Mike Duff has been back in the science news with the publication of one of his papers and a suitably hyped press release from Imperial College. The research does not actually propose a test of string theory, it merely uses some mathematical ideas in a way inspired by string theory to analyse the entanglement of qubits. Even so, the work is still pretty exciting because of this connection.

Duff’s work in this area began when he noticed that hyperdeterminants come up in the theory of entanglement and also as U-duality invariants defining the entropy of black holes in string theory. At the time, not many applications of hyperdeterminants were well-known, so their appearance in two parts of physics at the same time was taken as a sign that there may be some connection, Duff and his collaborators have been exploring this idea ever since.

The hyperdeterminant is a generalization of determinants to multi-dimensional matrices, or hypermatrices. For a 2x2x2 hypermatrix the hyperdeterminant is a homogeneous degree four polynomial in the 8 components of the hypermatrix, and is known as Cayley’s hyperdeterminant. It can be an invariant characterising the entanglement of three qubits, or an invariant of U-duality for a black hole.

If you look at one of his early papers on this you may notice that he actually cites one of my number theory papers, so you can see that I have some personal interest in this subject. He only cited me because he liked my formula for the hyperdeterminant in terms of Levi-Civita symbol which is

=

In fact the connection is much more interesting than that because my paper makes a link between hyperdeterminants and elliptic curves. Further work on this has shown that the next hyperdeterminant up for a 2x2x2x2 hypermatrix is related to the j-invariant and of course the j-invariant has been connected to the entropy of black holes too. This larger hyperdeterminant is a polynomial of degree 24, and here the number 24 is connected to the well-known importance of this number in string theory. I’d be happy to explain this to Duff over a pint if he wants to get in touch

Duff has taken his knowledge of these invariants in black hole entropy and applied it to count the number of possible states of entanglement for 4 qubits. That is what the latest paper is about. I think the real excitement is the idea that there may be some connection that is more than just a similarity of the mathematics. The question is, can the work be extended to much larger numbers of qubits in a way that makes string theory look like a theory of entanglement with the qubits playing the role of quantized information at a fundamental level? I don’t know if that is what Duff is thinking of, or if he has some deeper reason to expect something like that to be true, but it is much more interesting than the non-idea that this work provides a test for string theory.

**Update:** Here are some other articles worth linking to: Kea is covering hyperdeterminants as M Theory lessons 345, 346, 347, 348, 349, 350 and of course she has written earlier stuff that you can search for. Lubos has of course mentioned this topic before and even used my formula for the hyperdeterminant here. Another reasonable report on the latest findings can be found at Universe Today.

In fact the connection is much more interesting than that because my paper makes a link between hyperdeterminants and elliptic curves.This theory is worked out in full in the wonderful Green Book on multidimensional determinants by Gelfand et al. These are very serious mathematicians and they know the algebraic geometry. I have been blogging about this!

I know Kea. I must try to grok that associahedron stuff

Oh, I know I should write a paper classifying all qudit measures this way … maybe now that I have enough food to eat for a while I will make a start. The associahedron is just one simple example.

I started reading this paper yesterday. Lots of group theory stuff, and Young Tableaux work. This is not terribly different from something I have worked recently. The hyperdeterminant is the determinant of the Jordan Matrix algebra, or the F_4 automorphism. I was given a heads up on this yesterday, and already I find some interesting parallels here.

As for associahedra, I presume this involves the Stasheff polytope and octonions. I have covered that subject in considerable detail. This involves transformations between quantum braid groups, and I think classifies S-matrix systems.

“This involves transformations between quantum braid groups, and I think classifies S-matrix systems.” – I thought this was the reason for doing this kind of math, to explain the braidings, or effects of spin and other forces.

I read from Universe today:

“the classical interpretation of strings and branes is that they are quantum mechanical vibrating, extended charged black holes. – cool that they have arrived to that picture Black hole is kind of wormhole.

“realized that the mathematical description of the pattern of entanglement between three qubits resembles the mathematical description, in string theory, of a particular class of black holes.”

Reminds me on the earlier discussion of the (entangled) quantum world

One thing that I also have wondered about; quarks oscillate between up and down, where down has a mass twice as big. Charm and strange have also very different masses, as top-bottom quarks. How is the oscillation done and mass changed? Lagrangian energy oscillations (kinetic – potential?)?

