Equation regarding the Riemann tensor in the Cartan formalism

Question

I have a problem verifying the following equation (in three dimensions)

$$\epsilon_{abc} e^a\wedge R^{bc}=\sqrt{|g|}Rd^3 x$$

where $R$ is the Ricci scalar and $R^{bc}$ is the Ricci curvature

Attempt at a solution:

$$\epsilon_{abc} e^a\wedge R^{bc}=\epsilon_{abc} e_\mu^ae_\alpha^be_\beta^c R^{\alpha\beta}_{\nu\rho} dx^\mu\wedge dx^\nu\wedge dx^\rho$$

Now the idea is that the number of dimensions and the Levi-Civita tensor and the antisymmetry of the three-form forces the set $\{\alpha,\beta\}=\{\nu,\rho\}$. This will give the expression

\begin{align}\epsilon_{abc} e^a\wedge R^{bc}&=\epsilon_{abc} e_0^ae_1^be_2^c R^{12}_{12} dx^0\wedge dx^1\wedge dx^2+\\&\epsilon_{abc} e_0^ae_1^be_2^c R^{12}_{21} dx^0\wedge dx^2\wedge dx^1+\\&\epsilon_{abc} e_0^ae_2^be_1^c R^{21}_{21} dx^0\wedge dx^2\wedge dx^1+\\&\epsilon_{abc} e_0^ae_2^be_1^c R^{21}_{12} dx^0\wedge dx^1\wedge dx^2+({\rm cyclic\,permutations})\end{align}

The problem now is that the Ricci scalar is $R^{12}_{12}+R^{21}_{21}+({\rm cyclic\,permutations})$, so when counting the number of terms I obtain $2\sqrt{|g|}Rd^3 x$ which is wrong by a factor of 2. Can anyone see where I made a mistake?

This post imported from StackExchange Physics at 2015-10-11 18:32 (UTC), posted by SE-user user2133437

Comments

I don't know if you are intentionally overstacking the indices on the Riemann tensor, but in case not: $R^{\alpha\beta}{}_{\nu\rho}$ can be hacked R^{\alpha\beta}{}_{\nu\rho} or {R^{\alpha\beta}}_{\nu\rho}. The former risks breaking across line wraps, and the latter is semantically wrong, but they work in a pinch.

This post imported from StackExchange Physics at 2015-10-11 18:32 (UTC), posted by SE-user Chris White