**M ^{8}-H duality and tachyonic momenta**

Tachyonic momenta are mapped to space-like geodesics in H or possibly to the geodesics of X^{4} (see this, this and this). This description could allow to describe pair creation as turning of fermion backwards in time (see this). Tachyonic momenta correspond to points of de Sitter space and are therefore outside CD and would go outside the space-time surface, which is inside CD. Could one avoid this?

- Since the points of the twistor spaces T(M
^{4}) and T(CP_{2}) are in 1-1 correspondence, one can use either T(M^{4}) or T(CP_{2}) so that the projection to M^{4}or CP_{2}would serve as the base space of T(X^{4}). One could use CP_{2}coordinates or M^{4}coordinates as space-time coordinates if the dimension of the projection is 4 to either of these spaces. In the generic case, both dimensions are 4 but one must be very cautious with genericity arguments which fail at the level of M^{8}. - There are exceptional situations in which genericity fails at the level of H. String-like objects of the form X
^{2}× Y^{2}⊂ M^{4}⊂ CP_{2}is one example of this. In this case, X^{6}would not define 1-1 correspondence between T(M^{4}) or T(CP_{2}).Could one use partial projections to M

^{2}and S^{2}in this case? Could T(X^{4}) be divided locally into a Cartesian product of 3-D M^{4}part projecting to M^{2}⊂ M^{4}and of 3-D CP_{2}part projected to Y^{2}⊂ CP_{2}. - One can also consider the possibility of defining the twistor space T(M
^{2}× S^{2}). Its fiber at a given point would consist of light-like geodesics of M^{2}× S^{2}. The fiber consists of direction vectors of light-like geodesics. S^{2}projection would correspond to a geodesic circle S^{1}⊂ S^{2}going through a given point of S^{2}and its points are parametrized by azimuthal angle Φ. Hyperbolic tangent tanh(η) with range [-1,1] would characterize the direction of a time like geodesic in M^{2}. At the limit of η → +/- ∞ the S^{2}contribution to the S^{2}tangent vector to length squared of the tangent vector vanishes so that all angles in the range (0,2\pi) correspond to the same point. Therefore the fiber space has a topology of S^{2}.There are also other special situations such as M

^{1}× S^{3}, M^{3}× S^{1}for which one must introduce specific twistor space and which can be treated in the same way.

^{4})× T(CP

_{2}), where T(CP

_{2}) is identified as SU(3)/U(1)× U(1) characterizing the choices of color quantization axes.

- The definition of T(CP
_{2}) as the space of light-like geodesics from a given point of CP_{2}is not possible. One could also define the fiber space of T(CP_{2}) geometrically as the space of geodesics emating from origin at r=0 in the Eguchi-Hanson coordinates (see this) and connecting it to the homologically non-trivial geodesic sphere S^{2}_{G}r=∞. This relation is symmetric.In fact, all geodesics from r=0 end up to S

^{2}. This is due to the compactness and symmetries of CP_{2}. In the same way, the geodesics from the North Pole of S^{2}end up to the South Pole. If only the endpoint of the geodesic of CP_{2}matters, one can always regard it as a point S^{2}_{G}.The two homologically non-trivial geodesic spheres associated with distinct points of CP

_{2}always intersect at a single point, which means that their twistor fibers contain a common geodesic line of this kind. Also the twistor spheres of T(M^{4}) associated with distinct points of M^{4}with a light-like distance intersect at a common point identifiable as a light-like geodesic connecting them. - Geometrically, a light-like geodesic of H is defined by a 3-D momentum vector in M
^{4}and 3-D color momentum along CP_{2}geodesic. The scale of the 8-D tangent vector does not matter and the 8-D light-likeness condition holds true. This leaves 4 parameters so that T(H) identified in this way is 12-dimensional.The M

^{4}momenta correspond to a mass shell H^{3}. Only the momentum direction matters so that also in the M^{4}sector the fiber reduces to S^{2}. If this argument is correct, the space of light-like geodesics at point of H has the topology of S^{2}× S^{2}and T(H) would reduce to T(M^{4})× T(CP_{2}) as indeed looks natural.

