Quantum extended crystal PDE's

Abstract

Our recent results on {\em extended crystal PDE's} are generalized to PDE's in the category $\mathfrak{Q}_S$ of quantum supermanifolds. Then obstructions to the existence of global quantum smooth solutions for such equations are obtained, by using algebraic topologic techniques. Applications are considered in details to the quantum super Yang-Mills equations. Furthermore, our geometric theory of stability of PDE's and their solutions, is also generalized to quantum extended crystal PDE's. In this way we are able to identify quantum equations where their global solutions are stable at finite times. These results, are also extended to quantum singular (super)PDE's, introducing ({\em quantum extended crystal singular (super) PDE's}).