In 3d, one can write down the SO(N) Chern-Simons action to be S(A)=k192π∫MTr(AdA+23A3),
Let M′ be a bounding 4-manifold of M. We can always find such M′ since ΩSO3=0. Extend A to M′ and define S(A)=k192π∫M′Tr(F∧F),
One can use the above argument, together with the fact that Ωspin3=0 and the Rohlin theorem which implies that the signature of a closed spin 4-manifold is divisible by 16, to argue that for a spin 4-manifold, exp(iS) is well-defined for k∈Z.
I'm trying to derive the quantization condition of k using similar arguments as above, for 7d SO(N) Chern-Simons action (simply replace M by a 7-manifold, and A by 3-form ). The following facts may be helpful: ΩSO7=0, Ωspin7=0, σ(X)=(7p2(X)−p21(X))/45
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