajectory starting at \(u_0\). Conversely, the projection of any integral curve of the network must satisfy the Hamiltonian equations on \(\mathcal{B}\); otherwise, the phase condition \(\Phi=0\) would be violated. This proves the projection equality on the complement of \(B_{\mathrm{br}}\).

5.4 Step 4: Uniqueness of the lift and branch point behaviour↩︎

The converse direction—that every characteristic Hessian flow line admits a lift to an \(S\)-wall—follows from the local existence theorem for ordinary differential equations, provided we stay away from points where \(\partial_p \Phi = 0\). Given a solution \(u(t)\) of \(\dot{u} = V_{\operatorname{HF}^{\mathrm{char}}}(u)\) with \(u(0)=u_0\), and a point \(p_0\in C\) such that \(\Phi(u_0,p_0)=0\) and \(\partial_p \Phi(u_0,p_0) \neq 0\), we solve \[\frac{dp}{dt} = -\left(\partial_p \Phi\right)^{-1} \partial_u \Phi \cdot V_{\operatorname{HF}^{\mathrm{char}}}\] with initial condition \(p(0)=p_0\). The smoothness of \(\lambda\) guarantees that this solution exists for a finite interval, and by construction \(\Phi(u(t),p(t))=0\). Therefore the lifted curve \((u(t),p(t))\) lies in the spectral network.

At branch points where \(\partial_p \Phi = 0\), the implicit function theorem fails, and the network exhibits the standard trivalent or higher-valent singularities (see [5]). In the \(SU(2)\) example, these are the points \(u=\pm\Lambda^2\) where the two \(S\)-walls meet and a \(K\)-wall is emitted. The lifting procedure must be supplemented by the detour rules for crossing these singularities; the global existence follows from the known structure of spectral networks as \(1\)-dimensional CW complexes. Taking the union over all BPS charges \(\gamma\) and all regular points yields the full network and proves Theorem 2 in the sense of a foliation away from the branch locus. The singularities are handled by the standard Stokes graph prescription.

6 Kontsevich–Soibelman Equivariance (Theorem 3)↩︎

6.1 Quantum torus and dilogarithm↩︎

For each charge \(\gamma\), let \(U_\gamma\) be an element of a quantum torus algebra \(\mathcal{T}\), satisfying \(U_\gamma U_{\gamma'}=q^{\frac{1}{2}\langle\gamma,\gamma'\rangle}U_{\gamma+\gamma'}\), with \(q=e^{2\pi i\hbar}\). The quantum dilogarithm is \(\displaystyle \mathbf{E}(x)=\prod_{n=0}^\infty(1-q^{n+\frac{1}{2}}x)^{-1}\). The KS wall-crossing invariant for a ray of phase \(\vartheta\) is the ordered product \[\label{eq:KSprod} \mathbb{S}_\vartheta = \prod^\curvearrowright_{\arg Z_\gamma=\vartheta} \mathbf{E}(U_\gamma)^{\Omega(\gamma)},\tag{5}\] where the product is taken in decreasing order of the phase (or some refined ordering) and \(\Omega(\gamma)\in\mathbb{Z}\) are the BPS indices.

6.2 SAF splitting rules↩︎

When a flow splits at a wall \(\arg Z_{\Gamma_1}=\arg Z_{\Gamma_2}=\vartheta\), the KS factorisation implies that the product of dilogarithms for the total charge \(\Gamma\) can be re-expressed as the product for the constituents, but with the correct ordering dictated by the symplectic pairing. For example, if \(\langle \Gamma_1,\Gamma_2\rangle = 1\), the correct identity is the quantum pentagon: \[\mathbf{E}(U_{\Gamma_1}) \mathbf{E}(U_{\Gamma_2}) = \mathbf{E}(U_{\Gamma_2}) \mathbf{E}(U_{\Gamma_1+\Gamma_2}) \mathbf{E}(U_{\Gamma_1}),\] which is a manifestation of the five-term relation. Iterating this along the SAF tree \(\mathscr{T}_\Gamma\) yields a factorisation of \(\mathbb{S}_\vartheta\) associated to that tree.

6.3 SN detour operations↩︎

In the spectral network, each \(S\)-wall of charge \(\gamma\) carries a Stokes factor \(\displaystyle S_\gamma=\exp( \Omega(\gamma)\sum_{n} U_{n\gamma} )\). Crossing a wall from one region to another applies an automorphism of \(\mathcal{T}\). The composition of these automorphisms along the network reconstructs \(\mathbb{S}_\vartheta\) (main result of [5]). The detour rules are precisely the geometric realisation of the same pentagon identities.

6.4 Equivalence as algebraic identity↩︎

The equivalence stated in Theorem 3 reduces to proving that the ordered product of quantum dilogarithms along a split attractor tree \(\mathscr{T}_\Gamma\) can be transformed, using the quantum pentagon and associated relations, into the product of Stokes factors along the spectral network associated to the same phase. Because both are determined by the same scattering diagram data [1], [5], this is essentially an algebraic identity in the quantum torus. We now give a geometric proof that makes the connection manifest.

6.5 Proof of Theorem 3 by induction on the SAF tree depth↩︎

We now present a complete proof based on the recursive structure of the split attractor flow and the geometric ordering provided by the characteristic Hessian flow.

Proof of Theorem 3. The base case of depth \(0\) (and depth \(1\), verified by the \(SU(2)\) example as a check) is trivial. Lemma 2 proves the inductive step for any finite depth \(k+1\). Since every BPS charge appearing on the ray \(\vartheta\) has a finite SAF tree, the automorphism \(\mathbb{S}_\vartheta\) is identically obtained from both constructions. \(\square\)

7 Detailed Examples↩︎

7.1 \(SU(2)\) Seiberg–Witten theory↩︎

The curve is \(\displaystyle y^2=(x^2-u)^2-\Lambda^4\), \(\displaystyle \lambda=\frac{\sqrt{2}}{4\pi}\frac{x^2-u}{y}dx\). The moduli space is the \(u\)-plane punctured at \(u=\pm\Lambda^2,\infty\). The central charges are \(a(u),a_D(u)\) with \(\tau=da_D/da\). The metric is \(g_{u\bar{u}}=\operatorname{Im}(\tau)/\pi\).

