Wall-crossing, Hitchin systems, and the WKB approximation
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We consider BPS states in a large class of d=4, N=2 field theories, obtained
by reducing six-dimensional (2,0) superconformal field theories on Riemann
surfaces, with defect operators inserted at points of the Riemann surface.
Further dimensional reduction on S^1 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 0807.4723. It also yields a new method for computing the spectrum
using the combinatorics of triangulations of the Riemann surface.