Polylogarithmic Polygon Origami

Amplitudes in planar N=4 SYM are dual to light-like polygonal Wilson-loop expectation values. In many cases their perturbative expansion can be expressed in terms of multiple polylogarithms which also obey certain single-valuedness conditions or branch cut restrictions. The rigidity of this function space, together with a few other conditions, allows one to construct the six-point amplitude -- or hexagonal Wilson loop -- through at least five loops, and the seven-point amplitude through 3.5 loops. Then one can "fold" the polygonal Wilson loops into multiple degenerate configurations, expressing the limiting behavior in terms of simpler function spaces, and learning in the process about how amplitudes factorize.