Here are the 2+1D gravitational Chern-Simons action of the connection $\Gamma$ or spin-connection:

$$ S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{a} $$

$$ S=\int\omega\wedge\mathrm{d}\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega \tag{b} $$

Usual Chern-Simons theory is said to be topological, since $S=\int A\wedge\mathrm{d}A + \frac{2}{3}A\wedge A \wedge A$ does not depend on the spacetime metric.

(1) Are (a) and (b) topological or not?

(2) Do (a) and (b) they depend on the spacetime metric (the action including the integrand)?

(3) Do we have topological gravitational Chern-Simons theory then? Then, what do questions (1) and (2) mean in this context of being topological?

Here are the 2+1D gravitational Chern-Simons action of the connection $\Gamma$ or spin-connection:

$$ S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{a} $$

$$ S=\int\omega\wedge\mathrm{d}\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega \tag{b} $$

A usual Chern-Simons theory of 1-form gauge field is said to be topological, since $S=\int A\wedge\mathrm{d}A + \frac{2}{3}A\wedge A \wedge A$ does not depend on the spacetime metric.

(1) Are (a) and (b) topological or not?

(2) Do (a) and (b) they depend on the spacetime metric (the action including the integrand)?

(3) Do we have topological gravitational Chern-Simons theory then? Then, what do questions (1) and (2) mean in this context of being topological?

Here are the 2+1D gravitational Chern-Simons action of the connection $\Gamma$ or spin-connection:

$$ S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{1} $$

$$ S=\int\omega\wedge\mathrm{d}\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega \tag{1} $$

Usual Chern-Simons theory is said to be topological, since $S=\int A\wedge\mathrm{d}A + \frac{2}{3}A\wedge A \wedge A$ does not depend on the spacetime metric.

(1) Are they topological or not?

(2) Do they depend on the spacetime metric (the action including the integrand)?

(3) Do we have topological gravitational Chern-Simons theory then? What do (1) and (2) mean in this context?

Here are the 2+1D gravitational Chern-Simons action of the connection $\Gamma$ or spin-connection:

$$ S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{a} $$

$$ S=\int\omega\wedge\mathrm{d}\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega \tag{b} $$

Usual Chern-Simons theory is said to be topological, since $S=\int A\wedge\mathrm{d}A + \frac{2}{3}A\wedge A \wedge A$ does not depend on the spacetime metric.

(1) Are (a) and (b) topological or not?

(2) Do (a) and (b) they depend on the spacetime metric (the action including the integrand)?

(3) Do we have topological gravitational Chern-Simons theory then? Then, what do questions (1) and (2) mean in this context of being topological?