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Ok, so I am still trying to make my way through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Actions" (see http://citeseer.ist.psu.edu/viewdoc/download;jsessionid=807BE383780883ACB4CAB8BD48E8C90B?doi=10.1.1.128.1806&rep=rep1&type=pdf ), and I have a question (what's new). The authors start by showing how one can define the action over an arbitrary 3-manifold $M$ as $n\cdot S = \frac{k}{8\pi^2}\int_B \langle F\wedge F\rangle$ $(mod\textrm{ }1)$. From here they say that they need to get rid of the $n$-fold ambiguity in the definition of the action. So, to do this they add $-\langle \gamma^*(\omega),B\rangle$ to the action, where $\gamma:B\rightarrow BG$, and $B$ is a 4-manifold which has n copies of $M$ as its boundary. Does anyone know how they arrived at this being the correct term to add (cf. equation 3.4 - 3.5)? Thanks.

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You haven't said enough about what $\gamma$ is to get a complete answer. More or less the idea behind all of these things is that if $B$ is closed the integral you gave is an integer, so the correction term has to do with building a $4$-form on $B\cup -B$ and using the integrality of the integral there. – Charlie Frohman Apr 26 2011 at 12:45
According to the authors, $\gamma$ is just the classifying map from $B$ to $BG$. Could you elaborate more on what you were saying? Also, I had another post with questions about the same article, so I updated that page (see mathoverflow.net/questions/62002/…) with this question - to keep everything together - and so, thinking about closing this page. – Kevin Wray Apr 27 2011 at 14:04

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