A moonshine dialogue in mathematical physics by Michel Planat

Professor Planat,

Thank you for your kindness.I am not meaning to distract away from your essay but I felt that the quote was appropriate concerning your exploration(s). Everyone is in the promotion game one way or another and I appreciate your reference to the coincidence I discovered. Actually if it is true then physics = mathematics at the nexus involving the Monster, modular functions, QCD and gravity. As it is a 'razor sharp' coincidence (or curio) it is so good that it might be considered an 'open problem'. I will leave it at that and if someone wants to view how exceptonal the coincidence is they may review my comments on it at http://vixra.freeforums.org/isospin-gravitational-coupling-constant-and-ep-t386.html

I read your essay with great interest and although I did not comprehend some of the concepts I've come away from it with an increased sense of the use of dessins d'effants. Oh, and one last thing since I am here. The ratio is a QCD to gravity ratio and explicitly defines the 'gravitational coupling constant' on top of neutron star structure. Then it is a type of unification using non-abelian mathematics of gravity and QCD. Other definitions of the 'gravitational coupling constant' mix abelian and non-abelian math (i.e. proton and electron in same formula) which might be wrong, So it might be safer to say that the 'gravitational coupling constant' is not truly a measure of the weakness of gravity to the electromagnetic force (U1) but the measure of the weakness of gravity to the strong force (SU3).Because of the modular aspect of the coincidence involving 163 and the null vector relation of the Leech lattice this points directly to the Monster Group. Again look at Munafo's definition on his site.

Again Thank you, mark

Dear Dr. Planat,

Great and interesting read with a unique format. You introduced a couple of groups and models that definitely inspire some thought.

I loved the wrap up in your conclusion:

Math: Do you think that our mathematics is the real world?

Phys: As a provisional response, I offer you a quote of Stephen Hawking from his lecture "Godel and the end of the universe" [27]: In the standard positivist approach to the philosophy of science, physical theories live rent free in a Platonic heaven of ideal mathematical models...

Regards and good luck,

Ed Unverricht

I remember you from another one of these contests. If I remember you were advancing the Grothendieke cohomology of categories. You might look at my essay, for there is some informal overlap with this subject there. I think physical theories will be involved in the future with monoids, groupoids and categories.

I will read your essay as soon as possible. I am on travel right now, so it is a bit hard to read these.

Cheers LC

Greetings my friend,

I started to read your paper last night, and came to realize it requires some undivided attention - which it will get later today. I can mention that I also came upon HOTT recently, and I find their idea of univalent foundations fascinating - rooted as it is in a constructivist ideal, but with a firm calculational proof-checking basis.

You might find interesting the paper of my friend Franklin Potter, as he is also enamored of the Monster group and has some wonderful ideas; though I have yet to read his paper, I am sure it is worth checking out. As an aside; I was looking at info for Weyl E8, which he mentions, and ended up downloading Borcherd's paper proving Monstrous Moonshine, just the other night.

So I am eager to read what you have to say, and I will comment after.

All the Best,

Jonathan

Dear Michel,

I think there is no doubt that the prime numbers have significance for physics.

It's interesting in groups which you use that you get big integers. In my work I get small integers not exceeding over 6.

Another important link between your and my work are series of a1, a2 where you are adding 1. To add one, I often have in my work, but not with integers. Let me try to explain: If you have a physical appearance that is in relation to your a1 (y = 4372 / x; otherwise we can write y = 4371 / x 1 / x). That is happening as if 4372 is the limit of a process running in the segments of 1 / x to 4371 and this turns into another quality. For example attraction turns into repulsion. What is the physical meaning of integers a1, a2 is hard to say, but it is much easier to specify whole numbers (the exponents) 1-6 in the group of formula at the end of my essay.

Maybe you have not noticed my long answer to you on my site, because I have not put in the right place.

Best Regards,

Branko

Dear Michel,

I may have misread what you have said in your essay, and in that case I hope that my comment will be taken with that proviso in mind (and ignored or corrected, as applicable or practical).

