When one of the space-time dimension is compactified on S1, the QCD
exhibits the chiral phase transition at some critical radius. When we
further turn on a background θ term which depends on the S1 compactified
coordinate, a topological ordered phase appears at low energy via the
winding of θ. 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 ρ and ω mesons form a U(Nf) 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)_Nf and U(Nf)_N.
Description