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**Jerzy Krol**: *on* 10/5/12 at 9:51am UTC, wrote Lawrence, As far as I understand well, your idea relies on q-deforming the...

**Lawrence B. Crowell**: *on* 10/4/12 at 20:56pm UTC, wrote Jerzy, I think Torsten is on vacation. I wrote this on his blog site a...

**Sergey Fedosin**: *on* 10/4/12 at 6:01am UTC, wrote If you do not understand why your rating dropped down. As I found ratings...

**Jerzy Krol**: *on* 10/3/12 at 19:28pm UTC, wrote Dear Georgina, It is rather usual practice that answering the questions...

**Georgina Woodward**: *on* 10/3/12 at 3:29am UTC, wrote Dear Jerzy Krol, I'm not sure that answering a question with another...

**Hoang Hai**: *on* 9/19/12 at 14:11pm UTC, wrote Dear Very interesting to see your essay. Perhaps all of us are convinced...

**Jerzy Krol**: *on* 9/10/12 at 5:31am UTC, wrote Dear Ben, thanks a lot for your interest in my essay and exotic...

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FQXi FORUM

September 17, 2021

CATEGORY:
Questioning the Foundations Essay Contest (2012)
[back]

TOPIC: What if Natural Numbers Are Not Constant? by Jerzy Krol [refresh]

TOPIC: What if Natural Numbers Are Not Constant? by Jerzy Krol [refresh]

Mathematics, via model theory, gives us the possibility that natural numbers could be understood as varying objects. We analyze this from the point of view of physics were standard models of natural and real numbers are not always absolute or fixed. The extended equivalence principle appears covering the changes of the numbers. As the consequence strange exotic geometry emerges with which a kind of gravity is assigned. Taking such perspective, from the foundations of mathematics, sheds completely new light on the nature and construction of a theory of quantum gravity.

I work at Astrophysics and Cosmology Dept., University of Silesia, south of Poland. My mathematical studies determined the interest in model theory and exotic 4-geometries. Then, I completed my PHD at physics dept., titled 'Model-theoretic approach to quantum gravity' in 2005. I like and practise rock climbing and mountaneering. In fact, I think that both, mathematics and climbing, should go together.

Hi, Jerzy

If natural numbers is not constant,then dimensionless constants of physics are not constants?

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If natural numbers is not constant,then dimensionless constants of physics are not constants?

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Dear Yuri,

Your question seems to be well-posed only when you consider the title of my essay. However, the title does not suffice to validate the whole approach.

Besides, what you mean by fix value of 'constants of physics'? This is what you measure? Is this the same for you as the value of a specific number is? The constants of physics are elemnets of formal structure of numbers or rather they belong to physics?

Please, read the essay than we come back to the discussion.

Especially:

'there are constructions in mathematics which require different models of natural numbers. Some of such constructions are relevant to physics.'

Not all.

Jerzy

Your question seems to be well-posed only when you consider the title of my essay. However, the title does not suffice to validate the whole approach.

Besides, what you mean by fix value of 'constants of physics'? This is what you measure? Is this the same for you as the value of a specific number is? The constants of physics are elemnets of formal structure of numbers or rather they belong to physics?

Please, read the essay than we come back to the discussion.

Especially:

'there are constructions in mathematics which require different models of natural numbers. Some of such constructions are relevant to physics.'

Not all.

Jerzy

Dear Jerzy

My point of view you can read there

http://fqxi.org/community/forum/topic/1413

If you can find out common, we can discuss.

report post as inappropriate

My point of view you can read there

http://fqxi.org/community/forum/topic/1413

If you can find out common, we can discuss.

report post as inappropriate

Dear Jerzy,

what a great essay! Your method is very abstract but has a concrete result: you favour a spacetime model using exotic smoothness. We work together on this topic but now I began to understand it much better. what a great result!

I never thought that this kind of abstract math could be very used in physics.

Good look for the contest

Torsten

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what a great essay! Your method is very abstract but has a concrete result: you favour a spacetime model using exotic smoothness. We work together on this topic but now I began to understand it much better. what a great result!

