# NCG

## Seminar

### (2+1)-dimensional gravity in the geometrical and in the Chern-Simons formulation

Seminar Room 2, Newton Institute Gatehouse

#### Abstract

The talk relates the geometrical and the Chern-Simons formulation of (2+1)-gravity for Lorentzian signature and general cosmological constant and for the Euclidean case with negative cosmological constant. We establish a link between spacetime geometry and the description of phase space and Poisson structure in the Chern- Simons formalism. We discuss how the geometrical construction of spacetimes via grafting and infinitesimal Dehn twists along closed, simple geodesics gives rise to transformations on the phase space. We show that these transformations are generated via the Poisson bracket by the two canonical Wilson loop observables associated to the geodesic. For Lorentzian signature, we discuss the role of the cosmological constant as a deformation parameter in the Chern-Simons formalism. We show that grafting can be viewed as an infinitesimal Dehn twist with a formal parameter whose square is identified with the cosmological constant.