Complex Line Bundle on a Torus

Let \(E=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)\) be a complex torus.

A complex line bundle can be described by identifying fibers over lattice-equivalent points.

For example, consider the identification \((z,r)\sim (z+\tau, e^{-\pi i\tau-2\pi i z}r).\)

The scalar \(e^{-\pi i\tau-2\pi i z}\) is a transition factor: it tells us how the fiber coordinate $r$ changes when we move by the period $\tau$.

In the picture, the phase of this transition factor is shown as color on the torus.