I never thought that this kind of abstract math could be very used in physics.

Good look for the contest

Torsten

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Dear Doctor Król,

I tried my best to read your exceptionally well written essay, alas, due to my abject lack of training in the science of mathematics, I was unable to read it completely and I failed miserably to understand any of the part I did succeed in reading. I believe that one real Universe can only be occurring perpetually in one real here for one real now in one real dimension, once. I think Einstein was wrong when he assumed that there were three spatial dimensions. All real stuff has to always stay in one real dimension. It is possible to think that abstract stuff could be in three abstract dimensions, but then it would be difficult to determine how the abstract stuff was distributed. Perhaps the heavy abstract stuff would conveniently congregate in abstract dimension A, and abstract medium stuff would stay in abstract dimension B, and abstract light stuff would remain in abstract dimension C, but that seems mathematically unlikely to happen. Good luck in the contest.

report post as inappropriate

I tried my best to read your exceptionally well written essay, alas, due to my abject lack of training in the science of mathematics, I was unable to read it completely and I failed miserably to understand any of the part I did succeed in reading. I believe that one real Universe can only be occurring perpetually in one real here for one real now in one real dimension, once. I think Einstein was wrong when he assumed that there were three spatial dimensions. All real stuff has to always stay in one real dimension. It is possible to think that abstract stuff could be in three abstract dimensions, but then it would be difficult to determine how the abstract stuff was distributed. Perhaps the heavy abstract stuff would conveniently congregate in abstract dimension A, and abstract medium stuff would stay in abstract dimension B, and abstract light stuff would remain in abstract dimension C, but that seems mathematically unlikely to happen. Good luck in the contest.

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Dear Dr Fisher,

thanks for your kind interest in my essay and for your words. I think that abstract tools as used in mathematics, are just tools which enable us to express, sometimes, more than in ordinary language. Like electron microscope. In the same time, mathematics allows for detecting its own variation and makes it precise. It does not mean that all this is physically valid. But, sometimes, it is the case.

I wish you good luck too,

Jerzy

thanks for your kind interest in my essay and for your words. I think that abstract tools as used in mathematics, are just tools which enable us to express, sometimes, more than in ordinary language. Like electron microscope. In the same time, mathematics allows for detecting its own variation and makes it precise. It does not mean that all this is physically valid. But, sometimes, it is the case.

I wish you good luck too,

Jerzy

Dear Jerzy,

Great work! Although I am interested in exotic smoothness, I never found enough time to invest, but thanks to your essay and Torsten's, I feel that I am advancing. I like the connections with non-standard models of numbers. The chapter of exotic smoothness should receive more attention, both in mathematics and physics. Perhaps to blame is the lack of simple explicit examples. I am wondering if one can associate to each exotic smooth structure more "tangible" structures of other kind, geometric or topological. Could PL structures do this? Or maybe a metric tensor or other type of field, which appears regular in one smooth structure, but singular in the others.

Best regards,

Cristi Stoica

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Great work! Although I am interested in exotic smoothness, I never found enough time to invest, but thanks to your essay and Torsten's, I feel that I am advancing. I like the connections with non-standard models of numbers. The chapter of exotic smoothness should receive more attention, both in mathematics and physics. Perhaps to blame is the lack of simple explicit examples. I am wondering if one can associate to each exotic smooth structure more "tangible" structures of other kind, geometric or topological. Could PL structures do this? Or maybe a metric tensor or other type of field, which appears regular in one smooth structure, but singular in the others.

Best regards,

Cristi Stoica

report post as inappropriate

Dear Cristi,

I don't know Jerzy's opinion exactly but let me make exotic smoothness it more tangible.

Both proposals are true:

PL: There is a unique relation between PL and smooth, so that an exotic smoothness structure induces an "exotic" PL structure.

singular metric: An exotic smoothness structure has an extra 2-/3-handle pair in the handle decomposition. This pair is a saddle point, i.e. it has negative curvature but because of the vanishing of the vector fields is singular with finite ciurvature (semi-regular, inparticular a naked singularity). This fact must be seen in the metric too. In our older work we investigated it.

