  Frustrated spin systems on Kagome lattices have long been considered to be a
promising candidate for realizing exotic spin liquid phases. Recently, there
has been a lot of renewed interest in these systems with the discovery of
materials such as Volborthite and Herbertsmithite that have Kagome like
structures. In the presence of an external magnetic field, these frustrated
systems can give rise to magnetization plateaus of which the plateau at
$m=\frac{1}{3}$ is considered to be the most prominent. Here we study the
problem of the antiferromagnetic spin-1/2 quantum XXZ Heisenberg model on a
Kagome lattice by using a Jordan-Wigner transformation that maps the spins onto
a problem of fermions coupled to a Chern-Simons gauge field. This mapping
relies on being able to define a consistent Chern-Simons term on the lattice.
Using a recently developed method to rigorously extend the Chern-Simons term to
the frustrated Kagome lattice we can now formalize the Jordan-Wigner
transformation on the Kagome lattice. We then discuss the possible phases that
can arise at the mean-field level from this mapping and focus specifically on
the case of $\frac{1}{3}$-filling ($m=\frac{1}{3}$ plateau) and analyze the
effects of fluctuations in our theory. We show that in the regime of $XY$
anisotropy the ground state at the $1/3$ plateau is equivalent to a bosonic
fractional quantum Hall Laughlin state with filling fraction $1/2$ and that at
the $5/9$ plateau it is equivalent to the first bosonic Jain daughter state at
filling fraction $2/3$.
