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.

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