14h00: Joachim Kock
(Barcelona): Categories of graphs and
I will present some new formalisms for
Feynman graphs and
trees, motivated by operad theory, but expected also to be of interest
in quantum field theory and related subjects. One feature is the
of dealing with decoration and local structure, like for example
graphs for a particular quantum field theory, ribbon graphs, bipartite
graphs, etc.; binary trees, planar trees, or trees decorated by
primitive 1PI graphs.
15h30: Emily Burgunder
(Toulouse): Kontsevich graph
complexes and operads
On the one hand, Kontsevich showed that the Lie
homology of symplectic
vector fields can be computed via a certain graph homology.
been extended to operads via the work of Conant, Vogtmann and Mahajan.
On the other hand, Loday, Quillen and Tsygan have proven that the Lie
homology of the Lie matrices can be computed via cyclic homology,
can then be reinterpreted as graph homology. A
similar result due to
Procesi and Loday has been proven in the orthogonal case. I
these results to the operadic case by providing a functor from operads
Lie algebras with a group action of one of the groups : SP, O, or
and relate the Lie homology of these algebras with a graph complex. A
generalisation to Leibniz homology gives symmetric graphs.
17h15: Jean-Christophe Novelli
(Marne la Vallée): Polynomial
realizations of combinatorial Hopf algebras (I).
The heuristic notion of combinatorial Hopf
to a large class of examples of graded connected Hopf
bases on combinatorial families, and whose product
coproduct in a distinguished basis
described by combinatorial algorithms.
polynomial realization of such an algebra, we mean an
into an algebra of polynomials (commutative or not)
infinitely many variables, mapping the distinguished
basis to an
explicit family of polynomials. Thus the product
original algebra is induced by the ordinary product of
and it becomes obvious at this stage that it is
As for the coproduct, in most examples,
definition of the polynomials assumes an order on the variables
coproduct is then induced by the ordinal sum of alphabets. Since this
is trivially associative and compatible with multiplication,
algebra structure comes for free. However, this approach
seem to work for Hopf algebras of trees of the Connes-Kreimer
it is possible to replace the order on the alphabet
kind of binary relation for which an analog of the
can be defined. Then, these Hopf algebras of trees
realized as well. Joint work with Jean-Yves Thibon and Loïc Foissy.
october 15th 2010
9h00: Jean-Yves Thibon
la Vallée): Polynomial
realizations of combinatorial
Hopf algebras (II).
Following J-C. Novelli's talk.
10h45: Dirk Kreimer
& Boston University): Angles and
We discuss how the algebraic structures of
Feynman graphs determine analytic properties of amplitudes in physics.
Emphasis is put on the distinct roles played by scattering angles and
14h30: Walter Van Suijlekom
): Renormalization Hopf algebras for gauge
theories and BRST-symmetries.
The structure of the Connes-Kreimer renormalization
algebra is studied
for Yang-Mills gauge theories, with particular
emphasis on the
BRST-formalism. A coaction of the renormalization Hopf
algebra is defined on the
coupling constants and the fields. In this
context, BRST-invariance of
the action implies the existence of certain
Hopf ideals in the
renormalization Hopf algebra, encoding the (physical)
Slavnov-Taylor identities for
the coupling constants.