Feynman amplitudes, periods, and motives : international research conference on periods and motives : a modern perspective on renormalization : July 2-6, 2012, Institute de Ciencias Matemáticas, Madrid, Spain

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. Cover 1 Title page 4 Contents 6 Preface 8 A note on twistor integrals 10 1. Introduction 10 2. Linear Algebra 12 3. The Twistor Integral 14 4. Proof of theorem 1.1 16 References 19 Multiple polylogarithms and linearly reducible Feynman graphs 20 1. Introduction 20 2. Multiple Polylogarithms and Feynman Integrals 22 3. Linear Reducibility 28 4. Towards a Classification by Critical Minors 31 5. Conclusions 33 References 34 Comparison of motivic and simplicial operations in mod-l-motivic and étale cohomology 38 1. Introduction 38 2. Cohomology of the classifying space for a finite group 41 3. The total power operations: I 43 4. The total power operations: II 48 5. Comparison with the operadic definition of simplicial cohomology operations: properties of simplicial operations 54 6. Comparison between the motivic and simplicial operations 57 7. Cohomological operations that commute with proper push-forwards and Examples 60 References 63 On the Broadhurst-Kreimer generating series for multiple zeta values 66 1. Introduction 66 2. Period polynomials and the special depth filtration 67 3. Distributivity conjecture and Broadhurst-Kreimer dimensions 69 4. Shuffle subspaces of F 70 5. Proof of Theorem 3.4. 73 6. Proofs of Lemmas 5.2 and 5.3. 76 7. Multiple zeta values and their duals 78 References 85 Dyson–Schwinger equations in the theory of computation 88 1. Introduction 88 2. Primitive recursive functions and the Hopf algebra of flow charts 89 3. Flow charts, templates, and algorithms 94 4. Dyson–Schwinger equations in the Hopf algebra of flow charts 97 5. Operadic viewpoint 105 6. Renormalization of the halting problem 108 Acknowledgment 114 References 114 Scattering amplitudes, Feynman integrals and multiple polylogarithms 118 1. Introduction 118 2. Scattering amplitudes and Feynman integrals 119 3. Feynman integrals and multiple polylogarithms 121 4. Functional equations for multiple polylogarithms 124 5. The Hopf algebra of multiple polylogarithms and Feynman integrals 131 6. Conclusion 139 References 140 Equations D3 and spectral elliptic curves 144 1. Introduction 144 2. Determinantal differential equations 145 3. The Beukers-Zagier equation as a D2 equation 146 4. Modular D2 equations 147 5. Differential equations of type D3 152 6. Nondegenerate modular D3 equations 153 7. All solutions of the multiplicativity equations for D3 157 8. From D2’s to D3’s 159 References 161 Quantum fields, periods and algebraic geometry 162 1. Introduction 162 2. Graphs and algebras 162 3. Feynman Rules 166 4. Examples 169 5. Feynman rules from a Lie viewpoint 174 Acknowledgments 174 References 175 Renormalization, Hopf algebras and Mellin transforms 178 Motivation: The renormalization problem 178 1. Notations and preliminaries 179 2. Finiteness of renormalization by kinematic subtraction 181 3. Regularization and Mellin transforms 184 4. Hopf algebra morphisms and the renormalization group 188 5. Locality, finiteness and minimal subtraction 190 6. Dyson-Schwinger equations and correlation functions 193 7. Extensions towards \qft 200 8. Summary 202 Appendix A. The Hopf algebra of rooted trees 203 Appendix B. The Hopf algebra of polynomials 206 Appendix C. The Dynkin operator D=S*Y 207 References 209 Multiple zeta value cycles in low weight 212 1. Introduction 213 2. Combinatorial situation 214 3. Algebraic cycles 221 4. Parametric and combinatorial representation for the cycles: trees with colored edges 236 5. Bar construction settings 241 6. Integrals and multiple zeta values 251 References 255 Periods and Hodge structures in perturbative quantum field theory 258 1. Periods 258 2. Hodge structures 262 3. Picard-Fuchs equations 265 References 267 Some combinatorial interpretations in perturbative quantum field theory 270 1. Introduction 270 2. Dyson-Schwinger equations and the chord diagram expansion 271 3. Denominator reduction and special changes of variables 282 References 297 Back Cover 302 La 4e de couverture porte : "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."