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Level quantization of 7d $SO(N)$ Chern-Simons action

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In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be derived as follows:

Let $M^{\prime}$ be a bounding 4-manifold of $M$. We can always find such $M^{\prime}$ since $\Omega^{SO}_3=0$. Extend $A$ to $M^{\prime}$ and define $$S(A)=\frac{k}{192\pi}\int_{M^{\prime}}\text{Tr}(F \wedge F),$$ where $F$ is the curvature 2-form of $A$. We need $\exp(iS_M(A))$ to the independent of the choice of $M^{\prime}$, and the extension of $A$ from $M$ to $M^{\prime}$. Let $M^{\prime\prime}$ be another bounding manifold of $M$, then the difference of $S$ is $$\delta S = \frac{k}{192\pi}\int_{M^{\prime}\cup \bar{M}^{\prime\prime}}\text{Tr}(F \wedge F),$$ where $\bar{M}^{\prime\prime}$ denotes the orientation reversal of $M^{\prime\prime}$. $\delta S$ can be rewritten as $$\delta S = \frac{k\pi}{24}p_1(M^{\prime}\cup \bar{M}^{\prime\prime}) = \frac{k\pi}{8}\sigma(M^{\prime}\cup \bar{M}^{\prime\prime}),$$ where $p_1$ is the first Pontryagin number, and $\sigma$ is the signature of a 4-manifold. We also used the Hirzbruch signature theorem $\sigma(X)=p_1(X)/3$ for 4-manifolds $X$. Since $\sigma(X)$ is an integer, $exp(iS_M(A))$ is well-defined for $k$ equals multiples of 16.

One can use the above argument, together with the fact that $\Omega^{spin}_3=0$ and the Rohlin theorem which implies that the signature of a closed spin 4-manifold is divisible by 16, to argue that for a spin 4-manifold, $\exp(iS)$ is well-defined for $k\in \mathbb{Z}$.

I'm trying to derive the quantization condition of $k$ using similar arguments as above, for 7d $SO(N)$ Chern-Simons action (simply replace $M$ by a 7-manifold, and $A$ by 3-form ). The following facts may be helpful: $\Omega^{SO}_7=0$, $\Omega^{spin}_7=0$, $$\sigma(X) = (7p_2(X)-p_1^2(X))/45$$ for 8-manifold $X$.

This post imported from StackExchange Physics at 2014-09-15 21:05 (UCT), posted by SE-user Zitao Wang
asked Sep 15, 2014 in Theoretical Physics by Zitao Wang (165 points) [ no revision ]
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Hi Ryan, Thanks for pointing out. I'm being too naive there. The suitable analogous $SO(N)$ Chern-Simons action in 7d should be something like $S=\kappa \text{Tr} (A(dA^3)+\frac{8}{5}A^3(dA)^2+\frac{4}{5}A^2dAAdA+2A^5dA+\frac{4}{7}A^7)$, where $A$ is the metric connection, and I want to determine $\kappa$. The original problem I had in mind is to find a suitable 7d analogue of the 3d gravitational Chern-Simons action that would make the 7d Abelian Chern-Simons action topological. 

Perhaps one can define $A$ as the connection on a principal 3-bundle over some 3-group $G$ as in http://ncatlab.org/nlab/show/7d+Chern-Simons+theory#AbelianTheory for $B^3U(1)$. But I don't know of the 3-group corresponding to $SO(N)$. If this can be done, then we get a differential 3-form, and gravitational Chern-Simons action in its original form makes sense for the 7d case.

I think what you want is some integral characteristic class of the tangent bundle in 8d whose Chern-Weil density is a total derivative. The potential whose differential is the density is the Chern-Simons 7-form. If you look up the original math paper where the Chern-Simons invariant is defined, you can read about the general theory of secondary characteristic classes. Here is the paper http://www.jstor.org/discover/10.2307/1971013?uid=3739560&uid=2&uid=4&uid=3739256&sid=21104636509487

Yes. Using the transgression formula in eq.(3.5) and also eq.(5.13) of your reference, I was able to compute the Chern-Simons 7-form as given in my thread 2 $\frac{1}{2}TP_2(A) =\frac{1}{384\pi^4} \text{Tr} (A(dA^3)+\frac{8}{5}A^3(dA)^2+\frac{4}{5}A^2dAAdA+2A^5dA+\frac{4}{7}A^7)$ where $A$ is an $SO(7)$ connection. So I think the gravitational Chern-Simons action we are looking for in the bosonic case is $S=\int_M\frac{k}{384\pi^4} \text{Tr} (A(dA^3)+\frac{8}{5}A^3(dA)^2+\frac{4}{5}A^2dAAdA+2A^5dA+\frac{4}{7}A^7)$. The quantization of $k$ is a little bit tricky in this case, since $S = \int_{\tilde{M}}\frac{k}{12}(2p_2-p_1^2)$, which is not proportional to the signature of $\tilde{M}$, where $\tilde{M}$ is the bounding 8 manifold of $M$.

But there should be some general results about $p_1$ and $p_2$ for 8-manifolds. Presumably they can be arbitrary integers. Maybe for spin manifolds they have to be divisible by something.

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Thanks. This is really helpful.

Maybe to add that: the actual construction and theorem underlying this is in arXiv:1011.4735 This is a general machine that reads in an \((n+1)\)-cocylce \(\mu\) on an \(L_\infty\)-algebra \(\mathfrak{g}\)and spits out a fully local ("extended", "mult-tiered") \(n+1\)-dimensional Chern-Simons type Lagrangian for \(\mathfrak{g}\)-connections. 

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