Entanglement means 1/2 of masses? Are the quark masses opposite poles then? But as for BE-condensates, kaon-and matter-antimatter oscillations the symmetry is not perfect. This is the third number?

Yet observing the momentum of either of these particles is statistical (Bell). The result can vary over a range of values. But once an observation is made, we know with similar accuracy the momentum of the other particle which may be far away even on the other side of the universe.

This is also behind the synchronization (strings like entangled qubits?, quantized information as holography?) AND biology

http://www.technologyreview.com/blog/arxiv/25375/

Sorry for my unqualified platter, I am no mathematician either. Keas figures look so interesting, especielly the hexagon (Fermi surface?) with the center as a spike and center of oscillations. I have searched a figure like this long, but I can’t interpret this. The center is no zero, but a jump? To get the jump the oscillations have to be synchronized?

Ulla,

The Stasheff polytope is a curious object, and it has to do with an ordering of vertices and the operations they represent. The two dimensional Stasheff polytope is a pentagon or I_2(5) in group theory jargon. The three dimensions it is a bit more complicated. One takes three pentagons and fits them together so they form a “tent” of sorts. Repeat this and put these two tents together by matching their edgelinks. One can go into higher dimensions as well.

The one dimensional Stasheff polytope is just two edgelinks with two nodes, and where you connect the two nodes (vertices). The two original edgelinks are of the form o—o and define an elementary braid link. So this link is (ab) and with other such structures you can derive the Yang-Baxter relations. By connecting two edgelinks together there are now three nodes and there is an ordering ambiguity (ab)c or a(bc) which is described by the associator

[a, b, c] = a(bc) – (ab)c.

The Stasheff polytope is a graphical representation of this object, where here there are just three elements. With 4 elements the pentagon contains these elements in various associations, and for 5 elements there is the funny looking pentagonal “two-tents” construction and so forth.

The octonions are a nonassociative extension of the quaternions. Quaternions obey the multiplication rule

I^2 = j^2 = k^2 = -1

ijk = -1.

The quaternions define a Clifford algebra in 3 or 4 dimensions, where in four dimensions there is the inclusion of the unit 1 operation. Octonions extend this construction with 4 additional elements. There is a nonassociative multiplication rule given by the projective Fano plane. That is an interesting structure in its own right, but a bit beyond the scope of this post.

The connection to physics is treacherous ground in a way. The heterotic string obeys an E_8xE_8 system, where E_8 is an exceptional group representation of the octonions. Physically one can’t have currents that are nonassociative, or at least the physical meaning of that is odd. So operator product expansions (OPEs) are tricky. In a more general setting the octonions in 8 dimensions exist in the Jordan algebra J^3(O), which is defined by a matrix of the form

|x O O’*|

|O* y O” |

|O’ O”* z |

where I hope that turns out alright on the post. The system is 24 dimensional (3 octonions each in 8 dimensions) plus 3 from the diagonal scalars, where a constraint on the three scalars in a light cone condition gives a Lorentzian space in 26 dimensions. This connects the theory to the 26 dimensional bosonic string. A natural partition of the theory in 26 dimensions is 10 + 16 dimensions for the type II strings and the heterotic E_8xE_8 strings.

Nonassociativity enters in with the role of the F_4 group, which is the automorphism of this group. It defines a linear constraint Tr(M) and a quadratic element and a cubic element which is the determinant of the matrix. As this defines the determinant it means F_4 can diagonalize the matrix to compute eigenvalues. The cubic elements in the determinant are not necessarily associative, and further this defines a generalized Chern-Simons Lagrangian. The elements the matrix operates on are formed from U = uu^*, which themselves are elements of the algebra. The general matrix above then acts on this as MU = λU, for λ the eigenvalues determined by diagonalizing this with F_4. The element I is a density matrix and has a correspondence with quantum mechanics. Since these are density matrices they are idempotent U^2 = U, which means they exist on the Moufang plane — another bit to look up.

