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Consider an instance $h$ of the Gaussian free field on a simply connectedplanar domain with boundary conditions $-\lambda$ on one boundary arc and$\lambda$ on the complementary arc, where $\lambda$ is the special constant$\sqrt{\pi/8}$. We argue that even though $h$ is defined only as a randomdistribution, and not as a function, it has a well-defined zero contour lineconnecting the endpoints of these arcs, whose law is SLE(4). We construct thiscontour line in two ways: as the limit of the chordal zero contour lines of theprojections of $h$ onto certain spaces of piecewise linear functions, and asthe only path-valued function on the space of distributions with a naturalMarkov property.

A contour line of the continuum Gaussian free field