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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 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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure 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

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 12th 2012
   • (edited Jul 12th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have created a table _[[Isbell duality - table]]_ on pairs of entries about physics that are in algebra/geometry duality to each other. And I have included it into the relevant entries.

   I have created a table

   Isbell duality - table

   on pairs of entries about physics that are in algebra/geometry duality to each other.

   And I have included it into the relevant entries.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 13th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIs the duality precise enough that we should be able to take one entry, say, [[geometric quantization]] and dualize each part for an account of [[deformation quantization]] (sounds like this would better be called 'algebraic quantization')? For example, could we run through John's eight step [path](http://ncatlab.org/nlab/show/geometric+quantization#overview_17) to geometric quantization and dualize?

   Is the duality precise enough that we should be able to take one entry, say, geometric quantization and dualize each part for an account of deformation quantization (sounds like this would better be called ’algebraic quantization’)? For example, could we run through John’s eight step path to geometric quantization and dualize?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 13th 2012
   • (edited Jul 13th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Is the duality precise enough We discussed this a bit [here](http://nforum.mathforge.org/discussion/3346/geometric-quantization/?Focus=32610#Comment_32610) in a parallel thread. I think in the variants [[geometric quantization of symplectic groupoids]] $\leftrightarrow$ [[C-star algebraic deformation quantization]] the duality works out formally, by the central theorem there. In principle that seems to give a blueprint for how to similarly handle the duality in the higher case. I'll think about it. > deformation quantization (sounds like this would better be called 'algebraic quantization')? Yes, unfortunately history is not as systematic as my table is. :-) And more unfortunately, "algebraic quantization" in the literature refers to some niche approach to quantization of constrained systems.

   Is the duality precise enough

   We discussed this a bit here in a parallel thread. I think in the variants

   geometric quantization of symplectic groupoids $\leftrightarrow$ C-star algebraic deformation quantization

   the duality works out formally, by the central theorem there.

   In principle that seems to give a blueprint for how to similarly handle the duality in the higher case. I’ll think about it.

   deformation quantization (sounds like this would better be called ’algebraic quantization’)?

   Yes, unfortunately history is not as systematic as my table is. :-) And more unfortunately, “algebraic quantization” in the literature refers to some niche approach to quantization of constrained systems.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 19th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexThis issue arose again at the conference I've just attended, someone wondering how precise is the dualization between the columns of the table, worrying that deformation quantization is just formal, etc. Since you, Urs, have done much more on geometric quantization since the last exchange, particularly the 'higher' form, is there anything new to be said? How 'rough' is the duality between columns of the [[Isbell duality - table]]?

   This issue arose again at the conference I’ve just attended, someone wondering how precise is the dualization between the columns of the table, worrying that deformation quantization is just formal, etc.

   Since you, Urs, have done much more on geometric quantization since the last exchange, particularly the ’higher’ form, is there anything new to be said? How ’rough’ is the duality between columns of the Isbell duality - table?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 19th 2014
   • (edited Sep 19th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexLet me re-emphasize what I referred to in #3: while by default "deformation quantization" refers to formal deformation quantization, there is a "full" $C^\ast$-algebraic version. Regarding your question, indeed the duality has not been worked out much at all. Mind you, even the two sides of the duality are still waiting to receive more attention. Check out towards the end of the slides [Schenkel 14](http://ncatlab.org/nlab/show/AQFT#Schenkel14) (a talk given four days ago) for seeing for the first time somebody proposing to do what should have been done long ago: study the algebraic quantization of gauge theory via cosimplicial function algebras on moduli stacks of fields. This is precisely going in the right direction of what ought to be studied. But it is only beginning right now. Other than that, if I may just highlight again one fact we worked out which I am sort of fond of: the geometric quantization of a Poisson manifold as the holographic boundary theory of a 2d Poisson-Chern-Simons theory described in [[schreiber:master thesis Bongers]], [[schreiber:master thesis Nuiten]] I find beautifully gives a geometric-dual analog of the famous interpretation by Cattaneo-Felder of Kontsevich deformation quantization as the boundary of the pertrubative Poisson sigma-model. I don't presently have a way to phrase the duality/parallelity of these two constructions formally,but they "clearly" are the formal algebraic and the geometric aspects of the same story.

   Let me re-emphasize what I referred to in #3: while by default “deformation quantization” refers to formal deformation quantization, there is a “full” $C^\ast$-algebraic version.

   Regarding your question, indeed the duality has not been worked out much at all. Mind you, even the two sides of the duality are still waiting to receive more attention.

   Check out towards the end of the slides Schenkel 14 (a talk given four days ago) for seeing for the first time somebody proposing to do what should have been done long ago: study the algebraic quantization of gauge theory via cosimplicial function algebras on moduli stacks of fields. This is precisely going in the right direction of what ought to be studied. But it is only beginning right now.