Torsten

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I don't know Jerzy's opinion exactly but let me make exotic smoothness it more tangible.

Both proposals are true:

PL: There is a unique relation between PL and smooth, so that an exotic smoothness structure induces an "exotic" PL structure.

singular metric: An exotic smoothness structure has an extra 2-/3-handle pair in the handle decomposition. This pair is a saddle point, i.e. it has negative curvature but because of the vanishing of the vector fields is singular with finite ciurvature (semi-regular, inparticular a naked singularity). This fact must be seen in the metric too. In our older work we investigated it.

Torsten

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Dear Cristi,

Thanks a lot for your looking at my essay and for your interest. Sorry for some delay with answering - fortunately Torsten got the point of your questions - thanks Torsten for that!

Let me add that on one hand exotic R4 are ordinary smooth 4-manifolds, though curved, on the other they carry depth of topological and geometric oddities. These are: Casson handles and wild embeddings which might correspond to a state of QG (as Torsten beautifully explained in his essay); exotics also can be parts of string backgrounds which gives the fresh relation to existing theory of QG; or they refer to quasi-modular data which again refer to string theory and to the Seiberg-Witten theory data on them. This last can be a way to find a natural gravitational instanton assigned to exotics. Previously (1985), it was Witten who considered exotic spheres (though higher dimensional, i.e. 7, 11) as the most 'physical' examples of gravitational instantons. Currently, we are just working on this point with Torsten and trying to dig out gravitational instantons in exotic R4. That is funny, but before the contest I found and downloaded one of your singularity papers when we tried to make some progress in this exotic stuff.

Regarding the connection of exotic with foundations of mathematics there is something in it which we do not understand fully yet.

Thanks again, and now I will try to go through your essay.

Regards and best wishes,

Jerzy

Thanks a lot for your looking at my essay and for your interest. Sorry for some delay with answering - fortunately Torsten got the point of your questions - thanks Torsten for that!

Let me add that on one hand exotic R4 are ordinary smooth 4-manifolds, though curved, on the other they carry depth of topological and geometric oddities. These are: Casson handles and wild embeddings which might correspond to a state of QG (as Torsten beautifully explained in his essay); exotics also can be parts of string backgrounds which gives the fresh relation to existing theory of QG; or they refer to quasi-modular data which again refer to string theory and to the Seiberg-Witten theory data on them. This last can be a way to find a natural gravitational instanton assigned to exotics. Previously (1985), it was Witten who considered exotic spheres (though higher dimensional, i.e. 7, 11) as the most 'physical' examples of gravitational instantons. Currently, we are just working on this point with Torsten and trying to dig out gravitational instantons in exotic R4. That is funny, but before the contest I found and downloaded one of your singularity papers when we tried to make some progress in this exotic stuff.

Regarding the connection of exotic with foundations of mathematics there is something in it which we do not understand fully yet.

Thanks again, and now I will try to go through your essay.

Regards and best wishes,

Jerzy

Dear Jerzy, dear Torsten,

Thank you for your explanations. Good luck with your research.

Best regards,

Cristi

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Thank you for your explanations. Good luck with your research.

Best regards,

Cristi

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Dear Jerzy Krol,

In a tetrahedral-brane, if real numbers are representational as infinite points in an eigen-rotational string, their variability is uniform and expresses gravity as a tensor product, with the emergence of time within that system of eigen-rotation; whereas the variability of natural numbers is applicable for the holarchial clustering in, Coherently-cyclic cluster-matter universe model.

With best wishes,

Jayakar

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In a tetrahedral-brane, if real numbers are representational as infinite points in an eigen-rotational string, their variability is uniform and expresses gravity as a tensor product, with the emergence of time within that system of eigen-rotation; whereas the variability of natural numbers is applicable for the holarchial clustering in, Coherently-cyclic cluster-matter universe model.

