Joachim Kock:
Groupoids, and Faa di Bruno formulae for Green functions in
bialgebras of trees
In his work on gauge field theories, van Suijlekom discovered
that in the Connes-Kreimer Hopf algebra of Feynman graphs, a
certain infinite series Y satisfies the Faa di Bruno
comultiplication formula, dual to the formula for composition
of formal power series. Except for some denominators, Y is
the combinatorial Green function (sum of all graphs weighted
by inverses of symmetry factors).
In this talk I will explain a version of this result in the
setting of operadic trees, and give a very conceptual proof in
which the Faa di Bruno formula drops out as the homotopy
cardinality of an equivalence of groupoids. The use of
operadic trees rather than the usual combinatorial trees is
essential for the construction, and also allows the transfer
of the formula back to the realm of graphs.
The talk will start with the classical Faa di Bruno bialgebra,
and move on to the bialgebra of operadic trees; then I shall
explain some facts about groupoids, homotopy quotients and
homotopy cardinality, and finally put the things together in
an easy proof of the Faa di Bruno formula for operadic trees.
This is joint work with Imma Galvez and Andy Tonks.