Further studies of the notion of differentiable maps from Azumaya/matrix supermanifolds I. The smooth case: Ramond-Neveu-Schwarz and Green-Schwarz meeting Grothendieck

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In this sequel to works D(11.1) (arXiv:1406.0929 [math.DG]), D(11.2) (arXiv:1412.0771 [hep-th]), and D(11.3.1) (arXiv:1508.02347 [math.DG]), we re-examine --- and reformulate when in need --- several basic notions in super C∞-algebraic geometry as guided by the mathematical formulation of Ramond-Neveu-Schwarz fermionic strings and of Green-Schwarz fermionic strings from the viewpoint of Grothendieck on Algebraic Geometry. Two theorems that are the super counterpart of Theorem 3.1.1 and Theorem 3.2.1 of D(11.3.1) are proved. They unify the notion of "smooth maps from an Azumaya/matrix super smooth manifold with a fundamental module to a super smooth manifold" introduced in D(11.2), making it a complete super parallel to the setting for D-branes in the realm of algebraic geometry in D(1) (arXiv:0709.1515 [math.AG]) and D(2) (arXiv:0809.2121 [math.AG]), and in the realm of differential or C∞-algebraic geometry in D(11.1) and D(11.3.1). A prototypical definition of dynamical fermionic stacked D-brane world-volume on a space-time in the same spirit of RNS fermionic strings or GS fermionic strings is thus laid down. Similar to D(11.3.1), which paved the path to the construction of non-Abelian Dirac-Born-Infeld action (D(13.1) (arXiv:1606.08529 [hep-th])) and the standard action (D(13.3) (arXiv:1704.03237 [hep-th])) for fundamental bosonic stacked D-branes, the current notes shall serve the same for the construction of supersymmetric action for fundamental fermionic stacked D-branes of various dimensions --- a theme of another subseries of the D-project. A notion of "noncommutative C∞-rings" and "morphism" between them is introduced at the end as a byproduct.