Classical Polylogarithms for Amplitudes and Wilson Loops

We present a compact analytic formula for the two-loop six-particle MHV remainder function (equivalently, the two-loop light-like hexagon Wilson loop) in N = 4 supersymmetric Yang-Mills theory in terms of the classical polylogarithm functions Li_k with cross-ratios of momentum twistor invariants as their arguments. In deriving our result we rely on results from the theory of motives.Comment: 11 pages, v2: journal version, minor corrections and simplifications, additional details available at http://goo.gl/Cl0

Ideal webs, moduli spaces of local systems, and 3d Calabi-Yau categories

A decorated surface S is an oriented surface with punctures and a finite set of marked points on the boundary, such that each boundary component has a marked point. We introduce ideal bipartite graphs on S. Each of them is related to a group G of type A, and gives rise to cluster coordinate systems on certain spaces of G-local systems on S. These coordinate systems generalize the ones assigned to ideal triangulations of S. A bipartite graph on S gives rise to a quiver with a canonical potential. The latter determines a triangulated 3d CY category with a cluster collection of spherical objects. Given an ideal bipartite graph on S, we define an extension of the mapping class group of S which acts by symmetries of the category. There is a family of open CY 3-folds over the universal Hitchin base, whose intermediate Jacobians describe the Hitchin system. We conjecture that the 3d CY category with cluster collection is equivalent to a full subcategory of the Fukaya category of a generic threefold of the family, equipped with a cluster collection of special Lagrangian spheres. For SL(2) a substantial part of the story is already known thanks to Bridgeland, Keller, Labardini-Fragoso, Nagao, Smith, and others. We hope that ideal bipartite graphs provide special examples of the Gaiotto-Moore-Neitzke spectral networks.Comment: 60 page

Galois symmetries of fundamental groupoids and noncommutative geometry

We define motivic iterated integrals on the affine line, and give a simple proof of the formula for the coproduct in the Hopf algebra of they make. We show that it encodes the group law in the automorphism group of certain non-commutative variety. We relate the coproduct with the coproduct in the Hopf algebra of decorated rooted planar trivalent trees - a planar decorated version of the Hopf algebra defined by Connes and Kreimer. As an application we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. We give a criteria for a motivic iterated integral to be unramified at a prime ideal, and use it to estimate from above the space spanned by the values of iterated integrals. In chapter 7 we discuss some general principles relating Feynman integrals and mixed motives.Comment: 51 pages, The final version to appear in Duke Math.