$S^1$ reduction of 4D $\mathcal{N}=4$ Schur index and 3D $\mathcal{N}=8$ mass-deformed partition function

We study the compactification of 4D $\mathcal{N}=4$ SYM on $S^1$ from the viewpoint of the superconformal index. In the cases that the gauge group of the 4D SYM is $U(N)$ and $Usp(2N)$, the resulting 3D theory is believed to be the ABJM theory with the Chern-Simons level $k=1$ and $k=2$, respectively. This suggests that the small $S^1$ limit of the superconformal index of these 4D $\mathcal{N}=4$ SYMs is identical to the sphere partition function of the ABJM theories. Using a recently observed relation between the 4D and 3D R-charges for theories with twelve or more supercharges, we explicitly confirm this identity in the Schur limit of the 4D index. Our result provides a direct quantitative check of the relation between 4D $\mathcal{N}=4$ SYMs and 3D $\mathcal{N}=8$ ABJM theories.