2+1 gravity as a conformal gauge theory and some frontiers for Shape Dynamics

          @misc{ pirsa_PIRSA:12050069,
            doi = {10.48660/12050069},
            url = {https://pirsa.org/12050069},
            author = {Gryb, Sean},
            keywords = {},
            language = {en},
            title = {2+1 gravity as a conformal gauge theory and some frontiers for Shape Dynamics},
            publisher = {Perimeter Institute},
            year = {2012},
            month = {may},
            note = {PIRSA:12050069 see, \url{https://pirsa.org}}
          }
          

Abstract

I will start by showing that gravity, with positive cosmological constant in 2+1 dimensions, can be formulated as a theory of dynamic conformal spatial geometry. Exploiting the isomorphism between the isometry group of de Sitter space in D+1 dimensions and the conformal group in D dimensions, I will reinterpret the Chern--Simons formulation of 2+1 gravity as a gauge theory of a conformal connection. In Cartan's generalization of geometry, this connection represents an evolving spatial geometry locally modeled off the conformal sphere. After a suitable phase space reduction, we obtain shape dynamics. This remodeling explains, in 2+1 dimensions, the remarkable success of the York procedure for solving the initial value problem of general relativity and the uniqueness of the shape dynamics Hamiltonian. I will finish by speculating about possible connections between this work and the general shape dynamics program with holographic renormalization, AdS/CFT, and Horava gravity.