I have some confusions about Chern number of U(1) bundle on a torus. I list my understanding below and please help me to figure out the problem.
- A one-form $A$ can be defined over the whole torus.
- 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 $iA$.
- So the Lie-algebra valued local curvature two-form is $F = diA + iA \wedge iA = idA$
- 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 $iA \to iA +i d\Lambda (p)$ 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.