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 complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration 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 nforum 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).

Atrium > Mathematics, Physics & Philosophy: "Fundamental weight systems are quantum states"

Bottom of Page

1 to 8 of 8

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2021
    • (edited May 6th 2021)

    With David Corfield and Hisham Sati we are finalizing an article:

    \,

    \,

    The title refers to a proof presented, that all fundamental gl(n)-weight systems on horizontal chord diagrams are states with respect to the star-involution of reversion of strands.

    But the bulk of the article rephrases this theorem and its proof as a special case of a more general statement in geometric group theory: characterizing the (non/semi-)positive definite phases of the Cayley distance kernel on the symmetric group.

    Finally, a last section recalls from Sati, Schreiber 2019c the original motivation and interpretation of this result: Under Hypothesis H it proves, from first principles, that a pair of coincident M5-branes (transversal on a pp-wave background, as in the BMN matrix model) indeed do form a (bound) state. This is of course generally expected, but there did not use to be a theory to derive this from first principles.

    \,

    Comments are welcome. If you do take a look, please grab the latest version of the file from behind the above link.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2021

    This just made it into Dror Bar-Natan’s Pensieve:

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 17th 2021

    Example 3.11

    • should be λ rows(λ)\lambda_{rows(\lambda)} at end of first bracket
    • in 1), should be “labels to fill its first column”
    • in 2) should be “any labelling of the first column”
    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 17th 2021
    • (edited May 17th 2021)

    respectivey

    We close with briefly explaining

    ’by’ sounds better than ’with’

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2021

    Thanks, fixed now locally. I’ll upload after adding a pointer to Jucys’ theorem which we have re-derived…

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2022
    • (edited Oct 18th 2022)

    A few days back, Carlo Collari has posted a followup:

    claiming that the statement (that 𝔤𝔩 2\mathfrak{gl}_2 weight systems are quantum states) generalizes from the fundamental representation to all “irreducible Young diagram representations”.

    The idea of the proof is to observe:

    1. that the weight systems for these more general representations are obtained by pre-composing the fundamental weight system (which we proved is a state) by the corresponding Young symmetriser (p. 9);

    2. that the Young symmetrisers are star-selfadjoint and square to a multiple of themselves (Remark 16 and last lines on p. 11);

    3. that precomposition with such operators preserves states (Lemma 25).

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 16th 2022

    Don’t know if it’s a useful generalization, but our kernel is viewed in the context of a family on p.4 of

    • Iskander Azangulov, Viacheslav Borovitskiy, Andrei Smolensky, On power sum kernels on symmetric groups (arXiv:2211.05650)
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2022

    Thanks for highlighting. I have recorded the reference here.

1 to 8 of 8

  1.