  The superconformal index of a 4d gauge theory is computed by a matrix
integral arising from localization of the supersymmetric path integral on S^3 x
S^1 to the saddle point. As the radius of the circle goes to zero, it is
natural to expect that the 4d path integral becomes the partition function of
dimensionally reduced gauge theory on S^3. We show that this is indeed the case
and recover the matrix integral of Kapustin, Willet and Yaakov from the matrix
integral that computes the superconformal index. Remarkably, the superconformal
index of the "parent" 4d theory can be thought of as the q-deformation of the
3d partition function.
