Centre de Physique Theorique Ecole Polytechnique

Chern Simons theory on a lattice and a new description of 3-manifolds invariants. Abstract A new approach to the quantization of Chern-Simons theory has been developed in recent papers [1, 2, 3, 4]. It uses a " simulation " of the mod-uli space of flat connections modulo the gauge group which reveals to be related to a lattice gauge theory based on a quantum group. After a generalization of the formalism of q-deformed gauge theory to the case of root of unity, we compute explicitely the correlation functions associated to Wilson loops (and more generally to graphs) on a surface with punctures , which are the interesting quantity in the study of moduli space. We then give a new description of Chern-Simons three manifolds invariants based on a description in terms of the mapping class group of a surface. At last we introduce a three dimensional lattice gauge theory based on a quantum group which is a lattice regularization of Chern-Simons theory.