Arcadian Functor

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Marni D. Sheppeard

Wednesday, August 06, 2008

Quote of the Month

Regarding trinities, monsters and buckyball curves of genus 70, from John McKay at Lieven's blog:
Perhaps this is where to initiate a global discussion on $M$ and Witten’s 24 dim spin manifold $W$ for which he conjectures an effective $M$ action on its free loop space. I have suggested three approaches:
1. dKP (Carroll/Kodama) reduced using Norton’s replicability definition on Grunsky (=Neumann) coefficients so that the stress energy tensor = $-mn \cdot h[ \textrm{lcm}(m,n), \textrm{gcd} (m,n)]$.
2. There are 360 cusps in $MM$ (column $C$ of Conway-Norton Monstrous Moonshine). Is there an action on the cusps and elliptic marked points of all the Riemann surfaces of genus zero? A symplectic geometry? quotient?
3. Hirzebruch’s approach using Chern classes. Once Faber polynomials can be identified, we find we have replicable fns. It is then not far to $MM$ fns.
Any genuine attempts to decipher this comment are welcome.

7 Comments:

Anonymous Tony Smith said...

First, about Witten's 24-dim thing and the Monster:
About a year ago Ed Witten wrote an arxiv paper 0706.3359 entitled "Three-Dimensional Gravity Reconsidered". In it he discussed some things of more general interest (from my point of view) than physically unrealistic 3-dim gravity. Here are some excerpts :
"... we must describe a sequence of holomorphic CFT’s with c = 24k, k = 1, 2, 3, . . ..
For k = 1, it is believed [23] that there are precisely 71 holomorphic CFT’s with the relevant central charge c = 24. Of these theories, 70 have some form of Kac-Moody or current algebra symmetry extending the conformal symmetry. ...
we need a holomorphic CFT with c = 24 and no Kac-Moody symmetry.
Such a model was constructed nearly twenty-five years ago by Frenkel, Lepowsky, and Meurman, who also conjectured its uniqueness. The motivation for constructing the model was that it admits as a group of symmetries the Fischer-Griess monster group M – the largest of the sporadic finite groups. ...
The FLM interpretation is that 196884 is the number of operators of dimension 2 in their theory. One of these operators is the stress tensor, while
the other 196883 are primary fields transforming in the smallest non-trivial representation of M.
In our interpretation, the 196883 primaries are operators that (when combined with suitable anti-holomorphic factors) create black holes. ...
If ... the CFT’s dual to three-dimensional gravity
have M symmetry for all k ... , the M symmetry is invisible in classical General Relativity and acts on the microstates of black holes. ...".
The FLM Construction
Consider a holomorphic CFT with c = 24 ... The Leech lattice is an even, unimodular lattice of rank 24 with no vector of length squared less than 4.
One can construct a holomorphic theory of c = 24 by compactifying 24 chiral bosons Xi, i = 1, . . . , 24
using ...[the Leech]... even unimodular lattice. ...
FLM considered an orbifold by the Z2 symmetry Xi → −Xi ... this ... orbifold preserves modular invariance ... and gives the hoped-for theory with monster symmetry. ...".

Another way to look at this is from the point of view of James Lepowsky in math/QA/0706.4072 where he says:
"... One takes the torus that is the quotient of 24-dimensional Euclidean space modulo the Leech lattice ... The Monster is the automorphism group of the smallest nontrival string theory that nature allows ... Bosonic 26-dimensional space-time ...
"compactified" on 24 dimensions ...".

The FLM vertex operator construction is naturally bosonic,
and
is related to 2-dim Conformal Field Theories.
In 2-dim,
there is a boson-fermion correspondence between
bosonic Sine-Gordon soliton breathers
and
fermionic Thirring models
(see Coleman's book Aspects of Symmetry (Cambridge 1985) for details).

It turns out that there is also a fermionic structure corresponding to the bosonic FLM vertex operator structure - see the book
Spinor Construction of Vertex Operator Algebras, Triality, and E8 by Alex J. Feingold, Igor B. Frenkel, and John F. X. Ries (AMS 1991)
in which they say:
"... The Chevalley algebra is a commutative nonassociative algebra constructed on the direct sum of the three 8-dimensional representations of the Lie algebra D4. ... the Chevalley algebra gives a spinor construction for E8 ...[from]... two copies of the Chevalley algebra
...
The Griess algebra is a commutative nonassociative algebra constructed on the direct sum of the trivial representation and the minimal nontrivial representation of the Monster Group.
...
This provides a unified approach to the Chevalley, Griess and E8 algebras
...
There is a close analogy between our construction and that of the Monster vertex operator algebra ... ours is a fermionic construction using Clifford algebras while the Monster construction [of Frenkel, Lepowsky and Meurman in Vertex Operator Algebras and the Monster (Academic 1988)] is bosonic, using the Weyl algebra.
These two types of constructions have been studied in the physics literature, and when they yield isomorphic algebras one is said to have a "boson-fermion correspondence". ...".

