1. Original view about generalized imbedding space
The original generalization of imbedding space was basically following. Take imbedding space H=M4×CP2. Choose submanifold M2×S2, where S2 is homologically non-trivial geodesic sub-manifold of CP2. The motivation is that for a given choice of Cartan algebra of Poincare algebra (translations in time direction and spin quantization axis plus rotations in plane orthogonal to this plane plus color hypercharge and isospin) this sub-manifold remains invariant under the transformations leaving the quantization axes invariant.
Form spaces M4= M4\M2 and CP2 = CP2\S2 and their Cartesian product. Both spaces have a hole of co-dimension 2 so that the first homotopy group is Z. From these spaces one can construct an infinite hierarchy of factor spaces M4/Ga and CP
The hypothesis is that Planck constant is given by the ratio hbar= na/nb, where ni is the order of maximal cyclic subgroups of Gi. The hypothesis states also that the covariant metric of the Minkowski factor is scaled by the factor (na/nb)2. One must take care of this in the gluing procedure. One can assign to the field bodies describing both self interactions and interactions between physical systems definite sector of generalized imbedding space characterized partially by the Planck constant. The phase transitions changing Planck constant correspond to tunnelling between different sectors of the imbedding space.
2. Fractionization of quantum numbers is not possible if only factor spaces are allowed
The original idea was that the modification of the imbedding space inspired by the hierarchy of Planck constants could explain naturally phenomena like quantum Hall effect involving fractionization of quantum numbers like spin and charge. This does not however seem to be the case. Ga× Gb implies just the opposite if these quantum numbers are assigned with the symmetries of the imbedding space. For instance, quantization unit for orbital angular momentum becomes na where Zna is the maximal cyclic subgroup of Ga.
One can however imagine obtaining fractionization at the level of imbedding space for space-time sheets, which are analogous to multi-sheeted Riemann surfaces (say Riemann surfaces associated with z1/n since the rotation by 2π understood as a homotopy of M4 lifted to the space-time sheet is a non-closed curve. Continuity requirement indeed allows fractionization of the orbital quantum numbers and color in this kind of situation. Lifting up this idea to the level of imbedding space leads to the generalization of the notion of imbedding space.
3. Both covering spaces and factor spaces are possible
The observation above stimulates the question whether it might be possible in some sense to replace H or its factors by their multiple coverings.
- This is certainly not possible for M4, CP2, or H since their fundamental groups are trivial. On the other hand, the fixing of quantization axes implies a selection of the sub-space H4= M2× S2subset M4× CP2, where S2 is a geodesic sphere of CP2. M4=M4\M2 and CP2=CP2\S2 have fundamental group Z since the codimension of the excluded sub-manifold is equal to two and homotopically the situation is like that for a punctured plane. The exclusion of these sub-manifolds defined by the choice of quantization axes could naturally give rise to the desired situation.
- There are two geodesic spheres in CP2. Which one should choose or are both possible? For the homologically non-trivial godesic sphere corresponding to cosmic strings the isometry group is SU(2) subset SU(3). The homologically trivial S2 corresponds to a vacuum extremal and has isometry group SO(3)subset SU(3). A quantum criticality the value of Planck constant is undetermined. The vacuum extremal property is natural from the point of view of quantum criticality since in this case the value of Planck constant does not matter at all and one would obtain a direct connection with the vacuum degeneracy of Kähler action.
- The covering spaces in question would correspond to the Cartesian products M4na× CP2nb of the covering spaces of M4 and CP2 by Zna and Znb with fundamental group is Zna× Znb. One can also consider extension by replacing M2 and S2 with its orbit under Ga (say tedrahedral, octahedral, or icosahedral group). The resulting space will be denoted by M4×Ga resp. CP2×Gb. Product sign does not signify for Caretsian product here.
- One expects the discrete subgroups of SU(2) emerge naturally in this framework if one allows the action of these groups on the singular sub-manifolds M2 or S2. This would replace the singular manifold with a set of its rotated copies in the case that the subgroups have genuinely 3-dimensional action (the subgroups which corresponds to exceptional groups in the ADE correspondence). For instance, in the case of M2 the quantization axes for angular momentum would be replaced by the set of quantization axes going through the vertices of tedrahedron, octahedron, or icosahedron. This would bring non-commutative homotopy groups into the picture in a natural manner.
