An $Ar$ threesome: Matrix models, $2d$ conformal field theories, and $4d$ $N=2$ gauge theories

We explore the connections between three classes of theories: $Ar$ quiver matrix models, $d=2$ conformal $Ar$ Toda field theories, and $d=4$ $N=2$ supersymmetric conformal $Ar$ quiver gauge theories. In particular, we analyze the quiver matrix models recently introduced by Dijkgraaf and Vafa (unpublished) and make detailed comparisons with the corresponding quantities in the Toda field theories and the $N=2$ quiver gauge theories. We also make a speculative proposal for how the matrix models should be modified in order for them to reproduce the instanton partition functions in quiver gauge theories in five dimensions.