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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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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.

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 $k$.

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 \(\frac{1}{6}\)

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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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.

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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