Chern-Simons Improved Hamiltonians for Strings in Three Space Dimensions

TL;DR

This paper develops a general SO(2) gauge invariant Hamiltonian for strings in three dimensions, incorporating a Chern-Simons interaction that suggests the possibility of fractional statistics in string-like systems.

Contribution

It introduces a novel Hamiltonian framework for strings using gauge invariance and Chern-Simons interactions, linking extrinsic geometry to topological effects.

Findings

Constructed the most general SO(2) gauge invariant Hamiltonian for strings.

Revealed a long-range interaction mediated by a Chern-Simons gauge field.

Supported the idea of fractional statistics in three-dimensional string configurations.

Abstract

The Frenet equation governs the extrinsic geometry of a string in three-dimensional ambient space in terms of the curvature and torsion, which are both scalar functions under string reparameterisations. The description engages a local SO(2) gauge symmetry, which emerges from the invariance of the extrinsic string geometry under local frame rotations around the tangent vector. Here we inquire how to construct the most general SO(2) gauge invariant Hamiltonian of strings, in terms of the curvature and torsion. The construction instructs us to introduce a long-range (self-) interaction between strings, which is mediated by a three dimensional bulk gauge field with a Chern-Simons self-interaction. The results support the proposal that fractional statistics should be prevalent in the case of three dimensional string-like configurations.