  We define natural probability measures on cycle-rooted spanning forests
(CRSFs) on graphs embedded on a surface with a Riemannian metric. These
measures arise from the Laplacian determinant and depend on the choice of a
unitary connection on the tangent bundle to the surface.
  We show that, for a sequence of graphs $(G_n)$ conformally approximating the
surface, the measures on CRSFs of $G_n$ converge and give a limiting
probability measure on finite multicurves (finite collections of pairwise
disjoint simple closed curves) on the surface, independent of the approximating
sequence.
  Wilson's algorithm for generating spanning trees on a graph generalizes to a
cycle-popping algorithm for generating CRSFs for a general family of weights on
the cycles. We use this to sample the above measures. The sampling algorithm,
which relates these measures to the loop-erased random walk, is also used to
prove tightness of the sequence of measures, a key step in the proof of their
convergence.
  We set the framework for the study of these probability measures and their
scaling limits and state some of their properties.