So where do Stasheff polytopes come into the picture? They don’t so far except in the most trivial way with o—-o—-o. The pentagonal Stasheff polytope represents a quartic term, where there are 4 elements. The determinant of the matrix here is cubic, so in order to make sense of this you have to embed this into some larger 4×4 matrix. The F_4 would then be the automorphism on the subgroup J^3(O) of some larger group, call it J^4(O). So what would this be? That is not clear to me honestly. The thought occurred to me that somehow F_4 still might play a role here. A 4×4 would have 6 octonions, with 48 dimensions and with 4 scalars that is 52 dimensions — the size of F_4. So maybe the 4×4 octonionic system might be worked up as a way of looking at the complex (Hermitian plus anti-Hermitian) octonionic system corresponding to F_4. Physically this might connect to j-10 symbols and combinatorics. This would obey roles and “leaves” according to the I_2(5) Stasheff polytope. What about a J^5(O)? That would have 10 octonions (associators with 5 elements etc) and 5 scalars for a total of 85 elements. At this point the “Jordanumerology” seems to elude me, though I have not explored things in this direction much. The only thing which has occurred to me so far is to go to the quartic level to look at j-10 symbols and combinatorics.

An interesting excersize is to look at a dodecahedron and notice that three faces there form one of the tents of the 3-D Stasheff polytope. Then from there one could construct the Stasheff polytope from the by identifying combinations of the remaining 9 facets of the dodecahedron in triplets to form the other tent. Then repeat this for the upper tent moved to another location and so forth.

Phil,

When you mention, ” work be extended to much larger numbers of qubits in a way that makes string theory look like a theory of entanglement with the qubits playing the role of quantized information at a fundamental level?”, are you thinking very large numbers like 10^40 or very large entropies? Curious…

Taking a black hole the size of the observable universe and finding area of horizon in Planck units gives about 10^140, but the universe could be much larger or even infinite.

A quick ‘back of the envelope calculation’ for 10^140 Planck units yields 10^88 ‘degrees of freedom’ and associated 10^61 Planck units of mass-energy. Quantum entanglement for 10^88 ‘degrees of freedom’?Wow! Possibilities would probably be expo-exponential but finite. The holographic principle would be a player here as the very large numbers probably encode much more information (correlations, other) than what is immediately known (i.e. observables..). Very large numbers should be part of Quantum Information Theory.

Found an excellent read by Paul Davies that describes the current limitations of this, “The implications of a cosmological information bound for complexity, quantum information and the nature of physical law”. The link is,

http://cosmos.asu.edu/publications/chapters/chaitin_book.pdf

If it is true that the maxim “the graviton (or gravitational field) knows everything” then QIT will not be complete until a way of combining the gravitational interaction is somehow introduced, then perhaps the repeated blowups of exponentials can be tamed and errors can be eliminated. This then is another motivation for developing the theory of quantum gravity.

If you don’t know that the associahedra are the same thing as Stasheff polytopes, then I suspect you have not covered this subject in the kind of detail that I have in mind.

Phil

I’m always up for a pint

Mike

A beer sounds nice. Phil, have one on me.

I read a couple of times an article by Fomin and Reading:

arXiv:math/0505518v3

on associahedra. I am more interested in the Jordan algebra, primarlily because it connects with classical groups. However, the 14 vertices of the Stasheff polytope or associahedra might have something to do with the G_2 group or some SU(3) + 3 + bar-3 holonomy.

Hi Lawrence. Yeah, so the associahedra form a 1-ordinal operad, meaning that they are one dimensional in a categorical sense, whereas Jordan algebras will bring in higher dimensional structure. That is, the associahedra tile the real number points of genus zero moduli spaces, but we want to understand the larger number fields. The general polytopes are known to mathematicians but not really studied as yet. I talk quite a bit to Michael Rios, who is an expert on Jordan algebra matrix models and M theory, and he has a much clearer idea of how G2 comes into the picture.

Kea,

I must confess I read the Fomin & Reading review paper on this subject last year or so. I was left a bit pondering whether this had much to do with physics. I wrote a response above on this, which breaks some more of my thoughts out on this matter. The associahedra elements of the SP (Stasheff Polytope) are just combinatorial elements or one dimensional. I remember the term operad in connection to this, but forgot precisely what that is. The connection with Jordan matrix algebra seems to be in some sense categorical or according to some partial or forgetful functor that reduces the dimensionality of the “superspace” a physicists might be interested in.

