  Seven-particle scattering amplitudes in planar super-Yang-Mills theory are
believed to belong to a special class of generalised polylogarithm functions
called heptagon functions. These are functions with physical branch cuts whose
symbols may be written in terms of the 42 cluster A-coordinates on Gr(4,7).
Motivated by the success of the hexagon bootstrap programme for constructing
six-particle amplitudes we initiate the systematic study of the symbols of
heptagon functions. We find that there is exactly one such symbol of weight six
which satisfies the MHV last-entry condition and is finite in the $7 \parallel
6$ collinear limit. This unique symbol is both dihedral and parity-symmetric,
and remarkably its collinear limit is exactly the symbol of the three-loop
six-particle MHV amplitude, although none of these properties were assumed a
priori. It must therefore be the symbol of the three-loop seven-particle MHV
amplitude. The simplicity of its construction suggests that the n-gon bootstrap
may be surprisingly powerful for n>6.

High Energy Physics - Theory