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 eg, look at page 5 of https://math.berkeley.edu/~teleman/math/Riemann.pdf).

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


A 2-sheeted surface is defined by the equation that you write, $$ y^2=x(x-1)(x-a). $$ Set $x=2^{2/3}X+\frac{a+1}{3}$. Then the equation becomes $$ y^2=4X^3-h_2 X-h_3, $$ for some constants $h_2$ and $h_3$ which are easy to determine.

A torus is typically parametrized by a coordinate $z$ with identifications $$ z\sim z+1,\quad z\sim z+\tau, \qquad(1) $$ where $\tau$ is the modulus of the torus. We would then like to find functions $\wp(z)$ and $\mathcal{Y}(z)$ such that $$ \mathcal{Y}^2(z)=4\wp^3(z)-g_2 \wp(z)-g_3, $$ and such that both $\mathcal{Y}$ and $\wp$ are periodic, i.e. $$ \wp(z)=\wp(z+1)=\wp(z+\tau),\qquad(2) $$ and same for $\mathcal{Y}(z)$. Provisionally $g_i=h_i$, but we will change this a bit below. If we manage to find such functions, we can then try to identify points on the 2-sheeted Riemann surface and on the torus as $$ z\leftrightarrow (X,y)=(\wp(z),\mathcal{Y}(z)). $$

Let us discuss the function $\wp(z)$. Remember that there is a branch point in our surface at $\infty$. We would thus like $\wp(z)$ to take value $\infty$ only once, but all the values in a neighborhood of $\infty$ twice (since there is a branch-cut). In other words, we want $\wp(z)$ to have only one singularity on the torus (modulo identifications (1)), but this singularity must be a double pole.

So we would like to find functions with periodicity property (2), and one double pole modulo (2). If we have such a function $\wp(z)$ then $A\wp(z+B)$ also works. We can fix this freedom by requiring that the double pole is at $z=0$ and has coefficient $1$, i.e. $$ \wp(z)\sim z^{-2}, \quad(z\to 0). $$ It turns out that these requirements completely fix $\wp$(z), and this function is called the Weierstrass's $\wp$-function. (This letter is "p".) See this Wikipedia page. It is implicit in the notation that I use, but $\wp(z)$ depends also on $\tau$.

One can then check that if we take $\mathcal{Y}(z)=\wp'(z)$, then $$ \mathcal{Y}^2(z)=4\wp^3(z)-g_2 \wp(z)-g_3,\qquad (3) $$ where $g_i=g_i(\tau)$ are determined by $\tau$. Since we only have one parameter $\tau$, we cannot generically solve two equations $g_i(\tau)=h_i$. However, we can rescale $X$ and $y$ as $y\to \lambda^{1/2}y$ and $X\to \lambda^{1/3} X$, which changes $h_2$ and $h_3$. It is only the ratio $h_2^{3}/h_3^2$ that is invariant under such rescalings and we only need to solve for $$ h_2^3 h_3^{-2}=g_2^3(\tau)g_3^{-2}(\tau) $$ to determine what $\tau$ is, and find a rescaling of $y$ and $X$ which will bring the equation to the form (3). This will determine the mapping between the 2-sheeted surface and the torus.


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