Topological recursion, exact WKB analysis, and the (uncoupled) BPS Riemann-Hilbert problem

The notion of BPS structure describes the output of the Donaldson-Thomas theory of CY3 triangulated categories, as well as certain four-dimensional N=2 QFTs. Bridgeland formulated a certain Riemann-Hilbert-like problem associated to such a structure, seeking functions in the ℏ plane with given asymptotics whose jumping is controlled by the BPS (or DT) invariants. These appear in the description of natural complex hyperkahler metrics ("Joyce structures") on the tangent bundle of the stability space,and physically correspond to the "conformal limit". 

 

Starting from the datum of a quadratic differential on a Riemann surface X, I'll briefly recall how to associate a BPS structure to it, and explain, in the simplest examples, how to produce a solution to the corresponding Riemann-Hilbert problem using a procedure called topological recursion, together with exact WKB analysis of the resulting "quantum curve". Based on joint work with K. Iwaki.