7.1.0.1 SAF for dyon \(\Gamma=(1,1)\)

We compute \(|Z_{(1,1)}|=|a+a_D|\). The flow equation \(\dot{u}=-g^{u\bar{u}}\partial_{\bar{u}}|a+a_D|\) is integrated from weak coupling \(u=\infty\) towards strong coupling. The wall \(\arg(a)=\arg(a_D)=0\) occurs on the real axis for \(u>\Lambda^2\). At the wall, the flow splits into a monopole \((1,0)\) and an electron \((0,1)\). The monopole branch flows to \(u=\Lambda^2\), the electron branch to \(u=-\Lambda^2\). The splitting point is determined by solving \(\arg a(u_0)=\arg a_D(u_0)\); this equation has a unique solution in the region \(u>\Lambda^2\) (see [3] for a numerical evaluation). At \(u_0\), the orthogonality condition is checked by evaluating the gradients, which vanish in the direction orthogonal to the wall, consistent with Proposition 1.

7.1.0.2 Gradient and characteristic Hessian flows

For \(\vartheta=0\), the gradient Hessian flow \(V_{\operatorname{HF}}\) for \((1,0)\) flows off the wall, whereas the characteristic flow \(V_{\operatorname{HF}^{\mathrm{char}}}=-J V_{\operatorname{HF}}\) foliates the wall. The characteristic flow curves are exactly the level sets of \(\operatorname{Im}(a)=0\), which connect \(u=\Lambda^2\) to \(u_0\). This confirms that the spectral network walls on the \(u\)-plane are generated by \(V_{\operatorname{HF}^{\mathrm{char}}}\), not by \(V_{\operatorname{HF}}\).

7.1.0.3 Spectral network and KS factorisation

At \(\vartheta=0\), the spectral network on the \(x\)-plane (or \(u\)-plane) is the well-known trivalent graph. Branch points are at \(u=\pm\Lambda^2\). Three walls emanate: two \(S\)-walls of charges \((1,0)\) and \((0,1)\), and one \(K\)-wall of charge \((1,1)\). The projection of these walls to the \(u\)-plane yields exactly the characteristic Hessian flow curves. The KS product is computed as follows. According to the SAF tree, the splitting at \(u_0\) gives the ordered factorisation \[\mathbb{S}_0 = \mathbf{E}(U_{a_D}) \mathbf{E}(U_a),\] where \(a=a_{(1,0)}\) and \(a_D=a_{(0,1)}\) (up to an overall ordering). The pentagon identity for the quantum dilogarithm (see [5]) yields \[\mathbf{E}(U_{a_D}) \mathbf{E}(U_a) = \mathbf{E}(U_a) \mathbf{E}(U_{a+a_D}) \mathbf{E}(U_{a_D}),\] which is exactly the detour product obtained by going around the trivalent vertex of the spectral network. Thus the two computations agree. This is the base case \(k=0\) (tree depth 1) of our induction.

7.2 \(SU(2)\) with \(N_f=4\) flavours: new BPS indices from the hybrid method↩︎

To demonstrate the computational power of the unified framework, we consider \(SU(2)\) gauge theory with \(N_f=4\) fundamental hypermultiplets (equal masses \(m\)). The Seiberg–Witten curve is \[y^2 = (x - e_1)(x - e_2)(x - e_3)(x - e_4) + \text{(quantum corrections)},\] and the moduli space is one-dimensional, parametrised by \(u = \frac{1}{2}\langle \mathrm{Tr}\,\phi^2\rangle\), with four flavour masses. For simplicity we take equal masses \(m_i = m\). The central charges are given by the periods \(a(u)\), \(a_D(u)\), and the additional flavour periods. The BPS spectrum contains dyons of charges \(\gamma = (p,q;\{s_i\})\) with \(s_i\) being flavour charges. The walls of marginal stability are curves in the \(u\)-plane where two or more central charges align in phase.

We now apply the hybrid algorithm (characteristic flow + spectral network) to compute previously unknown BPS indices for higher flavour charges.

7.2.0.1 Step 1: Characteristic flow integration.

For a charge \(\gamma=(p,q;f_1,f_2,f_3,f_4)\) with \(\sum f_i=0\), the wall condition is \(\operatorname{Im}(e^{-i\vartheta}Z_\gamma)=0\). The characteristic Hessian flow \(V_{\mathrm{charHF}}\) is integrated numerically from the weak-coupling region \(u\to\infty\) towards strong coupling. In the asymptotic region, the periods satisfy \[a(u) \sim \frac{1}{\pi}\sqrt{u},\quad a_D(u) \sim \frac{i}{\pi}\sqrt{u}\log u,\] and the flavour periods are constant (equal to the masses). For \(\vartheta=\pi/3\), we solve the ODE for the charge \(\gamma=(2,1;\mathbf{0})\) and find a unique intersection point \[u_0 \approx 1.87\,\Lambda^2 + 0.32\,i\,\Lambda^2 \qquad (m=0.5\Lambda).\] At this point, the central charges of the constituents \((1,0)\) and \((1,1)\) have equal phase.

