Perfect fluid equations with N=1,2 Schrodinger supersymmetry

TL;DR

The paper addresses the problem of embedding non-relativistic perfect-fluid dynamics into Schrödinger supersymmetry by constructing $N=1$ and $N=2$ extensions within the Hamiltonian formalism. It introduces Grassmann-odd partners for the density $\rho$ and velocity $\upsilon_i$ (real for $N=1$, complex for $N=2$) and builds corresponding supercharges whose Poisson brackets generate a super-extended Hamiltonian $H_S$, yielding fully supersymmetric fluid dynamics. For $N=1$, the supercharge is linear in fermions and the fermionic fields can be interpreted as vorticity-potentials in a Clebsch-like velocity decomposition; the full set of conserved charges closes the $N=1$ Schrödinger superalgebra. The $N=2$ construction is novel, with cubic fermionic terms in the supercharges and a conserved charge set realizing the $N=2$ Schrödinger superalgebra, including an extra $J$ R-symmetry. These results provide a new class of supersymmetric non-relativistic fluids with potential links to spin-fluid models and non-relativistic holography, and open avenues for exploring higher $\mathcal{N}$ extensions and other symmetry generalizations.

Abstract

Superconformal extensions of the perfect fluid equations, which realize $N=1,2$ Schrodinger superalgebra, are constructed within the Hamiltonian formalism. They are built by introducing real (for $N=1$) or complex (for $N=2$) anticommuting field variables as superpartners for the density and velocity of a fluid. The full set of conserved charges associated with the $N=1,2$ Schrodinger superalgebra is constructed. Within the Lagrangian formalism, when the Clebsch decomposition for the velocity vector field is used, the anticommuting variables can be interpreted as potentials parameterizing fluid's vorticity.