3-dimensional Chern-Simons theories, with compact gauge group $G$, are determined by $H^4(BG)$. Looking at $U(1)$, with generator $c\_1^2\in H^4(BU(1))=\mathbb{Z}$ for 1st Chern class $c\_1$, there are infinitely many topological theories indexed by the level $k\in\mathbb{Z}$ (for class $k\cdot c\_1^2$). However, we can define other topological theories when an additional structure is put on the manifold. For a *spin structure*, this new "spin" U(1) Chern-Simons theory (in 3-dimensions) happens to exist for each *half-integer* $k\in\frac{1}{2}\mathbb{Z}$. Main reason behind this is that for your 3-manifold $Y$, a spin 4-manifold $X$ which bounds it will have *even* intersection-pairing.  

**[[Edit]]**: To expand on this, my source is Dijkgraaf/Witten. The "spin" topological theory is just taking your topological theory but requiring a choice of spin-structure. Given our functional $S=\frac{k}{4\pi^2}\int\_XF^2$ modulo 1, the spin structure makes this $=k\cdot(\text{even number})$, and so we can now enlarge the levels to $\frac{1}{2}\mathbb{Z}$.


My questions are (if they make sense): **Is there a structure on your manifold to have levels in $\frac{1}{n}\mathbb{Z}$ for some $n\ge 3$?** **Can you formulate "spin-c" Chern Simons?**


From a physical-perspective I would guess *no* to the first one, because at least in the case of half-integers it corresponds to fermions in nature (and fermions/bosons are what we live with). But then again there is the Fractional Quantum Hall Effect...



Analog of "Spin" Chern-Simons Theory
As David Roberts mentions in the comments above, indeed as one climbs up the [Whitehead tower of the orthogonal group](http://ncatlab.org/nlab/show/Whitehead%20tower#OfTheOrthogonalGroup) the higher [Pontraygin classes](http://ncatlab.org/nlab/show/Pontryagin%20class) become divisible by higher factors.


Notably in the next step, if you demand that also $\tfrac{1}{2}p\_1$ vanishes, hence that you have not only a Spin-structure but even a [String structure](http://ncatlab.org/nlab/show/String%20structure), then the second Pontryagian class becomes divisble by 6. 


In this case one can also divide the Lagrangian for 7-dimensional Chern-Simons theory by 6. This is discussed in 


Fiorenza, Sati, Schreiber, *[Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory](http://ncatlab.org/schreiber/show/7d+Chern-Simons+theory+and+the+5-brane)*.


