  The lattice construction of Euclidean path integrals has been a successful
approach of deriving 2+1D field theory dualities with a U$(1)$ gauge field. In
this work, we generalize this lattice construction to dualities with
non-Abelian gauge fields. We construct the Euclidean spacetime lattice path
integral for a theory with strongly-interacting $SO(3)$ vector bosons and
Majorana fermions coupled to an $SO(3)$ gauge field and derive an exact duality
between this theory and the theory of a free Majorana fermion on the spacetime
lattice. We argue that this lattice duality implies the desired infrared
duality between the field theory with an $SO(3)$ vector critical boson coupled
to an $SO(3)_1$ Chern-Simons gauge theory, and a free massless Majorana fermion
in 2+1D. We also generalize the lattice construction of dualities to models
with $O(3)$ gauge fields.
