# The value of Gravitational Chern Simons theory integration on some three manifolds

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Consider the 3d gravitational Chern Simons theory
$$S= \frac{k}{192 \pi} \int_{M_3} Tr(\omega d \omega + \frac{2}{3}\omega^3)$$
where $\omega$ is the spin-connection on $M_3$. For the theory to be well defined, $k$ has to be an integer. I am interested to know what is the precise value of this integral for certain $M_3$. For instance, when $M_3= \mathbb{R}^3$, the integration clearly vanishes.

(1): What about M_3= T^3 (the three torus with length R along each direction), S^3 (the three sphere with radius R) and RP^3= S^3/Z_2 (S^3 of radius R with anti-pode identified)?

(2): Are there compact closed manifolds that can distinguish all the k in Z class (i.e., for different k, e^{iS} yields different phases. )?

p.s.: We know that on a M_3 with bdry, the brdy can have chiral central charge c= k/2 hence probes the value of k.

Any results or references will be helpful.

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