A 3D Field-Theoretic Example for Hodge Theory

TL;DR

This paper demonstrates that a combined 3D Abelian gauge theory system serves as a concrete field-theoretic example of Hodge theory, establishing physical realizations of mathematical cohomological operators within the BRST formalism.

Contribution

It introduces a novel 3D field-theoretic model that exemplifies Hodge theory, connecting gauge symmetries with differential geometry at the algebraic level.

Findings

Establishes a 3D gauge system as an example of Hodge theory.

Provides physical realizations of de Rham cohomological operators.

First odd-dimensional (D=3) field-theoretic model for Hodge theory.

Abstract

We focus on the continuous symmetry transformations for the three ($2 + 1$)-dimensional (3D) system of a combination of the free Abelian 1-form and 2-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We establish that this combined system is a tractable field-theoretic model of Hodge theory. The symmetry operators of our present theory provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. Our present investigation is important in the sense that, for the first time, we are able to establish an odd dimensional (i.e. $D = 3$) field-theoretic system to be an example for Hodge theory (besides earlier works on a few interesting ($0 + 1$)-dimensional toy models as well as a set of well-known ${\mathcal N} = 2$ SUSY quantum mechanical systems of physical interest). For the sake of brevity, we have not taken into account the 3D Chern-Simon term for the Abelian 1-form gauge field in our theory which allows the mass and gauge-invariance to co-exist together.