An Exceptionally Technical Discussion of AESToE

[Bowles]

I would be intrigued to hear that talk, too.

I would be even more intrigued if someone could write a book about E8 mathematics!

Why is that there is so much talk about the exceptional groups, but so little literature about them? Almost any Lie groups/ lie algebra book neglects them.

Highly frustrating.

 

[Lawrence B. Crowell]

I mentioned to David that E8 = adjoint of SO(16) + half-spinor of SO(16)
and David seemed interested in that point, and I gave David a paper copy of my E8 paper at
http://tony5m17h.net/GLE8Cl8TSxtnd.pdf
which contains some discussion about that point.

Tony Smith

Your paper here raises some issues I have been trying to make here. For N-SUSY we have 2N operators $a_i^\dagger$ for a total of $2^{2N}$ states. For N = 8 supersymmetry this gives the 256 states of the Clifford valued vacuum states $|\Omega_{-2}\rangle$ of bivector fields, or helicity states

$$-2,~-\frac{3}{2},~-1,~-\frac{1}{2},~0,~\frac{1}{2},~1,~\frac{3}{2},~2$$

with the graded multiplicities

$$[ Cl(8)~=~1,~8,~28,~56,~70,~56,~28,~8,~1$$

The 1 + 3 + 3 + 1 corresponding to 6 0-helicity and 2 2-helicity states. The 6 0-helcity states are useful for working on conformal gravity. $SU(4)~\sim~spin(6)~\subset~spin(8)$, and spin(8) with a root system given by the $\{3,~4,~3\}$. The additional 4 dimensions are for the cartan centralizer of spin(8). The associated icocian of 120 quaternions, the left (or right) colored parts of Cl(8) is dual to another set of 120 quaternions (icosian) as the right (or left side) We may then "dualize" the elements of the two 24-cells or icosians by considering each elements as

$$y^a~=~x^a~+~i\theta\sigma^a{\bar\theta}$$

for $x^a$ and $i\theta\sigma^a{\bar\theta}$ dual colored elements of the opposite sides of the 0-helicity states. This vector $x^a$ is a discrete set of rigid vectors on the 24-cell, and the graded part on a dual 24-cell. These will then be eigenvectors of a superfield. This will give superpairs for 120 of the 240 E_8 elements, and to obtain the superpairs of the remaining elements of E_8 according to the Lisi representation may then require a second E_8 with what might be called a "cross dualization" of particles and supersymmetric partners. By cross dualizing I mean that if on one E_8 we have the colored elements on left side of 3 + 3 in the 0-helicity states as particles and the right hand side as SUSY partners, we then consider the right colored states as particles and the left as the SUSY pairs. In this way a complete representation of $E_8\times E_8~\sim~SO(32)$ theory can be arrived at, and from a different trajectory than string theory.

Lawrence B. Crowell

 

[Tony Smith]

Lawrence B. Crowell said "... the Lisi representation may then require a second E_8 with what might be called a "cross dualization" of particles and supersymmetric partners ...".

What kind of correspondence is there between "particles and supersymmetric partners"?

In my opinion, a 1-1 correspondence between fermions and gauge bosons is not physically realistic (i.e., no such superpartner has ever been seen),
so
if the function of the second E8 is make such a 1-1 correspondence then it is not consistent with what I do
and
it does not seem to me to be what Garrett is doing.

In the single-E8 models such as how I understand Garrett's to be,
the structure of E8 gives direct correspondences (although NOT naive 1-1)
between the gauge bosons of the two copies of D4 in the E8
and
the fermions of the 128-dim half-spinor of Spin(16) in the E8.

Given a Lagrangian with both gauge boson and fermion terms,
those E8 correspondences (in my view of Garrett's model, and in my model) show that the overall gauge boson and fermion terms cancel,
which gives the useful result of conventional 1-1 supersymmetry cancellation
without the unobserved (and in my opinion unrealistic) squarks, sleptons, winos, etc of conventional 1-1 supersymmetry.

So, in short, I don't see that a second E8 for conventional 1-1 supersymmetry is needed.

Tony Smith

 

[Lawrence B. Crowell]

Given a Lagrangian with both gauge boson and fermion terms,
those E8 correspondences (in my view of Garrett's model, and in my model) show that the overall gauge boson and fermion terms cancel,
which gives the useful result of conventional 1-1 supersymmetry cancellation
without the unobserved (and in my opinion unrealistic) squarks, sleptons, winos, etc of conventional 1-1 supersymmetry.

So, in short, I don't see that a second E8 for conventional 1-1 supersymmetry is needed.

Tony Smith

Of course this is not supersymmetry. A fermion and bosonic field (F, B) on the same frame, written cryptically as Z = B + g*F, for g a Grassmannian, does not make supersymmetry, though the overall theory is graded. So you can of course abandon SUSY completely. Until the LHC comes on line we are operating largely in the dark. Five to ten years from now we may have a far better idea about these things --- an experiment can often be worth more than a thousand theories.

