# BRST Symmetry for Torus Knot

## TL;DR

This paper develops a BRST symmetry framework for a particle constrained on a torus knot, extending the theory to first class constraints and constructing explicit nilpotent charges, thus advancing the quantization of such constrained systems.

## Contribution

It introduces the first BRST symmetry formulation for a particle on a torus knot, including the extension to first class constraints and explicit construction of BRST charges.

## Key findings

- Constructed nilpotent BRST/anti-BRST charges for the system.
- Extended the theory with Wess-Zumino terms to convert second class constraints to first class.
- Showed how various effective theories relate through generalized BRST transformations.

## Abstract

We develop BRST symmetry for the first time for a particle on the surface of a torus knot by analyzing the constraints of the system. The theory contains $2^{nd}$ class constraints and has been extended by introducing Wess-Zumino term to the convert it into a theory with first class constraints. BFV analysis of the extended theory is performed to construct BRST/anti-BRST symmetries for the particle on torus knot. The nilpotent BRST/anti-BRST charges which generate such symmetries are constructed explicitly. The states annihilated by these nilpotent charges consist the physical Hilbert space. We indicate how various effective theories on the surface of the torus knot are related through the generalized version of BRST transformation with finite field dependent parameters.