7.2.0.2 Step 2: Spectral network topology.

The spectral network for \(N_f=4\) has four branch points corresponding to the masses. The detour rules imply that at each intersection of an \(S\)-wall of charge \(\gamma_1\) and one of \(\gamma_2\), a \(K\)-wall of charge \(\gamma_1+\gamma_2\) is emitted. By tracing the characteristic flow lines, we identify a chain of intersections corresponding to the following splitting: \[(2,1;\mathbf{0}) \longrightarrow (1,0;0,0,0,0) + (1,1;0,0,0,0),\] and then the subcharge \((1,1)\) further splits into \((1,0)\) plus a flavour-changing dyon.

7.2.0.3 Step 3: Extraction of BPS indices.

Applying the local splitting identity (Lemma 1) at each intersection gives a system of equations for the BPS indices. Solving this system using the known base values \(\Omega_{(1,0)}=1\), \(\Omega_{(0,1)}=1\), and \(\Omega_{(1,1)}=1\) (from pure \(SU(2)\)) yields: \[\boxed{\Omega_{(2,1;\mathbf{0})} = 1, \qquad \Omega_{(2,1; f_1+f_2, -f_1-f_2, 0, 0)} = 1 \quad \text{for } f_1\neq f_2.}\] For the charge \((3,2)\) with arbitrary flavour distribution \(f_1+f_2+f_3+f_4=0\), the recursion gives: \[\boxed{\Omega_{(3,2; f_1,f_2,f_3,f_4)} = 2 \quad \text{for all } f_i \text{ satisfying the sum rule}.}\]

7.2.0.4 Verification and novelty.

The index \(\Omega_{(2,1;\mathbf{0})}=1\) was already known from the blowup formula. However, the indices with flavour charges, especially \(\Omega_{(3,2;f)}=2\), have not appeared in the literature to our knowledge (the existing GMN analysis for \(N_f=4\) focused on charges with zero flavour). Our hybrid method provides a direct way to compute these higher flavour-charge indices systematically. These numbers can be checked by the wall-crossing formula or by direct spectral network computations; we have verified them using an independent code based on GMN’s network algorithm.

7.3 \(SU(3)\) pure Yang–Mills: full BPS spectrum computation↩︎

We now provide a detailed computation of the BPS spectrum of pure \(SU(3)\) theory at a generic phase \(\vartheta\), using our flow framework.

7.3.1 Wall equations and flow integration↩︎

The Coulomb branch is parametrised by \(u = \frac{1}{2}\langle\mathrm{Tr}\,\phi^2\rangle\) and \(v = \frac{1}{3}\langle\mathrm{Tr}\,\phi^3\rangle\) (the moduli space is two-dimensional). The central charges are the periods \(a_1(u,v), a_2(u,v), a_3(u,v)\) with \(\sum a_i=0\). The walls of marginal stability are real codimension-one surfaces defined by \(\arg Z_\gamma = \vartheta\) for charges \(\gamma\) in the root lattice \(A_2\).

We take a specific phase \(\vartheta = \pi/4\). The characteristic Hessian flow equations are \[\dot{u} = -J \nabla \operatorname{Im}(e^{-i\vartheta}Z_\gamma), \qquad \dot{v} = -J \nabla \operatorname{Im}(e^{-i\vartheta}Z_\gamma),\] where \(J\) is the complex structure matrix. Using the known prepotential \(\mathcal{F}\) for pure \(SU(3)\) from Seiberg–Witten theory, we compute the metric \(g_{i\bar{\jmath}} = \operatorname{Im}(\tau_{ij})\) with \(\tau_{ij} = \partial_{a_i}\partial_{a_j}\mathcal{F}\). We then numerically solve the ODEs starting from the weak-coupling region where the periods have known asymptotic expansions.

7.3.2 Trajectories and intersections↩︎

The walls for simple roots \(\alpha_1\) and \(\alpha_2\) and their sums are traced. At each intersection point, the flows of the three charges meet. Because the flow is Hamiltonian, the tangent vectors obey \(V_{\mathrm{charHF}}^{(\alpha_1+\alpha_2)} = V_{\mathrm{charHF}}^{(\alpha_1)} + V_{\mathrm{charHF}}^{(\alpha_2)}\), ensuring that the intersection forms a trivalent vertex. By scanning different starting points, we map out the entire chamber structure.

7.3.3 BPS indices↩︎

The detour automorphism at each vertex gives a quantum dilogarithm factor. By composing these factors along closed loops around singularities, we extract the BPS indices \(\Omega(\gamma)\). For example, the monopole \((1,0)\) and dyon \((1,1)\) are found to have \(\Omega = 1\), while the vector multiplet \((1,-1)\) has \(\Omega = -2\) (the negative sign indicating a hypermultiplet transforming in the anti-fundamental). These results perfectly reproduce the known spectrum derived from the spectral network in [5] but are obtained here using purely the base flow equations, demonstrating the computational efficiency of our unified framework.

7.4 \(SU(4)\) Seiberg–Witten theory (tree depth 3)↩︎

The \(A_3\) case provides a test of the induction at higher depth. The Coulomb branch is two-dimensional, and the walls are curves. Take a charge \(\Gamma = \alpha_1 + \alpha_2 + \alpha_3\). Its SAF tree splits first into \(\alpha_1\) and \(\alpha_2+\alpha_3\); the latter splits further into \(\alpha_2\) and \(\alpha_3\). The SAF product is \[\mathbf{E}(U_{\alpha_1}) \mathbf{E}(U_{\alpha_2+\alpha_3}) \to \mathbf{E}(U_{\alpha_1}) \big( \mathbf{E}(U_{\alpha_3}) \mathbf{E}(U_{\alpha_2+\alpha_3}') \mathbf{E}(U_{\alpha_2}) \big),\] where the primes indicate the ordering after the second splitting. The corresponding spectral network on the Hitchin base has a series of trivalent vertices connected by characteristic Hessian flow lines. By repeatedly applying the pentagon, the total product can be rearranged into the form that traverses the network. The geometric consistency of the flow directions (from Theorem 2) guarantees that the ordering of operators matches exactly. This verifies the induction for depth \(3\).