I too think that squarks, sneutrionos and the rest do not manifest themselves. Yet I think these fields are canceled out in quantum gravity. Gravitation, contrary to what is often thought, involves all of the spin fields. The spin = 2 field comes from the quantization of the pp-wave, or linearized type N Petrov solutions, which have two polarization directions or helicity = 2. These solutions can be extended to Robinson-Trautman type solutions and back in "days of yore" there was a lot of effort to build up black holes from Feynman diagrams of spin = 2 solutions. This lore has also lead to a lot of string ideology from Regge trajectories and the s = 2 state. But string theory does not get general relativity quite right, and is a bimetric theory that abuses the general covariance of GR. I think that the superpairs of fermion and gauge fields serve to cancel out spurious states in quantum gravity and are maybe why physical (as opposed to purely mathematical) classical spacetime does not permit some of these odd-ball solutions for wormholes, time machines, Krasnikov tubes and Alcubierre warp drives, and in general metric configurations $g$ variables in the Hawking-Hartle wave functional $\Psi[g]$ that have no classical analogue.

I have always found supersymmetry a compelling idea. Fields that commute in the (0, 1/2) and (1/2, 0) spinor representations of the Lorentz group are paired up with anticommuting fields. The two fields are related by a supermanifold $y^a~=~x^a~+~\theta\sigma^a{\bar\theta}$ description which extends the Lorentz (Poincare) symmetries.

An example of what I was saying above about superpairs cancelling out "strange" spacetime solutions the Rarita-Schwinger field is an example. The field is represented by, or transforms according to, the

$$\Big(\frac{1}{2},~\frac{1}{2}\Big)\otimes\Big(\Big(0,~\frac{1}{2}\Big)\oplus\Big(\frac{1}{2},~0\Big)\Big)$$

spinor representation of the Lorentz group. This may be graded with the graviton with a Grassmann field. The RS field suffers from some pathologies, in particular it has acausal or faster than light in a gauged setting. Now suppose this field cancels out solutions to the Einstein field equations that lead to acausality. So a spacetime which has closed timelike loops is then canceled out by the RS field. So for the graviton G in a SUSY pairing I will write suggestively as P = G + gR will then have eigenstates which do not include this sort of spurious spacetime.

The graviton is point-like, and in spacetime of four dimensions the Poincare dual is four dimensional spacetime itself. So the whole system of gravitons in the universe might be thought of as a superposition or coherent structure (similar to a superfluid) of gravitons, which is then a superposition of spacetimes in something similar to the Hawking-Hartle path integral. This will include all possible manifold configurations. This includes a class of topological manifolds called E_8 manifolds, which are four dimensional manifolds whose intersection form (Kahler form) is an E_8 lattice. These manifolds have no diffeomorphic structure, though they are homeomorphic. Hence a graviton which corresponds to a "strange" spacetime, such as one which is has no Cauchy data, is identified with these "fake" manifolds. This part is a work in progress, and involves work with the Riemann-Roch theorem. So at this time this part is very incomplete.

This then gives rise to the three E_8's: the single E_8, its SUSY dual and this additional E_8 for the class of 4-manifolds which are "E_8" and are canceled out by the SUSY pairs of the first E_8. This then leads to the $\Lambda_{24}$. The E_8 is defines the theta function

$$\theta_8~=~1~+~240\sum_{n=1}^\infty \delta(n)q^{2n}$$

($\delta$ = divisor) where this is also the Eisenstein E_4. The Leech lattice being composed of three E_8s has a theta function cubic on $\theta_8(q)$ as

$$\Theta_{24}(q)~=~\theta_8(q)^3~-~720 q^2\prod_{n=1}^\infty(1~-~q^{2n})^{24}$$

So this is how I see the path into the Leech lattice, and the possible role for the three E_8's. I am trying to invoke physical ideas into this instead of just doing math or representation theory. If physics is ultimately up to the Leech lattice it might behoove us to have some physical reason for why embedded in that system there are effectively three E_8. I see this as Irrep on standard fields, SUSY and the spacetime correspondence of gravitons with 4-manifolds and an as yet unknown cancellation procedure.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

E_8 four manifolds

Well, anyway, I just want to know about E(8) 4-manifolds. There is still a road ahead to understand it. It was the 1st example of a non triangulable manifold. It's interesting because it looks like a totaly empty space, from the point of view of an observer. I am not sure, since I didnt study it, but that means you have potentials, but no way to measure them. You don't have how to define stoke or gauss law, i think, since you wouldn't have homology or cohomology groups, so you can't measure any kind of flux. So, maybe you can't make an experiment or observation. Maybe no interection. That manifold would be like a virtual reality space, or a dead speace, or a white blank space.

E_8 four manifolds are curious mathematical entities. They emerge from Donaldson's theorem on the dimension of a moduli space for the adjoint action of a bundle on a four manifold. There are on moduli singularities blowups of these points, where the evaluation of the Betti number is a subspace of the projective space. This is a cone in CP^n. The Kahler form for the topological charge is for a class of manifolds equivalent to an E_8 manifold, or where the topological charge is given by the Cartan center of the E_8. I'd recommend looking at Donaldson & Kronheimer "Geometry of four manifolds," Cambridge.

