D-branes and Azumaya/matrix noncommutative differential geometry,II: Azumaya/matrix supermanifolds and differentiable maps therefrom -- with a view toward dynamical fermionic D-branes in string theory

In this Part II of D(11), we introduce new objects: super-$C^k$-schemes and Azumaya super-$C^k$-manifolds with a fundamental module (or, synonymously, matrix super-$C^k$-manifolds with a fundamental module), and extend the study in D(11.1) ([L-Y3], arXiv:1406.0929 [math.DG]) to define the notion of `differentiable maps from an Azumaya/matrix supermanifold with a fundamental module to a real manifold or supermanifold'. This allows us to introduce the notion of `fermionic D-branes' in two different styles, one parallels Ramond-Neveu-Schwarz fermionic string and the other Green-Schwarz fermionic string. A more detailed discussion on the Higgs mechanism on dynamical D-branes in our setting, taking maps from the D-brane world-volume to the space-time in question and/or sections of the Chan-Paton bundle on the D-brane world-volume as Higgs fields, is also given for the first time in the D-project. Finally note that mathematically string theory begins with the notion of a differentiable map from a string world-sheet (a $2$-manifold) to a target space-time (a real manifold). In comparison to this, D(11.1) and the current D(11.2) together bring us to the same starting point for studying D-branes in string theory as dynamical objects.

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