I don't know string theory, but I do know about complex structures on 2-tori, also known as complex elliptic curves. Most of your questions were answered by levitopher, I'll just elaborate a bit on that part. The space of all complex structures on a topological torus is called the moduli space of elliptic curves. This means that points of this space correspond exactly to isomorphism classes of elliptic curves, where two elliptic curves are isomorphic if there exists a biholomorphic mapping between them (typically a point is singled out that has to be respected by the mapping, but that is not important).

It can be shown that every complex structure on a torus is obtained as a quotient of the complex plane modulo a lattice, i.e. a discrete subgroup of rank two of the plane, acting by translation: you roll up the plane in two independent directions.
An isomorphism is a multiplication by a complex number that induces a bijection on these lattices.

Now let $R_1,R_2$ be two generators of your lattice, hence two complex numbers. I assume that in the first part of the example they authors are thinking of two perpendicular generators $R_1$ and $iR_2$. In general, multiplication by a (nonzero) complex number doesn't change the isomorphism class of the corresponding complex torus, to we use it to scale one of the generators to 1, and we get a lattice generated by $1, R_2/R_1$. Conventionally this scaling is done in such a way that $\tau$ has positive imaginary part. The ratio $R_2/R_1$ is often denoted $\tau$.

Now two complex tori having the same $\tau$ have equivalent complex structures, but the converse doesn't quite hold yet. I think what we have now is the Teichmüller space, which is easy as a space itself, namely the complex upper half plane, but whose moduli interpretation is more technical, namely of complex structures on the torus up to only some complex isomorphisms (namely those isotopic to the identity). To go to the actual moduli space of complex structures, you have to factor out equivalent lattices: e.g. $1, \tau + 1$ generates the same lattice, and $\tau + 1$ corresponds to the same complex structure as $\tau$. This is essentially a change of basis, and all bases are obtained by applying elements of $SL_2(\Bbb Z)$ to a given set of generators. Note that this directly translates into an action on $\tau$ by Möbius transformations:

$$\begin{pmatrix} a & b \\ c & d\end{pmatrix}\tau = \frac{a\tau + b}{c\tau + d}$$

The quotient of the complex upper half plane (with coordinate $\tau$) under the action of $SL_2(\Bbb Z)$ is exactly the moduli space of complex structures on a topological torus.

This post imported from StackExchange Physics at 2015-06-05 09:45 (UTC), posted by SE-user doetoe