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arXiv · 2606.29190

Geometry of BPS Attractor, Hessian, and Spectral Flows

Qiang Wang
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Abstract

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 $|Z|$, the Hessian flow (HF) of $\operatorname{Im}(e^{-i\vartheta}Z)$, 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 K\"ahler structure; (ii) a precise lift--projection duality showing that the spectral network projects to the \emph{characteristic Hessian flow} (the Hamiltonian flow of $\operatorname{Im}(e^{-i\vartheta}Z)$) 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, $\Omega(n\alpha_1+m\alpha_2)=\frac{1}{n+m}\binom{n+m}{n}\binom{n+m}{n+1}$, 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, $Z_{\mathrm{disk}}^{SU(N)}(y) = \exp\!\,\Bigl( \sum_{\alpha\in\Phi_+} \sum_{k=1}^{\infty} \frac{1}{k}\binom{k+\mathrm{ht}(\alpha)-1}{\mathrm{ht}(\alpha)-1} e^{-k\langle\alpha,y\rangle} \Bigr) $, which reproduces the standard scattering diagram result and confirms the geometric framework.

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Qiang Wang. 2026-06-28. Geometry of BPS Attractor, Hessian, and Spectral Flows. https://arxiv.org/abs/2606.29190

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