With best wishes,

Jayakar

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Dear Jerzy,

I read your essay again and understand much of your previous work much better. Central point is your theorem 2: in my opinion it means that the appearance of a non-isomorphic model of natural numbers will lead (aurtomatically) to the exotic . How strong is this relation? Can it be reversed? What happens for having only a standard structure?

Great essay again!

Torsten

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I read your essay again and understand much of your previous work much better. Central point is your theorem 2: in my opinion it means that the appearance of a non-isomorphic model of natural numbers will lead (aurtomatically) to the exotic . How strong is this relation? Can it be reversed? What happens for having only a standard structure?

Great essay again!

Torsten

report post as inappropriate

Thanks Torsten for your questions and interest! I only briefly commented about these crucial issues in the essay.

Yes, indeed the inclusion of the non-isomorphic models of the numbers lead automatically to the construction of 'different', which, at this stage, are model-theoretic self-dual. This means, in particular, that, in their construction, one makes essential use of tools of...

view entire post

Yes, indeed the inclusion of the non-isomorphic models of the numbers lead automatically to the construction of 'different', which, at this stage, are model-theoretic self-dual. This means, in particular, that, in their construction, one makes essential use of tools of...

view entire post

Dear Torsten,

... in my last post 2 words are missing: 'smooth structures', so the sentence should read now:

'Yes, indeed the inclusion of the non-isomorphic models of the numbers lead automatically to the construction of 'different' smooth structures, which, at this stage, are model-theoretic self-dual.'

sorry for this omission...

... in my last post 2 words are missing: 'smooth structures', so the sentence should read now:

'Yes, indeed the inclusion of the non-isomorphic models of the numbers lead automatically to the construction of 'different' smooth structures, which, at this stage, are model-theoretic self-dual.'

sorry for this omission...

Dear Jerzy,

I find it really interesting how an idea which at first sight might seem totally bizarre can have a deep foundation in mathematics.

I found your exposition on indistinguishable non-isomorphic models reasonably clear, but it still boggles my mind. There is of course a rich history in which physicists began using mathematical methods that had until then no obvious physical application (Riemannian geometry and matrix algebra come to mind) and the use of these was ultimately justified by how they helped make testable predictions. I suspect that ultimately that will also determine whether the exotic mathematical methods you introduce will eventually become a standard part of physics. Thank you for stretching my mind with ideas which I believed were mathematically not possible, but evidently was mistaken to so believe.

All the best,

Armin

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I find it really interesting how an idea which at first sight might seem totally bizarre can have a deep foundation in mathematics.

I found your exposition on indistinguishable non-isomorphic models reasonably clear, but it still boggles my mind. There is of course a rich history in which physicists began using mathematical methods that had until then no obvious physical application (Riemannian geometry and matrix algebra come to mind) and the use of these was ultimately justified by how they helped make testable predictions. I suspect that ultimately that will also determine whether the exotic mathematical methods you introduce will eventually become a standard part of physics. Thank you for stretching my mind with ideas which I believed were mathematically not possible, but evidently was mistaken to so believe.

All the best,

Armin

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Dear Jerzy,

I thoroughly enjoyed reading your essay. It is without a doubt one of the deepest and most significant of the contributions to this contest. I have a few questions, some of which I think are interesting; I apologize in advance that others are merely reflective of my very limited knowledge of model theory.

First, concerning models and smoothness structures:

1....

view entire post

I thoroughly enjoyed reading your essay. It is without a doubt one of the deepest and most significant of the contributions to this contest. I have a few questions, some of which I think are interesting; I apologize in advance that others are merely reflective of my very limited knowledge of model theory.

First, concerning models and smoothness structures:

1....

view entire post

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Dear Ben,

thanks a lot for your interest in my essay and exotic 4-smoothness. Your questions are excellent. I mean they indicate the essence of the approach.

Let me try to answer them as far as I can.

A.1. In fact the 'classical' non-standard models of reals used by me are those found by A. Robinson in 1960s (nonstandard analysis). They contain infinitesimals and infinite...

view entire post

thanks a lot for your interest in my essay and exotic 4-smoothness. Your questions are excellent. I mean they indicate the essence of the approach.

