Canonical Coordinates

Back in M Theory Lesson 2 we looked at Mulase and Penkava's canonical coordinates for Riemann surfaces. As is well known, there are two important lattices in the plane: the square lattice, where a square can be glued to give a torus, and a hexagonal lattice.

For $\tau$ a sixth root of unity, there is a natural triangle with point $p$ given by the average of $\omega_1$, $\omega_2$ and $\omega_3$. A Strebel differential $Q$ is expressed in terms of the Weierstrass function as

$Q = \frac{1}{4 \pi^2} \frac{(\textrm{d}P^3)^2}{P^3 (1 - P^3)} $

exhibiting an obvious triality symmetry. Now given $n$ points $p_i$ on a Riemann surface, along with a set of $n$ positive real numbers, we get a unique Strebel differential $Q$. From this one generates a canonical triangulation of the surface. The real numbers $r_i$ are associated to contour integrals $\int \sqrt{Q}$ and one may choose a branch of the square root to make $r_i$ positive.