A field-theoretic model for Hodge theory

TL;DR

This paper shows that a 4D free Abelian 2-form gauge theory can serve as a field-theoretic model for Hodge theory, with symmetries corresponding to de Rham cohomological operators and discrete symmetries representing Hodge duality.

Contribution

It establishes a concrete realization of de Rham cohomology within a 4D gauge theory framework, linking mathematical operators to physical symmetries.

Findings

Symmetry transformations correspond to de Rham operators

Conserved charges obey cohomological algebra

Discrete symmetry realizes Hodge duality

Abstract

We demonstrate that the four (3 + 1)-dimensional free Abelian 2-form gauge theory presents a tractable field theoretical model for the Hodge theory where the well-defined symmetry transformations correspond to the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above continuous symmetry transformations, obey an algebra that is reminiscent of the algebra obeyed by the cohomological operators. The discrete symmetry transformation of the theory represents the realization of the Hodge duality operation that exists in the relationship between the exterior and co-exterior derivatives of differential geometry. Thus, we provide the realizations of all the mathematical quantities, associated with the de Rham cohomological operators, in the language of the symmetries of the present 4D free Abelian 2-form gauge theory.