Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 21\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 21\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