We present a compact analytic formula for the two-loop six-particle maximally helicity violating remainder function (equivalently, the two-loop lightlike hexagon Wilson loop) in N=4 supersymmetric Yang-Mills theory in terms of the classical polylogarithm functions Lik with cross ratios of momentum twistor invariants as their arguments. In deriving our formula we rely on results from the theory of motives.

We give a representation of the parity-even part of the planar two-loop six-gluon maximally helicity violating (MHV) amplitude of N ¼ 4 super-Yang-Mills theory, in terms of loop-momentum integrals with simple dual conformal properties. We evaluate the integrals numerically in order to test directly the Anastasiou-Bern-Dixon-Kosower/Bern-Dixon-Smirnov all-loop ansatz for planar MHV amplitudes. We find that the ansatz requires an additive remainder function, in accord with previous indications from strong-coupling and Regge limits. The planar six-gluon amplitude can also be compared with the hexagonal Wilson loop computed by Drummond, Henn, Korchemsky, and Sokatchev in arXiv:0803.1466. After accounting for differing singularities and other constants independent of the kinematics, we find that the Wilson loop and MHV-amplitude remainders are identical, to within our numerical precision. This result provides nontrivial confirmation of a proposed n-point equivalence between Wilson loops and planar MHV amplitudes, and suggests that an additional mechanism besides dual conformal symmetry fixes their form at six points and beyond.

In this note we further investigate the procedure for computing tree-level amplitudes in Yang-Mills theory from connected instantons in the B-model on P 3|4 , emphasizing that the problem of calculating Feynman diagrams is recast into the problem of finding solutions to a certain set of algebraic equations. We show that the B-model correctly reproduces all 6-particle non-MHV amplitudes. As a further check, we also show that n-particle amplitudes obtained from the B-model obey a number of properties required of gauge theory, such as having the correct soft and collinear gluon poles.