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I came across the following post regarding the boundary term in Chern-Simons theory (specifically the level quantization of the theory). I am new to differential forms so the following questions may be trivial, but either way:

  1. In equation (12) of the accepted answer, where did the self wedge product terms i.e. $F_{D}\wedge F_{D}$ go?

  2. Similarly, in (18), where did terms like $A_{S_{1}} \wedge dA_{S_{1}}$ go?

  3. What is the intuition behind the last statement: " the contributions from the adjacent parts of the different patches' boundaries cancel each other". How do I see this?

Qmechanic
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tumm
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1 Answers1

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  1. $F\wedge F$ is nonzero only if it has both a $\vec E = F_{0i}$ component and a $\vec B = F_{jk}$ component. However, if you look at their Eq. (10), $F_D$ only has the $F_{r\alpha}$ component and no others. Therefore $F_D\wedge F_D$ is zero.
  2. It's for the same reason - $A_{S_1}$ and $A_{S_2}$ by construction only have components along $S_1$ and $S_2$ respectively.
  3. The boundaries of patches like Eq. (20) and patches like Eq. (21) cancel out each other. So the total 3d manifold has no boundary. You can in fact show that this boundary-less manifold is a boundary of a 4d manifold. On the other hand, you can use Stokes theorem to reduce Eq. (17) to a CS form on the same boundary manifold. Therefore they are equivalent.
pathintegral
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