Torus cycle periods 1-forms!

Question

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!

Answer 1

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)
 

Comments

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

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