Duality between U× U and O× USp theories from Â₃=D̂₃

We study the duality between the ABJ(M) theory at Chern-Simons level $k=4$ and the orientifold ABJ theory at Chern-Simons level $k=1$ by using the S³ partition function. The partition function can be computed using the supersymmetric localization in terms of a matrix model, and we derive an ideal Fermi gas system by applying the Fermi gas formalism to the matrix model. By using this formalism, we show that the matrix model can be embedded in the partition function of Â₃ or D̂₃ quiver theories, and the equality of these quiver theories, which comes from Â₃=D̂₃, leads to exact relations of the matrix models. These exact relations can be used to check the identification of the flavor symmetries. The perfect agreement between the matrix models also implies that no decoupled sector is needed.