**Conformal confinement at the level of H**

The proposal of this, inspired by p-adic thermodynamics, is that all states are massless in the sense that the sum of mass squared values vanishes. Conformal weight, as essentially mass squared value, is naturally additive and conformal confinement as a realization of conformal invariance would mean that the sum of mass squared values vanishes. Since complex mass squared values with a negative real part are allowed as roots of polynomials, the condition is highly non-trivial.

M^{8}-H duality (see this and this) would make it natural to assign tachyonic masses with CP_{2} type extremals and with the Euclidean regions of the space-time surface. Time-like masses would be assigned with time-like space-time regions. It was found that, contrary to the beliefs held hitherto, it is possible to satisfy boundary conditions for the action action consisting of the Kähler action, volume term and Chern-Simons term, at boundaries (genuine or between Minkowskian and Euclidean space-time regions) if they are light-like surfaces satisfying also det{g_{4}}=0. Masslessness, at least in the classical sense, would be naturally associated with light-like boundaries (genuine or between Minkowskian and Euclidean regions).

**About the analogs of Fermi torus and Fermi surface in H ^{3}**

Fermi torus (cube with opposite faces identified) emerges as a coset space of E^{3}/T^{3}, which defines a lattice in the group E^{3}. Here T^{3} is a discrete translation group T^{3} corresponding to periodic boundary conditions in a lattice.

In a realistic situation, Fermi torus is replaced with a much more complex object having Fermi surface as boundary with non-trivial topology. Could one find an elegant description of the situation?

* 1. Hyperbolic manifolds as analogies for Fermi torus?*

The hyperbolic manifold assignable to a tessellation of H^{3} defines a natural relativistic generalization of Fermi torus and Fermi surface as its boundary. To understand why this is the case, consider first the notion of cognitive representation.

- Momenta for the cognitive representations (see this)define a unique discretization of 4-surface in M
^{4}and, by M^{8}-H duality, for the space-time surfaces in H and are realized at mass shells H^{3}⊂ M^{4}⊂ M^{8}defined as roots of polynomials P. Momentum components are assumed to be algebraic integers in the extension of rationals defined by P and are in general complex.If the Minkowskian norm instead of its continuation to a Hermitian norm is used, the mass squared is in general complex. One could also use Hermitian inner product but Minkowskian complex bilinear form is the only number-theoretically acceptable possibility. Tachyonicity would mean in this case that the real part of mass squared, invariant under SO(1,3) and even its complexification SO

_{c}(1,3), is negative. - The active points of the cognitive representation contain fermion. Complexification of H
^{3}occurs if one allows algebraic integers. Galois confinement(see this and this) states that physical states correspond to points of H^{3}with integer valued momentum components in the scale defined by CD.Cognitive representations are in general finite inside regions of 4-surface of M

^{8}but at H^{3}they explode and involve all algebraic numbers consistent with H^{3}and belonging to the extension of rationals defined by P. If the components of momenta are algebraic integers, Galois confinement allows only states with momenta with integer components favored by periodic boundary conditions.

^{3}would thus be replaced with an infinite discrete subgroup Γ. For a given P, the matrix coefficients for the elements of the matrix belonging to Γ would belong to an extension of rationals defined by P.

- The division of SO(1,3) by a discrete subgroup Γ gives rise to a hyperbolic manifold with a finite volume. Hyperbolic space is an infinite covering of the hyperbolic manifold as a fundamental region of tessellation. There is an infinite number of the counterparts of Fermi torus (see this). The invariance respect to Γ would define the counterpart for the periodic boundary conditions.
Note that one can start from SO(1,3)/Γ and divide by SO(3) since Γ and SO(3) act from right and left and therefore commute so that hyperbolic manifold is SO(3)\setminus SO(1,3)/Γ.

