Discrete and Continuous Symmetry Transformation Operators and Their Algebraic Structures: A $3D$ Field-Theoretic System

TL;DR

This paper explores the algebraic structures of symmetry transformations in a 3D gauge theory system, linking BRST formalism with de Rham cohomology, and introduces a pseudo-scalar field potentially relevant for cosmological models.

Contribution

It establishes a connection between BRST symmetries in a 3D gauge theory and de Rham cohomological operators, illustrating a physical realization of Hodge theory with novel pseudo-scalar fields.

Findings

Six continuous symmetry transformations identified

Algebraic structures mirror de Rham cohomology operators

Pseudo-scalar field with negative kinetic term discovered

Abstract

We discuss the discrete as well as the continuous symmetry transformations for a three $(2+1)$-dimensional $(3D)$ combined system of the free Abelian 1-form and 2-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism and establish their relevance in the context of the algebraic structures that are obeyed by the de Rham cohomological operators of differential geometry. In fact, our present field-theoretic system respects six continuous symmetry transformations and a couple of very useful discrete duality symmetry transformations. Out of the above six continuous symmetry transformations four are off-shell nilpotent (i.e. fermionic) in nature and two are bosonic. The algebraic structures, obeyed by the symmetry operators, are reminiscent of the algebra satisfied by the de Rham cohomological operators. Hence, our present $3D$ field-theoretic system provides a perfect example for Hodge theory where there is convergence of ideas from the physical aspects of the BRST formalism and mathematical ingredients that are connected with the cohomological operators of differential geometry at the algebraic level. One of the highlights of our present investigation is the appearance of a pseudo-scalar field in our theory (on the symmetry ground alone) which carries the negative kinetic term. Thus, it is one of the possible candidates for the ``phantom" fields of the cyclic, bouncing and self-accelerated cosmological models of the Universe.