  We establish a precise formal correspondence between a spinless p-wave chiral
fermionic superfluid and a spinless bosonic superfluid in a strong magnetic
field by accounting for the correct superfluid vorticity in the effective
theory of the former. In literature, the vorticity manifests itself
incompletely as the magnetic field. This paper demonstrates this substitution
can be understood as a truncation in the relevant expansion scheme, together
with field redefinitions. Those discarded components are critical to restoring
the Berry phase term in the effective theory, and the restoration incapsulates
these two systems in the same master Lagrangian. In addition to clarifying the
structure of the Berry phase, this formalism permits solving the bosonic system
on the lowest Landau level (LLL) based on the method of analogue. In
particular, we show that in the linear regime, the Maxwell equations governing
these systems are identical by reformulating the vortex crystal with an
auxiliary electromagnetic field. This approach offers a unified view for these
systems and yields solutions that are rotationally covariant, gauge invariants,
and physically interpretable.
