In some of the literature (for example, below Eq. (A3) of this paper), the following is claimed to be the Chern-Simons term in the Coulomb gauge:

\begin{equation}

2a_0(\partial_1a_2-\partial_2a_1)

\end{equation}

and it cited this paper, which below its Eq. (5) states that the above term is the Chern-Simons term in radiation gauge.

My questions are:

1. Is the radiation gauge the same as the Coulomb gauge, where $\partial_1a_1+\partial_2a_2=0$.

2. Why is the above term the Chern-Simons in the Coulomb gauge? The standard Chern-Simons term is

$$\epsilon_{\mu\nu\lambda}a_\mu\partial_\nu a_\lambda=a_0(\partial_1a_2-\partial_2a_1)+a_1(\partial_2a_0-\partial_0a_2)+a_2(\partial_0a_1-\partial_1a_0)$$

After integrating by parts it becomes

$$\epsilon_{\mu\nu\lambda}a_\mu\partial_\nu a_\lambda\rightarrow 2a_0(\partial_1a_2-\partial_2a_1)+2a_2\partial_0a_1$$

which still differs from $2a_0(\partial_1a_2-\partial_2a_1)$ by the last term that does not seem to vanish in the Coulomb gauge.