Séminaires Mathématique 2012-2013

  Mercredi 10 juillet, à 14h30, salle 114

Raimar Wulkenhaar (Mathematisches Institut, University of Münster) : Exact solution of the quartic matrix model and application to 4D noncommutative QFT

We study quartic matrix models with action trace (EM^2+(\lambda/4)M^4), where M is a Hermitean N\times N-matrix. The external matrix E encodes the dynamics, and \lambda is a scalar coupling constant. For E not a multiple of the identity matrix, we prove a universal algebraic recursion formula that gives all higher correlation functions in terms of the 2-point function and the distinct eigenvalues of E. The 2-point function satisfies a closed non-linear equation to be solved case by case. Renormalisability of the 2-point function implies renormalisability of the entire model and vanishing of the \beta-function.

As main application we prove that the noncommutative \phi^4_4-model is, in matrix basis and for \theta\to\infty, exactly solvable and non-trivial. All matrix correlation functions are expressed in terms of the solution of a non-linear integral equation, which exists by the Schauder fixed point theorem. Schwinger functions in position space have full Euclidean symmetry. They only depend on matrix correlation functions at coinciding indices per topological sector, and clustering is violated. Reflection positivity requires that the diagonal matrix 2-point function is a Stieltjes function. This is not the case for \lambda>0.

  Jeudi 4 juillet, à 14h30, salle 114

Valentin Ovsienko (Institut Camille Jordan, Lyon 1) : Algèbres classiques vues comme graduées commutatives

Les algèbres de Clifford ainsi que l’algèbre des octonions peuvent être vues comme algèbres graduées commutatives. Ce point de vue permet aborder plusieurs questions classiques différemment. Je donnerai un exemple : une définition de la trace d’une matrice à coefficients dans une algèbre de Clifford et le déterminant associé. J’expliquerai aussi une technique plus générale qui permet d’associer une algèbre Z_2^n-graduée à une forme cubique sur le corps à deux éléments.

  Mercredi 6 Mars, à 14h30, salle 114

Walter Van Suijlekom (Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, Nijmegen) : Gauge networks in noncommutative geometry

We introduce gauge networks as generalizations of spin networks and lattice gauge fields to almost-commutative manifolds. The configuration space of quiver representations (modulo equivalence) in the category of finite spectral triples is studied ; gauge networks appear as an orthonormal basis in a corresponding Hilbert space. We give many examples of gauge networks, also beyond the well-known spin network examples. Given a representation in the category of spectral triples of a quiver embedded in a spin manifold, we define a discretized Dirac operator on the quiver. We compute the spectral action of this Dirac operator on a four-dimensional lattice, and find that it reduces to the Wilson action for lattice gauge theories and a Higgs field lattice system. As such, in the continuum limit it reduces to the Yang-Mills-Higgs system. (joint with Matilde Marcolli)

  Mercredi 14 Novembre, à 14h30, salle 114

Luca Tomassini (Centre for Mathematical and Theoretical Physics) Roma & Dept. of Mathematics, University of Roma) : New space-time uncertainty relations for cosmological backgrounds

We start with a brief presentation of the basic ideas of the approach to spacetime uncertainty relations (STUR) advocated in "The quantum structure of spacetime at the Planck scale and quantum fields" (Comm. Math. Phys. Volume 172, N. 1 (1995)) by Doplicher, Fredenhagen and Roberts. Then, we derive new physically motivated STUR, extending previous work on Minkowski spacetime ( Class. Quantum Grav. 28 (2011), with Stefano Viaggiu. arXiv:1102.0894) to cosmological Friedman-Walker backgrounds. Finally, we discuss some new perspectives.

  Mardi 13 Novembre, à 14h30, salle 114

Pierre Martinetti (Department of Physics, University of Napoli Federico II and INFN, Napoli) : L’aspect métrique de la géométrie non-commutative : du problème de Monge au champ de Higgs

On verra comment le récemment découvert champ de Higgs et le très ancien problème des "déblais et remblais" de Monge sont reliés via la formule de distance de Connes en géométrie non-commutative. Plus précisément, on introduira une distance de Monge-Kantorovich sur l’espace des états d’une algèbre non-commutative, et on illustrera par divers exemples tirés de résultats récents sur le plan de Moyal les différences et similitudes entre cette distance de Monge-Kantorovich non-commutative et la formule de Connes.

  Mercredi 10 Octobre, à 14h30, salle 114

Valentin Bonzom (Perimeter Institute, Waterloo, Canada) : Random tensor models (al large N)

The study of random tensors generalizes random matrices to objects with d>2 indices. Remarkably, Feynman expansions in tensor models generate sums over triangulations of pseudo-manifolds in dimension d. Such models have been actively developed and solved in the past two years. I will introduce a suitable ensemble of random tensors and present the main results : universality at large N (large size of the tensor), a notion of continuum limit and existence of critical behaviors, and a new algebra which generalizes the Virasoro algebra found in matrix models and provides gluing rules for triangulations in dimension d.