A topological string: the Rasetti Regge Lagrangian, topological quantum field theory and vortices in quantum fluids

Abstract

The kinetic part of the Rasetti–Regge action IRR for vortex lines is studied and its relevance to string theory is established. It is shown that both IRR and the Polyakov string action IPol can be constructed with the same field Xμ. Unlike ING, however, IRR describes a Schwarz-type topological quantum field theory. Using generators of classical Lie algebras, IRR 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 H1(M, 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 3, and in higher dimensions they reduce to the Maurer–Cartan equations for so(m).