## Séminaire de Physique Théorique

**Geometrical (2+1)-gravity and the Chern-Simons formulation: spacetime geometry, physical observables and the cosmological constant**

Catherine Meusburger (Perimeter Institute, Canada)

Wednesday 23 January 2008 14:00 - Tours - Salle 1180 (Bât E2)

**Résumé :**

Due to the absence of local gravitational degrees of freedom, Einstein's theory of gravity in (2+1) dimensions can be formulated as a Chern-Simons gauge theory. The Chern-Simons formulation of (2+1)-dimensional gravity provides an efficient parametrisation of phase space and Poisson structure, which serves as a starting point for quantisation, as well as a complete set of gauge invariant Wilson loop observables. Its drawback is that it obscures the underlying spacetime geometry and thereby complicates the physical interpretation of the theory. In my talk I relate the geometrical and the Chern-Simons description of vacuum spacetimes of general genus and with general cosmological constant. I discuss how the geometry of the spacetime can be recovered from the variables parametrising the phase space in the Chern-Simons formalism and how changes of geometry manifest themselves as transformations on the phase space. I show that the two basic transformations which change the geometry of the spacetime, infinitesimal Dehn twists (earthquakes) and grafting along a closed, simple geodesic, are generated via the Poisson bracket by the two associated canonical Wilson loop observables. By introducing a description in which the cosmological constant plays the role of a deformation parameter, I demonstrate that these two transformations are closely related and that grafting can be viewed as an infinitesimal Dehn twist (earthquake) with a formal parameter whose square is minus the cosmological constant.

**Liens :**