# Lee Smolin's LQG may reproduce the standard model

1. Aug 11, 2006

### bananan

http://www.newscientist.com/channel/fundamentals/mg19125645.800
sounds like string theory, all particles from vibrations. how successful has this theory been in comparison to string theory, on particle physics?

"physical particles may seem very different from the space-time they inhabit, but what if the two are one and the same thing? New Scientist investigates

LEE SMOLIN is no magician. Yet he and his colleagues have pulled off one of the greatest tricks imaginable. Starting from nothing more than Einstein's general theory of relativity, they have conjured up the universe. Everything from the fabric of space to the matter that makes up wands and rabbits emerges as if out of an empty hat.

It is an impressive feat. Not only does it tell us about the origins of space and matter, it might help us understand where the laws of the universe come from. Not surprisingly, Smolin, who is a theoretical physicist at the Perimeter Institute in Waterloo, Ontario, is very excited. "I've been jumping up and down about these ideas," he says.

This promising approach to understanding the cosmos is based on a collection of theories called loop quantum gravity, an attempt to merge general relativity and quantum mechanics into a single consistent theory." ...

2. Aug 11, 2006

### marcus

He is evidently talking about work that first appeared this year. It is very much in the start-up stages.

Lee Smolin was invited to a special conference in July celebrating the 60th birthday of Gerard 't Hooft, and the talk he gave was about this new idea.
http://www1.phys.uu.nl/gerard60/program/
See title of Smolin's talk on 15 July
"Emergence of chiral matter from quantum gravity"

Many of the people at the conference were Nobel laureates or other big name theoreticians. I wonder how Smolin's idea of matter emerging from network states of the gravitational field went over!

It is a very novel approach, based on work that appeared in 2005 by a young Australian physicist named Sundance Bilson-Thompson.

It has been discussed in some PF threads, but very little has been written about it so far.

http://arxiv.org/abs/hep-ph/0503213

http://arxiv.org/abs/hep-th/0603022

the quantum state of the geometry of space is represented by a web or NETWORK
durable matter particles exist as twists and tangles ( braids) in this network.
semipermanent topological snarls in the geometry of space, in other words, represent matter.

In this approach the evolution of geometry (in interaction with matter) is conjectured to be explainable in terms of
elementary local "moves" by which the network modifies itself
(by breaking a link between a pair of nodes and reconnecting them some other way, or inserting a new node etc )
it modifies itself by quantum (loosely speaking probabilistic) local reconnections according to these "moves"
matter is special knots in the web that these moves can't usually untangle
only when two knots come together can something happen and them might cancel each other or turn into something else.
A quantum state of geometry is represented as a network of pure spatial relationship----what is between what, what is adjacent to what---a network or graph of nodes connected by links. It has not been proven that this SUCCEEDS as a quantum spacetime dynamics.
==============

that said, notice that what we're discussing is a small PRELIMINARY investigation of a possible quantum theory of space and matter----whether or not it can be brought to full conclusion, people are likely to learn something from investigating it.
AFAIK only two faculty and some 3 or 4 postdoc/gradstudent researchers have looked into this.
so far, half a dozen "man-years" or less
the payoff in interesting provocative results has been high IMHO given such a small investment

another thing one might remark is that a lot of what comes out in New Scientist is not especially meaningful.

If someone wanted to find out about this, I would probably not recommend the New Scientist. I'd suggest instead reading Sundance original paper and watching the video of his November 2005 talk at Perimeter Institute

If you want the video, go here
http://streamer.perimeterinstitute.ca:81/mediasite/viewer/FrontEnd/Front.aspx?&shouldResize=False
select "seminar series" from the menu at the left
click on "presenter" to get a list of seminar speakers alphabetical by first name
scroll down to "Sundance" click on the name and press "search"
You will get the November 16, 2005 talk---it is about an hour long.

or look on page 11 out of 24 in the whole listing of seminar talks
it will usually be about 13 from the end, so when there are 30 pages of talks it will be on page 17.

If you want a general overview by Lee Smolin, a recent paper is
http://arxiv.org/abs/hep-th/0605052

that would have references to other papers, including the two I mentioned

Last edited by a moderator: Apr 22, 2017
3. Aug 11, 2006

### alexsok

Thx for that. So is there a full version of this article elsewhere on the web?

