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• 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.

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