It seems to me that your view of the connection between mathematics and physics is that while there are obvious and pervasive parallels between the two, it is too difficult (or at least too early) to commit to a final and formal definition of the relationship.

Given your comprehensive knowledge of many areas of mathematics and related physics, this view merits thoughtful consideration by anyone interested in either subject (or their inter-relationship).

I chose not to take up specific issues which you mention as I am not sufficiently conversant in "your" subject, and instead focused on the "meta-message" conveyed by the essay as a whole. While reading your essay, it occurred to me that you may have an answer to certain questions that had intrigued me in the past, and if you permit, I would send you an email later to explore those issues (they are not strictly relevant to the concept motivating these essays, and so I prefer not to deal with it here).

Your essay receives my endorsement without reservation, and I will rate it over the weekend (favorably, of course).

I wish you continued success in your endeavors within FQXi, as well as in your academic work.

En

Quanta Magazine has a new article on moonshine titled "Mathematicians Chase Moonshine's Shadow" at link https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-shadow/#comment-337039

I intend to discuss this further, but your essay is a unique gift. I applaud your efforts. The connection between the Bell theorem and the Grothendiecke dessin d'enfant is most enlightening. You paper is definately a keeper, and I think you are onto something very deep. I give you the highest score on this.

Cheers LC

I gave you a 10 and put you up to 6.2, but not long after somebody gave you a 2. Too bad, I hope your paper sees favor with more people.

Cheers LC

Dear Michel,

Of course I am aware of orbifolds with respect to superstring theory. The vertex operator algebra with partition function p(q) =tr q^N = Π_{N}1/(1 - q^n) is related to the Dedekind eta function. The trace results in the power [p(q)]^{24} In this there is a module or subalgebra of SL(2,Z), eg S(Z) ⊂SL(2,C), that forms a set of operators S(z)∂_z. This module or subgroup is then over certain primes, such as either Heegner primes or maybe primes in the sequence for the monster group. This is of course related to the Kleinian groups and the compactification of the AdS_5.

The AdS_5 compactification issue is something I started to return to. I gave up on this after the FQXi contest over this because it did not seem to gather much traction. The AdS_5 = SO(4,2)/SO(4,1) is a moduli space. The Euclidean form of this S^5 =~ SO(6)/SO(5) is the moduli space for the complex SU(2) or quaterion valued bundle in four dimensions. The AdS_5 is then a moduli space, and the conformal completion of this spacetime is dual to the structure of conformal fields on the boundary Einstein spacetime. This moduli is an orbit space, and this is the geometry of quantum entanglements.

If this is the case then it seems we should be able to work out the geometry of 3 and 4 qubits according to cobordism or Morse theory. My idea is that the Kostant-Sekiguchi theorem has a Morse index interpretation. The nilpotent orbits N on an algebra g = h + k, according to Cartan's decomposition with [h,h] вЉ‚ h, [h,k] вЉ‚ k, [k,k] вЉ‚ h

N∩G/g = N∩K/k.

For map Ој:P(H) --- > k on P(H) the projective Hilbert space. The differential dОј = = П‰(V, V') is a symplectic form. The variation of ||Ој||^2 is given by a Hessian that is topologically a Morse index. The maximal entanglement corresponds to the ind(Ој).

In general orbit spaces are group or algebraic quotients. Given C = G valued connections and A = automorphism of G the moduli or orbit space is B = C/A. The moduli space is the collection of self or anti-self dual orbits M = {∇ \in B: self (anti-self) dual}. The moduli space for gauge theory or a quaternion bundle in 4-dimensions is SO(5)/SO(4), or for the hyperbolic case SO(4,2)/SO(4,1) = AdS_5. The Uhlenbeck-Donaldson result for the hyperbolic case is essentially a form of the Maldacena duality between gravity and gauge field.

