It’s high time we began a set of lessons in M theory. Today we will begin to look at the combinatorial structure of moduli spaces of Riemann surfaces, as studied by Mulase.

Consider the following fractional transformations on the upper half plane. T is the map taking z to z + 1. A fundamental region for this map is a strip of width 1. We take the strip between -1/2 and 1/2 on the real line. The map S transforms the inside of the unit circle to the outside via z –> -1/z. Note that SS = 1 and (TS)^3 = 1. A fundamental region for the group so generated is the part of the selected strip above the unit circle. Maps compose via matrix multiplication for fractional transformations. The group generated by the map S fixes i and TS fixes the point w = exp(pi i/3), a root of unity.

By gluing this region into a cylinder with 2 singular points we obtain M(1,1), the orbifold moduli of the one punctured torus. The J invariant gives J(i) = 1 and J(w) = 0. We also take J(i oo) = oo. Then it is possible to describe the equivalence between elliptic curves by the relation J(tau) = J(tau’), where tau is the complex parameter which characterises the curve. So the moduli M(1,1) is parameterised by either tau in the upper half plane, or by z in the Riemann sphere CP^1 without the points 0,1,oo, which is the moduli M(0,4), obtained via a quotient of H using gamma(2).

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## Mahndisa S. Rigmaiden said,

November 18, 2006 @ 10:43 pm

11 18 06

Teichmuller would be proud! And thanks for the lesson.

## Kea said,

November 18, 2006 @ 10:56 pm

Hi Mahndisa

I’m glad that at least you are here. It would be pretty lonely otherwise.

## Mahndisa S. Rigmaiden said,

November 18, 2006 @ 10:59 pm

11 18 06

Hello there:

Keep your chin up. Due to you and Matti, I have learned quite a bit and my next few posts will invoke some neat ideas. I will likely post them for Monday hopefully:)

BTW I am going over that series of notes by Mulase. The material is quite dense, but readable:)

## CarlB said,

November 19, 2006 @ 12:53 am

Do keep up the lessons. Anything with 3rd roots of unity can’t be useless.

I’m making little progress on book because the Hotel / casino I’m staying at in Curacao has both internet and Venezuelan TV with Spanish subtitled bad American movies.

It turns out that one can learn very useful foreign language phrases this way. If I just see this last movie a couple more times I will learn the Spanish for “Death to all the a**holes of the world,” something that you can use almost anywhere.

## Kea said,

November 19, 2006 @ 1:25 am

Hi Carl

Please tell us this useful phrase if you manage to work it out!