Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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!

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.