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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 8th 2019

starting a stub. Nothing here yet, but need to save.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 9th 2019
• (edited Nov 9th 2019)

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 11th 2019

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:

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 23rd 2019

added the original references on the fuzzy 4-sphere:

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 23rd 2019

added references on the fuzzy 6-sphere and higher:

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeDec 1st 2019

replaced the previous graphics (showing the chord diagrams coresponding to the various shape observables on the fuzzy 2-sphere) by a slightly improved version (here). Also adjusted the text slightly, but there remains much room for improvement.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJan 17th 2020

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.

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeJan 17th 2020

Is something odd going on with $j$ and $N$ in the description of the fuzzy 2-sphere and then later in the definition of $\rho_j$?

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJan 17th 2020

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.

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeJan 17th 2020

Post the fix, you still had a mistaken $j$, but I’ve fixed that now and another typo.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJan 17th 2020

Ah, I see. Thanks!

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeFeb 5th 2020

• Francesco Pisacane, $O(D)$-equivariant fuzzy spheres (arXiv:2002.01901)