Combinatorial Dyson-Schwinger equations, polymomial functors,
and operads of Feynman graphs
After briefly recalling the Connes-Kreimer Hopf algebra of
Feynman graphs and the Butcher-Connes-Kreimer Hopf algebra of
trees, as they appear in BPHZ renormalisation in Quantum Field
Theory, a main point of this talk is to explain the relationship
between the two via an intermediate bialgebra of P-trees, for P
a certain polynomial functor of primitive graphs, and to outline
some categorical interpretations resulting from this viewpoint.
The combinatorial Dyson-Schwinger equations of Bergbauer-Kreimer
take the form of polynomial fixpoint equations in groupoids,
X = 1 + P(X); the solution X (which is accordingly a W-type in
the sense of type theory) consists of certain nested graphs,
which are the P-trees, appearing as the operations of $\bar P$,
the free monad on P. The so-called overlapping divergences, a
main subtlety of BPHZ renormalisation, are interpreted as the
relations defining the operad of graphs as a quotient of $\bar P$.
In Quantum Chromodynamics one is interested in certain truncations
of the DSEs, required to generate sub-Hopf algebras. In the
abstract setting, truncation is interpreted as monomorphic natural
transformations between polynomial endofunctors, and the sub-Hopf
condition is satisfied by the cartesian natural transformations.