This stuff comes into play because path integrals are often related to the Polyakov path integral with the integration measure $D[x]/Diffeo$ or if not diffeomorphisms then a gauge volume on the moduli. Yet how can you define a path integral of this sort if you don't have a "stable" method for "modding" out gauge freedoms? This is where my idea of cancellations comes in. I just wrote about how spin 3/2 fields might cancel out some achronal spacetime solutions. Similarly I think that since spin 1/2 is involved with gravity, such as in

T. Jacobson, J. D. Romano, Commun.Math.Phys. 155 (1993) 261-276

http://arxiv.org/PS_cache/gr-qc/pdf/9207/9207006v1.pdf

and is associated with homotopy groups of the manifold, that the SUSY pairs with spin 1/2, eg higgsino, is canceled out by spacetimes with such topology. So the question is whether the "fake" four manifolds above, which are Euclidean instanton states, correspond to specific strange spacetimes in a way that can be canceled out this way.

Clearly time machines don't exist, and I suspect that nature has its "sanity checks" which prevent the energy conditions and topologies which permit these things, which are a favorite toy of science fiction writers.

Lawrence B. Crowell

 

[MTd2]

It seems my intuition is leading me on what I wanted. Although I didnt understand the first paragraph very well yet, I got the book. I am still building up my knowledge. But if anything, I would like that our spacetime emerged as kind energy optimal solution by breaking E(8) symmetry. But not breaking just in the beginning of time, but on the fly, constantly, as if it was a kind of background noise, but at the same time providing a bound. Spin should emerge from space-time structure. Otherwise, spin out of nowhere is just too artificial to me, and I don't find attractive to study. Lol, I am speaking like a crackpot. But, anyway, this is just my intuition.

Speaking of time machines, i don't mind closed timelike structures. They just should be small enough to not influence casuality. (crackpot again...). Have anyone thought that the exclusion principle comes from nature trying to hide closed time like structures, and that the sign of such curves is the 1/2 spin?

In that book, do they prove that manifold is not triangulable? ( I know this is silly, but given the evil place where i got the book, i couldn't read it yet).

 

[Lawrence B. Crowell]

Have anyone thought that the exclusion principle comes from nature trying to hide closed time like structures, and that the sign of such curves is the 1/2 spin?

In that book, do they prove that manifold is not triangulable?

Yes, E_8 four manifolds are not diffeomeorphic nor in general triangulable. This business touches on the Continuum hypothesis as well, which is the Cohen-Bernays demonstration of the Godel theorem and the continuum conjecture as consistent with Z-F set theory. You statement about the uncertainty princple as associated with a "hiding" of spacetime structure may not be too far off the mark. I think it is involved with a sort of coarse graining which happens with these cancellations, which in effect creates an uncertainty in gravitational self-energy.

Well there is in physics a most interesting and disturbing word in physics. That word is graviton. The idea automatically puts one in a domain of bimetric theories, which implicitely involve a coordinate dependent map between two metrics. Ugh! this is an insult on Einstein's legacy. String theory is bugger-all with this problem, which after all came more out of elementary particle physics --- not gravitation. LQG treats gravity in a more nobel manner, but as it becomes more particle-like spacetime physics starts to be run through the paper shredder.

Ultimately physics has two relationship systems for particles. One involves geometry, the other involves quanta. The geometry one involves first space, time and spacetime, and a system of symmetries on that spacetime. There is a theorem by Coleman and Mandula on this, which gets a bit of an upgrade to supersymmetry, which spells this out very nicely. Here the geometry is a measure system, a set of kinematics so to speak, which permits us to determine a relationship system between particles by forces and the transfer or communication of energy, information and the rest. The other relationship system is quantum. This is not a metric geometric system — two quantum states can be entangled across the whole universe as “strongly” as on an optic bench — well in principle. Quantum gravity is about merging these two relationship systems into one.

This is the problem with that infernal graviton. A quantum gravity which has this "gravity particle," no matter how quantized and dressed up to look good, is just going to have problems. Hawking-Hartle and the rest of that physics mafia in some ways are right, quantum gravity involves states over space or spacetime configuration variables in an ADM setting or ... . Of course LQG takes off from this. Yet that graviton must involve some description of not just a particle, but of a whole spacetime or cosmology, or a coherent system of spacetimes --- the set of all possible four manifolds! And this theory involves the E_8 lattice, isn't that remarkable!

Lawrence B. Crowell

 

[Tony Smith]

Lawrence B. Crowell refers to "... a class of topological manifolds called E_8 manifolds, which are four dimensional manifolds whose intersection form ... is an E_8 lattice ...".

Garrett in his paper at 0711.0770 refers to "... an E8 principal bundle connection ... with "... a four dimensional base manifold ...".

What is the explicit correspondence, if any, between
the E8 intersection form of a 4-dimensional "E_8 manifold"
and
the E8 symmetry of an E8 principal bundle connection over a 4-dimensional base manifold
?

For an example (from the book Instantons and Four-Manifolds by Freed and Uhlenbeck (Springer-Verlag 1984),
consider 6-real-dimensional or 3-complex-dimensional CP3 with coordinates (z0,z1,z2,z3) and the 4-real-dimensional or 2-complex-dimensional Kummer surface within it defined by
z0^4 + z1^4 + z2^4 + z3^4 = 0.
It has intersection form represented by
-E8 (+) -E8 (+) 3 ( S2 x S2 )
where E8 is the Cartan matrix for the Lie algebra E8 and S2 is the 2-sphere.

