Amplitudes at strong coupling as hyperk\"ahler scalars

Alday & Maldacena conjectured an equivalence between string amplitudes in AdS$_5 \times S^5$ and null polygonal Wilson loops in planar $\mathcal{N}=4$ super-Yang-Mills (SYM). At strong coupling this identifies SYM amplitudes with areas of minimal surfaces in AdS. For minimal surfaces in AdS$_3$, we find that the nontrivial part of these amplitudes, the \emph{remainder function}, satisfies an integrable system of nonlinear differential equations, and we give its Lax form. The result follows from a new perspective on `Y-systems', which defines a new psuedo-hyperk\"ahler structure \emph{directly} on the space of kinematic data, via a natural twistor space defined by the Y-system equations. The remainder function is the (pseudo-)K\"ahler scalar for this geometry. This connection to pseudo-hyperk\"ahler geometry and its twistor theory provides a new ingredient for extending recent conjectures for non-perturbative amplitudes using structures arising at strong coupling.