M Theory Lesson 198

Let's start with $(\mathbb{R}, \mathbb{C}, \mathbb{H})$ and the triple of Riemann surface moduli $(M(0,6), M(1,3), M(2,0))$, which have Euler characteristics $-6, - \frac{1}{6}, - \frac{1}{120}$ respectively. Observe that 120 is the number of elements in the icosahedral group, whereas 6 is the number of elements in $S_3$.

The triple of (orthogonal, unitary, symplectic) appeared in Mulase-Waldron T duality for partition functions over twisted graphs. Here, the unitary case is self dual, just like the Platonic tetrahedron. The real (orthogonal) case has half the number of matrix dimensions (punctures) as the quaternionic case, suggesting we associate the genus 1 moduli to $\mathbb{R}$ and the genus 0 moduli to $\mathbb{H}$. The dual graph to the cube is basically the 6 punctured sphere. This leaves the genus 2 moduli for the icosahedron and indeed the 120 in the Euler characteristic suggests a relation. Observe that without the octonions, one does not naturally encounter nonassociative structures in the triples, but such triples are also highly relevant to M Theory.

From a categorical perspective, one views these trinities as models of the category 3, the basic triangle, because they naturally form categories with only 3 objects and one natural map between any two objects. The collection of all such sets of three elements is the object 3 as an ordinal which counts cardinalities of sets, except that we have categorified the sets by making them categories! This is why it is not surprising to encounter grouplike cardinalities in the Euler characterstics of these models. (Actually, it is the orbifold structure of the moduli that gives them a groupoid character).