So,
my best guess is that McKay is looking at the geometry of the FLM vertex operator orbifold of 24-dim space modulo the Leech lattice,
or (equivalently by the fermion-boson correspondence)
the Feingold-Frenke-Ries spinor construction.

To my regret I don't understand enough about McKay's three approaches to say much about them,
but
it is a complicated field - for instance, in his 25 June 2008 review in Bull. AMS of Gannon's book "Moonshine Beyond the Monster", Borcherds (an expert in the field if anyone is) said
"... The most recent new idea about the monster is Witten’s suggestion of a relation between the monster and three dimensional gravity, which I am still toobaffled by to say anything useful about.
At the moment we can prove almost anything we want about the monster and moonshine (or could with a bit more effort) but are really short of good explanations for what is going on....".

Tony Smith







...

August 06, 2008 4:08 PM  
Blogger Kea said...

Wonderful, Tony! Since this is about Witten's j invariant ideas and black holes, maybe we can make progress in a different direction! It would be nice to understand what McKay is thinking about, though.

August 06, 2008 4:29 PM  
Anonymous Tony Smith said...

As to more about what McKay had on his mind, I still don't know much but here is a bit about Mckay's approach number 3, which mentioned Hirzebruch's approach.

Borcherds in his 25 June 2008 Bull AMS review of Gannon's book said:
"... Hirzeburch has suggested a “prize question” about the existence of a 24 dimensional manifold acted on by the monster, whose cohomology of twisted Dirac operators would give the monster vertex algebra. ...".

The Dirac operators seem to be to be like the fermionic spinor construction of Feingold, Frenkel, and Reis (see my previous comment).

As to their bosonic counterpart, FLM vertex operator algebra construction, Borcherds says:
"... nets of von Neumann algebras ... seem
to be closely related to vertex algebras ... Kawahigashi and Longo recently gave a construction of a net of von Neumann algebras with automorphism group the monster ... vertex algebras see ... unbounded operators but cannot handle orbifolds well, while von Neumann algebras can handle orbifolds but cannot see the unbounded operators. Perhaps there is some way to combine
vertex algebras and nets that can handle both orbifolds and unbounded operators, and therefore prove the generalized moonshine conjectures. ...".

My apologies for not getting much further so far.

Tony Smith

August 06, 2008 7:04 PM  
Blogger Kea said...

Perhaps there is some way to combine vertex algebras and nets that can handle both orbifolds and unbounded operators...

Hmmm. Well, that is exactly what we're trying to do, although I haven't come at this from these papers at all, and I'm sure Carl hasn't either.

August 06, 2008 7:14 PM  
Anonymous Tony Smith said...

Kea, about "... some way to combine vertex algebras and nets that can handle both orbifolds and unbounded operators...", said:

"... Hmmm. Well, that is exactly what we're trying to do, although I haven't come at this from these papers at all, and I'm sure Carl hasn't either. ...".

There are lots of ways to come at the same ultimate objective,
to build a physically realistic quantum model that is not only a nice AQFT but also gives the details of the Standard Model experimental results plus gravity:

1 - The category-theory-Grothendieck approach seems to be the most general

2 - The Monster-von Neumann algebra-vertex operator-24-dim manifold with Dirac operators- etc
approach is on a high (but-not-quite-so-high) level of generality, but gets more detailed structure because of the structure of the Monster

3 - My Clifford-algebra Cl(16) and E8 approach has a generalized hyperfinite II1 von Neumann algebra factor structure, but is not as general as 1 or 2. Its usefulness is looking at gauge group geometric structure which gives (following the general ideas of Armand Wyler) calculations of force strengths (roughly proportional to the volumes of the force groups) and particle masses, but it is more a limited local viewpoint (the algebra coming from completion of the union of a lot of local structures).

So, what would be nice would be a union of 1 and 2 and 3.

