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3 Matilde Marcolli California Institute of Technology, USA World Scientific N E W J E R S E Y L O N D O N S I N G A P O R E B E I J I N G S H A N G H A I H O N G K O N G TA I P E I C H E N N A I

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. FEYNMAN MOTIVES Copyright 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN ISBN ISBN (pbk) ISBN (pbk) Printed in Singapore.

5 Zvezda v Galaktike i voron na suku жivut, ne muqimy xekspirovsko i dilemmo i. No my my pozdni i stilь stare we i Vselenno i. Ona skvozь nas poet svo tosku. Yuri Manin (not yet quite) The Late Style

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7 Preface This book originates from the notes of a course on Geometry and Arithmetic of Quantum Fields, which I taught at Caltech in the fall of Having just moved to Caltech and having my first chance to offer a class there, I decided on a topic that would fall in between mathematics and theoretical physics. Though it inevitably feels somewhat strange to be teaching Feynman diagrams at Caltech, I hope that having made the main focus of the lectures the yet largely unexplored relation between quantum field theory and Grothendieck s theory of motives in algebraic geometry may provide a sufficiently different viewpoint on the quantum field theoretic notions to make the resulting combination of topics appealing to mathematicians and physicists alike. I am not an expert in the theory of motives and this fact is clearly reflected in the way this text is organized. Interested readers will have to look elsewhere for a more informative introduction to the subject (a few references are provided in the text). Also I do not try in any way to give an exhaustive viewpoint of the current status of research on the connection between quantum field theory and motives. Many extremely interesting results are at this point available in the literature. I try, whenever possible, to provide an extensive list of references for the interested reader, and occasionally to summarize some of the available results, but in general I prefer to keep the text as close as possible to the very informal style of the lectures, possibly at the cost of neglecting material that should certainly be included in a more extensive monograph. In particular, the choice of material covered here focuses mostly on those aspects of the subject where I have been actively engaged in recent research work and therefore reflects closely my own bias and personal viewpoint. In particular, I will try to illustrate the fact that there are two possible vii

8 viii Feynman Motives complementary approaches to understanding the relation between Feynman integrals and motives, which one may refer to as a bottom-up and a topdown approach. The bottom-up approach looks at individual Feynman integrals for given Feynman graphs and, using the parametric representation in terms of Schwinger and Feynman parameters, identifies directly the Feynman integral (modulo the important issue of divergences) with an integral of an algebraic differential form on a cycle in an algebraic variety. One then tries to understand the motivic nature of the piece of the relative cohomology of the algebraic variety involved in the computation of the period, trying to identify specific conditions under which it will be a realization of a very special kind of motive, a mixed Tate motive. The other approach, the top-down one, is based on the formal properties that the category of mixed Tate motives satisfies, which are sufficiently rigid to identify it (via the Tannakian formalism) with a category of representations of an affine group scheme. One then approaches the question of the relation to Feynman integrals by showing that the data of Feynman integrals for all graphs and arbitrary scalar field theories also fit together to form a category with the same properties. This second approach was the focus of my joint work with Connes on renormalization and the Riemann Hilbert correspondence and is already presented in much greater detail in our book Noncommutative geometry, quantum fields, and motives. However, for the sake of completeness, I have also included a brief and less detailed summary of this aspect of the theory in the present text, referring the readers to the more complete treatment for further information. The bottom-up approach was largely developed in the work of Bloch Esnault Kreimer, though in this book I will mostly relate aspects of this approach that come from my joint work with Aluffi. I will also try to illustrate the points of contact between these two different approaches and where possible new developments may arise that might eventually unify the somewhat fragmentary picture we have at the moment. This book only partially reflects the state of the art on this fast-moving subject at the specific time when these lectures were delivered, but I hope that the timeliness of circulating these ideas in the community of mathematicians and physicists will somehow make up for lack of both rigor and of completeness. Matilde Marcolli

9 Acknowledgments A large part of the material covered in this book is based on recent joint work with Paolo Aluffi and on previous joint work with Alain Connes, whom I thank for their essential contributions to the development of those ideas and results. I also thank Abhijnan Rej for conversations and collaboration on some of the topics in this book, and for providing detailed comments on an earlier draft of the manuscript. I thank the students of my Geometry and Arithmetic of Quantum Fields course, especially Domenic Denicola and Michael Smith, for their valuable comments and questions during the class. Many thanks are due to Paolo Aluffi for his very detailed and useful feedback on the final version of the manuscript. I also thank Arthur Greenspoon for a final proofreading of the text. This book was partly written during stays of the author at MSRI and at the MPI. I thank both institutes for the hospitality and for support. This work was partially supported by NSF grants DMS and DMS ix

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11 Contents Preface Acknowledgments 1. Perturbative quantum field theory and Feynman diagrams A calculus exercise in Feynman integrals From Lagrangian to effective action Feynman rules Simplifying graphs: vacuum bubbles, connected graphs One-particle-irreducible graphs The problem of renormalization Gamma functions, Schwinger and Feynman parameters Dimensional Regularization and Minimal Subtraction Motives and periods The idea of motives Pure motives Mixed motives and triangulated categories Motivic sheaves The Grothendieck ring of motives Tate motives The algebra of periods Mixed Tate motives and the logarithmic extensions Categories and Galois groups Motivic Galois groups vii ix xi

12 xii Feynman Motives 3. Feynman integrals and algebraic varieties The parametric Feynman integrals The graph hypersurfaces Landau varieties Integrals in affine and projective spaces Non-isolated singularities Cremona transformation and dual graphs Classes in the Grothendieck ring Motivic Feynman rules Characteristic classes and Feynman rules Deletion-contraction relation Feynman integrals and periods The mixed Tate mystery From graph hypersurfaces to determinant hypersurfaces Handling divergences Motivic zeta functions and motivic Feynman rules Feynman integrals and Gelfand Leray forms Oscillatory integrals Leray regularization of Feynman integrals Connes Kreimer theory in a nutshell The Bogolyubov recursion Step 1: Preparation Step 2: Counterterms Step 3: Renormalized values Hopf algebras and affine group schemes The Connes Kreimer Hopf algebra Birkhoff factorization Factorization and Rota-Baxter algebras Motivic Feynman rules and Rota-Baxter structure The Riemann Hilbert correspondence From divergences to iterated integrals From iterated integrals to differential systems Flat equisingular connections and vector bundles The cosmic Galois group

13 Contents xiii 7. The geometry of DimReg The motivic geometry of DimReg The noncommutative geometry of DimReg Renormalization, singularities, and Hodge structures Projective Radon transform The polar filtration and the Milnor fiber DimReg and mixed Hodge structures Regular and irregular singular connections Beyond scalar theories Supermanifolds Parametric Feynman integrals and supermanifolds Graph supermanifolds Noncommutative field theories Bibliography 207 Index 215