The bundles like U(1) and SU(3) and so on, how are they related to Calabi-Yau compactifications? For example, how does one relate Calabi-Yau compactifications to SU(2)? does the Calabi-Yau compactification give rise to SU(2)?
By the way, the group SU(2) is generated by the shifts of the torus along its own circles of Villarso and its rotations along the sphere. Accordingly, to generate a group SU(3), it will be necessary to move the Willarso circles of the three-dimensional torus and rotate this torus along the surface of the three-dimensional sphere.In this case, the group U(2) is generated by the proper isometries of the torus and its rotations along the surface of the sphere.
So it is so, but in my example, the dimension of the manifold is smaller. In addition, in my example, the mass of an electron can probably be interpreted as the latitude of the polar circle.
U(2) equals SU(2) x U(1), so it can arise as the isometry group of the manifold S^2 x S^1. But this is not the only way to obtain it.
It seems that I said this without thinking properly. Most likely, the latitude of the polar circle is responsible for the ratio of the coupling constants of the electromagnetic and weak interactions, and for the electron mass, another property of the toroidal shell should be chosen, for example, the length of the closed winding of the torus.In addition, in my example, the mass of an electron can probably be interpreted as the latitude of the polar circle.
Here I again did not think, but after thinking I realized that torus knots should be knitted simultaneously on the torus of time and on the torus of internal symmetries.However, it should be noted that the torus knot, which is responsible for the formation of mass, is formed not on the torus of internal symmetries, but on the torus of time.
It seems I guessed where the secret meaning of Koide's formula is buried. If we take the sectoral (90 degrees) area of the golden spiral as the lepton mass, then in the limit we get that three successive sectors of the golden spiral satisfy a condition very similar to the Koide formula. Indeed, it is easy to prove that the limit of the ratio of the square of the sum of three consecutive Fibonacci numbers to the sum of their squares is equal to the square of the golden ratio. And each Fibonacci number is equal to the side of the square in which the sector of the golden spiral is inscribed. It remains only to correct the formula taking into account the dependence of the area of the sector of the golden spiral on the area of the corresponding square.It turns out that the ratio of the masses of leptons is described by the Koide formula and is equal to 2/3, but its meaning is hidden from me...
Strange, but Wikipedia is silent about this wonderful property of Fibonacci numbers. Can anyone share a suitable math link?it is easy to prove that the limit of the ratio of the square of the sum of three consecutive Fibonacci numbers to the sum of their squares is equal to the square of the golden ratio
Villarso circles have the intrinsic property to make the toroidal shell on the sphere be placed between two arbitrary adjacent ribs scaled-down/up to symmetrical planes. If the rotational angle of the motion meridian circle does lie between two arbitrary adjacent ribs , then (quazi-)symmetry IS lost to a polygonal shape with three vertices attached by ridges to each other as well as to the indentation point. This spells that the vertical supports are located at the outer and inner equatorial circles of latitudinal bundle to transfer the binding fibers of the toroidal , in such a manner as to make the support reactive motions tangential to the shell middle surface.Maybe the following explanation will help you. The toroidal shell of a sphere is placed on a sphere between its polar circles, so if you draw on this shell a circle having a sphere radius that intersects two opposite points lying in two different polar circles, then one half of the circle is drawn on the outer side of the shell, and the other on internal. If we fix with the help of a needle an arbitrary point of the toroidal shell on the sphere, then it will still be able to rotate along the surface of the sphere, and if we also add the shell's own motion along Villarso circles (i.e., along the circles connecting opposite points of two polar circles ), then the complete group generated by these two groups is exactly the group SU(2 ). In turn, the extension of a group SU(2 ) to a group U(2 ) is due to the assumption of the complete group of proper motions of the torus and the complete group of rotation of the torus on the sphere.