For example, my local geometric calculations of force strengths and particle masses should (to unify 1 2 3) be consistent with the calculations done there.

One example of 1 (high-level) calculations is yours and Carl's, and my guess is that when they are completed they will be consistent in a nice way with my more local calculations (after all, the high-level global stuff and the local stuff should be able to connect in some sort of limit).

Another example is the work of Christian Beck (see hep-th/0207081 and hep-th/0105152 and his book Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields (World Scientific 2002)). He says:
"... Tchebyscheff maps ... are semi-conjugated to a Bernoulli shift with N symbols ...[and]... are distinguished by a minimum skeleton of higher-order correlations. In that sense they are closest to uncorrelated Gaussian white noise ... necessary for the quantization of standard model ...
We will now spatially couple the chaotic noise. This leads to chaotic noise objects, which, ... we will call 'chaotic strings'. ... . ... From a mathematical point of view, chaotic strings are 1-dimensional coupled map lattices of diffusively coupled Tchebyscheff maps. ...
Now let us discuss possible ways of spatially coupling the chaotic noise. ... physically it is most reasonable that the coupling should result from a Laplacian coupling ... This leads to coupled map lattices of the diffusive coupling form. ...
.. A ... physical interpretation would be that at each time step n a fermion-antifermion pair f_1, fbar_2 is spontaneously created in cell i by the field energy of the self-interacting potential. ... The interact with particles in neighbored cells by exchange of a ... gauge boson B_2, then they annihilate into another boson B_1 and the next ... creation of a particle-antiparticle pair ... takes place. ...
This can be symbolically described by the Feynman graph ... called a 'Feynman web' ... in this interpretation a is a ... standard model coupling constant, since it describes the strength of momentum exchange of neighored particles. At the same time, a can be regarded as an inverse metric in the 1-dimensional ... space, since it determines the strength of the Laplacian coupling. ... we will present ... numerical evidence that minima of the vacuum energy ... are observed for certain ... couplings a_i, and these ... couplings ... coincide with running standard model couplings ... the ... spectrum appears to reproduce the masses and coupling constants of the known quarks, leptons, and gauge bosons of the standard model ...
we obtain for [Higgs mass] m_H = 154 Gev ...[and]...
we ... get m_t = 164.5(2) GeV ... The corresponding pole mass [experimentally observed Tquark mass] M_t ...[is]...
M_t = 174.4(3) GeV. ...".

In short, Beck's global Tchebyscheff-Feynman quantum net gives results consistent with my local group geometry calculations (and with experiment).
If they are both consistent with experiment, and with each other,
it seems to me that they are just different aspects (somewhat global, and mostly local) of a really nice unified realistic physics model.

My opinion is that Beck's somewhat global stuff will also merge nicely with the more global general stuff that you and Carl are doing.

It is a shame that the establishment is more interested in preserving its money/job/grant power than in doing real work on stuff that actually is realistic and connected with experiments and observation.

For example, over on Peter Woit's blog about FQXi, Peter Woit mentions "... the decision to fund research that I [Peter Woit] would have thought couldn’t possibly qualify as “unlikely to be supported by conventional funding sources”. Linde, Vilenkin and Bousso are among the most prominent people in their fields, working at major institutions, giving high-profile talks at conferences, appearing regularly in the press, etc. ...".
That was followed by a comment by Belizean
"... Confusion about Templeton funding conventional research by established researchers in venerable institutions might be eliminated by the considering the foundation’s apparent objective: to purchase legitimacy. ...".
I think that Belizean is probably correct - that FQXi just wants to be accepted as part of the establishment, and has the money to buy its way in.

For example,
your general category theory work (1 above) has the potential to fulflill Grothendieck's dream of unifying the general (categories) with the mid-range (Monster structure) and with the local (Standard Model plus Gravity).
Now,
look at the category theory awards made by FQXi:

John Baez - UC Riverside - 131,865 for Categorifying Fundamental Physics
His proposal summary says "... Using categorification, we can phrase large portions of quantum mechanics in a purely discrete way. We want to know how far we can push this. ... We will ...[be]... incorporating a wide range of multi-media into our research, including videos of lectures and seminars made publicly available online. ...".
As Peter \Woit said about Linde et al, John Baez is "... among the most prominent people in ...[his].. field... , working at [a] major institution... , giving high-profile talks at conferences, appearing regularly in the press, etc. ...".