The existence of 14 verticies in the SP for three dim or the homotopy up to 5 did make me ponder whether or not G_2 plays some sort of role. G_2 is the automorphism group of the octonions.

Lawrence, my view is that the category is far more fundamental than the classical groups. And the 3d Stasheff polytope with 14 vertices is particularly interesting, but in a way that is not clear to anyone yet. For instance, Loday pointed out that a trefoil in R3 could be drawn on this polytope, with crossings on the three squares. Re the octonions, very roughly speaking, I would view these three squares as one half of a cube that describes the octonion units. This 3d Stasheff polytope appears as an axiom for tricategories, so it is relevant for octonions if we view the octonions as a 3d categorical structure. Since the whole operad of all associahedra (one in each dimension) is generated from dimensions 1, 2 and 3, we get all these polytopes for free (in some sense) whenever we get the 3d one.

Re superspaces: the recent twistor scattering results of Arkani-Hamed et al package amplitudes using SUSY, and this results in combinatorics which basically follow from the associahedra. That is, we can turn that around and start with the category theory, get the right recursion from there and FORGET the SUSY interpretation, which I do not view as physical.

Ok, so as a responsible citizen I am going to have to stop blogging for the rest of the day … Christchurch was just struck by a major Earthquake.

Kea,

I will have to stop as well. Associators, Stasheff polytopes and the homotopy system here is a sort of symbolic gadget for looking at higher systems of nonassociative objects. It is interesting there is a knot structure here (trefoil on the 3d SP), which seems to suggest some interesting interplay between Yang Baxter systems and associators. Then of course there are permutahedra, or mutations on I_2(5) and higher.

Yeah, I heard about the NZ earthquake just now. It is a bit of a big one 7.8. Good luck on that. And remember, while you have just had an earthquake, the US is having a much longer problem with balder-quacks. By that I mean lunatics who pass themselves off as smart “experts,” and who are nothing more than quacks who spew forth balderdash. An example is Glen Beck. I’d rather deal with an earthquake in the long run.

There is a nice summary about hyperdeterminants in Wikipedia.

Thanks, I wrote that

Haha, I like such dialogues whose a priori probability is not far from 1 in 6.8 billion. It happens to me often, too.

I found it helpful to relate hyperdeterminants to Freudenthal Triple Systems (see http://arxiv.org/abs/0812.3322 ) The eight components of the 2x2x2 hypermatrix can be put into the eight component FTS defined ove rthe Jordan Algebra C+C+C.

You can then extend the symmetry of the 2x2x2 hyperdeterminant by combining it with three 2 component vectors and putting that into the 14 components FTS defined over the Jordan algebra J3(R). The SL(2)xSL(2)xSL(2) invariance of the hyperdeterminant is then embedded in a larger symmetry that includes rotations of the cube. This makes sense of the observation that the principle minors of a 3×3 symmetric matrix have zero hyperdeterminant http://arxiv.org/abs/math/0604374 .

I am looking at this mathematics some, which I confess I am a bit unfamiliar with. I figure this must be some generalization of the linear, quadratic (trace) and cubic (determinant) invariant system in J^3(O). However, your system seems restricted to the reals, but I don’t know if that matters. The determinant of an 2×2 matrix is Sl(2,R)xSl(2,R) and by extension the hyperdeterminant of a 2x2x2 is Sl(2,R)xSl(2,R)xSl(2,R). So I presume by extension the if the rank is n and n^k that the hyperdeterminant is Sl(n+1,R)^k. If n = 15 this would then suggest to me that we have Sl(16,R) ~ Sl(2,O). The Jordan matrix and the cubic invariant of the F_4, the determinant is then an example of SL(2,O)xSL(2,O).