With your presentation of the П€-problem and the connection between the Bell theorem and Grothendieck's construction, you push this into moonshine group О"^+_0(2). This leads to the conclusion or conjecture, I am not entirely clear which, that the moonshine for the baby monster group is coincident with the the Bell theorem. The connection to the modular discriminant is interesting. This then gets extended to О"^+_0(5). Your statement on page 7 that g(q) = П†(q)^24 is much the same with what I wrote above. There is a bit here that I do not entirely follow, but the ideas are intriguing. I would be interested in knowing if the hyperbolic tilings of О"^+_0(5) have a bearing on the discrete group structure of AdS_5.

You may be familiar with Arkani-Hamed and Trka's amplitudhedron. The permutations arguments that you make give me some suspicion that this is related to that subject as well. This would be particularly the case is the О"^+_0(5) is related to the tiling and permutation of links on AdS_5 given that the isometry group of AdS_5 is SO(4,2) ~ SU(2,2) which can be called the twistor group. This is connected with Witten's so called "Twistor-string revolution."

Thanks for the paper reference. That looks pretty challenging to read. I am not quite at the level of a serious mathematician, though I am fairly good at math and well versed in a number of areas.

Cheers LC

Dear Michel,

A collaboration might be interesting. I have been pondering how it might be that ﾎ"^+_0(5) is related to the tiling and permutation of links on AdS_5. The quotient SO(4,2)/SO(4,1) = AdS_5 is not an entanglement group, at least not as I know, but this might have some relationship to entanglement. This might be through the ﾎ"^+_0(5). Particularly if this is related to Langlands in some way.

Cheers LC

This essay is excellent Michel..

You demonstrate well, how the very fundamentals of Math give rise to concepts and realities we know from Physics. I do think the Monster lurks behind a lot of orderly patterning that finds expression in the physical world; and I also affirm that it's not just too much Moonshine, that makes it look that way. A very high level discussion, but some humor too, which is nice.

The Monster made an appearance in my presentation at the 2nd Crisis in Cosmology Conference, in connection with my theory on the Mandelbrot Set and Cosmology, because of Witten's paper on 3-d gravity and BTZ Black Holes (which are 2-d) having a connection with the Monster group. But I feel strongly about the notion conveyed in my essay, that objects like the Mandelbrot Set and the Monster which arise from pure Math, must have some expression in real-world Physics.

All the Best,

Jonathan

Bonjour Michel,

very thoughtful essay. I like your use of dessins d'enfants to understand quantum theory. Your essay demonstrates the necessity to look into other areas of math instaed of the obvious ones.

More comments after a second reading.

Best

Torsten

PS: I also used dessins d'enfants in my work.... Thanks for bringing it to my attention.

Dear Michel,

in the last two years I went more deeply in hyperbolic geometric (hyperbolic 3-manifolds). Then I found many interesting relations to finite groups (of course much of it is also covered by a book of Kapovich "Hyperbolic 3-manifolds and discrete groups"). Together with my coauthor Jerzy, we calculated the partition function of a certain quantum field theory and found quasimodular behavior. Then we started to go into it more deeply and again found interesting relations to finite groups (Fuchsian groups). Then we managed to find a folaition of an exotic R^4 and this foliation is given by tessalation of a hyperbolic disk. Here, I found also your picture.

Your essay opened my eyes and it was like a missing link to fulfill another goal of us: to get a geometric description of quantum mechanics (right along your way).

For my there are many really deep thoughts in your essay and I certainly need moer time to grasp them.

Very good work,

Excited greetings

Torsten

Regarding Linas Vepstas and SL(2,Z)..

It was appreciated the link you sent to Linas' page the Modular Group and Fractals, and I agree there is a strong connection with other work, as you suggest. I guess you already know about the thesis of Tao Li, but I recently discovered this work following another thread, when I discovered the PhD thesis of Giulio Tiozzo, which you can find on his home page along with a link to other papers of interest.

All the Best,

Jonathan