Bob Coecke - Oxford - 89,981 for The Road to a New Quantum Formalism: Categories as a Canvas for Quantum Foundations
His proposal summary says "... The way we reason nowadays about quantum theory is still very `low-level', in terms of matices of so-called complex numbers. Recent work identified a high-level formal framework for quantum theory, resulting in purely diagrammatic languages for reason and compution, and also corresponding logics, that is, languages which a computer `understands'. By not making explicit the underlying `low-level pixels' there is still plenty of freedom to articulate the ideas behind other important foundational work on quantum theory. Therefore our approach can act as a canvas on which we can paint a variety of theories of physical reality. This would provide us with a true image of nature, rather than its pixels. ...".
It seems to me that Bob Coecke, far from trying to unify the local and global, is trying to sever ties with the local, and construct some sort of unconstrained "canvas".
Even if you take the position that his work might be useful,
he said on Peter Woit's blog that he "... found 'refuge' in a computer science department, and this grant enables me [Bob Coecke] to do foundations of physics stuff, and hire someone who otherwise would have a hard time getting a postdoc job given his field. ...".
I wondered whether he was really an impoverished refugee in an Oxford computer science department, so I looked at his Oxford web page where I found that he has 5 PhD students with 3 more starting in October 2008, in addition to being an acting supervisor for yet another PhD student.
Not only that, but he "works with" 3 postdocs at Oxford and with a long-term visitor and with 2 colleagues, and expects to have 2 more postdocs next year, with 2 more spending "a sabatical in our group next year".

It is clear to me that both John Baez's or Bob Coecke's category theory programs would be well-entrenched and well-funded whether or not they got FXQi grants,
and
further that their efforts seem to me to be more oriented toward flashy "... multi-media ... videos ..." and "... canvas[es] ... computer can understand ..."
than
toward the hard work of unifying global category stuff with mid-level Monster structure stuff and with low-level local E8 stuff.

Needless to say, I think that giving you a 100,000 grant would not only have saved FQXi a lot of money, but would have a lot more important in advancing human understanding of how physics works,
and that it is obvious to me that Belizean is correct,
that FQXi is just trying to buy its way into the private club that physics has become.

Tony Smith

PS - I am going to be away for several days, so I will go ahead now and say this even though this message may already be too long:
Borcherds (in his BullAMS review of Gannons' book) said:
"... A major problem with studying the monster vertex algebra is that it is constructed by splicing together two quite different pieces.
E8 is also sometimes constructed as a sum of two pieces:
a subalgebra D8, and a half spin representation S of D8.
...
The construction of the monster vertex algebra is formally very similar. In the case of E8 there is a more natural construction which writes
the E8 Lie algebra as just one piece, rather than the sum of two different pieces. For this, one just takes the vertex algebra of (the double cover of) the E8 lattice; the degree 1 piece of this is the E8 Lie algebra.
...
A major open problem is to find an analogous natural construction of the monster vertex algebra as just one piece ...".

This indicates to me that E8 structure is not only useful in making a local realistic physics model,
but also that the more general Monster structure inherits fundamental characteristics from E8 (after all, 3 E8 lattices sort of make a Leech lattice which gets you to the Monster),
so that studying how the E8 local model works in detail might be useful in building realistic Monster physics structures (moderately global)
which in turn might show how to build very general categorical structure that are realistic (a la Grothendieck).

PPS - Enjoy your new cafe job. I hope that all your customers are good people.

August 07, 2008 8:59 AM  
Blogger Kea said...

That's a lot to think about, Tony! Thanks. To be fair to FQXi, many of the projects that they funded, such as Bob's, are very worthwhile and probably useful. As for giving me 100,000. Sheesh. Thanks for the idea! I've been living off about 10,000 a year for a long time, so I could make 100,000 go a long way.

August 07, 2008 7:31 PM  
Blogger Mitchell said...

Carroll and Kodama begins: "The dKP hierarchy (= dispersionless limit of the KP hierarchy) and various reductions thereof (such as dNKdV) play an increasingly important role in topological field theory and its connections to strings and 2-D gravity... There are also connections to twistor theory..." (Something here on the noncommutative case of that dispersionless limit, by the way.)

Column C in Table 2 of Conway and Norton (1979) lists "cusp number" for each Monster "class", i.e. for a corresponding modular function, I assume. (Technical note, C&N's "classes" are usually individual conjugacy classes but are sometimes a union of two conjugacy classes, see p.314.)

August 13, 2008 5:48 PM  

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