M Theory Lesson 352

Some years ago we were considering noncommutative path integrals and higher categories. Now one version of Kapranov's classic work appears in this 2009 book. Here Hermitian matrices appear as a fundamental domain for the exponential map. (Kapranov promises further work on this subject, but all we find is one old paper).

Recall that the paths on a cube could be placed on a hexagon not unlike the three qutrit hexagons of recent posts. In what way do the $27$ possible paths for three qutrits get split up on the cube? We see that these $27$ paths are somewhat unnatural on the big cube, because they vary in length from $\sqrt{3}$ to $3$. However, they neatly fit into one corner of the cube. The point labeled $XYZ$ is reached by the six possible paths (of the degree $3$ hexagon) and stands for all six paths of length $\sqrt{3}$. Similarly, $XXY$ stands for three possible paths. The partition of the $10$ vertices is now $(1,6,3)$ for length squares of $(3,5,9)$ and path multiplicities $(6,3,1)$. That is, $27 = 6 + 18 + 3$. (This puts the real diagonal of a Jordan Hermitian object onto the vertices $XXX$, $YYY$ and $ZZZ$). A face of the cube is selected by eliminating one variable, picking out four three qutrit vertices and a total of eight paths (a basis for off diagonal octonions).