Loops, Knots, Gauge Theories and Quantum Gravity 2026, 36 - 2 + 1 Gravity

Taking 2 spatial and 1 temporal dimension enables testing the ideas of loop quantization, since even with 3 spatial dimensions, the Ricci tensor implies that spacetime is locally flat. The non-triviality of the Einstein theory comes from the topology of spacetime. The theory doesn't use local degrees of freedom. The Einstein action in 2 + 1 dimensions is written in first order form

(Eq. 9.66), using SO(2, 1) indices i, j, k. This is the Palatini action for the theory and the variables are real. The e.o.m are

(Eqs. 9.67-68). The canonical decomposition of the action consists of the pull-back of the connection Aⁱ_a to the 2D surface and the pull-back of the cotriad. Pulling back to the spatial slice turns the e.o.m. into constraints for those equations. It recovers Gauss law and a joint form of the diffeomorphism and Hamiltonian constraint of the 2 + 1 theory (p. 231). The system has twelve variables in phase space and six first class constraints, and exhibits no local degrees of freedom. The constraints are either linear or independent in momenta, which is much simpler than in 3 + 1 theory. The T-quantities constructed with the canonical variables have vanishing Poisson brackets with constraints and act as observables. This is due to the flatness of the connections. The 2 + 1 theory can be quantized as the 3 + 1 theory is. The loop representation of the T operators is also analogous to the 3 + 1 theory (p. 232). One can either promote the constraint equations to operators in loop space or use that the physical states in the connection representation have support on the moduli space of flat connections (p. 233). This implies that the space- and time-like cases in the connection representation seems to give rise to the same loop representation, but are distinct. The answer to the contradiction lies in the precise relationship between connection and loop representation: The time-like case is isomorphic to the symmetric connection representation, while that is not true for the space-like case (p. 235).