## M Theory Lesson 9

A while back kneemo mentioned this paper on the geometry of \$CP^{n}\$ and entanglement, based on the Fubini-Study metric. It’s really very nice. Think of \$CP^{2}\$ as a triangle, which is a manifold with corners in the jargon of 2-categories. This triangle is a projection of an octant on a 2-sphere. A generic internal point represents a 2-dimensional torus, but these tori degenerate to circles on the edges of the triangle and to points on the vertices. So an edge of the triangle is a 2-sphere, or rather a copy of \$CP^{1}\$. It’s all very heirarchical, just like a good motivic geometry should be.

The usual reduction to \$RP^{1}\$ from \$CP^{1}\$ uses antipodal points on the 2-sphere. This reduction works just as easily in the triangle picture. For higher dimensional projective spaces, triangles become higher dimensional simplices, just as one would expect.

Now the real question is: can we take what we know about real moduli for points on \$RP^{1}\$ (from Brown’s paper) and lift it to the complex case using this entanglement geometry, bearing in mind the operadic nature of the associahedra tilings for real moduli?

## 2 Responses so far »

1. 1

### L. Riofrio said,

Your emphasis on observables and monads is wonderful. Many researchers forget to relate their maths to reality, leading to elegant maths that predict nothing. I look forward to your posts in ‘007.

2. 2

### CarlBrannen said,

Time still remains to observe a Texas tradition, the eating of black eyed peas on the first new day of the year. In Texas, the first day of the New Year still has a little over 10 hours left in it.

As I heard it, the tradition is that you are to earn a dollar for each pea eaten. But before you give yourself digestive upset, you need to be aware that the tradition dates to a time when the dollar was valued at \$28 per ounce of gold. The approximate present value of a pea is about \$25.