Consider the 3d gravitational Chern Simons theory
S=k192π∫M3Tr(ωdω+23ω3)
where
ω is the spin-connection on
M3. 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
M3. For instance, when
M3=R3, 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.