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added a graphics (here) illustrating how what one might call the “shape observables” on the fuzzy 2-sphere (the integrals “$\int_{S^2_N} (R^{2k})$” for any choice of fixing the ordering ambiguity of the integrand) are encoded by chord diagrams, and are in fact $\mathfrak{sl}(2)$-weight system Vassiliev invariants.
To give this a little bit of a home, I added a minimum of text around that, and in order to give that text a bit of a home I added some further text with basics on the definition of the fuzzy 2-sphere.
am adding pointers on the fuzzy 3-sphere:
The fuzzy 3-sphere was first discussed (in the context of D0-brane-systems) in
Discussion in the context of M2-M5-brane bound states/E-strings:
added the original references on the fuzzy 4-sphere:
Harald Grosse, Ctirad Klimcik, P. Presnajder, On Finite 4D Quantum Field Theory in Non-Commutative Geometry, Commun. Math. Phys.180:429-438, 1996 (arXiv:hep-th/9602115)
Judith Castelino, Sangmin Lee, Washington Taylor, Longitudinal 5-branes as 4-spheres in Matrix theory, Nucl. Phys. B526:334-350, 1998 (arXiv:hep-th/9712105)
(via D5-branes)
added references on the fuzzy 6-sphere and higher:
Sanjaye Ramgoolam, Section 5 of: On spherical harmonics for fuzzy spheres in diverse dimensions, Nucl. Phys. B610: 461-488, 2001 (arXiv:hep-th/0105006)
Yusuke Kimura, On Higher Dimensional Fuzzy Spherical Branes, Nucl. Phys. B664 (2003) 512-530 (arXiv:hep-th/0301055)
What’s the $N$-dependence of the right normalization of the integral over the unit fuzzy 2-sphere by tracing?
Is it $\tfrac{1}{\sqrt{N^2 -1}}$ or just $\tfrac{1}{N}$?
I suppose it must be the former, but some authors use the latter.
Of course it depends on what one wants to do.
I’d be inclined to argue with the cross product formula for the volume element of the ordinary 2-sphere. Identifying that cross product differential with the commutator of the fuzzy functions gives the $\tfrac{1}{\sqrt{N^2 -1}}$-factor, but now the question is why not rescale that identification.
I was hoping the answer might be in
but if it is, I haven’t found it yet.
Is something odd going on with $j$ and $N$ in the description of the fuzzy 2-sphere and then later in the definition of $\rho_j$?
The first occurence of $\rho_j$ was a typo.
I have now fixed it, moved the discussion of normalizations to its own subsection Conventions and Normalizations and expanded a fair bit.
But maybe your comment already refers to these edits? Most equations are numbered now, please let me know which one you are looking at.
Post the fix, you still had a mistaken $j$, but I’ve fixed that now and another typo.
Ah, I see. Thanks!
added pointer to today’s
added pointer to today’s
added pointer to today’s
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