A contour line of the continuum Gaussian free field

Consider an instance of the Gaussian free field on a simply connected planar domain with boundary conditions on one boundary arc and on the complementary arc, where is the special constant . We argue that even though is defined only as a random distribution, and not as a function, it has a well-defined zero level line connecting the endpoints of these arcs, and the law of is . We construct in two ways: as the limit of the chordal zero contour lines of the projections of onto certain spaces of piecewise linear functions, and as the only path-valued function on the space of distributions with a natural Markov property. We also show that, as a function of is local (it does not change when is modified away from ) and derive some general properties of local sets.