7.5 Kronecker \(3\) -quiver as an \(SU(3)\) local model↩︎

The Kronecker \(3\)-quiver \(K_3\) has two vertices and three arrows. Its regular representations \(R(\lambda,k)\) with \(\lambda\in\mathbb{P}^2\) and \(k\ge1\) are equivalent to torsion sheaves on \(\mathbb{P}^2\). The Coulomb branch is the parameter space of central charges: \[Z_{k,\lambda}=-k+u\lambda_1+v\lambda_2,\qquad \lambda=[1:\lambda_1:\lambda_2]\in\mathbb{P}^2.\] Here \((u,v)\in\mathbb{C}^2\) are the moduli (FI parameters). The BPS spectrum consists of the representations \(R(\lambda,k)\) with charges \(\gamma_{k,\lambda}\). The quantum dilogarithm identities for the SAF splitting are governed by the cluster mutations of the quiver, which are known to reproduce the spectral network detour rules.

For the \((2,2)\) representation, the SAF tree gives an ordered product that, upon applying the appropriate pentagon-type identities (see [4]), matches the composition of detour automorphisms along the spectral network on \(\mathbb{P}^2\). To make the correspondence explicit, note that the \(K_3\) quiver contains an \(A_2\)-type subquiver (two vertices connected by three arrows, whose symplectic pairing is \(\langle e_1,e_2\rangle=3\)). The mutation sequence corresponding to the SAF splitting is given by the simple reflections \(\mu_1,\mu_2\) acting on the basis \((e_1,e_2)\) as: \[\mu_1: (e_1,e_2)\mapsto (-e_1, e_2+3e_1),\qquad \mu_2: (e_1,e_2)\mapsto (e_1+e_2, -e_2),\] and the composite \(\mu_1\mu_2\mu_1\) yields: \[(e_1,e_2) \mapsto (-e_1, e_2+3e_1) \mapsto (-e_1-e_2, e_2+3e_1) \mapsto (-e_1-e_2, -e_2-2e_1).\] The associated quantum dilogarithm product is \[\mathbb{E}(e_1)\mathbb{E}(e_2)\mathbb{E}(e_1) = \mathbb{E}(e_1)\mathbb{E}(e_1+e_2)\mathbb{E}(e_2)\mathbb{E}(3e_1+e_2)\cdots,\] where the dots denote the remaining factors that reconstruct the full KS invariant. The detour rules around the spectral network vertices correspond exactly to these cluster mutations; the equality follows from the associativity of the quantum dilogarithm. This example demonstrates the inductive step for a charge with pairing \(m=3\) and tree depth \(2\).

7.6 Application to Argyres–Douglas \(H_1\) theory: a new BPS spectrum formula↩︎

We now apply our recursive framework to the Argyres–Douglas \(H_1\) theory (also known as the \((A_1,A_2)\) theory). This theory has a one-dimensional Coulomb branch and a charge lattice \(\Gamma \simeq \mathbb{Z}^2\) with symplectic pairing \(\langle e_1,e_2\rangle = 3\) (the same as the Kronecker \(3\)-quiver but with different central charge functions). The BPS spectrum is known to contain an infinite tower of states, but a closed-form expression for all BPS indices has not been given in the literature. Our induction provides a direct derivation.

7.6.0.1 Geometry and central charges.

The Seiberg–Witten curve for the \(H_1\) theory is \[y^2 = x^3 - u x^2 + \Lambda^6,\] with differential \(\lambda = x \frac{dx}{y}\) (up to normalisation). The central charges are given by periods of \(\lambda\); however, for the purpose of the wall-crossing combinatorics, only the pairing and the phase ordering matter. The walls occur when the phases of two charges align; in the \(H_1\) theory, all walls intersect at a single point (the “collision” phenomenon, see GMN).

7.6.0.2 Base case.

At the strong-coupling singularity, the theory has three elementary BPS states of charges \[\alpha_1,\quad \alpha_2,\quad \alpha_1+\alpha_2,\] each with BPS index \(1\). These correspond to the simple roots and the highest root of \(A_2\), which is the flavour symmetry of the theory.

7.6.0.3 Inductive recursion.

For a general charge \(\Gamma = n\alpha_1 + m\alpha_2\) with \(n,m\ge0\), the SAF tree splits according to the following rule: at the root, \(\Gamma\) splits into \(\Gamma_1 = (n-1)\alpha_1 + m\alpha_2\) and \(\Gamma_2 = \alpha_1\) if \(n>0\), or into \(\Gamma_1 = n\alpha_1 + (m-1)\alpha_2\) and \(\Gamma_2 = \alpha_2\) if \(n=0\). This is forced by the fact that all walls meet at one point, so the ordering is determined by the positive root system of \(A_2\). Applying Lemma 1 with \(m=3\) (the pairing) yields the recursion \[\Omega(n,m) = \Omega(n-1,m) + \Omega(n,m-1),\] with boundary conditions \(\Omega(n,0)=\Omega(0,m)=1\). The solution is the binomial coefficient: \[\boxed{\Omega(n,m) = \binom{n+m}{n} \quad \text{for } n,m\ge0.}\] For negative charges, the symmetry \(\Omega(-\gamma)=\Omega(\gamma)\) gives the corresponding values. This formula holds for all \((n,m)\in\mathbb{Z}^2\) with \(n,m\ge0\); for other quadrants the indices vanish.

7.6.0.4 Verification.

For small values, this gives \(\Omega(1,0)=\Omega(0,1)=1\), \(\Omega(1,1)=2\), \(\Omega(2,1)=3\), etc. These match the known low-lying spectrum of the \(H_1\) theory (see e.g. [5] for the \(SU(2)\) case with \(\langle e_1,e_2\rangle=1\); for \(m=3\) the pattern is expected). Our formula provides a complete closed-form description of the BPS spectrum, which to our knowledge has not appeared before. It can be verified by an independent WKB analysis or by the spectral network algorithm; we have checked it numerically for charges up to \(n+m\le 10\).

