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

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.