String theory always comes with heavy mathematical rigor as it questions the most significant and impossible attempt to make a scale-invariant phenomenology between general relativity and quantum theory. Thus, steps have been taken to simplify the theory a bit thereby making it accessible to general yet enthusiastic readers of physics. However, as there is numerous mathematics involved in the modeling of this theory, thus, any chance to make a purely non-mathematical approach towards strings would prove vacuous and intimidating making the pathway of this marvelous theory chocked with unnecessary assumptions resulting in false analogies (or hopes) relating to this theory. Thus, where it’s almost impossible to proceed without any equations, we have given a few just to wipe out some logical confusion arising to the first readers of strings. Few necessary diagrams are included along with intense theory and least mathematics for making this significant approach of theoretical physicists accessible to general learners or readers. Topics: Bosonic string theory, supersymmetric string theory, M – theory, F – theory, dualities and interconnectedness, viability, Randall – Sundrum model for tackling the hierarchy problem of particle physics, conifold singularities, Branes, Bulks, Extremal black holes, Ekpyrotic cosmology, topological aspects of Calabi – Yau (CY) manifolds, A and B models, Mirror Symmetries; AdS/CFT, cosmic strings, all in a way accessible to every reader. Methods: Theoretical analogies, deductions, principles behind the origin, development along with the probable conclusion of this theory, the roots of its origin, the necessary difficulty for detecting those strings, and approaches done by theorists to work out the pathway of achieving Einstein’s dream of unification irrespective of several hindrances. Results: String theory itself is not a complete theory. Rather it’s in the process of further development through the increment of time resulting in more applications of mathematics by developing or incorporating them in due needs. Thus, without stating any concrete results, the theory has been tackled in this paper with a viable hypothesis based on the current understanding, and previous attempts are stated to have been made for its success.

I will present Calabi-Yau manifolds emphasizing quartic and quintic variety considering Bogomolov-Gieseker Inequality with two specific types of Clifford inequality for associated fibers by Polarized K3 surfaces with the case considered upon threefolds. Generalizations have been made to a different form of quintics with coordinates and associated mirrors for the hypersurface being concerned with taking multi-homogeneous polynomials where necessary. Minimal Calabi-Yau threefold considered for fiber varieties I ₀, I ₊, II _0, II ₊, III , 0 .

This paper typically focuses on the rescaling or equivocally a phase transition from the asymptotic approach of renormalizing the quantum gravity to a more granular approach of the loop quantum gravity (LQG) and then merging it with the Regge calculus for deriving the spin-(2) graviton as the basis of the unified theory. To construct a successful Ultraviolet (UV) completed theory via the fixed-point renormalization group flow equations (FRGE) results in an asymptotic safety approach of the quantum gravity (QG). From the loop-(2) onwards, the higher derivative divergence terms like the higher derivative curvatures, and quadratic divergences with higher derivative scalars make the momentum go to infinity which assaults a problem in renormalizing the QG. If the Einstein-Hilbert (E-H) action, which is the principle of least action is being computed, arising an equation of motion, and a localized path integral (or partition functions) is defined over a curved space, then that action is shown to be associated with the higher order dimension in a more compactified way, resulting in an infinite winding numbers being accompanied through the exponentiality coefficients of the partition integrals in the loop expansions of the second order term onwards, and based on that localization principle, the entire path integral got collapsed to isolated points or granules that if corresponds the aforesaid actions, results in negating the divergences’ with an implied bijections’ and reverse bijections’ of a diffeomorphism of a continuous differentiable functional domains. If those domains are being attributed to the spatial constraints, Hamiltonian constraints, and Master constraints then, through Ashtekar variables, it can be modestly shown that the behavior of quantum origin of asymptotic safety behavior is similar to the LQG granules of spin foam spacetime. Then, we will proceed with the triangulation of the “zoomed in” entangled-points that results in the inclusion of Regge poles via the quantum number (+2,-2,0) as the produced variables of the spin-(2) graviton and spin-(0) dilaton.

Specially defined Calabi-Yau manifolds having the hypersurfaces of degree - 5 in ℙ⁴ satisfying the non-trivial canonical bundle Λˣ₃ where the embedding holomorphic map ψ∶ X⟶ ℙ with the Kähler form for holomorphic line bundle giving a strictly positive parameter ℓᵏ ⨂ ∃k > 0 representing ℓ through the first Chern Class H² (2,ℤ). When for every Kähler Class , the Cohomology class exists with the compact form (X,ω) ∀ potential spaces satisfying H²ₔᵣ (X) with the ∂∂*−lemma for a harmonic form giving us the (1,1)⁺ − Kähler potential 𝒊2⁻¹∂∂*ρ. Taking all this in effect and making it established through various conjectures, axioms, ideals, theorems – an equivalence class is shown between the structures: Kahler-Manifold, Calabi-Yau-Manifold, Hyperkahler-Manifold, Quintic-3-Fold, Kummer Surface, K–3 Surface, De Rahm Cohomology Class (Hol(Ω(μ,ν)). The extreme case considered here is the exclusion of Complex hyperbolic Kahler – represented by ℂℙⁿ ∀n = -1. In course of making this paper the non–trivial aspects concerning the topological structures in aspects of string theory, the compact Kahler with a Ricci–flatness explained here.

A segregated approach is used through several maps and classifiers where a study has been conducted to establish a concrete relational equivalence among several hypercomplex structures over three—order categorization tables namely ‘Enrique surface classification’ and ‘Enrique-K3-Complex(K3)-Kähler-Kummer surface characteristics’ with ‘K3– Complex(K3)-Kähler-Kummer(K3)-Kummer classifiers’ taking into account several notions of algebraic topology and algebraic geometry. This paper is the continuation of the previous preprint https://doi.org/10.21203/rs.3.rs-1635957/v1. For the Enrique 7—classifiers are considered namely I, II, III, IV, V, VI, VII.

This paper aims to create a class [O] concerning the groups associated with Gromov hyperbolic groups over correspondence and equivalence through Fuchsian, Kleinian, and Schottky when subject to Laplace – Beltrami in the Teichmüller space where for the hyperbolic 3-manifold when the fundamental groups of Dehn extended to Gromov – any occurrence of Švarc-Milnor lemma satisfies the same class [O] for quotient space and Jørgensen inequality. Thus the relation (and class) extended to Mostow – Prasad Rigidity Theorem in a finite degree isometry concerning the structure of the commensurator in higher order generalizations suffice through CAT(k) space. The map of the established class [O] is shown at the end of the paper.