M Theory Lesson 9

A while back kneemo mentioned this paper on the geometry of $C{P}^n$ and entanglement, based on the Fubini-Study metric. It's really very nice. Think of $C{P}^2$ 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 $C{P}^1$. It's all very heirarchical, just like a good motivic geometry should be.

The usual reduction to $R{P}^1$ from $C{P}^1$ 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 $R{P}^1$ (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?