Wednesday, August 10, 2011

Theory Update 99

Recall that a $2$-ordinal is a pair $(m,n)$ of ordinary $1$-ordinals, denoted by a two level tree with $m + n$ leaves. A one level tree with $m$ leaves is usually used as a symbol for the associahedron of real dimension $m - 2$. That is, the vertices of an associahedron are given by a binary planar tree obtained by expanding the higher valent nodes until all nodes are trivalent. A one level tree can be turned into a two level tree by adding a single root edge as a base level. In this way, there are two sets of associahedra that naturally occur in the $2$-ordinals, one for $m$ and one for $n$. The other $2$-ordinals denote more complicated polytopes, whose vertices are similarly obtained. Since the associahedra tile the real points of the moduli spaces for Riemann spheres with $k$ marked points, and the associahedra are also used to count terms in real $N = 4$ SYM scattering amplitudes, the $2$-ordinals fill in the gaps, much like the complex numbers fill in the gaps between the real and imaginary axes in the plane.


  1. There is no need to worry about this $N = 4$ supersymmetry business. $N$ just counts the dimension of a number field, or a quantum information space. What could be simpler?

  2. Is there a standard well-known (to category theorists) way to describe this octonion-related structure:

    There are 7 independent E8 lattices.
    They correspond to the 7 imaginary octonions.
    start with an E8 lattice (or a superposition of all 7 of them)
    replace each basis element with its corresponding E8 lattice
    then repeat the process
    to get a branching tree-like structure.

    What can you say about such a structure?


  3. Why that's an interesting comment, Tony! In the following post I mention the Kostant $31 \times 8$ construction for the $248$d representation, using the finite field $F_{32}$. These $SL(2)$ matrices can be thought of as $SL(2,F)$ matrices for $F$ some large 'field' like the bioctonions $CO$. In the bioctonions it is interesting to take the seven units $ie_{k}$ (for $i$ the complex unit and $k \in 1,\cdots,7$ for $O$) to get $7$ nice elements of $SL(2,CO)$.

    What does it mean to replace such a unit by the lattice? I suppose it's like taking matrices of matrices, branching upwards to infinite dimension. That sounds like an operad, and they are all about trees after all. But it especially reminds me of the famous seven trees in one mapping.

  4. But I digress. As you point out, the $E_8$ lattice can be defined using integral octonions. We really like integral matrices in M theory, because they are more basic than more general numbers. If we then worked with $HO$, the quaternioctonions or whatever they are called, we could have units $u_{i}e_{k}$ for $i \in 1,2,3$ and $k$ the $7$ units of above. That would be like tripling the $i$ for $CO$ and give three copies of $E_8$! The field $HO$ is of course $32$ dimensional over the reals. We can allow any crazy field we like. In fact, it seems wrong to think of flat space as anything other than a collection of numbers.

  5. Yes, I think that the integral structures are fundamental
    and that smooth structures are approximations useful for calculation.

    It is indeed "... like taking matrices of matrices, branching upwards to infinite dimension ..."
    "7 trees in one" sounds like what I need for visualization.

    Blass mentions that it is related to X = 1 + X2
    reminds me of the iterated maps of fractal Mandlbrot things.

    If you do iterated maps of nested E8 lattices
    do you end up with complicated-looking fractal things
    (due to the 7 trees in one process)
    instead of just a (relatively dull) succession of branchings ?


  6. I'm not sure yet how they come into $E_8$, but fractals appear naturally in M theory, for a number of reasons, such as the existence of attractor ribbon templates (containing knotted periodic orbits). Or, by filling a two dimensional space with a curve, one is in a sense interpolating the dimension from one to two.

    The 'approximation' to smoothness is far more profound. We really are talking about emergent spaces. Nothing, no continuum, no set, nothing, exists without good measurement reasons. That is why I often call it Constructive Number Theory. That is, the continuum is redefined, through the creation of these number labeled trees, fractals, ribbons etc.

  7. Thanks very much for very helpful thoughts.

    What it seems like to me is
    curve filling 8-dim space (sort of like Hilbert's curve and Gray Codes)
    described by Octrees (tree with each node having 8 children)
    that are Linear Octrees (where maximum depth is fixed a priori as in computer fractal generation).
    Wikipedia says "A linear octree can be represented by a linear array instead of a tree data structure".

    Do you know any more detailed Octree or Linear Octree references than Wikipedia ?


    PS - Is your Constructive Number Theory closely related to Surreal Numbers ?

  8. Hi Tony, I am afraid I have not worked much at all with the octonions. That is kneemo's speciality. Of course, they turn out to be intimately related to higher categories, so things will converge eventually to one basic set of ideas, much as they did in Newton's time.

    Yes, the surreals are one handy inspiration for me, although I am referring to a much vaster, as yet non existent, Number Theory. With categorical foundations, one can finally rid oneself of annoying problems with infinite cardinals, so the surreals become more natural than the usual reals (recalling that in a topos there is more than one version for the reals, showing that they are not 'basic'). Their binary nature is related to the binary nature of qubit logic, but QG will use a whole hierarchy of logics.


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