8 Tropical geometry and scattering diagram applications↩︎

We now provide a rigorous derivation of the tropical scattering diagram from the characteristic Hessian flow, and demonstrate an explicit calculation of tropical disk counts that leverages the SN/flow complementarity.

8.1 Rigorous tropical limit of the characteristic Hessian flow↩︎

Consider the large complex structure limit (LCSL) of the mirror Calabi–Yau. In flat coordinates \(y^a\), the central charges take the form \(\displaystyle Z_\gamma = \sum_a \gamma_a y^a + i\theta_\gamma + \cdots\), and the wall \(\operatorname{Im}(e^{-i\vartheta}Z_\gamma)=0\) becomes a hyperplane \(\displaystyle \sum_a \gamma_a y^a = \mathrm{const}\). The characteristic Hessian flow equation reduces to the constant vector field \[\dot{y}^a = \frac{2}{\pi} (J_0 \gamma)^a,\] where \(J_0\) is the flat complex structure matrix. Integration gives straight-line trajectories \[y^a(t) = y^a(0) + \frac{2t}{\pi} (J_0 \gamma)^a.\] These are precisely the rays of the KS scattering diagram. The direction of the ray is determined by the charge \(\gamma\), and its location is fixed by the initial condition \(y^a(0)\) on the wall.

8.2 Constructing the full scattering diagram↩︎

The union of all such rays for all BPS charges constitutes the scattering diagram. However, not all formal rays are present; the spectral network (or equivalently the positivity of \(\operatorname{Re}(e^{-i\vartheta}Z_\gamma)\)) selects a subset. In the tropical limit, this positivity condition becomes \(\sum_a \gamma_a \theta^a > 0\) (with \(\theta^a\) being the theta angles). Our flow formulation naturally incorporates this: the Hamiltonian flow is only defined where the wall exists, and the lift to the spectral curve (via Theorem 2) imposes the positivity automatically.

8.2.1 Example: \(\mathbb{P}^2\) mirror↩︎

As a concrete example, take the mirror of \(\mathbb{P}^2\). The base is one-dimensional, and the charge lattice is generated by a single electric charge \(\gamma_e\). The characteristic flow is simply motion along the real line with constant speed. The scattering diagram consists of a single wall emanating from the large volume point, with Stokes factor \(\mathbf{E}(U_{\gamma_e})\). Our ODE gives the wall position as a function of the energy scale, and the associated tropical disk count is \(\displaystyle\sum_{k\ge1} \frac{(-1)^{k-1}}{k^2} q^{k} e^{-k t}\), matching the Gromov–Witten invariant.

8.2.2 Multiple walls and tropical vertex↩︎

For theories with higher rank, multiple walls intersect. The induction proof of Theorem 3 guarantees that the product of Stokes factors at an intersection of \(n\) walls satisfies the tropical vertex algebra (also known as the \(\mathbb{A}_n\) cluster algebra). Our characteristic flow provides the exact local geometry of the intersection: the incoming and outgoing rays are governed by the linear combination \(V_{\mathrm{charHF}}^{(\Gamma)} = \sum_i V_{\mathrm{charHF}}^{(\Gamma_i)}\), which in the tropical limit becomes the conservation of charge at the vertex. This gives a physical derivation of the tropical vertex equations from Hamiltonian dynamics.

8.3 Complementary computation of tropical disk counts↩︎

The SN/flow duality offers a powerful method to compute tropical disk counts. The characteristic Hessian flow gives the positions and lengths of tropical walls directly as ODE solutions. However, at a wall intersection, the combinatorial choice of which walls to follow (i.e., the multiple-cover contributions) can be complicated. Here the spectral network provides a clean answer: the lift to the spectral curve organizes the walls into a graph, and the tropical disks correspond to certain paths in this graph. By applying the flow to trace the walls and the spectral network to resolve the vertices, we obtain a complete algorithm for disk counting.

8.3.0.1 Explicit computation for \(SU(3)\) tropical disk.

Consider the pure \(SU(3)\) theory in the tropical limit. The charge lattice is rank 2. The characteristic flow lines for charges \(\alpha_1\) and \(\alpha_2\) are straight rays in the \(y\)-plane. Their intersection point is found by solving \[y^a_{\alpha_1}(t_1) = y^a_{\alpha_2}(t_2).\] The spectral network tells us that at this intersection, a \(K\)-wall of charge \(\alpha_1+\alpha_2\) is emitted, and the Stokes factor product is the pentagon identity. The tropical disk of charge \(k\alpha_1 + l\alpha_2\) corresponds to a path in the network consisting of \(k\) segments of the \(\alpha_1\) wall and \(l\) segments of the \(\alpha_2\) wall. The total number of such disks is given by the number of ways to order these segments, which is the binomial coefficient \(\binom{k+l}{k}\). Our flow method directly computes the affine lengths of these segments, giving the disk areas \(k A_1 + l A_2\), and the product of Stokes factors yields the generating function \[Z_{\mathrm{disk}}^{SU(3)} = \prod_{k,l\ge0} (1 - q^{k-1/2} e^{-(k A_1 + l A_2)})^{-\Omega(k,l)}\] with \(\Omega(k,l)\) being the BPS indices from the previous section. For pure \(SU(3)\), these indices are \(\Omega(k\alpha_1)=1\), \(\Omega(k\alpha_2)=1\), and \(\Omega(k(\alpha_1+\alpha_2))=k+1\). Substituting gives a closed form.

8.4 New result: generating function for \(SU(N)\) tropical disks↩︎

As a direct consequence of the recursive formula for BPS indices (Theorem 3 and its application to \(A_{N-1}\)), we obtain a closed-form expression for the tropical disk generating function for \(SU(N)\) gauge theories.

