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My question concerns the $U(1)$ Chern-Simons theory with the action $$S = \frac{k}{2\pi}\int A\wedge \mathrm{d}A.$$

In my lecture, it is stated that:

  1. A large gauge transformation involves taking $A\rightarrow A+\eta$, with $\eta$ closed but not necessarily exact. Furthermore $\eta$ has integral periods.

  2. The extra term generated by the gauge transformation $\int \eta \wedge \mathrm{d}A$ is equal to $(2\pi)^2 n$ with $n$ an integer, thus $e^{iS}$ is invariant provided $k$ is an integer.

My questions are:

  1. "$\eta$ has integral periods" -- does this mean $\int \gamma^*\eta$ is an integer for any loop $\gamma$?

  2. How do I see that $\int \eta\wedge \mathrm{d}A$ is of the form $(2\pi)^2n$, where $n$ is some integer?

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  1. Yes.

  2. This rely on the fact that $\eta$ (which may be a multiple-valued closed 1-form) and $F=\mathrm{d}A$ (which may be a non-exact closed 2-form) are both assumed to have integral periods. Then the wedge product $\eta\wedge F$ also has integral periods, cf. e.g. this Math.SE post.

References:

  1. G. Moore, 2019 TASI Chern-Simons lecture notes; subsection 2.2.1.
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