The differential of all two-loop MHV amplitudes in N=4 Yang-Mills theory

We present an explicit analytic calculation of the differential of the planar n-particle, two-loop MHV scattering amplitude in N = 4 super Yang-Mills theory. The result is expressed only in terms of the polylogarithm functions Li-k(-x), for k = 1, 2, 3, with arguments x belonging to the special class of dual conformal cross-ratios known as cluster X-coordinates. The surprising fact that these amplitudes may be expressed in this way provides a striking example of the manner in which the cluster structure on the kinematic configuration space underlies the structure of amplitudes in SYM theory.