On Geometric Quantization of Universes and Gravitational Constant Variety*

The geometric quantization commutes with (dimensional) reduction, exactly. Reduction theory is crucial in geometry and theoretical physics, which reconciles the abstract concept of symmetry of a system with the practical implication of changes of variables to simplify the system. The geometric quantization can precisely recover the topological geometric monodromy.

I. this paper discovers several fundamental monodromies, which are independent on experimental measurements with dimension.

1. gravitational constant G= 2/3.

2. reduced Planck constant h= 2π√3.

3. Boltzmann constant K_B = 8√3. Clearly, gcd(h, K_B) = 2√3.

4. Escape velocity v_esc = √5/2. 

II. this paper discovers that in 2-D case of circle S¹, Length l_n/ Area a_n = 2/3 √3 = G√3 as 2π√3 / π(√3)² =  2/√3= 2/3 √3= G√3, it means G=2/3. However, in 3-D sphere or polytopes, Surface s_n / Volume v_n = √3, it means that for 3-D case, G√3= √3, then G=1, which shows a non-trivial gravity. Here we compute all 3-D polytopes, as Cube, Sphere, Tetrahedron, Octahedron, Dodecahedron, Icosahedron.

III. we construct a T-Duality Courant Algebroid, where the standard Courant Algebroid is of form: TM= TM + T^*M, the T-Duality Courant Algebroid is of form: DT-TM= TM - T^*M. The physical realizations of Courant Algebroid and T-Duality Courant Algebroid have essential differential representations: for Courant Algebroid, 2n ± √3 + 1/(2n ± √3); for T-Duality Courant Algebroid, 2n ± √5 - 1/(2n ± √5)). But if n=1, we have: 

2± √3 + 1/(2± √3) = 2± √5 - 1/(2± √5) = 4. This monodromy 4 is just maximal limit light velocity rather than measurement value 3x...km/s.

VI. From the T-Duality Courant Algebroid, we discover that the escape phenomena is just original image of the Black- Holes, which will completely solve the analytical representation of the Black-Hole problems. As a start point of escape mutation, the escape velocity √5/2 arises extremel deformation orbits: √5/2 + 1/2 = φ = 1.618, √5/2 - 1/2 = φ -1 = 0.618. We found that φ² = 1 + φ Eq.:

φ² - φ - 1 = 0, there exist two roots: φ = √5/2 ± 1/2, which are Real Number solutions, i.e. the Black-Hole is observable or visual. However for normal deformation of the universe, there exists a domain: √3 → 1/√3, there exist complex roots: β = 1/2 ± i√3, which is not observable or visual, its equation is β² - β +1 = 0.

Conclusion: Black-Hole is a escaping Universe, new Universe is a return of BH.