Three dimensional quotient singularity and 4d $\mathcal{N}=1$ AdS/CFT correspondence

We systematically study the AdS/CFT correspondence induced by D3 branes probing three dimensional Gorenstein quotient singularity ${\mathbb{C}}^3\mathrm{/}G$. The field theory is given by the McKay quiver, which has a vanishing NSVZ beta function assuming that all the chiral fields have the $U(1{)}_R$ charge $\frac{2}{3}$. Various physical quantities such as quiver Hilbert series, superconformal index, central charges, etc are computed, which match exactly with those computed using the singularity. We also study the relevant deformation of those theories and find the dual geometry, therefore generate many new interesting AdS/CFT pairs. The quiver gauge theory defined using finite subgroups of $SO(3)$ group has some interesting features, for example, its Seiberg duality behavior is quite interesting.