# Analog of “Spin” Chern-Simons Theory

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.
[]: 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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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. –  David Roberts Jun 27 '12 at 6:45
Could you please explain how your "spin" U(1)-CS theory is defined? Does it come from an action functional? –  Konrad Waldorf Jun 27 '12 at 7:07
Spin structures corresponds to square roots of canonical bundle may be you need higher degree roots of it. –  Alexander Chervov Jun 27 '12 at 9:01
@Alexander: Isn't this only true for Riemann surfaces, whereas Chris talks about spin structures on 3-manifolds? –  Konrad Waldorf Jun 27 '12 at 9:25
@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. –  Alexander Chervov Jun 28 '12 at 5:50

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.