  A gauge invariant reformulation of nonrelativistic fermions in background
magnetic fields is used to obtain the Laughlin and Jain wave functions as exact
results in Mean Field Theory (MFT). The gauge invariant framework trades the
U(1) gauge symmetry for an emergent holomorphic symmetry and fluxes for
vortices. The novel holomorphic invariance is used to develop an analytical
method for attaching vortices to particles. Vortex attachment methods
introduced in this paper are subsequently employed to construct the Read
operator within a second quantized framework and obtain the Laughlin and Jain
wave functions as exact results entirely within a mean-field approximation. The
gauge invariant framework and vortex attachment techniques are generalized to
the case of spherical geometry and spherical counterparts of Laughlin and Jain
wave functions are also obtained exactly within MFT.
