LIPN - 7030 UMR CNRS

Equipe CALIN
GDR CNRS Renormalisation
Bures sur Yvette, le 27 Février 2012.

IHES, 35 Route de Chartres  91440 Bures-sur-Yvette, France .

Rencontre organisée par Gérard H.E. DUCHAMP, Maxim KONTSEVICH, Gleb KOSHEVOY  et HOANG NGOC MINH

Combinatorics of Mathematical Renormalization : a special day

Les algèbres de Hopf combinatoires et diagrammatiques conduisent souvent à des calculs effectifs en Renormalisation mais nous aurons également besoin de mieux comprendre ces calculs par l'intermédiaire de leurs représentations géométriques.

Programme

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Liste des participants

Programme

 10h30-10h30 Accueil des participants 10h30-11h30 Christian Brouder Quantum field theory on curved spacetimes : merging Borcherds' and the causal approaches There are two successful approaches to renormalized quantum field theory on curved spacetimes: i) the causal approach initiated by Brunetti and Fredenhagen [Commun. Math. Phys. 208 (2000) 623-61] and further developed by themselves, Hollands, Wald and Dütsch; ii) a Hopf algebraic approach recently proposed by Borcherds [Alg. Numb. Theor. 5 (2011) 627-58]. The causal approach is analytic, it uses non-linear functionals and microlocal analysis. Borcherds' approach is more geometric and algebraic, it uses Hopf algebra bundles and a Gaussian condition for Feynman measures. A common framework will be presented to merge these two approaches. 11h30-12h30 Nikolay Nikolov Operadic structures in the renormalization The operads naturally appear in the description of composition of formal power series of vector variables. This is the intrinsic reason why operads appear related to the renormalization group and its action. We study this relation in a joint work with Jean-Louis Loday. On the combinatorial level our construction gives an operadic interpretation of the group attached to the Connes-Kreimer Hopf algebra. We can introduce also an operad structure on the function spaces spanned by Feynman amplitudes. All these operads, the combinatorial and the functional one, give rise to groups that are related by morphisms induced by operadic morphisms. I will also discuss the generalization of the Callan–Symanzik equation that appears as a Maurer-Cartan equation on the constructed groups. Such a generalized Callan–Symanzik equation was suggested by the author as a recursive system of cohomological equations that determine basic differential anomalies in the renormalization of Feynman diagrams. 12h30-14h30 Lunch 14h30-15h30 Frédéric Patras On the BWH factorization for non Rota-Baxter schemes For renormalization schemes (i.e. for given regularization +  subtraction maps) whose subtraction map satisfies the Rota-Baxter (RB) identity, the Bogoliubov recursion yields automatically a Birkhoff-Wiener-Hopf decomposition of Feynman rules characters in the Hopf algebra picture. When the RB property is lost, the Bogoliubov recursion still makes sense. We analyse the corresponding combinatorics and show how it relates to the recently introduced exponential renormalization method. Based on joint works with K. Ebrahimi-Fard. 15h30-16h30 Olivier Bouillot Renormalization of multitangent functions and applications After some reminders on multizeta values, we will introduce the multitangent functions. Then, we will see a profound link with multizeta values: the reduction into monotangent functions. After, we will study in details renormalization of divergent multitangent functions and will see what will be the renormalisation contribution relatively to multizeta values. Finally, we will introduce some conjectures concerning the links between multizeta values and multitangent functions. 16h30-17h00 Tea/cofee Break 17h00-18h00 Vincel Hoang Ngoc Minh $\phi$-deformed shuffle bialgebras and bases in duality In order to extend the Schützenberger factorization in the context of $\phi$-deformed shuffle bialgebras, we give an explicite construction of bases in duality via the convolutional CQMM theorem. 18h00 Pierre Cartier A few concluding words

Début

Liste des participants

Van Chiên Bui (Paris 13)
Christian Brouder (CNRS-Paris 6)
Pierre Cartier (IHES)
Viêt Dang (Paris7)
Gérard Duchamp (Paris 13)

Joachim Kock (Barcelona)
Maxim
Kontsevich (IHES)
Gleb Koshevoy (IHES)
Jean-Yves Enjalbert (Paris 13)
Sylvia Goodenough (Paris 13)
Vincel Hoang Ngoc Minh (Lille 2/Paris 13)
Frédéric Menous (Paris 11)
Nikolay M. Nikolov (IHES)
Nguyen Hoang (Paris 13)

Frédéric Patras (CNRS-Nice)
Karol Penson (Paris 6)