BRST and Related Superfield Approach to a Few Interesting Models of Point Particles and a Model of Bosonic String: A Brief Review

TL;DR

This review explores the BRST symmetry, CF restrictions, and superfield approaches in various 1D and 2D diffeomorphism invariant particle and string models, highlighting their gauge structure and singular Lagrangians.

Contribution

It introduces a new interacting scalar relativistic particle model with electromagnetic background and demonstrates the universal nature of CF restrictions using the modified supervariable approach.

Findings

CF-type restrictions are universal across models.

1D models are described by singular Lagrangians respecting gauge symmetry.

The equivalence of diffeomorphism and gauge symmetries is established.

Abstract

In this brief review-cum-research article, we discuss a few key aspects of the off-shell nilpotent (anti-)BRST symmetry transformations, Curci-Ferrari (CF) type restriction(s), coupled Lagrangians/Lagrangian densities, etc., for the 1D diffeomorphism (i.e. reparameterization) invariant models of $(i)$ a non-relativistic and non-SUSY free particle, $(ii)$ a scalar (i.e. non-SUSY) relativistic free particle, $(iii)$ a spinning (i.e. SUSY) relativistic free particle, and $(iv)$ a 2D diffeomorphism invariant model of a specific bosonic string theory within the framework of BRST and related supervariable/superfield approach. We take up a new 1D diffeomorphism invariant model of an interacting scalar relativistic particle with the electromagnetic field. The latter is treated as a constant background variable and is, therefore, independent of the evolution parameter. We show that the universal nature of the CF-type restriction is maintained in the case of this new model, too. We exploit the modified Bonora-Tonin supervariable/superfield approach (MBTSA) to BRST formalism in the context of the 1D and 2D diffeomorphism invariant theories to prove the universal nature of the CF-type restriction(s). We further demonstrate that the 1D diffeomorphism invariant models of all kinds of particles are described by the singular Lagrangians. This singular nature has, to the best of our knowledge, not been explicitly shown elsewhere. Thus, these 1D models also respect the gauge symmetry transformations which are generated by the first-class constraints that exist on them. The equivalence of the 1D diffeomorphism and gauge symmetries is also established under specific conditions..