4. Aug 12, 2006

### bananan

Thanks Marcus,

incidentally, do you have an opinion on the Heim- Dröscher Theory, and its possible connections with LQG/spinfoam, perhaps its forumla for mass of elementary particles can be re-derived from LQG, to give a boost of LQG over its enemy, lubos

Last edited by a moderator: Apr 22, 2017
5. Aug 12, 2006

### CarlB

I saw his lecture on this a few months ago at the May APS meeting in Dallas, Texas. It was not well received largely because (a) it makes no new predictions, and (b) they don't have any clue on how to get neutrinos to work, and (c) getting 3 generations seemed iffy.

The primary complaint that arose in the question and answer section was that the theory suggested that there should be an infinite number of generations and had great difficulty explaining why there appear to be exactly three generations, particularly, exactly three neutrinos.

I guess I should admit that I have a pony in this race.

Carl

6. Aug 12, 2006

### bananan

has string theory been more successful in this regard? (points abc) what's the pony you have in this race?

7. Aug 12, 2006

### Dcase

From 'Generic predictions of quantum yjroties of garvity; referenced 8-11 "If you want a general overview by Lee Smolin, a recent paper is" by marcus
http://arxiv.org/abs/hep-th/0605052

For item 6 - the emergence of matter from quantum geometry -
from my perspective - sounds as though ‘quantum geometry’ is a form of energy - and may relate to string vibrations?

Smolin may yet unify LGQ and causal spin networks with the various string theories?

I truly hope that he or someone is able to do this.

8. Aug 12, 2006

### Kea

:rofl:

9. Aug 13, 2006

### CarlB

The idea behind "particle physics" is that as a physical object is made more and more energetic, it becomes simpler because it is broken into smaller and smaller parts that can be treated alone. For example, on a cold morning your car might have extravagant patterns of frost on it. As your car warms up, these change to dewdrops, which are simpler because the water molecules aren't stuck together any more. Warming further, the dewdrops evaporate into the air, which is simpler yet. Heat that water up higher, it disassociates into atoms. Hotter, and it becomes nuclei and electrons. Hotter yet, and you get neutrons, protons and electrons. Hotter still, and you presumably get up and down quarks and electrons. As the material gets hotter, it splits into simpler subparts, conversely, as it cools, it condenses back again.

But at the level of the electrons and up and down quarks, the splitting/condensation sequence somehow gets complicated. As you add more energy to the system, instead of continuing to break into smaller parts, one finds that these particles are "point particles", and in adding energy one must suddenly deal with a whole plethora of unusual particles from the other two generations. The point here is that this is contrary to our expectations that things should get simpler as we heat them up. Instead, what is happening is that the extra energy allows new degrees of freedom to show up.

I assume that the three generations of elementary fermions must be made up of simpler particles, or "preons". Lee and Sundance's idea is similar in that they also assume preons, and their preons have some similarities to mine in that both ideas assume that the quarks and leptons are unified and are composites more or less formed from triples of subparticles.

Their idea is that the quarks and leptons are each made up of three subparticles which they call Helons, $$H_0, H_+, H_-,$$ where the suffix gives the electric charge in units of 1/3 e. These are braid groups, more or less. The three generations arise from differing complexity in the number of crossings of the braid group. They are unable to explain why there are exactly three generations or why they have the relative masses.

I model the subparticles based on a type of Clifford algebra called a "geometric algebra", a theory which was started by David Hestenes 25 years ago. This is different from braids in that it is a heck of a lot simpler to do calculations. If you don't want to learn a bunch of math, you can think of a great example of a Geometric Algebra as being the set of functions that map from space time to the set of 4x4 matrices that you could describe by taking sums of products of Dirac matrices.

My first splitting/condensation is that I break the quarks and leptons into their "chiral" halves. That is, I assume that the electron is a composite particle composed of two parts, a left handed electron and a right handed electron. These two parts convert back and forth into each other with a probability that is proportional to the mass (more or less, neutrinos are a bit odd). The energy required to break, for example, an electron is approximately infinite (i.e. Plank mass) because the chiral states are massless and therefore travel at speed c.

This division of quark or lepton into two "constituents" is similar in that it is assuming that the quarks and leptons are composite particles that are "condensed" from simpler particles, but this layer of condensation is different from the ones following in two ways. First, the binding energy is "infinite", and second, there is a phase change here in that mass appears. The unbound particles travel at c while the bound particles travels have mass and travel at slower speeds. I like this way of going to the next preon state because it is compatible with the quarks and leptons being point particles (at energies smaller than the Plank mass), and it is the natural division of the quarks and leptons into subparts (i.e. the weak interactions treat the chiral particles very differently). This method treats mass as just another interaction between particles.

