M Theory Lesson 70

Since it seems to be Mad Moonshine blogging week, let's take a quick look at Gannon's paper The algebraic meaning of genus 0, mentioned back in Week 233.

After a nice review (for physicists) of the Moonshine theorem, Gannon gets around to discussing the braid group $B_3$. Recall that $B_3$ gives the modular group $PS{L}_2\left(\mathbb{Z}\right)$ when quotiented by its centre. Now $B_3$ is the fundamental group for the space $S{L}_2\left(\mathbb{Z}\right)\setminus S{L}_2\left(ℝ\right)$, which looks like the complement of the trefoil knot.

Even more amazing, $B_3$ is the mapping class group for an extended moduli space $M_{1,1}^{ext}$ of 1-punctured tori marked with a state $v$, which naturally appears in rational CFT. For conformal weight $k$, this group acts on (some convenient) characters $\chi (\tau ,v)$ via

${\sigma }_1.\chi (\tau ,v)=e^{\frac{-2\pi ik}{12}}\chi (\tau +1,v)$

${\sigma }_2.\chi (\tau ,v)=e^{\frac{-2\pi ik}{12}}\chi (\frac{\tau }{1-\tau },\frac{v}{(1-\tau {)}^k})$

where we recognise the usual action of modular generators $T$ and $S$ on $\tau$. Carl Brannen will just love those 12th roots. Naturally, we would like to compare all this to Loday's trefoil on the K4 polytope, with crossings on the three squares. Since this polytope is dual to Mulase's 6-valent ribbon vertex cell decomposition of ${ℝ}^3$, it must somehow describe the trefoil complement space, and the triality of the j-invariant would be lifted to a triality for these squares. Oh, perhaps we could use the torus that we made out of two such polytopes as cylinders based on honeycomb geometries. That is, draw the trefoil on this torus, which is two glued copies of the planar annulus (replacing the 2-sphere), each bounded inside by a central hexagon. On each annulus, the square tiles correspond to three principal directions in the honeycomb plane. Or maybe not!