Hopf Algebras in Renormalization Theory: Locality and Dyson-schwinger Equations from Hochschild Cohomology

In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion. CONTENTS Introduction and acknowledgments 1 1. Rooted trees, Feynman graphs, Hochschild cohomology and local counterterms 2 1.1. Motivation 2 1.2. Basic definitions and notation 4 1.3. The Hopf algebra of rooted trees 4 1.4. Tree-like structures and variations on a theme 5 1.5. Hochschild cohomology of bialgebras 6 1.6. Finiteness and locality from the Hopf algebra 8 2. Hopf subalgebras and Dyson-Schwinger equations 13 2.1. Hopf subalgebras of decorated rooted trees 13 2.2. Combinatorial Dyson-Schwinger equations 15 2.3. Applications in physics and number theory 20 2.4. Final remarks 27 References 27 INTRODUCTION AND ACKNOWLEDGMENTS The relevance of infinite dimensional Hopf and Lie algebras for the understanding of local quantum field theory has been established in the last couple of years. Here, we focus on the role of the 1-cocycles in the Hochschild cohomology of such renormalization Hopf algebras. After an introductory overview which recapitulates the well-known Hopf algebra of rooted trees we exhibit once more the crucial connection between 1-cocycles in the Hochschild cohomology of the Hopf algebra, locality and the structure of the quantum equations of motion. For the latter, we introduce combinatorial DysonSchwinger equations and show that the perturbation series provides Hopf subalgebras indexed only by the order of the perturbation. We then discuss assorted applications of such equations which focus on the notion of self-similarity and transcendence. 1 2 C. BERGBAUER AND D. KREIMER This paper is based on an overview talk given by one of us (D. K.), extended by a more detailed exhibition of some useful mathematical aspects of the Hochschild cohomology of the relevant Hopf algebras. It is a pleasure to thank the organizers of the 75ème Rencontre entre Physiciens Théoriciens et Mathématiciens for organizing that enjoyable workshop. D. K. thanks Karen Yeats for discussions on the transcendental nature of DSEs. C. B. acknowledges support by the Deutsche Forschungsgemeinschaft. He also thanks Boston University and the IHES for hospitality. 1. ROOTED TREES, FEYNMAN GRAPHS, HOCHSCHILD COHOMOLOGY AND LOCAL COUNTERTERMS 1.1. Motivation. Rooted trees store information about nested and disjoint subdivergences of Feynman graphs in a natural way. This has been used at least implicitly since Hepp’s proof of the BPH subtraction formula [22] and Zimmermann’s forest formula [37]. However it was only decades later that the algebraic structure of the Bogoliubov recursion was elucidated by showing that it is essentially given by the coproduct and the corresponding antipode of a Hopf algebra on rooted trees [24, 9]. The same result can be formulated more directly in terms of a very similar Hopf algebra on 1PI Feynman graphs [10]. We start with the description in terms of rooted trees which serves as a universal role model for all Hopf algebras of this kind. For instance, the subdivergences of the φ3 diagram in six spacetime dimensions