Mapping double cover over a torus

[aheight]

I am interested in plotting contours (and integrals) over the algebraic function $w^2-(1-z^2)(1-k^2 z^2)=0$ on it's normal Riemann surface, a torus. Anyone here interested in helping me with this? I have the basic idea just the details I'm having problems with. Would be a nice educational tool I think.

 

[lavinia]

I am interested in plotting contours (and integrals) over the algebraic function $w^2-(1-z^2)(1-k^2 z^2)=0$ on it's normal Riemann surface, a torus. Anyone here interested in helping me with this? I have the basic idea just the details I'm having problems with. Would be a nice educational tool I think.

For me this would be a challenge but if you are willing to teach me I would like to work it through with you.

 

[aheight]

For me this would be a challenge but if you are willing to teach me I would like to work it through with you.

Hi lavinia,

It's a two step process: (1) Map the torus to a rectangular region in say the $\xi-\eta$ plane, (2) Using (4) Schwartz-christoffel transforms, map a double covering of the complex plane onto the region in (1). These are entirely doable if care is taken to manage which sheets of the multi-valued Schwartz-christoffel transform $\displaystyle f(z)=\int_0^z \frac{1}{\sqrt{(1-w^2)(1-w^2/a^2)}}dw$ are begin integrated over. The problem arises when we begin using the associated elliptic functions to carry out the integrations. These function are in my opinion not very well presented leading to confussion and frustration. But we can avoid that by performing the integrations directly in a very unambiguous fashion. That is, if we wish to map the upper half-plane conformally using a multi-valued contour function, then we certainly cannot encounter any discontinuities there. It's certainly easy to devise a case where there are discontunities: use branch 1 on one section and branch 2 in another section of the half-plane. That of course would not lead to a conformal mapping. The standard convention is to integrate over the principle-value sheet but that also can lead to ambiguities when we're using multi-valued functions. We want to then integrate over one analytically-continuous sheet of the function $\frac{1}{\sqrt{(1-w^2)(1-w^2/9)}}$. And it doesn't matter which one and we don't want to just blindly say, "use the principal value sheet". We want to use something more absolute and definitive. To get some idea of the complexity of this, consider what the real part of that function looks like. It's shown in the first picture.
Now where exactly is the principle-valued branch? Oh we can get to it, but still, it's not obvious. Consider a single-valued section of that plot which is not analytically continuous in the upper half plane shown in the second plot. It's obvious that if I integrate over that second plot, it's not going to be analytic (conformal) along the splits. But I can extract an analytically-continuous and single-valued section of that function in the upper half plane. That is shown in the third picture and that is the type of (real) surface we need to integrate over to get a conformal mapping of the upper half plane onto a rectangle in the $\xi-\eta$ plane . Now how do we devise a function that will implement this integration flawlessly?

 

[lavinia]

I am interested in plotting contours (and integrals) over the algebraic function $w^2-(1-z^2)(1-k^2 z^2)=0$ on it's normal Riemann surface, a torus. Anyone here interested in helping me with this? I have the basic idea just the details I'm having problems with. Would be a nice educational tool I think.

Ok. For starters how do you get the torus from $w^2-(1-z^2)(1-k^2 z^2)=0$? Is this just a branch of the function?

I am also unsure exactly what you want to integrate. What are the pictures of?

What I understand:

- The integral $\displaystyle f(z)=\int_0^z \frac{1}{\sqrt{(1-w^2)(1-w^2/a^2)}}dw$ maps the upper half plane onto a rectangle.
- One can tile the plane by changing the integration paths to wind an integer number of times around the singularities at $1$ and $a$ and that four adjacent rectangles form a fundamental domain for the doubly periodic inverse function. This function defines a ramified two fold cover of the Riemann sphere by the torus.
- All of the meromorphic functions on a complex torus are ratios of translated theta functions.

 

[aheight]

Ok. For starters how do you get the torus from $w^2-(1-z^2)(1-k^2 z^2)=0$? Is this just a branch of the function?

The algebraic function $w^2-(1-z^2)(1-k^2 z^2)=0$ for say $k=1/3$ has genus 1 so it's normal Riemann surface is a torus. Not sure if the general case for any $k$ is a torus as well. That's what I initially wanted to study but we already have an explicit means of creating the normal Riemann surface for the function $w^2(1-z^2)(1-k^2z^2)-1=0$ or $w=\frac{1}{\sqrt{(1-z^2)(1-k^2z^2)}}$ so I thought it would be better to first study this function first.

I am also unsure exactly what you want to integrate. What are the pictures of?

The pictures are the real component of the function $w=\frac{1}{\sqrt{(1-z^2)(1-k^2z^2)}}$. Those are the surfaces we need to integrate over to compute the function $f(z)=\displaystyle \int_0^z \frac{1}{\sqrt{(1-z^2)(1-k^2z^2)}} dz$. Now, this integral is "conveniently" expressed in terms of built-in elliptic functions in Mathematica, the software I use however there are some problems: Mathematica does not use the correct branch of the function to compute some of the elliptic integral values. So I thought it would be better if I just integrate it directly and bypass these functions although would still use them just to compare results. Doing it manually this way I think, also gives us a better understanding of what exactly is going on.

Perhaps it would be best if I summarize the method more explicitly then you can see where I'm going with this. Will do so later.

 

[aheight]

Let's start with mapping the torus to a 4-quadrant rectangle in the $\xi-\eta$ plane. We'll do that by mapping the four color-coded sections of the torus below to the associated rectangular colors in the $\xi-\eta$ plane. How do we do this? (I may not have the color associations just right, numbers may not be quite right also but qualitatively this is how it looks--will need to actually do the mapping to get everything perfect). How about we just focus on this part first? Would be interesting to map an $\xi-\eta$ coordinate system onto the torus. For example, what would horizontal and vertical lines in the yellow section of the $\xi-\eta$ plane look like on the torus? What is the significance of the borders between the colors? How are these related to the branch points of the function? Would be nice to map all of that onto the torus, basically map a coordinate system onto the torus showing everything we need to know about the underlying square root function.