WKB expansions, resurgence and BPS structures (after Iwaki-Nakanishi)

Link: https://bbb.mpim-bonn.mpg.de/b/gae-nhq-dzk [3]

We introduce the WKB method, which was originally initiated as the method for obtaining approximate solutions of the Schrödinger equation in the semiclassical limit in quantum mechanics, and later evolved via the Borel resummation method into the so-called exact WKB analysis. We describe how the Voros symbols encode the monodromy data and how that evolves along mutations of the Stoke graphs. On the other hand, the moduli spaces of meromorphic quadratic differentials on Riemann surfaces gives the stability spaces on a triangulated category of quivers with potential. The associated BPS structures count finite-length trajectories. Our main goal is to show that the corresponding Riemann—Hilbert problem can be partially solved via exact WKB analysis. If time permits, we will briefly describe how the method of topological recursion reconstructs WKB solutions.