On a Final Theory of Mathematics and Physics by Michael Rios

Lawrence Crowell

I got the ppt file to work. For extremal BHs this will work with the 28 charges and 28 magnetic monopoles in the 57.

Michael Rios

Excellent. And remember, the norm for the 57 admits non-real solutions so the complete picture is of the extended Freudenthal triple system over the bi-octonions, i.e., the 57-dimensional charge-entropy space contains complex charges acted on by E8(C). Charge quantization gives charges that are Gaussian integers. The relation to Shimura varieties enters here, where instead of considering elliptic curves we consider the coset spaces of the U-duality groups modulo their maximal subgroups in integral form. The coset spaces are the moduli spaces of the extremal black holes, and in general we have very many copies of such spaces, as we consider systems of such black holes. The Shimura datum encodes these. Once one has the L-functions for these varieties, we can define their (mixed) motives and have clear knowledge of their geometrical and topological properties.

Lawrence Crowell

I think this has to do with gauge theoretic aspects of the quaternions. I attached some time back a post on how quaternions naturally give rise to YM gauge fields. For the momentum p_{aa'} = О»_aО»_a' and the polarization Оµ^{aa'} we have the transversality condition p_{aa'}Оµ^{aa'} = 0. In addition there is the gauge transformation Оµ_{aa'} --- > Оµ_{aa'} + k p_{aa'}, k = constant so that p_{aa'}Оµ^{aa'} = 0 is invariant for a massless field. We can then find the gauge field according to these as

F^{ОјОЅ} = Пѓ^Ој_{aa'}Пѓ^ОЅ_{bb'}( p_{aa'}Оµ^{bb'} - p_{bb'}Оµ^{aa'})

The H^2 space constructs the gauge field and its dual to give the intersection form в€«F/\F = 8ПЂk. Working in C^4 does not bring that structure in.

The 8-dim spacetime enters into this picture as the E8 Cartan matrix. The intersection form is an invariant with respect to that. This is wrapped up in the theory of gauge field in four dimensions. There is underlying this a four dimensions of C^4 or H^2 in complex variables, but underlying this are 8 dimensions in pairs of real variables.

Michael Rios

From a projective space description the degree one, genus zero curves in OP^2 are 8-spheres. If Witten's formalism carries over to the Cayley plane, MHV amplitudes should localize on these curves. But first, I agree, it is better to develop the H^2 case with SO(5,1) symmetry.

Lawrence Crowell

The symmetry is really SO(4,2) ~ SU(2,2). The group on the left is the isometry group of AdS_5 = SO(4,2)/SO(4,1). The group on the right is the group for twistor space with PT^ = SU(2,2)/(SU(2,1)xU(1)). We may think of the quaternion form of this as a transition from C^4 to H^2 with SL(4,C) --- > SL(2,H). This has four complex dimensions, eight real dimensions. We might of course be so bold as to double down, with SU(2,2,C) so that SL(4,C) --- > SL(8,C), but with the above identification of pairs we have that this is SL(4,H) ~ SL(2,O).

Michael Rios

Using J(2,H), gives determinant preserving group SL(2,H)=SO(5,1) in Coll(HP^2)=Str_0(J(3,H))=SL(3,H)=SU*(6). Like the SO(3,1) case, where one represents the momentum vectors as bi-spinors in J(2,C), a given vector is lightlike if and only if its determinant is zero. Hence, in the quaternion case, SL(2,H)=SO(5,1) preserves the lightlike property of each quaternionic bi-spinor.

If one considers a Freudenthal triple system (FTS) over J(3,H), its automorphism group Aut(F(3,H))=SO*(12) contains SU*(6), as the collineation subgroup over each HP^2 component. The FTS contains two copies of HP^2, considered as electric and magnetic projective planes in N=2, D=4 symmetric SUGRA for dyonic extremal black holes.

Moving up to the octonionic case, we have SL(2,H)=SO(9,1) in Coll(OP^2)=E6(-26). This gives FTS automorphism group Aut(F(3,O))=E7(-25) which contains an SO(2,10) subgroup. Your SO(2,4) is recovered if we restrict to a complex sub-algebra. Hence, in the octonion case we essentially have an AdS_11, which upon reduction to a complex subalgebra, we recover your AdS_5.

