$N=1$ fermionic D3-branes in RNS formulation I. $C^\infty$-Algebrogeometric foundations of $d=4$, $N=1$ supersymmetry, SUSY-rep compatible hybrid connections, and $\widehat{D}$-chiral maps from a $d=4$ $N=1$ Azumaya/matrix superspace

As the necessary background to construct from the aspect of Grothendieck's Algebraic Geometry dynamical fermionic D3-branes along the line of Ramond-Neveu-Schwarz superstrings in string theory, three pieces of the building blocks are given in the current notes: (1) basic $C^\infty$-algebrogeometric foundations of $d=4$, $N=1$ supersymmetry and $d=4$, $N=1$ superspace in physics, with emphasis on the partial $C^\infty$-ring structure on the function ring of the superspace, (2) the notion of SUSY-rep compatible hybrid connections on bundles over the superspace to address connections on the Chan-Paton bundle on the world-volume of a fermionic D3-brane, (3) the notion of $\widehat{D}$-chiral maps $\widehat{\varphi}$ from a $d=4$ $N=1$ Azumaya/matrix superspace with a fundamental module with a SUSY-rep compatible hybrid connection $\widehat{\nabla}$ to a complex manifold $Y$ as a model for a dynamical D3-branes moving in a target space(-time). Some test computations related to the construction of a supersymmetric action functional for SUSY-rep compatible $(\widehat{\nabla}, \widehat{\varphi})$ are given in the end, whose further study is the focus of a separate work. The current work is a sequel to D(11.4.1) (arXiv:1709.08927 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]) and is the first step in the supersymmetric generalization, in the case of D3-branes, of the standard action functional for D-branes constructed in D(13.3) (arXiv:1704.03237 [hep-th]).