Regarding a possible duality between (2+1)D gravity and Chern-Simons Theory

Consider the local Lorentz vector $omega^a$ constructed from the spin connection via the Hodge dual: $omega^a=frac12epsilon^{abc}e_comega^c{}_{ab}$, where $e_a=eta_{ab}e^b{}_mumathrm dx^mu$. Essentially since the triad and this spin vector are on the same footing, we can construct a Chern-Simon formulation of 3D gravity. In the following we work with a negative cosmological constant for simplicity, but the generalisation to arbitrary cosmological constant merely involves a slightly different algebra of generators (e.g. $mathfrak{iso}(2, 1)$ for $Lambda=0$).

Introduce the connections transforming in the adjoint representation of the Lorentz group $mathrm{SO}(2,1)$ : $$ mathfrak{so}(2,1) : [J_a,J_b]=epsilon_{ab}{}^cJ_c A^a_pm=omega^apmfrac{e^a}{ell} mathbf A_pm=A^a_pm J_a $$ where $ell$ is the cosmological length scale. Additionally, the matrix representation of the Hodge dual of the Ricci tensor is $$ mathbf R=R^aJ_a=mathrm domega+omegawedgeomega $$ Finally, since we are trying to construct an alternative formulation of 2+1D gravity, we can form a scalar action by integrating the Chern-Simons form of $mathbf A$ over a 3-manifold: $$ I_mathrm{CS}[mathbf A]=mathbf A+mathrm dmathbf A +frac23mathbf{Awedge Awedge A} $$ Via a routine computation, $$ mathrm{Tr}(I[mathbf A_+]-I[mathbf A_-])=frac2ellmathrm{Tr}left[2ewedgemathbf R+frac{2}{3ell^2}ewedge ewedge e-mathrm d(omegawedge e)right] $$

Tracing over all the generators via the Killing form definition $mathrm{tr}(J_aJ_b)simkappa(J_a,J_b)equiveta_{ab}$, one ends up with the following: $$ mathrm{Tr}(I[mathbf A_+]-I[mathbf A_-])=frac2ellleft(frac12sqrt{-g}R+frac1{ell^2}sqrt{-g}right)mathrm d^3 x-frac2ellmathrm d(omega^awedge e_a) $$

Thus the following is equivalent to the Einstein-Hilbert action, up to an irrelevant total derivative term: $$ S_mathrm{EH}approxfrac{ell}{16pi G}int_{mathcal M}mathrm{Tr}(I[mathbf A_+]-I[mathbf A_-]) $$

This is a concrete manifestation of gauge-gravity duality in 3 dimensions, relating a pure gravitational theory to a Chern-Simons gauge theory. A nice heuristic way I like to motivate this is that the symmetry algebra obeyed by the Killing vectors of $mathrm{AdS}_3$ is $mathfrak{so}(2, 2)$, which shows clean chiral bisplitting as $mathfrak{so}(2, 2)congmathfrak{so}(2, 1)oplusmathfrak{so}(2, 1)$, and these factors correspond to $I[mathbf A_+]$ and $I[mathbf A_-]$. As it turns out, the connections are flat on-shell.

One may compute, for instance, infinitesimal surface charges, phase space for $mathrm{AdS}_3$, asymptotic symmetries (and hence the integrated surface charge) in either formalism, and see that they match classically. So we can attack the problem of 3D gravity through two angles, and we find that e.g. computing the conserved charges is easier in the CS formulation, while computing the asymptotic symmetry algebra is easier in gravity proper.

However, as Alan Garbarz remarks, this classical equivalence is only due to the invertibility of the dreibein, which may not be the case in the quantum theory if one starts probing beyond the saddle points. A naïve perturbative analysis would appear that the quantum theory is immediately valid, but the question of its validity still lingers due to non-perturbative effects. Classically and perturbatively, however, this duality is exact.

Witten (https://arxiv.org/abs/0706.3359) gives more evidence for the non-perturbative breaking of this duality, including:

• Diffeomorphisms in 3D gravity are translated into gauge transformation of the connections. However, this only works for gauge transformations in the component connected to the identity, and so is not satisfied for a class of diffeomorphisms which may become important in the quantum theory
• There is no a priori reason to sum over topologies in the Chern-Simons formulation

While these two features can be put in by hand, it raises issues of naturalness. Nonetheless, the Chern-Simons - 3D Gravity duality is very powerful and often serves as a foundation for exactly solvable models of 3D gravity.