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

Radosław Antoni Kycia

TL;DR

The paper develops an analogue of the Poincaré lemma for the codifferential by introducing a cohomotopy operator that singles out anticoexact forms and yields a direct-sum decomposition with coexact components on star-shaped regions. This framework, linked to Clifford algebras via the Dirac operator, enables systematic, local solutions to exterior differential systems arising in physics, including vacuum and massive Dirac–Kähler equations as well as Maxwell and Kalb–Ramond theories. The method clarifies how fields decompose into coexact and anticoexact parts, provides constructive integral representations, and reveals gauge structures and potential reconstructions within a unified Dirac–Clifford perspective. Overall, the approach offers a practical, locality-respecting toolkit for analyzing field equations in classical and string-inspired settings, with explicit connections to de Rham theory on suitable manifolds.

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-Kähler equation, coupled Maxwell-Kalb-Ramond system of equations occurring in a bosonic string theory and its reduction to the Dirac equation.

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

TL;DR

The paper develops an analogue of the Poincaré lemma for the codifferential by introducing a cohomotopy operator that singles out anticoexact forms and yields a direct-sum decomposition with coexact components on star-shaped regions. This framework, linked to Clifford algebras via the Dirac operator, enables systematic, local solutions to exterior differential systems arising in physics, including vacuum and massive Dirac–Kähler equations as well as Maxwell and Kalb–Ramond theories. The method clarifies how fields decompose into coexact and anticoexact parts, provides constructive integral representations, and reveals gauge structures and potential reconstructions within a unified Dirac–Clifford perspective. Overall, the approach offers a practical, locality-respecting toolkit for analyzing field equations in classical and string-inspired settings, with explicit connections to de Rham theory on suitable manifolds.

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-Kähler equation, coupled Maxwell-Kalb-Ramond system of equations occurring in a bosonic string theory and its reduction to the Dirac equation.

Paper Structure

This paper contains 13 sections, 14 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Anticoexact forms
  3. (Anti)coexact vs. (anti)exact forms
  4. Relation to Clifford algebras
  5. Application to equations of physics
  6. The vacuum Dirac equation
  7. Massive Dirac equation
  8. Massless Dirac equation with source
  9. Maxwell equations
  10. Kalb-Ramond equations
  11. Cohomotopic fermionic harmonic oscillator
  12. Conclusions
  13. Relation to de Rham theory

Key Result

Theorem 1

(The Poincaré lemma for codifferential) For a star-shaped region $U$, if $\delta \omega=0$ for $\omega \in \Lambda^{k}(U)$, then there exists $\alpha \in \Lambda^{k+1}(U)$ for $k<n=dim(U)$, such that $\omega=\delta \alpha$.

Figures (1)

  • Figure 1: The action of $d$, $H$, $\delta$ and $h$ on $\Lambda$. Here $1<r<n-1$.

Theorems & Definitions (28)

  • Theorem 1
  • Definition 1
  • Proposition 1
  • Proof
  • Proposition 2
  • Proof
  • Proposition 3
  • Proof
  • Definition 2
  • Lemma 1
  • ...and 18 more