# 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

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 $\frac{1}{6}$

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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