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The U(1) bundle on a torus is a important math setup for a lot of physics problems. Since I am awkward on this subject and many of the physics material doesn't give a good introduction. I like to put some of my understanding here and please help me to check whether they are right or wrong.
1. A one-form ##A## can be defined over the whole torus.
2. To define a connection one-form for this bundle, we need a Lie-algebra valued one-form on the torus. So I can simply define this form by adding an ##i## to ##A## as ##iA##.
3. So the Lie-algebra valued local curvature two-form is ##F = diA + iA \wedge iA = idA##
4. If there is no continuous section can be found. Both ##iA## and ##F## can only be well defined on local charts. In the overlapping parts ##{U_i} \cap {U_j}## the transition function is ##{t_{ij}}(p) = \exp [i\Lambda (p)]##. So we have the transition ##iA \to iA + id\Lambda (p)## and ##F \to t_{ij}^{ - 1}Ft_{ij}^{ - 1} = F## and the Chern number is found as
##\int_T {\frac{i}{{2\pi }}trF = } \int_T {\frac{i}{{2\pi }}F} \in Z##
So ##F## is exactly ##idA## everywhere? is this right ?
If so, by directly appling stokes theorem
##\int_T {F = - i} \int_T {dA = } \int_{\partial T} {A = 0} ##
I found the Chern number must be 0? What is wrong here? Please help.
1. A one-form ##A## can be defined over the whole torus.
2. To define a connection one-form for this bundle, we need a Lie-algebra valued one-form on the torus. So I can simply define this form by adding an ##i## to ##A## as ##iA##.
3. So the Lie-algebra valued local curvature two-form is ##F = diA + iA \wedge iA = idA##
4. If there is no continuous section can be found. Both ##iA## and ##F## can only be well defined on local charts. In the overlapping parts ##{U_i} \cap {U_j}## the transition function is ##{t_{ij}}(p) = \exp [i\Lambda (p)]##. So we have the transition ##iA \to iA + id\Lambda (p)## and ##F \to t_{ij}^{ - 1}Ft_{ij}^{ - 1} = F## and the Chern number is found as
##\int_T {\frac{i}{{2\pi }}trF = } \int_T {\frac{i}{{2\pi }}F} \in Z##
So ##F## is exactly ##idA## everywhere? is this right ?
If so, by directly appling stokes theorem
##\int_T {F = - i} \int_T {dA = } \int_{\partial T} {A = 0} ##
I found the Chern number must be 0? What is wrong here? Please help.
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