Hopf Term, Loop Algebras and Three Dimensional Navier-Stokes Equation

TL;DR

This paper explores the mathematical structure of three-dimensional perfect fluid dynamics, revealing connections to the Hopf term, loop algebras, and string theory, with implications for vortex filament motion and related algebraic frameworks.

Contribution

It introduces a novel description of 3D perfect fluid dynamics using the Hopf term and loop algebras, linking fluid motion to string theory concepts and algebraic structures.

Findings

Fluid dynamics described by Hopf term of nonlinear sigma model

Poisson brackets given by loop algebra, including Virasoro and O(3) current algebra

Analysis of 2D case and $w_{1+ abla}$ structure

Abstract

The dynamics of the 3 dimensional perfect fluid is equivalent to the motion of vortex filaments or "strings". We study the action principle and find that it is described by the Hopf term of the nonlinear sigma model. The Poisson bracket structure is described by the loop algebra, for example, the Virasoro algebra or the analogue of O(3) current algebra. As a string theory, it is quite different from the standard Nambu-Goto string in its coupling to the extrinsic geometry. We also analyze briefly the two dimsensional case and give some emphasis on the $w_{1+\infty}$ structure.