# Torus cycle periods 1-forms!

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Hi, I have the following question.

Reading about Seiberg-Witten theory one comes to realize that the $(a,a_D)$ are given by the quantities

$\omega_A = \oint_A \lambda$ and $\omega_B = \oint_B \lambda$respectively. Ok, although I am not sure about what the physical motivation is, what I want to ask is why if our curve is, for example $y= \sqrt{ (x-x_1)(x-x_2)(x-x_3)}$, this one form $\lambda$is defined as $\frac{dx}{y}$.

More generally, why the cycles of the torus are defined like this. It certainly looks like a contour integral but I would like to ask a mathematical motivation for their specific form and maybe where to read about it explicitly.

Thanks a lot!

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The isomorphism between the torus $\mathbb{C}/\Lambda$ and the elliptic curve is given by the Weierstrass $\wp$function associated to the lattice $\Lambda$. It is induced by the map $\mathbb{C} \to \mathbb{P}^2$ , $z \mapsto [\wp(z):\wp'(z):1]$. Because $x=\wp(z)$ and $y=\wp'(z)$$\frac{dx}{y}=\frac{\wp'(z)dz}{\wp'(z)}=dz$ (the 1-form$dz$ is translation invariant and thus descends on $\mathbb{C}/\Lambda$).

Now, if we take a basis $(A,B)$ of the omology group $H_1(E,\mathbb{Z})$, then what you denoted by $\omega_A$ and $\omega_B$ form precisely a basis of $\Lambda$. This basis is induced by the homeomorphism $\mathbb{C}/\Lambda \simeq S^1 \times S^1$ each corresponding to one circle. A place to learn about this is J.S. Milne's http://www.jmilne.org/math/Books/ectext.html (Elliptic Curves)

answered May 29, 2014 by (120 points)

Hi, thanks a lot for your answer, I think it will help me a lot. Since I am struggling to find time to read all this very analytically, could you let me know exactly what part of the book is relevant?

Ok, thanks a lot

@conformal_gk Its mostly in chapter III (EC over complex numbers)

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