Supervariable approach to particle on a torus knot: A model for Hodge theory

TL;DR

This paper models a particle on a torus knot using supervariable approach, revealing BRST symmetries and their algebraic structure, thus providing a physical realization of Hodge theory.

Contribution

It introduces a supervariable framework to analyze symmetries of a particle on a torus knot, connecting physical symmetries with differential geometry.

Findings

Identification of BRST and anti-BRST symmetries with nilpotency

Discovery of novel (anti-)co-BRST, bosonic, and ghost scale symmetries

Algebra of conserved charges mirrors de Rham cohomological operators

Abstract

We analyze a particle constrained to move on a $(p,q)$-torus knot within the framework of supervariable approach and deduce the BRST as well as anti-BRST symmetries. We also capture the nilpotency and absolute anti-commutativity of (anti-)BRST symmetries in this framework. Further, we show the existence of some novel symmetries in the system such as (anti-)co-BRST, bosonic, and ghost scale symmetries. We demonstrate that the conserved charges (corresponding to these symmetries) adhere to an algebra which is analogous to that of de Rham cohomological operators of differential geometry. As the charges (and the symmetries) find a physical realization with the differential geometrical operators, at the algebraic level, the present model presents a prototype for Hodge theory.