[Flots de Ricci et leurs propriétés d'intégrabilité en deux dimensions]
Dans cet article, nous résumons les principaux résultats sur les flots de Ricci en physique et en mathématique. En théorie des champs, dans les modèles sigma bidimensionnels, les flots de Ricci décrivent la renormalisation de la métrique de l'espace cible. En tant que tels ils permettent de comprendre le problème de la condensation de tachyons hors de la couche de masse, et de la sélection du vide en cordes fermées dans le régime gravitationnel faible. Dans le contexte de la géométrie différentielle, ils offrent le moyen de déterminer des métriques canoniques sur les variétés riemanniennes et de faire des progrès dans leur classification (conjecture de géométrisation). Notre attention sera plus particulièrement portée aux déformations géométriques de basse dimension, pour lesquelles on découvre une riche structure algébrique. A deux dimensions, nous montrons que le flot de Ricci est intégrable ; ceci grâce à une algèbre de dimension infinie avec un noyau de Cartan antisymétrique qui incorpore la variable de déformation dans son système de racines. Les déformations de surfaces bidimensionnelles contrôlent également le flot de Ricci sur des variétés à trois dimensions ainsi que leur décomposition en facteurs premiers. L'exérèse des singularités potentielles le long de cycles évanescents est cependant nécessaire. Nous discutons enfin quelques exemples simples parmi lesquels les solitons de Ricci, et dressons un catalogue d'applications à d'autres systèmes physiques.
We review the main aspects of Ricci flows as they arise in physics and mathematics. In field theory they describe the renormalization group equations of the target space metric of two-dimensional sigma models to lowest order in the perturbative expansion. As such, they provide an off-shell approach to the problem of tachyon condensation and vacuum selection in closed string theory in the weak gravitational regime. In differential geometry they introduce a systematic framework to find canonical metrics on Riemannian manifolds and make advances towards their classification by proving the geometrization conjecture. We focus attention to geometric deformations in low dimensions and find that they also exhibit a rich algebraic structure. The Ricci flow in two dimensions is shown to be integrable using an infinite-dimensional algebra with antisymmetric Cartan kernel that incorporates the deformation variable into its root system. The deformations of two-dimensional surfaces also control the Ricci flow on 3-manifolds and their decomposition into prime factors by applying surgery prior to the formation of singularities along shrinking cycles. A few simple examples are briefly discussed including the notion of Ricci solitons. Other applications to physical systems are also listed at the end.
Mot clés : Flot de Ricci, Intégrable, Géométrisation
@article{CRPHYS_2005__6_2_175_0,
author = {Ioannis Bakas},
title = {Ricci flows and their integrability in two dimensions},
journal = {Comptes Rendus. Physique},
pages = {175--184},
publisher = {Elsevier},
volume = {6},
number = {2},
year = {2005},
doi = {10.1016/j.crhy.2004.12.003},
language = {en},
}
Ioannis Bakas. Ricci flows and their integrability in two dimensions. Comptes Rendus. Physique, Volume 6 (2005) no. 2, pp. 175-184. doi : 10.1016/j.crhy.2004.12.003. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2004.12.003/
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