Amplitudes at strong coupling as hyperkähler scalars

Abstract

Alday & Maldacena conjectured an equivalence between string amplitudes in AdS5×S5 fixed by null polygonal boundaries in Minkowski-space with both amplitudes and Wilson loops in planar N=4 super-Yang-Mills (SYM). At strong coupling this leads to an identification of SYM amplitudes with areas of minimal surfaces in AdS. Together with Gaiotto, Sever & Vieira, they introduced a `Y-system' for computing this area. We first establish a correspondence between Y-systems and twistor spaces that will apply more generally, and which, in the cases considered here determine a geometry on the space of kinematic data. In the case of minimal surfaces in AdS3 with boundaries on null polygons with 4k+2 edges, we show that the geometry in question is a split signature pseudo-hyperkähler structures and that the remainder function for the amplitude is a Plebanski scalar that generates the geometry. This geometry leads to explicit overdetermined completely integrable systems of differential equations for the area, and we also give its Lax system.

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