Wall-crossing, Hitchin systems, and the WKB approximation

We consider BPS states in a large class of d=4d=4, N=2N=2 field theories, obtained by reducing six-dimensional (2,0)(2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S1S1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich–Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in Gaiotto et al. [40]. It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.