## Quantum Gravity Lesson 1

##### 30 November 2018

Some of you may be wondering how Monster Moonshine gets a signature of (27,3) in higher Super Yang-Mills theories. Allow me to explain.

As we know, gauge groups and spacetimes are not fundamental in quantum gravity. Quantum mechanics tells us to start with a discrete set of measurement observables, and continuous things like complex Lie algebras are concretely built up as a limit of discrete geometries, much as SU(2) comes from the (universal for computation) B3 representation in the fusion category for Fibonacci anyons. Lattices for Lie algebras are a good place to find the essential discrete data.

For E8, we saw that the 240 roots are given by two copies of Kapranov’s permutoassociahedron in dimension 3. Think of the ambient three dimensional vector space as a qutrit space, so that we take tensor products for total state spaces. Three qubits gives a 27 dimensional space, corresponding to three copies of the 240 roots. The 196560 points needed for the Leech lattice come from 720 (three times 240) times 273, where 273 is Rob Wilson’s octonion Leech lattice factor, equal to 1 + 16 + 256. We now think of this as a set of cubes (since polytopes are everywhere): one point, one 4-cube, and one 8-cube. It helps to look at these cubes as lifts of cubes in dimensions 3 and 7. As we have seen, the 3-cube carries Cohl Furey’s complex octonion charges for the Standard Model particles (negative charges), which we view as three ribbon strands in the anyon/dyon picture. (This is all published already). The full theory of course has a Langlands structure to it, with electric magnetic duality. Now you can probably figure out the missing details yourself. Look for thee time directions in the three qubits that label the cube, the anyons underlying leptons and quarks.