  When one of the space-time dimension is compactified on $S^1$, the QCD
exhibits the chiral phase transition at some critical radius. When we further
turn on a background $\theta$ term which depends on the $S^1$ compactified
coordinate, a topological ordered phase appears at low energy via the winding
of $\theta$. We discuss what kind of theories can describe the physics near the
critical point by requiring the matching of topological field theories in the
infrared. As one of the possibilities, we propose a scenario where the $\rho$
and $\omega$ mesons form a $U(N_f)$ gauge theory near the critical point. In
the phase where the chiral symmetry is restored, they become the dual gauge
boson of the gluon related by the level-rank duality between the three
dimensional gauge theories, $SU(N)_{N_f}$ and $U(N_f)_{-N}$.