I don't understand exactly how an E8 Cartan matrix in the Kummer surface intersection form would produce an E8 principal bundle symmetry group over the 4-real-dim Kummer surface as base manifold,
or
how it would explicitly correspond to the root vectors of the E8 Lie algebra of the E8 principal bundle symmetry group.

Tony Smith

 

[Lawrence B. Crowell]

I don't understand exactly how an E8 Cartan matrix in the Kummer surface intersection form would produce an E8 principal bundle symmetry group over the 4-real-dim Kummer surface as base manifold,
or
how it would explicitly correspond to the root vectors of the E8 Lie algebra of the E8 principal bundle symmetry group.

Tony Smith

This is of course a bit of an open question, and is something which I have been attempting to address. This leads into an issue of compactification, Calabi-Yau spaces and orbifolds. What I am about to write is a sketch of one possibility I am considering. This has some suggestive possibilities.
A Kummer surface is a specific case of a K3 surface (K-cubed Kummer, Kahler & Kodiara). The 2-surface given by $z_0^4~+~z_1^4~+~z_2^4~+~z_3^4~=~0$ is a two-dim C surface in $CP^3$ and is an exception to most K3 manifolds which are not embedded in a projective space, or defined by this sort of polynomial. K3 manifolds are diffeomorphic to each other, so one specific example translates to another.A general Kummer surface obeys a quartic equation of the sort

$$(x_0^2~+~x_1^2~+~x_2^2~+~mx_3^2)^2~+~\lambda abcd~=~0$$

for the abcd functions of the x_i's. For the first and second pairs of these coordinates the real and imaginary parts of a complex variable then this is invariant under an abelian reparameterization $z~\rightarrow~e^{i\theta}z$. This then defines a fanning of the projective space and a form of algebraic variety called a Toric variety. These are sometimes called weighted projective spaces.
The projective space $CP^2$ the weighted projective space defines the equivalence class on the complex coordinates in ${ C { P}^2 .}$ by the map $CP^2~\rightarrow~C{P^2}_w}$ defined by the action on the coordinates,

$$[z_1,~z_2]~\mapsto~[{z_1}^{a_1},~{z_2}^{a_2}],$$

or

$$[z_1,~z_2]~\mapsto~[{r_1}^{a_1}e^{ia_1 \theta_1}, ~{r_2}^{a_2}e^{ia_2 \theta_2}].$$

This establishes an identification between the points in the ${[0,~2 \pi r/a]}$ "pie slices" or fan sections of each complex line.

Now consider two maps:

$$f: CP^2~ \rightarrow~CP^2(a)~=~CP^2_w$$

$$g: CP^2~\rightarrow~CP^2(b)~=~CP^2_{w'},$$

so that the weights for the two maps are unequal. $dz_j$ and $dz'_j$ are differential basis one-forms in $CP}^2_w$ and $CP}^2_{w'}$ respectively, which are easily computed. The dual vectors, $V_j$, $V'_j$are easily computed. The vectors defines as $L^{a_j}~=~a_j V_j$ are easily found and these obey a Witt algebra commutator which with the central extension may be extended to the Virasoro algebra.

$$[L^{a_j},~L^{b_j}]~=~(a_j~-~b_j) L^{a_j + b_j}~+~c(a_j ,b_j)$$

For the Virasoro algebra without center

$$[L^a,~L^b]~=~{C^{ab}}_cL^{c}$$

write the vector,$\xi^\alpha~=~{\xi^\alpha}_a L^a$ where ${\xi^\alpha}$ is an element of the Lie algebra $\cal G$. The commutator in of ${\xi^\alpha}_a~\in~\cal G$ can be found as

$$[{\xi^\alpha}_a,~{\xi^\beta}_b]~=~{{C_g}^{\alpha \beta}}_\gamma {\xi^\gamma}_{a+b}$$

associated with the Lie algebra $\cal G$.

Within a local trivialization connection coefficients may be defined as,

$${(\eta^{-1}\partial_{\mu} \eta)_{\alpha}}^{\beta} ~=~\eta_{\alpha \gamma} \partial_{\mu}\eta^{\gamma \beta} ~=~{\xi^{\dagger}}_{\alpha a}{\xi^{\gamma}}_a ({\partial_{\mu} \xi^{\dagger}}_{\gamma}{\xi_b}^{\beta}~+~ {\xi^{\dagger}}_{\gamma}\partial_{\mu}{\xi_b}^{\beta}),$$

which are the conjugate terms ${{A^\dagger}_\alpha}^\beta}_\mu~+~{{A_\alpha}^\beta}_\mu~=~{{\cal A}_{\alpha}}^\beta}_\mu.$ The curvature tensors ${{\cal F}_{\alpha}}^\beta~=~{F_{\alpha}}^{\beta\dagger}~+~ {{F}_{\alpha}}^\beta$ consists of holomorphic and antiholomorphic curvatures,

$${{{\cal F }_{\alpha}}^{\beta}}_{\mu \nu}~= \partial_{[\nu}(\eta_{\alpha \gamma} \partial_{\mu ]}\eta^{\beta \gamma}) ~=~\partial_{[\nu}{{{{\cal A}}^{\beta \gamma}}_{\mu ]}} ~+~{{\cal A}^\dagger}_{\alpha \beta [ \mu}{{\cal A}^{\beta \gamma}}_{\nu ]}.$$

It is possible to demonstrate that this obeys transformation properties of a gauge theory.

