  The Moore-Read state, one of the leading candidates for describing the
fractional quantum Hall effect at filling factor $\nu{=}5/2$, is a paradigmatic
$p$-wave superconductor with non-Abelian topological order. Among its many
exotic properties, the state hosts two collective modes: a bosonic density wave
and a neutral fermion mode that arises from an unpaired electron in the
condensate. It has recently been proposed that the descriptions of the two
modes can be unified by postulating supersymmetry (SUSY) that relates them in
the long-wavelength limit. Here we extend the SUSY description to construct
wave functions of the two modes on closed surfaces, such as the sphere and
torus, and we test the resulting states in large-scale numerical simulations.
We demonstrate the equivalence in the long-wavelength limit between SUSY wave
functions and previous descriptions of collective modes based on the
Girvin-MacDonald-Platzman ansatz, Jack polynomials, and bipartite composite
fermions. Leveraging the first-quantized form of the SUSY wave functions, we
study their energies using the Monte Carlo method and show that realistic
$\nu{=}5/2$ systems are close to the putative SUSY point, where the two
collective modes become degenerate in energy.
