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.
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.
posted by Kea | 11:33 AM
2 Comments:
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,
Matti
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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