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

Abstract

ABSTRACT. 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

Graphons and renormalization of large Feynman diagrams

“…For overlapping sub-divergencies, there are some challenges but the corresponding tree representation could be a linear combination of decorated rooted trees. We refer the reader to [1,12,23,24] for further details in this issue.…”

“…The completion of the Hopf algebra H with respect to this topology is the extended Hopf algebra H = n≥0 H n which has elements of the form n≥0 Γ n such that Γ n ∈ H n . It is shown that solutions of combinatorial DSEs belong to this completed Hopf algebra [1,19,25,26,40,41]. On the other side, for each n, X n is a finite Feynman diagram which guarantees the finiteness of the terms Y n s. Each Y n could be constructed from Y n−1 by a growing (not generally uniform) attachment graph sequence.…”

“…At the third stage, thanks to the addressed grading structure on G simple [0,1] and the coproduct (3.2), the required antipode on H simple could be defined inductively.…”

“…For each n ≥ 1, make a finite subset V (Γ n , ρ n ) in the closed interval [0, 1] with respect to the Feynman diagram Γ n . This set contains |V (t Γn )| nodes in [0,1] which is the number of vertices in the rooted tree t Γn corresponding to the graph Γ n . These nodes are selected by projections of the vertices in the graph t Γn under a fixed injective poset type embedding map ρ n from the tree t Γn (as a poset) in the interval [0, 1].…”

“…Each term X n is produced on the basis of the terms with the lower degrees and furthermore, terms X n s play the role of generators for a Hopf sub-algebra associated to the equation DSE [1,18,19,25,26,40,41]. This presentation of the solution X, which belongs to the completion of H [[α]] with respect to the n-adic topology, has been applied as a starting point to build some new mathematical structures which are capable to encode non-perturbative parameters [33,35,36].…”