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

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