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I know that in 3D, there is a gravitational CS action given by

$$S = \int d^3 x \, \sqrt{-g} \, \epsilon^{\mu\nu\rho} \Big( R_{\mu\nu}{}^{ab} \omega_{\rho ab} + \frac23 \omega_{\mu}{}^{ab} \omega_{\nu b c}\omega_\rho{}^c{}_a \Big) \,, $$

whose field equation read the Cotton tensor

$0 = C_{\mu\nu} = \epsilon_\mu{}^{\tau\rho} \nabla_\tau S_{\rho \nu}$ ,

where $S_{\mu\nu}$ is the Schouten tensor

$S_{\mu\nu} = R_{\mu\nu} - \frac14 g_{\mu\nu} R \,.$

I am wondering if there is a five dimensional gravitational Chern-Simons action, and does it have a field equation that is defined by a special tensor such as the Cotton tensor?

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2 Answers 2

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One can try $$ CS_5= \int{ \rm tr}\{ \omega R^2-\frac 12 R \omega^3 +\frac 1{10} \omega^5 \} $$ where the $R$ and $\omega$ are 2-form and 1-form matrices. This is the analogue of the gauge-field CS term, but we would have
$d(CS_5)={\rm} tr\{ R^3\}$ but I think this is zero since we only have Pontryagin form in dimensions that are multiples of four. If $R$ is skew asymmetric matrix, as is the curvature tensor, then ${\rm tr}(R^{2n+1})=0$ for any integer $n$.

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The Chern-Simons Lagrangian under the action of the coboundary operator $d$ is the standard QFT or gravitational Lagrangian in $4$-dimensions.

If you want to imagine a $6$ dimensional spacetime, $5$ space plus time, there are then $5$ electric fields and $5$ magnetic fields. The Weyl tensor in this space would have the gravito-electric and magnetic fields $$ E_{\alpha\beta}~=~C_{\mu\alpha\beta\nu}U^\mu U^\nu,~B_{\alpha\beta}~=~*C_{\mu\alpha\beta\nu}U^\mu U^\nu $$ for $*$ the Hodge dual star action on the first antisymmetric pair of indices. This would then be a $5\times 5$ traceless matrix. The Petrov representation of gravity would then be not $3\times 3$ but $5\times 5$. You could then have a CS-like Lagrangian that reproduces the Lagrangian here under $d$.

The CS Lagrangian is a case of the Hopf fibration. The quaternion level is the inclusion map $S^3~\hookrightarrow~S^7~\rightarrow~S^4$. This is on a deep level how a three dimensional space is tied as a knot in seven dimensions. The CS Lagrangian and the standard Lagrangian living respectively in $S^3$ and $S^4$ are then in $S^7$ dual to each other. If you want to have something consistent with the Hopf fibration then the octonion level is $S^7~\hookrightarrow~S^{15}~\rightarrow~S^8$ In this case one might consider the $AdS_7\times S^4$ dual to $AdS_4\times S^7$ in the $AdS/CFT$ correspondence. In that case you could consider doing something with $7$ dimension spacetime physics.

So keep kicking around the idea, and I might as well. It might also be in the literature. I could be interesting, or it could be wrong.

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