  We conformally weld (via "quantum zipping") two boundary arcs of a Liouville
quantum gravity random surface to generate a random curve called the
Schramm-Loewner evolution (SLE). We develop a theory of quantum fractal
measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and
analyze their evolution under welding via SLE martingales. As an application,
we construct the natural quantum length and boundary intersection measures on
the SLE curve itself.
