  A variety of analytical approaches have been developed for the study of
quantum spin systems in two dimensions, the notable ones being spin-waves,
slave boson/fermion parton constructions, and for lattices with one-to-one
local correspondence of faces and vertices, the 2D Jordan-Wigner (JW)
fermionization. Field-theoretically, JW fermionization is implemented through
Chern-Simons (CS) flux attachment. For a correct fermionization of lattice
quantum spin-$1/2$ magnets, it is necessary that the fermions obey mutual
bosonic (anyonic) statistics under exchange - this is not possible to implement
on arbitrary 2D lattices if fermionic matter couples only to the lattice gauge
fields. Enlarging the gauge degrees of freedom to include the dual lattice
allows the construction of consistent mutual Chern-Simons field theories. Here
we propose a mutual CS theory where the microscopic (spin) degrees of freedom
are represented as lattice fermionic matter additionally coupled to specific
combinations of dual lattice gauge fields that depend on the local geometry. We
illustrate the use of this method for understanding the properties of a
honeycomb Kitaev model subjected to a strong Zeeman field in the $z$-direction.
Our CS gauge theory framework provides an understanding why the topological
phase is degraded at lower (higher) critical fields for the ferro- (antiferro-)
magnetic Kitaev interaction. Additionally, we observe an effectively
one-dimensional character of the low-excitations at higher fields in the
$z$-direction which we also confirm by spin-wave calculations.
