Surgery and bordism groups in quantum partial differential equations.II: Variational quantum PDE's

This is the second part of a work devoted to the interplay between surgery, integral bordism groups and conservation laws, in order to characterize the geometry of PDE's in the category $\mathfrak{Q}_S$ of quantum (super)manifolds. In this paper we will consider variational problems, in the category $ {\mathfrak Q}_S$, constrained by partial differential equations. We get theorems of existence for local and global solutions. The characterization of global solutions is made by means of integral bordism groups. Applications to some important examples of the Mathematical Physics, as quantum super-black-hole solutions of quantum super Yang-Mills equations, are discussed in some details. Quantum supermanifolds allow us to unify, at the quantum level, the four fundamental forces, (gravitational, electromagnetic, weak-nuclear, strong-nuclear), in an unique geometric structure. The geometric theory of PDE's, built in the category ${\mathfrak Q}_S$ of quantum supermanifolds, gives us the right mathematic tool to describe quantum phenomena also at very high energy levels, where quantum-gravity becomes dominant.