**Contemporary Mathematics**

Volume: 648;
2015;
289 pp;
Softcover

MSC: Primary 11; 14; 16; 32; 81;

**Print ISBN: 978-1-4704-2247-9
Product Code: CONM/648**

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**Electronic ISBN: 978-1-4704-2727-6
Product Code: CONM/648.E**

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# Feynman Amplitudes, Periods and Motives

Share this page *Edited by *
*Luis Álvarez-Cónsul; José Ignacio Burgos-Gil; Kurusch Ebrahimi-Fard*

This volume contains the proceedings of the International Research
Workshop on Periods and Motives—A Modern Perspective on
Renormalization, held from July 2–6, 2012, at the Instituto de
Ciencias Matemáticas, Madrid, Spain.

Feynman amplitudes are integrals attached to Feynman diagrams by
means of Feynman rules. They form a central part of perturbative
quantum field theory, where they appear as coefficients of power
series expansions of probability amplitudes for physical
processes. The efficient computation of Feynman amplitudes is pivotal
for theoretical predictions in particle physics.

Periods are numbers computed as integrals of algebraic differential
forms over topological cycles on algebraic varieties. The term
originated from the period of a periodic elliptic function, which can
be computed as an elliptic integral.

Motives emerged from Grothendieck's “universal cohomology
theory”, where they describe an intermediate step between
algebraic varieties and their linear invariants (cohomology). The
theory of motives provides a conceptual framework for the study of
periods. In recent work, a beautiful relation between Feynman
amplitudes, motives and periods has emerged.

The articles provide an exciting panoramic view on recent
developments in this fascinating and fruitful interaction between pure
mathematics and modern theoretical physics.

#### Readership

Graduate students and research mathematicians interested in modern theoretical physics and algebraic geometry.

# Table of Contents

## Feynman Amplitudes, Periods and Motives

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- A note on twistor integrals 110 free
- Multiple polylogarithms and linearly reducible Feynman graphs 1120
- Comparison of motivic and simplicial operations in mod-𝑙-motivic and étale cohomology 2938
- 1. Introduction 2938
- 2. Cohomology of the classifying space for a finite group 3241
- 3. The total power operations: I 3443
- 4. The total power operations: II 3948
- 5. Comparison with the operadic definition of simplicial cohomology operations: properties of simplicial operations 4554
- 6. Comparison between the motivic and simplicial operations 4857
- 7. Cohomological operations that commute with proper push-forwards and Examples 5160
- References 5463

- On the Broadhurst-Kreimer generating series for multiple zeta values 5766
- 1. Introduction 5766
- 2. Period polynomials and the special depth filtration 5867
- 3. Distributivity conjecture and Broadhurst-Kreimer dimensions 6069
- 4. Shuffle subspaces of ℱ 6170
- 5. Proof of Theorem 3.4. 6473
- 6. Proofs of Lemmas 5.2 and 5.3. 6776
- 7. Multiple zeta values and their duals 6978
- References 7685

- Dyson–Schwinger equations in the theory of computation 7988
- 1. Introduction 7988
- 2. Primitive recursive functions and the Hopf algebra of flow charts 8089
- 3. Flow charts, templates, and algorithms 8594
- 4. Dyson–Schwinger equations in the Hopf algebra of flow charts 8897
- 5. Operadic viewpoint 96105
- 6. Renormalization of the halting problem 99108
- Acknowledgment 105114
- References 105114

- Scattering amplitudes, Feynman integrals and multiple polylogarithms 109118
- 1. Introduction 109118
- 2. Scattering amplitudes and Feynman integrals 110119
- 3. Feynman integrals and multiple polylogarithms 112121
- 4. Functional equations for multiple polylogarithms 115124
- 5. The Hopf algebra of multiple polylogarithms and Feynman integrals 122131
- 6. Conclusion 130139
- References 131140

- Equations D3 and spectral elliptic curves 135144
- 1. Introduction 135144
- 2. Determinantal differential equations 136145
- 3. The Beukers-Zagier equation as a D2 equation 137146
- 4. Modular D2 equations 138147
- 5. Differential equations of type D3 143152
- 6. Nondegenerate modular D3 equations 144153
- 7. All solutions of the multiplicativity equations for D3 148157
- 8. From D2’s to D3’s 150159
- References 152161

- Quantum fields, periods and algebraic geometry 153162
- Renormalization, Hopf algebras and Mellin transforms 169178
- Motivation: The renormalization problem 169178
- 1. Notations and preliminaries 170179
- 2. Finiteness of renormalization by kinematic subtraction 172181
- 3. Regularization and Mellin transforms 175184
- 4. Hopf algebra morphisms and the renormalization group 179188
- 5. Locality, finiteness and minimal subtraction 181190
- 6. Dyson-Schwinger equations and correlation functions 184193
- 7. Extensions towards \qft 191200
- 8. Summary 193202
- Appendix A. The Hopf algebra of rooted trees 194203
- Appendix B. The Hopf algebra of polynomials 197206
- Appendix C. The Dynkin operator D=S*Y 198207
- References 200209

- Multiple zeta value cycles in low weight 203212
- Periods and Hodge structures in perturbative quantum field theory 249258
- Some combinatorial interpretations in perturbative quantum field theory 261270
- Back Cover Back Cover1302