Jianqiang Zhao
In this paper we shall define the analytic continuation of the multiple polylogarithms by using Chen's theory of iterated path integrals and compute the monodromy of all multiple logarithms explicitly.
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We give a simple axiomatic description of the degree 0 part of the polylogarithm on abelian schemes and show that its realisation in analytic Deligne cohomology can be described in terms of the Bismut-Köhler higher analytic torsion form of the Poincaré bundle.
We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf algebra.
It's well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall {\em explicitly} determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications: First we study of the limit mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple {\em poly}logarithms. Then following Deligne and Beilinson we …
We give a conceptual proof of the fact that the realisation of the degree zero part of the polylogarithm on abelian schemes in analytic Deligne cohomology can be described in terms of the Bismut-Köhler higher analytic torsion form of the Poincaré bundle. Furthermore, we provide a new axiomatic characterization of the arithmetic Chern character of the Poincaré bundle using only invariance properties under isogenies. For this we obtain a decomposition result for the arithmetic Chow group of independent interest.
We investigate the elliptic analogs of multi-indexed polylogarithms that appear in the theory of the fundamental group of the projective line minus three points as sections of a universal nilpotent bundle with regular singular connection. We use an analytic uniformisation to derive the fundamental nilpotent De Rham torsor of a single elliptic curve in terms of a double Jacobi form introduced by Kronecker. We then extend this result to any smooth family, relatively to the base, i.e., to the moduli stack $M_{1,2}$ over $M_{1,1}$. Everything relies on explicit formulas that turn out to be algebra…
In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops. These quantities are elements of stuffle and shuffle algebras implying algebraic relations being widely independent of the special quantities considered. They are supplemented by structural relations. The generalizations are given in terms of generalized harmonic sums, (generalized) cyclotomic sums, and sums containing in addition binomial and inverse-binomia…
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.
We give a simple explicit construction of the Grassmannian n-logarithm, which is a multivalued analytic function on the quotient of the Grassmannian of generic n-dimensional subspaces in 2n-dimensional coordinate complex vector space by the action of the 2n-dimensional coordinate torus. We study Tate iterated integrals, which are homotopy invariant integrals of 1-forms dlog(rational functions). We introduce the Hopf algebra of integrable symbols related to an algebraic variety, which controls the Tate iterated integrals We give a simple explicit formula for the Tate iterated integrals related …
Holomorphic functions are amazing because their values in an ever so small disk in the complex plane completely determine the function values at arbitrary points in their maximum possible domain. The process of extending such a function beyond its initial domain is called analytic continuation. We attempt to make this theoretic result tractable by computers. In the present article, we first prove that any algorithm for analytic continuation can generally not depend on finitely many function values only, without closer inspection of the function itself. We then derive a computable local bound o…
We show that the hemisphere partition function for certain U(1) gauged linear sigma models (GLSMs) with D-branes is related to a particular set of Mellin-Barnes integrals which can be used for analytic continuation to the singular point in the Kähler moduli space of an $h^{1,1}=1$ Calabi-Yau (CY) projective hypersurface. We directly compute the analytic continuation of the full quantum corrected central charge of a basis of geometric D-branes from the large volume to the singular point. In the mirror language this amounts to compute the analytic continuation of a basis of periods on the mirror…
We perform the analytic continuation of solutions to the hypergeometric differential equation of order $n$ to the third regular singularity, usually denoted $z=1$, with the help of recurrences of their Mellin--Barnes integral representations. In the resonant case, there are necessarily logarithmic solutions. We apply the result to Picard-Fuchs equations of certain one--parameter families of Calabi--Yau manifolds, known as the mirror quartic and the mirror quintic.
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