The action of a Lie pseudogroup $\mathcal{G}$ on a smooth manifold$M$induces a prolonged pseudogroup action on the jet spaces$J$^{$n$}of submanifolds of$M$. We prove in this paper that both the local and global freeness of the action of $\mathcal{G}$ on$J$^{$n$}persist under prolongation in the jet order$n$. Our results underlie the construction of complete moving frames and, indirectly, their applications in the identification and analysis of the various invariant objects for the prolonged pseudogroup actions.

We find some integrability conditions for low-dimensional manifolds to admit metrics with nonnegative scalar curvature. In particular, we solve the positive action conjecture in general relativity in the affirmative.

We give close formulas for the counting functions of rational curves on complete intesection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the Prepotential.

We define a notion of stability for chiral ring of four dimensional N= 1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N= 1 field theory derived from D3 branes probing a three-fold singularity X, and show that the K stability which implies the existence of Ricci-flat conic metric on X is equivalent to the stability of chiral ring of the corresponding field 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 ofdifferentiable maps from an Azumaya/matrix supermanifold with a fundamental module to a real manifold or supermanifold'. This allows us to introduce the notion offermionic 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 C^ k -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.