Constraint Dynamics and Gravitons in Three Dimensions

TL;DR

This work analyzes the fully non-linear constraint structure of three-dimensional gravity with a gravitational Chern-Simons term and a cosmological constant in the dreibein formulation. Using Dirac’s constraint method, it derives primary and secondary constraints, computes their algebra, and classifies them into first- and second-class sectors. The main result is that there is exactly one propagating graviton degree of freedom per spatial point, independent of the cosmological constant, and that the theory is not generally equivalent to a Chern-Simons gauge theory, except under specific identifications. These findings corroborate certain linear analyses predicting a persistent bulk DOF and raise questions about how bulk dynamics relate to boundary CFT features at the critical point $|a mu l|=1$.

Abstract

The complete non-linear three-dimensional Einstein gravity with gravitational Chern-Simons term and cosmological constant are studied in dreibein formulation. The constraints and their algebras are computed in an explicit form. From counting the number of first and second class constraints, the number of dynamical degrees of freedom, which equals to the number of propagating graviton modes, is found to be 1, "regardless of" the value of cosmological constant. I note also that the usual equivalence with Chern-Simons gauge theory does "not" work for general circumstances.