A Topological String: The Rasetti-Regge Lagrangian, Topological Quantum Field Theory, and Vortices in Quantum Fluids

TL;DR

This paper explores the Rasetti-Regge vortex line action, linking it to topological quantum field theory and string theory, and examines how vortex dynamics depend on the topology of the worldsheet in various dimensions.

Contribution

It introduces a topological quantum field theory formulation of vortex lines using the Rasetti-Regge action and generalizes it to higher dimensions with topological dependence.

Findings

Rasetti-Regge action links to string theory models.

Vortex dynamics depend on the topology of the worldsheet.

Equations reduce to classical geometrical equations in specific cases.

Abstract

The kinetic part of the Rasetti-Regge action I_{RR} for vortex lines is studied and links to string theory are made. It is shown that both I_{RR} and the Polyakov string action I_{Pol} can be constructed with the same field X^mu. Unlike I_{NG}, however, I_{RR} describes a Schwarz-type topological quantum field theory. Using generators of classical Lie algebras, I_{RR} is generalized to higher dimensions. In all dimensions, the momentum 1-form P constructed from the canonical momentum for the vortex belongs to the first cohomology class H^1(M,R^m) of the worldsheet M swept-out by the vortex line. The dynamics of the vortex line thus depend directly on the topology of M. For a vortex ring, the equations of motion reduce to the Serret-Frenet equations in R^3, and in higher dimensions they reduce to the Maurer-Cartan equations for so(m).