Periods in Quantum Field Theory and Arithmetic: ICMAT, Madrid, Spain, September 15-December 19, 2014

Description

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. Preface Contents Perturbative Quantum Field Theory Meets Number Theory 1 Introduction 2 Residues of Primitively Divergent Amplitudes 2.1 Periods in Position Space Renormalization 2.2 Vacuum Completion of 4-Point Graphs in 4 2.3 Primitive Conformal Amplitudes 3 Double Shuffle Algebra of Hyperlogarithms 4 Formal Multizeta Values 4.1 Shuffle Regularized MZVs 4.2 Hopf Algebra of Motivic Zeta Values 5 Single-Valued Hyperlogarithms. Applications 6 Outlook References Some Open Problems on Feynman Periods 1 The Leading Feynman Period 2 Generalized Feynman Periods 2.1 Convergence 2.2 General Properties and Relations 2.3 Families of Graphs with Polylogarithmic Periods 2.4 Further Remarks References Periods and Superstring Amplitudes 1 Introduction 2 Periods on calM0,N calM0,N calM0,N calM0,N 3 Volume Form and Period Matrix on calM0,N calM0,N calM0,N calM0,N 4 Motivic and Single–valued Multiple Zeta Values 5 Motivic Period Matrix Fmathfrakm Fmathfrakm Fmathfrakm Fmathfrakm 6 Open and Closed Superstring Amplitudes 7 Complex Versus Iterated Integrals References The Number Theory of Superstring Amplitudes 1 Introduction 1.1 The Disk Amplitude 2 The α'-Expansion from the Drinfeld Associator 2.1 Background on MZVs and the Drinfeld Associator 2.2 Deforming the Disk Integrals 2.3 Four- and Five-Point Examples 2.4 Higher Multiplicity 3 Motivic MZVs and the α'-Expansion 3.1 Matrix-Valued Approach to Disk Amplitudes 3.2 Motivic MZVs 3.3 Cleaning up the α'-Expansion 4 Further Directions and Open Questions 4.1 The Closed String at Genus Zero and Single-Valued MZVs 4.2 The Open String at Genus One and Elliptic MZVs 4.3 The Closed String at Higher Genus References Overview on Elliptic Multiple Zeta Values 1 Introduction 1.1 Multiple Zeta Values 1.2 The Algebra of A-elliptic Multiple Zeta Values 1.3 Structure of the Article 1.4 Note Added in Print 2 Multiple Polylogarithms and the Drinfeld Associator 2.1 Iterated Integrals 2.2 Multiple Polylogarithms 2.3 The Drinfeld Associator 3 Multiple Elliptic Polylogarithms and the Elliptic KZB Associator 3.1 An Elliptic Analogue of the KZ-Equation 3.2 The Elliptic KZB Associator 3.3 Multiple Elliptic Polylogarithms 4 The Algebra of A-elliptic Multiple Zeta Values 4.1 Definition of A-elliptic Multiple Zeta Values 4.2 Relations Between A-elliptic Multiple Zeta Values 4.3 The Dimension of the Space of A-elliptic Multiple Zeta Values 5 Elliptic Multiple Zeta Values and Iterated Eisenstein Integrals 5.1 Reminder on Iterated Eisenstein Integrals 5.2 The Differential Equation for A-elliptic Multiple Zeta Values 5.3 Restoring the Constant Terms of A-elliptic Multiple Zeta Values 5.4 Explicit Formulae for A-elliptic Multiple Zeta Values 5.5 A Special Algebra of Derivations References The Elliptic Sunrise 1 Introduction 2 Basic Properties of the Massive Sunrise Integral 3 The Differential Equation in Two Dimensions 4 The Massive Sunrise Integral in Two Dimensions 5 The Massive Sunrise Integral Around Four Dimensions 6 Conclusions References Polylogarithm Identities, Cluster Algebras and the mathcalN = 4 Supersymmetric Theory 1 Introduction 2 The Maximally Supersymmetric Theory 3 Kinematics 4 Introduction to Cluster Algebras 5 The Cluster Algebra for mathbbG(k,n) 6 Poisson Brackets 7 Elements of Projective Geometry 8 Polylogarithm Identities 9 Open Questions References Multiple Eisenstein Series and q-Analogues of Multiple Zeta Values 1 Introduction 2 Outlook and Related Work 3 Multiple Eisenstein Series 3.1 Multiple Zeta Values and Quasi-shuffle Algebras 3.2 Multiple Eisenstein Series and the Calculation of Their Fourier Expansion 4 Multiple Divisor-Sums and Their Generating Functions 4.1 Brackets 4.2 Derivatives and Subalgebras 5 Bi-Brackets and a Second Product Expression for Brackets 5.1 Bi-Brackets and Their Generating Series 5.2 Double Shuffle Relations for Bi-Brackets 5.3 The Shuffle Brackets 6 Regularizations of Multiple Eisenstein Series 6.1 Formal Iterated Integrals 6.2 Shuffle Regularized Multiple Eisenstein Series 6.3 Stuffle Regularized Multiple Eisenstein Series 6.4 Double Shuffle Relations for Regularized Multiple Eisenstein Series 7 q-Analogues of Multiple Zeta Values 7.1 Brackets as q-Analogues of MZV and the Map Zk 7.2 Connection