The Poincare lemma for codifferential, anticoexact forms, and applications to physics

TL;DR

This paper develops a homotopy theory for codifferential operators on Riemannian manifolds, introducing a cohomotopy operator that decomposes differential forms and offers new solutions to physics equations like Dirac and Maxwell systems.

Contribution

It introduces a novel homotopy framework for codifferential operators, enabling form decomposition and new solution methods for fundamental physics equations.

Findings

Decomposition of differential forms into coexact and anticoexat parts.

Application of the method to vacuum Dirac-Kähler and Maxwell-Kalb-Ramond equations.

New approaches to solving exterior differential systems in physics.

Abstract

The linear homotopy theory for codifferential operator on Riemannian manifolds is developed in analogy to a similar idea for exterior derivative. The main object is the cohomotopy operator, which singles out a module of anticoexact forms from the module of differential forms defined on a star-shaped open subset of a manifold. It is shown that there is a direct sum decomposition of a differential form into coexact and anticoexat parts. This decomposition gives a new way of solving exterior differential systems. The method is applied to equations of fundamental physics, including vacuum Dirac-K\"{a}hler equation, coupled Maxwell-Kalb-Ramond system of equations occurring in a bosonic string theory and its reduction to the Dirac equation.