# Renormalization at the confluence of analysis, algebra and geometry

17-21 sept. 2012

Centre International de Rencontres Mathématiques (www.cirm.univ-mrs.fr) - 163 avenue de Luminy, Marseille (France)

### http://renorm.sciencesconf.org

Renormalisation has recently experienced a tremendous boost of interests and results and is a very active area of research at the cross-road of many ﬁelds in 1. Mathematics, on the one hand, such as - algebra via Hopf algebras, tree structures, combinatorics, - analysis via Mould calculus, pseudodiﬀerential calculus, distributions, non- linear PDEs, - geometry via Hamiltonian geometry, noncommutative geometry, geometric quantisation, Poisson structures, algebraic geometry, - number theory via zeta and polyzeta functions and motives, 2. Theoretical Physics, on the other hand, such as - perturbative quantum ﬁeld theory for particle physics, condensed matter and cosmology, - perturbative quantum ﬁeld theories on non commutative space-times, - the various renormalisation groups, as for instance those “`a la Wilson” and Stueckelberg-Petermann, - constructive quantum ﬁeld theories, - string theories. In spite of their diversity, these concepts all serve the common drive to make sense of a renormalised theories, but their very diﬀerences call for a confrontation and possibly a uniﬁcation of the various points of views. Historically, quantum ﬁeld theory oﬀered, and still does, a high degree of precision when the theory is compared with experiments, yet the underlying mathematical mechanisms remain mysterious. In spite of great ad- vances in both the understanding of some algebraic mechanisms with pioneering work by Connes and Kreimer and some analytic mechanisms with decisive work by Brunetti and Fredenhagen, the mathematical picture available at this stage is still far from com- plete. Connes and Kreimer in 1998 shed light on the Hopf algebra structures underlying the combinatorics of Feynman rules used by physicists to compute Feynman integrals in momentum space arising in correlation functions in a euclidean perturbative quantum ﬁeld theory, opening the way for new algebraic tree-like structures and their relation to issues of analytic nature. Suijlekom then investigated gauge invariance in (perturbative) quantum Yang-Mills theories vs. renormalization using Hopf algebras. Interesting links of methodological nature have emerged with Mould calculus and multizeta values, inso- far as Hopf algebras prove to be a very useful tool to deal with divergences in general. This can be illustrated by the implementation of Abel type theorems for the non com- mutative generating series of multizeta values to obtain the counter-terms eliminating divergence of multizeta values. Moreover, the action of the diﬀerential Galois group of polylogarithms on their asymptotic expansions, in various comparison scales, allows to remove the ambiguities due to the implementation of regularization procedures. In the perturbative approach to quantum ﬁeld theory, amplitudes typically deliver multiple zeta values as found by Kreimer and Broadhurst (1999) and with further convincing evidence recently found by Brown (2009). Hence multiple zeta functions should be of motivic origin as indicated by Bloch, Esnault and Kreimer (2006) and also Connes and Marcolli in their joint book (2008). Motives therefore seem to oﬀer an appropriate framework to understand these links better. A subtle mixture of microlocal analysis with algebraic, combinatorial and cohomo- logical methods was developed by Brunetti and Fredenhagen in 1996 to put real space renormalisation on ﬁrm mathematical grounds on the basis of a reﬁning of the Epstein- Glaser approach. Indeed, the emphasis on the local aspects allowed them to deliver the ﬁrst renormalization prescription to all orders on ﬁxed curved Lorentzian space-times. Hollands and Wald in 2002, following a suggestion of Fredenhagen, implemented further the local viewpoint by merging it with the notion of general covariance, which resulted in the possibility to ﬁx the renormalization ambiguities via the conceptually clean use of functors and natural transformations. This point of view oﬀered also new insights on the renormalisation group in curved space-times. Along the same line of thought, a central part of the recent work by Costello on the one hand based on the Polchinski-Wilson renormalization group concept together with that of eﬀective actions and Borcherds work on the other hand, which is similar to the Brunetti-Fredenhagen version of the Epstein-Glaser approach . Kontsevich has also recently presented his own view of real space renormalization techniques using vertex algebras. On the other hand, these al- gebras have been used by Tamarkin in his attempt to renormalization. These two last approaches seems related to the theory recently developed by Hollands and Wald for the treatment of operator product expansions. Another interesting connection between diﬀerent approaches, namely that of Connes-Kreimer with that of Epstein-Glaser has been recently put forward by Bergbauer, Brunetti and Kreimer on the one hand and by Keller and Fredenhagen on the other, but a lot still remains to do for a complete identiﬁcation. Surprisingly, as shown by work by Guo, Paycha and Zhang, renormalisa- tion techniques occur also in the far ﬁeld of combinatorics when counting the number of integer points on a cone or generalizing the Euler-Maclaurin formula to cones, two issues which already have a long history independent of the renormalisation issue. Multiple zeta values at nonpositive integers investigated by Manchon and Paycha on the one hand and Guo and Zhang on the other hand, are such renormalized sums on cones. This link to combinatorics on cones oﬀers a new testing ground to understand the renormalisation group from another perspective and leads to interesting issues which could interest both the mathematical physics and the combinatorics communities. Also, new ideas emerging from the theory of strings on renormalization for quantum ﬁeld theories call for a better understanding and a confrontation with other points of views. Finally, tremendous progress was done on quantum field theories of non-relativistic matter interacting with relativistic bosons. This workshop aims at bringing together several communities working on renormalization so that they can exchange their techniques and points of view.

*Discipline scientifique*:

**Mathématiques**

Lieu de la conférence