# 2-sheeted Riemann surface with 2 branch cuts and Torus

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)?

• – Prahar May 28 '19 at 19:58

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.