topological actions

Question

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 $\omega \in H^4(BG,\mathbb{Z})$?

Answer 1

They are trying to define the Chern-Simons action over a manifold $M$ by writing it as the integral of $\int F \wedge F$ over a bounding manifold $B$. When the bundle is nontrivial, they consider a more general cochain and show that there exists a $B$ over which the bundle extends such that $\partial B$ = $n$ copies of $M$. So, you can define

$$ n S = \int_B F \wedge F $$

But, because actions enter into imaginary exponentials in the path integral, this is really only defined mod 1 (once you reenter all the coefficients that I omitted). So, the action $S$ is only defined mod $1/n$.

They show how the second formula resolves the ambiguity in the text that follows, but it's probably best to think of it as a differential character or in terms of differential cohomology. A more rigorous presentation might be http://arxiv.org/abs/hep-th/9111004.pdf .

Update:

Let me try another explanation. We know from the above that $$ n S_{CS} = \int_B F \wedge F \quad \mbox{mod 1} $$

Thus, $$ S_{CS} = \frac{1}{n}\int_B F \wedge F + \frac{q(B,E)}{n} $$ with $q(B,E) \in \mathbb{Z}$. The simplest guess is that $q = 0$, but it's easy to see that the resulting action is not independent of the choice of $B$. In particular, we would want, for closed $B$, that $\frac{1}{n} \int_B F \wedge F \in \mathbb{Z}$, but it's only in $\frac{1}{n}\mathbb{Z}$.

So, the goal is to choose a $q(B,E)$ such that the action makes sense. Since you want something that is an integer when applied to a closed $B$, it's not too hard to guess something like DV's action.

Answer 2

I don't think they actually find the constant you refer to in the update. As they remark, the choice of $\omega$ which maps to $\frac{k}{8\pi^2}Tr(F\wedge F)$ under $H^4(BG,\mathbb Z)\rightarrow H^4(BG, \mathbb R)$ is only defined up to torsion. The Chern-Simons invariant depends on the choice of $\omega$. At least that's what Dan Freed seems to indicate at the end of the appendix in this paper . He also has a follow up paper which discusses more the Chern-Simons theory where $G$ isn't necessarily simply connected.

Answer 3

Takefumi Nosaka has written a paper with Eri Hatakenaka that describes connections between DW and quandle cocycle invariants.