to Other q-Analogues References A Dimension Conjecture for q-Analogues of Multiple Zeta Values 1 Introduction 2 q-analogues of MZVs and Bi-Brackets 2.1 Bi-Brackets as q-Analogues of MZVs 2.2 Bi-Brackets and Quasi-modular Forms 3 Computational Evidences for the Conjectures References Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota–Baxter Algebras 1 Introduction 2 Convergence Domain for q-MZVs 3 Rota–Baxter Algebra 4 q-Analogs of Hoffman algebras 5 q-Stuffle relations 6 Jackson's Iterated q-Integrals 7 q-Shuffle Relations 8 Duality Relations 9 The General Type G q-MZVs 10 Numerical Data 11 Conclusions References q-Analogues of Multiple Zeta Values and Their Application in Renormalization 1 Introduction 2 Linear Relations Among Multiple Zeta Values 2.1 Double Shuffle Relations 2.2 Duality and Derivation Relations 3 q-Analogues of Multiple Zeta Values 3.1 From Jackson's Integral to Rota–Baxter Operators 3.2 Algebraic Setting of Double Shuffle Structure 3.3 Duality and q-Shuffle 4 q-Renormalization of Multiple Zeta Values 4.1 Quasi-shuffle Renormalization Problem 4.2 Shuffle Renormalization Problem 4.3 Comparison of Different Solutions to the Renormalization Problem References Vertex Algebras and Renormalization 1 The General Concept of OPE in the Axiomatic QFT 2 Huygens Locality and Vertex Algebras 3 Operads and OPE Algebras 4 Connection with the Renormalization Theory References Renormalization and Periods in Perturbative Algebraic Quantum Field Theory 1 Introduction 2 Functionals 3 The S-Matrix and Time-Ordered Products 4 Distributional Residues and Periods 5 Renormalization Group Flow 6 Conclusion References Symmetril Moulds, Generic Group Schemes, Resummation of MZVs 1 Introduction 2 Hopf Algebras 3 Generic Bialgebras 4 Symmetril Moulds and Generic Group Schemes 5 Resummation of MZVs 6 A New Resummation Process References Mould Theory and the Double Shuffle Lie Algebra Structure 1 Introduction 2 Definitions for Mould Theory 3 Lie Subalgebras of `3́9`42`"̇613A``45`47`"603AARI 4 Dictionary with the Lie Algebra and Double Shuffle Framework 5 The Group `3́9`42`"̇613A``45`47`"603AGARI 6 The Mould Pair pal/pil and Ecalle's Fundamental Identity 7 The Main Theorem References On Some Tree-Indexed Series with One and Two Parameters References Evaluating Generating Functions for Periodic Multiple Polylogarithms via Rational Chen–Fliess Series 1 Introduction 2 Preliminaries 2.1 Chen–Fliess Series 2.2 Bilinear Realizations of Rational Chen–Fliess Series 3 Evaluating Periodic Multiple Polylogarithms 3.1 Periodic Multiple Polylogarithms 3.2 Periodic Multiple Polylogarithms with Non-periodic Components 3.3 The Dendriform Setting 4 Examples 5 Conclusions References Arborified Multiple Zeta Values 1 Introduction 2 Shuffle and Quasi-Shuffle Hopf Algebras 3 The Butcher-Connes-Kreimer Hopf Algebra of Decorated Rooted Trees 4 Simple and Contracting Arborification 5 Arborification of the Map mathfraks 6 Poset Multiple Zeta Values References Lie Theory for Quasi-Shuffle Bialgebras 1 Introduction 2 Quasi-Shuffle Algebras 3 Quasi-Shuffle Bialgebras 4 Lie Theory for Quasi-Shuffle Bialgebras 5 Endomorphism Algebras 6 Canonical Projections on Primitives 7 Relating the Shuffle and Quasi-Shuffle Operads 8 Coalgebra and Hopf Algebra Endomorphisms 9 Coderivations and Graduations 10 Decorated Operads and Graded Structures 11 Structure of the Decorated Quasi-Shuffle Operad 12 The Quasi-Shuffle Analog of the Descent Algebra 13 Lie Theory, Continued References Galois Action on Knots II: Proalgebraic String Links and Knots 1 Introduction 2 Proalgebraic Braids and Infinitesimal Braids 2.1 The GT-Action 2.2 The GRT-Action 2.3 Associators 2.4 The Motivic Galois Group 3 Proalgebraic Tangles and Chord Diagrams 3.1 The GT-Action 3.2 The GRT-Action 3.3 Associators 4 Main Results 4.1 Proalgebraic String Links 4.2 Proalgebraic Knots References On Distribution Formulas for Complex and l-adic Polylogarithms 1 Introduction 2 Complex Distribution Relations 3 l-adic Case (general) 4 l-adic Polylogarithms (Review) 5 Distribution Relations for tildeχmz 6 Homogeneous Form 7 Translation in Kummer–Heisenberg Measure 8 Inspection of Special Cases References On a Family of Polynomials Related to ζ(2,1)=ζ(3) 1 Introduction 2 Multiple Polylogarithms 3 Special Polynomials 4 A General Family of Polynomials 5 Polynomials Related to the Alternating MZV Identity References