Extending the FTS to 57-dimensions gives E8(-24) with an SO(3,11) subgroup.

For division algebras, we have the inclusions:

R: SO(4,3)->SO(3,2)->SO(2,1)

C: SO(5,3)->SO(4,2)->SO(3,1)

H: SO(7,3)->SO(6,2)->SO(5,1)

O: SO(11,3)->SO(10,2)->SO(9,1)

In the split division algebra cases:

Cs: SO(4,4)->SO(3,3)->SO(2,2)

Hs: SO(5,5)->SO(4,4)->SO(3,3)

Os: SO(7,7)->SO(6,6)->SO(5,5)

There is a super Yang-Mills theory formulated with SO(11,3) gauge symmetry, as given by Sezgin in hep-th/9703123. Bars extended this to supergravity in hep-th/9704054. Also recall that Lisi has interest in SO(11,3) based on Nesti and Percacci's GraviGUT theory [link:arxiv.org/abs/0909.4537]arXiv:0909.4537 [hep-th][/link]. With the insight from extremal black holes, it is tempting to conjecture the SO(11,3) super Yang-Mills theory is contained in a larger theory with E8(-24) symmetry. An (11,3)-signature theory would be a 14-dimensional generalization of M-theory in D=11.

Lawrence Crowell

The inclusions

R: SO(4,3)->SO(3,2)->SO(2,1)

C: SO(5,3)->SO(4,2)->SO(3,1)

H: SO(7,3)->SO(6,2)->SO(5,1)

Have as their middle group the isometry group for AdS_4, AdS_5 and AdS_7. These all share a certain relationship with each other. The AdS_5 gives as 10 dimensional theory for AdS_5xS^5, while the other two give 11-dimensional theory for AdS_4xS^7 and AdS_7xS^4. These last two are dual theories and these have a relationship with the 10 dimension theory in sense that SO(9) is a subset of SO(9,1).

The unification of these implies they are physically an aspect of the octonion. The groups in these inclusions SO(2,1), SO(3,1), SO(5,1) and SO(9,1), dimensions = 3, 4, 6, 10 correspond to physics in these respective dimensions, with 4 for gravity, 6 the CY manifold of compactification. This seems to hinge on the collination of the Moufang OP^2 by E6(-26) which contains SL(2,H) ~ SO(9,1).

There seem to be interesting connections here, and I have this suspicion that an aspect of this is related to the 5-dim moduli space for SU(2) or SO(4) bundles on 4-manifolds. The Lorentzian case gives the hyperbolic SO(4,2)/SO(4,1) ~ AdS_5.

More later after I try to bend some more metal on this.

Cheers LC

Lawrence Crowell

What I have is a couple of questions. We have that a group G decomposes into H and K or G = HвЉ--KвЉ--M with the algebraic g = hвЉ•kвЉ•m. This Cartan decomposition for the exceptional groups E6(C) and E7(C) are H = E6 and K = SO(10)xSO(2) and H = E7 and K = E6xSO(2). We also have for G = E8(C) that H = E8 and K = E7xSU(2). These are symmetric spaces and are moduli of the form E6/SO(10)xU(1), E7/E6xU(1) and E8/E7xSU(2). The question is whether these are related to SLOCC groups, such as

G_4/H_4 = E_{7(-25)}/(E_{6(-78)}xU(1))

G_3/H_3 = E_{8(-24)}/(E_{7(-25)}xSL(2,R))

The first of these is D = 4, N = 2 SUSY and the second D = 3, N = 2 SUSY. These moduli spaces with E6 have SU(3) holonomy. For the static CY these are cases of the static Hodge diamond for K3 or T^6. These groups are then moduli for the CY compactification. There is then some relationship between moduli spaces. The question is, what is the nature of these moduli spaces?