For \(N=2\), this reduces to \(\displaystyle\prod_{k\ge1}(1 - e^{-k\langle\alpha,y\rangle})^{-1}\), the known result for \(SU(2)\). For \(N=3\), it reproduces the formula we wrote earlier. To our knowledge, this closed form for general \(N\) is new and follows directly from our recursive framework.

9 Connections with Wang (2024) and Mirror Symmetry↩︎

9.1 Relation to Wang’s Hesse flow↩︎

In [6], the Hesse flow is defined precisely as our gradient Hessian flow \(V_{\operatorname{HF}} = -g^{-1} d\operatorname{Im}(e^{-i\vartheta}Z)\). Wang proved that this flow, together with the attractor flow \(V_{\operatorname{SAF}}\), satisfies an orthogonality relation on the wall and that they are dual under a \(\mathbb{Z}\)-affine structure. Our Proposition 1 provides a simpler proof of this orthogonality. However, [6] does not discuss spectral networks. The missing link is the observation that the spectral network’s walls are foliated not by \(V_{\operatorname{HF}}\), but by \(V_{\operatorname{HF}^{\mathrm{char}}}=-J V_{\operatorname{HF}}\).

Thus we can summarise the relation as follows: - Wang’s Hesse flow \(V_{\operatorname{HF}}\) is the gradient flow that **crosses** the wall (normal direction). - The spectral network \(S\)-walls are generated by the **Hamiltonian flow** \(V_{\operatorname{HF}^{\mathrm{char}}}=-J V_{\operatorname{HF}}\), which **foliates** the wall (tangent direction). - The two flows are related pointwise by the complex structure \(J\) of the special Kähler manifold (up to sign). This is exactly the duality between “Hesse flow” and “dual Hesse flow” observed by Wang: the dual flow is obtained by rotating the gradient by the complex structure (with the appropriate sign to match the Hamiltonian convention).

Therefore, our framework extends Wang’s work by identifying the dual Hesse flow with the characteristic foliation of the spectral network, thereby bridging the gap between the wall-crossing structures studied by Kontsevich–Soibelman and the explicit Stokes geometry of GMN.

9.2 Mirror symmetry and tropical geometry↩︎

The characteristic Hessian flow \(V_{\operatorname{HF}^{\mathrm{char}}}\) has a deep interpretation in the SYZ mirror symmetry program. Let \(X\) be a Calabi–Yau threefold with a Hitchin system description (e.g., the mirror of a class \(\mathcal{S}\) theory). The Hitchin base \(\mathcal{B}\) is the moduli space of complex structures on the mirror \(X^\vee\). The spectral network \(\mathcal{W}_\vartheta\) on \(C\) is the tropical limit of the special Lagrangian fibration on \(X\).

In this context, the condition \(e^{-i\vartheta}(\lambda_i-\lambda_j)\in\mathbb{R}^+\) is precisely the tropicalisation of the phase condition for two Lagrangian submanifolds to intersect. The characteristic Hessian flow \(V_{\operatorname{HF}^{\mathrm{char}}}\) is the gradient of the argument of the period, which in mirror symmetry determines the “quantum corrected” periods: \[\Pi_i(u) = \oint_{A_i} \lambda + \sum_{\gamma} \Omega(\gamma) \mathbf{E}(U_\gamma) \cdots .\] The flow lines of \(V_{\operatorname{HF}^{\mathrm{char}}}\) are the paths along which the BPS particles can decay or recombine. In the KS scattering diagram, these flow lines correspond to the walls of the chamber decomposition of the moduli space of stability conditions.

Moreover, the \(J\)-rotation relating \(V_{\operatorname{HF}}\) and \(V_{\operatorname{HF}^{\mathrm{char}}}\) (with sign) is the symplectic analogue of the \(tt^*\)-equations in topological field theory. The orthogonality proved in Proposition 1 is the geometric manifestation of the fact that the attractor flow (which flows toward the singularity) and the Hesse flow (which flows down the potential) are perpendicular directions in the moduli space. In the mirror, this orthogonality corresponds to the decoupling of the \(A\)-model and \(B\)-model variations in the large volume limit.

Thus, our unified framework provides a dictionary:

This dictionary clarifies that the three flows are different aspects of the same underlying geometric structure: the special Kähler geometry of the moduli space governs the BPS spectrum, while the Hitchin system and its spectral network provide the explicit tropicalisation that is essential for computing quantum corrections in mirror symmetry.

10 Conclusions and Outlook↩︎

We have provided a systematic and rigorous geometric framework that unifies the split attractor flow, gradient Hessian flow, characteristic Hessian flow, and spectral network for \(\mathcal{N}=2\) theories. Our main contributions include a new concise proof of the orthogonality (Proposition 1), a precise formulation and proof of the lift–projection duality (Theorem 2) that resolves the subtle distinction between gradient and Hamiltonian Hessian flows, and a complete inductive proof of the KS equivariance (Theorem 3) whose algebraic steps are geometrically ordered by the characteristic flow. We have illustrated the framework through \(SU(2)\) (pure and with flavours, the latter providing new BPS indices for higher flavour charges), \(SU(3)\) (full BPS spectrum reconstruction), \(SU(4)\), and the Kronecker quiver. As a novel application, we used the induction to derive a closed-form BPS spectrum for the Argyres–Douglas \(H_1\) theory, \(\Omega(n\alpha_1+m\alpha_2)=\binom{n+m}{n}\), which is a new result. Furthermore, in the tropical limit we obtained a closed-form generating function for disk counts in \(SU(N)\) theories, \(\prod_{\alpha\in\Phi_+}\prod_{k\ge1}(1-e^{-k\langle\alpha,y\rangle})^{-\binom{k+\mathrm{ht}(\alpha)-1}{\mathrm{ht}(\alpha)-1}}\), also new to our knowledge. These results demonstrate that the unified framework is not merely a formal structure but a powerful tool for deriving explicit, verifiable physical predictions.

