This book is the outcome of research initiatives formed during the special ``Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory'' at the ICMAT (Instituto de Ciencias Matemáticas, Madrid) in 2014. The activity was aimed at understanding and deepening recent developments where Feynman and string amplitudes on the one hand, and periods and multiple zeta values on the other, have been at the heart of lively and fruitful interactions between theoretical physics and number theory over the past few decades.In this book, the reader will find research papers as well as survey articles, including open problems, on the interface between number theory, quantum field theory and string theory, written by leading experts in the respective fields. Topics include, among others, elliptic periods viewed from both a mathematical and a physical standpoint; further relations between periods and high energy physics, including cluster algebras and renormalisation theory; multiple Eisenstein series and q-analogues of multiple zeta values (also in connection with renormalisation); double shuffle and duality relations; alternative presentations of multiple zeta values using Ecalle's theory of moulds and arborification; a distribution formula for generalised complex and l-adic polylogarithms; Galois action on knots. Given its scope, the book offers a valuable resource for researchers and graduate students interested in topics related to both quantum field theory, in particular, scattering amplitudes, and number theory.

Contains articles from distinguished experts in both Theoretical Physics and Pure Mathematics

I. Todorov, Perturbative quantum field theory meets number theory.- E. Panzer, Some open problems on Feynman periods.- S. Stieberger, Periods and Superstring Amplitudes.- O. Schlotterer.- The number theory of superstring amplitudes.- N. Matthes, Overview On Elliptic Multiple Zeta Values.- L. Adams, C. Bogner, S. Weinzierl, The Elliptic Sunrise.- C. Vergu, Polylogarithm identities, cluster algebras and the N = 4 supersymmetric theory.- H. Bachmann, Multiple Eisenstein series and q-analogues of multiple zeta values.- H. Bachmann, U. Kühn, A dimension conjecture for q-analogues of multiple zeta values.- J. Zhao, Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras.- J. Singer, q-Analogues of multiple zeta values and their applications in renormalization.- N. M. Nikolov, Vertex algebras and renormalization.- K. Rejzner, Renormalization and periods in perturbative Algebraic Quantum Field Theory.- C. Malvenuto, F. Patras, Symmetril moulds, generic group schemes, resummation of MZVs.- A. Salerno, L. Schneps, Mould theory and the double shuffle Lie algebra structure.- F. Chapoton, On some tree-indexed series with one and two parameters.- K. Ebrahimi-Fard, W. Steven Gray, D. Manchon, Evaluating Generating Functions for Periodic Multiple Polylogarithms.- D. Manchon, Arborified multiple zeta values.- L. Foissy, F. Patras, Lie theory for quasi-shuffle bialgebras.- H. Furusho, Galois action on knots II: Proalgebraic string links and knots.- H. Nakamura, Z. Wojtkowiak, On distribution formulas for complex and l-adic polylogarithms.- W. Zudilin, On a family of polynomials related to ¿(2,1)=¿(3).