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Lelouch
Lelouch
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Hi all. The question I have should be a rather simple one, but I just can't think it through.

So the Chern-Simons action reads \begin{equation} S = \int_M {\rm tr} (A\wedge dA + \frac{2}{3} A\wedge A \wedge A) \end{equation} where $M$ is 3-fold, and similarly for higher dimensional manifold.

Now, my question is:

**since $A$, the connection 1-form is only defined patch by patch, what do we really mean by doing the integration? **

It would be understandable if I write \begin{equation} S = \int_M {\rm tr} \left[(A-A_0)\wedge d(A-A_0) + \frac{2}{3} (A-A_0)\wedge (A-A_0) \wedge (A-A_0)) \right ] \end{equation} where $A_0$ is some reference connection, since $A-A_0$ is globally defined 1-form valued in ${\rm Lie}G$, or the bundle in question is trivial.

I see that under gauge transformation (or different chart), \begin{equation} CS(A^g) - CS(A) = d\alpha(A,g) + Q(g) \end{equation} where $Q(g)$ is closed. But I don't know how I can infer the validity of doing the integration from this gauge transformation.

Thank you!

Hi all. The question I have should be a rather simple one, but I just can't think it through.

So the Chern-Simons action reads \begin{equation} S = \int_M {\rm tr} (A\wedge dA + \frac{2}{3} A\wedge A \wedge A) \end{equation} where $M$ is 3-fold, and similarly for higher dimensional manifold.

Now, my question is:

**since $A$, the connection 1-form is only defined patch by patch, what do we really mean by doing the integration? **

It would be understandable if I write \begin{equation} S = \int_M {\rm tr} \left[(A-A_0)\wedge d(A-A_0) + \frac{2}{3} (A-A_0)\wedge (A-A_0) \wedge (A-A_0)) \right ] \end{equation} where $A_0$ is some reference connection, since $A-A_0$ is globally defined 1-form valued in ${\rm Lie}G$, or the bundle in question is trivial.

I see that under gauge transformation (or different chart), \begin{equation} CS(A^g) - CS(A) = d\alpha(A,g) + Q(g) \end{equation} where $Q(g)$ is closed. But I don't know how I can infer the validity of doing the integration from this gauge transformation.

Thank you!

Hi all. The question I have should be a rather simple one, but I just can't think it through.

So the Chern-Simons action reads \begin{equation} S = \int_M {\rm tr} (A\wedge dA + \frac{2}{3} A\wedge A \wedge A) \end{equation} where $M$ is 3-fold, and similarly for higher dimensional manifold.

Now, my question is:

**since $A$, the connection 1-form is only defined patch by patch, what do we really mean by doing the integration? **

It would be understandable if I write \begin{equation} S = \int_M {\rm tr} \left[(A-A_0)\wedge d(A-A_0) + \frac{2}{3} (A-A_0)\wedge (A-A_0) \wedge (A-A_0)) \right ] \end{equation} where $A_0$ is some reference connection, since $A-A_0$ is globally defined 1-form valued in ${\rm Lie}G$.

I see that under gauge transformation (or different chart), \begin{equation} CS(A^g) - CS(A) = d\alpha(A,g) + Q(g) \end{equation} where $Q(g)$ is closed. But I don't know how I can infer the validity of doing the integration from this gauge transformation.

Thank you!

Source Link
Lelouch
Lelouch
  • 857
  • 7
  • 15

How to understand Chern-Simons action

Hi all. The question I have should be a rather simple one, but I just can't think it through.

So the Chern-Simons action reads \begin{equation} S = \int_M {\rm tr} (A\wedge dA + \frac{2}{3} A\wedge A \wedge A) \end{equation} where $M$ is 3-fold, and similarly for higher dimensional manifold.

Now, my question is:

**since $A$, the connection 1-form is only defined patch by patch, what do we really mean by doing the integration? **

It would be understandable if I write \begin{equation} S = \int_M {\rm tr} \left[(A-A_0)\wedge d(A-A_0) + \frac{2}{3} (A-A_0)\wedge (A-A_0) \wedge (A-A_0)) \right ] \end{equation} where $A_0$ is some reference connection, since $A-A_0$ is globally defined 1-form valued in ${\rm Lie}G$, or the bundle in question is trivial.

I see that under gauge transformation (or different chart), \begin{equation} CS(A^g) - CS(A) = d\alpha(A,g) + Q(g) \end{equation} where $Q(g)$ is closed. But I don't know how I can infer the validity of doing the integration from this gauge transformation.

Thank you!