Projet ANR-06-BLAN-0311 – RepSurfaces : Espaces de modules de représentations de groupes de surfaces : espaces de Teichmüller généralisés et géométrie
Programme : Programme blanc (BLANC) 2006
Année de financement : 2006
Coordinateur du projet : CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE - DELEGATION REGIONALE ILE-DE-FRANCE SECTEUR SUD (CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE - DELEGATION REGIONALE ILE-DE-FRANCE SECTEUR SUD)
Partenaires : Centre national de la recherche scientifique; Université de Grenoble Alpes; Université de Toulouse III - Paul Sabatier
Montant : 174000.0 euros
Date de début : 2006
Durée : 36 mois
Résumé : The main focus of this project is the moduli space M(S,G) of representations in a Lie group G of the fundamental group of a closed surface S.   When G is compact (for instance SU(2)), the corresponding moduli space is compact, acted upon ergodically by the mapping class group. Moreover, the choice a complex structure on the surface identifies M(S,G) with a space of holomophic objects. The space M(S,G) is the classical phase space for 3D-gauge field theory. In the beginning of the 90's, various schemes for the quantisation of M(S,G) showed unexpected links between various fields of mathematics: invariants of 3-manifolds, representations of quantum groups, symplectic and algebraic geometry.   On the other hand, in the 80's, Thurston explained the importance of Teichmüller space in understanding the geometry of 3-manilfolds. Teichmüller space is one of the components of the space M(S,G) when G=PSL(2,R). When compared to SU(2), we see that PSL(2,R) is at the other end of the spectrum: it is the most non compact real form of PSL(2,C). From the moduli spaces point of view, this has important consequences: Teichmüller spaces have some highly different features from M(S,G) when G is compact. It is non compact and contractible, the mapping class group acts properly and the quotient has a holomorphic interpretation. Teichmüller space itself carries a natural complex structure. It also turns out that Teichmüller space has a very rich geometry unveiled by Thurston and his followers. From the physicists point of view, Teichmüller theory should play the role of the classical space for 2+1-gravity.  In the last years, partly motivated by the interest of physicists, new perspectives have been opened and the purpose of this project is to give our team the opportunity to participate to this new thrust.   The project has four sides, which we describe now briefly. This four sides are rather independent but strongly linked, they are related to different fields of mathematics. The central object of research will be M(S,G) when G is non compact. A: The other components. When G=PSL(2,R), the space M(S,G) has other components than Teichmüller spaces. As of today, very little is known about these "exotic components". In the first part of the project, we propose to understand their geometric properties. More specifically, we are interested in their compactfications and the action of the mapping class group on these components . B: 3 dimensional geometry of spaces of constant curvature. A special subset of M(S,PSL(2,C)), the set of quasi-Fuchsian representations plays a central role in hyperbolic geometry. A similar set plays a role in the study of Anti de Sitter geometry. The second group of questions addressed here is to parametrise these subsets of representations (when G=PSL(2,C) or G=SO(2,2)) by geometric objects from Thurston's theory (such as laminations etc.) Some of these questions are very old, but recent ideas involving in particular the inclusion of singularities  or "particles"- opened a new perspective on the subject.  C: Quantisation of Teichmüller spaces. Recently, a natural algebra of observables has been quantised by L.Checkhov and V.Fock. This quantisation lead to the construction of new invariants for hyperbolic 3-manifolds. However, the relation with classical objects (mainly volumes) remains obscure.  	D: Generalisations of Teichmüller theory. One can describe Thurston-Teichmüller theory as a dictionary between various fields: dynamical systems, geometry, combinatorics, integrable systems. Rather surprisingly, it appears now that some special components of M(S,PSL(n,R)) provide a similar dictionary. However many questions remains open in the subject.