Duality between $U\times U$ and $O\times USp$ theories from $A_{3}=D_{3}$

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 $S^{3}$ 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 $\hat{A}_{3}$ or $\hat{D}_{3}$ quiver theories, and the equality of these quiver theories, which comes from $\hat{A}_{3}=\hat{D}_{3}$, 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.