I am reading papers about topological massive gravity (TMG) in 3-dimensional spacetime. I come across two kinds of formalism to describe TMG. In the first kind, the gravitational Chern-Simons (CS) term is constructed by Christoffel connection $\Gamma$ which reads \begin{equation} S_1=\int \Gamma \wedge d\Gamma +\frac23\Gamma\wedge\Gamma\wedge\Gamma \end{equation} The other involves spin-connection $\omega$ \begin{equation} S_2=\int \omega\wedge d \omega+\frac23\omega\wedge\omega\wedge\omega \end{equation}

My questions are:

  1. Are those two kinds of formalism are equal? By equal I mean $S_1=S_2$ up to some boundary term?

  2. What the differences between those two formalisms? When should we use the first one or the second?


Topologically massive gravity formulated in terms of $\Gamma$ or $\omega$ is the same; for example in Maloney's paper on the geometric microstates of the three-dimensional black hole, he uses the $\Gamma$ formulation, and Witten's talk on revisiting $2+1$-dimensional gravity includes the Chern-Simons term with the spin connection $\omega$, but they are both describing the same theory.

One subtlety I'd like to point out, when performing reformulations of fields, is it can modify it in a non-trivial way. In particular, gravity in $2+1$ dimensions is considered trivial, but when formulated in terms of a gauge field out of the connection and vierbein,

$$A = \begin{pmatrix} \omega & e\\ -e & 0 \end{pmatrix}$$

allowing a non-invertible $e$ and thus non-classical configurations. It turns out this makes the Einstein-Hilbert action ill-defined unless the coupling is quantised.

Since you seem interested in topological massive gravity, and since it wasn't mentioned in the papers you listed, I would like to point out it has been conjectured by Witten and others that if the Frenkel-Lepowsky-Meurman conjecture is true, it may be that the dual $k=1$ CFT is the theory whose symmetry is described by the Monster group - I don't know if this avenue is of interest to you.

To explicitly show the equivalence, it seems rather messy. As a starting point, note the relation between the spin connection $\omega$ and the affine connection $\Gamma$ as,

$$\Gamma^\lambda_{\mu\nu} = e^\lambda_A \partial_\mu e^A_\nu + e^\lambda_A e^B_\nu {\omega_\mu}^A_B$$

where capital Roman indices denote the orthonormal basis and Greek indices the coordinate basis. We can substitute this into the Chern-Simons term for $\Gamma$, which is explicitly in index notation,

$$\mathcal L \sim \epsilon^{\lambda\mu\nu} \Gamma^\rho_{\lambda \sigma} \left( \partial_\mu \Gamma^\sigma_{\rho\nu} + \frac{2}{3} \Gamma^\sigma_{\mu\kappa} \Gamma^\kappa_{\nu\rho}\right),$$

using the definition of the wedge product and exterior derivative. Using the relation between the connections, we thus have,

$$\mathcal L \sim \epsilon^{\lambda\mu\sigma}\left(e^\rho_A \partial_\lambda e^A_\nu + e^\rho_A e^B_\sigma {\omega_\lambda}^A_B \right) \bigg[ \partial_\mu \left( e^\sigma_A \partial_\rho e^A_\nu + e^\sigma_A e^B_\nu {\omega_\rho}^A_B\right)+ \bigg.\\ + \bigg. \frac{2}{3} \left(e^\sigma_A \partial_\mu e^A_\kappa + e^\sigma_A e^B_\kappa {\omega_\mu}^A_B \right) \left( e^\kappa_A \partial_\nu e^A_\rho + e^\kappa_A e^B_\rho {\omega_\nu}^A_B\right)\bigg].$$

At this point it is a matter of tedious manipulation, which should hopefully show an equivalence to the theory in terms of the spin connection.

  • $\begingroup$ Thank you for your answer very much! It is a good starting point. $\endgroup$ – phys_student Nov 26 '16 at 1:15
  • $\begingroup$ @phys I've updated my answer with confirmation. $\endgroup$ – JamalS Dec 1 '16 at 22:20

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