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In this review we discuss the relevance of the Hochschild co homology 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 f or the understanding of local quantum field theory has been established in the l ast 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 wel l-known Hopf algebra of rooted trees we exhibit once more the crucial connection b etween 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 co mbinatorial DysonSchwinger equations and show that the perturbation series p ovides Hopf subalgebras indexed only by the order of the perturbation. We then discuss assorted applications of such equations which focus on the notion of s elf-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 aspe ct of the Hochschild cohomology of the relevant Hopf algebras. It is a pleasure to thank the organizers of the75ème Rencontre entre Physiciens Th éoriciens et Math́ ematiciensfor organizing that enjoyable workshop. D. K. thanks Karen Yeat s for discussions on the transcendental nature of DSEs. C. B. acknowledges suppo rt 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 su bdivergences of Feynman graphs in a natural way. This has been us d at least implicitly since Hepp’s proof of the BPH subtraction formula [22] a nd Zimmermann’s forest formula [37]. However it was only decades later that t e algebraic structure of the Bogoliubov recursion was elucidated by showing that i t is essentially given by the coproduct and the corresponding antipode of a Hopf alg ebra on rooted trees [24, 9]. The same result can be formulated more directly in te rms of a very similar Hopf algebra on 1PI Feynman graphs [10]. We start with the des cription 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

Hopf algebras in renormalization theory: locality and Dyson–Schwinger equations from Hochschild cohomology