- Monopole flux tubes can be regarded as M4 deformations of cosmic strings representable as Cartesian products of string world sheet X2⊂ M4 and 2-surface Y2⊂ CP2. Partonic 2-surfaces would appear as "ends" of 2-sheeted monopole flux tubes. If the holomorphic realization of holography makes sense, the space-time surfaces are complex algebraic surfaces. In the simplest situation the 2-D cross section of a cosmic string is a complex surface of CP2. A more general option is as a complex algebraic curve in E2× CP2.
- Riemann-Roch theorem (see this) allows to define geometric genus (see this) of a complex algebraic curve in CP2 as
g= (d-1)(d-2)/2-s ,
where s is the number of singularities, which are cones and as a special case cusps (infinitely sharp cones). According to the Wikipedia article, this formula generalizes to algebraic surfaces in higher than 2-D complex manifolds, or at least projective space.
From this one can conclude for the (d-1)(d-2)/2 ≥ g for partonic 2-surfaces as complex surfaces in CP2, there are always singularities. For s=0, g=0 allows d=0 and d=1. For s=0, g=1 allows d=3 related to elliptic functions. Already for g=2 one has s≥ 1. The genera g=(d-1)(d-2)/2 are special in that they also allow s=0.
- It is known (see this) that for s=0 the topological genus, algebraic genus and arithmetic genus are identical. This might be relevant for the definition of genus for the p-adic counterparts of partonic 2-surfaces, where the topological genus does not make sense. This could make g ∈ {0,1, (d-1)(d-2)2} cognitively special. It would seem that p-adic variants of g=2 partonic 2-surfaces do not make sense unless one can eliminate the singularities by a deformation of Y2 to a complex 2-surface in E2× Y2. One should also be able to represent g>0 surfaces as surfaces in E2× CP2, where CP1 corresponds to either d=1 of d=2.
It is instructive to consider the CP2 option and its generalization in more detail from the perspective of weak and strong interactions and family replication phenomenon.
- g=0 option is the most natural one for cosmic strings and allows polynomials of degree d=1 and d=2. d=1 would correspond to the homologically non-trivial geodesic sphere of CP2 and d=2 a more complex surface. For the homologically non-trivial sphere only the Kähler form would contribute to the vertex related to the splitting of the cosmic string. This could explain why the generation of hadronic and gluonic monopole strings does not lead to a parity violation.
For d=2 and g=0 induced electroweak fields are non-vanishing and parity violations are predicted. Could photons and gluons correspond to cosmic strings with cross section as d=1 surface of CP2? Could parity violating weak bosons relate to cosmic strings with a d=2 spherical cross section so that the difference between strong and weak interactions would reduce to algebraic geometry?
- The genus g=1 could be also realized for cosmic strings with d=3 to which elliptic functions. In this case, the induced weak fields would be present for the CP2 option. This does not conform with the idea that parity breaking effects do not depend on the genus (generation of fermion).
Could the deformations of partonic 2-surfaces in M4 degrees of freedom come in rescue? For partons as complex 2-surfaces in E2× S2 ⊂ E2× CP2, S2 homologically non-trivial geodesic sphere, no charged weak fields would be present. If this picture is correct the deformations in E2 degrees of freedom would distinguish between fermion families but the difference should be subtle. I do not know whether the formula for algebraic surfaces in projective spaces still holds true.
- Genus g=2 partonic 2-surface in CP2 would have at least one singular point. Is this physically acceptable? Is it possible to avoid the singularity for the Y2 ⊂ E2× S2 ⊂ E2× CP2 option? Blowing up of the singularities by removing a small disk of S2 around the singularity and gluing back a disk of E2× S2 is what comes to mind. Blowup, in particular a blowup at a given point of complex manifold, such as a cone singularity of complex surface, is described in the Wikipedia article (see this).
Topologically this means construction of a connected sum with the projective space CP1 by removing a small disk around the singularity. The realization of this operation would now occur in E2× Y2. If the genus g= d(d-1)/2-s is preserved in the blowup so that one would obtain non-singular representatives also in g=2 case. Obviously the formula for the genus would not hold anymore.