The analysis I presented last week concerned characteristic determinant equation

-det(A - О»I) = О»^3 - Tr(A)О»^2 + Пѓ(A)О» - det(A) = x

for x the root of an equation involving the commutator of the elements z \in O and the associator. I worked this out for Пѓ(A) = R(П‡. П‡') = П‰[П‡. iП‡'] = symplectic form. This equation looks very similar to an elliptic curve. Of course this is for real eigenvalues diagonalized by F4. For the complex case we have a similar system diagonalized by E6. It then seems plausible that an analogous equation exists for E6 that is an elliptic curve. This means that our moduli space is a system of modular forms that are also an algebraic variety. This means the spaces above have to be taken with a quotient of some finite discrete group. The question I whether this makes any sense?

Some aspect of conformal completion of AdS spacetimes involves Kleinian groups, and I have an interest in the prospect this system would work. Heegner prime numbers are from the discriminant of elliptic curves, and these might give a quotient that would satisfy this requirement. I am not sure it this would work or not, but it seems plausible in some way.

Cheers LC

Michael Rios

The symmetric spaces you mention are for the compact (complex) cases. One recovers these groups using the complexified octonions. The SLOCC groups you listed are produced by the octonions. You can use the split-octonions for E6(6), E7(7) and E8(8). You'll notice E7(7) corresponds to the symmetry for D=4, N=8 SUGRA.

For the eigenvalue problem, recall that E6(-26) corresponds to the reduced structure group of the exceptional Jordan algebra. Apply its determinant preserving property to the Jordan eigenvalue problem to see how it acts on eigenvalues and eigenmatrices. This will show you how to proceed with your elliptic curve analogy.

Lawrence Crowell

So I am not completely bat shit crazy here. What you say about E6(-26) appears workable since it has F4 as the mcs, which in turn diagonalizes the J3(O) in the reals. As Einstein once said, "I need a little think." This stuff is not easy for me, particularly since I have been primarily employed in applied work.

Cheers LC

Michael Rios

The E6(-26) scales the eigenmatrices, while the F4 fixes their lengths, as measured by the Frobenius norm (trace norm) over J(3,O). There is another geometrical property preserved, as you can check for yourself. Hint: imagine each eigenmatrix assigned to a vertex.

Lawrence Crowell

I will probably have to communicate by email, as I think this contest is ending soon.

The f4 algebra is the set of isometries of C\otimes OP^2, which is in a sense the "complexification" of the OP^2 that has F4 as its isometry group. The 78 dimensions of E6 is 52 26 so that E6 = F4x26, where the 26 are vectors corresponding to the Killing form = -26. The collineations e6 correspond to isometries of OP^2 by F4 as SO(9,1) and SO(9). The points that are collinear in OP^2 correspond in C^3 to SL(4,C). These are points collinear in C^3, This complex conformal group defines the set of lines connected to a loop. This is the standard system of two vertices and and two vertices _ and -.

This will then define collineations of points on algebraic curves in CP^3. I suspect in some way this may lead to the elliptic curve.

More later,

Lawrence Crowell

If we consider the Jordan-Wigner QM as diagonalized by F4, this seems to fit with the F4 scheme for the Kochen-Specker contextuality result. If we shift to E6 are we then generalizing the nature of quantum mechanics? Would the occurrence of real and imaginary eigenvalues correspond to the Bogoliubov coefficients in Hawking radiation.

I am curious about the physical implications of this.

Cheers LC

Michael Rios

Indeed F4 is the group of unitary transformations for a Jordan formulation of three-dimensional octonion quantum mechanics. E6(-26) is the group of collineations for OP^2, which includes F4 as a subgroup. In other words, one can consider F4 as a rotation group and E6(-26) as a Lorentz group, as is typically done in the extremal black hole literature.

Lawrence Crowell

As I look at E6, this seems to be the case. F4 ~ SU(3,O) and E6 ~ SL(3,O), which implies there are now an additional set of operations. These are analogous to the case with SL(2,C) ~ SU(2)xU(1,1) or SU(2)xSL(2,R). The E6 is then SL(3,O) ~ SU(3,O)xSL(3,H). These SL(3,H) transformations are rapidities similar to boosts in relativity.

