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

+ 2 like - 0 dislike
583 views

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.

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysi$\varnothing$sOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.