Université
Blaise Pascal

Laboratoire de mathématiques UMR 6620

GDR CNRS "Renormalisation"

Projet ANR "Modunombres"

Clermont- Ferrand, October 14/15 th. 2010

Laboratoire de mathématiques UMR 6620

GDR CNRS "Renormalisation"

Projet ANR "Modunombres"

Clermont- Ferrand, October 14/15 th. 2010

Feynman
graphs in physics, combinatorics,

homological algebra and category theory

Schedule:

Thursday, October 14th 2010

14h00: Joachim Kock (Barcelona): Categories of graphs and trees.

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 ease 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. This has 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, which can then be reinterpreted as graph homology. A similar result due to Procesi and Loday has been proven in the orthogonal case. I extend these results to the operadic case by providing a functor from operads to Lie algebras with a group action of one of the groups : SP, O, or SL, and relate the Lie homology of these algebras with a graph complex. A generalisation to Leibniz homology gives symmetric graphs.

Tea/Coffee break

17h15: Jean-Christophe Novelli (Marne la Vallée): Polynomial realizations of combinatorial Hopf algebras (I).

The heuristic notion of combinatorial Hopf algebra refers to a large class of examples of graded connected Hopf algebras, bases on combinatorial families, and whose product and coproduct in a distinguished basis are described by combinatorial algorithms. By a polynomial realization of such an algebra, we mean an embedding into an algebra of polynomials (commutative or not) usually in infinitely many variables, mapping the distinguished basis to an explicit family of polynomials. Thus the product of the original algebra is induced by the ordinary product of polynomials, and it becomes obvious at this stage that it is associative. As for the coproduct, in most examples, the definition of the polynomials assumes an order on the variables and the coproduct is then induced by the ordinal sum of alphabets. Since this operation is trivially associative and compatible with multiplication, the Hopf algebra structure comes for free. However, this approach does not seem to work for Hopf algebras of trees of the Connes-Kreimer family. But it is possible to replace the order on the alphabet by another kind of binary relation for which an analog of the ordinal sum can be defined. Then, these Hopf algebras of trees can be realized as well. Joint work with Jean-Yves Thibon and Loïc Foissy.

9h00: Jean-Yves Thibon (Marne la Vallée): Polynomial realizations of combinatorial Hopf algebras (II).

Following J-C. Novelli's talk.

Tea/Coffee break

10h45: Dirk Kreimer (IHES & Boston University): Angles and Scales.

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 energy scales.

Lunch

14h30: Walter Van Suijlekom (Radboud): Renormalization Hopf algebras for gauge theories and BRST-symmetries.

The structure of the Connes-Kreimer renormalization Hopf 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.

homological algebra and category theory

Schedule:

Thursday, October 14th 2010

14h00: Joachim Kock (Barcelona): Categories of graphs and trees.

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 ease 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. This has 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, which can then be reinterpreted as graph homology. A similar result due to Procesi and Loday has been proven in the orthogonal case. I extend these results to the operadic case by providing a functor from operads to Lie algebras with a group action of one of the groups : SP, O, or SL, and relate the Lie homology of these algebras with a graph complex. A generalisation to Leibniz homology gives symmetric graphs.

Tea/Coffee break

17h15: Jean-Christophe Novelli (Marne la Vallée): Polynomial realizations of combinatorial Hopf algebras (I).

The heuristic notion of combinatorial Hopf algebra refers to a large class of examples of graded connected Hopf algebras, bases on combinatorial families, and whose product and coproduct in a distinguished basis are described by combinatorial algorithms. By a polynomial realization of such an algebra, we mean an embedding into an algebra of polynomials (commutative or not) usually in infinitely many variables, mapping the distinguished basis to an explicit family of polynomials. Thus the product of the original algebra is induced by the ordinary product of polynomials, and it becomes obvious at this stage that it is associative. As for the coproduct, in most examples, the definition of the polynomials assumes an order on the variables and the coproduct is then induced by the ordinal sum of alphabets. Since this operation is trivially associative and compatible with multiplication, the Hopf algebra structure comes for free. However, this approach does not seem to work for Hopf algebras of trees of the Connes-Kreimer family. But it is possible to replace the order on the alphabet by another kind of binary relation for which an analog of the ordinal sum can be defined. Then, these Hopf algebras of trees can be realized as well. Joint work with Jean-Yves Thibon and Loïc Foissy.

Friday,
october 15th 2010

9h00: Jean-Yves Thibon (Marne la Vallée): Polynomial realizations of combinatorial Hopf algebras (II).

Following J-C. Novelli's talk.

Tea/Coffee break

10h45: Dirk Kreimer (IHES & Boston University): Angles and Scales.

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 energy scales.

Lunch

14h30: Walter Van Suijlekom (Radboud): Renormalization Hopf algebras for gauge theories and BRST-symmetries.

The structure of the Connes-Kreimer renormalization Hopf 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.