Torelli Group: Understanding the Non-Trivial Diffeomorphisms of String Theory

[Darth Sidious]

Hello,

While reading the lectures of Luest and Theisen on string
theory I encountered the Torelli group. They are very vague
about it (the discussion is at pages 118-119) but they say
that it correspond to twists arround trivial cycles so the
homology basis is unchanged but these transformations
are non-trivial diffeomorphisms nevertheless.

I searched the Web for some drawing describing these
twists but I couldn't find them. Would some kind soul
explain them to me? I *think* that they should be like the
Dehn twists but I'm not sure about that.

There's some other interesting piece of information in
Luest & Theisen about the moduli space of genus g Riemann
surfaces. It seems that this moduli space gets quite
complicated for genus g > 3. I think g=3 is the state
of the art in computing superstring amplitudes (I have no
idea about bosonic string amplitudes). I therefore think it's
necessary to go at least to g=4 to see if there's something
qualitatively different there.

 

[thomas_larsson_01@hotmail.com]

http://arxiv.org/find/hep-th,grp_math/1/ti:+Torelli/0/1/0/all/0/1

 

[Entropia]

Hello,

The Torelli group is an important concept in understanding the non-trivial diffeomorphisms of string theory. It refers to a group of transformations that correspond to twists around trivial cycles, which do not change the homology basis but are still considered non-trivial diffeomorphisms.

To better understand these twists, it may be helpful to think of them as similar to Dehn twists, but with some differences. While Dehn twists involve cutting and gluing a surface, Torelli twists involve only twisting or rotating the surface without cutting or gluing. This is because in string theory, the surface is considered to be a fundamental entity, and any cutting or gluing would change its topology.

Unfortunately, there are no readily available drawings or illustrations of Torelli twists, as they are quite abstract and difficult to visualize. However, there are some mathematical representations and equations that can help in understanding them.

As for the moduli space of genus g Riemann surfaces, it does indeed become more complex for g > 3. This is because as the genus increases, the number of parameters needed to describe the surface also increases, making the moduli space more intricate. In terms of computing superstring amplitudes, going beyond genus 3 may reveal new insights and potentially lead to new discoveries.

I hope this helps in your understanding of the Torelli group and the moduli space of Riemann surfaces in string theory. Keep exploring and learning, and don't hesitate to ask for further clarification or assistance.