My second splitting/condensation is to break the chiral particles into three subparticles that I am now calling "snuarks". You can get my snuarks by taking Lee and Sundance's helons, and (a) splitting them into left and right halves, and (b) splitting $$H_0$$ into a particle / antiparticle pair. Consequently, I have eight snuarks while they have 3 helons, but as far as preon models go, the two models are fairly similar. For a while I thought that I could get an algebraic model that would underlie their theory, after splitting the $$H_0$$ into two particles, and after combining the right and left handed snuarks. The basic problem with getting my version to line up with theirs is that I have complex numbers (or things that act like complex numbers anyway), and I can't see how to get that in their theory.

My final splitting is to take the snuarks, and break each of them up into two particles that are assumed to be the truly elementary particles which I call "binons". In terms of the Geometric algebra, these are "primitive idempotents", which is what the mathematicians use when they want to say "$$\rho^2 = \rho$$ and you can't make them any simpler". In terms of the Dirac matrices, the binons are the set of all possible density matrices, which is why I am constantly harping on density matrices on physics forums. In standard quantum mechanics, the spinor wave states are fundamental and the density matrices are derived from these. In my version of QM, these are reversed. My version is much more elegant, as you can see by reading the short discussion in https://www.physicsforums.com/showthread.php?t=124904 but this simplicitly and elegance only happens with spin-1/2 density matrices, which is why I use them. This gives a geometric foundation for the elementary particles which is a great feature of my theory.

I call them "binons" because their quantum numbers of Clifford algebra primitive idempotents are binary (look up the "spectral decomposition theorem" for Clifford algebras to see a proof). As with any symmetry class, there are a bunch of different ways to choose good quantum numbers. You need 8 different types of binons to get the quarks and leptons. There quantum numbers are therefore $$(\pm 1, \pm 1, \pm 1),$$ where the signs are chosen independently. Quantum numbers are additive. We can assume that the first of these quantum numbers depends on orientation, that is, it is the usual spin-1/2.

Binons are bound together by a potential energy. Since the binons are represented by Clifford algebra numbers, it is natural to use those numbers to define the potential energy. The definition is very simple. One adds together the Clifford algebra numbers, and then computes the "absolute value squared" of the sum. For the Dirac algebra, you could define the "absolute value squared" as the function which takes a matrix and gives the sum of the absolute squares of all its entries. That is a rather ugly definition (since it depends on the choice of representation etc.), but it turns out to be compatible (at least for the usual representations that physicists use) with the unique natural definition. The scale of the potential energy is the Plank mass. It is the potential energy that determines how binons have to be combined to make low energy particles and it is a great success of this theory that one can derive the structure of the quarks and leptons from first principles this way, with such a simple definition.

The snuarks are composed of two binons that are "compatible" in that their direction of travel is identical. In this, my theory is similar to the old "zitterbewegung" theory. The direction in which a chiral spin-1/2 particle travels is completely determined by its spin orientation, so the requirement that the snuark be compatible in their direction of travel is equivalent to requiring that their first quantum number be the same.

This leaves the other two quantum numbers arbitrary. I suppose that the next quantum number is "weak isospin". There are four cases: (+1+1, +1-1, -1+1, -1,-1). These four cases divide into a weak isospin doublet (+1+1, -1-1), and two weak isospin singlets (+1-1), (-1+1). The usual weak isospin quantum numbers are obtained by dividing these values, (2,-2,0,0) by 4. This simple derivation gives both the correct SU(2) symmetry and the correct pattern of representations, a great success of this idea.

The three snuarks that make up half of a quark or lepton are oriented in different directions. It turns out that this is necessary from the way that the potential energy is defined; otherwise there would be no bound states with energies less than Plank mass scale. So my version of preons has that when you successivley break an electron up into its components you first find 2 chiral states, then 6 snuarks, and finally 12 binons. That three snuarks are bound this way gives rise to an SU(3) symmetry, which is a great success of this theory.

As Clifford algebraic numbers, the snuarks can be multiplied by complex constants. That means that there are multiple ways of combining them together. Since this is a theory based on (pure) density matrices, one finds the number of ways that one can combine three snuarks together by solving the basic pure density matrix equation: $$\rho^2 = \rho.$$ It is a great success of this theory that when you do this, you obtain three solutions, which correspond to the three generations of elementary particles.

