We study the geometric description of BPS states in supersymmetric theories
with eight supercharges in terms of geodesic networks on suitable spectral
curves. We lift and extend several constructions of Gaiotto-Moore-Neitzke from
gauge theory to local Calabi-Yau threefolds and related models. The
differential is multi-valued on the covering curve and features a new type of
logarithmic singularity in order to account for D0-branes and non-compact
D4-branes, respectively. We describe local rules for the three-way junctions of
BPS trajectories relative to a particular framing of the curve. We reproduce
BPS quivers of local geometries and illustrate the wall-crossing of finite-mass
bound states in several new examples. We describe first steps toward
understanding the spectrum of framed BPS states in terms of such "exponential
networks."Comment: 82 pages, 60 figures, typos fixe
