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 bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics 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 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 homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory kan 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 natural nforum nlab nonassociative 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 string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory 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
    • CommentTimeJan 14th 2020

    It finally dawned on us that the analysis here from Hypothesis H exhibits holographic gauge/gravity duality as concrete mathematical duality. Now on p. 4 of v2.

    Comments are welcome.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 15th 2020

    This is a case of Koszul duality? I see at this MO answer, an instance is given as

    1. the relation between the homotopy groups of a topological space and its (co)homology groups

    In your case, how does the duality between homology and cohomology produce that shift of dimension from bulk to boundary, if that makes sense?

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 15th 2020

    Is term ’co-observable’ as used in physics new with you? It seems computer scientists use it in a coalgebraic setting for the supervisory control of discrete-event systems.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2020
    • (edited Jan 15th 2020)

    Yeah, my first thought was whether this relates to holography as Koszul duality. But at this point, all we see is plain linear duality of Hopf algebras.

    In this context, the bulk/boundary correspondence is very much brought about by the simple fact that round chord diagrams and Jacobi diagrams have a bulk and a boundary.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2020
    • (edited Jan 15th 2020)

    Yes, “co-observable” is something I just made up, in need of a term. But they do form a co-algebra, as any homology does!

    A more fancy terminology for the spectrum of co-observables is: The motive of the phase space:

    Let PhasePhase be a phase space, and EE a ring spectrum, in the given ambient cohesive \infty-topos. Then EE-valued observables are [Σ Phase,E][\Sigma^\infty Phase, E], while EE-value co-observables are EΣ PhaseE \wedge \Sigma^\infty Phase. But the geometric suspension spectrum Σ Phase\Sigma^\infty Phase, that’s the motive of phase space.