I have a two-part question:

(1) First and foremost: I have been going through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Field Theories." (See Link) Here they give a general definition for the Chern-Simons action for a general $3$-manifold $M$. My question is if anyone knows of any follow-up to this, or notes about their paper?

(2) To those who know the paper: They say that they have no problem defining the action modulo $1/n$ (for a bundle of order $n$) as $n\cdot S = \int_B Tr(F\wedge F)$ $(mod 1)$, but that this has an $n$-fold ambiguity consisting of the ability to add a multiple of $1/n$ to the action - What do they mean here? Also, later on they re-define the action as $S = 1/n\left(\int_B Tr(F\wedge F) - \langle \gamma^\ast(\omega),B\rangle\right)$ $(mod 1)$ - How does this get rid of the so-called ambiguity?

Basically my question is if anyone can further explain the info between equations 3.4 and 3.5 in their paper. Thanks.

**Update:** I'm fine with re-defining the action as $S = 1/n\left(\int_B Tr(F\wedge F) - \langle \gamma^\ast(\omega),B\rangle\right)$ $(mod 1)$. But, does anyone know how they came to discover that this is what to add to the action to remove the ambiguity in the previous definition the action? I mean, if you only know $S$ modulo $1/n$, and if you think it's $S_0 = \int_B Tr(F\wedge F)$ it means that the real action is $S = S_0 + a_{M,B}\big/n$, where $a_{M,B}$ is some integer that possibly depends on $M$, $B$ and the extension of $E$ over $B$. So to remove the ambiguity you have to find this integer for the
specific data, but how do they FIND the expression for this integer; that is, how do they calculate the $a_{M,B}$ to be $\langle \gamma^\ast(\omega),B\rangle$. where $\omega \in H^4(BG,\mathbb{Z})$?