TITLE: Analog of "Spin" Chern-Simons Theory
QUESTION [9 upvotes]: 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...

REPLY [8 votes]: As David Roberts mentions in the comments above, indeed as one climbs up the Whitehead tower of the orthogonal group the higher Pontraygin classes 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, 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.