Grothendieck meeting [Wess & Bagger]: [Supersymmetry and supergravity: IV, V, VI, VII, XXII] (2nd ed.) reconstructed in complexified $\mathbb{Z}\mathrm{/}2$-graded $C^{\mathrm{\infty }}$-Algebraic Geometry, I. Construction under trivialization of spinor bundle

Forty-six years after the birth of supersymmetry in 1973 from works of Julius Wess and Bruno Zumino, the standard quantum-field-theorists and particle physicists’ language of superspaces', supersymmetry’, and supersymmetric action functionals in superspace formulation' as given in Chapters IV, V, VI, VII, XXII of the classic on supersymmetry and supergravity: Julius Wess & Jonathan Bagger: Supersymmetry and Supergravity (2nd ed.), is finally polished, with only minimal mathematical patches added for consistency and accuracy in dealing with nilpotent objects from the Grassmann algebra involved, to a precise setting in the language of complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry. This is completed after the lesson learned from D(14.1) (arXiv:1808.05011 [math.DG]) and the notion of $d=3+1$, $N=1$ towered superspaces’ as complexified $\mathbb{Z}\mathrm{/}2$-graded $C^{\mathrm{\infty }}$-schemes, their distinguished sectors, and purge-evaluation maps first developed in SUSY(1) (= D(14.1.Supp.1)) (arXiv:1902.06246 [hep-th]) and further polished in the current work. While the construction depends on a choice of a trivialization of the spinor bundle by covariantly constant sections, as long as the transformation law and the induced isomorphism under a change of trivialization of the spinor bundle by covariantly constant sections are understood, any object or structure thus defined or constructed is mathematically well-defined. The construction can be generalized to all other space-time dimensions with simple or extended supersymmetries. This is part of the mathematical foundation required to study fermionic D-branes in the Ramond-Neveu-Schwarz formulation.