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

Euler complexes and geometry of modular varieties

There is a mysterious connection between the multiple polylogarithms at N-th roots of unity and modular varieties. In this paper we "explain" it in the simplest case of the double logarithm. We introduce an Euler complex data on modular curves. It includes a length two complex on every modular curve. Their second cohomology groups recover the Beilinson-Kato Euler system in K_2 of modular curves. We show that the above connection in the double logarithm case is provided by the specialization at a cusp of the Euler complex data on the modular curve Y_1(N). Furthermore, specializing the Euler complexes at CM points we find new examples of the connection with geometry of modular varieties, this time hyperbolic 3-folds.Comment: Dedicated to Joseph Bernstein for his 60th birthday. The final version. Some corrections were made. To appear in GAFA, special volume dedicated to J. Bernstei

Polylogarithms, regulators and Arakelov motivic complexes

We construct an explicit regulator map from the weigh n Bloch Higher Chow group complexto the weight n Deligne complex of a regular complex projective algebraic variety X. We define the Arakelovweight n motivic complex as the cone of this map shifted by one. Its last cohomology group is (a version of) the Arakelov Chow group defined by H. Gillet. and C.Soule. We relate the Grassmannian n-logarithms (defined as in [G5]) to geometry of the symmetric space for GL_n(C). For n=2 we recover Lobachevsky's formula for the volume of an ideal geodesic tetrahedron via the dilogarithm. Using the relationship with symmetric spaces we construct the Borel regulator on K_{2n-1}(C) via the Grassmannian n-logarithms. We study the Chow dilogarithm and prove a reciprocity law which strengthens Suslin's reciprocity law for Milnor's K_3 on curves.Comment: Version 3: It is the final version, as it will appear in JAMS. 71 pages, 12 figure

Geometry of the trilogarithm and the motivic Lie algebra of a field

We express the Aomoto trilogarithm explicitely via classical trilogarithm and investigate the algebraic-geometric structures behind this: different realuzations of the weight three motivic complexes. Using this results we give an explicit motivic construction of the Grassmannian 4-logarithm and Borel regulator map for K_7(C).Comment: This is a paper from the proceedings of Jerusalem conference on Regulator