Boundary reparametrizations and six-point functions on the AdS$_2$ string

We compute the tree-level connected six-point function of identical scalar fluctuations of the AdS$_2$ string worldsheet dual to the half-BPS Wilson line in planar ${\cal N}=4$ Super Yang-Mills. The calculation can be carried out analytically in the conformal gauge approach, where the boundary reparametrization mode of the string plays a crucial role. We also study the analytic continuation of the six-point function to an out-of-time-order configuration, which is related to a 3-to-3 scattering amplitude in flat space. As a check of our results, we also numerically compute the six-point function using the Nambu-Goto action in static gauge, finding agreement with the conformal gauge answer.