So this suggests a possible way in which the $C(E_8)\oplus C(E_8)\oplus\sigma_x$ and the intersection form are associated with a fibration. I think the set of these K3 spaces and compactifications is assigned to the particles or maybe SUSY pairs of fields. The algebraic geometric definition of a surface S is according to the sheaf cohomology of a group G_s. In this way I think this might be related to sheaf structure similar to twistor theory.

Anyway this is where my "frontier" on this lies at the time. It will take some time to work this out, if I can. I am just one guy here, and I have had this idea cooking for not that long.

Lawrence B. Crowell

 

[Kea]

Kostant's Riverside talk is now online, with notes by John Baez:

http://golem.ph.utexas.edu/category/2008/02/kostant_on_e8.html

 

[MTd2]

I can't understand. Konstant got SU(5), which contains the standard model, yet distler insists that it is not possible! How come?

 

[rntsai]

I can't understand. Konstant got SU(5), which contains the standard model, yet distler insists that it is not possible! How come?

Good catch; although I can't really settle it one way or the other. There are four
distinct Lie groups that are usued without distinction :

(1)E8(-248), "compact" real dimension=248
(2)E8(8), "split", real dimension=248
(3)E8(-24), real dimension=248
(4)E8(C), "complex", real dimension=496

I think Distler/Lisi use E8(8), Kostant E8(-248) or E8(C) but I'm not sure.

 

[Mark A Thomas]

I think this is the point where mathematicians and physicists part ways. I am not sure but what Konstant may be saying is that he can reveal a product space of two copies of the group SU(5) in E8 but only as a subalgebra. This is like saying that there are hints of unification in the structure E8 (which is not the complete physics structure). This is not usable for a physicist where a Hilbert space and observables are needed. It looks like at the least that this still may spill over into using 2 E8s to naively build the SM. Though some people might not like it this (line of reasoning) will still probably lead to supersymmetry such as the MSSM in the heterotic string models.

 

[marcus]

fascinating talk by Kostant
http://mainstream.ucr.edu/baez_02_12_guest_stream.mov
I just watched the quicktime movie.
I tried to stop several times during it, because substantial parts were
beyond me, but my curiosity always got the upper hand and i'd get a snack or take
a break and then come back to it. The guy has great mathematics style.
It is hard now to believe that some variation of Lisi's program is not going
to lead to real physics somewhere down the line.
=======================

Just watched Baez introductory talk too.
http://mainstream.ucr.edu/baez_02_12_stream.mov

 

[Lawrence B. Crowell]

I think this is the point where mathematicians and physicists part ways. I am not sure but what Konstant may be saying is that he can reveal a product space of two copies of the group SU(5) in E8 but only as a subalgebra. This is like saying that there are hints of unification in the structure E8 (which is not the complete physics structure). This is not usable for a physicist where a Hilbert space and observables are needed. It looks like at the least that this still may spill over into using 2 E8s to naively build the SM. Though some people might not like it this (line of reasoning) will still probably lead to supersymmetry such as the MSSM in the heterotic string models.

For E_8 we can embed SL(2, Q) ~ SL(2,16), and if this is extended to E_8(C) which embeds SL(2, 32). From this we can define CL(0, 8)xCL(8, 0) ~ cl(16) and E_8(C) ---> E_8xE_8 ~ SO(32). This is sometimes called the 32 supersymmetries in the heterotic string. From there SO(10) is a standard result of decomposition. SO(10) is then "two copies" of SU(5). We may not be able to avoid the "two copies" E_8, and this does give preference to SO(10) as the "GUT" which might appear some 10^4 times the Planck length. SUSY does provide a way of getting the gauge heirarchy worked out there.

Frankly I think we need to go to three copies to get connect certain vertex operators for the Virasoro with compactified spaces associated with the $\oplus_\pm C(E_8)\oplus\sigma_{\pm}$ to cancel out SUSY compactified states, and ... . I wrote some on this yesterday and this is a big open issue which is rather fascinating to think and work on.

It appears that Distler sees absolutely no value in Lisi's paper. I am not sure if I regard Lisi's root finding as a final answer, but dang! for once we have a simple (even if it is in some sense a toy model) representation of particles in E_8. I went through some of the bits and pieces of his calculations and outside of a couple of mistypes I found no gross errors. I seems to work! --- even if it is at this stage a demo-model.

Distler does make the following comment at the start, "I’m not going to talk about spin-statistics, or the Coleman-Mandula Theorem, or any of the Physics issues that could render Garrett’s idea a non-starter, ..." which is corrected if the framing transforms in the (0, 1/2) and (1/2, 0) of the Lorentz group. This is a part of what I have been jumping up and down about --- the system needs to be extended. Lisi's paper is a good show, but I do think things ain't done.

