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...

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