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In many classical papers about Chern-Simons theory (see, e.g. [1]), it is claimed that the Chern-Simons theories with gauge group $G$ are classified by an element of $k\in H^4(BG,\mathbb Z)$, the so-called "level". (I'm not assuming here that $G$ is simply connected, or even connected)

Now, I have reasons to believe that this description is not quite accurate. Rather, I suspect that only the levels subject to a certain positivity condition yield actual quantum field theories. The exact positivity condition I have in mind can be formulated by saying that the bilinear form on $\mathfrak g$ which is the image of $k$ under the Chern-Weil homomorphism should be positivie definite . (Here, the Chern-Weil homomorphism goes from $H^*(BG)$ to $Sym(\mathfrak g^*)$)

Question: Is there any reference on Chern-Simons theory in which the above positivity requirement on the level is noted?


Reference: [1] R. Dijkgraaf and E. Witten, Topological Gauge Theories And Group Cohomology, Commun. Math. Phys. 129, 393 (1990).

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