Let me try to answer them as far as I can.

A.1. In fact the 'classical' non-standard models of reals used by me are those found by A. Robinson in 1960s (nonstandard analysis). They contain infinitesimals and infinite...

view entire post

Dear

Very interesting to see your essay.

Perhaps all of us are convinced that: the choice of yourself is right!That of course is reasonable.

So may be we should work together to let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.

Why we do not try to start with a real challenge is very close and are the focus of interest of the human science: it is a matter of mass and grain Higg boson of the standard model.

Knowledge and belief reasoning of you will to express an opinion on this matter:

You have think that: the Mass is the expression of the impact force to material - so no impact force, we do not feel the Higg boson - similar to the case of no weight outside the Earth's atmosphere.

Does there need to be a particle with mass for everything have volume? If so, then why the mass of everything change when moving from the Earth to the Moon? Higg boson is lighter by the Moon's gravity is weaker than of Earth?

The LHC particle accelerator used to "Smashed" until "Ejected" Higg boson, but why only when the "Smashed" can see it,and when off then not see it ?

Can be "locked" Higg particles? so when "released" if we do not force to it by any the Force, how to know that it is "out" or not?

You are should be boldly to give a definition of weight that you think is right for us to enjoy, or oppose my opinion.

Because in the process of research, the value of "failure" or "success" is the similar with science. The purpose of a correct theory be must is without any a wrong point ?

Glad to see from you comments soon,because still have too many of the same problems.

Regards !

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

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Very interesting to see your essay.

Perhaps all of us are convinced that: the choice of yourself is right!That of course is reasonable.

So may be we should work together to let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.

Why we do not try to start with a real challenge is very close and are the focus of interest of the human science: it is a matter of mass and grain Higg boson of the standard model.

Knowledge and belief reasoning of you will to express an opinion on this matter:

You have think that: the Mass is the expression of the impact force to material - so no impact force, we do not feel the Higg boson - similar to the case of no weight outside the Earth's atmosphere.

Does there need to be a particle with mass for everything have volume? If so, then why the mass of everything change when moving from the Earth to the Moon? Higg boson is lighter by the Moon's gravity is weaker than of Earth?

The LHC particle accelerator used to "Smashed" until "Ejected" Higg boson, but why only when the "Smashed" can see it,and when off then not see it ?

Can be "locked" Higg particles? so when "released" if we do not force to it by any the Force, how to know that it is "out" or not?

You are should be boldly to give a definition of weight that you think is right for us to enjoy, or oppose my opinion.

Because in the process of research, the value of "failure" or "success" is the similar with science. The purpose of a correct theory be must is without any a wrong point ?

Glad to see from you comments soon,because still have too many of the same problems.

Regards !

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

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Dear Jerzy Krol,

I'm not sure that answering a question with another question is really an answer to the original question. Though your "what if" is one that I have never considered before, which makes it interesting. Perhaps I should marvel that such modelling can be done allowing problems to be looked another way. I hope you get some more readers who can properly appreciate your work. Good luck in the competition.

Kind regards Georgina.

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I'm not sure that answering a question with another question is really an answer to the original question. Though your "what if" is one that I have never considered before, which makes it interesting. Perhaps I should marvel that such modelling can be done allowing problems to be looked another way. I hope you get some more readers who can properly appreciate your work. Good luck in the competition.

Kind regards Georgina.

report post as inappropriate

Dear Georgina,

It is rather usual practice that answering the questions give rise to some others. If these others are closer to somebodys imagination and understanding is often a matter of subjective point of view. The objective part is the commection of the beginning and the final results - this can be quite non-trivial. Thanks for your remarks however I do not believe that my work will be very popular. Anyway, I have presented what I think is crucial for geometry and gravity and which is the result of working in the subjects for many years.

Thanks and good luck to you,

Jerzy

It is rather usual practice that answering the questions give rise to some others. If these others are closer to somebodys imagination and understanding is often a matter of subjective point of view. The objective part is the commection of the beginning and the final results - this can be quite non-trivial. Thanks for your remarks however I do not believe that my work will be very popular. Anyway, I have presented what I think is crucial for geometry and gravity and which is the result of working in the subjects for many years.