The derivation in the previous paragraph requires solving equations which have 3x3 matrices. One of the side effects of this is that one obtains a new way of expressing the masses of the leptons. Having done this, one finds that there are remarkable patterns in the lepton masses. In particular, it appears that there are discrete symmetries that define the hierarchy of the generations, and also the hierarchy from the neutrinos to the charged leptons. This is a great success of the theory and is discussed here: https://www.physicsforums.com/showthread.php?t=117787

Carl

10. Aug 13, 2006

### kneemo

Hi CarlB

Density matrices (primitive idempotents) certainly do provide a nice geometrical picture of quantum processes. This is because primitive idempotents give points in projective space. Of course, the question of which projective space, depends on the underlying division algebra you're working with as well as the dimension of your Hilbert space. In the finite dimensional complex case, in which the Hilbert space is $$\mathbb{C}^n$$, we recover points in the projective space $$\mathbb{CP}^{n-1}$$ (see http://math.ucr.edu/home/baez/octonions/node8.html"). The most natural way they arise, is as you've noted, in the spectral decomposition of Hermitian operators:

$$\Phi=\lambda_1P_1+...+\lambda_nP_n$$.

Density matrix formalism plays an integral role in matrix models for string theory. In matrix models, fluctuations of D0-branes (preons) are given by scalar fields $$\Phi_m$$ (m=1,...,d) (Hermitian matrices), which are in the adjoint representation of the unbroken $$U(n)$$ gauge symmetry group.

One diagonalizes a scalar field $$\Phi$$ by a $$U(n)$$ gauge transformation yielding:

$$\Phi = U\left(\begin{array}{ccc}\lambda_1 & & 0 \\ & \ddots & \\ 0 & & \lambda_n \end{array}\right)U^{\dagger}$$.

The real eigenvalue $$\lambda_i$$ describes the classical position of D0-brane i. The unitary matrix $$U$$ with entries $$U_{ij}(i\neq j)$$ describe fluctuations about classical spacetime arising from the short open strings connecting D0-branes i and j.

Of course the diagonalization is equivalent to the spectral decomposition in terms of primitive idempotents. So recalling the decomposition:

$$\Phi=\lambda_1P_1+...+\lambda_nP_n$$.

we can see a Hermitian operator expanded as a linear combination of primitive idempotents, which in turn can be associated with n fluctuating D0-branes (preons).

Last edited by a moderator: Apr 22, 2017
11. Aug 13, 2006

### CarlB

I've been trying to get away from the concepts of both Hilbert space and the apparently arbitrary choice of division algebra in favor of a purely algebraic approach.

The ideals of a Clifford algebra naturally form a copy of the complex numbers without any need to assume them, provided that the algebra has an odd number of canonical basis vectors. The Pauli algebra provides a good example of this. For example, the matrix

$$\rho_x = \frac{1}{2}\left(\begin{array}{cc}1&1\\1&1\end{array}\right)$$

is the density matrix for spin-1/2 in the +x direction. Any product of the form

$$\rho_x M \rho_x$$

will be a complex mutiple of $$\rho_x.$$ For example:

$$\rho_x\left(\begin{array}{cc}a+b&c-id\\c+id&a-b\end{array}\right)\rho_x$$

$$= (a+c)\; \rho_x$$

where a, b, c, and d are complex constants. Now the above was written in the usual representation of the Pauli algebra. If you translate it into purely geometric (density matrix) form it is somewhat more elegant:

$$\rho_x(a + b\rho_z + c\rho_x + d\rho_y)\rho_x = (a+c)\;\rho_x$$

In the above, a and c are "complex" numbers in that they are of the form

$$a = a_R + a_I\sigma_x\sigma_y\sigma_z$$

The underlying idea here is to eliminate unphysical gauge freedom. But the whole essence of Hilbert space implies a gauge freedom, so I'm avoiding Hilbert space. The density matrix formalism provides a way of doing this.

By the way, the method I'm using to extend the density matrix formalism to more complicated cases than spin-1/2 is Schwinger's measurement algebra. There is an interesting paper out by LP Horwitz on the use of the quaternions as the division algebra:
http://www.arxiv.org/abs/hep-th/9702080

The book by Schwinger, "Quantum Kinematics and Dynamics" has the arbitrary choice of complex numbers as the division algebra. I think that the way I'm doing it, relying on the density matrices themselves to define the division algebra, is far more natural and geometric.

