Tetrahedral equation of Zamolodchikov
I encountered in Facebook a link to very interesting article by John Baez telling about tetrahedral equation of Zamolodchikov - Zamolodchikov is one of the founders of conformal field theories. I should have been well-aware about this equation. Already because I worked some time ago with the question how non-associativity and language might emerge from fundamental physics [A(BC) is different from (AB)C: language is excellent example]. The illustrations in the article of Baez are necessary to obtain some idea about what is involved and I strongly recommend them.
From the illustrations of the text of Baez one learns that a the 2-D surface in 4-D space-time is deformation known as third Reidermeister move. Physicists talk about Yang-Baxter equation (YBE) and it says that it does nothing for the topology. YBE tells that it does nothing to the quantum staet.
One can however assume that "doing nothing" is replaced with what is called 2-morphism. "Kind of gauge transformation" takes place would be the physicist's first attempt to assign to this something familiar. The outcome is unitarily equivalent with the original but not the same anymore. This actually requires a generalization of the notion of group to quantum group. Braid statistics emerges: the exchange of braid strands brings in phase or even non-commutative operation on two braid state.
Tetrahedral equation generalizes "Yang-Baxterator" so that it is not an identity anymore but becomes what is called 2-morphism. One however obtains an identity for two different combinations of 4 Reidemeister moves performed for 4 strands instead of 3. To make things really complicated one could give up also this identity and consider next level in the hierarchy.
What makes this so interesting that in 4-D context of TGD that also 2-knots formed by 2-D objects (such as string world sheets and partonic 2-surfaces) in 4-D space-time become possible. Quite generally: D-2 dimensional things get knotted in D dimensions. I have proposed that 2-knots could be crucial for information processing in living matter. Knots and braids would represent information such as topological quantum computer programs, 2-knots information processing such as developing of these programs.
In TGD one would something much more non-trivial than Reidermeister moves. The ordinary knots could really change in the operations represented by 2-knots unlike in Reidermeister moves. 2-knots/2-braids could represent genuine modifications of 1-knots since the reconnections at which knot strands can go through each other could open the knot partially or make it more complex (remember what Alexander the Great did to open the Gordion knot). The process of forming of knot invariant means gradual opening of knot in systematic stepwise manner. This kind of process could take in 4-D and be represented by string world sheet and corresponding evolution of quantum state in Zero Energy Ontology (ZEO) would represent opening of knot.
One of the basic questions in consciousness theory is whether problem solving could have as a universal physical or topological counterpart. A crazy question: could opening of 1-knot - a process defining 2-knot- serve as the topological counterpart of problem solving and give rise to its quantal counterpart in ZEO? Or could Reidermeister moves transforming trivial knot to manifestly trivial form correspond to problem solving. It would seem that the Alexandrian manner to solve problems is what happens in the real world;-).
What about higher-D knots? 4-D space-time surfaces can get knotted in 6-D space-times. If the twistorialization of TGD by lifting space-time surfaces to 6-D surface in the product of twistor spaces of Minkowski space and CP2 makes sense then space-time surfaces have representations as 4-surfaces in their 6-D twistor space. Could space-time surfaces get 4-knotted in twistor-space? If so, poor space-time surface - classical world - would be in really difficult situation!;-). By the way, also light-like 3-surfaces representing parton orbits could get knotted at the 5-D boundaries of 6-D twistor space regions assignable to space-time regions with Euclidian or Minkowskian signature!
For a summary of earlier postings see Links to the latest progress in TGD.