More Moonshine
Tony Smith has some interesting comments on Witten's 24 dimensional manifold, naturally conjuring up Leech lattices and the like. What thoughts would a categorical M theorist have about this?
Firstly, 24 equals $3 \times 8$, which means we are really only looking at three dimensions over the octonions, and three categorical dimensions is already fairly tricky when it comes to 3-groups or groupoid structures. Higher dimensional manifolds are categorically messy objects, but simple (usually rational) surfaces arising from ribbon graphs and Fuchsian groups are more natural. Now the buckyball trinity involves surfaces of genus 0, 3 and 70. It seems a shame, therefore, to go hunting for surfaces of genus 24, when we already have surfaces of genus 3, and the 24 vertices of the permutohedron ($S_{4}$) are already lurking in the shadows.
This encoding of high classical dimensions down into surface structures (conformal field theory) is the beauty of moonshine. And since we look at the trinities as aspects of ternary logic, we expect to find not group actions, but 3-groups and groupoids. For example, the trinity $(E_8, E_7, E_6)$ (as for many group triples) can be viewed as a groupoid on three objects, each group corresponding to an object of the category.
Aside: I start my new cafe job in half an hour.
Firstly, 24 equals $3 \times 8$, which means we are really only looking at three dimensions over the octonions, and three categorical dimensions is already fairly tricky when it comes to 3-groups or groupoid structures. Higher dimensional manifolds are categorically messy objects, but simple (usually rational) surfaces arising from ribbon graphs and Fuchsian groups are more natural. Now the buckyball trinity involves surfaces of genus 0, 3 and 70. It seems a shame, therefore, to go hunting for surfaces of genus 24, when we already have surfaces of genus 3, and the 24 vertices of the permutohedron ($S_{4}$) are already lurking in the shadows.
This encoding of high classical dimensions down into surface structures (conformal field theory) is the beauty of moonshine. And since we look at the trinities as aspects of ternary logic, we expect to find not group actions, but 3-groups and groupoids. For example, the trinity $(E_8, E_7, E_6)$ (as for many group triples) can be viewed as a groupoid on three objects, each group corresponding to an object of the category.
Aside: I start my new cafe job in half an hour.
posted by Kea | 8:05 AM
4 Comments:
Sorry, but I cannot resist one more comment before I go out of town and am away from the net for about a week or so.
Kea, over on Tommaso's blog you said "... that the Fourier transform was an arithmetic operator in a von Neumann algebra implementation of Langlands duality, in the same sense that Witten’s j invariant is. ...".
That seems to me to be closely related to the Tchebyscheff-Feynman quantum net structure that Christian Beck uses in hep-th/0207081 and hep-th/0105152 and his book Spatio-Temporal Chaos and Vacuum Fluctuations of Quantized Fields (World Scientific 2002) to calculate
"... [Higgs mass] m_H = 154 Gev ...[and]...
m_t = 164.5(2) GeV ... The corresponding pole mass [experimentally observed Tquark mass] M_t ...[is]...
M_t = 174.4(3) GeV. ...".
(described more fully in my previous comment on another thread here).
PS - Too bad that Lubos and Jacques seem (from their comments on Peter Woit's blog) hostile or oblivious (respectively) to such interesting stuff.
It seems to me that they are happiest when they are tearing down somebody else's models, rather than doing the hard work of building their own realistic models. I think that would make them true "critics" about whom Teddy Roosevelt said:
""It is not the critic who counts:
not the man who points out how the strong man stumbles or where the doer of deeds could have done better.
The credit belongs to the man who is actually in the arena,
whose face is marred by dust and sweat and blood,
who strives valiantly,
who errs and comes up short again and again,
because there is no effort without error or shortcoming,
but
who knows the great enthusiasms, the great devotions,
who spends himself for a worthy cause;
who, at the best, knows, in the end, the triumph of high achievement,
and who, at the worst, if he fails, at least he fails while daring greatly,
so that his place shall never be with those cold and timid souls who knew neither victory nor defeat."
Kea: well done for getting Professor Jacques Distler to take at least a cursory interest in your work. Latest two comments from Tommaso's blog post:
98. Kea - August 6, 2008
Come on, I’m sure you’ve looked at Witten’s recent work on 3d gravity. I was merely using a short description of it.
99. Jacques Distler - August 6, 2008
His work on 3D Gravity?
Then I know what “J-Invariant” you’re talking about. But, no, it’s not “Witten’s J-Invariant.”
However, I have no idea what “a von Neumann algebra implementation of Langlands duality” might be, let alone what such a beast might have to do with Witten’s work on 3D gravity.
Care to elucidate?
(I wish you fun exchanges when you respond to Professor Jacques Distler ... hope you're not too tired to respond effectively after your cafe job, though.)
Hi all. I'm afraid I'm very tired, but I did leave one comment just now over on Tommaso's blog. No doubt Lubos will shortly laugh at how moronic it is, and Carl will probably provide further links which will not be read. Anyway, I'm going to bed.
Well, I've been waiting for Distler and Motl to say what an idiot I am over on Tommaso's blog, but they must have given up - pikers.
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