A moonshine dialogue in mathematical physics by Michel Planat

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

Dear Jonathan,

many thanks for the links also from my side. I know very well the work of McMullen (and also used it to understand the quantum fluctuations of geometry).

Certainly I have to look in your essay, next point on my list.

Best

Torsten

Dear Michel,

thanks for your words.

You are absolutely right, this conclusion is strange. Actually I used the wrong tense and interschange math and physics. The corect statement is:

"the relation to math was mainly caused by the simple calculable problems in physics"

I think then it made more sense.

Thanks for the quote. Yes it is my intention. Our new paper about foliations of exotic R^4 gives also a relation to quantum field theory (we found a factor III_1 algebra which is typical for a QFT)

My remarks about dessins d'enfants were a little bit cryptic. A central point in the construction of the foliation is the embedding of a tree in a hyperbolic disk (here one has a Belyi pair i.e. a polynomial). A central point in the 4-manifold theory is the infinite tree giving a Casson handle. Of course one has finite subtrees. Here comes the dessins d'enfants into play: the embedding of these finite trees are described by this structure.

Currently we try to relate this Casson handle to Connes-Kreimer renormalization theory. If our feeling is true then the action of the absolute Galois group (central for the dessins d'enfants) must be related to the cosmic Galois group.

Of course the whole approach must be related to the interpretation of quantum mechanics too. Even in your essay you presented this relation. Certainly I have to go more deeply into your ideas.

Best

Torsten

Dear Michel,

this brilliant essay needs more point. You got a 9 from me.

Torsten

Michel,

A fun but abstruse journey where symbolic characters guide us through the connective maze of math and physics.

"A mathematician is a blind man in a dark room looking for a black cat that isn't there." Schrodinger's Cat Thought Experiment? You connect math, physics and other disciplines in multiple meanings of PSI, the proposed 20,000 year link of math, graphical embedding and Riemann surfaces -- world sheet.

It seems to be a Rubrik"s Cube of learned references.

Quite clever.

Jim

Cher Michel,

What a fun essay! Because I share your interest in primes, modular arithmetic, and sphere packing -- with the attendant mysterious roles of numbers 12 and 24 -- you might be interested in this draft mishmash of research that I hope to develop someday if I live long enough.

To the current topic, while I agree with your mathematical interpretation of Bell's theorem, I think that adding a time parameter changes the game for physical applications, as explored in my essay here.

You get my highest mark just for the sheer exuberance of your exposition, and the passion that shines through the symbols.

All best,

Tom

Bonjour Michel,

Another nice essay from you, although once again I did not understand everything, for lack of mathematical knowledge.

I also believe that numbers are at the root of everything, if you have time you can take a look at my essay, there are a few intriguing equations in it.

Et ça fait plaisir de voir d'autres Français sur ce forum !

A+,

Patrick

Dear Michel,

Very nice conversation between math and phys. The dialogue is captivating and full of interesting and deep connections. In particular, the connections between dessins d'enfant and Bell inequalities, and the moonshine group, are very intriguing. Good luck in the contest!

Best wishes,

Cristi Stoica