Also the orbifolds M4/Ga× CP2/Gb can be allowed as also the spaces M4/Ga× (CP2×Gb) and (M4×Ga)× CP2/Gb. Hence the previous framework would generalize considerably by the allowance of both coset spaces and covering spaces.
- The full imbedding space is union of these spaces intersecting at the quantum critical manifold and configuration space is union over configuration spaces with different choices of quantum critical manifold (choice of quantization axes) so that Poincare and color invariance are not lost.
What could be the interpretation of these two kinds of spaces?
- Jones inclusions appear in two varieties corresponding to M:N<4 and M:N=4 and one can assign a hierarchy of subgroups of SU(2) with both of them. In particular, their maximal Abelian subgroups Zn label these inclusions. The interpretation of Zn as invariance group is natural for M: N< 4 and it naturally corresponds to the coset spaces. For M:N=4 the interpretation of Zn has remained open. Obviously the interpretation of Zn as the homology group defining covering would be natural.
- M:N=4 should correspond to the allowance of cosmic strings and other analogous objects. Does the introduction of the covering spaces bring in cosmic strings in some controlled manner? Formally the subgroup of SU(2) defining the inclusion is SU(2) would mean that states are SU(2) singlets which is something non-physical. For covering spaces one would however obtain the degrees of freedom associated with the discrete fiber and the degrees of freedom in question would not disappear completely and would be characterized by the discrete subgroup of SU(2).
For anyons the non-trivial homotopy of plane brings in non-trivial connection with a flat curvature and the non-trivial dynamics of topological QFTs. Also now one might expect similar non-trivial contribution to appear in the spinor connection of M2×Ga and CP2×Gb. In conformal field theory models non-trivial monodromy would correspond to the presence of punctures in plane.
- For factor spaces the unit for quantum numbers like orbital angular momentum is multiplied by na resp. nb and for coverings it is divided by this number. These two kind of spaces are in a well defined sense obtained by multiplying and dividing the factors of H by Ga resp. Gb and multiplication and division are expected to relate to Jones inclusions with M:N< 4 and M:N=4, which both are labelled by a subset of discrete subgroups of SU(2).
- The discrete subgroups of SU(2) with fixed quantization axes possess a well defined multiplication with product defined as the group generated by forming all possible products of group elements as elements of SU(2). This product is commutative and all elements are idempotent and thus analogous to projectors. Trivial group G1, two-element group G2 consisting of reflection and identity, the cyclic groups Zp, p prime, and tedrahedral, octahedral, and icosahedral groups are the generators of this algebra.
By commutativity one can regard this algebra as an 11-dimensional module having natural numbers as coefficients ("rig"). The trivial group G1, two-element group G2 generated by reflection, and tedrahedral, octahedral, and icosahedral groups define 5 generating elements for this algebra. The products of groups other than trivial group define 10 units for this algebra so that there are 11 units altogether. The groups Zp generate a structure analogous to natural numbers acting as analog of coefficients of this structure. Clearly, one has effectively 11-dimensional commutative algebra in 1-1 correspondence with the 11-dimensional "half-lattice" N11 (N denotes natural numbers). Leaving away reflections, one obtains N7. The projector representation suggests a connection with Jones inclusions. An interesting question concerns the possible Jones inclusions assignable to the subgroups containing infinitely manner elements. Reader has of course already asked whether dimensions 11, 7 and their difference 4 might relate somehow to the mathematical structures of M-theory with 7 compactified dimensions. One could introduce generalized configuration space spinor fields in the configuration space labelled by sectors of H with given quantization axes. By introducing Fourier transform in N11 one would formally obtain an infinite-component field in 11-D space. Both M4 and CP2 would give N11 factor.
- How do the Planck constants associated with factors and coverings relate? One might argue that Planck constant defines a homomorphism respecting the multiplication and division (when possible) by Gi. If so, then Planck constant in units of hbar0 would be equal to na/nb for H/Ga× Gb option and nb/na for H×(Ga× Gb) with obvious formulas for hybrid cases. This option would put M4 and CP2 in a very symmetric role and allow much more flexibility in the identification of symmetries associated with large Planck constant phases.