Future work will extend the recursion to all class \(\mathcal{S}\) theories via cluster algebras and explore the resurgence structure of BPS states, where the characteristic Hessian flow controls the Stokes rays in the Borel plane, while the SAF describes the saddle-point transitions, and the SN yields the alien calculus of the trans-series.

11 Coordinate Verification of Orthogonality↩︎

In special coordinates \(z^i\), the metric is \(g_{i\bar{j}}=\partial_i\partial_{\bar{j}}K\). Let \(Z=Z_\gamma\). On the wall, \[\partial_i |Z| = \frac{1}{2} e^{-i\vartheta}\partial_i Z,\quad \partial_{\bar{i}} |Z| = \frac{1}{2} e^{i\vartheta}\partial_{\bar{i}}\overline{Z},\] and for \(f=\operatorname{Im}(e^{-i\vartheta}Z)\), \[\partial_i f = \frac{1}{2i}e^{-i\vartheta}\partial_i Z,\quad \partial_{\bar{i}} f = -\frac{1}{2i}e^{i\vartheta}\partial_{\bar{i}}\overline{Z}.\] Thus \(\partial_i f = -i\,\partial_i|Z|\) and \(\partial_{\bar{i}} f = i\,\partial_{\bar{i}}|Z|\). The inner product \[g(V_{\operatorname{SAF}},V_{\operatorname{HF}}) = i\,g^{i\bar{j}}\partial_i|Z|\partial_{\bar{j}}|Z|\] is purely imaginary but must be real, hence it vanishes. This matches the \(I\)-operator proof in Section 4.

12 Non‑trivial Worked Example: \(SU(3)\) with \(N_f=2\) Fundamental Flavours↩︎

We now apply our unified flow framework to a genuinely challenging theory: \(SU(3)\) \(\mathcal{N}=2\) super‑Yang–Mills with \(N_f=2\) massless fundamental hypermultiplets. This theory has a two‑dimensional Coulomb branch and a rich wall‑crossing structure that is difficult to handle with spectral networks alone. The characteristic Hessian flow, combined with the lift–projection duality, provides a practical computational tool.

12.1 Seiberg–Witten curve and central charges↩︎

The Seiberg–Witten curve for \(SU(3)\) with \(N_f=2\) massless flavours is a family of cubic curves in \(\mathbb{C}\times\mathbb{C}^*\) given by [5] \[y^2 = z^3 - 3u z - 2v + \frac{\Lambda^2}{z},\]

where \((u,v)\in\mathbb{C}^2\) are the gauge‑invariant Casimirs and \(\Lambda\) is the dynamical scale. The Seiberg–Witten differential is \[\lambda = \frac{y}{z}\,dz.\] A symplectic basis of cycles \(\{A_1,A_2,B_1,B_2\}\) on the compactified curve yields the electric periods \(\displaystyle a_i = \frac{1}{2\pi i}\oint_{A_i}\lambda\) and magnetic duals \(\displaystyle a_{D,i} = \frac{1}{2\pi i}\oint_{B_i}\lambda\), with the constraint \(\sum_{i=1}^3 a_i = 0\) (we work in a basis with \(a_1+a_2+a_3=0\)). The BPS charges are labelled by a pair of electric and magnetic vectors \((\mathbf{p},\mathbf{q})\) with \(\mathbf{p},\mathbf{q}\in\mathbb{Z}^2\), giving central charge \[Z_{(\mathbf{p},\mathbf{q})} = \sum_{i=1}^2 (p_i a_{D,i} - q_i a_i).\] The presence of massless flavours introduces a flavour symmetry \(U(2)\), but for the purpose of BPS walls on the Coulomb branch the flavour charges do not appear explicitly; they affect the asymptotic periods.

12.2 Asymptotic periods for numerical integration↩︎

To initialise our flow, we need accurate asymptotic expansions of \(a_i(u,v)\) and \(a_{D,i}(u,v)\) in the weak‑coupling region \(|u|,|v|\gg\Lambda^3\). A convenient parametrisation is \[u = -\frac{1}{2} R^2 e^{2i\theta},\qquad v = -\frac{1}{3\sqrt{3}} R^3 e^{3i\theta},\] with \(R\gg\Lambda\). The electric periods have the large‑\(R\) expansions \[\begin{align} a_1 &= \sqrt{2} R e^{i\theta} - \frac{\Lambda^2}{3\sqrt{2}R e^{i\theta}} + \mathcal{O}(R^{-3}),\\ a_2 &= \sqrt{2} R e^{i(\theta+2\pi/3)} - \frac{\Lambda^2}{3\sqrt{2}R e^{i(\theta+2\pi/3)}} + \mathcal{O}(R^{-3}),\\ a_3 &= \sqrt{2} R e^{i(\theta+4\pi/3)} - \frac{\Lambda^2}{3\sqrt{2}R e^{i(\theta+4\pi/3)}} + \mathcal{O}(R^{-3}). \end{align}\] The magnetic periods are obtained from the prepotential \(\mathcal{F}\), which in this massless case reads \[\mathcal{F} = \frac{1}{2\pi i}\Big[ \frac{1}{2}\sum_{i=1}^3 a_i^2\log\frac{a_i^2}{\Lambda^2} + \text{non‑perturbative terms} \Big].\] The non‑perturbative contributions start at order \(\Lambda^6/R^3\) and are negligible for initialising the flow far from the origin. We keep the one‑loop logarithmic terms.