Thanks and good luck to you,

Jerzy

If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is and was the quantity of people which gave you ratings. Then you have of points. After it anyone give you of points so you have of points and is the common quantity of the people which gave you ratings. At the same time you will have of points. From here, if you want to be R2 > R1 there must be: or or In other words if you want to increase rating of anyone you must give him more points then the participant`s rating was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

Sergey Fedosin

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Sergey Fedosin

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Jerzy,

I think Torsten is on vacation. I wrote this on his blog site a couple of days ago. It took me a while to get back to this. I have been working some on how Yangians work in this. The Chern-Simons Lagrangian and the appearance of knot crossing equations seems to suggest there is a dual gauge form of the Yang-Baxter equation underlying this.

A Lie algebra gl(N) is a set of NxN...

view entire post

I think Torsten is on vacation. I wrote this on his blog site a couple of days ago. It took me a while to get back to this. I have been working some on how Yangians work in this. The Chern-Simons Lagrangian and the appearance of knot crossing equations seems to suggest there is a dual gauge form of the Yang-Baxter equation underlying this.

A Lie algebra gl(N) is a set of NxN...

view entire post

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Lawrence,

As far as I understand well, your idea relies on q-deforming the UEA of gl(N) and then recovering YB eqs from the double commutator. Since the YB represent the knots (braid group) then one calculates the Wilson lines on these knots (links), by the use of the CS lagrangian on a 3-manifold M, however the principal bundle is with G = SL(2,C).

So, we have YM (3d) on M with G = SL(2,C) and (dual?) YB description as above. We also embed B3 into SL(2,R) and represent in PSL(2,R) (B3 embeds in PSL(2,Z)).

Now, Yangian structure is recovered in the diagrams (BCFW) by their SL(2,Z) symmetry. And, probably, you state that this indicates that this (SYM, 4-d, N=4) theory should be somehow dual to 3-d CS on M? In order to achieve it you thicken the 3-slices in R4 and glue them together to have R4?

If it is so, that would require more arguments; certainly, the difficulty is my only partial understanding of your approach. We know that the idea of duality is very rich and fruitful though sometimes overinterpreted. For example the agreement of correlation functions in different QFTs does not necessary mean that the theories describe the same (physical) realm. I am not saying that your very interesting ideas are overinterpreted but without further insights, not only algebraic agreement, it is difficult to me to understand them well. On the other hand it would be very interesting if this works indeed. Probably I should take a look into some papers cited in your essay.

Thanks for your very interesting proposal!

Wishes,

Jerzy

As far as I understand well, your idea relies on q-deforming the UEA of gl(N) and then recovering YB eqs from the double commutator. Since the YB represent the knots (braid group) then one calculates the Wilson lines on these knots (links), by the use of the CS lagrangian on a 3-manifold M, however the principal bundle is with G = SL(2,C).

So, we have YM (3d) on M with G = SL(2,C) and (dual?) YB description as above. We also embed B3 into SL(2,R) and represent in PSL(2,R) (B3 embeds in PSL(2,Z)).

Now, Yangian structure is recovered in the diagrams (BCFW) by their SL(2,Z) symmetry. And, probably, you state that this indicates that this (SYM, 4-d, N=4) theory should be somehow dual to 3-d CS on M? In order to achieve it you thicken the 3-slices in R4 and glue them together to have R4?

If it is so, that would require more arguments; certainly, the difficulty is my only partial understanding of your approach. We know that the idea of duality is very rich and fruitful though sometimes overinterpreted. For example the agreement of correlation functions in different QFTs does not necessary mean that the theories describe the same (physical) realm. I am not saying that your very interesting ideas are overinterpreted but without further insights, not only algebraic agreement, it is difficult to me to understand them well. On the other hand it would be very interesting if this works indeed. Probably I should take a look into some papers cited in your essay.

Thanks for your very interesting proposal!

Wishes,

Jerzy

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