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

TL;DR

The paper studies a 3D BRST-quantized model combining free Abelian 1-form and 2-form gauge fields to realize the de Rham cohomology algebra at the algebraic level.It identifies six continuous symmetries (four off-shell nilpotent and two bosonic) and a pair of discrete dualities, establishing a two-to-one mapping to the differential geometry operators $d$, $\\delta$, and $\\Delta$ via the symmetry algebra.Curci-Ferrari type restrictions $B_\\mu + \\bar{B}_\\mu + \partial_\\mu \phi = 0$ and ${\\cal B}_\\mu + \\bar{\\cal B}_\\mu + \partial_\\mu \widetilde{\phi} = 0$ underpin the equivalence of the coupled Lagrangians, enforce absolute anticommutativity, and yield a unique bosonic generator $s_\\omega$ with $s_\\omega = s_b s_d + s_d s_b$ and $s_\\omega + s_{\\bar{\\omega}} = 0$ on the CF-restricted submanifold.Additionally, the theory features a massless pseudo-scalar with a negative kinetic term, interpreted as a phantom field with potential cosmological relevance, and the work provides a concrete 3D Hodge-theoretic realization that connects BRST physics to differential geometry.

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.