- There is a deep connection between the topology and geometry of the Fermi manifold as a hyperbolic manifold. Hyperbolic volume is a topological invariant, which would become a basic concept of relativistic topological physics (see this).
The hyperbolic volume of the knot complement serves as a knot invariant for knots in S

^{3}. Could this have physical interpretation in the TGD framework, where knots and links, assignable to flux tubes and strings at the level of H, are central. Could one regard the effective hyperbolic manifold in H^{3}as a representation of a knot complement in S^{3}?Could these fundamental regions be physically preferred 3-surfaces at H

^{3}determining the holography and M^{8}-H duality in terms of associativity (see this and this). Boundary conditions at the boundary of the unit cell of the tessellation should give rise to effective identifications just as in the case of Fermi torus obtained from the cube in this way.

*2. De Sitter manifolds as tachyonic analogs of Fermi torus do not exist*

Can one define the analogy of Fermi torus for the real 4-momenta having negative, tachyonic mass squared? Mass shells with negative mass squared correspond to De-Sitter space SO(1,3)/SO(1,2) having a Minkowskian signature. It does not have analogies of the tessellations of H^{3} defined by discrete subgroups of SO(1,3).

The reason is that there are no closed de-Sitter manifolds of finite size since no infinite group of isometries acts discontinuously on de Sitter space: therefore these is no group replacing the Γ in H^{3}/Γ (see this).

* 3. Do complexified hyperbolic manifolds as analogs of Fermi torus exist?*

The momenta for virtual fermions defined by the roots defining mass squared values can also be complex. Tachyon property and complexity of mass squared values are not of course not the same thing.

- Complexification of H
^{3}would be involved and it is not clear what this could mean. For instance, does the notion of complexified hyperbolic manifold with complex mass squared make sense. - SO(1,3) and its infinite discrete groups Γ act in the complexification. Do they also act discontinuously? p
^{2}remains invariant if SO(1,3) acts in the same way on the real and imaginary parts of the momentum leaves invariant both imaginary and complex mass squared as well as the inner product between the real and imaginary parts of the momenta. So that the orbit is 5-dimensional. Same is true for the infinite discrete subgroup Γ so that the construction of the coset space could make sense. If Γ remains the same, the additional 2 dimensions can make the volume of the coset space infinite. Indeed, the constancy of p_{1}• p_{2}eliminates one of the two infinitely large dimensions and leaves one.Could one allow a complexification of SO(1,3), SO(3) and SO(1,3)

_{c}/SO(3)_{c}? Complexified SO(1,3) and corresponding subgroups Γ satisfy OO^{T}=1. Γ_{c}would be much larger and contain the real Γ as a subgroup. Could this give rise to a complexified hyperbolic manifold H^{3}_{c}with a finite volume? - A good guess is that the real part of the complexified bilinear form p• p determines what tachyonicity means. Since it is given by Re(p)
^{2}-Im(p)^{2}and is invariant under SO_{c}(1,3) as also Re(p)• Im(p), one can define the notions of time-likeness, light-likeness, and space-likeness using the sign of Re(p)^{2}-Im(p^{2}) as a criterion. Note that Re(p)^{2}and Im(p)^{2}are separately invariant under SO(1,3).The physicist's naive guess is that the complexified analogs of infinite discrete and discontinuous groups and complexified hyperbolic manifolds as analogs of Fermi torus exist for Re(P

^{2})-Im(p^{2})>0 but not for Re(P^{2})-Im(p^{2})<0 so that complexified dS manifolds do not exist. - The bilinear form in H
^{3}_{c}would be complex valued and would not define a real valued Riemannian metric. As a manifold, complexified hyperbolic manifold is the same as the complex hyperbolic manifold with a hermitian metric (see this) and this) but has different symmetries. The symmetry group of the complexified bilinear form of H^{3}_{c}is SO_{c}(1,3) and the symmetry group of the Hermitian metric is U(1,3) containing SO(1,3) as a real subgroup. The infinite discrete subgroups Γ for U(1,3) contain those for SO(1,3). Since one has complex mass squared, one cannot replace the bilinear form with hermitian one. The complex H^{3}is not a constant curvature space with curvature -1 whereas H^{3}_{c}could be such in a complexified sense.

For a summary of earlier postings see Latest progress in TGD.

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