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I am encountering this kind of problem in physics. The problem is like this:
Some quantity ##A## is identified as a potential field of a ##U(1)## bundle on a space ##M## (usually a torus), because it transforms like this ##{A_j}(p) = {A_i}(p) + id\Lambda (p)## in the intersection between patches ##{U_i} \cap {U_j}##. So there should be a ##U(1)## bundle corresponding to this potential field and a Chern number is defined to characterize this bundle (or the physical system)
##c = \frac{i}{{2\pi }}\int_T F ##
Where ##F## is the field strength ##F = dA##.
One may change the physical system smoothly so that the potential field also changes smoothly. Let's say the system is changing with ##t##, and the potential field becomes a smooth function ##A(t)##, while the transform rule ##{A_j}(p) = {A_i}(p) + id\Lambda (p)## is still valid.
So the question is, is this smooth change of system possible to change the Chern number ##c## ?
Some quantity ##A## is identified as a potential field of a ##U(1)## bundle on a space ##M## (usually a torus), because it transforms like this ##{A_j}(p) = {A_i}(p) + id\Lambda (p)## in the intersection between patches ##{U_i} \cap {U_j}##. So there should be a ##U(1)## bundle corresponding to this potential field and a Chern number is defined to characterize this bundle (or the physical system)
##c = \frac{i}{{2\pi }}\int_T F ##
Where ##F## is the field strength ##F = dA##.
One may change the physical system smoothly so that the potential field also changes smoothly. Let's say the system is changing with ##t##, and the potential field becomes a smooth function ##A(t)##, while the transform rule ##{A_j}(p) = {A_i}(p) + id\Lambda (p)## is still valid.
So the question is, is this smooth change of system possible to change the Chern number ##c## ?
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