Robert Coquereaux   Web pages

Physicien théoricien au  Centre de Physique Théorique  (CPT)
Case 907. Luminy.  13288. Marseille.
France.

Curriculum Vitae & Publications list

Fusion graphs

To every Dynkin diagram G and to every positive integer k (level), one associates a fusion category that can be described either in terms of integrable representations of affine Lie algebras, or in terms of a particular class of representations of quantum groups at roots of unity. Fusion by fundamental integrable representations can be described by graphs called "fusion graphs". These fusion categories somehow generalize, at the quantum level, the theory of representations of Lie groups and Lie algebras, and the process of "fusion" replaces the usual notion of tensor product of representations. The web pages that are given (here) show fusion graphs of type (G,k) and describe some of their properties, for all types of Dynkin diagrams (of small rank), and for some chosen values of the level.

Fusion graphs for quantum modules and quantum subgroups

The previous monoidal categories somehow generalize, at the quantum level, the theory of representations of Lie groups and Lie algebras. Classically, every finite subgroup of a Lie group possesses its own theory of representations, and the set obtained by considering sum of irreducible representations is a module over the representation ring associated with the chosen Lie group. The same phenomenon exists in a quantum framework: for every fusion category of type (G,k) one can consider "module-categories" (not to be confused with modular categories) on which the previous one (which is actually monoidal) acts. Action of (G,k) on a chosen module can itself be described by a graph. We refer (here) to the lectures given at the school of Bariloche 2000, and in particular to the contribution of A. Ocneanu describing the classification of quantum modules and quantum subgroups, for G=SU(2), SU(3), and SU(4). Our own web pages (there) summarize these classifications but give other examples (without pretending to be exhaustive) for other choices of G.

In physics, one knows how to associate particular quantum field theories (conformal field theories of type WZW) to fusion categories defined by the couple (G,k) where G is a Dynkin diagram and k is a non negative integer. Module-categories relative to the corresponding different possible choices of "quantum subgroups" or "quantum modules", along with their fusion graphs, possess their own interpretation in the framework of boundary conformal field theories. In particular, one knows, for every one of them, how to define and calculate a modular invariant quantity that is interpreted as a partition function.

Some of my own papers in html (hypertext), ps or pdf

Videos : Espaces fibrés et interactions fondamentales (1h.) : Lecture given at the M.S.H, Paris, 2005. Conference "Fibrés, fibrations et connexions" (coll. Les archives audiovisuelles de la recherche).

Livre HyperTexte - HyperText Book:

FIBER BUNDLES AND CONNECTIONS

Table of contents - Table des matières

Ce livre est une introduction à la théorie des espaces fibrés. Il est destinés aux jeunes (et aux moins jeunes) voulant découvrir les structures mathématiques qui se situent à la base de notre compréhension des théories physiques contemporaines. Il s'adresse donc, a priori , à un public d'étudiants en physique théorique mais il peut également constituer une introduction aux idées en question pour les jeunes mathématiciens, et plus généralement pour les curieux possédant un bagage mathématique suffisant (niveau requis égal ou supérieur à celui qu'on acquiert , en France, dans les classes de Mathématiques Spéciales ou à l'issue du DEUG A).

L'accent est essentiellement mis sur la théorie des espaces fibrés et celle des connexions, ainsi que sur la géométrie riemanienne, mais les deux premiers chapitres contiennent un résumé de notions générales concernant les variétés différentiables, les groupes et les algèbres de Lie. Le dernier chapitre est une introduction à la géométrie non commutative.

Le style est volontairement informel. Si vous cherchez à lire un résumé hyper-dense, ceci n'est pas pour vous.

Dans sa forme actuelle, l'ouvrage ne contient aucune description de la géométrie des varétés graduées (super-géométrie). Ce thème sera ultérieurement incorporé à l'ouvrage.

La première version html a été mise sur l'internet en mai 1997.

La version html actuellement disponible est la version 3.00 (mai 2002).

La version ps actuellement disponible est la version 3.00 (mai 2002).

Mots clefs : variétés, groupes de Lie, algèbres de Lie, actions de groupes, espaces homogènes, espaces fibrés, connexions, dérivées covariantes, courbure, champs de jauge, Yang-Mills, Relativité Générale, interactions fondamentales, gravité, géométrie non commutative.

Keywords : manifolds, Lie groups, Lie algebras, group actions, homogeneous spaces, fiber bundles, connections, covariant derivatives, curvature, gauge fields, Yang-Mills, General Relativity, fundamental interactions, gravity, noncommutative geometry.

Centre de Physique Théorique		Centre International de Rencontres Mathématiques
Clef de l'univers accrochée à la lune		Associahedron K3
About the artist

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CIMPA-UNSA-UNESCO - ARGENTINA SCHOOL :

BARILOCHE 2000

Quantum symmetries in theoretical physics and mathematics

Simetrías cuánticas en física teórica y en matemática

Symétries quantiques en physique théorique et en mathématiques

Available lectures (December 2001) ( download ) : N. Andruskiewitsch, M. Dubois-Violette, D. Evans, A. Ocneanu, O. Ogievetsky, J.B. Zuber.

Lecturers (photos) : N. Andruskiewitsch, M. Dubois-Violette, D. Evans, A. Ocneanu, O. Ogievetsky, N. Reshetikhin, M. Rosso, A. Varchenko, S.L. Woronowicz, J.B. Zuber.

  Classification of SL(2,C) and SL(3,C)  quantum subgroups : Poster presented at the Bariloche school (January 2000) by  A. Ocneanu  in relation with the classification of  SU(3) lattice integrable models (P. Di Francesco - J.B. Zuber ).

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Warning : The following comments are missing and the bookmarks are often rather old... This material will be improved !

Mathematics & Theoretical Physics : comments and bookmarks

Quantum field theory	Elementary particles
Classical and quantum gravity	Cosmology
Differential and riemannian geometry	Lie groups, etc.
Quantum groups and quantum groupoids	Non Commutative Geometry
Comments on Physics, Mathematics, Life, the Universe and Everything