M Theory Lesson 23

Observe that the knots of Lesson 22 are not the same as the knots that appear inside 3-manifolds in Jones-Witten CSFT. Rather they are like the objects in the 1-operad case, which are the boundary circles for Riemann surfaces. The full 2-operad diagram, bounded by 3-spheres, is a 4-dimensional space.

Some people spend a lot of time worrying about 4-manifold invariants. For example, will we ever find a combinatorial formulation of the Seiberg-Witten invariants? The usual idea here is to consider a spin foam formulation along the lines of the Crane-Yetter classical invariant for 4-manifolds. A more careful use of higher categorical structures should improve the performance of the spin foam geometry. What if we used our 2-operad combinatorics?

The 3-spheres alone form the usual 4-discs 1-operad, which is well understood. Think of the internal 3-spheres as the leaves of a 1-level Batanin tree. The second level leaves must effectively attach labels to each internal 3-sphere. This must be associated with the embedded link.


2 Responses so far »

  1. 1

    Matti Pitkanen said,

    Dear Kea,

    for few years ago I had a short encounter with 2-operads as I tried to understand what they could mean in TGD framework but could not get out anything concrete. Nevertheless, I cannot avoid the feeling that they might provide insights about the formulation of TGD.

    First some background. 3-D light-like 3 orbits of partonic 2-surfaces are basic objects and there is 4-dimensional space-time surface associated with each of them and identified as a preferred extremum of so called Kaehler action defining classical physics as an exact part of quantum theory.

    *The quantum dynamics at light-like 3-surfaces is dictated by almost topological QFT (metric creeps in via light-likeness as a surface of M^4xCP_2) defined by Chern-Simons action for induced Kahler form. By light-likeness this is only possible dynamics. Light-likeness implies a generalization of the conformal symmetries of string theory since 3-D light-like surfaces are metrically 2-D.

    *p-Adicization of the theory leads to the notion of number theoretic braids associated with light-like 3-surfaces. Braiding S-matrix is associated with these braids. Ordinary Feynman diagrams generalize: vertices are now partonic two-surfaces at which the ends of light-like 3-surfaces meet. Number theoretic braids replicate at the vertices. There is no summation over generalized Feynman diagrams: single minimal diagram associated with particular reaction identified as a maximum of Kahler function characterizes the reaction. Coupling constant evolution follows from number theory at the level of three theory from the assignement of p-adic prime to given parton.

    There are good arguments suggesting that the entire quantum TGD is basically dictated by a hierarchy of number fields starting from hyper-octonions (sub-space of complexified octonions with Minkowskian metric) and ending up with discrete number fields and even finite fields. It might be that this hierarchy might relate with operad hierarchy.

    *At the lowest level are orbits of point like particles identified as strands of number theoretical braid. This reflects the facts that quantum measurements are characterized by a finite resolution and that cognition is discrete. This kind of discretization differs from standard ones in that it brings in number theory (hierarchy of algebraic extensions of rationals and p-adics, Galois groups as symmetries, etc..).

    *At the next level (real numbers) stringy anti-commutation relations for fermionic oscillator operators (commutators can vanish only at 1-D curve of partonic 2-surface). Finite measurement resolution implies effective non-commutative geometry and strings are replaced with points of number theoretic braids.

    *2-D partons (complex numbers) are basic dynamical objects if one restricts the consideration to Fock states. Instead of full S-matrix, only vertices of generalized Feynman diagrams are defined by a discretized version of stringy N-point functions assigned to the number theoretic braids. Partonic 2-surfaces are analogous to closed stringy world sheets and target spaces of string models can be seen as fictive constructs brought in by vertex operator construction.

    *The 3-D light-like orbits of partons defining maxima of Kaehler function are responsible for the infinite ground state degeneracy analogous to spin glass degeneracy. Therefore basic dynamical objects are genuinely 3-dimensional with light-like dimension bringing in spin glass degeneracy.

    *Next level corresponds to space-time surfaces and hyper-quaternions as maximal associative sub-manifold of hyper-octonions. The classical interior dynamics of space-time surface defines classical correlates for quantum dynamics necessitated by basic assumption of quantum measurement theory. Classical theory is indeed exact part of quantum theory.

    *The highest level of hierarchy is non-associative 8-D hyper-octonionic space and space-time surface can be regarded either as a surface in this space or in M^4xCP_2. In the proposed number theoretic vision associativity dictates both the classical and quantum dynamics of TGD Universe completely.

    Best Regards,


  2. 2

    Kea said,

    Hi Matti. Well, I’m glad you can see some connections here. Personally, of course, I am looking at this a different way. Off on my own tangent, perhaps, but we seem to making a little progress.

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