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

Question

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

Comments

Hi Zitao. I'm confused by your question. In your 3d example, when you write $p_1$ you must mean that $A$ is the metric connection. In 7d, what "metric" 3-form do you have in mind?

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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. 

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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.

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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

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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 $M$.

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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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Witten said on page 1 of http://arxiv.org/pdf/hep-th/9609122.pdf that for spin manifolds, the first Pontryagin class in divisible by 2 in a canonical way. This implies that for 7d, the thermal hall conductivity for the fermionic case is half that of the bosonic case. Not sure how he reached this conclusion and if there are other general relations.

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A basic remark is that \(7p_2(X)-p_1^2(X)\)has to be a multiple of \(45\), so it put constraints on \(p_1, p_2\). For instance, \(p_2=7, p_1=2\) is a possible choice...This will be in fact a more evident choice, if the A-genus would be zero, which would give : \(\hat{A}_2 = (-4p_2 + 7 p_1^2)/5760 = 0\).... 

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But the point is that $2p_2-p_1^2$ runs through $\mathbb{Z}$, so the constraint you mentioned does not put any extra constraint on $k$.

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oh wait, my last comment was wrong, it has to be multiples of 5. Sorry. So this does put some constraint on k.

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Regarding the comment above on lifting to a higher group:

yes, what we discuss in section 4.5 of arXiv:1201.5277 is 7d Chern-Simons theory not on 1-form SO(N)-gauge fields but on 1- and 2-form gauge fields for what is called the "String 2-group" extension of SO(N). The argument is that by arXiv:1202.2455 this is the correct choice if the 7d CS theory is supposed to be that appearing from the nonabelian 1-loop term in the CS term of 11d supergravity (actually in the full story its a "\(\mathrm{String}^a\)"-2-group that matters, see the article for the details.).

And on these StringSO(N)-2-group 2-form connections, the prefactor in question is \(\frac{1}{6}\)

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Thanks. This is really helpful.

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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.