A 2-Sheeted Riemann surface, with 2 branch cuts has a genus 1. A ring torus also has a genus 1 (In fact, section 13.4 of John Terning's book, modern supersymmetry and dynamics and duality claims that by joining branch cuts on one sheet with branch cuts on other gives a torus. For example, look at page 5 of this pdf.

My question is, is there a conformal mapping which maps 2-sheeted Riemann surface with 2 branch cuts to a torus? Is $y^2=x(x−1)(x−a)$ the "mapping" (here the branch points of the Riemann surface are at $0$, $1$, and infinity)?

Note: If there is any more information needed to make the question clear please let me know, I will try (I am not from a mathematical background). As a physicist, I understand torus as a rectangle on a patch of complex plane with periodic boundary conditions (or a rectangle where the opposite sides are identified with each other.)

Also, does a torus have translational symmetry (In the sense that for example, any point z in the complex plane, is similar to $z + c$ where $c$ is some constant)?


1 Answer 1


The trouble is that you can form a conformal structure on the torus not just from a rectangle, but more generally from a parallelogram, with periodic boundary conditions. Furthermore, different choices of parallelograms may give two conformal structures on the torus which are not conformally equivalent.

For example, any two tori formed from two rectangles are conformally equivalent if and only if the length/width ratios of those rectangles are equal. And when you bring in parallelograms, there are still a lot more conformally inequivalent possibilities.

In fact there is an entire "moduli space" of conformal structures on the torus. For each value of $a$, the Riemann surface defined by the equation $y^2 = x(x-1)(x-a)$ is just one of those points.

So when you read a statement like you referred to, there are two things to keep in mind.

First, the statement can be understood from a purely topological perspective, namely that the Riemann surface in question is homeomorphic to the torus.

Second, the statement can be understood from a conformal perspective, namely that the Riemann surface in question represents one particular point on the conformal moduli space of the torus.

There is an whole subfield of conformal geometry called Teichmuller theory, where one analyzes the moduli space of conformal structures on a given topological surface.


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