- Since all quark genera g ≤ 2 appear in strong interactions, which do not violate parity, one should have a way of constructing g>0 surfaces from the homologically non-trivial sphere CP1⊂ CP2 with n=1 complex surface in E2⊂ CP1. Addition of handles should be the way. These surfaces would be associated with quarks, gluons and mesons, which all would correspond to 2-sheeted monopole fluxe tubes.
This operation should be possible also for the d=2 complex sphere carrying induced weak gauge fields. The predicted higher families of weak bosons as analogs of mesons could be obtained from d=2 monopole flux tubes. The existence of strong and weak interactions would reflect the existence of d=1 and d=2 complex spheres of CP2. In particular, one obtains non-singular g=2 fermions. Also leptons could correspond to d=2 spheres.
What can one say about the homological (Kähler magnetic) charge of a partonic 2-surface with a given genus. At least homological charges +/- 1 and +/- 2 should be realized for the partonic 2-surfaces. For about 4 decades ago, my friend Lasse Holmström, who is a mathematician, gave me as a gift a Bulletin of American Mathematical Society containing articles about 4-D topology and also about topology of CP2. At page 124 there were interesting results related to the realization of homologically non-trivial 2-surfaces in CP2, in particular there were conditions on the minimal genus of these surfaces.
The basic result was that a surface with homology charge h can be realized as a surface with genus g=(h-1)(h-2)/2 and there are no known realizations with a smaller genus. For d=h, this sequence would correspond to the sequence g= (d-1)(d-2)/2 for complex surfaces without singularities. This correlation between genus and homology charge troubled me since in the TGD framework h∈{+/- 1,+/- 2} should be possible for all genera. The addition of handles to d=1,2 complex spheres of CP1⊂ CP1⊂ E2 would solve the problem. An interesting question is whether the sequence 0,1,6,10,15,... of homologically special genera could have a physical interpretation and perhaps predict a hierarchy of analogs of strong and weak interactions.
About the number of complex deformations of a given partonic 2-surface
It is interesting to ask about how many deformations a given partonic 2-surface represents as a complex surface in E2× CP1, where CP1 corresponds to the surface of CP2 with d∈{1,2}. For the deformations of CP1 with d=1,2, one can express E2 complex coordinate as a meromorphic function of CP1 complex coordinate. More generally, one can consider the partonic 2-surface in E2× S2 as a surface with given genus g and consider the complex deformations of this surface. The dimension of the space of these deformations is of obvious physical interest if generalized holomorphy is accepted.
In the case of a pole, the E2 point would go to infinity so that poles are not allowed. If the notion of Hamilton-Jacobi structure (see this) makes sense, one can slice M4 also using closed partonic 2-surfaces with complex coordinates so that meromorphic functions with poles are allowed. In TGD, rational functions with rational coefficients of corresponding polynomials are favoured.
These functions can be characterized by so-called principal divisors expressible as formal superpositions D=∑ νkPk. Here Pk are the singular points (zeros for νk>0 and poles for νk<0). One can assign also to complex one-forms divisors: this kind of divisor is known as canonical divisor and is unique apart from addition of principal divisor, which corresponds to a multiplication of the 1-form with a meromorphic function. The degree of the divisor can be defined as deg(D)= ∑ νk.
Riemann-Roch theorem applies also to algebraic surfaces such as complex surfaces in E2× CP1, and allows to get grasp about the numbers of the surfaces obtained as deformations of CP1 with a given divisor D for a surface with a given genus g. These numbers correspond to the dimensions of the linear spaces of rational functions, whose poles are not worse than the coefficients of D, where Pk are the singular points (zeros for nk>0 and poles for nk<0). The Riemann-Roch formula reads as
l(D)-l(K-D)=deg(D)-g+1 .
Here l(D) is the dimension of the space of meromorphic functions h for which all the coefficients of (h)+D are non-negative (no poles). The term -l(K-D) is a correction term present only for low degrees deg(D) defining the analog of polynomial degree characterizing the winding number of h. Because l(K-D) is a dimension of vector space, it cannot be negative and vanishes for large enough degrees. For large values of deg(D) the formula reads therefore as l(D)=deg(D)-g+1.
See the article About Platonization of Nuclear String Model and of Model of Atoms or the chapter with the same title.
For a summary of earlier postings see Latest progress in TGD.
For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.
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