For polynomial cubic differentials of degree at most 3, the new spectral core determines all spectral network degenerations and produces a BPS structure satisfying the Kontsevich-Soibelman wall-crossing formula.
Riemann-Hilbert problems from rank 3 WKB spectral networks
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We extract cluster structures and establish spectral coordinates from rank 3 WKB spectral networks $\mathcal W(\varphi,\vartheta)$ when zeros of $\varphi(z)$ are almost on a line in the complex plane. Then, we provide solutions to the Riemann-Hilbert problems (cf. arXiv:1611.03697) defined by these WKB spectral networks, using the spectral coordinates. As an application, we embed spaces of framed polynomial cubic differentials, associated with these WKB spectral networks, into spaces of stability conditions, adopting the approach of arXiv:1302.7030.
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Spectral networks for polynomial cubic differentials
For polynomial cubic differentials of degree at most 3, the new spectral core determines all spectral network degenerations and produces a BPS structure satisfying the Kontsevich-Soibelman wall-crossing formula.