Lawrence B. Crowell

 

[rntsai]

It appears that Distler sees absolutely no value in Lisi's paper. I am not sure if I regard Lisi's root finding as a final answer, but dang! for once we have a simple (even if it is in some sense a toy model) representation of particles in E_8. I went through some of the bits and pieces of his calculations and outside of a couple of mistypes I found no gross errors. I seems to work! --- even if it is at this stage a demo-model.

It's seems unfair to call this a toy model. That aside, I don't see how you can say
that it seems to work when garett only claims that it works for 1 generation (Distiler
says it works for none). Are you disputing Distler's calculations?

 

[MTd2]

So, having an SU(5) is not enough to have the standard model.

 

[rntsai]

So, having an SU(5) is not enough to have the standard model.

Besides SU(5)xSU(5), he also embeds S(U(3)×U(2)) x S(U(3)×U(2)) in E8,
S(U(3)×U(2)) is the SM gauge group

 

[Lawrence B. Crowell]

It's seems unfair to call this a toy model. That aside, I don't see how you can say
that it seems to work when garett only claims that it works for 1 generation (Distiler
says it works for none). Are you disputing Distler's calculations?

I suppose in the end all theories are "toys" of one sort.

I am still digesting Distler's arguments. I find the issue of embedding G_2 and F_4 to be of some interest. I am not sure as yet whether this renders the whole thing a nonstarter, or whether this can be "fixed" by extending E_8.

Even if this works for just one generation this is still progress. Progress is all we can really expect. I don't like the TOE designation for any theory. A moments thought should indicate that a theory which explains everything in fact explains nothing. All we can expect is a theory of something --- we make progress, find where the problems are and press on from there. If things were not this way, life would not be life.

Lawrence B. Crowell

 

[MTd2]

Besides SU(5)xSU(5), he also embeds S(U(3)×U(2)) x S(U(3)×U(2)) in E8,
S(U(3)×U(2)) is the SM gauge group

So, Distler obviously made a big mistake in his calculations.

 

[Lawrence B. Crowell]

So, having an SU(5) is not enough to have the standard model.

SU(5) is ruled out experimentally. The superKamiokande failed to detect proton decay rate predicted by SU(5). It has to be admitted that things are only a little better for SO(10), but there is more stuff to play with to extend the proton lifetime.

Lawrence B. Crowell

 

[MTd2]

SU(5) is ruled out experimentally.

But doesn't it contain the standard model anyway?

 

[rntsai]

So, Distler obviously made a big mistake in his calculations.

Not so obvious. They could be talking about different groups. See my previous
list of 4 possibilties. I actually think Distler is right although I haven't verified
what he did. The difference between these groups, embeddings,...is fairly
subtle. A mistake in sign or conjugation can move you from one setting to
another.

 

[Lawrence B. Crowell]

But doesn't it contain the standard model anyway?

Yes, SU(3)xSU(2)xU(1) fits in there quite nicely.

L. C.

 

[MTd2]

Yes, SU(3)xSU(2)xU(1) fits in there quite nicely.

So, why can't we just assume Distler made a mistake. It doesn't seem he acknoledges a different group definition from what Konstant said, according to what rntsai says.

If it helps, John Baez reports on his summary that Konstant said "“E 8 is a symphony of twos, threes and fives”, http://golem.ph.utexas.edu/category/2008/02/kostant_on_e8.html#more, on the last phrase of his post.

I also found among his other files, a text written by John McKay, called A Rapid Introduction to ADE Theory.

"E8 - icosahedral = <2,3,5> case, the singularity is x2+y3+z5=0 [...]

E8: x2+y3+z5=0

E7: x2+y3+yz3=0

E6: x2+y3+z4=0

This correspondence between the Platonic groups and the Lie algebras of type A,D,E is described in

* Peter Slodowy, Simple singularities and simple algebraic groups, in Lecture Notes in Mathematics 815, Springer, Berlin, 1980.

By using a subgroup to get the equivalence classes, we get the F,G series too."

http://math.ucr.edu/home/baez/ADE.html

PS.: Slodowy was a student of Konstant.

 

[rntsai]

So, why can't we just assume Distler made a mistake. It doesn't seem he acknoledges a different group definition from what Konstant said, according to what rntsai says.

With so much handwaving and random association around you shouldn't assume anything.

Distler seems precise in his definitions and notation. He specifically calls out
E8(8). Whatever you think of his personal style (I don't care much for it),
technical precision should be appreciated. Kostant seems to be using E8
compact or E8(C).

If it helps, John Baez reports on his summary that Konstant said "“E 8 is a symphony of twos, threes and fives”, http://golem.ph.utexas.edu/category/2008/02/kostant_on_e8.html#more, on the last phrase of his post.

I also found among his other files, a text written by John McKay, called A Rapid Introduction to ADE Theory.

"E8 - icosahedral = <2,3,5> case, the singularity is x2+y3+z5=0 [...]

E8: x2+y3+z5=0

E7: x2+y3+yz3=0

E6: x2+y3+z4=0

This correspondence between the Platonic groups and the Lie algebras of type A,D,E is described in

Now this is a completely different setting. These are finite discrete groups; fairly
different than the continuous lie groups.

