Conformally inequivalent tori (for one-loop string diagrams)

[nrqed]

This is driving me nuts. Hope someone can explain this to me.

To classify conformally inequivalent tori, one introduces the parameter tau. Tori with tau =tau +1 and tau = -1/tau are conformally equivalent.

Now, consider two conformally inequivalent tori. Let's say one has tau =i and the other one has tau = i +1/2.

Zwiebach calls the first one a "rectangular torus".

How can we visualize the torus with tau = i+1/2 ?

Zwiebach draws a picture in figure 23.18 but I don't get it.The figure indicates that if we move along one cycle of a torus, we end up shifted a certain amount in the perpendicular direction. I don't understand how we can not, for any torus, simply go along a cycle and get back to the same position. That seems trivially possible for any torus!

Hope someone can clarify this.

Thanks

 

[BenTheMan]

The parameter tau is a vector in the complex plane. You know that you can represent a torus by the complex plane, modded out by some lattice symmetry or something. Then you can set one leg of the torus to be 1 along the real axis (i.e. $$e^{0i\pi}$$) and then the other vector (the structure modulus or the volume modulus) to be tau.

The torus you described $$\tau = i + 1/2$$ has one leg along the real axis of length 1, and one leg which has and endpoint at (1/2, 1). (This is hard to describe with words!) Let me try to do a John Baez drawing...

Im
  i|   /  tau
   |  /
   | /
   |/
   |=======>____ Re
      1/2   1

The drawing is crude, but hopefully the idea is clear. (Also note, typically we normalize tau to have length of one.)

Now tau can take on any value, and every value of tau (uniquely, I think) specifies a torus.

Now to the second question. I think that "travel along a cycle" means "travel in a direction perpendicular to tau", which means that you're not going parallel to the real axis.

Does this make any sense?

 

[nrqed]

Hi Ben,

Thanks a lot for the reply!

And for the drawing !

Yes. I understand the definition of tau. What confuses me is the connection with actually moving along the torus.

I guess that we must imagine that there is a chart mapping the points on the torus to the tau plane. My problem is that if I visualize a torus (any torus), I don't see why we cannot map it to a rectangle in the tau plane. I mean, what prevents us from using a chart such that tau = i ? What prevents us from using a mapping such that when the lines of constant real parts and constant imaginary parts are simply aligned with the "cycles" around the torus? Do you see what I mean?

I know that there is something I am missing. The value of tau has a deeper signification that the mapping of the points on the torus to the compex plane, it seems.

 

[nrqed]

Oh wait... I get it now

I was thinking about it the wrong way. Now it's clear.

It helped me to formulate my question and then my response to your reply.
So thanks again!

 

[xepma]

I struggled with the same question a while back but I didn't explore it further. And now I read this topic, and I can't come up with a convincing picture, haha.

So, well, could you explain it to me perhaps?