# Positivity for the level of Chern-Simons theory

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).

• Belated comment... doesn't this fail for ABJM models?
– user21299
Feb 12, 2020 at 22:09
• @alexarvanitakis: Could you please elaborate why you think that it fails in those models? Feb 12, 2020 at 23:53
• Perhaps I came across the wrong way: I was genuinely asking if it does fail and whether you considered this. The reason is the following: in ABJM theories (which are Chern-Simons plus some matter), the gauge group can be a product, say $U(N)\times U(N)$, and the Chern-Simons part is $k (S_{CS}[A_{left}]-S_{CS}[A_{right}])$ where $A_{left/right}$ are gauge fields for each $U(N)$ factor, and $k$ here is an integer (this is the naive physicist writing of the CS action of course). I can't say off the top of my head if this $k$ corresponds to a positive class $H^4(B(U(N)\times U(N)),\mathbb Z)$
– user21299
Feb 13, 2020 at 1:40