Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,353 answers , 22,789 comments
1,470 users with positive rep
820 active unimported users
More ...

  Torus cycle periods 1-forms!

+ 3 like - 0 dislike
863 views

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!

asked May 28, 2014 in Theoretical Physics by conformal_gk (3,625 points) [ no revision ]

1 Answer

+ 3 like - 0 dislike

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 ahalanay (120 points) [ no revision ]

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)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar\varnothing$sicsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...