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.

In this paper we study motivic amplitudes -objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free from the ambiguities inherent in any attempt to choose particular functional representatives. We find that the cluster structure on the kinematic configuration space Conf n (P 3 ) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude A M 7,2 and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as cluster X -coordinates on Conf n (P 3 ). We also find intriguing relations between motivic amplitudes and the geometry of generalized associahedrons, to which cluster coordinates have a natural combinatoric connection. For example, the obstruction to A M 7,2 being expressible in terms of classical polylogarithms is most naturally represented by certain quadrilateral faces of the appropriate associahedron. We also find and prove the first known functional equation for the trilogarithm in which all 40 arguments are cluster X -coordinates of a single algebra. In this respect it is similar to Abel's 5-term dilogarithm identity.

In type IIB light-cone superstring field theory, the cubic interaction has two pieces: a delta-functional overlap and an operator inserted at the interaction point. In this paper we extend our earlier work hep-th/0204146 by computing the matrix elements of this operator in the oscillator basis of pp-wave string theory for all µp + α ′ . By evaluating these matrix elements for large µp + α ′ , we check a recent conjecture relating matrix elements of the light-cone string field theory Hamiltonian (with prefactor) to certain three-point functions of BMN operators in the gauge theory. We also make several explicit predictions for gauge theory.

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We apply the dressing method to construct new classical string solutions describing various scattering and bound states of magnons. These solutions carry one, two or three SO(6) charges and correspond to multi-soliton configurations in the generalized sine-Gordon models.

It has recently been proposed that the D-instanton expansion of the open topological B-model on P 3|4 is equivalent to the perturbative expansion of N = 4 super Yang-Mills theory in four dimensions. In this note we extend the results of hep-th/0402016 and recover the gauge theory results for all n-point googly amplitudes by computing the integral over the moduli space of curves of degree n − 3 in P 3|4 .