Surgery and bordism groups in quantum partial differential equations, II: Variational quantum PDEs

Abstract

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 PDEs in the category Q S of quantum (super)manifolds. (Part I is quoted in Ref. [A. Prástaro, Surgery and bordism groups in quantum partial differential equations, I: The quantum Poincaré conjecture, Nonlinear Anal. TMA, in press ( doi:10.1016/j.na.2008.11.077)].) In this paper we will consider variational problems, in the category 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 Mathematical Physics, as quantum super-black-hole solutions of quantum super-Yang–Mills equations, are discussed in some detail. 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 PDEs, built in the category 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.