Functional Renormalization Group Analysis of $O(3)$ Nonlinear Sigma Model and Non-Abelian Bosonization Duality

It is known that the $U(2)$ Wess-Zumino-Witten model is dual to the free fermion theory in two dimensions via non-Abelian bosonization. While it is decomposed into the $SU(2)$ Wess-Zumino-Witten model and a free compact boson, the former is believed to be equivalent to the $O(3)$ nonlinear sigma model with the theta term at $\theta=\pi$. In this work, we reexamine this duality through the lens of non-perturbative 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 a 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.