Braids and anyons in the TGD framework are discussed here. Braid statistics has an interpretation in terms of rotations as homotopies at a 2D plane of the spacetime surfaces instead of rotations in M^{4}. One can use M^{4} coordinates for the M^{4} projection of the spacetime surface.
As a matter of fact, arbitrary isometry induced flows of H can be lifted to rotations as flows along the lifted curve at the spacetime surface and for manysheeted spacetime the flows, which correspond to identity in H can lead to a different spacetime sheet so that the braid groups structure emerges naturally (see here).
The representations of H isometries at the level of WCW act on the entire 3surface identifiable as a generalized pointlike particle and by holography on the entire spacetime surface. The braid representations of isometries act inside the spacetime surface. This suggests a generalization of the notions of gravitational and inertial masses so that they apply to all conserved charges. Generalization of Equivalence Principle would state that gravitational and inertial charges are identical.
The condition that the Dirac operator at the level of H has tangential part equivalent to the Dirac operator for induced spinors, implies that the conserved isometry currents of H are conserved along the flow lines of corresponding Killing vector fields and proportional to the Killing vectors lifted/projected to the spacetime surface. This has an interpretation as a local hydrodynamics conservation law analogous to the conservation of ρ v^{2}/2+p along a flow line.
One can ask whether the 2dimensionality, which makes possible nontrivial and nonAbelian homotopy groups, is really necessary for the notion of the braid group in the TGD framework. As a matter of fact, the conditions are not expected to be possible for all conserved charges, and the intuitive guess that they hold true only for Cartan algebra representing maximal set of commuting observables would provide a spacetime correlate of the Uncertainty Principle. If so, the spacetime surface would depend on the choice of quantization axes. This conforms with quantum classical correspondence. For instance, the Cartan algebra of rotation group would act on a plane so that the effective 2dimensionality of braid group and quantum group representations would hold true.
This view has some nice consequences.
 If the spacetime surface is nsheeted, the rotation of 2π can take the particle to a different spacetime sheet, and only n foldrotation brings it back to its original position. The formula for fractional Hall conductivity is the same as in the case of integer Hall effect except that the 1/ℏproportionality is replaced with 1/ℏ_{eff}proportionality in TGD framework (see this).
 Degeneracy of fermion states also makes nonAbelian braid statistics possible. Since the Galois group acts as a symmetry group, the degeneracy would be naturally associated with the representations of the Galois group. Galois singletness of the manyanyon states guarantees reduces braid statistics to ordinary statistics for these. Galois confinement is proposed to be a central element of quantum biology (see this and this).
Braid statistics could also relate to the problem created by BoseEinstein and Fermi statistics.
 The problem is that manyboson and manyfermion states are maximally entangled so that state function reduction is in the QFT framework possible only for the entanglement between fermions and bosons.
In the TGD framework the situation is even more difficult since all elementary particles can be constructed from quarks. The replacement of pointlike particles with 3surfaces however forces us to reconsider the notion of particle identity. Number theoretic definition of identity applying to cognitive representations is attractive.
 The intuitive proposal is that Galois representations can entangle and that the reduction of entanglement is possible. In particular, the decomposition of extension to a hierarchy of extensions with Galois groups forming a hierarchy of normal subgroups allows the notions of cognitive measurement cascade \cite{btart/SSFRGalois}.
 A more rigorous basis for the intuition emerges from the TGD view about braiding. The Galois group can be always represented as a subgroup of a suitable symmetric group S_{n}. S_{n} allows braidings and therefore induces a braiding of the Galois group. The discrete subgroups of symmetry groups of TGD could allow representation as a Galois group of the spacetime surface. They could also allow braiding defined by the lift of the continuous isometry flow to the spacetime surface. This suggests that the notion of a quantum group could allow a geometric interpretation in terms of the braiding based on the manysheeted submanifold geometry.
 The Galois group is in general nonAbelian and the braided Galois group would define braid statistics allowing higherD representations. This would also make possible a nonmaximal entanglement and the reduction of entanglement for the tensor products would be possible..
See the the chapter TGD and condensed matter or the article
with the same title.
