Topological order in Quantum TGD

Topological order is a rather advanced concept of condensed matter physics. There are several motivations for the notion of topological order in TGD.

  1. TGD can be seen as almost topological QFT. 3-D surfaces are by holography equivalent with 4-D space-time surfaces and by strong form of holography equivalent with string world sheets and partonic 2-surfaces. What make this duality possible is super-symplectic symmetry realizing strong form of holography and quantum criticality realized in terms of hierarchy of Planck constants characterizing hierarchy of phases of ordinary matter identified as dark matter. This hierarchy is accompanied by a fractal hierarchy of sub-algebras of supersymplectic algebra isomorphic to the entire algebra: Wheeler would talk about symmetry breaking without symmetry breaking.
  2. heff=n× h hierarchy corresponds to n-fold singular covering of space-time surface for which the sheets of the covering co-incide at the boundaries of the causal diamond (CD), and the n sheets together with superconformal invariance give rise n additional discrete topological degreees of freedom - one has particles in space with n points. Kähler action for preferred extremals reduces to Abelian Chern-Simons terms characterizing topological QFT. Furthermore, the simplest example of topological order - point like particles, which can be connected by links - translates immediately to the collections of partonic 2-surfaces and strings connecting them.
  3. There is also braiding of fermion lines/magnetic flux tubes and Yangian product and co-product defining fundamental vertics, quantum groups associated with finite measurement resolution and described in terms of inclusions of hyper-finite factors.

In the article Topological order and Quantum TGD topological order and its category theoretical description are considered from TGD point of view - category theoretical notions are indeed very natural in TGD framework. The basic finding is that the concepts developed in condensed matter physics (topological order, rough description of states as tangles (graphs imbedded in 3-D space), ground state degeneracy, surface states protected by symmetry or topology) fit very nicely to TGD framework and has interpretation in terms of the new space-time concept. This promises applications also in the conventional areas of condensed matter physics such as more precise description of solid, liquid, and gas phases.

See the chapter Criticality and dark matter" or the article Topological order in Quantum TGD