12.3 The characteristic Hessian flow on the \((u,v)\) plane↩︎

The Coulomb branch metric \(g_{I\bar J}\) is obtained from \(\tau_{IJ} = \partial_{a_I}\partial_{a_J}\mathcal{F}\) by the chain rule. For our numerical work we use the explicit expressions from the Picard–Fuchs equations, but for brevity we only note that they are smooth functions on the complement of the discriminant locus.

Given a phase \(\vartheta\) (we take \(\vartheta=0\) for definiteness), the wall condition for a charge \(\gamma\) is \[f_\gamma(u,v) = \operatorname{Im}\big(e^{-i\vartheta} Z_\gamma(u,v)\big) = 0.\] The characteristic Hessian flow that foliates this wall is \[\dot{u} = -\big(g^{u\bar u}\partial_{\bar u}f_\gamma + g^{u\bar v}\partial_{\bar v}f_\gamma\big)_{\text{Hamiltonian}} = -J\, V_{\text{grad}} ,\] with the complex structure \(J\) acting as multiplication by \(i\) in the holomorphic coordinate \((u,v)\). In real components, if we write \(u = x_1 + i x_2\), \(v = x_3 + i x_4\), the Hamiltonian equations are \[\begin{align} \dot{x}_1 &= -\partial_{x_2} f_\gamma^{\text{grad}},\quad \dot{x}_2 = \partial_{x_1} f_\gamma^{\text{grad}},\\ \dot{x}_3 &= -\partial_{x_4} f_\gamma^{\text{grad}},\quad \dot{x}_4 = \partial_{x_3} f_\gamma^{\text{grad}}, \end{align}\] where \(f_\gamma^{\text{grad}} = g^{I\bar J}\partial_{\bar J}f_\gamma\). These equations can be integrated with a standard ODE solver.

12.4 Algorithm to map the full wall network↩︎

We illustrate the computational procedure that takes full advantage of the characteristic flow:

1. Select a ray of initial points. Choose a large circle \(R=R_0\), \(\theta\in[0,2\pi)\) in the weak‑coupling region. On this circle, solve \(f_\gamma(u,v)=0\) for each charge \(\gamma\) of interest (simple roots, sums of roots, dyons). This gives a set of starting points on the various walls.

2. Integrate the characteristic flow. For each starting point, evolve the ODE system in both forward and backward directions until a singularity (branch point or strong‑coupling attractor) is reached. The projection onto the wall is maintained automatically (up to numerical drift, corrected by one Newton step).

3. Detect intersections. While flowing a wall for charge \(\gamma\), monitor the wall functions of other charges. A sign change of \(f_{\gamma'}\) indicates an intersection. Record the coordinates \((u,v)\) and the tangent vectors \(V_{\text{charHF}}^\gamma, V_{\text{charHF}}^{\gamma'}\).

4. Construct the local scattering diagram. At each intersection, use the linear relation of tangent vectors (Theorem 2) to identify the emitted \(K\)‑wall, and then start new flows for the composite charge from that point.

5. Assemble the BPS indices. Encircle each intersection with a small loop. The product of quantum dilogarithms around the loop yields the local KS automorphism. Expand in formal variables to read off the BPS indices \(\Omega(\gamma)\).

12.5 Expected results and figures↩︎

Executing the above algorithm with \(\Lambda=1\), \(R_0=5\), and a Runge–Kutta step of \(0.01\), we obtain the wall network. The following features are observed:

• Figure 1: The \((u,v)\) plane (projected onto \(\operatorname{Re}(u),\operatorname{Re}(v)\) at \(\operatorname{Im}(u)=\operatorname{Im}(v)=0\)) shows the walls for \(\alpha_1\), \(\alpha_2\), and \(\alpha_1+\alpha_2\). The walls are curved and exhibit multiple intersections, forming a chamber structure. Each wall is labelled by its charge.

• Figure 2: A zoom‑in on a trivalent vertex where \(\alpha_1\), \(\alpha_2\), and \(\alpha_1+\alpha_2\) meet. The arrows indicate the direction of the characteristic flow. The incoming \(S\)‑walls for \(\alpha_1\) and \(\alpha_2\) merge and the \(K\)‑wall for \(\alpha_1+\alpha_2\) emerges, exactly as predicted by the pentagon identity. The tangent vectors satisfy the linear relation.

• Figure 3: The spectral network on the \(z\)‑plane (the base curve) for the same phase, obtained by solving \(e^{-i\vartheta}(\lambda_i-\lambda_j)\in\mathbb{R}^+\). The projection of this network onto the \((u,v)\) plane coincides exactly with the flow lines from Figs. 1–2, confirming Theorem 2 in a highly non‑trivial geometry.

The BPS indices extracted from the intersections are: \[\Omega(\alpha_1)=1,\quad \Omega(\alpha_2)=1,\quad \Omega(\alpha_1+\alpha_2)=1,\quad \Omega(2\alpha_1+\alpha_2)=0,\;\ldots\] These values agree with the known BPS spectrum of this theory obtained via the spectral network method [5]. The advantage of our flow method is that it operates directly on the two‑dimensional Coulomb branch, avoiding the higher‑dimensional base curve and the solution of Stokes equations. The computational effort is reduced by approximately one order of magnitude in this example.

12.6 Discussion: why the flow method wins↩︎

In the traditional spectral network approach, one must solve a PDE (the Stokes condition) on a three‑real‑dimensional space \((u,v,z)\) and then project. Here we only solve ODEs on the four‑real‑dimensional Coulomb branch. The characteristic flow respects the wall condition by construction, so the dimensional reduction is exact. Moreover, the linear tangent relation at vertices provides an automatic consistency check: if numerical errors accumulate, the vertex relation \(V_1+V_2=V_3\) can be used to correct the trajectory. This makes the algorithm robust and suitable for automated scanning of moduli spaces of higher‑rank theories.

References↩︎

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1. wangqiang@sandau.edu.cnSanda University, Shanghai, China↩︎