Riemann-Hilbert problems from rank 3 WKB spectral networks

Abstract: 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.