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nLab > Latest Changes: fuzzy sphere

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2019

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

    v1, current

    • 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 “∫ S N 2(R 2k)\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 𝔰𝔩(2)\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.

    diff, v3, current

    • 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:

    diff, v5, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2019

    added the original references on the fuzzy 4-sphere:

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2019

    added references on the fuzzy 6-sphere and higher:

    diff, v8, current

    • 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.

    diff, v9, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2020

    What’s the NN-dependence of the right normalization of the integral over the unit fuzzy 2-sphere by tracing?

    Is it 1N 2−1\tfrac{1}{\sqrt{N^2 -1}} or just 1N\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 1N 2−1\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.

    diff, v15, current

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 17th 2020

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

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2020

    The first occurence of ρ j\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 jj, but I’ve fixed that now and another typo.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2020

    Ah, I see. Thanks!

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeFeb 6th 2020

    added pointer to today’s

    diff, v20, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2021

    added pointer to today’s

    • Anwesha Chakraborty, Partha Nandi, Biswajit Chakraborty, A note on spectral triple with real structure on fuzzy sphere (arXiv:2111.03012)

    diff, v23, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2022

    added pointer to today’s

    • Denjoe O’Connor, Brian P. Dolan, Exceptional fuzzy spaces and octonions [arXiv:2210.14754]

    diff, v24, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2023

    added pointer to today’s

    • Samuel Kováčik, Juraj Tekel, Fuzzy Onion as a Matrix Model [arXiv:2309.00576]

    diff, v25, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2024

    added pointer to:

    diff, v28, current

1 to 16 of 16

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