There is only one problem with this, and that is that the number of canonical basis vectors for the usual spacetime is 4, and that does not contain a complex unit. (That is, there is no element of the Clifford algebra CL(3,1) that squares to -1 and commutes with everything in the algebra). For this reason, but more importantly because I need it to get the right complexity for the binons, I have to assume one hidden dimension.

My suspicion for the hidden dimensions in string theory is that the large number they require comes from the fact that they are not assuming preons. The preons add hidden degrees of freedom, it is natural that those degrees of freedom would show up as compacted dimensions in a theory that ignored them.

This is very informative. The reason I started working on particle physics again after a break of 25 years is because I bought a copy of Polchinski's two volume textbook and wasn't satisfied with it. But I didn't get to a part that got into matrix models.

By restricting myself to density matrices, what I am doing is only working with the operators and avoiding the things that they operate on, (other than each other). Well, most of the time, but sometimes it is useful to work in state vector form, but I always keep an eye on how this relates to the density operator form.

Carl

Last edited: Aug 13, 2006
12. Aug 13, 2006

### Kea

And matrix model D0-branes (preons) show up as objects in the ribbon graph description of moduli spaces (for Riemann surfaces with punctures). This is best understood in terms of labelled (metric) ribbon graphs, where the labels on edges take values in the positive real numbers. The use of twisted ribbons for nonorientable surfaces allows a consideration of the quaternionic ensemble.

The application of T-duality essentially means a restriction to the Hermitean case. See Mulase's papers, including the related paper
http://www.math.ucdavis.edu/~mulase/courses/2001modulich01.pdf which looks especially at the theory of elliptic curves.

In Smolin's picture, invariance under loop-addition moves would also indicate the presence of a duality. A ribbon vertex with three legs is typically dual to a triangle, which sort of introduces a loop.

Last edited: Aug 13, 2006
13. Aug 13, 2006

### Kea

...oh, and there is an association between the local pieces of $\mathcal{M}_{g,n} \times \mathbb{R}_{+}^{n}$ and the Stasheff associahedra of planar trees. This is in Mulase's papers. In the elliptic curve paper it is observed that the moduli $\mathcal{M}_{1,1}$ is like a cylinder with 2 corners. Corners are the reason one needs to do extended (2-categorical) TFTs, such as those studied by Pfeiffer and Lauda, but here the surface is a moduli space. The cylinder is what one gets by gluing the lines $Re(\tau) = \frac{1}{2}$ and $Re(\tau) = - \frac{1}{2}$ above the radius 1 semicircle, where $\tau$ is the usual parameter for the upper half plane. This picks out points with $y$ coordinates $\textrm{exp}(\pm 2 \pi i / 3)$.

Considering the special points $0,1, \infty$ on $\mathbb{P}^{1}$, then the inverse image under the $j$ invariant gives for $j^{-1}(0)$ one of $\textrm{exp}(\pm 2 \pi i / 3)$ (multiplicity 3 = trivalent vertex) and for $j^{-1}(\infty)$ the three puncture points $0,1, \infty$, each of which gets something Mulase calls a bigon (multiplicity 2) etc.

I think that this is all really about operads, which are introduced in
On operad structures of moduli spaces and string theory
http://www.citebase.org/fulltext?format=application/pdf&identifier=oai:arXiv.org:hep-th/9307114
Kimura, Stasheff, Voronov

This paper has a particularly nice theorem, namely, to quote: String vertices exist.

So, although the interpretation does change, it would appear that CarlB's analysis is on a fairly solid theoretical footing.

Last edited by a moderator: Apr 22, 2017
14. Aug 14, 2006

### kneemo

Eigenmatrices

Indeed, it is desirable to think in terms of operators only, so that one can study operators acting on operators, yielding more operators (the regular representation). If we wish our operators to be Hermitian, we want to find a product that enables the composition of a Hermitian operator with another Hermitian operator, to produce a Hermitian operator. Pascual Jordan, John von Neumann and Eugene Wigner did this in 1934 [1] and classfied all the "Jordan algebras" (observable algebras) over the finite dimensional division algebras. The maximal octonionic case (3x3 Hermitian matrices) was the most subtle however, as it cannot be recovered by assigning the Jordan product to an associative algebra. For this reason it is called the exceptional Jordan algebra (for a matrix model see Smolin's http://arxiv.org/abs/hep-th/0104050").

