2026/06/28 by Qiang Wang
#math-ph #hep-th #math.MP
We provide a systematic and rigorous geometric framework that relates three structures naturally associated to BPS central charges in N=2 supersymmetric gauge theories: the split attractor flow (SAF) of |Z|, the Hessian flow (HF) of Im(e-iϑZ), and the spectral network (SN) on the base curve of the Hitchin fibration. Our main contributions are: (i) a concise proof of orthogonality between SAF and gradient Hessian flow using only the Kähler structure; (ii) a precise lift--projection duality showing that the spectral network projects to the characteristic Hessian flow (the Hamiltonian flow of Im(e-iϑZ)) on the Hitchin base, clarifying a crucial distinction; (iii) a complete proof of the Kontsevich--Soibelman (KS) equivariance by induction on the SAF tree depth, with the geometric ordering provided by the characteristic Hessian flow. We illustrate the framework with detailed and nontrivial examples: SU(2) pure and Nf=4 (including BPS indices for higher flavour charges), SU(3) pure (full BPS spectrum reconstruction), SU(4), the Kronecker 3-quiver, and we apply the induction to derive a closed-form BPS spectrum for the Argyres--Douglas H1 theory, Ω(nα1+mα2)=(1)/(n+m)\binomn+mn\binomn+mn+1, which is known from Cecotti--Vafa and serves as a strong consistency check of our geometric recursion. In the tropical limit we obtain an explicit generating function for disk counts in SU(N) gauge theories, ZdiskSU(N)(y) = exp ( ∑α∈Φ+ ∑k=1∞ (1)/(k)\binomk+ht(α)-1ht(α)-1 e-k⟨α,y⟩ ) , which reproduces the standard scattering diagram result and confirms the geometric framework.