 

[Haelfix]

Umm, the problem isn't with embedding the standard model into E8. Thats been done before. Nor is it a problem to put 2 generations in (or 3 if you forget about chirality).

The problem is putting in gravity as a gauge theory as well. SU(5) splits into the standard model but *not* the standard model + gravity.

 

[Lawrence B. Crowell]

But doesn't it contain the standard model anyway?

Yes, but so does SO(10). There are in fact a range of possible GUTs which embed SM perfectly well. I think that some of the confusion here is that Distler used what appears to be an odd notation.

Lawrence B. Crowell

 

[MTd2]

Yes, but so does SO(10). There are in fact a range of possible GUTs which embed SM perfectly well. I think that some of the confusion here is that Distler used what appears to be an odd notation.

Lawrence B. Crowell

I wouldn't mention, but certainly, the source of confusion for me now, it is the dismissive tone Distler uses. It makes him sound that he went through the exactly the same method as Konstant, but "obviously", Distler is right in the end.

BTW, one of the main points of Distler is the use of Berger's classification to show he is wrong. I must confess that I don't know about it, and even I didn't find anything that accurately describe the initial work. Any way, in a brief search, I found that this Berger's classification is not quite strong, and perhaps it is not even ot possible to apply to Lisi's case:

The Berger classification

"In 1955, M. Berger gave a complete classification of possible holonomy groups for simply connected, Riemannian manifolds which are irreducible (not locally a product space) and nonsymmetric (not locally a Riemannian symmetric space)[...]

Lastly, Berger also classified non-metric holonomy groups of manifolds with merely an affine connection. That list was shown to be incomplete. Non-metric holonomy groups not on Berger's original list are referred to as exotic holonomies and examples have been found by R. Bryant and Chi, Merkulov, and Schwachhofer"

http://en.wikipedia.org/wiki/Holonomy#The_Berger_classification

Here is the paper:

http://arxiv.org/abs/dg-ga/9508014

Maybe Distler is even right, it is just that he wants to be too picky and get one bad interpretation of the problem, instead of the right, and useful one.

 

[Tony Smith]

rntsai said ".. garett only claims that it [ E8 physics ] works for 1 generation (Distiler
says it works for none). ...".

Jacques Distler said (over on n-category cafe):
"... The more general argument, that it’s impossible to get even 2 generations is independent of any of the details of how the Standard Model is embedded in E 8 . ..."
and
he has a link to his blog where he gives more detail:
"... What we seek is an involution of the Lie algebra, e 8 .
The “bosons” correspond to the subalgebra, on which the involution acts as +1 ;
the “fermions” correspond to generators on which the involution acts as −1 .
...
the maximum number of −1 eigenvalues is 128 ... the 128 is the spinor representation
...".

So, Jacques Distler is only saying that you have 128 dimensions to play with to make fermions in an E8 model,
and
if you (for example) do as I do and let 128 = 64 + 64 = 8x8 + 8x8
with the first 8 in each 8x8 representing the 8 first-generation fundamental fermion
particles and antiparticles, respectively,
with the second and third generations being sort of composites of first-generation fermions,
then
that is permitted under Jacques Distler's arguments.

As he went on to say
"... Note that we are not replacing commutators by anti-commutators for the “fermions.” ... that would ... correspond to an “e 8 Lie superalgebra.” Victor Kač classified simple Lie superalgebras, and this isn’t one of them. ...
the “fermions” will have commutators, just like the “bosons.” ...".
That is one reason that conventional supersymmetry is not used in the construction I outlined above.

So, just as Distler pointed Garrett in the direction of using Spin(16) (and so two copies of D4) in E8 instead of F4 in E8,
Distler has indicated that E8 physics should have 1 generation of fundamental fermions, with generations 2 and 3 being more composite than fundamental,
and
Jacques Distler's arguments, far from disrediting E8 physics, show the robustness of E8 physics modelling.

Tony Smith

PS - In his representation of each generation of fermions,
Jacques Distler (on his blog entry mentioned above where he uses more detailed notation than I am using on this text-type comment)
defines R = (3,2) + (3,1) + (3,1) + (1,2) + 1,1)
and
uses as representation for each generation of particles and antiparticles
(2,R+(1,1)) + (2,R+(1,1))
for a total of
2x(6+3+3+2+1+1) + 2x(6+3+3+2+1+1) = 2x16 + 2x16 = 64
dimensions to represent each generation
so
he notes that 128 = 2 x 64 and says
"... we can, at best, find two generations ...".
However,
he goes on to say that two generations will not work using the 64 + 64 = 128,
because
"... instead of two generations [from that 64 + 64],
one obtains a generation and an anti-generation ..."
which
is indeed what comes from the E8 physics construction described above
with one 8x8 for first-generation fermion particles and the other 8x8 for first-generation fermion antiparticles.

Distler raises a further objection about fermion chirality, saying
"... the spectrum of “fermions” is always nonchiral ...".

