# Quantized coefficients of Chern-Simons action and F $\wedge$ F $\wedge \dots$

We know that for U(1) gauge field Chern-Simons action in 2+1 Dim(ension), we have an action $$S=\alpha \int A \wedge dA$$ with $\alpha=k/(4\pi)$ for a proper level quantization. Here $k$ is the level of the theory.

For U(1) gauge field action in 3+1 Dim(ension), we have an action $$S=\alpha \int F \wedge F$$ with $\alpha=\theta/(4\pi)$ (or $\alpha=\theta/(2\pi)^2$?) for a proper coefficient. However, the $\theta$ needs not to have level quantization.

Q1: How about other odd $(d+1)$-simensional spacetime Chern-Simons action: $$S=\alpha \int A \wedge dA \wedge dA \wedge \dots$$ what is $\alpha=$? for a proper level quantization? Is this $\frac{k}{2(2\pi)^{d/2}}$?

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Q2: How about other even $(d+1)$-simensional spacetime Chern-Simons action: $$S=\alpha \int F \wedge F \wedge \dots$$ what is $\alpha=$? for a proper coefficient? (However, the $\theta$ needs not to have level quantization.) Is this $\frac{\theta}{(2\pi)^{(d+1)/2}}$?

• If you use any Ref, please do share with me. Thank you.
• Edit: $\theta$ needs not to have level quantization. But what is the proper coefficient for U(1) field? – user32229 May 7 '14 at 14:28

## 1 Answer

At least for even dimensions the normalization is that $$S = \frac{(i/2\pi)^n}{n! }\int \operatorname{tr} F^{\wedge n}$$ is an integer. I learned this from Gauge Fields, Knots, and Gravity by Baez and Muniain. They do not give a proof but you can find one in From Calculus to Cohomology by Madsen and Tornehave. Since the odd dimensional case is a surface term for the even dimensional case I believe that the normalization is the same but I am not entirely sure about this.