Topological strings: Why is the complex structure for $T^2$ denoted as $\tau$ in string theory?

Question

In these notes by Vafa on topological string theory he says in page 7 that the moduli of the 2-torus can be repackaged into two quantities: $$A=iR_1/R_2 \,\,\,\,\,\,\,\,\, \tau=iR_2/R_1$$ where $A$ describes the overall area of the torus or its size and $\tau$ describes its complex structure or its shape.

1. Why $A$ measures the area?
2. Why is $\tau$ describing the complex structure of $T^2$? The complex structure of $T^2$ which is Kahler is a tensor $J$. What is its relation to this $\tau$? And what has the complex structure to do with the shape of $T^2$? I would assume that the cohomology class of the Kahler form only as to do with the area.
3. Later he says that this is an example of mirror symmetry in string theory. Why? Mirror symmetry relates two different CYs. Here we only have different moduli of $T^2$
4. Finally, which parameters actually correspond to the moduli space of $T^2$? Both $A,\tau$ only $A$ or only $\tau$?

This is a quite mathematical question but it is in the heart of string theory.

Answer 1

I will attempt to answer with very little string theory background - because your questions seem oriented towards this basic case rather then the theory in general.

1. First, a correction. On page 7 of that article, it defines $A=iR_1R_2$, not $R_1/R_2$. So, since the torus is flat, $A$ is $i$ times the usual area $R_1R_2$.

2. As you say, a complex structure is a map $J$ such that $J^2=-1$. It comes from thinking of the complex structure on $\mathbb{C}$, where $iz=i(x+iy)=-y+ix$, so it exchanges the roles of the two coordinates. If the torus is a rectangular region of $\mathbb{C}$ with the opposite sides identified, the complex structure is a "rotation+flip", and changes the appearance of the rectangle. Since $\tau$ is the ratio of the two sides of the rectangle, it tells us something about the shape of the torus [some clarity below].

3. The torus is a CY manifold in 1 dimension, so the symmetry $A\leftrightarrow\tau$ is a map between two CY manifolds. He equates this with T-duality $A\leftrightarrow 1/A$, which is closely related to mirror symmetry.

4. Well, to be clear we are talking about the torus metrics, which are completely specified by $R_1$ and $R_2$. (This is not the same as "the moduli space of $\mathbb{T}^2$" because that would mean more or less structure, depending on the context. To a topologist, the moduli space of tori is 0-dimensional, since there is only one 2d topological surface with genus 1). That means just $A$ wouldn't cut it - there would be pairs $(R_1,R_2)$ with the same $A$ but different sizes. If you include $\tau$ (linearly independent from $A$), then you can break that degeneracy. So the moduli space is parametrized by either the pair $(R_1,R_2)$ or $(A,\tau)$. (He does say that for more general tori you need to consider real parts for $A$ and $\tau$, so the moduli space would be bigger).

[Some Clarity] In case that wasn't clear - consider the complex structure of $\mathbb{C}$, the imaginary unit $i$. It's action on the edges is

$(R_1,R_2)\rightarrow (-R_2,R_1)$

So what happens to $A$ and $\tau$ under this map?

$$A=iR_1R_2\rightarrow A'=-iR_2R_1$$ $$\tau=iR_2/R_1\rightarrow \tau'=i(R_1)/(-R_2)$$

So $A$ doesn't tell us anything about the complex structure, because under that map we just get $A\rightarrow -A$. However, $\tau\rightarrow -1/\tau$, so $\tau$ tells "how wide" and "how long" the torus is (at least, the ratio of these), which is the complex structure.

Answer 2

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.

Answer 3

Could the strings themselves, be the torus just observed in 1 dimension and might explain why it’s thought to be a “string”. Instead the “string” is the top of the torus vibrating and spinning. I thought if this because the vibrating strings reminded me of the top of a vortex from Birds Eye view, but it needs its own energy source so I thought a torus. Thoughts? Edit: just a physics geek, I don’t have a degree in anything