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We propose that the complete planar S-matrix of N=4 super Yang-Mills -including all N^kMHV partial amplitudes to all loops - is equivalent to thecorrelation function of a supersymmetric Wilson loop in twistor space.Remarkably, the entire classical S-matrix arises from evaluating thecorrelation function in the self-dual sector, while the expansion of thecorrelation function in powers of the Yang-Mills coupling constant provides theloop expansion of the amplitudes. We support our proposal with explicitcomputations of the n particle NMHV and NNMHV trees, the integrands of the1-loop MHV and NMHV amplitudes, and the n particle 2-loop MHV amplitude. Thesecalculations are performed using the twistor action in axial gauge. In thisgauge, the Feynman diagrams of the correlation function are the planar duals ofthe usual MHV diagrams for the scattering amplitude. The results are presentedin the form of a sum of products of dual superconformal invariants in(momentum) twistor space, and agree with the expressions derived inarXiv:1009.1854 directly from the MHV rules. The twistor space Wilson loop is anatural supersymmetric generalization of the standard Wilson loop used tocompute MHV amplitudes. We show how the Penrose-Ward transform can be used todetermine a corresponding supersymmetrization on space-time.

The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space