# Gauge field for holomorphic line bundle on the complex 2-torus

Hi there,

let $X = \mathbb{C}/\Lambda$ the complex 2-torus. The lattice can always be taken as $\Lambda = \mathbb{Z} + \tau \mathbb{Z}$ with $\tau \in \mathbb{C}$ and $\text{Im} \left( \tau \right) > 0$. Any holomorphic line bundle on $X$ is then described as the quotient $\left(\mathbb{C} \times \mathbb{C} \right) / \hat{\Lambda}$, where the action of $\hat{\Lambda}$ can always be taken to be given by the group 1-cycle $$e_1 \left( z \right) = 1, \qquad e_\tau \left( z \right) = e^{-2 \pi i \left( m + b \right)}$$ In this equation $m$ is the degree of the line bundle under consideration, and $b \in \mathbb{C} / \Lambda$ gives all the additional information needed to write down the divisor class of this very bundle. This class can e.g. be represented by $$D = \left( m+1 \right) \left[ 0 \right] + \left( -1 \right) \left[ b - \frac{m}{2} \right] \in \text{Div} \left( \mathbb{C} / \Lambda \right)$$ where square brackets indicate points in $\mathbb{C}/ \Lambda$.

I am now wondering what the gauge field $A \left( z \right)$ for these holomorphic line bundles is. In particular I am looking for an explicit expression of $A$ in terms of $b$ and $m$. Does someone of you happen to know this expression?

Maybe this is a standard result and I just missed it when searching the literature. In this case I would be very grateful for a hint on where to find and read about this result.

Thank you very much!

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