Derived Moduli Spaces of Nonlinear PDEs II: Variational Tricomplex and BV Formalism

TL;DR

This paper explores the derived geometric structures of nonlinear PDEs, focusing on the variational tricomplex and BV formalism, to deepen understanding of their solution spaces and boundary conditions.

Contribution

It introduces a derived enhancement of the de Rham complex for nonlinear PDEs and analyzes its implications for the BV formalism with boundary considerations.

Findings

Developed a derived geometric framework for nonlinear PDEs

Analyzed the impact on the BV formalism with boundary conditions

Provided new insights into the functional differential calculus on solution spaces

Abstract

This paper is the second in a series of works dedicated to studying non-linear partial differential equations via derived geometric methods. We study a natural derived enhancement of the de Rham complex of a non-linear PDE via algebro-geometric techniques and examine its consequences for the functional differential calculus on the space of solutions. Applications to the BV-formalism with and without boundary conditions are discussed.