The Jordan algebras over the quaternions and octonions are very exciting to work with as the quaternions are noncommutative and the octonions are noncommutative and nonassociative. However, when it comes to Hermitian operators in these cases, we are not guaranteed that all eigenvalues are real. In fact, we can have quaternionic and octonionic eigenvalues [2]. This is likely a reason why quantum mechanics over such division algebras didn't go mainstream.

In [2], the authors proposed the Jordan eigenvalue problem:

$$A\circ V=\lambda V$$.

In essence, we are considering eigenvalues under the regular representation of the Jordan algebra. The corresponding matrices $$V$$ are called eigenmatrices. In [2], it turns out we can get real eigenvalues for Hermitian matrices over the quaternions and octonions, when we restrict the eigenmatrices to be primitive idempotents (hence points in projective space).

The complex case is much easier to study, as the eigenvalues for Hermitian matrices are real for the Hilbert space eigenvalue problem. This facilitates the process of bridging the Hilbert space formalism and the purely operator formalism. Here's a little theorem from my thesis:

Theorem.
Let $$A$$ be an $$n\times n$$ Hermitian matrix over $$\mathbb{C}$$ with a complete orthonormal set of eigenvectors $$v_1,v_2,...,v_n$$ and corresponding eigenvalues $$\lambda_1,\lambda_2,...,\lambda_n\in\mathbb{R}$$. Then the set of matrices $$W_{ij}=\frac{1}{2}(v_iv_j^*+v_jv_i^*)$$ for $$i\leq j$$ is a complete orthogonal set of Hermitian eigenmatrices of $$A$$ with corresponding eigenvalues $$\lambda_{ij}=\frac{1}{2}(\lambda_i+\lambda_j)$$.

Here's a cool lemma that follows.

Lemma.
For Hermitian eigenmatrices $$P_i=W_{ij}(i=j)$$, $$P_i$$ satisfy $$P_i^2=P_i$$ and $$tr(P_i)=1$$. That is, the $$P_i$$ are primitive idempotents.

Proof.
Expanding we have $$P_i=\frac{1}{2}(v_iv_i^*+v_iv_i^*)=v_iv_i^*$$. Squaring this and invoking orthonormality of eigenvectors yields $$P_i^2=v_iv_i^*v_iv_i^*=v_iv_i^*=P_i$$ and $$tr(P_i)=v_i^*v_i=1$$.

In summary, over the complex field we can see that solving the eigenvalue problem in the state vector formalism leads to the solution of the eigenvalue problem in the operator formalism. Moreover, we see that the set of density eigenmatrices are in one-to-one correspondence with the set of eigenvectors (as well as their corresponding eigenvalues). This begs the question, "what is the physical interpretation of the non-density Hermitian eigenmatrices (with their real eigenvalues)?" Certainly, for the non-density eigenmatrices, we lose the nice geometrical interpretation as points in projective space--which may complicate the cool ribbon graph connection that Kea nicely noted .

[1] Pascual Jordan, John von Neumann, Eugene Wigner, On an algebraic
generalization of the quantum mechanical formalism, Ann. Math. 35
(1934), 29-64.

[2] Tevian Dray, Corinne A. Manogue, The Exceptional Jordan Eigenvalue Problem, http://arxiv.org/abs/math-ph/9910004"

Last edited by a moderator: Apr 22, 2017
15. Aug 14, 2006

### Kea

Cool stuff, kneemo! I knew I needed to get into those exceptional algebras for some reason!

Oh, but the usual complex moduli only need ordinary flat ribbons. We can then do twisted ribbons (quaternionic) for the interesting case which includes noncommutativity ... the mathematicians are still working all this out ... and, let's see, for octonionic? Good category theory question!

Hang on a minute ... we only need associahedra up to (roughly) the dimension of the moduli space, so the $\mathbb{P}^{1}$ case only needs simple polygons. Going up higher in dimension should give us a 'basic template' for the quaternionic and octonionic cases. After all, the associativity breaking etc. is exactly what associahedra and permutohedra are all about!

What if we took a triple of ribbons for the quaternionic case, using the idea of hyperkahler geometry? Does this make sense? And then add twists...hmm...

