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

+ 5 like - 0 dislike

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.

Most recent comments show all comments

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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