Feynman Motives

by Matilde Marcolli

Overview

This book presents recent and ongoing research work aimed at understanding the mysterious relation between the computations of Feynman integrals in perturbative quantum field theory and the theory of motives of algebraic varieties and their periods. One of the main questions in the field is understanding when the residues of Feynman integrals in perturbative quantum field theory evaluate to periods of mixed Tate motives. The question originates from the occurrence of multiple zeta values in Feynman integrals calculations observed by Broadhurst and Kreimer.Two different approaches to the subject are described. The first, a “bottom-up” approach, constructs explicit algebraic varieties and periods from Feynman graphs and parametric Feynman integrals. This approach, which grew out of work of Bloch-Esnault-Kreimer and was more recently developed in joint work of Paolo Aluffi and the author, leads to algebro-geometric and motivic versions of the Feynman rules of quantum field theory and concentrates on explicit constructions of motives and classes in the Grothendieck ring of varieties associated to Feynman integrals. While the varieties obtained in this way can be arbitrarily complicated as motives, the part of the cohomology that is involved in the Feynman integral computation might still be of the special mixed Tate kind. A second, “top-down” approach to the problem, developed in the work of Alain Connes and the author, consists of comparing a Tannakian category constructed out of the data of renormalization of perturbative scalar field theories, obtained in the form of a Riemann-Hilbert correspondence, with Tannakian categories of mixed Tate motives. The book draws connections between these two approaches and gives an overview of other ongoing directions of research in the field, outlining the many connections of perturbative quantum field theory and renormalization to motives, singularity theory, Hodge structures, arithmetic geometry, supermanifolds, algebraic and non-commutative geometry.The text is aimed at researchers in mathematical physics, high energy physics, number theory and algebraic geometry. Partly based on lecture notes for a graduate course given by the author at Caltech in the fall of 2008, it can also be used by graduate students interested in working in this area.

Product Details

ISBN-13:	9789814304481
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/05/2010
Edition description:	New Edition
Pages:	236
Product dimensions:	5.90(w) x 8.90(h) x 0.70(d)

Table of Contents

Preface vii

Acknowledgments ix

1 Perturbative quantum field theory and Feynman diagrams 1

1.1 A calculus exercise in Feynman integrals 1

1.2 From Lagrangian to effective action 6

1.3 Feynman rules 9

1.4 Simplifying graphs: vacuum bubbles, connected graphs 12

1.5 One-particle-irreducible graphs 14

1.6 The problem of renormalization 18

1.7 Gamma functions, Schwinger and Feynman parameters 20

1.8 Dimensional Regularization and Minimal Subtraction 21

2 Motives and periods 25

2.1 The idea of motives 25

2.2 Pure motives 28

2.3 Mixed motives and triangulated categories 34

2.4 Motivic sheaves 37

2.5 The Grothendieck ring of motives 38

2.6 Tate motives 39

2.7 The algebra of periods 44

2.8 Mixed Tate motives and the logarithmic extensions 45

2.9 Categories and Galois groups 49

2.10 Motivic Galois groups 50

3 Feynman integrals and algebraic varieties 53

3.1 The parametric Feynman integrals 54

3.2 The graph hypersurfaces 60

3.3 Landau varieties 65

3.4 Integrals in affine and projective spaces 67

3.5 Non-isolated singularities 71

3.6 Cremona transformation and dual graphs 72

3.7 Classes in the Grothendieck ring 76

3.8 Motivic Feynman rules 78

3.9 Characteristic classes and Feynman rules 81

3.10 Deletion-contraction relation 84

3.11 Feynman integrals and periods 93

3.12 The mixed Tate mystery 94

3.13 From graph hypersurfaces to determinant hypersurfaces 97

3.14 Handling divergences 112

3.15 Motivic zeta functions and motivic Feynman rules 115

4 Feynman integrals and Gelfand-Leray forms 119

4.1 Oscillatory integrals 119

4.2 Leray regularization of Feynman integrals 121

5 Connes-Kreimer theory in a nutshell 127

5.1 The Bogolyubov recursion 128

5.1.1 Step 1: Preparation 128

5.1.2 Step 2: Counterterms 128

5.1.3 Step 3: Renormalized values 129

5.2 Hopf algebras and affine group schemes 130

5.3 The Connes-Kreimer Hopf algebra 133

5.4 Birkhoff factorization 135

5.5 Factorization and Rota-Baxter algebras 137

5.6 Motivic Feynman rules and Rota-Baxter structure 139

6 The Riemann-Hilbert correspondence 143

6.1 From divergences to iterated integrals 143

6.2 From iterated integrals to differential systems 145

6.3 Flat equisingular connections and vector bundles 146

6.4 The “cosmic Galois group” 147

7 The geometry of DimReg 151

7.1 The motivic geometry of DimReg 151

7.2 The noncommutative geometry of DimReg 155

8 Renormalization, singularities, and Hodge structures 167

8.1 Projective Radon transform 167

8.2 The polar filtration and the Milnor fiber 170

8.3 DimReg and mixed Hodge structures 173

8.4 Regular and irregular singular connections 176

9 Beyond scalar theories 185

9.1 Supermanifolds 185

9.2 Parametric Feynman integrals and supermanifolds 190

9.3 Graph supermanifolds 199

9.4 Noncommutative field theories 201

Bibliography 207

Index 215