The bloodhound commenter nosy snoopy has found this paper by Schlesinger. It is a remarkably clear exposition of the notion of a tower of quantisations, in terms of deformed Turing machines. All of String theory fits onto the lowest rungs of this tower. On a related note, Never Ending Books informs us of a new paper on the Manin-Marcolli cave, inspired by Plato.

The physical principles of M theory also include the n-alities that arise from the Machian nature of inertia. Carl Brannen has made great progress undertanding inertia for the first rung. See this PF post about the particle masses. They are organised according to the primes 2 and 3. This number 3 is the 3 that we have been discussing in recent posts. In the world of 3, one would work with trialgebras, rather than bialgebras; trioperads rather than bioperads. Naturally that is a master idempotent-swapping trioperad.

Physically, we observe that the classical reality that emerges from considering all scales is nothing short of a Mathematical God. The unreasonable effectiveness of mathematics is no longer unreasonable.

Behold, I come quickly: hold that fast which thou hast, that no man take thy crown.

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## kneemo said,

March 29, 2007 @ 4:32 am

Naturally that is a master idempotent-swapping trioperad.M-theory spoiler:

In octonionic projective space, SO(9) (more precisely Spin(9)) is the subgroup of F4 that can swap two idempotents while keeping the third fixed.

Pierre Ramond advocates that one should equate the projective space SO(9) with the SO(9) light-cone little group in eleven

dimensions. What do you think?

## Kea said,

March 29, 2007 @ 5:13 am

Hi kneemo. Yes, I should have mentioned the swap-two business. I guess that from a stringy perspective what you suggest makes sense.

## Anonymous said,

March 29, 2007 @ 4:23 pm

kneemo said referring to hep-th/0102012 “… Pierre Ramond advocates that one should equate the projective space SO(9) with the SO(9) light-cone little group in eleven dimensions …”.

A few months later, Ramond extended that paper in hep-th/0112261, where his abstract said in part:

“… The search for exceptional structures specific to eleven dimensions leads us to exceptional groups in the description of space-time. One specific connection, through the coset F4/SO(9), may provide a generalization of eleven-dimensional supergravity. Since this coset happens to be the projective space of the Exceptional Jordan Algebra, its charge space may be linked to the fundamental degrees of freedom underlying M-theory. …”.

I generalized from F4 to E6 (which has some properties like a complexified version of F4) to construct a string theory model that is directly representative of the Standard Model plus Gravity.

I was blacklisted from putting it on arXiv, but I did put it on the web on the cdsweb.cern.ch preprint server as an external (EXT) paper EXT-2004031.

It can be found at cdsweb.cern.ch/record/730325 and can be downloaded from there.

It is entitled E6, Strings, Branes, and the Standard Model

and its abstract states:

“… E6, an exceptional Lie algebra, contains generators with spinor fermionic characteristics, providing a way to include fermions in string theory without 1-1 boson-fermion supersymmetry. E6 graded structure E6 = g(-2) + g(-1) + g(0) + g(1) + g(2) can be used to construct a physically realistic representation of the 26-dimensions of string theory in which strings are interpreted as world-lines in a Many-Worlds quantum picture. This paper describes the construction step-by-step. The resulting physical interpretation is consistent with the physics model described in physics/0207095. …”.

Later that year, in October 2004, CERN (perhaps due to pressure from the Cornell arXiv?) terminated its EXT service, so that I no longer have such a place to post my work.

Tony Smith

PS – As to ternary products, the octonions have a natural triple product that is related to triality and the 24-dimensional Leech lattice. See, for example, the book “Introduction to Octonion and Other Non-Associative Algebras in Physics”, by Susumu Okubo (Cambridge 1995).

## kneemo said,

March 29, 2007 @ 7:42 pm

Very cool Tony!

Thanks for the link. I see you included an online discussion with Lubos and Urs about the possible D-brane description of the exceptional Jordan algebra (EJA). They mention the usual Chan-Paton gauge groups that arise for N coincident branes such as U(N), SO(2N), USp(2N). The U(3) resembles the EJA structure most closely, as it is the isometry group of CP^2. In the (twistor)projective space description these three branes are not actually coincident, but rather form a 3D projective basis for CP^2. Hence we can associate the branes with three orthogonal idempotents of the Jordan algebra h_3(C).

What Gunaydin teaches us in hep-th/0502235 is that h_3(C) describes the charge space of an N=2, d=5 extremal black hole with moduli space G/H=SL(3,C)/U(3). He also gave constructions for the other degree three Jordan algebras h_3(R), h_3(H), h_3(O) with moduli spaces SL(3,R)/SO(3), SU*(6)/USp(6) and E6(-26)/F4. In terms of collineation and isometry groups, the moduli spaces are just M_5=Coll(KP^2)/Isom(KP^2).

In d=4 and d=3, the octonionic moduli space M_5=E6(-26)/F4 is enhanced to M_4=E7(-25)/E6xU(1) and M_3=E8(-24)/E7xSU(2), respectively, and the relevant tangent spaces are upgraded to a 56D Freudenthal triple system (FTS) and a FTS plus an extra coordinate (57D charge-entropy space).

## kneemo said,

March 29, 2007 @ 8:22 pm

Tony,

Here are some links to KITP video talks about Jordan algebras and extremal black holes:

Dr. Murat Günaydin, Penn State

Minimal Unitary Representations of U-Duality Groups, Conformal Quantum Mechanics and Extremal Black Holes

Dr. Boris Pioline, LPTHE Paris

The Quantum Attractor Mechanism

You’ll hear Pioline mention that a string description is not yet known.

## Carl Brannen said,

March 29, 2007 @ 8:39 pm

When I posted the mean spirited PF calculation on the powers of 5/2 I figured it would set me up for an extra couple months in purgatory but I couldn’t resist. I’ve been too ashamed to look over there to see if the thread got locked in response.