# The Poincare lemma, antiexact forms, and fermionic quantum harmonic oscillator

## TL;DR

This paper explores the properties of the homotopy operator in the Poincaré lemma, develops an abstract operator calculus, and applies it to a fermionic quantum harmonic oscillator-like eigenvalue problem, with implications for complex manifold theory.

## Contribution

It introduces an abstract operator calculus based on the homotopy operator and applies it to formulate differential equations and analyze fermionic quantum harmonic oscillators.

## Key findings

- Constructed an abstract operator calculus with the exterior derivative as an abstract derivative.
- Presented an eigenvalue problem resembling the fermionic quantum harmonic oscillator.
- Explored the dual complex to the Dolbeault bicomplex on complex manifolds.

## Abstract

The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincar\'{e} lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of the eigenvalue problem that resembles the fermionic quantum harmonic oscillator is presented. The second part presents the dual complex to the Dolbeault bicomplex generated by the homotopy operator on complex manifolds.