Editor’s Choice (Research Article)

General Relativity and Gravitation

, Volume 44, Issue 7, pp 1615-1636

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Shape dynamics in 2 + 1 dimensions

  • Timothy BuddAffiliated withInstitute for Theoretical Physics, Utrecht University Email author 
  • , Tim KoslowskiAffiliated withPerimeter Institute for Theoretical Physics


Shape Dynamics is a formulation of General Relativity where refoliation invariance is traded for local spatial conformal invariance. In this paper we explicitly construct Shape Dynamics for a torus universe in 2 + 1 dimensions through a linking gauge theory that ensures dynamical equivalence with General Relativity. The Hamiltonian we obtain is formally a reduced phase space Hamiltonian. The construction of the Shape Dynamics Hamiltonian on higher genus surfaces is not explicitly possible, but we give an explicit expansion of the Shape Dynamics Hamiltonian for large CMC volume. The fact that all local constraints are linear in momenta allows us to quantize these explicitly under a certain assumption on the kinematic Hilbert space, and the quantization problem for Shape Dynamics turns out to be equivalent to reduced phase space quantization. We consider the large CMC-volume asymptotics of conformal transformations of the wave function. We then discuss the similarity of Shape Dynamics on the 2-torus with the explicitly constructible strong gravity Shape Dynamics Hamiltonian in higher dimensions.


Canonical general relativity General relativity in 2 + 1 dimensions Shape dynamics Dirac quantization