The cubic system or J^3(O) physically is a Chern-Simons Lagrangian system, and this is under the action of a coboundary operator

Dω_3[A] – -> F^2

where D is a covariant co-boundary operator, ω_3[A] the CS Lagrangian and F^2 the typical Lagrangian for a YM field. The structure of the theory is given by equivalently the Freudenthal product system or triality, and the automorphism of the F_4 on J^3(O). The quartic term determined by the boundary operation on the CS Lagrangian would seem to be then one level higher Sl(2,O)^2 – -> Sl(2,O)^3. If this is the case this suggests some deeper relationship between physics on a space and that on the boundary of that space, such as the celebrated AdS/CFT.

There are different ways you can try to extend to quaternions or octonions. To put quaternions in a 2x2x2 hypermatrix it may be better to consider it as a 2x2x2x2x2x2 hypermatrix of complex numbers with a suitable pattern of entries. This is because a quaternion is like a 2×2 matrix of complec numbers.

The ideas about Chern-Simons is very interesting. This could relate to Witten’s more recent take on 3d quantum gravity in which the black hole entropy comes from the J-function. The U-duality group in 3D should be E8 after a reduction by 8 dimensions from M-theory. The E8 invariants on the adjoint rep are related to the 2x2x2x2 hyperdeterminant and this hyperdeterminant is related to a J-invariant through my number theory work. Some peices fit together but it’s not a complete jigsaw yet.

The pairing of numbers, real pairs to complex number, complex pairs to quaternions, quaterion pair to octonions, and so forth is being emulated this way as hyper-matrices. Each one of these pairing involves an Sl(2,R), or for k there is an SL(2^k, R), where K is the Cayley number. A hyper-determinant on 2x2x2 of octonions is then this “squared,” or 2x2x2x2x2x2. with (SL(2,R)^2)^3 = SL(2,R)^6. Sl(2, R) is an interesting group, for it is the isometry group of conformal quantum mechanics.

This seems to connect up with AdS_n structure as well. A metric for a charged black hole in an AdS_4 spacetime with a black hole is

ds^2/L^2 = r^2(-Adt^2 + dx^idx_i) + (r^2A)^{-1}dr^2, 1/L^2 = AdS curvature

where A = (1 + m^2/r^4 – q^2/r^2) is a BPS metric for a black hole in the AdS. For r – -> infinity this recovers a straight AdS_4 spacetime, where at the horizon this is an AdS_2xS^2 spacetime. The symmetry of the AdS_4 is broken. AdS_n = O(2, n-1)/O(1,n-1) (a hyperbolic version of S^n = O(n+1)/O(n)) and where for n = 2 then

AdS_2 = O(2,1)/O(1,1) ~ SL(2,R)

Hence the AdS_2 is the symmetry group of conformal quantum mechanics. By contrast the AdS_4 that is recovered for r – -> oo there are 2 + 1 dimensional conformal symmetries. Thus the black hole breaks the conformal symmetries of the vacuum, where the vacuum modes with k ~ m exhibits broken Lorentz symmetries, and there is left s = ½ conformal QFT. So the reduction of the number of conformal symmetries appears to be reflected in the degree (if I am to use that term) of these determinants. The full AdS_4 (AdS_n in general?) theory is then a hyperdeterminant, but then maybe on the horizon there is a reduction here, say from 2x2x2 to 2×2.

Such problem usually have the typical action

S = (1/2k^2)∫d^4x sqrt{-g}(R + 2/L^2 – L^2F^2).

Yet to look at this in detail with Lorentz breaking we need to break the Ricci scalar into

R = TrK^2 – (TrK)^2 + C δW/δg_{ij} g_{ijkl} δW/δg_{kl}

For W an action term computed on the 3-spatial surface. This is similar to what Horava did with some of his analysis with the anisotropy of time t – >b^zt for z = 3 and the breakdown of Lorentz symmetry at the UV limit.

The Chern Simon Lagrangian enters into this picture with this renormalization group flow picture. Here we are looking at a group flow from the horizon at r = oo to the BH horizon. The attractor point at the IR limit then should be the Einstein field equations themselves. The CS lies in the determinant (automorphism of F_4 on the J^3(O)) and there is an RG flow in this theory to the boundary with this reduction from a higher system — maybe a hyperdeterminant.

Yes, E8(-24) and E8(8) are U-duality groups for homogeneous supergravities in D=3 (hep-th/0512296). These are symmetry groups for the single variable extensions of the Freudenthal triple systems over J(3,O) and J(3,Os), where O and Os denote the octonions and split-octonions.

