Quantum gravity in three dimensions, Witten spinors and the quantisation of length

Understand

In this paper, I investigate the quantisation of length in euclidean quantum gravity in three dimensions.

• The starting point is the classical hamiltonian formalism in a cylinder of finite radius.
• At this finite boundary, a counter term is introduced that couples the gravitational field in the interior to a two-dimensional conformal field theory for an SU(2) boundary spinor, whose norm determines the conformal factor between the fiducial boundary metric and the physical metric in the bulk.
• The equations of motion for this boundary spinor are derived from the boundary action and turn out to be the two-dimensional analogue of the Witten equations appearing in Witten's proof of the positive mass theorem.