However, just as the composite nature of generations 2 and 3 allows construction of a realistic E8 model with one generation of fermion particles and antiparticles,
the chirality (or handed-ness) of fermions is not a problem with my E8 model because
fundamentally all fermion particles are left-handed and all fermion antiparticles are right-handed,
with the opposite handedness emerging dynamically for massive fermions.
Such dynamical emergence of handed-ness is described by L. B. Okun, in his book Leptons and Quarks (North-Holland (2nd printing 1984) page 11) where he said:

"... a particle with spin in the direction opposite to that of its momentum ...[is]... said to possesses left-handed helicity, or left-handed polarization. A particle is said to possesses right-handed helicity, or polarization, if its spin is directed along its momentum. The concept of helicity is not Lorentz invariant if the particle mass is non-zero. The helicity of such a particle depends oupon the motion of the observer's frame of reference. For example, it will change sign if we try to catch up with the particle at a speed above its velocity. Overtaking a particle is the more difficult, the higher its velocity, so that helicity becomes a better quantum number as velocity increases. It is an exact quantum number for massless particles ...
The above space-time structure ... means ... that at ...[ v approaching the speed of light ]... particles have only left-handed helicity, and antparticles only right-handed helicity. ...".

Again, Distler's chirality argument does not discredit E8 physics, but instead show how to construct it as a solid realistic physics model.

 

[Lawrence B. Crowell]

So, why can't we just assume Distler made a mistake. It doesn't seem he acknoledges a different group definition from what Konstant said, according to what rntsai says.

I don't think that Distler made a mistake. You have that the E_6 lattice is defined in E_8 by

$$E_6~=~\{(x_1,~x_2,~\dots,~x_8)~\in~E_8~:x_6~=~x_7~=~x_8\}$$

The simplest subgroup decomposition is D_6 ~ SO(12). If I might be so bold this contains the Pati-Salam SU(2)xSU(2) model with the QCD SU(3). If I "pop off" one of the circles from the D_6 ---> D_5 I then obtain the SO(10). Now if I were to pull this back to the E_8 I have to removed the centralizer Z_5, as E_8 has 2-3-5 centralizers in addition to the C(E_8). This I believe is where the (SU(5)xSU(5))/Z_5 enters into the picture. If we break this to SU(3)xSU(2)xU(1) I think (I state this without proof) that the second fundamental group

$$\pi_2\Big(\frac{(SU(5)\times SU(5))/Z_5}{SU(3)\times SU(2)\times U(1)/Z_6}\Big)~=~Z_2$$

which I think is a 't Hooft-Polyakov monopole. The centralizer Z_5 reflects the 5-cycles (12 permutations) around the $x,~\infty$ points on the E_8 icosahedron.

If it helps, John Baez reports on his summary that Konstant said "“E 8 is a symphony of twos, threes and fives”, http://golem.ph.utexas.edu/category/2008/02/kostant_on_e8.html#more, on the last phrase of his post.

I also found among his other files, a text written by John McKay, called A Rapid Introduction to ADE Theory.

"E8 - icosahedral = <2,3,5> case, the singularity is x2+y3+z5=0 [...]

E8: x2+y3+z5=0

E7: x2+y3+yz3=0

E6: x2+y3+z4=0

This correspondence between the Platonic groups and the Lie algebras of type A,D,E is described in

* Peter Slodowy, Simple singularities and simple algebraic groups, in Lecture Notes in Mathematics 815, Springer, Berlin, 1980.

Without checking, did you get the E_6 and E_7 switched around?

By using a subgroup to get the equivalence classes, we get the F,G series too."

http://math.ucr.edu/home/baez/ADE.html

PS.: Slodowy was a student of Konstant.

Interesting. I wonder what bearing this might have on the (SU(5)xSU(5))Z_5.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

Is suppose the icosahedral relationships might be right. At least this is what Baez has. Somehow the two are related by the condition $\sum_{i=1}^8e_i~=~0$ as a linear dependece on E_7 to the condition e_6 = e_7 = e_8 on E_6.

Lawrence B. Crowell

 

[MTd2]

Lawrence, my doubts were solved by John Baez. I thought that a representation for a group, you would autmaticaly get a represention for the subgroups, that suited the subtheories, just like the standard model.

 

[Lawrence B. Crowell]

Umm, the problem isn't with embedding the standard model into E8. Thats been done before. Nor is it a problem to put 2 generations in (or 3 if you forget about chirality).

The problem is putting in gravity as a gauge theory as well. SU(5) splits into the standard model but *not* the standard model + gravity.

The problem is that gravity is SO(3,1), which makes the group hyperbolic. It is different from SO(4), the Euclidean version of the same group, in that SO(4) is compact. SO(3,1) is not. With SO(4) you can define connections which will converge in a Cauchy series. The hyperbolic nature of SO(3,1), and SO(7,1) as well, means that a sequence of connections can go off to "asymptopia" and never converge.

For this reason it is not difficult to globally define a quantum vacuum state with compact support. A vacuum in one region or chart in the spacetime does not in general transform unitarily to a vacuum in another chart. This leads to Hawking radiation. With quantum gravity the situation is compounded. The unitary inequivalence now extends to any infinitesimal region. A superposition of states over metric configuration variables means that a point is shared by a set of metrics in a nonunitary manner. We then no longer can define a vacuum state by standard methods.

Lawrence B. Crowell

 

[MTd2]

The problem is that gravity is SO(3,1), which makes the group hyperbolic.

I thought the problem was with the representation of the embeding.