M Theory Lesson 62

Last November, in the pre maths blogger days, we started with Mulase's lectures on moduli spaces of Riemann surfaces. In particular, let us look once more at the $S_3$ action on the Riemann sphere $\mathbb{CP}^1$. The real axis is the equator, with the point $\frac{1}{2}$ sitting opposite the point at infinity. The dihedral action helps define a compactified form of the moduli space for the once punctured torus $M_{1,1}$ (elliptic curve), which was described by a glued region of the upper half plane, sitting above the unit circle. The j invariant gives the mapping from the 3-punctured Riemann sphere to the complex plane which respects the dihedral action on the equatorial triangle, and the torus orbifold is the quotient space.

The j invariant is used to obtain Grothendieck's ribbon diagram from the inverse image of the interval $[0,1]$, so both sphere and torus moduli are essential in understanding the ribbon for the 3-punctured sphere. Recall that these are the only moduli of real dimension 2. In the six dimensions of twistors there are three complex moduli, namely $M_{0,6}$, $M_{1,3}$ and $M_{2,0}$ which have (respectively) orbifold Euler characteristics of $-6$, $- \frac{1}{6}$ and $- \frac{1}{120}$. Octonion analogues of such moduli will be easier to understand using n-operad combinatorics, because non-commutative and non-associative geometry is a tricky business.