To recover the complex form of E8 requires the use of the exceptional Jordan C*-algebra, J(3,Oc), the complexification of the exceptional Jordan algebra, consisting of 3×3 Hermitian matrices over the bioctonions (arXiv:1005.3514 [hep-th]). One merely constructs an FTS for J(3,Oc) and extends the structure by a single complex variable, yielding a 57 dimensional (complex) space which E8(C) acts on. The extended FTS constructions for J(3,O) and J(3,Os) become special cases of this construction, since the octonions and split-octonions are contained the bioctonions.

The FTS over J(3,Oc) also has immediate applications in Duff’s study of three qubit entanglement. Specifically, the eight component FTS defined over the Jordan Algebra C+C+C, is contained in the FTS over J(3,Oc), where C+C+C is the subalgebra of all diagonalized elements in J(3,Oc). Hence, since the J(3,Oc) has E7(C) symmetry, one can relate three qubit entanglement classes to orbits of E7(C).

Kneemo, thanks for the info and references, this is very useful. There is one thing I don’t understand. In what way can E8(C) act on a 57 dimensional space? I thought the smallest representaion of E8 was 248 dimensions. Is it just some subgroup that acts on this space or is there some fancy non-linear action?

I too am wondering about this as well. E_8(C) or what I interpret as OxC is constructed from the J^3(O) in reals. With the constraint condition on the diagonal reals, and a pairing on the remaining 26 dimensions this is 52 dimensions. The replacements of the diagonals with complex (antihermitian) elements jumps us from 26 dimensions to 38, and these (the Hermitian and antiHermitian) together with a G_2 gives the OxC in 78 dimenions. So I am not sure where 57 comes into the pciture.

I intend to write more later today or tomorrow on this. Kneemo has directed us to some interesting references.

Yes, the action is non-linear. Gunaydin, Koepsell and Nicolai provide the details in hep-th/0008063, and the interpretation of the 57 dimensional space as a ‘charge-entropy’ space for D=3 extremal black holes is given by Gunaydin in hep-th/0502235. From a Jordan algebraic perspective, the 57 dimensional space is an FTS with an extra coordinate. E8 preserves a generalized lightcone on the 57 dimensional space, where the corresponding norm is constructed using the E7 quartic norm of the FTS.

In hep-th/0008063, Gunaydin et al. note that the 57 dimensional norm can take complex values when the E7 norm is negative valued. Hence, it is necessary to complexify the representation space to get a realization of E8(C) on the extended FTS. This is one reason I approached the construction using the exceptional Jordan C*-algebra in arXiv:1005.3514, making it possible to study the exceptional N=2 and N=8 supergravities in a single framework. This hints at a new physical theory, which has such supergravities as limits.

Phil, your first link does not work.

Ah yes, that Holtz and Sturmfels paper is pretty cool. I was looking at that matrix last year in connection to three qubit entanglement, but I have not thought much about the FTS.

I fixed the link, thanks. I must remind myself how the FTS connection goes and write it out.

I added to my blog a little article about possible application of hyper-determinants to the field equations characterizing n:th order criticality which is very relevant notion in TGD Universe due to the infinite-D vacuum degeneracy of Kahler action. These equations are formally multilinear equations in the variation and under special conditions also genuinely multilinear equations. For local action principle genuine multilinearity typically fails and one obtains non-linear terms.

In TGD framework effective 2-dimensionality however implies genuine multilinearity since the deformations of space-time surface are expressible as non-local functionals of those for partonic 2-surfaces and their tangent space data.

This relates also closely to integrability and absence of local divergences in the functional integral. Hence infinite-D hyper-determinants defined as generalizations of Gaussian determinants might be very relevant for characterizing the occurrence of higher order phase transitions in TGD Universe.

See this.

With this your need to explore the MSSM – world would vanish….

I like this.

