  BPS Wilson loops in supersymmetric gauge theories have been the subjects of
active research since they are often amenable to exact computation. So far most
of the studies have focused on loops that do not intersect. In this paper, we
derive exact results for intersecting 1/8 BPS Wilson loops in N=4
supersymmetric Yang-Mills theory, using a combination of supersymmetric
localization and the loop equation in 2d gauge theory. The result is given by a
novel matrix-model-like representation which couples multiple contour integrals
and a Gaussian matrix model. We evaluate the integral at large N, and make
contact with the string worldsheet description at strong coupling. As an
application of our results, we compute exactly a small-angle limit (and more
generally near-BPS limits) of the cross anomalous dimension which governs the
UV divergence of intersecting Wilson lines. The same quantity describes the
soft anomalous dimension of scattering amplitudes of W-bosons in the Coulomb
branch.
