Towards Cohomology of Renormalization: Bigrading the Combinatorial Hopf Algebra of Rooted Trees

Abstract:

The renormalization of quantum field theory twists the antipode of a noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of primitive divergences. The Hopf algebra of undecorated rooted trees, ℋ _R , generated by a single primitive divergence, solves a universal problem in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the cocommutative subalgebra ℋ_ladder of pure ladder diagrams and the Connes–Moscovici noncocommutative subalgebra ℋ_CM of noncommutative geometry. These three Hopf algebras admit a bigrading by n, the number of nodes, and an index k that specifies the degree of primitivity. In each case, we use iterations of the relevant coproduct to compute the dimensions of subspaces with modest values of n and k and infer a simple generating procedure for the remainder. The results for ℋ_ladder are familiar from the theory of partitions, while those for ℋ_CM involve novel transforms of partitions. Most beautiful is the bigrading of ℋ _R , the largest of the three. Thanks to Sloane's superseeker, we discovered that it saturates all possible inequalities. We prove this by using the universal Hochschild-closed one-cocycle B ₊, which plugs one set of divergences into another, and by generalizing the concept of natural growth beyond that entailed by the Connes–Moscovici case. We emphasize the yet greater challenge of handling the infinite set of decorations of realistic quantum field theory.