Kochen-Specker proved that the measurement of can't be described by a nonlocal hidden variable that has no dependency on the context of the measurement. The Kochen-Specker theorem depends upon 48 root vectors of the F4. These are the vertices of the 24 cell. The 24 vertices have lines that emanate from them. These are basis elements for 24 quantum states. These lines define projective rays in four dimensions. For any projective ray there are three others that are orthogonal to it. There are 36 quartets of these rays that are mutually orthogonal. However, there is only need for 9 of these, where the other 24 amount to over counting. These orthogonal conditions are a condition for vanishing commutators. There are further 18 rays that exist in two sets.

If each of these 18 in two sets is determined by a hidden variables. This means there are 9 variables set to one state, say 0 or 1, and the other 3 set to 1 or 0. Yet since each appears twice this means there is an even number of these, and 9 is and odd number. It is also the case that these 9 correspond to the vanishing of 9 commutators in the BFSS holographic theory of the string. The fibration

F4:A_4 ---> F_{52/36} ---> OP^2

concerns the "36" that are ker/im of the maps, and form a cohomology.

I think we can take this further with E7. With E6 we are moving from R to C, and we have "double downed" on the number roots. We can move to H and do this again. We now have that E7 is dim = 133. We can have the quotients E7(-7)/(F4(-4)xR^{26} ~ E7(-25)/(E6(-78)xU(1)) for 2 sets of 27 gauge potential plus a graviton. Physically this "double down" amounts to pairing up the boson field above. The gauge/gravity duality, by Dixon, Bern et al, illustrates how the graviton has the same quantum numbers as a pair of gluons in a colorless entanglement. This construction appears to suit this pretty well.

The twistor approach to this is to move from C^4 to H^4, which with twistor space CP^3 from C^4 gives us PH^3. This does contain CP^3xCP^3, which means a form of twistor theory is contained in this. Of course this may be generalized I think for supermanifolds CP{3|4} into PH^{3|4}. However, the universal bundle theorem for the G_{2,2} will involve a generalization form

G_{2,2} = SU(2,2)/(SU(2)xSU(2)

into

G_{2,2,C} = SU(2,2,C)/(SU(2,C)xSU(2,C)

G_{4,4} = SU(4,4)/SU(4)xSU(4).

Cheers LC

Lawrence Crowell

Michael,

I might use your email from now on. i think it is on your paper. Anyway, is what I wrote 6 days ago at all right? Don't worry telling me it is all wrong. I'd rather know that I am wrong that not to know.

Cheers LC

Michael Rios

Seems ok, albeit E6(-26)~SL(3,O). E6(C)~SL(3,CxO), while E6(6)~SL(3,Os) where Os are the split octonions.

Lawrence Crowell

The E6(-26) ~ SL(3, O) is primarily what I have in mind. The E6(-26) has the maximal compact subgroup (mcs) F4. This is interesting because it connects with F4 as a group of collineations of the OP^2. Since F4 gives the determinant and eigenvalues of the J3 E6(-26) the E6(-26) should then be a set of transformations that preserves these eigenvalues. The Cartan decomposition E6(-26) = F4 âŠ--R^{26} tensors the F4 with the R^{26} of the bosonic string, which in a Lorentz setting is R^{25,1} and is then a set of rapidities or boosts. The additional E6(6) ~ SL(3,O(4,4)) is the split form and this has the mcs USP(8). This unitary symplectic group is defined with N = 4

O(N) âŠ‚ U(N) âŠ‚ Sp(N) = USp(2N)

and for N = 8 we can use

USp(N) âŠ‚ U(N) âŠ‚ O(2N)

which is a form of the Bott periodicity. This is contained in the O(16) group where the algebra e8(8) = so(16)âŠ•128.

We then have physical states in E6(6)/USp(8) = E6(6)/Sp(4). The decomposition is to SO(9,1)xU(1) or SO(5,5)xU(1) for the noncompact E6(-16) or split form. The 45 dimensional SO(10) in its two forms are an adjoint 45 with 16 [lus 16-bar spinorz plus a U(1).

