Suppose $C$ is a 3-form, and $G$ is a 4-form defined by $G = dC$. Also, $M_{11}$ is an 11-dimensional manifold (without a boundary), $W_{6}$ is a 6-dimensional submanifold of $M_{11}$ and $D_{\epsilon}W_6 = -S_{\epsilon}W_{6}$ is the 4-sphere bundle over $W_6$.
Further, suppose $\rho$ is a 0-form and $e_{2}^{1}$ is a 2-form. Under a variation,
$$\delta C = -d(\rho e_{2}^{1})$$
I want to compute the variation $\delta S_{CS}$ in the Chern-Simons integral
$$S_{CS} = -\lim_{\epsilon\rightarrow 0}\int\limits_{M_{11}\backslash D_{\epsilon}W_6} C \wedge G \wedge G$$
Apparently, the correct answer is
$$\delta S_{CS} = -\lim_{\epsilon \rightarrow}\int\limits_{S_{\epsilon}W_6}\rho e_{2}^{1}\wedge G \wedge G$$
But what I get is something else. Here is my detailed derivation.
$$\delta S_{CS} = -\lim_{\epsilon\rightarrow 0}\left[\int\limits_{M_{11}\backslash D_\epsilon W_6}\delta C \wedge G \wedge G + 2 \int\limits_{M_{11}\backslash D_\epsilon W_6}\delta G \wedge C \wedge G\right]$$
From integration by parts
$$\int\limits_{\partial(M_{11}\backslash D_\epsilon W_6)}\delta C \wedge C \wedge G = \int\limits_{M_{11}\backslash D_\epsilon W_6}d(\delta C \wedge C \wedge G) = \int\limits_{M_{11}\backslash D_\epsilon W_6} \delta dC \wedge C \wedge G - \int\limits_{M_{11}\backslash D_\epsilon W_6} \delta C \wedge G \wedge G$$
So it should follow that
$$\delta S_{CS} = -\lim_{\epsilon\rightarrow 0}\left[2\int\limits_{\partial(M_{11}\backslash D_\epsilon W_6)}\delta C \wedge C \wedge G + 3\int\limits_{M_{11}\backslash D_\epsilon W_6}\delta C \wedge G \wedge G\right]$$
As $M_{11}\backslash D_{\epsilon}W_6$, for finite $\epsilon$ cannot support an 11-form, the second integral vanishes inside the limit, and we are left with
$$\delta S_{CS} = -\lim_{\epsilon\rightarrow 0}\left[2\int\limits_{\partial(M_{11}\backslash D_\epsilon W_6)}\delta C \wedge C \wedge G\right]$$
Substituting $\delta C = -d(\rho e_2^1)$ we get
$$\delta S_{CS} = +\lim_{\epsilon\rightarrow 0}\left[2\int\limits_{\partial(M_{11}\backslash D_\epsilon W_6)} d(\rho e_2^1) \wedge C \wedge G\right]$$
Now,
$$d(\rho e_2^1 \wedge (C\wedge G)) = d(\rho e_2^1) \wedge C\wedge G + \rho e_2^1 \wedge d(C\wedge G)$$
and $\partial(\partial(M_{11}\backslash D_\epsilon W_6)) \equiv 0$, so
$$\delta S_{CS} = -\lim_{\epsilon\rightarrow 0}\left[2\int\limits_{\partial(M_{11}\backslash D_\epsilon W_6)}\rho e_2^1 \wedge G \wedge G\right]$$
As a last step, using $\partial(M_{11}\backslash D_\epsilon W_6) = -S_{\epsilon} W_6$, one gets the final expression
$$\delta S_{CS} = +\lim_{\epsilon\rightarrow 0}\left[2\int\limits_{S_\epsilon W_6}\rho e_2^1 \wedge G \wedge G\right]$$
This is off by a sign and a factor of 2.
What seems to be wrong in the derivation here?
Physics background: These algebraic manipulations are inspired by a calculation of the M5-brane anomaly in M-theory, perhaps discussed first in a paper by Freed, Minasian, Harvey and Moore (http://arxiv.org/abs/hep-th/9803205). It has been pointed out in some follow-up papers that there are an odd number of minus signs involved, and that the original paper may have overlooked one sign. (Of course the M5-brane anomaly is seen to cancel, but the cancellation is a bit involved and requires a careful understanding of minus signs and factors. Hence this question.)
This post imported from StackExchange Mathematics at 2015-08-09 22:38 (UTC), posted by SE-user leastaction
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