Rigid Rotor as a Toy Model for Hodge Theory

TL;DR

This paper demonstrates that a rigid rotor model exhibits BRST, anti-BRST, and co-BRST symmetries, forming an algebra analogous to de Rham cohomology, thus serving as a toy model for Hodge theory.

Contribution

It applies the superfield approach to establish nilpotent symmetries in a rigid rotor, illustrating a physical realization of Hodge theory through symmetry algebra.

Findings

Identifies BRST and anti-BRST symmetries preserving the kinetic term and action.

Derives co-BRST symmetries preserving gauge-fixing and Lagrangian.

Shows the symmetry algebra mirrors de Rham cohomological operators.

Abstract

We apply the superfield approach to the toy model of a rigid rotor and show the existence of the nilpotent and absolutely anticommuting Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations, under which, the kinetic term and action remain invariant. Furthermore, we also derive the off-shell nilpotent and absolutely anticommuting (anti-) co-BRST symmetry transformations, under which, the gauge-fixing term and Lagrangian remain invariant. The anticommutator of the above nilpotent symmetry transformations leads to the derivation of a bosonic symmetry transformation, under which, the ghost terms and action remain invariant. Together, the above transformations (and their corresponding generators) respect an algebra that turns out to be a physical realization of the algebra obeyed by the de Rham cohomological operators of differential geometry. Thus, our present model is a toy model for the Hodge theory.