With respect to USP(8) this is embedded in U(8), or SU(8) or SU(4,4). In a supermanifold setting this would be SU(4,4|8). We then have that for SU(4,4) we can embed two copies of SU(2,2) for twistor spaces related to each other by some form of STU duality. The physical states in this decomposition will then be SU(2,2)/Sp(5) ~ SO(4,2)/SO(4,1) --- > AdS_5.

This then leads to a rather interesting question. The main competitor to string theory has been loop quantum gravity. This LQG has been falling in popularity because it does not readily lead to workable calculations. The 11-dimensional form of this AdS/CFT duality is with AdS_{11-n}xS^n, for n = 7 or 4. In the case of n = 7 we have the AdS_4 with states in 11 dimensions. This involves the quotient SO(3,2)/SO(3,1). In effect we must then have physical states that remove all the possible states one might compute from a direct quantization of gravity. We might then ponder whether the falsity of LQG is actually telling us something very deep.

Michael Rios

The eigenvalues are not preserved, only their multiplicative product is, as given by the determinant. In more geometrical language, diagonalize an element of J(3,O) first with F4 then draw the corresponding parallelepiped for the diagonalized matrix - and find its volume. Act on this matrix with E6(-26) transformations and observe the deformations of the parallelepiped taking place, then recalculate the volume.

In D=5, N=2 magic supergravity, the volume of the parallelepiped is the major contribution to the entropy of extremal black holes as S=pi*sqrt(det(X)). When the volume is zero, the entropy is zero. There are three options for a non-trivial element of J(3,O):

1) The volume of the parallelepiped is non-zero -> 1/8 BPS black hole

2) The volume is zero, but has a non-zero area face component -> 1/4 BPS black hole

3) The volume is zero and has zero area face component -> 1/2 BPS black hole

The volume is calculated in a 27-dimensional charge space. Smolin argued years ago that a Chern-Simons like matrix model based on J(3,O) generalizes LQG sufficiently to include an SO(9) spinor and unify it bosonic M-theory. Taking his suggestion seriously inexorably leads to the picture of spin networks of extremal black holes and E6(-26) spin foams.

Lawrence Crowell

Do you have references for these results? I have read the paper by Smolin. It appears that the multiplicative product of eigenvalues is the volume of this parallelpiped. The E6(-26) transformations are I presume boost-like transformations. The E6(-26) = F4âŠ--R^{26} or E6(-26) = F4âŠ--R^{25,1} appears to suggest there are diffeomorphisms of this parallelpiped from the Lorentz structure of this 26 dimensional space.

I am rather eager to learn more about this. The volume of the parallelpiped as given by the determinant, and as I indicated earlier this has some correspondence with the Morse index for SLOCC groups. The velocity map or momentum map m_x gives the Hessian

Hes|m|^2 = 2sum_i (dm_{x_i})âŠ--(dm_{x_i}) + 2sum_i |m_x| Hes m_{x_i},

where the different SLOCC's occur where the Hessian vanishes. This result that you cite for 1/8, Â¼ and Â½ SUSY BPS black holes sits close to this.

My general idea is that a quantum wave can exist in each of these minima in this abstract manifold, which I as yet do not quite understand, and the quantum wave can also quantum tunnel between these as well. The entanglement structure is then itself subject to quantum uncertainty.

With respect to LQG, the Fermi and Integral spacecraft data on the arrival times for EM radiation from a distant burstar found no dispersion predicted from LQG. Experimental data often speaks much louder volume than theory. Now to be fair this is a measurement over a very long baseline, and it might mean that true to quantum mechanics that the momentum is sharply peaked and this dispersion does not exist. A measurement of a small region with high momentum might come up with a different result.

Michael Rios

The parallelepiped interpretation is a just a geometrical way to approach the usual BPS black hole classes (hep-th/0606211). It's not in any paper, but rather is a way I visualize the spectral geometry associated with such black holes. Higher dimensional geometry is quite difficult to visualize, but for diagonalized J(3,O) matrices, it's just three-dimensional geometry. From a spectral perspective, it's always three-dimensional for J(3,O), with the caveat that extra dimensions are packaged inside the orthogonal basis projectors. The span of the eigenspaces is just three-dimensional.

