Rigorously Computed Enumerative Norms as Prescribed through Quantum Cohomological Connectivity over Gromov – Witten Invariants

Novikov ring which has been used as a coefficient over the closed symplectic manifold that when encountered through an extension, Big one, from the ordinary cohomology to the quantum cohomology, then the ‘fuzzy’ quantum nature could be more precisely described by the ‘quantum cup product’ inducing the variations from the Riemann sphere connecting two points as analogous to the J – Holomorphic curves or Pseudoholomorphic curves between two points as in symplectic manifolds. Poincare duality as interpreted over the curves makes the associativity over two bubble–manifolds through Gromov connections which later makes a non–local invariance over the Gromov – Witten Invariants. Complex graded Novikov ring being associated over Poincare duality finds its way through the Riemann surface over genus – 0 and marked points – K, through the perturbed Cauchy – Riemann Equation. For the n – point Gromov – Witten Invariants, n=3 is taken for small quantum cohomology and n greater than or equal to 4 for big quantum cohomological models. This proves essential in establishing the duality (Topological) between Heterotic SO(32), Heterotic E8*E8 with Type II-A supersymmetric strings in M – Theory.