Geometry of BPS Attractor, Hessian, and Spectral Flows

We provide a systematic and rigorous geometric framework that relates three structures naturally associated to BPS central charges in $\mathcal{N}=2$ supersymmetric gauge theories: the split attractor flow (SAF) of $\mathrm{\mid }Z\mathrm{\mid }$, the Hessian flow (HF) of $\mathrm{Im}(e^{-i\vartheta }Z)$, 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 \emph{characteristic Hessian flow} (the Hamiltonian flow of $\mathrm{Im}(e^{-i\vartheta }Z)$) 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 $N_f=4$ (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 $H_1$ theory, $\mathrm{\Omega }(n{\alpha }_1+m{\alpha }_2)=\frac{1}{n+m}\left(\genfrac{}{}{0px}{}{n+m}{n}\right)\left(\genfrac{}{}{0px}{}{n+m}{n+1}\right)$, 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, $Z_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{k}}^{SU(N)}(y)=\exp \text{\hspace{0.17em}⁣}\left({\sum }_{\alpha \in {\mathrm{\Phi }}_{+}}{\sum }_{k=1}^{\mathrm{\infty }}\frac{1}{k}\left(\genfrac{}{}{0px}{}{k+\mathrm{h}\mathrm{t}(\alpha )-1}{\mathrm{h}\mathrm{t}(\alpha )-1}\right)e^{-k⟨\alpha ,y⟩}\right)$, which reproduces the standard scattering diagram result and confirms the geometric framework.