Abstract:

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 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.

References:

C. Meusburger, Geometrical (2+1)-gravity and the Chern-Simons
formulation: Grafting, Dehn twists, Wilson loop observables and the
cosmological constant,
Commun. Math. Phys. 273 (2007) 705–754.

C. Meusburger, Grafting and Poisson structure in (2+1)-gravity with
vanishing cosmological constant,
Commun. Math. Phys. 266 (2006) 735--775,