I am reading up on this mathematical physics. Much of it looks pretty familiar. Something occurred to me yesterday which I have been pondering for a while. Maybe Kneemo and Philip can shed some light on this. The relevant dimensions are 26, 16, 11, and 10. The ten dimensional supersymmetric theory is 10 dimensions, 9 space plus time, which defines the 10 dimensional Supersymmetric Yang Mills (SYM) theory. This theory is a nongravitational theory which describes open strings, and the reason for 10 dimensions is that the Virasoro algebra has an anomaly cancellation property in this number of dimensions. The theory is not entirely satisfactory, for it is not renormalizable outside of compactification. We jump to the other end of these number to 26, which is the dimensionality of the bosonic string, where again this cancels anomalies at this dimension. However, these are vital to understand much about string theory. The relationship between the 26 and 10 dimensions involves the number of supercharges (charges which define supersymmetric fields), which contains 8 charges plus their superpairs — 16 in total. This involves an interesting relationship between Clifford algebras and the Cayley numbers 1, 2, 4, 8, whereby if you add two to these you get 3, 4, 6, and 10. For a Cayley number n the supersymmetric theory is an so(n+1,1) group action on the Cayley or Moufang plane (the subspace where Trace(V) = 0 and we can define a density matrix in quantum mechanics). Again there is some machinery here which I will avoid. The paper by [link: http://arxiv.org/abs/math-ph/9910004%5D Dray and Manogue [/link] (I hope this link command works) breaks some of this mathematics out. The Freudenthal Triple System (FTS) defines a 3-cycle which constructs a 10 dimensional theory. So given the Cayley number n = 8 (for supercharges) the theory is a CL_{9,1} = R[16](+)R[16]. (here the term (+) means oplus.

Now what about our 11 dimensions? That comes about from a 4-cycle. The FTS is due to an automorphism on the Jordan matrix algebra which defines a sum, trace of quadratic elements and a determinant. As with vector spaces the determinant of a matrix gives its eigenvalues. We can go one bit higher, a hyperdeterminant. The 3-cycle is a rule on fields (ψ*ψ)ψ =/= 0 then ψ(ψ*ψ)ψ = 0, which is a cohomology. The 4-cycle takes this into a spinor-vector rule with the product , which involves an antisymmetric system of elements which appear to define a hyperdeterminant. The result is this defines a Clifford algebra CL_{10,1} = R[32]. So this in a rule of thumb is where we add 3 to each of the Cayley numbers. I refer you to a paper by [link: http://arxiv.org/abs/1003.3436%5D Baez and Huerta[/link] for this how 3 and 4 cycles determine Clifford algebras on Cayley numbers, with Clifford dimensions 3, 4, 7, and 11..

So what we have is a nice system in 10 dimensions, which is dual to something in 16 = 8+8 dimensions. In group theory this is SO(10) and E_8xE_8, where the last part is the infamous heterotic string, or closed string which carries 24 field elements that contain the “graviton.” The SO(10) is our more well behaved (well except for renormalization) open string theory (eg type II) which describes things like the nuclear interaction. We also have this 4-cycle stuff, which pops us up one dimension and completes in some low energy approximation this thing we call M-theory. Within this structure for N = 4 supersymmetry the AdS/CFT theorem may be derived. This says the isometries of the boundary of an AdS spacetime contains the conformal structure of a quantum field theory. The structure of this appears more generally involved with the relationship between the 3 and 4-cycles, or equivalently the determinant (F_4 automorphism) and the hyperdeterminant. There is also some deep topological relationship between them. I think that the paper “Freudenthal triple classifcation of three-qubit entanglement” Borsten,1 Dahanayake, M. J. Duff, H. Ebrahim, W. Rubens is important in this regards and a comparison with the recent paper by Duff et all on 4Q-bits.

[…] Pint With Mike Duff By Philip Gibbs In a recent post I cheekily suggested that I could tell Mike Duff some interesting things about […]

I do not know if this is related or not but since square roots and 24 seem to be involved here is something that is interesting: “On Liouville’s Twelve Squares Theorem” by Kenneth S. Williams

http://www.math.carleton.ca/~williams/papers/pdf/321.pdf

Mark,

I can see this has a lof of θ-function type of analysis. This is an aspect of what I work with, particularly in regards to discrete systems with the conformal completion of AdS spacetimes and Heisenberg group work. I can’t say off had whether the particular theorem proved here is of much relevance to physics, or physics at this time.