Last edited: Aug 14, 2006
16. Aug 14, 2006

### kneemo

I'm thinking the triple of ribbons arises in all the Jordan algebras of degree 3, that is, all the 3x3 Hermitian Jordan algebra cases. The quaternionic and octonionic cases are likely the ones with twists. I'll sketch a little something.

In the quaternionic 3x3 case, we get 3 vertices (D0-branes) connected by projective lines, producing a kind of triangle in the 8D space HP^2 (with isometry group Sp(3)). Each line is a copy of HP^1 (with isometry group Sp(2)). The resulting "loop" will thus have dimension 4. This is likely dual to the ribbon vertex you mentioned. To make the connection to hyperkähler manifolds, we only need a Riemannian manifold of dimension 4x2=8 or 4x3=12 with holonomy group contained in Sp(2) or Sp(3). And since the holonomy group is a subgroup of the isometry group, it's likely we'll find a hyperkähler manifold somewhere (maybe the 8D HP^2 has holonomy group Sp(2)?).

In the octonionic case, we get a triangle in the space OP^2 (which has isometry group F4, E6(-26) as a collineation group, E7(-25) as a conformal group, and E8(-24) as a quasiconformal group). The triangle is made of 8D OP^1 lines, making the "loop" 8-dimensional. (Perhaps there is a G2 manifold in the loop somewhere.)

HP^2 and OP^2 describe microscopic BPS black holes in the recent string literature (see http://arxiv.org/abs/hep-th/0512296"). The connection is not explicit however, as stringy people haven't learned about eigenmatrices and their relation to ribbon graphs yet.

Last edited by a moderator: Apr 22, 2017
17. Aug 14, 2006

### Kea

Cool, cool, cool!

Really? Oh, dear.

18. Aug 14, 2006

Staff Emeritus
I am so happy the two of you, a couple of my big heroes, are sharing. May you be blessed with a breakthrough!

19. Aug 14, 2006

### Kea

I'm so happy that kneemo is here ... maybe this is a breakthrough! Well, I'm just having fun. Look, if we take kneemo's idea and we think about complex moduli spaces modelled on $\mathbb{P}^{3}$ then, guess what? There are only three 3-dimensional (complex) moduli that matter, namely

$$\mathcal{M}_{1,3} \hspace{2cm} \mathcal{M}_{0,6} \hspace{2cm} \mathcal{M}_{2,0}$$

According to Mulase, these have orbifold Euler characteristics of, respectively (if I've calculated right - I'm not usually up this late)

$$- \frac{1}{6} \hspace{2cm} - 6 \hspace{2cm} - \frac{1}{120}[/itex] so ... let's see ... that might not be the most interesting thing ... must sleep ... Last edited: Aug 14, 2006 20. Aug 14, 2006 ### kneemo Thanks for the words of support sA. Kea, I like the -6 Euler characteristic, and so do string theorists. Following the arguments in [1], I'll explain why. Begin with the adjoint representation of [tex]E_8$$ and break it down to $$SU(3)\times E_6$$. Under this subgroup, we have the decomposition:

$$(8,1)\oplus(3,27)\oplus(\overline{3},\overline{27})\oplus(1,78)$$.

The first and last terms are the adjoint representations of $$SU(3)$$ and $$E_6$$, respectively. The two middle terms are tensor products of the three-dimensional representations of $$SU(3)$$ with 27-dimensional representations of $$E_6$$.

The difference of massless modes with distinct chirality in $$(3,27)$$ is given by:

$$n_{27}^L-n_{27}^R=\mathtr{index}(iD_3^{(6)})=\frac{1}{48}(2\pi)^3\int tr_3F\wedge F\wedge F=\frac{-\chi(K)}{2}$$

The Dirac operator and trace are taken in the vector bundle associated to the three-dimensional rep of $$SU(3)$$. The massive modes are recovered from combining modes of opposite chirality, where the absolute value of $$\frac{-\chi(K)}{2}$$ gives the number of particle generations. This is why string theorists are so interested in Calabi-Yau's with Euler characteristic +/-6.

You recovered Euler characteristic -6 (=$$\chi(\mathfrak{M}_{0,6})=(-1)^{6-1}(6-3)!$$) for $$\mathfrak{M}_{0,6}$$ (the moduli space of Riemann surfaces of genus 0 with 6 marked points), which plugs nicely into the index formula and gives three particle generations. :surprised

[1] Candelas, Horowitz, Strominger, Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985), pp. 46-47.

Last edited: Aug 14, 2006