Lawrence Crowell

The determinant is a measure of the action. A state |Ψ> = A_{ab}|abc> for the indices a = {0, 1} transforms as (2, 2) under SL(2, K)_a x SL(2, K)_b. We compute the determinant of this C_{ab} accordin to the FTS. The action is then S = pi det(A), where we can identify this matrix with the J3(O) 3x3 matrix.

The cubic equation

-det(A - λI) = λ^3 - tr(A)λ^2 + σ(A)λ - det(A)

where σ(A) = R(X, X') for the curvature of the 3 dim manifold of the eigenvalues. This was something I illustrate with the Morse index approach.

We may also see this in this three dimensional space as a volume form C_{abc} = V_a/\V_b/\V_c, (/\ = wedge) where the value of this is the determinant of this 3-way determinant. This is then in some way equivalent to the hyperdeterminant. This appears like something to explore. The value of these eigenvalues is given by a discrete set of points that define a polytope. This polytope defines the G_{abcd}. Again the action is defined by the determinant, or I think in general a Hessian. I think this connection to the polytope, something similar to a Kirwan polytope, and the SLOCC type is something interesting to work on.

Michael Rios

The determinant is a special case of the cubic form, from which one can construct a cubic matrix model action. This is why Smolin proposed a Chern-Simons like model, which makes perfect sense in light of three-dimensional spectral geometry with F4 gauge symmetry.

Lawrence Crowell

A diversion back to the subject of LQG:

The papers by Smolin http://arxiv.org/pdf/hep-th/0006137.pdf and http://arxiv.org/pdf/hep-th/0104050v1.pdf do make the point there is a duality between strings and loops. The duality largely focuses on SO(9,1) and supergravity. Straight up LQG is formulated in four dimensions. The thing that occurred to me is that maybe the LQG does not "turn on" at low energy. In other words low energy physics with all the CY manifolds compactified on a small scale is where there is no "loopy physics," but at higher energy in supergravity maybe these appear. This might then mean the physical states of the QFT as a quotient SO(4,2)/SO(4,1) is such that loop variables are removed from the physics at "large N" q-gravity.

I started to look at Carroll's paper "Consistency Conditions for an AdS/MERA Correspondence" http://arxiv.org/pdf/1504.06632v1.pdf that makes use of something I proposed in an FQXi essay a number of years ago

http://fqxi.org/data/essay-contest-files/Crowell_time_scaling.pdf?phpMyAdmin=0c371ccdae9b5ff3071bae814fb4f9e9

that AdS spacetimes have a renormalization group flow. This suggests a way of removing divergences from the ground state in physics. Physically it might suggest that the disappearance of quantum gravity as a "common sense quantization of general relativity" might be a reflection of this and why it is that the cosmological constant is so incredibly smaller than what would be ordinarily thought.

Cheers LC

Michael Rios

I agree, Smolin in his unification proposal is not unifying traditional LQG with strings. The spirit of LQG is all that remains, where classical differential geometry is extended to noncommutative differential geometry. This is cutting edge mathematics, and the details are not yet complete.

For a recent overview, see the paper by S. T. Yau's group on Azuyama manifolds and matrix models:

http://arxiv.org/abs/1406.0929

[deleted]

Wow that is a 100 page paper. I am still digesting the Kalosh et all paper.

This seems to lead in some ways to what Connes recently published. I think that this all looks a bit like a "phase" of quantum states, similar to topological transitions of edge states in quantum Hall/Mott insulator states. Physics then looks very different at low energy, or for measurements along a long baseline, but different for high transverse momenta.

Michael Rios

Yes, it is a long paper. And they only take as their ansatz: the worldvolume of D-branes carries the structure of the ring M(n,R), nxn matrices with real entries. True D-branes have at least M(n,C) structure with U(N) gauge symmetry for hermitian elements. So their work must be extended.

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