Normalization of the Chern-Simons level in $SO(N)$ gauge theory

Question

In a 3d SU(N) gauge theory with action $\frac{k}{4\pi} \int \mathrm{Tr} (A \wedge dA + \frac{2}{3} A \wedge A \wedge A)$, where the generators are normalized to $\mathrm{Tr}(T^a T^b)=\frac{1}{2}\delta^{ab}$, it is well known the Chern-Simons level $k$ is quantized to integer values, i.e. $k \in \mathbb{Z}$.

My question is about the analogous quantization in $SO(N)$ gauge theories (A more standard normalization in this case would be $\mathrm{Tr}(T^a T^b)=2\delta^{ab}$ ). Some related subtleties are discussed in a (rather difficult) paper by Dijkgraaf and Witten Topological Gauge Theories and Group Cohomology, but I am not sure about the bottom line.

Does anyone know how to properly normalize the Chern-Simons term in $SO(N)$ gauge theories, or know a reference where this is explained?

Answer 1

Let me normalize the action as $$S=\frac{k}{4\pi}\int\langle A\wedge dA + \frac{1}{3} A\wedge[A\wedge A]\rangle$$ for $\langle,\rangle$ being the Killing form. This coincides with your normalization for $SU(N)$.

Variation of the Chern-Simons action under a gauge transformation $g:M\rightarrow G$ is given by $$S\rightarrow S + \frac{k}{24\pi}\int_{g_*[M]} \langle\theta\wedge[\theta\wedge\theta]\rangle,$$ where $\theta\in\Omega^1(G;\mathfrak{g})$ is the Maurer-Cartan form (Proposition 2.3 in http://arxiv.org/abs/hep-th/9206021). The last term is also called the Wess-Zumino term. Therefore, $\exp(iS)$ is invariant if $$\frac{k}{24\pi}\int_{[C]} \langle\theta\wedge[\theta\wedge\theta]\rangle\in2\pi\mathbf{Z}$$ for $[C]$ the generator of $H_3(G;\mathbf{Z})$.

For $G=SO(N)$, the homology is generated by $SO(3)\subset SO(N)$, and that term can be computed as follows. As you say, $$\frac{1}{24\pi}\int_{SU(2)} \langle\theta\wedge[\theta\wedge\theta]\rangle=2\pi,$$ but $SU(2)\rightarrow SO(3)$ is a 2:1 local diffeomorphism, so $$\frac{1}{24\pi}\int_{SO(3)} \langle\theta\wedge[\theta\wedge\theta]\rangle=\pi.$$

Therefore, the level $k$ in this case has to be even. See also appendix 15.A in the conformal field theory book by Di Francesco, Mathieu and Senechal.