Alday & Maldacena conjectured an equivalence between string amplitudes in AdS5 ×S5 and null polygonal Wilson loops and a duality with amplitudes for planar N = 4 super-Yang-Mills (SYM). At strong coupling this identifies SYM amplitudes with (regularized) areas of minimal surfaces in AdS. They reformulated the problem as a Hitchin system and in collaboration with Gaiotto, Sever & Vieira they introduced a Y-system and a thermodynamic Bethe ansatze (TBA) expressing the complete integrability that could in principle be used to solve for the amplitude. This lecture will review the parts of this material that we need and use them to identify new geometric structures on the spaces of kinematics for super Yang-Mills amplitudes/null polygonal Wilson loops. In AdS3, we find that the nontrivial part of these amplitudes at strong coupling, the remainder function, is the (pseudo-)K ̈ahler scalar for a (pseudo-)hyper-Kaher geometry. It satisfies an integrable system and we give its its Lax form. The result follows from a new perspective on Y-systems more generally as defining the natural twistor space associated to the hyperkahler geometry of the Hitchin moduli space for these minimal surfaces. These connections in particular allows us to prove that the amplitude at strong coupling satisfies the Plebanski equations for a hyperKahler scalar for these pseudo-hyperk ̈ahler and related geometries. These hyperkahler geometries go beyond the semiflat examples with a nontrivial TBA. These new structures underpinning the N=4 SYM amplitudes will be important beyond strong coupling.
