Quantum extended crystal PDE's

TL;DR

This paper generalizes the theory of extended crystal PDEs to quantum supermanifolds, analyzes obstructions to global solutions, and studies stability, with applications to quantum super Yang-Mills equations.

Contribution

It introduces a geometric framework for quantum extended crystal PDEs, extending stability and obstruction results to quantum supermanifolds and singular PDEs.

Findings

Obstructions to global quantum smooth solutions identified.

Stability of quantum PDE solutions at finite times established.

Applications to quantum super Yang-Mills equations detailed.

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}).