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).


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.