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I have noticed that there are two common choices for the coefficients in defining the Chern-Simons lagrangian:

$ S(A) = \frac{k}{8\pi^2}\int_M Tr(AdA + 2/3 A^3) $ and $ S(A) = \frac{k}{4\pi}\int_M Tr(AdA + 2/3 A^3). $ What is going on here, why the two different choices? In both cases, the parameter $k$ is always integer, correct?

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In the first case, a gauge transformation changes S(A) by an integer. In the second case, by $2\pi$ times an integer. The second case is a useful normalization for physicists, who care about the behavior of $exp(iS(A))$. The first case is probably a more sensible convention for doing differential geometry, but I don't actually know where it's used.

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