Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Monday, January 01, 2007

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?


Blogger 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.

January 01, 2007 8:30 PM  
Blogger 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.

January 02, 2007 8:45 AM  

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