  It is known that the $SU(2)$ Wess-Zumino-Witten model is dual to the free
fermion theory in two dimensions via non-Abelian bosonization. Additionally,
the $SU(2)$ Wess-Zumino-Witten model is believed to be equivalent to the $O(3)$
nonlinear sigma model with the theta term. In this work, we reexamine this
duality through the lens of renormalization group (RG) flow. We analyze the RG
flow structure of the $O(3)$ nonlinear sigma model with the theta term in two
dimensions using the functional renormalization group. Our results reveal a
nontrivial fixed point with a nonzero value of the topological coupling. The
scaling dimensions (critical exponents) at this fixed point suggest the
realization of duality between the $O(3)$ nonlinear sigma model with the theta
term and the free fermion theory, indicating that these models belong to the
same universality class.