   Other than that, if I may just highlight again one fact we worked out which I am sort of fond of: the geometric quantization of a Poisson manifold as the holographic boundary theory of a 2d Poisson-Chern-Simons theory described in master thesis Bongers (schreiber), master thesis Nuiten (schreiber) I find beautifully gives a geometric-dual analog of the famous interpretation by Cattaneo-Felder of Kontsevich deformation quantization as the boundary of the pertrubative Poisson sigma-model. I don’t presently have a way to phrase the duality/parallelity of these two constructions formally,but they “clearly” are the formal algebraic and the geometric aspects of the same story.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 21st 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIs it possible to illustrate >ordinary quantum mechanics is a boundary field theory of a 2d Poisson-Chern-Simons theory TQFT with examples from QM 101? So, >what you took to be a free particle/plane wave can be thought of a free X in a 2d Poisson-Chern-Simons theory TQFT? And >what you took to be a particle in a potential well can be thought of as Y in a 2d Poisson-Chern-Simons theory TQFT?

   Is it possible to illustrate

   ordinary quantum mechanics is a boundary field theory of a 2d Poisson-Chern-Simons theory TQFT

   with examples from QM 101? So,

   what you took to be a free particle/plane wave can be thought of a free X in a 2d Poisson-Chern-Simons theory TQFT?

   And

   what you took to be a particle in a potential well can be thought of as Y in a 2d Poisson-Chern-Simons theory TQFT?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 21st 2014
   • (edited Sep 21st 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThis is the topic of _[[Poisson holography]]_, which has become section 1.2.11.4 of [dcct](https://dl.dropboxusercontent.com/u/12630719/dcct.pdf#page=179) (p. 179)

   This is the topic of Poisson holography, which has become section 1.2.11.4 of dcct (p. 179)

8. • CommentRowNumber8.
   • CommentAuthorarnaud
   • CommentTimeMay 19th 2016
   • (edited May 19th 2016)
   • PermaLink
   Author: arnaud
   Format: MarkdownItex**How ’rough’ is the duality between columns of the Isbell duality - table?** I resume the question of David Corfield. Is there really, i.e. formally, an isbell duality in QFT ? It is still not clear. I can understand that it exists a functor (or a 2-functor) from the category of observables to the set of endomorphisms (or 2-endomorphisms) in a groupoid, but what is the spectre-cospectre adjunction in this context? What could mean the monad of this adjunction ? Is there a particular article or theorem dealing with that ? Is someone currently working on the topic ?

   How ’rough’ is the duality between columns of the Isbell duality - table?

   I resume the question of David Corfield. Is there really, i.e. formally, an isbell duality in QFT ? It is still not clear. I can understand that it exists a functor (or a 2-functor) from the category of observables to the set of endomorphisms (or 2-endomorphisms) in a groupoid, but what is the spectre-cospectre adjunction in this context? What could mean the monad of this adjunction ? Is there a particular article or theorem dealing with that ? Is someone currently working on the topic ?

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMay 19th 2016
   • (edited May 19th 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexJust as I already said by email, the table matches geometric against algebraic aspects of QFT, in the evident way, but there is no claim that there is a formal adjunction doing this. For most entries in the table there is not even a full mathematical formalization available yet. In the title of the page that the entry sits in, I use "Isbell duality" as the word for the duality between geometry and algebra. If it seems confusing, I'll rename the entry to "duality between geometry and algebra".

   Just as I already said by email, the table matches geometric against algebraic aspects of QFT, in the evident way, but there is no claim that there is a formal adjunction doing this. For most entries in the table there is not even a full mathematical formalization available yet.

   In the title of the page that the entry sits in, I use “Isbell duality” as the word for the duality between geometry and algebra. If it seems confusing, I’ll rename the entry to “duality between geometry and algebra”.

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 19th 2016
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItex>If it seems confusing, I’ll rename the entry to “duality between geometry and algebra”. Given we introduce [[Isbell duality]] as that which mediates between [[higher algebra]] and [[higher geometry]], changes would seem to have to go further. Perhaps it should remain as an aspiration, especially if some parts have been formalized and the duality shown, such as you say for >[[geometric quantization of symplectic groupoids]] and [[C-star algebraic deformation quantization]] Perhaps a bright young PhD student could take this aspiration further :)

    If it seems confusing, I’ll rename the entry to “duality between geometry and algebra”.

    Given we introduce Isbell duality as that which mediates between higher algebra and higher geometry, changes would seem to have to go further. Perhaps it should remain as an aspiration, especially if some parts have been formalized and the duality shown, such as you say for

    geometric quantization of symplectic groupoids and C-star algebraic deformation quantization

    Perhaps a bright young PhD student could take this aspiration further :)