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...

  • $\begingroup$ You might be interested in considering fractional Pontryagin classes which turn up in higher gauge theory. There's certainly some Chern-Simons stuff there. For example: ncatlab.org/nlab/show/differential+string+structure and ncatlab.org/nlab/show/differential+fivebrane+structure. $\endgroup$ – theHigherGeometer Jun 27 '12 at 6:45
  • $\begingroup$ Could you please explain how your "spin" U(1)-CS theory is defined? Does it come from an action functional? $\endgroup$ – Konrad Waldorf Jun 27 '12 at 7:07
  • $\begingroup$ Spin structures corresponds to square roots of canonical bundle may be you need higher degree roots of it. $\endgroup$ – Alexander Chervov Jun 27 '12 at 9:01
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    $\begingroup$ @Alexander: Isn't this only true for Riemann surfaces, whereas Chris talks about spin structures on 3-manifolds? $\endgroup$ – Konrad Waldorf Jun 27 '12 at 9:25
  • $\begingroup$ @Konrad Waldorf okay you are probably right I forgot the details. Any way it might be relevant if 3-fold = 2-fold \cross S^1. $\endgroup$ – Alexander Chervov Jun 28 '12 at 5:50

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.


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